Dynamic Behavior of Materials, Volume 1

Conference Proceedings of the Society for Experimental Mechanics Series

Jamie Kimberley · Leslie Elise Lamberson Steven Mates  Editors

Dynamic Behavior of Materials, Volume 1 Proceedings of the 2018 Annual Conference on Experimental and Applied Mechanics

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

More information about this series at http://www.springer.com/series/8922

Jamie Kimberley • Leslie Elise Lamberson • Steven Mates Editors

Dynamic Behavior of Materials, Volume 1 Proceedings of the 2018 Annual Conference on Experimental and Applied Mechanics

123

Editors Jamie Kimberley Department of Mechanical Engineering New Mexico Tech Socorro, NM, USA

Leslie Elise Lamberson Drexel University Philadelphia, PA, USA

Steven Mates National Institute of Standards and Technology Gaithersburg, MD, USA

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-319-95088-4 ISBN 978-3-319-95089-1 (eBook) https://doi.org/10.1007/978-3-319-95089-1 Library of Congress Control Number: 2018957609 © The Society for Experimental Mechanics, Inc. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Dynamic Behavior of Materials represents one of the eight volumes of technical papers presented at the 2018 SEM Annual Conference and Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics and held in Greenville, SC, on June 4–7, 2018. The complete proceedings also include volumes on Challenges in Mechanics of Time-Dependent Materials; Advancement of Optical Methods & Digital Image Correlation in Experimental Mechanics; Mechanics of Biological Systems & Micro- and Nanomechanics; Mechanics of Composite, Hybrid & Multifunctional Materials; Fracture, Fatigue, Failure and Damage Evolution; Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems; and Mechanics of Additive and Advanced Manufacturing. Each collection presents early findings from experimental and computational investigations on an important area within experimental mechanics. Dynamic behavior of materials is one of these areas. The Dynamic Behavior of Materials track was initiated in 2005 and reflects efforts to bring together researchers interested in the dynamic behavior of materials and structures and provide a forum to facilitate technical interaction and exchange. In the past years, this track has represented an ever-growing area of broad interest to the SEM community, as evidenced by the increased number of papers and attendance. The contributed papers span numerous technical divisions within SEM, which may be of interest not only to the dynamic behavior of materials community but also to the traditional mechanics of materials community. The track organizers thank the authors, presenters, organizers, and session chairs for their participation, support, and contribution to this track. We are grateful to the SEM TD chairs for cosponsoring and/or co-organizing the sessions in this track. They would also like to acknowledge the SEM support staff for their devoted efforts in accommodating the large number of paper submissions this year, making the 2018 Dynamic Behavior of Materials track successful. Socorro, NM, USA Philadelphia, PA, USA Gaithersburg, MD, USA

Jamie Kimberley Leslie Elise Lamberson Steven Mates

v

Contents

1

Error Analysis for Shock Equation of State Measurements in Polymers Using Manganin Gauges . . . . . . . . . . . Jennifer L. Jordan and Daniel Casem

2

Ballistic Impact Experiments and Quantitative Assessments of Mesoscale Damage Modes in a Single-Layer Woven Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christopher S. Meyer, Enock Bonyi, Bazle Z. Haque, Daniel J. O’Brien, Kadir Aslan, and John W. Gillespie Jr

3

A Novel Approach for Plate Impact Experiments to Obtain Properties of Materials Under Extreme Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bryan Zuanetti, Tianxue Wang, and Vikas Prakash

1

9

19

4

Effect of the Ratio of Charge Mass to Target Mass on Measured Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. C. Taylor, W. G. Szymczak, H. U. Leiste, and W. L. Fourney

5

Fracture and Failure Characterization of Transparent Acrylic Based Graft Interpenetrating Polymer Networks (Graft-IPNs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balamurugan M. Sundaram, Ricardo B. Mendez, Hareesh V. Tippur, and Maria L. Auad

43

Dynamic Crack Branching in Soda-Lime Glass: An Optical Investigation Using Digital Gradient Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balamurugan M. Sundaram and Hareesh V. Tippur

51

A Hybrid Experimental-Numerical Study of Crack Initiation and Growth in Transparent Bilayers Across a Weak Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sivareddy Dondeti and Hareesh V. Tippur

57

6

7

27

8

Inelastic Behavior of Tungsten-Carbide in Pressure-Shear Impact Shock Experiments Beyond 20 GPa . . . . . Z. Lovinger, C. Kettenbeil, M. Mello, and G. Ravichandran

65

9

Mechanical Response and Damage Evolution of High-Strength Concrete Under Triaxial Loading . . . . . . . . . . . Brett Williams, William Heard, Steven Graham, Bradley Martin, Colin Loeffler, and Xu Nie

69

10

Heterodyne Diffracted Beam Photonic Doppler Velocimeter (DPDV) for Pressure-Shear Shock Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Mello, C. Kettenbeil, M. Bischann, Z. Lovinger, and G. Ravichandran

73

An Optimization-Based Approach to Design a Complex Loading Pattern Using a Modified Split Hopkinson Pressure Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suhas Vidhate, Atacan Yucesoy, Thomas J. Pence, Adam M. Willis, and Ricardo Mejia-Alvarez

77

11

12

Development of “Dropkinson” Bar for Intermediate Strain-Rate Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bo Song, Brett Sanborn, Jack Heister, Randy Everett, Thomas Martinez, Gary Groves, Evan Johnson, Dennis Kenney, Marlene Knight, and Matthew Spletzer

81

13

Radial Inertia Effect on Dynamic Compressive Response of Polymeric Foam Materials . . . . . . . . . . . . . . . . . . . . . . . Bo Song, Brett Sanborn, and Wei-Yang Lu

85

vii

viii

Contents

14

Examining Material Response Using X-Ray Phase Contrast Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. J. Jensen, B. Branch, F. J. Cherne, A. Mandal, D. S. Montgomery, A. J. Iverson, and C. Carlson

89

15

History Note: Machining, Strain Gages, and a Pulse-Heated Kolsky Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Rhorer, S. Mates, E. Whitenton, and T. Burns

95

16

Improved Richtmyer-Meshkov Instability Experiments for Very-High-Rate Strength and Application to Tantalum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Michael B. Prime, William T. Buttler, Saryu J. Fensin, David R. Jones, Ruben Manzanares, Daniel T. Martinez, John I. Martinez, Derek W. Schmidt, and Carl P. Trujillo

17

Mechanical Characterization and Numerical Material Modeling of Polyurea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 James LeBlanc, Susan Bartyczak, and Lauren Edgerton

18

Full-Scale Testing and Numerical Modeling of Adhesively Bonded Hot Stamped Ultra-High Strength Steel Hat Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Y. B. Liu, D. Cronin, and M. Worswick

19

Mechanical Characterization of ZrO2 Rich Glass Ceramic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Balamurugan M. Sundaram, Jamie T. Westbrook, Charlene M. Smith, and John P. Finkeldey

20

Microstructure Characterization of Electrodeposited Nickel Tested at High Strain Rates . . . . . . . . . . . . . . . . . . . . . 119 Jonathan P. Ligda, Daniel Casem, and Heather Murdoch

21

The Flow Stress of AM IN 625 under Conditions of High Strain and Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Rajesh K. Ananda-Kumar, Homar Lopez-Hawa, Wilfredo Moscoso-Kingsley, and Viswanathan Madhavan

22

Proton Radiography of Reverse Ballistic Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Brady Aydelotte, Michael Golt, Brian Schuster, Jason Allison, Frank Cherne, Matthew Freeman, Johnny Goett III, Brian Hollander, Brian Jensen, Julian Lopez, Fesseha Mariam, Michael Martinez, Jason Medina, Christopher Morris, Levi Neukirch, Adam Pacheco, Mary Sandstrom, Alexander “Andy” Saunders, Tamsen Schurmann, Amy Tainter, Zhaowen Tang, Dale Tupa, Joshua Tybo, Wendy Vogan-McNeil, Carl Wilde, and John Wright

23

The Effect of ECAE on the Ballistic Response of AZ31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Tomoko Sano and Phillip Jannotti

24

Development of an Interferometer and Schlieren Based Measurement Technique for Resolving Cavitation Pressure Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Selda Buyukozturk, Alexander K. Landauer, and Christian Franck

25

Quasi-Static and Dynamic Poisson’s Ratio Evolution of Hyperelastic Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Brett Sanborn and Bo Song

26

Revisit of Dynamic Brazilian Tests of Geomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Brett Sanborn, Elizabeth Jones, Matthew Hudspeth, Bo Song, and Scott Broome

27

Interface Chemistry Dependent Mechanical Properties in Energetic Materials Using Nano-Scale Impact Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Ayotomi Olokun, Chandra Prakash, I. Emre Gunduz, and Vikas Tomar

28

Optimization of an Image-Based Experimental Setup for the Dynamic Behaviour Characterization of Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Pascal Bouda, Delphine Notta-Cuvier, Bertrand Langrand, Eric Markiewicz, and Fabrice Pierron

29

High Strain Rate Response of Adhesively Bonded Fiber-Reinforced Composite Joints: A Computational Study to Guide Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Suraj Ravindran, Subramani Sockalingam, Addis Kidane, Michael Sutton, Brian Justussone, and Jenna Pang

30

Pressure-Shear Plate Impact Experiments on Soda-Lime Glass at Pressures Beyond 20 GPa . . . . . . . . . . . . . . . . . 163 C. Kettenbeil, M. Mello, T. Jiao, R. J. Clifton, and G. Ravichandran

Contents

ix

31

Dynamic Mechanical Response of T800/F3900 Composite Under Tensile and Compressive Loading. . . . . . . . . 167 Yogesh Deshpande, Peiyu Yang, Jeremy Seidt, and Amos Gilat

32

Experimental Investigation of Rate Sensitive Mechanical Response of Pure Polyurea . . . . . . . . . . . . . . . . . . . . . . . . . . 173 K. Srinivas, C. Lakshmana Rao, and Venkitanarayanan Parameswaran

33

Experimental Study on Dynamic Fracture Response of Al6063-T6 Under High Rates of Loading. . . . . . . . . . . . 181 Anoop Kumar Pandouria, Purnashis Chakraborty, Sanjay Kumar, and Vikrant Tiwari

34

Ballistic and Material Tests and Simulations on Ultra-High Performance Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Sidney Chocron, Alexander Carpenter, Nikki Scott, Oren Spector, Alon Malka-Markovitz, Zev Lovinger, and Doron Havazelet

35

Mechanical Behavior of Ta at Extreme Strain-Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Daniel Casem, Daniel Magagnosc, Jonathan Ligda, Brian Schuster, and Timothy Walter

36

Constitutive Modeling of Polyamide Split Hopkinson Pressure Bars for the Design of a Pre-stretched Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A. Bracq, G. Haugou, and H. Morvan

37

Investigating the Mechanical and Thermal Relationship for Epoxy Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Michael Harr, Paul Moy, Timothy Walter, and Kevin Masser

38

A Novel Auxetic Structure with Enhanced Impact Performance by Means of Periodic Tessellation with Variable Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 M. Taylor, L. Francesconi, A. Baldi, X. Liang, and F. Aymerich

39

On the Response of Polymer Bonded Explosives at Different Impact Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Suraj Ravindran, Addis Tessema, and Addis Kidane

40

Localized Microstructural Deformation Behavior of Dynamically Deformed Pure Magnesium . . . . . . . . . . . . . . . 225 Peter Malchow, Suraj Ravindran, and Addis Kidane

41

Energy Absorption Characteristics of Graded Foams Subjected to High Velocity Loading . . . . . . . . . . . . . . . . . . . . 229 Abigail Wohlford, Suraj Ravindran, and Addis Kidane

42

Residual Structural Capacity of a High-Performance Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 George Vankirk, William Heard, Andreas Frank, Mike Hammons, and Jason Roth

43

Dynamic Mode II Fracture Response of PMMA Within an Aquatic Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Vivianna Gomez, Ian Delaney, Rodrigo Chavez, and Veronica Eliasson

44

An Image-Based Inertial Impact Test for the High Strain Rate Properties of Brittle Materials . . . . . . . . . . . . . . . 243 Lloyd Fletcher and Fabrice Pierron

45

An Image-Based Approach for Measuring Dynamic Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Lloyd Fletcher, Leslie Lamberson, and Fabrice Pierron

46

The Effect of in-Plane Properties on the Ballistic Response of Polyethylene Composites . . . . . . . . . . . . . . . . . . . . . . . 251 Julia Cline

47

Storage and Loss Moduli of Low-Impedance Materials at kHz Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Wiroj Nantasetphong, Zhanzhan Jia, M. Arif Hasan, Alireza V. Amirkhizi, and Sia Nmeat-Nasser

48

Effects of Pressure and Strain Rate on the Mechanical Behavior of Glassy Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Abigail Wohlford, Timothy Walter, Daniel Casem, Paul Moy, and Addis Kidane

49

The Role of Texture on the Strain-Rate Sensitivity of Mg and Mg Alloy AZ31B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Nathan Briggs, Moriah Bischann, and Owen T. Kingstedt

50

Shock Compaction of Al Powder Examined by X-Ray Phase Contrast Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 A. Mandal, M. Hudspeth, B. J. Jensen, and S. Root

51

Shock Compression Response of Model Polymer/Metal Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 David Bober, Yoshi Toyoda, Brian Maddox, Eric Herbold, Yogendra Gupta, and Mukul Kumar

x

Contents

52

High-Strain Rate Interlaminar Shear Testing of Fibre-Reinforced Composites Using an Image-Based Inertial Impact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 J. Van Blitterswyk, L. Fletcher, and F. Pierron

53

Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys in Shear Using In-Situ Synchrotron Radiation X-Ray Diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Christopher S. Meredith, Zachary Herl, and Marcus L. Young

54

Developing an Alternative to Roma Plastilina #1 as a Ballistic Backing Material for the Ballistic Testing of Body Armor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Randy Mrozek, Tara Edwards, Erich Bain, Shawn Cole, Eugene Napadensky, and Reygan Freeney

55

IBII Test for High Strain Rate Tensile Testing of Adhesives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 A. Guigue, L. Fletcher, R. Seghir, and F. Pierron

56

Two Modified Digital Gradient Sensing with Higher Measurement Sensitivity for Evaluating Stress Gradients in Transparent Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Chengyun Miao and Hareesh V. Tippur

57

Quantitative Visualization of Sub-Micron Deformations and Stresses at Sub-Microsecond Intervals in Soda-Lime Glass Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Chengyun Miao and Hareesh V. Tippur

58

Microstructural Effects in the High Strain Rate Ring Fragmentation of Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Sarah Ward, Christopher Braithwaite, and Andrew Jardine

59

Uncertainties in Low-Pressure Shock Experiments on Heterogeneous Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Tracy J. Vogler, Matthew Hudspeth, and Seth Root

60

Effects of Fluid Viscosity on Wave Propagation Through Submerged Granular Media . . . . . . . . . . . . . . . . . . . . . . . . 331 Hrachya Kocharyan and Nikhil Karanjgaokar

61

Numerical Study of the Failure Mechanism of Ceramics during Low Velocity Impact Used in Protective Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Constantine (Costas) Fountzoulas and Raymond E. Brennan

62

Influence of High Strain Rate Transverse Compression on the Tensile Strength of Polyethylene Ballistic Single Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Frank David Thomas, Daniel Casem, Tusit Weerasooriya, Subramani Sockalingam, and John W. Gillespie Jr

63

The Utility of 3D Digital Image Correlation for Characterizing High-Rate Deformation . . . . . . . . . . . . . . . . . . . . . . 345 Phillip Jannotti

64

Characterization of Dynamic Deformation and Failure of Novel Light Weight Steel Alloy . . . . . . . . . . . . . . . . . . . . 351 T. R. Walter, P. Moy, T. Sano, and K. Limmer

65

Dynamic Fragmentation of MAX Phase Ti3 SiC2 from Edge-On Impact Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 355 P. Forquin, N. Savino, L. Lamberson, M. Barsoum, and M. Morais

66

Application of High-Speed DIC to Study Damage of Thin Membranes Under Blast . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 P. Razavi, H. Tang, K. Pooladvand, M. E. Ravicz, A. Remenschneider, J. J. Rosowski, J. T. Cheng, and C. Furlong

Chapter 1

Error Analysis for Shock Equation of State Measurements in Polymers Using Manganin Gauges Jennifer L. Jordan and Daniel Casem

Abstract Piezoresistive manganin gauges undergo a change in resistance as a function of applied stress and have been used in shock experiments to measure both longitudinal and transverse stress. Careful studies are required to calibrate the gauges where the experimental conditions are well-known. In this paper, a series of shock experiments on PMMA, where manganin gauges are used as input, propagated, and transverse gauges to measure shock velocity and stress will be analyzed to understand the sources and magnitude of error. Discussion of error propagation through the experiment will be provided. Keywords Manganin gauge · Error analysis · PMMA

1.1 Introduction Manganin is a copper-manganese-nickel alloy that is sensitive to applied hydrostatic pressure and is used in fluid pressure cells and as gauges for shock wave experiments [1]. The manganin is packaged for use in shock wave experiments between polymer sheets creating a gauge package that is approximately 100 μm thick [2] and has been used to characterize the Hugoniot response of a variety of materials, including metals [3], polymers [3–8], composites [9], and biological materials [10]. Manganin gauges have been used experimentally in both a longitudinal (perpendicular to the shock wave propagation direction) and lateral (parallel to the shock wave propagation direction). The calibration of commercial manganin gauges for longitudinal applications that is used by most current researchers is that conducted by Rosenberg, Yaziv, and Partom on Vishay Micro-Measurements LM-SS-125CH-048 gauges [11]. This calibration divides the response of the gauge into two parts, less than 1.5 GPa and greater than 1.5 GPa, which correspond to the elastic and plastic response of the manganin, respectively. The calibration experiments rely on knowledge of the Hugoniot of the test material because the expected pressure is determined from the impact velocity and impedance matching. Rosenberg, et al. found that the calibration for LM-SS-125CH-048 gauges in a longitudinal configuration is: R for σ < 1.5 GP a R0

(1.1)

R + 0.278 for σ > 1.5 GP a R0

(1.2)

σ = 51.3

σ = 39.45

They cited the difference between the experimental pressure and the curve to be ± 2%. Although Rosenberg, et al. [11] conducted more than 60 experiments, they are not presented tabularly in the paper, so it is impossible to recreate the fit and include experimental error on both parameters to determine the error in the fitting parameters. Although Vantine, et al. also do not include tabular data for their custom gauge, they too cite the calibration accuracy to ± 2%.

J. L. Jordan () Shock and Detonation Physics, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] D. Casem US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_1

1

2

J. L. Jordan and D. Casem

Bourne and Rosenberg [7] have compared VISAR (Velocity Interferometry System for Any Reflector) and manganin gauges and found negligible differences except when the rise times are very fast, where the gauge exhibited ringing. For conducting materials, two features are observed – a dip prior to the arrival of the wave and ringing in the very fast rise time experiments [2]. There have been studies conducted to understand the effects of introducing the gauge package into the experimental system. Appleby-Thomas, et al. [3] performed a careful study on deviation of the lateral gauge orientation from perpendicular to the shock, and they observed a two wave structure resulting from the looped wire gauge element, increased rise time, and decreased peak stress with increasing gauge misalignment. Appleby-Thomas, et al. [4] investigated the effect of modifying the encapsulation of lateral stress gauges using epoxy, a dry joint, or suspension in a castable material. They found that, although encapsulation affected the initial rise and overshoot, it did not affect the gradient behind the rise, which was attributed to a material response [4]. In this paper, the results of experiments presented in Reference [12] will be reanalyzed to investigate the sources of error in the experiments focusing on the input longitudinal gauges as representative examples of manganin gauge analysis. The calibrations requiring linear fits are analyzed using a linear fit method that takes into account error in both the x and y coordinates.

1.2 Results and Discussion The experiments described in this paper have already been presented in detail in Reference [12]. A brief summary is presented here for ease of reference. Four experiments were conducted on two different PMMA samples – Rohm and Haas Type II UVA sheet (referred to as RH) and Polycast Poly II UVT, MIL-PRF-6425E sheet (referred to the paper as Poly). Three of the experiments were symmetric impact at nominally 150, 300, and 600 m/s. The fourth experiment employed a soda lime glass impactor to achieve higher pressures. The purpose of this paper is to understand the sources of error associated with the manganin gauges shown in Fig. 1.1, namely the input longitudinal gauge. The error in an experiment has two sources – random error and systematic error. Random error results from making repeated measurements with researcher judgement, fluctuating conditions, and other small disturbances [13] (Table 1.1). In order to combine independent random errors, for example when using mass and dimensions to determine density, the square root of the sum of squares is used [13]:  εfR

=

∂f ∂x

2

 εx2

+

∂f ∂y

2  12 εy2

(1.3)

where f = f (x, y) and ε is the error. Equation (1.1) can be expanded to include any number of independent measured parameters. Systematic error results from the calibration of the instrument and is typically considered to be half of the smallest division [13]. Since the systematic errors are not likely to compensate for each other, they’re combined algebraically [13]:  εfS =

Fig. 1.1 Schematic of the experiments described in detail in Reference [12]

∂f ∂x



 εx +

∂f ∂y

 εy

(1.4)

1 Error Analysis for Shock Equation of State Measurements in Polymers Using Manganin Gauges

3

Table 1.1 Shock velocity and error results for experiments described in Reference [12], where Poly4 is not included because the second gauge failed and did not have a time of arrival Sample Poly1 Poly2 Poly3 RH1 RH4 RH5 RH6

Shock velocity (mm/μs) 2.894 3.060 3.298 2.892 3.101 3.228 3.293

Random error (mm/μs) 0.020 0.005 0.018 0.046 0.014 0.012 0.014

Systematic error (mm/μs) 0.0018 0.0020 0.0023 0.0015 0.0017 0.0018 0.0019

Total error (mm/μs) 0.020 0.006 0.018 0.046 0.014 0.012 0.014

Finally, the random and systematic errors can be combined using the square root of the sum of squares [13]:

εf =

   2 12 2 εfR + εfS

(1.5)

The shock velocity in the material was determined from the arrival time at a manganin gauge mounted between a thin driver plate and the front surface of the sample (input) and a second manganin gauge mounted between two samples (transmitted), as shown in Fig. 1.1. The arrival time was determined by reading the time at which the gauge trace deviated from the baseline. There is possibility for variations between people taking this measurement, since the deviation from the baseline is open to interpretation. In this case, all the measurements were conducted by the same person. The systematic error in the time of arrival measurements was assumed to be ± 0.001 μs, which was the accuracy of the time measurement. The sample thickness was measured using a micrometer, with three measurements per sample. The systematic error in the thickness measurement was assumed to be ½ of the finest division, which was ± 0.0005 mm for the micrometer used. The shock velocity can be determined from the thickness and the arrival times at both gauges: Us =

x t1 − t2

(1.6)

where x is the thickness of the sample, t1 is the arrival time at the second gauge, and t2 is the arrival time at the first gauge. The random error is given by:  R εU S

=

1 t1 − t2

2

 εx2

+

x (t1 − t2 )2

2

 εt21

+

x −(t1 − t2 )2

1

2

2

εt22

(1.7)

Determining the stress in the sample from the raw voltage data is a multi-step process. Prior to the experiment, the calibration between the voltage and change in resistance is determined using a decade resistance box to vary the resistance into the Dynasen pulsed power supply measuring the associated change in voltage. The input resistances converted to the change in resistance over the initial resistance ( R R0 ) and the measured voltages are shown in Fig. 1.2. Typically, standard least squares fitting is used when there’s only error in the dependent variable and the independent variable is known well. However, a bivariate fit should be used when there are errors in both variables. Since there are errors in the measurement of the input resistance and the output voltage, the bivariate fit of York [14, 15] as described by Cantrell [16] was used and is described briefly below. In this method,

Wi βi Vi m=

Wi βi Ui

(1.8)

is solved iteratively by assuming a starting value for the slope, m, and calculating Wi =

wxi wyi wxi + m2 wyi

(1.9)

4

Fig. 1.2

J. L. Jordan and D. Casem

R R0

versus voltage for several gauge calibrations

Ui = xi − x

(1.10)

Vi = yi − y

(1.11)

and  βi = Wi

Ui mVi + wyi wxi

(1.12)

and changing the value of the guess until the difference is approximately zero to the desired degree of accuracy. In Eqs. (1.7, 1.8, 1.9, and 1.10) above, x and y are the measured values, wxi and wyi are the weighting factors assumed to be the inverse of the variance of the variable, i.e. 12 and 12 , respectively, and σxi

σyi

Wi xi x=

Wi

(1.13)

Wi yi y=

Wi

(1.14)

The equations above assume that the correlation between x and y errors is zero. The error in the slope, m, and intercept, b, can be determined by σm2 =



1  2 Wi βi −β 

std.err m =

σm2

S n−2

(1.15)

1 Error Analysis for Shock Equation of State Measurements in Polymers Using Manganin Gauges

5

Table 1.2 Slope calculated with standard least squares and bivariate methods, where input corresponds to the input gauge, output corresponds to the output gauge, and trans corresponds to the transverse gauge Least squares [/V] 0.1012 ± 0.0023 0.0947 ± 0.0012 0.0798 ± 0.0036 0.0935 ± 0.0039 0.0942 ± 0.0022 0.0792 ± 0.0023 0.0936 ± 0.0066

RH1 Poly1-input Poly1-output Poly1-trans Poly2–4, RH4–6 input Poly2–4, RH4–6 output Poly2–4, RH4–6 trans

Bivariate [/V] 0.1012 ± 0.0022 0.0946 ± 0.0006 0.0798 ± 0.0017 0.0935 ± 0.0016 0.0941 ± 0.0009 0.0792 ± 0.0012 0.0932 ± 0.0031

Difference 0.0026% 0.0032% 0.0258% 0.0146% 0.1243% 0.0036% 0.3966%

and 2  x + β σm2   S std.err b = σb2 n−2

(1.16)

Wi βi β=

Wi

(1.17)

σb2 =

1 Wi

where

and S=



[yi − (mxi + b)]2

(1.18)

Using the method described above, the slope for each set of data points shown in Fig. 1.2 was determined with the intercept set to zero, and the slopes are given in Table 1.2 in comparison with the slope determined from standard least squares fitting. It can be seen that there is minimal difference between the two slopes, so in this case, standard least squares would be an appropriate method of determining the slope. However, should either measurement have larger error associated with it, then the bivariate method would be a more appropriate choice. The raw data in the form of voltage versus time for the experiments are shown in Fig. 1.3a. The data converted to R R0 using the factors determined in Table 1.2 are shown in Fig. 1.3b. The next step is to determine the stress from the change in resistance, which is accomplished using published calibrations for longitudinal gauges (Micromeasurements LM-SS-210AW-048/SP60) [11, 17]. The issue with these sets of calibration data is that the complete data sets are not published, only the fitted equations and graphs of the data. For the longitudinal gauges, the authors cite the difference between individual points and the fitted curve is ± 2%; however, this does not mean that the error in the coefficients in Eqs. 1.1 and 1.2 is ± 2%. Since the error in the parameters cannot be calculated with the available data, it was assumed that adding 2% error to the pressure calculation to the error propagated from the resistance measurements, would be a sufficient estimation of the error. In order to determine the average voltage, a weighted average was used with the error associated with each data point. The error in the average pressure is the standard deviation of the data points used in the average, which takes into account the large scatter between neighboring data points due to the noise in the voltage measurement. The average pressures and the calculated particle velocity, up , from the pressure, P, initial density, ρ 0 , and shock velocity given in Table 1.1 are shown in Table 1.3 and plotted in Fig. 1.2. The particle velocity is calculated from the Rankine-Hugoniot jump conditions: P = ρ0 Us up

(1.19)

with the associated error propagation according to Eq. (1.3):

εuRp

⎡ ⎤1  2 2  2 2  1 P P 2 2 2 ⎦ ⎣ = εP + ερ0 + εUs ρ0 Us ρ0 Us2 ρ02 Us

(1.20)

6

Fig. 1.3 (a) Voltage, (b)

J. L. Jordan and D. Casem

R R0 ,

and (c) Stress versus time for input gauges in PMMA. The darker lines are the Poly samples

Table 1.3 Pressure, calculated particle velocity, and density for experiments described in Reference [12] Sample Poly1 Poly2 Poly3 Poly4 RH1 RH4 RH5 RH6

Density (g/cm3 ) 1.207 1.207 1.207 1.207 1.19 1.19 1.19 1.19

Pressure (GPa) 0.29 ± 0.02 0.51 ± 0.01 1.22 ± 0.01 2.02 ± 0.03 0.19 ± 0.01 0.54 ± 0.01 1.20 ± 0.01 1.90 ± 0.02

Particle velocity (km/s) 0.084 ± 0.004 0.138 ± 0.002 0.307 ± 0.004 0.055 ± 0.003 0.147 ± 0.003 0.314 ± 0.003 0.486 ± 0.006

1.3 Conclusions The major sources of error in manganin gauge experiments come from the resistance calibration, the published pressure calibrations, and the noise in the voltage traces. The resistance calibration in these experiments had minimal error. The noise in the voltage traces contributed largely to the error in the average pressure. The published pressure calibrations do not provide enough detail to propagate error through the calculations, i.e. there is no published error for the calibration factors in Eqs. (1.1 and 1.2).

1 Error Analysis for Shock Equation of State Measurements in Polymers Using Manganin Gauges

7

References 1. Special Use Sensors – Manganin Pressure Sensor. 2014 [cited 2017 10/5]; Available from: http://www.vishaypg.com/docs/11524/manganin. pdf 2. Bourne, N.K.: On the shock response of piezoresistive gauges. Meas. Sci. Technol. 15(2), 425–431 (2004) 3. Appleby-Thomas, G.J., et al.: On the effects of lateral gauge misalignment in shocked targets. Rev. Sci. Instrum. 83(6), 063904-1–063904-8 (2012) 4. Appleby-Thomas, G.J., et al.: On the interpretation of lateral manganin gauge stress measurements in polymers. J. Appl. Phys. 108(3), 0335241–033524-9 (2010) 5. Appleby-Thomas, G., Hazell, P., Stennett, C.: The variation in lateral and longitudinal stress gauge response within an RTM 6 epoxy resin under one-dimensional shock loading. J. Mater. Sci. 44(22), 6187–6198 (2009) 6. Bat’Kov, Y.V., Novikov, S.A., Fishman, N.D.: Shear stresses in polymers under shock compression. AIP Conf. Proc. 370(1), 577–580 (1996) 7. Bourne, N., Rosenberg, Z.: Manganin gauge and VISAR histories in shock-stressed polymethylmethacrylate. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 455(1984), 1259–1266 (1999) 8. Bourne, N.K., Millett, J.C.F., Goveas, S.G.: The shock response of polyoxymethylene and polyethylene. J. Phys. D. Appl. Phys. 40(18), 5714–5718 (2007) 9. Appleby-Thomas, G., et al.: Shock propagation in a cemented tungsten carbide. J. Appl. Phys. 105(6), 064916 (2009) 10. Appleby-Thomas, G., et al.: The high strain-rate behaviour of selected tissue analogues. J. Mech. Behav. Biomed. Mater. 33, 124–135 (2013) 11. Rosenberg, Z., Yaziv, D., Partom, Y.: Calibration of foil-like manganin gauges in planar shock wave experiments. J. Appl. Phys. 51(7), 3702– 3705 (1980) 12. Jordan, J.L., Casem, D., Zellner, M.: Shock response of Polymethylmethacrylate. J. Dyn. Behav. Mater. 2(3), 372–378 (2016) 13. Forbes, J.W.: Shock wave compression of condensed matter: a primer Publisher Springer Science and Business Media Berlin, Germany (2013) 14. York, D.: Least-squares fitting of a straight line. Can. J. Phys. 44(5), 1079–1086 (1966) 15. York, D., et al.: Unified equations for the slope, intercept, and standard errors of the best straight line. Am. J. Phys. 72(3), 367–375 (2004) 16. Cantrell, C.: Technical note: review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems. Atmos. Chem. Phys. 8(17), 5477–5487 (2008) 17. Rosenberg, Z., Moshel, G.: Revisiting the calibration of manganin gauges for lateral stress measurements in shock-loaded solids. J. Appl. Phys. 115(10), 103511 (2014)

Chapter 2

Ballistic Impact Experiments and Quantitative Assessments of Mesoscale Damage Modes in a Single-Layer Woven Composite Christopher S. Meyer, Enock Bonyi, Bazle Z. Haque, Daniel J. O’Brien, Kadir Aslan, and John W. Gillespie Jr

Abstract In this work, we investigated the mesoscale impact and perforation damage of a single layer, woven composite target transversely impacted below and above the ballistic limit by a rigid projectile sized on the order of a tow width. To visualize mesoscale impact damage in woven composites, a thin translucent composite target was used, which provided access to both impact and back-face surfaces. High-resolution photography was used to visualize mesoscale damage, and impact and residual velocity data relative to the location of projectile impact on weaving architecture were quantified. It was found that impact on a tow-tow crossover requires more energy to perforate than impact on a matrix-rich interstitial site or on adjacent, parallel tows. Mesoscale damage in thin, woven composites was characterized for impact velocities below and above the ballistic limit. Four mesoscale damage modes were identified: transverse tow cracks, tow-tow delamination, 45◦ matrix cracks, and punch- shear. These damage modes were observed both on the surface and inside the composites. High-resolution images of these damage modes were quantified in digital damage maps whereby the output of color intensity correlated with the quantity and type of material damage. Digital maps generated for select specimens revealed characteristic damage patterns in woven fabric composites including a diamond pattern in matrix cracking and a cross pattern in tow–tow delamination. It was found that the greatest extent and quantity of mesoscale damage occurs for impact velocity just below the ballistic limit. Keywords Transverse crack · Delamination · Mesoscale · Damage · Impact

2.1 Introduction Length scales associated with the weaving architecture of woven fabric composite materials play an important role in the energy dissipation mechanisms of composites. “Macroscale” is the structural length scale of composites, “mesoscale” is the fiber-tow or yarn length scale of woven composites, and “microscale” is the fiber length scale of composites. Under transverse ballistic impact, the weaving architecture provides additional energy dissipation mechanisms over laminated, unidirectional composites. Kinetic energy of projectiles impacting fiber reinforced polymer composites is dissipated through macroscale damage modes and microscale damage mechanisms. Energy dissipating macroscale damage modes include crush, punch-shear, compression-shear, tension-shear, transverse tow damage and delamination, and in-plane matrix damage. Energy dissipating microscale damage mechanisms include fiber fracture, fiber-matrix interface debonding, matrix cracking, and frictional sliding. The weaving architecture of textile composites provides mesoscale energy dissipation mechanisms including primary tow matrix cracking, tension-shear and compression-shear tow failure, elastic deformation of secondary tows [1], additional delamination resistance due to the out-of-plane undulation [2], tow-tow and tow-matrix delamination [3], and coupled tension and bending due to local undulation of interwoven tows [4]. Impact kinetic energy is dissipated at the

C. S. Meyer () University of Delaware, Center for Composite Materials, Newark, DE, USA US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA e-mail: [email protected] E. Bonyi · K. Aslan Morgan State University, Department of Civil Engineering, Baltimore, MD, USA B. Z. Haque · J. W. Gillespie Jr University of Delaware, Center for Composite Materials, Newark, DE, USA D. J. O’Brien US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_2

9

10

C. S. Meyer et al.

mesoscale in woven composites by mesoscale damage modes: transverse tow cracking, tow-tow delamination, 45◦ matrix cracking, and punch-shear damage [5, 6]. Mesoscale damage modes are comprised of microscale damage mechanisms. Thus, mesoscale energy dissipation and damage come from the accumulation of energy dissipating microscale damage mechanisms. Impactor dimension relative to the composite is another important length scale, and the impactor nose shape influences how kinetic energy is transmitted into the composite. Mesoscale damage modes and mechanisms occur at the length scale of the individual tows so impactor diameter on the order of tow width may influence the macroscopic damage modes such as punch-shear and crush. A projectile diameter less than the fabric tow width may impact on individual tows, tow-tow crossovers, and interstitial resin pockets; however, larger diameter projectiles will impact on multiple tows. A mesoscale impact model considering projectile diameter is necessary to understand the effect of weaving architecture on the impact damage behavior of woven fabric composites [5]. Thickness of the composite is another length scale that must be considered in the study of impact damage modes and mechanisms. In previous works, Haque et al. [7–9] studied and modeled the ballistic impact response of different composite laminates. In those works, the ballistic impact is described with three phenomenologically different phases: (1) penetration; (2) transition; and (3) perforation. Each of these phases is dominated by different deformation and damage mechanisms. During the penetration phase, there is little deformation in the material ahead of the projectile up to the target back face. Penetration is dominated by through-thickness compression, fiber crush, fiber-matrix shear, and material flow from the annulus surrounding the projectile-target contact region. As the impact event enters the transition phase, the material surrounding and ahead of the projectile begins to experience transverse shear deformation resulting in delamination and the appearance of a “deformation cone” ahead of the projectile moving at a slower velocity than the projectile. In the perforation phase, the projectile and target move at the same relative velocity, and deformation is dominated by compression-shear ahead of the projectile and in-plane tension-shear around the projectile. The extent to which any given target exhibits each of the three phases of penetration, transition, and perforation is determined by the relative target and projectile geometries. Very thick laminates are dominated by penetration while very thin targets are dominated by perforation. For penetration mechanics of composite materials, a thick composite is herein defined as one in which penetration, transition, and perforation occurs; a thin composite is defined as one in which only perforation occurs. An intermediate thickness composite (between thick and thin) can be defined which shows little or no penetration but shows both the transition and perforation phases. Perforation damage mechanisms can be isolated by ballistic testing of a thin composite laminate, and in the limit, by testing a single layer woven fabric composite laminate. Lomov et al. [4] noted that single-ply composites are seldom tested (or used), but local variations due to lamina nesting, random tow placement, and interpenetration and overlapping of tows in multi-layer composites contributes to error in strain field characterization. The work by Lomov et al. [4] suggests that experiments on a single ply can provide insight into the material behavior by minimizing sources of error related to the interaction between multiple plies. Impact kinetic energy is another important consideration in composite impact and damage, so projectile impact on composite targets must also consider impact velocity as an important variable. The ballistic limit velocity of a projectiletarget pair, VBL , is a function of target material properties, thickness, and weaving architecture, and of projectile mass, length and diameter, and nose shape. Ballistic limit velocity is related to and sometimes called V50 , which is a statistical measure defined as the projectile impact velocity for which at least 50% of ballistic impacts perforate the target and the remainder rebound or lodge in the target [10]. The theories of transverse impact on a 1D fiber [11–16] and a 2D membrane [17] provide important insight into the perforation mechanics of a single layer, thin composite plate. When a projectile transversely impacts a thin composite target, an implosion wave (axial stress wave) travels outward from the projectile impact location, and a transverse deformation cone develops such that the edge or kink point of the cone propagates out from the point of impact, satisfying the conservation of momentum and energy. The transverse deformation cone propagates with a cone wave-front velocity that depends on the axial wave speed of the target material and the projectile impact velocity. The implosion or axial wave speed of the target depends only on the target material properties. The mechanism of momentum and energy transfer to the target depends on the implosion wave speed and the cone wave speed. With increasing projectile impact velocity, the rate of momentum and energy transfer to the target increases. As the impact velocity increases to the ballistic limit velocity, the total momentum of the projectile is transferred to the target. Impact velocities greater than the ballistic limit exceed the composite target’s capacity to dissipate energy without catastrophic failure in the neighborhood of the projectile, so the projectile perforates and exits the target with some residual velocity in the direction of impact. All transverse impacts of composite materials with projectile velocities exceeding the ballistic limit include the perforation phase, regardless of composite thickness. Therefore, understanding ballistic impact damage necessitates studying the perforation phase. To isolate and examine the damage modes and mechanisms present in the perforation phase up to and exceeding the ballistic limit velocity, the present study examined a composite plate made from a single ply of woven S-2

2 Ballistic Impact Experiments and Quantitative Assessments of Mesoscale Damage Modes in a Single-Layer Woven Composite Table 2.1 Average S-2 glass/ SC-15 composite properties

Density Thickness Areal density

avg

ρC = 1.758 ± 0.03 HC = 0.887 ± 0.024 AD = ρ C HC = 1.559

11 g/cm3 mm kg/m2

glass fiber. A single ply composite ensures that neither the penetration nor transition phases will be present, but only the perforation phase will occur. To investigate the perforation dominated mesoscale damage modes and mechanisms associated with thin woven fabric composites, ballistic impact experiments were conducted with a 5.6 mm (.22 caliber) right circular cylindrical steel projectile impacting a single layer, plain-weave S-2 Glass/SC15 composite. To investigate kinetic energy, momentum transfer, and mesoscale damage, impact velocities up to and exceeding the ballistic limit velocity were recorded, impact locations relative to tow-tow crossovers were determined, and mesoscale damage was characterized and quantified using high-resolution optical photography.

2.2 Experimental Setup Vacuum-assisted resin transfer molding was used to produce single-ply composite panels from plain-weave (PW) S-2 glass fabric (5×5 tows/inch, areal density of one ply is 744 g/m2 (24 oz./yd2 ), AGY 463-AA-2BL, 30 ends) infused with SC15 epoxy resin (Applied Poleramic, Benicia, CA). A two-part cure under vacuum was followed, first at 35 ◦ C (95 ◦ F) for 24 h and then temperature ramped up at 0.5 ◦ C/min to 115 ◦ C (239 ◦ F) and was held for 3 h. A wet saw was used to machine ballistic test specimens that were 304 mm × 304 mm (12 inch × 12 inch) in-plane dimensions. Properties of the composite are provided in Table 2.1. A smooth bore, helium gas gun was used to propel steel right circular cylindrical projectiles into the fixtured targets. The fixture clamped the targets between two 360 mm square by 6.4 mm thick aluminum plates with 203 mm (8 inch) diameter central holes. Impact and residual velocity were measured by radar. Mechanical properties of the composites may be found in a prior publication [5]. Post-experimental high-resolution images were captured using a Phase One IQ180 80 MP CCD digital back and DT RCam reprographic camera, Phase One 120 mm macro lens, Kaiser RSD adjustable copy stand, and Huion light box backlight and two movable incandescent front lights. The full resolution of these images is 10,328 x 7760 pixels and a spatial resolution of 300 dots per inch. The smallest size that can be resolved is 84 μm, sufficient to distinguish multiple transverse cracks across an average 5080 μm tow width. Image resolution allowed ImageJ [18] processing to correlate projectile impact location to either a “center impact (A)” on fiber-rich tow-tow crossovers or an “off-center impact (B)” on adjacent parallel tows or matrix-rich interstitial regions within the PW architecture. Highresolution images were inspected and damage was counted and translated into digital damage maps with MATLAB.

2.3 Ballistic Results Impact velocity, VI , and residual velocity, VR , are provided in Table 2.2, along with the relative impact locations, either A = 188 m/s), but “center impact (A)” or “off-center impact (B).” Ballistic limit velocity was calculated for center impact (V50 B there was insufficient data for calculating ballistic limit velocity for off-center impact (V50 ) in accordance with MIL-STD662F [10]. Hence, the ballistic limit equations by Lambert and Jonas [19] and Haque and Gillespie [20] were used to fit the A A data and determine a ballistic limit velocity for center impact VBL,LJ = 188 m/s, VBL,H G = 187 m/s and off-center impact B B VBL,LJ = 167 m/s, VBL,H G = 166 m/s. These data and fits are plotted in Fig. 2.1a for center impact and in Fig. 2.1b for off-center impact. Since the targets were thin, single-layer composites projectiles either perforated or rebounded, so the data in Fig. 2.1 is separated into complete perforation or no perforation.

2.4 Quantitative Investigation of Mesoscale Damage Damage mechanisms were investigated for impact velocities close to the ballistic limit using high-resolution images for a non-perforation at an impact velocity slightly less than the ballistic limit velocity, target 5158–22, VI = 162 m/s, VR = −26 m/s, shown in Fig. 2.2, and for a complete perforation at an impact velocity a slightly more than the ballistic

12

C. S. Meyer et al.

Table 2.2 Perforation data of single layer PW S-2 Glass/SC-15 composites, for center impact, A, and off-center impact, B Target ID 5159–1 5159–2 5159–3 5159–4 5159–5 5159–6 5159–7 5159–8 5159–9 5159–10 5159–11 5159–12 5159–13 5159–14 5159–15 5155–16 5155–17 5159–18 5155–19 5155–20 5155–21 5158–22 5158–23 5158–24 5155–25 5158–6 5155–10

Impact velocity VI , m/s 104 125 166 214 293 200 188 198 186 153 488 472 439 355 239 195 186 177 181 174 167 162 166 215 282 173 152

Residual velocity VR , m/s −14 −19 −25 134 257 129 −25 113 121 −24 N/A 448 412 322 191 137 119 −32 102 62 84 −26 −26 175 254 −97 N/A

Relative impact locations B A A A B A A B B A N/A B A A A B B A A B B A A B B A A

limit velocity, target 5155–17, VI = 186 m/s, VR = 119 m/s, shown in Fig. 2.3. White lines mark approximate tow widths and interstitial, matrix-rich areas between fiber tows in Figs. 2.2 and 2.3. Transverse tow cracking, tow-tow delamination, and 45◦ matrix cracking are damage modes that occur in the mesoscale [5, 6], the scale of a tow-width, as seen in Figs. 2.2 and 2.3. In ballistic impact, transverse cracks initiate and propagate due to primary tow [21] in-plane tensile loading, which is spread by in-plane shear. Shear lag redistributes this loading as transverse cracks form, leading to formation of multiple, roughly parallel transverse cracks across tow widths, as seen in Figs. 2.2 and 2.3. Transverse cracks propagate parallel to the fiber direction and may spread across multiple unit cells, as seen in Figs. 2.2 and 2.3. Transverse cracks propagate through the tow thickness until intersecting orthogonal tows, then the cracks turn and propagate between orthogonal tows becoming tow-tow delamination. Accumulation of transverse cracks reduces the composites capacity for spreading load by in-plane shear and causes primary tows to be loaded preferentially. Primary tows stretch in tension and form an expanding pyramid of deformation, and in-plane shear and bending at the expanding pyramid wave-front lead to progressively larger diamonds of 45◦ matrix cracks in matrix-rich interstitial regions, as may be seen in Figs. 2.2 and 2.3. Sufficiently energetic impacts cause punch-shear damage as the projectile punches through the composite, causing tension-shear fracture of fibers and matrix around the projectile annulus, fiber-matrix interface debonding, and matrix pulverization and ejection. The reader is referred to other works by the authors for additional discussion on these mesoscale damage modes [5, 6].

2.5 Qualitative Investigation of Mesoscale Damage The high-resolution images captured for several specimens had their mesoscale damage modes quantified. The images were inspected for transverse cracks, tow-tow delamination, and 45◦ matrix cracks, and the quantity of each damage mode was counted for each unit cell in a 41x41 grid of unit cells across the 203 mm (8 inch) diameter (each unit cell is 5.08 mm square). The quantities of each damage mode were placed into damage maps to provide at-a-glance visual representations of

2 Ballistic Impact Experiments and Quantitative Assessments of Mesoscale Damage Modes in a Single-Layer Woven Composite

a

13

500 Complete perforation, center Impact No perforation, center Impact

450 400 350

VR, m/s

300 250 200 Lambert-Jonas Equation p p 1/p VR=β(VI - VBL )

150

β =0.960, p=2.70, VBL=188 m/s

100

2

r =0.993

50

Haque-Gillespie Equation 2 2 2 1/2 VR=[ VT + β *(ζVI - VBL )]

VT=103 m/s, β =0.976, ζ =1.03, VBL=187 m/s

0

2

r =0.996

-50

0

50

100

150

200

250

300

350

400

450

500

VI, m/s

b 500

Complete perforation, off-center impact No perforation, off-center impact

450 400 350

VR, m/s

300 250 200 Lambert-Jonas Equation p p 1/p VR=β(VI - VBL )

150

β=0.989, p=2.67, VBL=167 m/s 2

r =0.990

100

Haque-Gillespie Equation 2 2 2 1/2 VR=[VT + β*(ζVI - VBL )]

50

VT=63.6 m/s, β=0.994, ζ=1.05, VBL=166 m/s

0 -50

2

r =0.984

0

50

100

150

200

250

300

350

400

450

500

VI, m/s Fig. 2.1 Ballistic limit analysis of perforation data presented in Table 2.1 with determination of VBL for (a) center impact and for (b) off-center impact using Lambert-Jonas [19] and Haque-Gillespie [20] ballistic limit equations

the damage modes. Fig. 2.4 provides damage maps for (a) transverse cracks, (b) tow-tow delamination, and (c) 45◦ matrix cracks. In Fig. 2.4, color bars are provided such that the color intensity in the figure can give the viewer an at-a-glance visual estimate of the quantity of each damage type. In this way, it is seen that damage increases up to a maximum at the ballistic limit and decreases thereafter as the damage occurs under higher energy density. The front face damage is quantified in Table 2.3. In Table 2.3, tow-tow delamination total represents total number of unit cells with any percentage of delamination, while other damage totals are actual numbers of cracks counted per unit cell and totaled across the specimen.

14

C. S. Meyer et al.

Fig. 2.2 Damage mechanisms in an off-center impact, non-perforated target, 5158–22, VI = 162 m/s, VR = −26 m/s with (a) transverse tow cracking, (b) tow-tow delamination, and (c) 45◦ matrix cracking primary tows and interstitial matrix-rich areas between woven tows indicated by white lines

Fig. 2.3 Damage mechanisms in an off-center impact, completely-perforated target 5155–17, VI = 186 m/s, VR = 119 m/s with (a) transverse tow cracking, (b) tow-tow delamination, (c) 45◦ matrix cracking, and (d) punch-shear damage and interstitial matrix-rich areas between woven tows indicated by white lines

2 Ballistic Impact Experiments and Quantitative Assessments of Mesoscale Damage Modes in a Single-Layer Woven Composite

15

Fig. 2.4 Digital damage maps of (a) transverse cracks, (b) tow-tow delamination, and (c) 45◦ matrix cracks for three impact velocities, one less than (Left, VI = 104 m/s), one near to (Middle, VI = 174 m/s), and one greater than (Right, VI = 239 m/s) the ballistic limit velocity

2.6 Conclusions Ballistic impact experiments were conducted on a single-layer woven composite with a projectile scaled on the order of a single tow width. Mesoscale damage modes occur at the length scale of a single tow width and these damage modes include transverse cracks, tow-tow delamination, and 45◦ matrix cracks. Distinct ballistic limit velocities were found for

16

C. S. Meyer et al.

Table 2.3 Damage quantification: total numbers of cracks, total number of delaminated unit cells, and percentages of area with each damage type Target ID 5159–1 5159–10 5155–20 5159–8 5159–15 5155–25 5159–14 5159–12

Transverse cracks Total % of area damaged 953 28.3 1236 34.6 1302 31.3 1252 31.3 541 13.3 346 7.9 529 13.3 279 7.6

45◦ Matrix cracks Total % of area damaged 71 3.5 378 15.9 204 7.7 247 8.7 105 4.5 58 2.5 78 3.7 28 1.5

Tow-tow delamination Total % of area damaged 287 17.1 57 15.3 802 47.7 302 18.0 77 4.6 61 3.6 79 4.7 51 3.0

two different impact locations indicating, as expected, greater energy is required to perforate at the fiber-rich tow overlaps and less energy is required to perforate fiber-poor and matrix-rich interstitial regions. This result is expected because of the differences in strength and load distribution, hence differences in damage evolution and energy dissipation, between these two impact locations. This mesoscale effect would not be seen for projectiles scaled larger than a tow width since the energy is distributed over a wider area and dissipation is averaged over these distinct fiber-rich and matrix-rich regions. Three mesoscale damage modes were qualitatively investigated using high-resolution optical inspection of a back-lit, thin, translucent plain weave glass fiber reinforced epoxy composite. These three mesoscale damage modes were also quantified using high-resolution imaging and were graphically depicted in damage maps to provide at-a-glance visual identification of this damage. From these damage maps, it was found that the extent of damage increases with increasing impact velocity and is a maximum at the ballistic limit velocity, and the extent of damage then decreases for impact velocities beyond the ballistic limit. Inspection of damage maps also reveals that tow-tow delamination is concentrated in primary tows and nearby secondary tows, as revealed by the cross-pattern of tow-tow delamination. Inspection of damage maps also reveals that 45◦ matrix cracks are concentrated in quadrants separated by the cross of primary tows. These ballistic impact data, highresolution qualification and quantification of mesoscale damage, and digital damage maps provide useful data for gleaning new understanding of the time-resolved process of damage evolution in perforation of thin composites and are useful for finite element model validation. Acknowledgements Research was sponsored by the U.S. Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Army Research Laboratory or the U.S. Government. Thanks to Molla Ali of University of Delaware, Center for Composite Materials for help with material characterization. Thanks to Zuhal Onuk, Bridgit Kioko, Oreoluwa Adesina, and Carisse Lansiquot of Morgan State University for help with MATLAB scripting and damage counting. Thanks to Nebiyou Getinet and Jian Yu of the U.S. Army Research Laboratory for help with conducting ballistic experiments.

References 1. Gower, H.L., Cronin, D.S., Plumtree, A.: Ballistic impact response of laminated composite panels. Int. J. Impact Eng. 35, 1000–1008 (2008) 2. Karkkainen, R.L.: Dynamic micromechanical modeling of textile composite strength under impact and multi-axial loading. Compos. Part B Eng. 83, 27–35 (2015) 3. Karkkainen, R.L., Mcwilliams, B.: Dynamic micromechanical modeling of textile composites with cohesive interface failure. J. Compos. Mater. 46(18), 2203–2218 (2012) 4. Lomov, S.V., et al.: Full-field strain measurements for validation of meso-FE analysis of textile composites. Compos. Part A Appl. Sci. Manuf. 39(8), 1218–1231 (2008) 5. Meyer, C.S., et al.: Mesoscale ballistic damage mechanisms of a single-layer woven glass/epoxy composite. Int. J. Impact Eng. 113(November 2017), 118–131 (2018) 6. Bonyi, E., et al.: Assessment and quantification of ballistic impact damage of a single-layer woven fabric composite. Int. J. Damage Mech. (2018) 7. Gama, B.A., Gillespie, J.W.: Finite Element Modeling of Impact, Damage Evolution and Penetration of Thick-Section Composites. Int. J. Impact Eng. 38, 181–197 (2011) 8. Haque, B.Z., Gillespie Jr., J.W.: Penetration and Perforation of Composite Structures. Mech. Eng. Res. J. 9(March), 37–42 (2013) 9. Jordan, J.B., Naito, C.J., Haque, B.Z.: Progressive damage modeling of plain weave E-glass/phenolic composites. Compos. Part B Eng. 61, 315–323 (2014) 10. Military Test Method Standard MIL-STD-662F, V50 Ballistic Test for Armor, DOD, 1997

2 Ballistic Impact Experiments and Quantitative Assessments of Mesoscale Damage Modes in a Single-Layer Woven Composite

17

11. Rakhmatulin, K.H.A.: Oblique impact at a large velocity on a flexible fiber in the presence of friction (in Russian). Prikl Mat Mekh. 9, 449–462 (1945) 12. Rakhmatulin, K.H.A.: Impact on a flexible fiber (in Russian). Prikl Mat Mekh. 11, 379–382 (1947) 13. Rakhmatulin, K.H.A.: Normal impact at a varying velocity on a flexible fiber (in Russian). Uchenye Zap. Moskovosk gos Univ. 4, 154 (1951) 14. Rakhmatulin, K.H.A.: Normal impact on a flexible fiber by a body of given shape (in Russian). Prikl Mat Mekh. 16, 23–24 (1952) 15. Smith, J.C., McCracking, F.L., Schiefer, H.F.: Stress-strain relationships in yarns subjected to rapid impact loading. Part V: wave propagation in long textile yarns impacted transversely. J. Res. Natl. Bur. Stand. (1934). 60(5), 701–708 (1955) 16. Smith, J.C., McCrackin, F.L., Schiefer, H.F.: Stress-strain relationships in yarns subjected. Text. Res. J. 28, 288–302 (1958) 17. Leigh Phoenix, S., Porwal, P.K.: A new membrane model for the ballistic impact response and V50performance of multi-ply fibrous systems. Int. J. Solids Struct. 40(24), 6723–6765 (2003) 18. Schneider, C.A., Rasband, W.S., Eliceiri, K.W.: NIH image to ImageJ: 25 years of image analysis. Nat. Methods. 9(7), 671–675 (2012) 19. Lambert, J.P., Jonas, G.H.: Towards standardization in terminal ballistics testing: velocity representation. BRL-R-1852, Aberdeen Proving Ground, MD (1976) 20. Haque, B.Z., Gillespie, J.W.: A new penetration equation for ballistic limit analysis. J. Thermoplast. Compos. Mater. 28(7), 950–972 (2015) 21. Naik, N.K., Shrirao, P.: Composite structures under ballistic impact. Compos. Struct. 66, 579–590 (2004)

Chapter 3

A Novel Approach for Plate Impact Experiments to Obtain Properties of Materials Under Extreme Conditions Bryan Zuanetti, Tianxue Wang, and Vikas Prakash

Abstract In this paper we present a novel approach to conduct normal plate impact experiments at elevated temperatures up to 1000 ◦ C. To enable this approach, custom adaptations are made to the breech-end of the single-stage gas-gun at Case Western Reserve University. These adaptations include a precision-machined steel extension piece, which is strategically designed to mate the existing gun-barrel by providing a high tolerance match to the bore and keyway. The extension piece contains a vertical cylindrical heater-well, which houses a resistive coil heater attached to a vertical stem with axial/rotational degrees of freedom. The assembly enables thin metal specimens held at the front-end of a heat-resistant sabot to be heated uniformly across the diameter to the desired test temperatures. Using the configuration, symmetric normal plate impact experiments are conducted on 99.6% tungsten carbide (no binder) using a heated (room temperature to 650 ◦ C) WC flyer plate and a room temperature WC target plate at impact velocities ranging from 233 to 248 m/s. The measured free-surface particle velocity profiles are used to obtain the elastic/plastic behavior of the impacting WC plates as well as the temperaturedependent shock impedance of the flyer. The results indicate a dynamic strength of approximately 6 GPa for the WC used in the present study (strain-rates of about 105 ), and a decreasing flyer plate longitudinal impedance with increasing temperatures up to 650 ◦ C. Keywords Normal plate impact · Incipient plasticity · Elevated temperatures · Tungsten carbide · Hugoniot elastic limit · Longitudinal impedance

3.1 Introduction It is well understood that dynamic response of most engineering materials is pressure, temperature, and rate sensitive. In this regard, investigations aimed towards exploration of material behavior (i.e. plasticity, strength and failure) at thermomechanical extremes can be experimentally challenging, and thus, knowledge of the dynamic response of materials in certain thermomechanical regimes is still limited by the deficiency of reliable experimental data. Consequently, there is an opportunity for contributions from research in the realm of experimental techniques and novel diagnostics. Traditionally, light gas-guns and/or explosively driven plate impact experiments have been utilized to study the dynamic inelasticity and other phenomenon such as spallation, or phase transformation that occur with very high strain rates (105 – 107 /s), or combinations of high stresses (hydrostatic and/or deviatoric) under dynamic loading conditions [1–3]. These tools have also been adapted to include heating elements which enable investigations of material behavior to extend into thermomechanical extremes. These adaptations usually involve the addition of an induction, or of a resistive heater element to the target-end of the gas-gun [4–6]. Though these adaptations have been shown to be experimentally feasible, the approach inherently leads to special experimental challenges, which require careful considerations. Some of these challenges include: (1) heating the target using an induction coil heating system or a resistive heater, subjects various elements of the target holder and/or the alignment-fixture to differential thermal expansion, requiring remotely controlled alignment adjustment tools with continuous feedback for maintaining parallelism of the target and flyer plates; (2) heating the target plate in combined pressure-shear plate impact configuration requires the fabrication of heat-resistant optical holographic gratings on the target free surface for the measurement of the free surface particle velocity motion during the experiments; (3) a limitation on the stress-states that can be imparted on the sample in certain plate impact configurations that involve sandwiched specimens between hard elastic target plates, because of possible thermal softening of heated target plates that must remain elastic during impact to allow unambiguous interpretation of the experimental results from the measured particle velocity profiles;

B. Zuanetti · T. Wang · V. Prakash () Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_3

19

20

B. Zuanetti et al.

and (4), differential thermal expansion of the target assembly, particularly for targets that utilize an optical window, since precise tolerances between the sample, bond layer, coating(s), and window are difficult to maintain over large temperature ranges [7]. Despite advances in our previous state of understanding regarding the dynamic behavior of materials, significant questions still remain unclear. A few of these important questions are: What is the origin of enhanced strain-rate sensitivity observed in metals? What is the cause for the observed weak thermal softening in metals near their liquid-solidus transition temperature at high loading rates? Will the evolution of dislocation structures continue to play a major role in controlling the shearing resistance of metals near their melting temperature? And, how can existing dynamic plasticity models be updated in order to fully correlate the observed mechanical response of metals over large strain-rate, pressure, and temperature ranges? The objective of the current research is to address some of these important questions by extending the capability of the current plate impact experiments to obtain experimental data under thermomechanical extremes. The present paper discusses a novel approach for performing elevated temperature plate impact investigations utilizing a breech-end sabot heater system, which enables thin metal specimens held at the front-end of a heat-resistant sabot to be heated to temperatures in excess of 1000 ◦ C, prior to firing, thus allowing high temperature normal and/or combined pressureshear plate impact experiments to be conducted while alleviating several of the aforementioned experimental challenges. We present results from three normal plate impact elevated temperature experiments conducted on 99.6% pure tungsten carbide (WC), in which the WC flyer plates are heated to temperatures in the range from 23 to 650 ◦ C. The approach enables the Hugoniot Elastic Limit (HEL) of WC, corresponding to its dynamic strength under uniaxial strain, to be obtained, and also estimation of the temperature-dependent longitudinal impedance of the heated WC flyer plate. The results indicate a HEL ∼ 6.0 GPa for WC at strain-rates of about 105 /s, and a decrease in the longitudinal impedance of WC by about 22% as the WC flyer temperatures are increased from 23 to 650 ◦ C.

3.2 Experimental Procedure The schematic of the elevated temperature normal plate impact experiment is shown in Fig. 3.1. To conduct the elevated temperature experiments, the specimen (flyer plate) on the sabot, at the breech end of the gun barrel, is heated by a resistive heater housed in a custom designed extension piece made from SAE 4340 steel. The extension piece (shown in red) is approximately 610 mm in length and contains an 82.5 mm diameter cylindrical bore with a 6.35 mm × 3.17 mm key-way. The vertical cylindrical heater-well, which is an integral part of the steel extension piece, has a diameter of 76.2 mm and houses a 54 mm diameter 800 W resistive coil heater rated for a maximum operating temperature of 1200 ◦ C in an oxidizing environment (100 mTorr vacuum). The choice of resistive coil heating is dictated by the desired temperature range as well as the heating rates it enables when compared with other commercially available resistive heating elements. The heater-head is attached to a vertical stem with over a 100 mm of axial reach, which allows the heater to be safely drawn-in and stored within the heater well prior to firing the gas-gun. Moreover, to enable elevated temperature oblique plate impact experiments, a full 360◦ of rotational freedom is provided to the heater head so as to align it parallel to an inclined flyer plate on the sabot. A heat-resistant sabot is designed strategically to minimize heat transfer to the sabot body (by conduction) from the heated specimen plate end of the sabot, this is shown schematically in Fig. 3.2. Traditionally, sabots used in plate impact experiments are machined almost entirely from aluminum or Lexan; in the present application, however, precautions must be taken to prevent thermal expansion as well as thermal softening of the sabot so as to avoid seizure of the sabot in the gun barrel. The new sabot design comprises an aluminum cap adhered to an aluminum tube 200 mm in length and 82.5 mm in diameter using two-part 5-minute epoxy. The cap accommodates a sealing O-ring and an alignment key, which slides in the slotted keyway in the gun barrel to prevent rotation of the sabot during its acceleration down the gun barrel [8]. The length of the aluminum tube is strategically selected to enable impacts with tilts 1000. That is, in this case, for V > 0.03 m/sec. The target plate exceeds this velocity 0.13 ms after first motion. Therefore, the effect of Re < 1000 can be ignored. Given that Cd = 1, and V = 4.873 m/sec, then D = 2.56 kg-m/sec2 . The mass of the aluminum plate is 10.01 kg. Therefore the maximum possible deceleration is 0.256 m/sec2 . The Gage 314 target plate reaches its peak velocity about 3.1 ms after it starts to move, Fig. 4.7. If this deceleration acted on the plate for this whole time (which it does not, since the drag force depends on V2 ) it would reduce the peak velocity by 0.008 m/sec. Thus the effect of air drag on peak velocity is negligible. Air drag night have a measurable, though not necessarily significant, effect on the motion of the plate over longer times. However, times significantly after the time of Vmax are not of interest here. Hence, air drag can be ignored in all cases that are of interest to here.

4.5 Target Mass – Charge Mass Effects The mass of the target relative to the mass of the explosive enters into the problem of determining the impulse applied to the target by the rising soil and explosion products in at least two different ways. First, the test takes place in a gravity field and this affects the initial velocity and, hence the measured impulse. This can be shown intuitively to be so by examining an extreme case: E.g., imagine a shallowly buried charge with a mass of a few grams being detonated under a target whose mass is multiple tons. There will be no measurable initial target velocity and therefore, zero impulse will be calculated, even though an impulse was actually applied to the target. The second issue in which the mass of the target and the mass of the explosive are important is fluid – structure interaction. I.e., the response of the target to the loading modifies the loading on the target and, in fact, generally reduces the loading by recoiling from it. This was explored by G. I. Taylor [8] for blast waves in free air operating on a responding target and extended to the compressible air blast case by Kambouchev, et al. [9] Unfortunately, the case where the target loading is applied by a mixture of soil, water and air plus explosive products, which is initiated by a shallowly buried explosive in water – saturated soil does not appear to be amenable to a closed form analytical solution. Never the less, if the other conditions of explosive events are the same, i.e., depth of burial of the charge (DoB), height of the target (HoT) above the surface of the soil, size of target, soil properties, etc., there is, a consistent, power law relationship between the charge mass to target mass ratio (m/M) and the reduced impulse (I/M = V), as measured by the target’s peak velocity, at least if the range of values of the ratio m/M is not too large. In this paper, this relationship is developed, based on test data, to provide a heuristic tool to estimate the impulse that one would expect to measure if either the charge or target mass is changed between two otherwise identical tests.

4.6 Discussion Annex 4.2 shows the test data used in this analysis. The peak actual velocity V of the target is simply the peak measured target impulse or momentum (M*V) normalized by the target mass (MV/M = V). As stated above, the charge mass can also be normalized in the same way, i.e., (m/M). The peak target velocity (V) is plotted against the normalized charge mass (m/M) from Annex 4.2 in Fig. 4.11. In all cases shown, for all combinations of DoB and HoT and all charge sizes, the exponent in the power law equation of the line fitted to the data is less than 1. For a given target mass and loading conditions, it would seem reasonable to assume that if the charge weight were doubled, all else being the same, the impulse applied to a given target would also double. Or, conversely, if the charge mass were held the same and the target weight were doubled, the impulse would still be the same. This is, in fact, the case. However, experimental results, Fig. 4.11, show that the measured target velocity does not increase linearly with (m/M), but rather increases less than linearly. The exponents range from 0.77 to 0.97, depending on the conditions of the tests, i.e.,

34

L. C. Taylor et al.

Fig. 4.11 Target velocity vs. charge to target mass ratio

the depth of burial and height of target. Most are between 0.8 and 0.9. If the impulse actually applied to the targets were known, it could be represented simply by MV, and the exponent, in all cases, would be 1, unity. There appear to be at least two possible reasons for this exponent, in every case, to be less than one. One is fluid structure interaction: When the target recoils from the loading by moving or deforming significantly it reduces the load being applied to the target. A lighter target or a very deformable one recoils more quickly from the loading and thereby reduces the effective loading and thus reduces the resulting velocity of the target. It does this in two ways: The velocity of the target reduces the relative velocity of the material hitting the target and a lighter target is farther from the explosion at any particular time during the event. However, [10] shows that, at least under the conditions of HoT, DoB, soil conditions, etc., when a particular set of plastically deforming targets is compared with a non-deforming target of the same mass and geometry, loaded essentially identically, all targets, whether deforming and non-deforming, very quickly assumed essentially the same upward velocity in spite of the deformable targets’ deformation. This plastic deformation would be expected to relive some of the loading due to its deformation, if fluid structure interaction were a major factor. Thus, it would appear that, over the range of charge weights and target weights considered in this paper, fluid structure interaction is probably not as significant as the charge vs. target weight effect discussed below. This latter, and, we believe, more important, cause for the exponent to be less than 1 is an artifact of the weight of the target relative to the weight of the charge that is the origin of the load on the target. To use an extreme illustration, if one imagines a shallowly buried charge of a few grams detonated under a target whose mass is multiple tons, no initial target velocity will be measured and therefore, zero impulse will be calculated, i.e., zero V, even though an impulse was actually applied to the target. The target velocity (V) that is measured in a test with a free flying target is not the result of just the applied force (F); it is the result of the total force applied to the plate: (F – Mg), where g is the acceleration due to gravity. Consider the setup depicted in Fig. 4.12 below with a rigid plate suspended above the ground which will be subjected to a short, time-dependent force Fp in the upward (positive) direction.

4 Effect of the Ratio of Charge Mass to Target Mass on Measured Impulse

35

Fig. 4.12 Description of suspended plate loading in experiments

In this figure z = zp is used to denote the location (height) of the platform and z = z0 is the initial equilibrium plate location zp (0) = z0 . The other forces acting on the platform are those due to gravity (acting downward) and the force of the support holding the platform up. That is:  Fp (t) =

P (x, t) dx

(4.2)

p

Fg = −Mg  Fs =

Mg − Fp (t) Fp (t) ≤ Mg and zp (t) = z0 0 Fp (t) ≥ Mg or zp (t) > z0 

FT = F p + F g + F s =

Fp (t) ≤ Mg and zp (t) = z0 0 Fp (t) − Mg Fp (t) ≥ Mg or zp (t) > z0

(4.3)

(4.4)

(4.5)

Under the assumption that the loading occurs over a relatively short time tI (e.g., between 2 and 4 msec) an impulse will be imparted to the platform, which according to the impulse-momentum law is tI I=

FT (t)dt = Mv I

(4.6)

0

causing the platform to move upward with a velocity vI . Note that the impulse defined in (5) is derived from the total force on the platform, not just the force due to the explosive loading. In particular, if the platform mass is sufficiently large, e.g. M ≥ Fp (t)/g, then the total force will remain zero, and the experiment will not be able to measure the loading from the explosive. In this sense, we actually desire the impulse due to the target loading only, that is. tI Ip =

Fp (t)dt.

(4.7)

0

In general, this difference between total and loading forces should be taken into consideration as long as Fp (t) is not a Dirac delta function, taking on an infinitely large value over an infinitesimally small time period. If, for example, Fp (t) is a “step” function.  Fp (t) =

Fp0 0 < t ≤ tI t > tI 0

(4.8)

36

L. C. Taylor et al.

Fig. 4.13 Hypothetical force as a function of time

then it follows from (5) and (6) that   I = Fp0 − Mg tI = Ip − Mgt I

(4.9)

resulting in an apparent decrease in impulse of MgtI . If the assumption that Fp (t) is a step function (or increases instantaneously to a value larger then Mg) then the simple correction MgtI could be added to the observed impulse I to attain Ip . This has further implications in cases when Fp (t) “ramps up” over a non-zero time interval before attaining a value larger than Mg. For example, consider the situation depicted in Fig. 4.13. Here tM is the first time such that Fp (tM ) = Mg. To compare this case with the previous step function case, suppose that the value for Ip , that is the integral of Fp (t) is the same, so that Ip = Fp0 tI . In this case, the “observed” impulse will be tM tI tI I = FT (t)dt = Fp (t)dt − Mg (tI − tM ) = Ip − Fp (t)dt − Mg (tI − tM ) 0

tM

0

,

(4.10)

< Ip − Mgt M − Mg (tI − tM ) = Ip − Mgt I

which is less than the observed impulse due to a step function (9) (or one in which tM = 0). In this case the correction should be slightly less thanMgtI . For example, if Fp (t) is approximately linear fort ≤ tM , then the correction would be Ip = I + Mgt I − Mgt M /2

(4.11)

Unfortunately, an accurate value for tM is very difficult to ascertain, and is likely to be very small, in any case. The rate of rise of the force applied the target plate is very high. Hence, as a practical matter, it may be better to simply useMgtI as the correction. This analysis, at least in part, explains why a target of larger mass produces an observed impulse that is smaller than that expected, based on a test with a target of smaller mass. Further, Eq. (4.9) above suggests that the measured impulse depends not only on the applied impulse, but also on how “impulsive” the load was, i.e., how long the loading lasted. We plan to explore this effect in the near future. Figure 4.11 shows that over a reasonably wide range of test conditions, there is an empirical power law relationship I /M = Vmax = K(m/M)n ; K, n = empirical constants

(4.12)

4 Effect of the Ratio of Charge Mass to Target Mass on Measured Impulse

37

between the peak measured velocity of the target and reduced charge size (m/M) in which, in all cases, n < 1. This equation combines the effects of fluid – structure interaction and effect of the relative weight of the charge and target. Most, though not all, of the exponents are between 0.8 and 0.95. The test cases available are not all well distributed in terms of m/M and the total number of tests for some combinations of parameters is quite small, so these values of the empirical constants cannot be considered determined with great precision. Never the less, the values of the exponents show a consistent pattern of all being less than, but not far below unity.

4.7 Results Equation (4.12) can be used to estimate the peak velocity (impulse) to be expected in a planned test based on past tests at the same or scaled conditions of DoB, HoT, target size and soil properties, but a different charge weight or target weight or even when both are different. Since I /M = Vmax = K(m/M)n

(4.13)

I = MK(m/M)n .

(4.14)

so that

Then, I



n −n m = (M/m) K(m/M) = (M/m) K(M/m)

(4.15)

so that I



1−n m = K(M/m) .

(4.16)

When planning tests with different explosive weights and different target weights, but scaled test conditions of DoB, HoT, target size and soil material and conditions, or when estimating the expected impulse of another test with a different target mass or charge mass or both, one can say the following: (V2 /V1 ) = K(m2 /M1)n /K(m1 /M1 )n = ((m2 /M2 ) / (m1 /M1 ))n

(4.17)

(V 2/V 1) = (m2 M1 /M2 m1 )n

(4.18)

If there are not enough prior test data to provide the empirical exponent “n” in the above equation, for the conditions of the planned tests, an estimate of 0.85–0.9 is more likely give a better estimate of the expected test result than using unity. Acknowledgements The data used in this paper were generated in multiple test programs under the sponsorship of many different sponsors over the course of 14 years. They are too numerous to mention by name. Never the less, the authors wish to thank them all. The paper would not have been possible without a large amount of data. We also wish to thank the many under graduate and graduate students who carried out many of the test in the Dynamic Effects Laboratory at the University of Maryland over this time period. Finally, we particularly wish to thank Mr. Collin Pecora of the Army Research Laboratory, Aberdeen, MD for making available some relatively recent large scale test data from tests conducted in their VIMF.

Charge type 104 3 × RP-80 116 3 × RP-80 117 3 × RP-80 118 3 × RP-80 121 3 × RP-87 122 3 × RP-83 128 3 × RP-83 131 3 × RP-83 132 3 × RP-83 137 3 × RP-83 138 DETA 314 DETA 318 DETA 379 DETA T1 DETA O P 1 DETA O P 2 DETA 14.3–1 DETA 14.3–2 DETA Test 1 DETA Test 5 DETA T1 DETA Steel DETA 110 3 × RP-80 111 3 × RP-87 112 3 × RP-83 129 3 × RP-83 135 3 × RP-80 136 3 × RP-80 140 3 × RP-80 143 3 × RP-80 114 3 × RP-80 123 3 × RP-80 125 3 × RP-83 115 3 × RP-80 119 3 × RP-87 120 3 × RP-83 126 3 × RP-83

Chg mass [g] 0.609 0.609 0.609 0.609 0.207 3.333 3.333 3.333 3.333 3.333 0.609 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 9.3 4.4 4.4 0.609 0.207 3.333 3.333 0.609 0.609 0.609 0.609 0.609 0.609 3.333 0.609 0.207 3.333 3.333

Test conditions small scale tests

Test name Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Gage Gage Gage IT-2 Vel 2015.07.09 2015.11.17 Impulse Impulse Low press Low press Deform able 2015.03.23 Navy 2 Navy 2 Navy 2 Navy 2 Navy 2 Navy 2 Navy 2 Navy 2 Navy 4 Navy 4 Navy 4 Navy 5 Navy 5 Navy 5 Navy 5

Annex 4.1 Target mass [g] 1440 1525 1525 1525 1406 1406 6809 12,072 4517 19,190 1525 10,010 10,010 10,010 10,020 9931 9903 10,070 10,070 10,128 23,749 11,320 23,680 1440 1440 1440 6809 325 325 1106 1108 1440 2110 12,072 1525 1525 1440 2881.7

Target dimensions 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 20.6 × 20.6 in 20.6 × 20.6 in 20.6 × 20.6 in 8 in × 8in 19 in Dia 19 in Dia 19 in Dia 19 in Dia 19 in Dia 19 in Dia 14.3 in Dia 14.3 in Dia 19 in Dia 19 in Dia 15 in × 18 in 19 in Dia 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 6 in Dia 8 in Dia 8 in Dia 6 in Dia 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in 8 in × 8in

Mass ratio Chg/target 0.0004229 0.0003993 0.0003993 0.0003993 0.0001472 0.0023706 0.0004895 0.0002761 0.0007379 0.0001737 0.0003993 0.0004395 0.0004395 0.0004395 0.0004391 0.0004431 0.0004443 0.0004370 0.0004370 0.0004344 0.0003916 0.0003887 0.0001858 0.0004229 0.0001438 0.0023146 0.0004895 0.0018738 0.0018738 0.0005506 0.0005496 0.0004229 0.0002886 0.0002761 0.0003993 0.0001357 0.0023146 0.0011566 Target/chg 2364.532 2504.105 2504.105 2504.105 6792.271 421.842 2042.904 3621.962 1355.236 5757.576 2504.105 2275.177 2275.177 2275.177 2277.273 2257.045 2250.682 2288.580 2288.580 2301.818 2553.656 2572.727 5381.818 2364.532 6956.522 432.043 2042.904 533.662 533.662 1816.092 1819.376 2364.532 3464.696 3621.962 2504.105 7367.150 432.043 864.596

Reduced target size [m/kgˆ(1/3)] 2.3973 2.3973 2.3973 2.3973 3.4351 1.3603 1.3603 3.5029 3.5029 3.5029 2.3973 2.9451 2.9451 2.9451 2.9451 2.9451 2.9451 2.2166 2.2166 2.9451 2.2949 2.3251 2.9451 2.3973 3.4351 1.3603 1.3603 1.7980 2.3973 2.3973 1.7980 2.3973 2.3973 1.3603 2.3973 3.4351 1.3603 1.3603 DoB [in.] 0.19 0.19 0.19 0.19 0.13 0.33 0.33 0.33 0.33 0.33 0.19 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.51 0.39 0.39 0.38 0.27 0.67 0.67 0.34 0.34 0.34 0.34 0.19 0.19 0.33 0.19 0.13 0.33 0.33

DoB [m] 0.0048 0.0048 0.0048 0.0048 0.0033 0.0084 0.0084 0.0084 0.0084 0.0084 0.0048 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0130 0.0099 0.0099 0.0097 0.0069 0.0170 0.0170 0.0086 0.0086 0.0086 0.0086 0.0048 0.0048 0.0084 0.0048 0.0033 0.0084 0.0084

Reduced DoB [m/Kgˆ(1/3)] 0.0569 0.0569 0.0569 0.0569 0.0558 0.0561 0.0561 0.0561 0.0561 0.0561 0.0569 0.0605 0.0605 0.0605 0.0605 0.0605 0.0605 0.0605 0.0605 0.0605 0.0616 0.0605 0.0605 0.1139 0.1159 0.1139 0.1139 0.1019 0.1019 0.1019 0.1019 0.0569 0.0569 0.0561 0.0569 0.0558 0.0561 0.0561 HoT [in.] 0.76 0.76 0.76 0.76 0.53 1.33 1.33 1.33 1.33 1.33 0.76 1.58 1.58 1.58 1.58 1.58 1.58 1.58 1.58 1.58 2.03 1.58 1.58 0.76 0.53 1.33 1.33 0.76 0.76 0.76 0.76 0.38 0.38 0.67 0 0 0 0

HoT [m] 0.0193 0.0193 0.0193 0.0193 0.0135 0.0338 0.0338 0.0338 0.0338 0.0338 0.0193 0.0401 0.0401 0.0401 0.0401 0.0401 0.0401 0.0401 0.0401 0.0401 0.0516 0.0401 0.0401 0.0193 0.0135 0.0338 0.0338 0.0193 0.0193 0.0193 0.0193 0.0097 0.0097 0.0170 0 0 0 0

Reduced HoT [m/Kgˆ(1/3)] 0.2277 0.2277 0.2277 0.2277 0.2276 0.2262 0.2262 0.2262 0.2262 0.2262 0.2277 0.2449 0.2449 0.2449 0.2449 0.2449 0.2449 0.2449 0.2449 0.2449 0.2452 0.2449 0.2449 0.2277 0.2276 0.2262 0.2262 0.2277 0.2277 0.2277 0.2277 0.11387 0.11387 0.11393 0 0 0 0

38 L. C. Taylor et al.

Navy 5 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF

127 Test 1 Test 3 Test 4 Test 4a Test 5 Test 6 Test 7 Test ARL SB-1a SB-1b SB-1c SB-1d SB-1e SB-1f SB-1g SB-1i SB-1j SB-2a SB-2b SB-2c SB-2d SB-2e SB-2f SB-2a SB-2b SB-2c SB-2d SB-2e SB-2f

3 × RP-83 TNT flat TNT flat TNT flat TNT flat TNT flat TNT flat TNT flat TNT flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat C-4 flat 3.333 4554 4536 4545 4545 2334 4545 2334 7484 4000 4000 4000 4000 4000 4000 4000 4000 4000 8000 8000 8000 8000 8000 8000 8000 8000 8000 8000 8000 8000

6809.5 12,505,540 12,505,540 12,505,540 11,852,370 11,852,370 11,852,370 11,534,850 11,534,850 10,927,040 10,927,040 10,927,040 10,927,040 10,927,040 10,927,040 9307.715 9307.715 9307.715 10,927,040 10,927,040 10,927,040 2,792,315 2,792,315 2,792,315 10,927,040 10,927,040 10,927,040 2,792,315 2,792,315 2,792,315

8 in × 8in 8 × 9.25 ft 8 × 9.25 ft 8 × 9.25 ft 6 × 12 ft 6 × 12 ft 6 × 12 ft 6 × 12 ft 6 × 12 ft 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 60 in Sq 0.0004895 0.0003642 0.0003627 0.0003634 0.0003835 0.0001969 0.0003835 0.0002023 0.0006488 0.0003661 0.0003661 0.0003661 0.0003661 0.0003661 0.0003661 0.0042975 0.0042975 0.0042975 0.0007321 0.0007321 0.0007321 0.0028650 0.0028650 0.0028650 0.0007321 0.0007321 0.0007321 0.0028650 0.0028650 0.0028650

2043.054 2746.056 2756.953 2751.494 2607.782 5078.136 2607.782 4942.095 1541.268 2731.760 2731.760 2731.760 2731.760 2731.760 2731.760 232.693 232.693 232.693 1365.88 1365.88 1365.88 349.039 349.039 349.039 1365.88 1365.88 1365.88 349.039 349.039 349.039

1.3603 2.1892 2.1921 2.1907 2.1907 2.7356 2.1907 2.7356 1.8551 0.0960 0.0960 0.0960 0.0960 0.0960 0.0960 0.0960 0.0960 0.0960 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762 0.0762

0.33 4 8 4 4 3.175 4 3.175 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.0084 0.1016 0.2032 0.1016 0.1016 0.0806 0.1016 0.0806 0.1016 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508

0.0561 0.0613 0.1228 0.0613 0.0613 0.0608 0.0613 0.0608 0.0519 0.0320 0.0320 0.0320 0.0320 0.0320 0.0320 0.0320 0.0320 0.0320 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254 0.0254

0 16 16 8 8 0 16 6.35 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

0 0.4064 0.4064 0.2032 0.2032 0 0.4064 0.1613 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064 0.4064

0 0.2452 0.2455 0.1227 0.1227 0 0.2453 0.1216 0.2078 0.2560 0.2560 0.2560 0.2560 0.2560 0.2560 0.2560 0.2560 0.2560 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032 0.2032

4 Effect of the Ratio of Charge Mass to Target Mass on Measured Impulse 39

Test name Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Navy 1 + 6 Gage Gage Gage IT-2 Vel 2015.07.09 2015.11.17 Impulse Impulse Low press Low press Deformable 2015.03.23 Navy 2 Navy 2 Navy 2 Navy 2 Navy 2 Navy 4 Navy 4 Navy 4 Navy 5 Navy 5 Navy 5 Navy 5 Navy 5

Annex 4.2

104 116 117 118 121 122 128 131 132 137 138 Gage 314 Gage 318 Gage 379 T1 Old plate 1 Old plate 2 14.3–1 14.3–2 Test 1 Test 5 T1 Steel 110 111 112 129 135 114 123 125 115 119 120 126 127

Charge mass [g] 0.609 0.609 0.609 0.609 0.207 3.333 3.333 3.333 3.333 3.333 0.609 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 9.3 4.4 4.4 0.609 0.207 3.333 3.333 0.609 0.609 0.609 3.333 0.609 0.207 3.333 3.333 3.333

Test results, small scale tests Target mass [g] 1440 1525 1525 1525 1406 1406 6809 12,072 4517 19,190 1525 10010.78 10010.78 10010.78 10,020 9931 9903 10069.75 10069.75 10,128 23,749 11,320 23,680 1440 1440 1440 6809 325 1440 2110 12,072 1525 1525 1440 2881.7 6809.5

Mass ratio Chg/target 0.0004229 0.0003993 0.0003993 0.0003993 0.0001472 0.0023706 0.0004895 0.0002761 0.0007379 0.0001737 0.0003993 0.0004395 0.0004395 0.0004395 0.0004391 0.0004431 0.0004443 0.0004370 0.0004370 0.0004344 0.0003916 0.0003887 0.0001858 0.0004229 0.0001438 0.0023146 0.0004895 0.0018738 0.0004229 0.0002886 0.0002761 0.0003993 0.0001357 0.0023146 0.0011566 0.0004895 Target/chg 2364.532 2504.105 2504.105 2504.105 6792.271 421.842 2042.904 3621.962 1355.236 5757.576 2504.105 2275.177 2275.177 2275.177 2277.273 2257.045 2250.682 2288.580 2288.580 2301.818 2553.656 2572.727 5381.818 2364.532 6956.522 432.043 2042.904 533.662 2364.532 3464.696 3621.962 2504.105 7367.150 432.043 864.596 2043.054

Peak vel [m/sec] 5.696 6.158 4.742 5.638 1.882 22.348 6.597 3.133 8.293 2.052 6.074 4.873 5.275 5.009 5.425 4.478 6.106 5.367 5.521 4.179 4.352 4.972 1.957 6.118 2.435 23.970 6.439 17.279 8.073 5.950 5.394 10.973 3.851 39.395 22.861 16.474

Total impulse [kg-m/sec] 8.202 9.391 7.232 8.598 2.646 31.421 44.918 37.816 37.458 39.379 9.264 48.778 52.806 50.143 54.355 44.471 60.470 54.047 55.598 42.328 103.361 56.286 46.334 8.809 3.506 34.517 43.844 5.616 11.626 12.555 65.115 16.733 5.873 56.728 65.879 112.179

Specific impulse [kg-m/sec]/kg 13467.716 15420.057 11875.276 14118.119 12780.391 9427.218 13476.746 11345.829 11238.415 11814.802 15211.023 11085.995 12001.390 11396.037 12353.306 10107.029 13743.200 12283.492 12635.824 9619.896 11114.117 12792.385 10530.468 14465.182 16935.818 10356.241 13154.609 9221.399 19089.745 20616.053 19536.428 27476.337 28369.691 17020.140 19765.743 33657.178

40 L. C. Taylor et al.

IT tests IT tests IT tests IT tests IT tests IT tests Deformable Deformable Deformable Deformable Deformable Deformable F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 F S 2004 VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF VIMF

IT-4 AL-1 IT-4 AL-2 IT-4 STREP IT-4 ST IT-5 AL IT-5 ST T2 T3 T5 M2 T6 M16 Test 1 Test 3 Test 4 Test 4a Test 5 Test 6 Test 7 Test ARL SB-1a SB-1b SB-1c SB-1d SB-1e SB-1f SB-1 g SB-1i SB-1j SB-2a SB-2b SB-2c SB-2d SB-2e SB-2f SB-3a SB-3b SB-3c SB-3d SB-3e SB-3f

4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 0.55 4.4 4554 4536 4545 4545 2334 4545 2334 7484 4000 4000 4000 4000 4000 4000 4000 4000 4000 8000 8000 8000 8000 8000 8000 4000 4000 4000 4000 4000 4000

10,010 10,010 23,410 23,410 10,010 23,410 11,260 11,260 5498 11,349 1244 10,540 12,505,540 12,505,540 12,505,540 11,852,370 11,852,370 11,852,370 11,534,850 11,534,850 10,927,040 10,927,040 10,927,040 10,927,040 10,927,040 10,927,040 9307.7154 9307.7154 9307.7154 10,927,040 10,927,040 10,927,040 2792314.6 2792314.6 2792314.6 10,927,040 10,927,040 10,927,040 1861543.1 1861543.1 1861543.1

0.0004396 0.0004396 0.0001880 0.0001880 0.0004396 0.0001880 0.0003908 0.0003908 0.0008003 0.0003877 0.0004421 0.0004175 0.0003642 0.0003627 0.0003634 0.0003835 0.0001969 0.0003835 0.0002023 0.0006488 0.0003661 0.0003661 0.0003661 0.0003661 0.0003661 0.0003661 0.0042975 0.0042975 0.0042975 0.0007321 0.0007321 0.0007321 0.0028650 0.0028650 0.0028650 0.0003661 0.0003661 0.0003661 0.0021488 0.0021488 0.0021488

2275.000 2275.000 5320.455 5320.455 2275.000 5320.455 2559.091 2559.091 1249.545 2579.318 2261.818 2395.455 2746.056 2756.953 2751.494 2607.782 5078.136 2607.782 4942.095 1541.268 2731.760 2731.760 2731.760 2731.760 2731.760 2731.760 232.693 232.693 232.693 1365.880 1365.880 1365.880 349.039 349.039 349.039 2731.760 2731.760 2731.760 465.386 465.386 465.386

7.233 7.335 3.366 3.814 4.838 2.307 4.120 4.059 7.557 4.102 4.608 3.668 3.596 4.053 6.167 7.027 4.191 3.955 3.503 6.452 2.136 2.174 1.770 1.788 1.907 1.804 17.862 18.648 17.507 3.359 3.469 3.311 10.713 11.718 11.796 2.026 2.228 2.150 9.510 10.952 9.888

72.399 73.423 78.794 89.278 48.424 54.015 46.390 45.703 41.546 46.555 5.732 38.658 44975.236 50688.486 77121.472 83289.507 49673.283 46879.224 40411.614 74424.216 23336.152 23750.816 19342.155 19542.556 20840.703 19711.422 166.252 173.572 162.952 36700.152 37908.646 36179.787 29914.158 32720.163 32937.130 22141.855 24342.930 23495.188 17703.149 20387.625 18406.954

16454.390 16686.933 17907.622 20290.429 11005.498 12276.102 10543.147 10387.145 9442.190 10580.621 10421.463 8785.952 9875.985 11174.710 16968.421 18325.524 21282.469 10314.461 17314.316 9944.444 5834.038 5937.704 4835.539 4885.639 5210.176 4927.856 41.563 43.393 40.738 4587.519 4738.581 4522.473 3739.270 4090.020 4117.141 5535.464 6085.733 5873.797 4425.787 5096.906 4601.738

4 Effect of the Ratio of Charge Mass to Target Mass on Measured Impulse 41

42

L. C. Taylor et al.

References 1. Fourney, W.L., Leiste, H.U., Bonnenberger, R., Goodings, D.J.: Explosive impulse on plates Fragblast. Int. J. Blasting Fragmentation. 9(1), 1–17 (2005) 2. Taylor L. C., , Skaggs, R. R., Gault, W., Vertical impulse measurements of mines buried in saturated sand, Fragblast, Int. J. Blasting Fragmentation, 9 1, 2005, pp 19–28 3. Bergeron, D.M., Trembley, J.E.: Canadian research to characterize mine blast output, 16th International Symposium on the Military Aspects of Blast and Shock, Oxford, UK (2000) 4. Hlady, S.L.: Effect of soil parameters on land mine blast, 18th International Symposium on the Military Aspects of Blast and Shock, Bad Reichenhall, Germany (2004) 5. Gniazdowski, N.: The vertical impulse measurement facility maintenance and inspection manual, ARL Technical Report (2004) 6. Skaggs, R.R., Watson, J., Adkins, T., Gault, W., Canami, A., Gupta, A.D.: Blast loading measurements by the vertical impulse measurement fixture (VIMF). US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD (2007) 7. Hoerner, S.F.: Fluid Dynamic Drag: Theoretical, Experimental and Statistical Information, Hoerner Fluid Dynamics (1965) 8. Taylor, G.I.: In: Batchelor, G.K. (ed.) Aerodynamics and the Mechanics of Projectiles and Explosions, The Scientific papers of Sir Geoffrey Ingram Taylor, Vol. III. Cambridge University Press, Cambridge (1963) 9. Kambouchev, N., Noels, L., Radovitsky, R.: Nonlinear compressibility effects in fluid-structure interaction and their implications on the airblast loading of structures. J. Appl. Phys. 100, 063519 (2006) 10. Taylor, L.C., Leiste, H.U., Fourney, W.L.: Comparison of the response of deforming and non-deforming targets to the explosion of a charge buried in saturated sand, Proceedings of the 8th International Conference on Shock & Impact Loads on Structures, Adelaide, Australia, 2–4 December 2009

Chapter 5

Fracture and Failure Characterization of Transparent Acrylic Based Graft Interpenetrating Polymer Networks (Graft-IPNs) Balamurugan M. Sundaram, Ricardo B. Mendez, Hareesh V. Tippur, and Maria L. Auad

Abstract IPNs are made of two or more polymer networks, each polymerized in the presence of the other/s. They can be suitable alternatives to traditional polymers made from single monomer as desirable characteristics of the constituent polymers can be engineered into IPNs. In this study, an acrylic-based transparent graft Interpenetrating Polymer Networks or simply graft-IPNs were processed and their mechanical properties in general and fracture/failure behaviors in particular were characterized. Good optical transparency, high fracture toughness, and high stiffness were among the attributes targeted in the graft-IPNs for potential transparent armor applications. The graft-IPNs were synthesized by sequential polymerization of compliant elastomeric polyurethane (PU) phase and a stiff acrylate-based copolymer (CoP) phase to generate crosslinks (or, ‘grafts’) between the two networks. A series of such graft-IPNs were synthesized by varying the ratios of CoP:PU. Uniaxial tension tests were performed on the resulting IPNs to measure the elastic modulus and strength whereas mode-I fracture toughness was measured under both quasi-static and dynamic loading conditions. A Hopkinson pressure bar was used in conjunction with an optical technique called Digital Gradient Sensing (DGS) and ultrahigh-speed photography to measure the fracture behavior during stress wave loading. The results show significant enhancements in the crack initiation toughness for some of the graft-IPN compositions relative to the constituents as well as commercially procured PMMA and polycarbonate (PC) sheet stocks. Besides the optical transparency, the increase in fracture toughness is attributed to the grafts or crosslinks generated between the PU and CoP networks. Keywords Digital gradient sensing · Transparent material · Interpenetrating polymer networks (IPNs) · Dynamic fracture · Material characterization

5.1 Introduction Optically transparent structural materials made from polymers or their blends are highly suitable for both civilian and military applications requiring impact resistance such as safety enclosures, aircraft canopies, protective eyewear, automotive windows, and helmet visors etc. Polymers or their blends due [1, 2] to their low density, superior fracture toughness and high energy absorption are used as substitutes for transparent ceramics and glasses. One such family of polymer systems of interest is interpenetrating polymer networks or IPNs [3]. These polymer systems comprise of two or more networks which are at least partially interlaced on a molecular scale without being chemically crosslinked and hence inseparable unless individual crosslinks are broken. In this work, the mechanical characterization of newly synthesized transparent graft-IPNs [4] are discussed including their response to dynamic loading. The dynamic impact properties need to be fully understood if these materials should find an application in need of high impact resistance. Thus, in this study the mechanical behavior, including quasi-static and dynamic fracture behaviors of transparent acrylic/polyurethane graft-IPNs are discussed. Quasi-static tension and fracture tests are carried out first followed by dynamic impact/fracture tests using a modified Hopkinson pressure bar loading apparatus [5]. The measurements under dynamic loading are made using a full-field optical technique called Digital Gradient Sensing

Currently with Science and Technology Division, Corning Incorporated. B. M. Sundaram () · H. V. Tippur Department of Mechanical Engineering, Auburn University, Auburn, AL, USA e-mail: [email protected] R. B. Mendez · M. L. Auad Department of Chemical Engineering, Auburn University, Auburn, AL, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_5

43

44

B. M. Sundaram et al.

(DGS) in conjunction with ultrahigh-speed photography which has been established for a variety of transparent materials including polymers [2] and glass [6, 7]. These are described in the following sections starting with a brief description of DGS. This is followed by a brief description of graft-IPN synthesis. Material characterization details including quasi-static tension and fracture testing along with their results are detailed next followed by the dynamic characterization.

5.2 Digital Gradient Sensing (DGS) Figure 5.1 shows a schematic of the experiment setup for transmission-mode DGS [8] technique. In this method, a planar surface with speckle pattern (black and white spray painted random dots) is photographed through a planar transparent specimen. A broad-spectrum illumination is used for recording the gray scales on the target. The speckle pattern is first recorded through the specimen in a no-load/undeformed state to obtain a reference image. When loaded, the non-uniform stresses in the specimen affect the local refractive index. Additionally, the specimen thickness changes non-uniformly due to the Poisson effect. A combination of these two effects, known as the elasto-optic effect, cause the light rays to deflect from their initial path when they propagate through the specimen. The speckles are photographed again through the specimen in this loaded/deformed state. The local deviations of light rays can be quantified by correlating speckle images in the deformed and undeformed states to obtain in-plane speckle displacement components on the target plane. The local angular deflections are related to the gradients of in-plane normal stresses as [8], φx;y

 ∂ σxx + σyy . = ±Cσ B ∂ (x; y)

Fig. 5.1 The schematic of the experimental setup for Digital Gradient Sensing (DGS) technique

(5.1)

5 Fracture and Failure Characterization of Transparent Acrylic Based Graft Interpenetrating Polymer Networks (Graft-IPNs)

45

The immediate advantage of stress gradient measurements is that they can be numerically integrated using an integration scheme such as the Higher order Finite difference based Least-squares Integration (HFLI) [9] to obtain stress fields [10].

5.3 Material Synthesis The graft-IPNs were synthetized in a single-step procedure using polyurethane and the copolymer phases whose constituents were mixed separately at room temperature [4, 11]. To prepare the copolymer phase (CoP), MMA and the BisGMA resin were mixed at a mass ratio of 90:10 MMA:BisGMA. Thermal initiator was added for polymerization. Next, the diisocyanate was added to the PTMG/TRIOL mixture, with an additional amount of DCH added subsequently to generate crosslinks between networks. Once PU and CoP phases were prepared separately, they were mixed together and catalyzers for both the phases were added. Finally, all samples were poured into the molds and placed in an oven at 40 ◦ C for 17 h after which the samples were transferred to a water bath at 60 ◦ C for 24 h and at 80 ◦ C for another 24 h. Thus, prepared g-IPNs were removed from the molds and machined to final test samples. These steps were repeated by varying the ratio of CoP: PU from 100:10 to 60:40 by wt.% to obtain a family of such g-IPNs with an intent to optimize the CoP: PU ratio (see Ref. [12] for details).

5.4 Quasistatic Tests 5.4.1 Elastic Modulus The uniaxial tension tests were performed using Instron 4665 UTM in displacement control mode (0.015 mm/s). Specimens were made to dumbbell shape with a cross-section of 3 mm x 2.8 mm using CNC machine and three samples of each composition were tested. The strains were measured using an extensometer attached to the specimen. Engineering stressstrain plots were developed based on the measured load and strain. Elastic modulus was computed from the slope of the stress-strain curve at 1 million frames per sec) was used in conjunction with DGS and a Hopkinson pressure bar to load V-notched SLG plates to investigate the crack branching phenomenon. The experimental parameters were controlled such that a single mode-I crack that initiated at the V-notch tip propagated through the glass plate before branching into two prominent mixedmode daughter cracks. The optical measurements of angular deflection fields that represent stress gradients in two orthogonal in-plane directions were obtained. Using higher order finite-difference based least-squares integration (HFLI) scheme, stress invariant fields (σ xx + σ yy ) were evaluated near dynamically propagating crack-tip throughout the branching process. Keywords Digital gradient sensing · Crack branching · Soda-lime glass · Dynamic fracture · Hopkinson bar

6.1 Introduction Crack branching, one of the characteristics of dynamic brittle fracture, has been studied for several decades [1, 2]. Early experimental studies used crack speeds to indirectly infer the crack driving force and propose crack growth/branching criteria [3, 4]. Experiments have consistently noted an increase in fracture surface roughness prior to branching. This increase in roughness is often connected with crack path instability and branching seen in brittle solids. There have also been several numerical investigations on dynamic crack branching phenomenon [5–7]. Yet, there are inconsistencies between experimental observations and theoretical predictions, such as limiting speed, branching angle, etc., that motivates further investigation of this phenomenon using full-field optical methods. However, full-field non-contact optical measurement of deformations or stresses to visualize and quantify various fracture parameters during dynamic crack growth leading to branching in brittle solids has been limited to polymers and sorely lacking for brittle materials such as glass or ceramics. In light of these, the current work intends to overcome this experimental challenge posed by soda-lime glass (SLG) to examine dynamic crack growth and branching by extending the full-field optical mapping technique called Digital Gradient Sensing (DGS) [8]. With this background, the current work briefly describes the optical method of DGS as applied to study glass. Subsequently, the experimental details of implementing DGS in conjunction with ultrahigh-speed photography and impact loading are presented. Subsequently, crack branching, the associated crack-tip features, optical measurements and integrated stress fields are shown. A comprehensive analysis of measured angular deflections to locate the crack-tip, evaluate stress intensity factors (SIFs) and velocity histories, quantify surface roughness and plausible branching mechanism are reported.

Currently with Science and Technology Division, Corning Incorporated, Corning, NY, USA. B. M. Sundaram () Department of Mechanical Engineering, Auburn University, Auburn, Al, USA e-mail: [email protected] H. V. Tippur Department of Mechanical Engineering, Auburn University, Auburn, AL, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_6

51

52

B. M. Sundaram and H. V. Tippur

6.2 Digital Gradient Sensing (DGS) Figure 6.1 shows a schematic of the experimental setup for transmission-mode DGS [8]. In this method, a planar surface with speckle pattern (black and white spray painted random pattern) is photographed through a planar transparent specimen. Ordinary white light illumination is used for recording the gray scales on the target. The speckle pattern is first recorded through the specimen at no-load/undeformed state to obtain a reference image. When loaded, the non-uniform stresses affect the local refractive index of the specimen. Additionally, the specimen thickness changes non-uniformly due to the Poisson effect. A combination of these two effects, known as the elasto-optic effect, cause the light rays to deflect from their initial path when they propagate through the specimen. The speckle pattern is photographed again through the specimen in this loaded/deformed state. The local deviations of light rays can be quantified by correlating speckle images in the deformed and undeformed states to obtain in-plane speckle displacement components on the target plane. The local deviations of light rays in two orthogonal planes can be quantified using speckle displacements and the optical gap between the specimen and speckle target planes. The local angular deflections are related to the gradients of in-plane normal stresses as [8], φx;y

 ∂ σxx + σyy = ±Cσ B ∂ (x; y)

(6.1)

Using a pin-hole camera approximation, the angular deflections measured at the target plane can be mapped back to the yo specimen plane using the mapping function, tan θ = Ly = L+ , where y and yo are specimen and target plane coordinates (horizontal or vertical) [9, 10], respectively. Glass produces extremely small angular deflections when compared to other materials such as transparent polymers [11]. Hence, to increase the measurement sensitivity,  needs to be large. This in turn produces difficulties in terms of insufficient light intensity and increased sensitivity to random vibrations which may occur during experiments.

Fig. 6.1 The schematic of the experimental setup for transmission-mode Digital Gradient Sensing (DGS) technique

6 Dynamic Crack Branching in Soda-Lime Glass: An Optical Investigation Using Digital Gradient Sensing

53

6.3 Experimental Details The schematic of the experiment setup used for dynamic fracture tests is shown in Fig. 6.2 [12]. A Hopkinson pressure bar (long-bar) apparatus was used to load the sample. A 6 feet long, 1-inch diameter long-bar with a wedge shaped tip was held against an unconstrained specimen on an adjustable platform. A 12-inch-long, 1-inch diameter striker held inside the barrel of a gas-gun was used to impact the long-bar. Both the long-bar and the striker were of the same diameter and made of maraging steel. The striker was launched towards the long-bar using a gas-gun. When the striker contacted the long-bar, a compressive stress wave propagated along the bar before transmission into the specimen in contact. A commercially available SLG sheet with 3/16 thickness was machined to 6 × 4 in-plane dimensions with a 40◦ V-notch (see Fig. 6.3). The notch was extended using a diamond impregnated circular saw. The transmitted compressive wave forced the wedge open thereby initiating a crack in the extended notch tip. A target plate decorated with random black and white speckles was placed behind the specimen at a distance (∼1222 mm) to measure angular deflections of light rays using DGS in the region-of-interest. The speckle pattern was photographed through the specimen using a Kirana 05M ultrahigh-speed digital camera at >1 million frames per second. The camera is capable of recording 180 images at 924 × 768 pixels resolution. When the striker contacted the long-bar, a trigger signal initiated the recording of the fracture event at a preset delay (time for the stress-wave to propagate through the long-bar). The recorded images were correlated to obtain speckle displacements, using which angular deflections of light rays were evaluated as a 2-D array of data at spatial positions in the field-of-view.

Fig. 6.2 Experiment setup (top view) used to study dynamic fracture of soda-lime glass

54

B. M. Sundaram and H. V. Tippur

Fig. 6.3 Specimen geometry and associated loading configuration used to produce dynamic mode-I crack growth in SLG

4”

5/16”

6

40º

initial notch-tip

Fig. 6.4 Photographs of three fractured specimens (2 samples) showing consistent fracture pattern

6.4 Crack Growth Morphology The photographs of the corresponding fractured specimens (2 samples) are shown in Fig. 6.4. The crack upon initiation at the initial notch propagates first as a mode-I mother crack towards the middle of the specimen before branching into two daughter cracks. The daughter cracks then propagate almost symmetrically relative to the mother crack until they approach the specimen edge which perturbs crack paths due to stress wave reflections.

6.5 Optical Measurements The recorded reference and deformed images were correlated using ARAMIS image analysis software to obtain contours of constant angular deflections around a propagating crack-tip [13]. The details are avoided here for brevity. The resulting contours in Fig. 6.5 represent the angular deflections of light rays at three select time instants (t = 0 μs represents the time

6 Dynamic Crack Branching in Soda-Lime Glass: An Optical Investigation Using Digital Gradient Sensing

55

Fig. 6.5 Angular deflection contour plots (contour interval = 5 × 10−6 rad) for a SLG subjected to dynamic wedge loading. The arrowhead (in the top left image) shows the crack growth direction. The overlaid white lines suggest the approximate crack path. (t = 0 correspond to the crack initiation at original notch tip)

Fig. 6.6 (σ xx + σ yy ) contours (1.5 MPa intervals) obtained by integration Scheme [14]. The overlaid white line suggests the approx. Crack path. (t = 0 μs in these corresponds to crack initiation at the original notch tip)

at which the crack initiated). They represent the stress gradients [8] in two mutually perpendicular directions in the x-z and y-z planes with (x, y) denoting the in-plane coordinates of the specimen.

6.6 Stress Measurements A full-field integration scheme called the Higher-order Finite-difference-based Least-squares Integration, or HFLI, demonstrated in Ref. [14] was adopted to compute (σ xx + σ yy ) from the two orthogonal stress gradient fields and are shown in Fig. 6.6 as contours. In Fig. 6.6a, the mode-I mother crack, propagating self-similarly, is surrounded by stress contours expected for (σ xx + σ yy )(∝f (V) KId (t) r−1/2 cos (θ /2)) based on LEFM with estimated maximum tensile stress occurring at the crack-tip location. The stresses reduce to zero and change over to compressive values away from the crack-tip. After the crack branches, two mixed-mode daughter cracks are produced with two distinct stress singularities seen in Fig. 6.6b.

56

B. M. Sundaram and H. V. Tippur

6.7 Crack Branching Measured stress gradient fields can be analyzed to evaluate fracture parameters such as instantaneous crack-tip position, velocity and SIFs. Using SIF history and elastic modulus, one could obtain strain energy release rate histories. By monitoring these parameters throughout the crack growth (including branching) and comparing it with fracture surface roughness plausible mechanisms for crack branching based on microcracking and changes to the process zone size can be theorized (see Ref. [15] for details).

6.8 Summary In this work, the dynamic crack growth and branching in SLG were investigated. The optical full-field measurement method of DGS in conjunction with ultrahigh-speed photography (>1 million fps) was used to make real-time measurements. Unlike polymers, deformation in glass are extremely small before failure which makes it exceedingly challenging to make realtime measurements. This has been overcome by using very large (∼1.2 m)  and thus increasing the sensitivity of the DGS method. This made it necessary to include a mapping function to transform the displacements in ‘target plane’ to ‘specimen plane’ as the perspective effects can no longer be approximated due to very large . The optically mapped crack-tip fields have helped to both visualize and quantify crack growth behavior in the pre- and post-crack branching regimes. The measured orthogonal stress gradients near daughter cracks were used to evaluate (σ xx + σ yy ) using integration scheme reported previously by the authors. Thus, obtained stress fields closely match with the expected fields based on LEFM. Further, using the measured orthogonal stress gradient fields, various fracture parameters were evaluated including a plausible crack branching mechanism. Acknowledgement The authors would like to thank the U.S. Army Research Office for supporting this research through grants W911NF-16-10093 and W911NF-15-1-0357 (DURIP).

References 1. Yoffe, E.H.: The moving Griffith crack. Lond. Edinb. Dublin Philos. Mag. J. Sci. 42(330), 739–750 (1951) 2. Shand, E.B.: Experimental study of fracture of glass: I, the fracture process. J. Am. Ceram. Soc. 37(2), 52–60 (1954) 3. Kobayashi, A.S., Mall, S.: Dynamic fracture toughness of Homalite-100. Exp. Mech. 11, 11–18 (1978) 4. Dally, J.W.: Dynamic photoelastic studies of fracture. Exp. Mech. 19, 349–381 (1979) 5. Marder, G., Gross, S.: Origin of crack tip instabilities. J. Mech. Phys. Solids. 43(1), 1–48 (1995) 6. Bobaru, F., Zhang, G.: Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int. J. Fract. 196(1), 59–98 (2015) 7. Zavattieri, P.D.: Study of dynamic crack branching using intrinsic cohesive surface with variable initial elastic stifness. GM Research & Development Centre. R&D-9650 (2003) 8. Periasamy, C., Tippur, H.V.: Measurement of orthogonal stress gradients due to impact load on a transparent sheet using digital gradient sensing method. Exp. Mech. 53(1), 97–111 (2013) 9. Sundaram, B.M., Tippur, H.V.: Full-field measurement of contact-point and crack-tip deformations in soda-lime glass. Part-I: quasi-static loading. Int. J. Appl. Glas. Sci. 9(1), 114–122 (2018) 10. Sundaram, B.M., Tippur, H.V.: Full-field measurement of contact-point and crack-tip deformations in soda-lime glass. Part-II: stress wave loading. Int. J. Appl. Glas. Sci. 9(1), 123–136 (2018) 11. Sundaram, B.M., Tippur, H.V.: Dynamic mixed-mode fracture behaviours of PMMA and polycarbonate. Eng. Fract. Mech. 176, 186–212 (2017) 12. Sundaram, B.M., Tippur, H.V.: Dynamic crack growth normal to an interface in bi-layered materials: an experimental study using digital gradient sensing technique. Exp. Mech. 56, 37–57 (2016) 13. Sundaram, B.M., Tippur, H.V.: Dynamics of crack penetration vs. branching at a weak interface: an experimental study. J. Mech. Phys. Solids. 96, 312–332 (2016) 14. Miao, C., Sundaram, B.M., Huang, L., Tippur, H.V.: Surface profile and stress field evaluation using digital gradient sensing method. Meas. Sci. Technol. 27(9), 095203 (16pp) (2016) 15. Sundaram, B.M., Tippur, H.V.: Dynamic fracture of soda-lime glass: A full-field optical investigation of crack initiation, propagation and branching. 120, 132–153 (2018)

Chapter 7

A Hybrid Experimental-Numerical Study of Crack Initiation and Growth in Transparent Bilayers Across a Weak Interface Sivareddy Dondeti and Hareesh V. Tippur

Abstract Transparent layered structures are of importance to both the military and civilian communities. Their applications include but not limited to lightweight transparent armor, automotive windshields and canopies, personnel shields and visors as well as electronic displays. The introduction of adhesive interlayers is a low-cost approach for developing mechanically resilient multilayered lightweight structures. However, a rigorous mechanics based design of such architectures requires tailoring interfaces (layer thickness, adhesive properties, number of layers, interface location, etc.). Among the multitude of issues involved in this regard, grasping the mechanics of dynamic crack growth across interfaces is of paramount importance. In this context, this work builds on optical investigations of Sundaram and Tippur (J Mech Phys Solids 96:312–332, 2016) who reported dynamic crack-interface interactions related to crack penetration vs. crack branching at a weak interface when the interface was oriented perpendicular to the incoming mode-I crack in an otherwise homogeneous bilayer. A major finding of this work was that a slowly growing crack with a lower stress intensity factor penetrated the interface and grew into the next layer without branching. On the contrary, a fast growing crack with a higher stress intensity factor debonded the interface ahead of its arrival at the interface and hence branched into the interface and subsequently into the next layer as (two) mixedmode daughter cracks creating higher fracture surface area. In order to exploit this observation and gain further insight into crack growth in multilayered structures, a hybrid experimental-numerical approach that mimics the complexities observed in the bilayer experiments is attempted. This includes optical measurement of the force histories imposed on the bilayer during impact loading of a V-notched PMMA sample impacted by a long-rod with wedge shaped tip matching the notch. Digital Gradient Sensing (DGS) method has been utilized in conjunction with ultrahigh-speed photography followed by optical data analysis to visualize and quantify the force histories. The measured force histories along with other previously determined interface and PMMA characteristics are used as input parameters into a finite element model that includes cohesive elements to benchmark the experiments. Thus validated computational model will be used to investigate a variety of parameters far too complex to emulate experimentally in multilayer architectures. Keywords Digital gradient sensing · Cohesive zone modeling · Interface · Transparent layered structures · Dynamic fracture · Crack path selection

7.1 Introduction Crack penetration and deflection phenomena play a significant role, particularly, when a brittle layered-structure is subjected to complex dynamic loading. Early work by Cook [2] suggested the possibility of strength and toughness enhancement of such systems by optimizing the ratio between the adhesive strength of the interface and the cohesive strength of the brittle phase. Most reports on this topic studied by Tippur et al. [3] that deal with dynamic interfacial crack growth along the interface have brought to light several previously unknown aspects of fracture mechanics including high crack speeds and the possibility of crack propagation at intersonic speeds. The existing studies on this subject generally deal with the interfaces inclined to the crack propagation direction where deflection/penetration mechanics of a single crack-tip are analyzed. Rosakis, et al. [4] have used optical methods to visualize crack-tip fields and crack-tip location during highly transparent fracture events. Sundaram and Tippur [1] investigated interface interaction during crack propagation by varying the initial position of the interface in a bilayer.

S. Dondeti · H. V. Tippur () Department of Mechanical Engineering, Auburn University, Auburn, AL, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_7

57

58

S. Dondeti and H.V. Tippur

A few numerical investigations using finite element analysis, in conjunction with stiffness degradation techniques conducted by Timmel, et al. [5] or using cohesive elements by Needleman, et al. [6], and peridynamic simulations [7] have also been reported on crack growth across interfaces. The stiffness degradation or element erosion techniques usually suffer from mesh refinement issues. The analyses based on cohesive elements, on the other hand, are common; for example, Needleman, et al. [6] studied the crack growth in an elastic solid across an interface and into an elastic-viscoplastic solid. The results indicated that, higher strength interface promoted crack penetration, whereas weaker one caused the crack deflection. Parmigiani, et al. [8] also introduced crack ahead of the interface and studied deflection/penetration with cohesive-zone model by including both strength and toughness parameters simultaneously. In a more recent study, Liu, et al. [9] studied strain rate effects on growth dynamics of a crack-tip lodged into an interface perpendicular to the crack. The possibilities of crack growth through the interface as well as its bifurcation along the interface before entering the next layer are examined in their work. Although Sundaram and Tippur [1] could explain the differences in the observed penetrations vs. branching phenomena in bilayered specimens using optically measured fracture parameters, extending the same to more complex scenarios involving multilayered architectures involving different layer geometries and mechanical characteristics is a challenging task. Further, conducting experiments is tedious and likely impossible. Hence a hybrid experimental-numerical study of crack initiation and branching can address some design and optimization aspects of the problem such as suitable number of layers, layer thickness, spatial gradation of interfaces, their locations and strength within the architecture, etc. One of the potential approaches to do this would be to use an experimentally validated finite element analysis utilizing cohesive zone models supplemented by suitable measurements. Thus, in this work a hybrid experimental-numerical approach is followed first by measuring force histories acting on a PMMA bilayer optically. Subsequently a finite element model based on cohesive zone elements is developed to mimic the experimental observations of Sundaram and Tippur [1] as a starting point.

7.2 Specimen Preparation A bilayer specimen (Fig. 7.1) was prepared from two 50 mm × 170 mm strips of cast PMMA of 8.6 mm thickness. The two strips were roughened using 400 grit sandpaper and joined along 8.6 mm × 170 mm faces by using transparent acrylic adhesive Weldon-16. This produced a discrete plane of weakness in an otherwise homogenous PMMA specimen. The resulting interface was 100 μm thick and its mode-I crack initiation toughness was approximately 50% of the virgin material [10]. A 400 V-notch was machined into the first layer at the mid-span of the longer edge of the specimen. The depth of the V-notch was 19 mm from the free edge extended by 2 mm as shown. Fig. 7.1 Bi-layered PMMA with a normally oriented interface at 28 mm from the extended crack-tip

Interface 19 mm

o

150 mm

40

28 mm

100 mm

7 A Hybrid Experimental-Numerical Study of Crack Initiation and Growth in Transparent Bilayers Across a Weak Interface

59

Fig. 7.2 The schematic of experimental setup for Digital Gradient sensing (DGS) technique to determine planar stress gradients

7.3 Digital Gradient Sensing (DGS) A schematic representation of the experimental setup for transmission-mode DGS technique which is used to measure angular deflections is shown in the Fig. 7.2. In transmission-mode DGS [11], a random speckle pattern on the target plate is photographed through a planar, optically transparent object being studied. The gray scales on the target plate are illuminated using an ordinary white light. The speckle pattern is first photographed to obtain a reference image in the undeformed state of the specimen. That is, point P on the target plane (x0 -y0 plane) is captured by the camera through point O on the specimen plane (x-y plane). Upon loading, the non-uniform state-of-stress changes the refractive index of the specimen in the crack-tip vicinity. In addition, Poisson effect produces a change in the thickness. A combination of these, commonly referred to as the elasto-optical effect, causes the light rays to deflect from their original path as they propagate in the crack-tip vicinity. The speckle pattern is once again photographed through the specimen in the deformed state. After deformation, neighboring point Q on the target plane is recorded by the camera through point O on the specimen plane. The angular deflections of the light rays can be quantified by correlating speckle images in the deformed and reference states. Using digital image correlation, displacements δ x and δ y are evaluated and the corresponding angular deflections of the light rays φ x and φ y in two orthogonal planes (x-z and y-z planes, the z-axis coinciding with the optical axis of the setup and x-y being the specimen plane coordinates) can be computed by knowing the distance between the specimen plane and the target plane. The angular deflections of light rays proportional to the gradients of the in-plane normal stresses are shown as φx;y

 ∂ σx + σy = ±Cσ B , ∂x; y

where Cσ is the elasto-optic constant of the material, B is its initial thickness and σ x and σ y denote the thickness-wise averages.

7.4 Experimental Setup The force history experienced by the PMMA bilayer subjected to transient loading was measured by using Digital Gradient Sensing (DGS [6]) method used in conjunction with ultrahigh-speed photography. The V-notch in the bilayer was subjected to dynamic loading using a Hopkinson pressure bar. The loading end of the bar was wedge shaped, matching the V-notch

60

S. Dondeti and H.V. Tippur

Data Acquisition System

Striker

Computer

Putty

Long Bar

Barrel Gas-gun Cylinder

Speckle Target Specimen

Strain Gauge

Delay Generator

High Energy Flash Lamps

Lamp Control Unit

Kirana-05M Ultrahigh Speed Camera

Fig. 7.3 Schematic of the Experimental Setup

made in the specimen. A striker held inside a gas-gun barrel was aligned co-axially with the Hopkinson pressure bar. The striker was 305 mm long, and 25.4 mm diameter and Hopkinson bar was 1830 mm long, 25.4 mm diameter, both made AL 7075-T6. Upon launching the striker by the gas-gun, collision between the striker and the Hopkinson bar generated a stress wave which subsequently loaded the faces of the V-notch. The optical measurements were made using Kirana-05 M ultrahigh speed 10 bit camera with 924x768 pixels resolution operating at 200,000 frames per second (Fig. 7.3).

7.5 Optical Measurements The force histories on the bilayer PMMA were measured by recording the speckle images in the undeformed/reference and deformed states. The angular deflections of light rays in two mutually perpendicular planes representing gradients of (σ x + σ y ) in the x- and y-directions were measured by correlating speckle images in the deformed state with the one in the reference state. Representative contours of orthogonal angular deflections in the vicinity of the loading point and the notch tip/crack-tip are shown in Fig. 7.4 for two time instants after the start of the impact. (Time t = 0 denotes the time at which the striker impacted the Hopkinson pressure bar.) By approximating these optical measurements as if they were due to line-load acting on the edge of a planar sheet, the force histories on the V-notch of specimen were extracted. The details are not provided here for brevity. Thus obtained normal force perpendicular to the wedge was obtained from the resultant angular deflection contours at different time instants and is shown in Fig. 7.5. Evidently, the impact force history includes a ramp loading from zero to a max. Value of ∼1000 N over ∼50 μs followed by constant load magnitude for the next 70 μs or so before starting to unload. This force history was used in the cohesive element based FE model to simulate crack growth behavior in bilayers.

7.6 FE Simulations (Work-in-Progress) A 2D plane stress model employing cohesive elements was created to study dynamic crack propagation in the specimen as shown in Fig. 7.1. The PMMA has a Young’s modulus, E = 3500 MPa, the Poisson’s ratio, v = 0.35 and mass density, ρ = 9.98 × 10−7 kg/mm3 . A bilinear traction separation law was used to model the normal and shear traction with cohesive elements. The 2D model with 22,580 linear quadrilateral elements of type COH2D4 cohesive elements with zero thickness

7 A Hybrid Experimental-Numerical Study of Crack Initiation and Growth in Transparent Bilayers Across a Weak Interface

61

Fig. 7.4 Angular deflection contour plots proportional to stress-gradients in the x- and y- directions

Normal Force, N

1200 1000 800 600 400 200 0 200

250

300 350 Time, μs

400

450

Fig. 7.5 Impact load history extracted from optical measurements

were used to simulate crack propagation along the weak interfaces, as well as from the sharpened crack-tip to the end of the specimen perpendicular to the weak interface, as shown in Fig. 7.6. Additionally, 480,698 linear triangular elements of type CPS3 were used to simulate bulk material (PMMA). The cohesive stresses (σ c and τ c ) and fracture energy (G), for bulk materials were assigned as, σ c,bulk = 60 Mpa, τ c,bulk = 30 Mpa and (Gbulk )I = (Gbulk )II = 0.4 N/mm and for interface, σ c,interface = 30 Mpa, τ c,interface = 15 Mpa and (Ginterface )I = 0.1 N/mm, (Ginterface )II = 0.2 N/mm were considered. To start with, these parameters were used for Mode-I and Mode-II conditions, and will be altered in the next phase of the work. The preliminary results show that the crack propagated until the weak interface and then deflected along the weak interface, as shown in the inset of Fig. 7.7. The interface debonded well before the crack tip arrived at the interface, as shown in Fig. 7.8. These results are in agreement with the earlier experimental studies [1]. The crack propagation will be studied next with different interface locations. Further, the deflection of the crack into the bulk material in the second layer, after propagation along the interface, is yet to be simulated.

62 Fig. 7.6 Finite element model configuration

S. Dondeti and H.V. Tippur

Interface Cohesive Elements with zero thickness

Bulk Material Elements

Fig. 7.7 Finite element model and preliminary results

Fig. 7.8 Finite element model with debonded interface results

7.7 Summary A combined experimental-finite element method is being pursued for understanding the intricacies of crack growth behavior in elastically homogeneous brittle bilayers containing a distinct plane of weakness perpendicular to an incoming dynamically growing mode-I crack. Previous experimental studies of Sundaram and Tippur [1] have been supplemented in this work by force history measurements on the bilayers using Digital Gradient Sensing (DGS) in conjunction with ultrahigh-speed photography and a Hopkinson pressure bar loading device. Finite element models incorporating cohesive elements along the prospective crack growth based on the experiments have been built. Measured fracture energies and cohesive characteristics for the bulk and interfacial regions have been incorporated into the model. Preliminary results are encouraging and show that a mode-I crack growing towards an interface debonds the interface before its arrival at the interface resulting in crack branching seen in experiments. Further effort to simulate various other experimental observations namely crack penetration into the next layer with or without branching and the conditions for the same are being currently pursued.

7 A Hybrid Experimental-Numerical Study of Crack Initiation and Growth in Transparent Bilayers Across a Weak Interface

63

Acknowledgement Partial support of this research through Army Research Office grants W911NF-16-1-0093 and W911NF-15-1-0357 (DURIP) are gratefully acknowledged.

References 1. Sundaram, B.M., Tippur, H.V.: Dynamics of crack penetration vs. branching at a weak interface: an experimental study. J. Mech. Phys. Solids. 96, 312–332 (2016) 2. Cook, J., Gordon, J.E.: A mechanism for the control of crack propagation in all-brittle systems. Proc. R. Soc. A: Math. Phys. Eng. Sci. 282(1391), 508–520 (1964) 3. Tippur, H.V., Rosakis, A.J.: Quasi-static and dynamic crack growth along bimaterial interfaces: a note on crack tip field measurements using coherent gradient sensing. Exp. Mech. 31, 243–251 (1991) 4. Chalivendra, V.B., Rosakis, A.J.: Interaction of dynamic mode-I crack with inclined interfaces. Eng. Fract. Mech. 75(8), 2385–2397 (2008) 5. Timmel, M., Kolling, S., Osterrieder, P., DuBois, P.A.: A finite element model for impact simulation with laminated glass. Int. J. Impact Eng. 34(8), 1465–1478 (2007) 6. Siegmund, T., Fleck, N.A., Needleman, A.: Dynamic crack growth across interface. Int. J. Fract. 85(4), 381–402 (1997) 7. Hu, W., Wang, Y., Yu, J., Yen, C.F., Bobaru, F.: Impact damage on a thin glass plate with a thin polymer backing. Int. J. Impact Eng. 62, 152–165 (2013) 8. Parmigiani, J.P., Thouless, M.D.: The roles of toughness and cohesive strength on crack deflection at interfaces. J. Mech. Phys. Solids. 54(2), 266–287 (2006) 9. Liu, L.G., Ou, Z.C., Duan, A.G.P., Huang, F.L.: Strain-rate effects on deflection/penetration of crack terminating perpendicular to bimaterial interface under dynamic loading. Int. J. Fract. 167, 135–145 (2011) 10. Sundaram, B.M., Tippur, H.V.: Dynamic crack growth normal to an Interface in bi-layered materials: an experimental study using digital gradient sensing technique. Exp. Mech. 56(1), 37–57 (2016) 11. Periasamy, C., Tippur, H.V.: A full-field digital gradient sensing method for evaluating stress gradients in transparent solids. Appl. Opt. 51(12), 2088–2097 (2012)

Chapter 8

Inelastic Behavior of Tungsten-Carbide in Pressure-Shear Impact Shock Experiments Beyond 20 GPa Z. Lovinger, C. Kettenbeil, M. Mello, and G. Ravichandran

Abstract Pressure-shear plate impact (PSPI) tests commonly use Tungsten-Carbide (WC) as anvil plates, sandwiching the tested material. The common use of WC in these tests is due to its high impedance and high strength, allowing to reach high pressures, with an elastic response, enabling a straightforward analysis of the tested material. Recent modifications of a powder gun facility at Caltech have enabled pressure-shear plate impact experiments (PSPI) to reach higher velocities with corresponding higher pressures and strain rates. Entering this regime, the inelastic behavior of WC has to be taken into account to extract the response of the tested material. In this work we examine the inelastic behavior of WC in the pressureshear set-up via numerical simulations and PSPI experiments. The 3D numerical simulations enabled to study effects of friction, slip and tilt on the measured signals and so their sensitivity to the material strength and failure behavior. A material model was calibrated in relation to the experimental results. Keywords Pressure-shear · Tungsten-carbide · Dynamic strength · High pressure · High strain rates

8.1 Introduction Tungsten-Carbide (WC) is commonly used in high-pressure applications due its extreme hardness (9 on Moss scale), and its high strength and impedance. The sintered WC is also used with different percentages of cement, commonly Cobalt, in the range of 3–15%, mainly attributing to higher ductility [1]. Pressure-shear plate impact (PSPI) experiments commonly use WC as anvil plates [2]. The common use of WC in these tests is due to its high impedance and high strength, allowing to reach high pressures maintaining an elastic response and enabling a straight forward analysis of the tested material. Its high melting point also allows reaching high temperatures which are intrinsic under shock heating. Conducting pressure-shear impact experiments above the WC’s Hugoniot elastic limit (HEL), namely ∼5–7 GPa, requests to take into account its inelastic behavior to extract the response of the tested material. One can find in the literature various works which characterized the behavior of WC at high pressures, high strains and high strain rates: Shock experiments, measuring the material’s Equation of State, [3, 4], Diamond Anvil Cell tests to examine its behavior at high pressures [5, 6], and plate impact tests to learn its dynamic behavior under impact loading, e.g. [7–9]. In [9] Clifton and Jiao conducted a symmetric pressure PSPI experiment on WC, with the main goal of extracting its inelastic response for the analysis of pressure shear tests above 9 GPa [10, 11]. Pushing for PSPI experiments above 20 GPa, the inelastic response becomes crucial, undergoing higher strains, strain rates, pressures, and temperatures. Further comprehension is needed to calibrate a reliable material model which will allow extrapolating its use to higher pressures. The shear strength of the tested material in the PSPI experiment is extracted from the measured free surface velocities. In this work, we examined via 3D numerical simulations of pressure-shear experiments the effects of the actual impact characteristics and the material behavior on the measured free surface velocity signals. Properties of friction, slip, and tilt were examined first to differentiate them from the effects of the material behavior. Free surface velocity signal sensitivities were then examined in relation to the evolving material strength. A material model was calibrated in relation to the experimental results, examining the dominance and contribution of the effects of strain, strain rate, pressure and possible damage.

Z. Lovinger () · C. Kettenbeil · M. Mello · G. Ravichandran California Institute of Technology, Pasadena, CA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_8

65

66

Z. Lovinger et al.

8.2 Results The examination of tilt in the numerical simulations clearly reveals an early wave which is then followed by a response, quite similar to that achieved it an impact case having no tilt. Figure 8.1 shows the effect of different tilt angles on the transverse free surface velocity signal, in comparison with the experimental signal measured in [9]. As the measure of tilt is significant in pressure shear experiments, these results demonstrate the ability to extract the actual tilt in such tests, in a quantitative manner, from the level of the precursor wave. The effect of friction in PSPI experiments is crucial. The angle at which the impact is conducted is designed to ensure full contact between the impactor and the target, and enable the measurement of the shear response. The effect of friction was examined on the measured transverse velocity, by changing the friction coefficient in the model. Figure 8.2 shows the transverse free surface velocity signal with different friction values. In these simulations, a friction coefficient of 0.2 was enough to ensure full contact during the impact in the shear direction. Furthermore, the imprint of slip can be well detected, which has significant importance in the analysis of such signals in order to differentiate such behavior from actual material softening. Comparison with experimentally measured pressure-shear signals enabled calibration of a material model for WC, which demonstrates a dominant strain hardening effect. Figure 8.3 shows the comparison of the normal signal in a PSPI experiment reported in [10], reaching ∼30 GPa and the simulation result using the calibrated model, accounting mainly for strain hardening of the WC. Additional effects of strain rate hardening, pressure hardening and damage softening were examined in the process of calibration in relation to the material behavior in the PSPI experiments.

Fig. 8.1 The effect of tilt angle on the transverse free surface velocity signal: simulations Vs experiment [9]

Fig. 8.2 The effect of friction/slip on the transverse free surface velocity signal: Simulations Vs experiment [9]

8 Inelastic Behavior of Tungsten-Carbide in Pressure-Shear Impact Shock Experiments Beyond 20 GPa

67

Fig. 8.3 Normal free surface velocity signal in PSPI experiment [11]: experiment Vs simulation

References 1. Mandel, K., Radajewski, M., Krüger, L.: Strain-rate dependence of the compressive strength of WC–Co hard metals. Mat. Sci. Eng. A. 612, 115–122 (2014) 2. Frutschy, K.J., Clifton, R.J.: High-temperature pressure-shear plate impact experiments using pure tungsten carbide impactors. Exp. Mech. 38, 116–125 (1998) 3. Dandekar, D.P., Grady, D.: Shock equation of state and dynamic strength of tungsten carbide. AIP Conf. Proc. 620, 783 (2002) 4. Litasov, K.D., Shatskiy, A., Fei, Y., Suzuki, A., Ohtani, E.: Pressure-volume-temperature equation of state of tungsten carbide to 32 GPa and 1673 K. J. Appl. Phys. 108, 053513 (2010) 5. Getting, I.C., Chen, G., Brown, J.A.: The strength and rheology of commercial tungsten carbide Cermets used in high-pressure apparatus. In: Liebermann, R.C., Sondergeld, C.H. (eds.) Experimental Techniques in Mineral and Rock Physics. Pageoph Topical Volumes. Birkhäuser, Basel (1993) 6. Amulele, G., Manghnani, M.H., Marriappan, S., Hong, X.: Compression behavior of WC and WC-6%Co up to 50 GPa determined by synchrotron x-ray diffraction and ultrasonic techniques. J. Appl. Phys. 103, 113522 (2008) 7. Grady, D.: Shock wave compression in brittle solids. Mech. Mat. 29(3), 181–203 (1998) 8. Millet, J.C.F., Bourne, N.K., Dandekar, D.P.: Lateral stress measurements and shear strength in shock loaded tungsten carbide. J. Appl. Phys. 96(7), 3727–3732 (2004) 9. Clifton, R.J., Jiao, T.: Pressure and strain-rate sensitivity of an elastomer:(1) pressure-shear plate impact experiments; (2) constitutive modeling, in elastomeric polymers with high rate sensitivity. In: Barsoum, R.G. (ed.) Elastomeric Polymers with High Rate Sensitivity, pp. 17–64. Elsevier, Oxford (2015) 10. Jiao, T., Kettenbeil, C., Ravichandran, G., Clifton, R.J.: Experimental investigation of the shearing resistance of soda-lime glass at pressures of 9 GPa and strain rates of 106 s−1 . In: Chau, R., Germann, T., Lane, M. (eds.) Shock Compression of Condensed Matter 2017. American Institute of Physics, Melville (2018) 11. Kettenbeil, C., Mello, M., Jiao, T., Clifton, R.J., Ravichandran, G.: Pressure-shear plate impact experiment on soda-lime glass at a pressure of 30GPa and strain rate of 4·107 s−1 . In: Chau, R., Germann, T., Lane, M. (eds.) Shock Compression of Condensed Matter 2017. American Institute of Physics, Melville (2018)

Chapter 9

Mechanical Response and Damage Evolution of High-Strength Concrete Under Triaxial Loading Brett Williams, William Heard, Steven Graham, Bradley Martin, Colin Loeffler, and Xu Nie

Abstract Current weapons effects modeling efforts rely heavily on quasi-static triaxial data sets. However, there are fundamental knowledge gaps in the current continuum modeling approach due to limited experimental data in the areas of dynamic effects and damage evolution. Arbitrary scalar values used for damage parameters have experimentally unverified mathematical forms that often do not scale to different geometries, stress states, or strain rates. Although some preliminary tests have been performed through dynamic triaxial compression experiments, the results are difficult to interpret due to changes in specimen diameter and length-to-diameter ratio. In this study, a high-strength concrete (f’c ∼130 MPa) was investigated under triaxial loading conditions at confining pressures up to 300 MPa. Three cylindrical specimen sizes were used to determine size effects, including 50 × 114 mm, 25 × 50 mm, and 25 × 13 mm. For a limited number of specimens, X-Ray Computed Microtomography (XCMT) scans were conducted. It was noted that size and length-todiameter ratio have substantial effects on the experimental results that must be understood to determine dynamic effects based on specimen geometries used in dynamic triaxial compression experiments. Additionally, by quantifying pore crushing and crack development under a variety of triaxial loading conditions, future multi-scale modeling efforts will be able to incorporate systematically defined damage parameters that are founded on experimental results. Keywords Triaxial loading · High-strength concrete · Damage · Aspect ratio · Micro-CT

9.1 Introduction When developing continuum models, material properties need to be understood for a wide range of stress states. Many concrete modeling efforts rely heavily on a suite of quasi-static confined experiments on 50 × 114 mm cylindrical specimens as detailed by Williams et al. [1]. However, specimen geometries must be substantially different to satisfy diameter restrictions and stress equilibrium requirements for Kolsky bar experiments. For example, prior work has been published with cylindrical specimen sizes of 19 × 13 mm [2]. In future work, 25 × 13 mm specimens will be used to maintain a length-to-diameter ratio of 1:2. Before these data can be used in modeling efforts, additional quasi-static experiments are required to develop a baseline dataset to account for modified specimen geometries. Although the effects of scaling and length-to-diameter ratios have been thoroughly characterized for unconfined compression tests [3], these parameters have not been investigated for confined testing conditions.

B. Williams () U.S. Army Engineer Research and Development Center, Vicksburg, MS, USA Southern Methodist University, Dallas, TX, USA e-mail: [email protected] W. Heard · S. Graham U.S. Army Engineer Research and Development Center, Vicksburg, MS, USA B. Martin Air Force Research Laboratory, Eglin AFB, Valparaiso, FL, USA C. Loeffler · X. Nie Southern Methodist University, Dallas, TX, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_9

69

70

B. Williams et al.

9.2 Experiments For this study, a high-strength self-consolidating concrete, BBR9, was selected as the material of interest. This concrete was developed by the US Army Engineer Research and Development Center using the following constituent materials: crushed limestone sand, type I/II cement, grade 100 slag, microsilica, and high-range water-reducing admixture. Traditional unconfined compression tests were performed on 75 × 150 mm cylindrical specimens yielding a compressive strength of 130 MPa. The traditional suite of quasi-static confined experiments was conducted using 50 × 114 mm cylindrical specimens to form a baseline for scaled datasets. To investigate the effects of scaling, one triaxial experiment was conducted at 200 MPa confining pressure using reduced specimen dimensions (25 × 50 mm) while maintaining a length-to-diameter ratio of 2:1. Furthermore, effects of changes in length-to-diameter ratios were investigated by performing one triaxial experiment on a scaled specimen (25 × 13 mm) with a length-to-diameter ratio of 1:2. For observation of damage evolution, X-Ray Computed Microtomography (XCMT) was conducted on specimens before and after loading as shown in Fig. 9.1.

9.3 Results Triaxial experiments were performed on BBR9 using the conventional specimen geometry (50 x 114 mm) as shown in Fig. 9.2a. In these experiments, specimens are first loaded hydrostatically until the desired confining pressure is reached (maintaining a principal stress difference of zero). These experiments are axis-symmetric (σ2 = σ3 ) due to radial confining

Fig. 9.1 Micro-CT results comparing concrete porosity (a) before and (b) after triaxial loading (200 MPa confining pressure, 10% axial strain)

Fig. 9.2 BBR9 triaxial compression results for (a) 50 × 114 mm specimens with confining pressures ranging from 10–300 MPa and (b) three different specimen geometries with fixed confining pressure of 200 MPa

9 Mechanical Response and Damage Evolution of High-Strength Concrete Under Triaxial Loading

71

pressure applied on the outer circumference of the cylindrical specimens. Subsequently, the axial load is increased while radial confining pressure is maintained at a constant value. Axial strain is measured using LVDTs and radial strain is measured using a spring-arm radial deformeter. These tests verify that concrete becomes more ductile as confining pressure increases. After conducting baseline experiments, the results from three different specimen geometries were compared as illustrated in Fig. 9.2b. Examining triaxial data for specimen geometries of 50 × 114 mm and 25 × 50 mm under 200 MPa confining pressure reveals that size effects may be present, even when length-to-diameter ratios are approximately the same (2:1). Testing of the 25 × 50 mm specimen was stopped at the same principal stress difference (deviatoric stress), 370 MPa, to observe damage evolution as previously shown in Fig. 9.1. After going through the specified loading cycle in the ductile failure region, it is noted that pore collapse is prevalent. These damage mechanisms are substantially different from macrocracking observed in a similar material under unconfined loading conditions [4]. An additional triaxial experiment was conducted under 200 MPa confining pressure using a 25 × 13 mm specimen. In addition to size effects noted earlier, this final test indicates that length-to-diameter ratio also has an effect on material behavior under triaxial compression.

9.4 Conclusion Preliminary studies have revealed that size effects must be considered when interpreting quasi-static triaxial experimental results. Furthermore, effects from length-to-diameter ratios must also be understood when using specimens with non-standard aspect ratios. In order to isolate rate effects in triaxial compression experiments, a series of tests must be conducted both under quasi-static and dynamic (Kolsky bar) testing conditions using the same specimen geometry. Additionally, scaled experiments using 2:1 length-to-diameter ratio are needed to correlate the data to historical datasets that have been used for model calibration. Future work will be focused on conducting additional experiments on 25 × 50 mm and 25 × 13 mm specimens under various confining pressures yielding brittle, quasi-brittle, and ductile failures. Additional XCMT scans will be performed in conjunction with all triaxial experiments so that damage evolution can be quantified for the purpose of informing damage parameters within continuum-based models. The work described in this document was funded under the US Army Basic Research Program under PE 61102, Project T22, Task 02 “Material Modeling for Force Protection” and was managed and executed at the US Army ERDC. Permission to publish was granted by the Director, Geotechnical and Structures Laboratory.

References 1. Williams, E.M., Graham, S.S., Reed, P.A., Rushing, T.S.: Laboratory characterization of Cor-Tuf concrete with and without steel fibers. U.S. Army Enineer Research and Development Center. (2009) 2. Mondal, A.B., Chen, W., Martin, B., Heard, W.: Dynamic Triaxial Compression Experiments on Cor-Tuf Specimens, in: Dynamic Behavior of Materials, vol. 1, pp. 245–249. Springer International Publishing, Cham (2013). https://doi.org/10.1007/978-3-319-00771-7_30 3. Ozyildirim, C., Carino, N.J.: Chapter 13: Concrete Strength Testing. In: Significance of Tests and Properties of Concrete and ConcreteMaking Materials, pp. 125–125–16. ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428–2959 (2006). https://doi.org/10.1520/STP37731S 4. Loeffler, C., Williams, B.A., Heard, W.F., Martin, B., Nie, X.: 3-D damage characterization in heterogeneous materials. In: Society of Engineering Mechanics, pp. 1–3. US Army ERDC, Orlando (2016)

Chapter 10

Heterodyne Diffracted Beam Photonic Doppler Velocimeter (DPDV) for Pressure-Shear Shock Experiments M. Mello, C. Kettenbeil, M. Bischann, Z. Lovinger, and G. Ravichandran

Abstract We present details on the design and validation of a heterodyne diffracted beam photonic Doppler velocimeter (DPDV) for pressure-shear plate impact (PSPI) shock experiments. The fiber optic interferometer collects symmetrically diffracted 1st order beams produced by a thin, specular, metallic grating deposited on the rear surface of the impacted target plate and separately interferes each of these beams with a reference beam of a slightly increased wavelength. The resulting interference signals contain an upshifted carrier signal with a constant frequency at zero particle velocity. Signal frequency content in recorded waveforms from PSPI experiments is extracted using a moving-window DFT algorithm and then linearly combined in a post- processing step to decouple and extract the normal and transverse velocity history of the rear target surface. The 0th order (normally reflected) beam can also be interfered in a separate heterodyne PDV configuration to obtain an additional, independent measurement of the normal particle velocity. An overview of the DPDV configuration is presented along with a discussion of the interferometer sensitivities to transverse and normal particle velocities. Results from a normal impact experiment conducted on Y-cut quartz are presented as experimental validation of the technique. Keywords Heterodyne · Diffraction · Pressure-shear · Transverse particle velocity · Y-cut quartz

10.1 Introduction Pressure-shear plate impact (PSPI) experiments have traditionally relied on free space beam interferometers such as the transverse displacement interferometer (TDI) and normal displacement interferometer (NDI) or normal velocity interferometer (NVI), to measure transverse and normal velocities at the rear surface of the target plate [1]. Alternative interferometer schemes feature a dual beam VISAR arrangement [2] and a recently developed all fiber-optic TDI-NDI configuration [3]. We present details on the development, and experimental validation of a heterodyne diffracted beam photonic Doppler velocimeter (DPDV) system for simultaneous measurement of normal and transverse particle velocity components in pressure-shear plate impact (PSPI) shock experiments. A source probe produces a thin collimated beam which is normally incident to the rear surface of the target plate. The 0th order (normally reflected) beam is collected by the same source probe, and a pair of fiber-optic side probes receive the symmetrically diffracted 1st order beams. Each beam is then passed through a fiber-optic circulator and then combined with a reference beam of a slightly higher wavelength to create a heterodyne interference signal with an upshifted carrier frequency at zero particle velocity. The frequency content encoded within the recorded DPDV waveforms corresponds to a scaled linear combination (sum or difference) of the normal and transverse particle velocity components. A moving-window discrete Fourier transform (DFT) algorithm is applied to extract the DPDV signal frequencies f+ (t) and f− (t) from the recorded signals, which also contain the constant user-selected carrier frequency. Normal and transverse particle velocity components are subsequently decoupled in a post-processing algorithm through addition or subtraction and appropriate scaling of the extracted signal frequencies. The normally reflected (0th order) beam is also interfered in a heterodyne PDV arrangement to obtain an additional independent measurement of the normal particle velocity. The fiber-optic DPDV system has been configured with a powder gun capable of achieving impact velocities of 1.8 km/s and has enabled PSPI shock impact investigations on the strength and failure properties of novel materials at shear strain rates approaching 108 s−1 .

M. Mello () · C. Kettenbeil · M. Bischann · Z. Lovinger · G. Ravichandran California Institute of Technology, Pasadena, CA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_10

73

74

M. Mello et al.

x2 , u˙

r orde

3 2

th

+n

1

th

-n

x1 , u˙

2

Circulators 2 1

probe

Single Mode Fiber-Optic Cables

1

X

3 2

ord

refle c (0 th ted bea orde m probe r)

1

er

3 2

probe

flyer target

Impact Chamber

Oscilloscope

1

X

Drive Group 1 x 4 coupler (splitter)

V0

“down barrel” PDV probe

Polarization-maintaining Fiber-Optic Cables

Source Laser (λS) Fiber-Optic Patches

3 (2 beams supressed) Attenuators

A

A

A

A

Sensing Group D

1x2 coupler

4km D Delay 1x2 coupler

Reference Group Reference Laser (λR) 1 x 2 coupler (splitter)

Power Monitors Photodetector Photodetector

P

P

1x2 coupler 1x2 coupler

Reference beams

A A

Fig. 10.1 DPDV optical configuration for the combined measurement of normal and transverse particle velocity history in PSPI shock experiments

10.2 Interferometer Configuration The fiber-optic DPDV arrangement is comprised of the drive, reference, and sensing groups as depicted in Fig. 10.1. The source light is produced by a KOHERAS BOOSTIK model E15 fiber laser by NKT Photonics with a Lorentzian linewidth of 0.1 kHz, an adjustable center wavelength between 1550 nm and 1570 nm, and a maximum output power of 2 W. The drive group hardware elements include a 1 × 4 fiber-optic splitter, fiber-optic circulators, attenuators, and fiber-optic probes. Fiber optic side probes are designed to efficiently collect the diffracted beams even as they rotate due to tilt and decenter from the optical axis as a consequence of accumulated normal displacement of the target plate rear surface. Reference group hardware elements depicted in Fig. 10.1 generate the reference beams, which are interfered with the drive group source beams diffracted and reflected by the target. Sensing group hardware elements depicted in Fig. 10.1 interfere the collected source light with the reference light and convert the resulting interference signals into digitized fringe records for analysis.

10.3 Interferometer Measurement Sensitivity and Analysis of Recorded Waveforms DPDV measurement sensitivity is derived by invoking the two beam, time-averaged interference equation given by ±

I (t) =

IS±

+ IR±

  2π ± ± ± + 2 IS IR cos (u1 (t)(1 − cos θn ) ± u2 (t)sin θn ) + 2π(νS − νR )t − φ ) . λS

(10.1)

Here IS± and IR± respectively represent the time-averaged background intensity of the source and reference beams and the ± symbols designate interference of the n = +1 or n = −1 diffracted order with its respective reference beam [1].

10 Heterodyne Diffracted Beam Photonic Doppler Velocimeter (DPDV) for Pressure-Shear Shock Experiments

75

Application of a moving-window discrete Fourier transform (DFT) algorithm to the pair of recorded DPDV waveforms extracts the encoded signal frequencies corresponding to f + (t) =

1 (u˙ 1 (t) (1 + cosn ) + u˙ 2 (t)sinn ) + (νS − νR ) λS

(10.2)

f + (t) =

1 (u˙ 1 (t) (1 + cosn ) − u˙ 2 (t)sinn ) + (νS − νR ) λS

(10.3)

where λS represents the wavelength of the source laser, u˙ 1 , u˙ 2 are the normal and transverse particle velocity components, ν S and ν R correspond to the optical frequencies of the source and reference laser light beams, and φ ± is a constant arbitrary phase term in each interference signal. Subtracting Eq. (10.3), from Eq. (10.2) and substituting from the grating equation sinn = nλ/d yields an expression for the transverse particle velocity expressed in terms of the grating pitch (d), diffraction order (n), given by

u˙ 2 (t) =

d  + f (t) − f − (t) 2n

(10.4)

The frequency scaling factor d/2n effectively represents the fundamental measurement sensitivity of the DPDV to changes in transverse velocity and is equivalent to the sensitivity of a TDI [1]. Using the 1st order beams from a 400 lines/mm grating in the current DPDV configuration results in a transverse velocity measurement sensitivity of 1.25 m/s/MHz. Addition of the two DPDV signal frequencies given by Eqs. (10.2 and 10.3) yields an expression for the normal particle velocity in terms of the measured signal frequencies and the independently measured carrier frequency (ν C = ν S − ν R ) given by

u˙ 1 (t) =

 + λS f (t) + f − (t) − 2νC . 2 (1 + cosn )

(10.5)

The scaling factor λS /2(1 + cos n ) represents the fundamental measurement sensitivity of the DPDV to changes in normal velocity. DPDV is therefore, (1 + cos n ) times more sensitive to changes in normal velocity compared to a standard PDV, which has a sensitivity of λS /2 [4]. Using the 1st order beams from a 400 lines/mm grating in the current DPDV configuration results in a normal velocity measurement sensitivity of 0.434 m/s/MHz, which represents a 1.78× increase over the sensitivity of the PDV when using the 0th order beam at the same source wavelength.

10.4 Experimental Validation – Normal Impact of Y-Cut Quartz A normal plate impact experiment was conducted using a borosilicate glass flyer plate and a single crystalline Y-cut quartz target plate as a means of validating the new DPDV system. Single crystalline Y-cut quartz was selected as a target material because of the strong anisotropic coupling exhibited between longitudinal and transverse particle motion when impacted along its x2 axis. Quasi-longitudinal (QL) and quasi-transverse (QT) waves emerge as the only nonzero eigenvalues of the elasto-acoustic tensor which governs the problem [1]. Sharp velocity jumps registered in the normal and transverse directions upon arrival of the QL and QT waves at the rear surface of the target present an ideal scenario for evaluating various attributes of the DPDV system such as predicted velocity measurement sensitivities and the optical heterodyne feature for automatic, accurate detection of sharp velocity reversals. A 300 nm thick, 400 lines/mm (d = 2.5 μm) gold diffraction grating was fabricated onto the backside of the polished Y-cut quartz target plate using standard photolithography procedures in a cleanroom environment. The grating was deposited on the target plate with its grooves aligned parallel to the x1 axis of the Y-cut quartz substrate. The grating produced sharp, specular 0th order and 1st order diffraction beams along 1st order ◦ diffraction angles n = ± 38.32 per the grating equation. No additional diffraction orders are produced by a 400 lines/mm grating at the source wavelength λ = 1550). DPDV probes were physically aligned to the ±1st order diffraction angles by optimizing the light return from each diffracted beam until the maximum light returns were achieved. Normal and transverse velocity profiles measured at the rear surface of the Y-cut quartz target plate using the DPDV system are plotted in Fig. 10.2. The velocity profiles were obtained from the acquired fringe records using a Hamming window of 50 ns with the window shifted by a 50 ps time step for every analysis. The dashed lines represent the predicted

76

M. Mello et al.

200

200

longitudinal transverse predicted

longitudinal 150

100

100

transverse 50 0

b transverse velocity (m/s)

free surface velocity (m/s)

a

80

predicted experiment

60 40 20 0

0 -20 0

0.5

1.0

Time (μs)

1.5

2.0

0

20

40

60

80

100 120 140 160 180 200

normal velocity (m/s)

Fig. 10.2 (a) Measured longitudinal and transverse velocity profiles compared to predicted values based on the measured impact speed of V0 = 212 m/s (b) Orthogonality of the measured velocity jumps during the arrival of the QL wave, and the QT wave

velocity jumps of each respective motion component. The measured steady initial and final transverse particle velocity levels are in excellent agreement with theory while the measured normal velocity jumps are within 6–7% of their respective predicted values. The observed deviation is attributed to a small impact tilt angle of ∼1mrad, which caused the QL wave to deviate from the x2 axis (quartz crystal y-axis) by an amount consistent with the observed deviation. The observed deviation is also partly attributed to uncertainties in the values of the elastic constants of Y-cut quartz and borosilicate glass.

References 1. Kim, K., Clifton, R.J., Kumar, P.: A combined normal- and transverse-displacement interferometer with an application to impact of Y-cut quartz. J. Appl. Phys. 48, 4132–4139 (1977) 2. Chhabildas, L.C., Sutherland, H.J., Asay, J.R.: A velocity interferometer technique to determine shear-wave particle velocity in shock-loaded solids. J. Appl. Phys. 50, 5196 (1979) 3. Zuanetti, B., Wang, T., Prakash, V.: A compact fiber optics-based heterodyne combined normal and transverse displacement interferometer. Rev. Sci. Instrum. 88, 033108 (2017) 4. Dolan, D.H.: Accuracy and precision in photonic Doppler velocimetry. Rev. Sci. Instrum. 81, 053905 (2010)

Chapter 11

An Optimization-Based Approach to Design a Complex Loading Pattern Using a Modified Split Hopkinson Pressure Bar Suhas Vidhate, Atacan Yucesoy, Thomas J. Pence, Adam M. Willis, and Ricardo Mejia-Alvarez

Keywords Split Hopkinson pressure bar · Blast-induced traumatic brain injury · Optimization · Impact · Finite element simulation

11.1 Introduction The split Hopkinson pressure bar (SHPB) technique is used to characterize the mechanical response of a material during impact loading when a single stress wave pulse passes through that material [1]. The SHPB setup consists of two long bars: an incident bar and a transmission bar. The specimen, which needs to be characterized, is placed between these two bars. A striker propelled from a gas gun hits the incident bar generating a stress wave that propagates through the incident bar. A part of this wave is transmitted to the specimen and the transmission bar while the remaining part of the wave reflects back into the incident bar. By measuring the incident, transmitted, and reflected waves, the mechanical properties of the specimen are determined for high-strain-rate deformations. Additionally, SHPB can be used as an actuator to generate impact loading. Sarntinoranont et al. [2] proposed the use of SHBP in generating an impulsive overpressure loading to study the effect of sudden over-pressurization on a brain tissue. This loading technique was used to investigate the tissue injury mechanism that causes blast-induced traumatic brain injury (bTBI). However, the pressure loading experienced by the brain tissue due to a blast exposure is extremely complex and highly dynamic, which cannot be approximated as a single overpressure pulse [3]. Thus, the traditional SHPB technique cannot be used to replicate this complex loading profile. Our study presents a novel approach to design an actuator based on the SHPB, which can ultimately be used to generate highly dynamic loading profiles. The principal idea here is to build the incident bar by joining multiple rods, made up of different materials, to increase the dynamic nature of the incident pulse. When a stress wave propagates through this multimaterial incident bar, the difference in the elastic properties of the materials on either side of the interface where two rods are coupled creates an impedance mismatch for wave propagation. Due to the impedance mismatch, part of the wave transmits into the next rod, while the other part reflects back. This transmission and reflection repeat at every intermediate interface, which generates a complex stress wave at the other end of the bar because of the wave interference. With this understanding, SHPB can be used to generate a desired complex loading profile by simply modifying the incident bar.

S. Vidhate () · A. Yucesoy · T. J. Pence · R. Mejia-Alvarez Department of Mechanical Engineering, Michigan State University, East Lansing, MI, USA e-mail: [email protected] A. M. Willis Department of Mechanical Engineering, Michigan State University, East Lansing, MI, USA Department of Neurology, San Antonio Military Medical Center, Sam Houston, TX, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_11

77

78

S. Vidhate et al.

11.2 Methods The parameters affecting the modified SHPB actuator design are the length and material selection of the rod components that make up the incident bar. These design parameters determine the stress wave profile generated by the modified SHPB actuator. Based on the desired loading profile, the parameters need to be optimized such that the loading generated by the actuator is same as the desired loading profile. This optimization is achieved by coupling the finite element analysis of the modified SHPB with an optimization algorithm. A finite element (FE) code in MATLAB (MathWorks Inc, Natick, MA, USA) was developed to simulate the modified SHPB. The FE simulation was performed by solving a one-dimensional wave equation for an elastic medium using the method of characteristics [4]. An optimization algorithm, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) [5], is used to minimize the Root-Mean-Square Error (RMSE) between desired loading and the loading generated by the actuator. This RMSE is computed in the frequency domain instead of the time domain to emphasize the dynamic nature of the loading patterns. Firstly, NSGA-II initializes the optimization process by generating a random set of parameter values. RSME is then computed for these new random designs using finite element simulations. Based on the performance of these designs, NSGA-II further modifies the parameters to create new designs that are yet again evaluated by computing the RMSE. This process is iterated until optimal parameters are determined, which eventually minimize the error between desired loading profile and the loading profile obtained by the modified SHPB actuator.

11.3 Results To test the methodology proposed above, an optimization problem is established. For this test problem, pressure loading that was based on the brain tissue’s response to a blast wave [3] was chosen as the desired loading profile. The incident bar was simulated as a multi-material bar made by joining 10 different rod components. Figure 11.1 shows the comparison between desired loading profile and the loading profile generated by the optimized SHPB actuator.

30 Desired loading profile Loading profile generated by modified SHPB

Pressure amplitude (kPa)

25

20

15

10

5

0 0

0.5

1

1.5

2

2.5

3

Frequency (kHz) Fig. 11.1 Comparison of loading profiles in frequency domain

3.5

4

4.5

5

11 An Optimization-Based Approach to Design a Complex Loading Pattern Using a Modified Split Hopkinson Pressure Bar

79

The error between these loading profiles can be reduced further by increasing the number of individual rod components used to build the multi-material incident bar. However, such reduction in error increases the total length of the modified SHPB setup, which is usually a limiting aspect of the experimental setup.

References 1. Chen, W., Song, B.: Split Hopkinson (Kolsky) Bar Design, Testing and Applications. Springer, Boston (2011) 2. Sarntinoranont, M., Lee, S.J., Hong, Y., King, M.A., Subhash, G., Kwon, J., Moore, D.F.: High-strain-rate brain injury model using submerged acute rat brain tissue slices. J. Neurotrauma 29, 418–429 (2012) 3. Tan, X.G., Przekwas, A.J., Gupta, R.K.: Computational modeling of blast wave interaction with a human body and assessment of traumatic brain injury. Shock Waves 27, 889–904 (2017) 4. Bedford, A., Drumheller, D.S.: Introduction to Elastic Wave Propagation. Wiley, Chichester (1996) 5. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002)

Chapter 12

Development of “Dropkinson” Bar for Intermediate Strain-Rate Testing Bo Song, Brett Sanborn, Jack Heister, Randy Everett, Thomas Martinez, Gary Groves, Evan Johnson, Dennis Kenney, Marlene Knight, and Matthew Spletzer

Abstract A new apparatus – “Dropkinson Bar” – has been successfully developed for material property characterization at intermediate strain rates. This Dropkinson bar combines a drop table and a Hopkinson bar. The drop table was used to generate a relatively long and stable low-speed impact to the specimen, whereas the Hopkinson bar principle was applied to measure the load history with accounting for inertia effect in the system. Pulse shaping technique was also applied to the Dropkinson bar to facilitate uniform stress and strain as well as constant strain rate in the specimen. The Dropkinson bar was then used to characterize 304L stainless steel and 6061-T6 aluminum at a strain rate of ∼600 s−1 . The experimental data obtained from the Dropkinson bar tests were compared with the data obtained from conventional Kolsky tensile bar tests of the same material at similar strain rates. Both sets of experimental results were consistent, showing the newly developed Dropkinson bar apparatus is reliable and repeatable. Keywords Kolsky bar · Dropkinson bar · Intermediate strain rate · Tensile stress-strain curve

12.1 Introduction The material response at intermediate strain rates is of great interest to automotive industries and electronic packaging [1, 2]. However, the development of intermediate-strain-rate experimental techniques has significantly lagged behind the desire of material properties at intermediate strain rates. Current intermediate-strain-rate testing techniques are mainly based on servohydraulic system/drop table or Hopkinson bar methods. Either servohydraulic system or drop table for intermediatestrain-rate testing is an open loop system with inertia, or wave propagation, neglected. In addition to system inertia, specimen inertia has not been well discussed or properly addressed in current servohydraulic test frames or drop-table-based tests. Hopkinson-bar based apparatus accounts for inertia (or wave propagation) effect in both test apparatus and specimen. However, the gas-gun driven Hopkinson bars generate a relatively short duration of loading, which is insufficient to deform the specimen to relatively large deformation at intermediate strain rates, although long Hopkinson bar systems have been recently developed [3–5]. A hybrid concept that combines a servohydraulic test frame (or a drop table) and a Hopkinson bar has also been proposed for intermediate strain rate testing [6, 7]. In this research, we developed a similar hybrid apparatus – Dropkinson bar – which combines a drop table and a Hopkinson bar. The new Dropkinson bar is capable of (1) generating a stable intermediate impact speed with a long duration (1–2 ms or even longer); (2) curtailing system inertia effect (no ringing in force history); (3) minimizing specimen inertia effect (uniform stress and strain in the specimen); and (4) facilitating constant strain-rate deformation in specimen (pulse shaping technique). The Dropkinson bar was utilized to characterize 304L stainless steel and 6061-T6 aluminum for verification.

12.2 The Dropkinson Bar Design The Dropkinson bar was developed in tension mode as shown in Fig. 12.1. Upon the free drop of the carriage, the impactor attached to the bottom of the carriage impacts the impact plate at its center. The impact plate then transfers the impact load to

B. Song () · B. Sanborn · J. Heister · R. Everett · T. Martinez · G. Groves · E. Johnson · D. Kenney · M. Knight · M. Spletzer Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_12

81

82

B. Song et al.

Fig. 12.1 Photograph of the Dropkinson bar

the tensile specimen attached to the edge through an adapter. Such an impact load subjects the specimen to dynamic tension. The tensile stress wave then transmits into the vertical Hopkinson bar through the threads between the specimen and the bar end. The strain gages on the surface of the Hopkinson bar record the full load/stress history applied to the specimen. A custom-made laser extensometer, which followed the same concept as presented in Ref. [8], was implemented to the Dropkinson bar for specimen deformation measurement. The tensile stress-strain response is thus obtained.

12.3 Experimental Verification A conventional Kolsky tension bar and the Dropkinson bar were used to characterize the same materials (304L stainless steel and 6061-T6 aluminum) with identical specimen geometry at the similar strain rates (∼600 s−1 ) for verification of the new Dropkinson bar apparatus. Five experiments were repeated with each apparatus and the average curve from the five experiments was used to represent the tensile stress-strain curve of the material. Figure 12.2 compares the averaged stressstrain curves of 304L stainless steel (Fig. 12.2a) and 6061-T6 (Fig. 12.2b) obtained from the Dropkinson bar and conventional Kolsky tension bar experiments. The flow stress obtained from the Dropkinson bar tests is approximately 3% higher than that obtained from the conventional Kolsky tension bar tests. However, when accounting for standard deviation during plastic deformation, the stress-strain responses obtained from the Dropkinson bar tests are consistent with those obtained from the conventional Kolsky tension bar tests for both materials. Therefore, the newly developed Dropkinson bar demonstrates the capability for mechanical characterization of materials at intermediate strain rates with high reliability and repeatability.

12 Development of “Dropkinson” Bar for Intermediate Strain-Rate Testing

83

Fig. 12.2 Comparison of tensile stress-strain curves obtained from the Dropkinson bar and conventional Kolsky Tension bar tests. (a) 304L stainless steel; (b) 6061-T6 aluminum

12.4 Conclusions A new apparatus – “Dropkinson Bar” – has been developed for material property characterization at intermediate strain rates. This Dropkinson bar combines a drop table and a Hopkinson bar to generate a relatively long and stable low-speed impact to the specimen and to measure the load history. Pulse shaping technique was applied to the Dropkinson bar to facilitate uniform stress and strain as well as constant strain rate in the specimen. The Dropkinson bar was successfully verified through comparing the tensile stress-strain data for 304L stainless steel and 6061-T6 aluminum to those obtained from conventional Kolsky tensile bar tests at similar strain rates. Acknowledgement Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in this article do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

References 1. Lim, S.J., Huh, H.: Fracture loci of DP980 steel sheet for auto-body at intermediate strain rates. Int. J. Automot. Technol. 18, 719–727 (2017) 2. Jing, J., Gao, F., Johnson, J., Liang, F.Z., Williams, R.L., Qu, J.: Brittle versus ductile failure of a lead-free single solder joint specimen under intermediate strain rate. IEEE Trans. Compon. Packag. Manuf. Technol. 1, 1456–1464 (2011) 3. Gilat, A., Matrka, T.A.: A new compression intermediate strain rate testing apparatus. EPJ Web Conf. 6, 39002 (2010) 4. Luo, H., Cooper, W.L., Lu, H.: Effects of particle size and moisture on the compressive behavior of dense Eglin sand under confinement at high strain rates. Int. J. Impact Eng. 65, 40–55 (2014) 5. Song, B., Syn, C.J., Grupido, C.L., Chen, W., Lu, W.-Y.: A long split Hopkinson pressure (LSHPB) for intermediate-rate characterization of soft materials. Exp. Mech. 48, 809–815 (2008) 6. LeBlanc, M.M., Lassila, D.H.: A hybrid technique for compression testing at intermediate strain rates. Exp. Mech. 20, 21–24 (1996) 7. Petiteau, J.-C., Othman, R., Guégan, P., Le Sourne, H., Verron, E.: A drop-bar setup for the compressive testing of rubber-like materials in the intermediate strain rate range. Strain. 50, 552–562 (2014) 8. Nie, X., Song, B., Loeffler, C.M.: A novel splitting-beam laser extensometer technique for Kolsky tension bar experiment. J. Dyn. Behav. Mater. 1, 70–74 (2015)

Chapter 13

Radial Inertia Effect on Dynamic Compressive Response of Polymeric Foam Materials Bo Song, Brett Sanborn, and Wei-Yang Lu

Abstract Polymeric foams have been extensively used in shock isolation applications because of their superior shock or impact energy absorption capability. In order to meet the shock isolation requirements, the polymeric foams need to be experimentally characterized and numerically modeled in terms of material response under shock/impact loading and then evaluated with experimental, analytical, and/or numerical efforts. Measurement of the dynamic compressive stress-strain response of polymeric foams has become fundamental to the shock isolation performance. However, radial inertia has become a severe issue when characterizing soft materials. It is even much more complicated and difficult to address the radial inertia effect in soft polymeric foams. In this study, we developed an analytical method to calculate the additional stress induced by radial inertia in a polymeric foam specimen. The effect of changing profile of Poisson’s ratio during deformation on radial inertia was investigated. The analytical results were also compared with experimental results obtained from Kolsky compression bar tests on a silicone foam. Keywords Poisson’s ratio · Radial inertia · Polymeric foam · Dynamic compression · Kolsky compression bar

13.1 Introduction Polymeric foams have been extensively used in shock isolation applications because of their superior shock or impact energy absorption capability. Experimentally, it is challenging when characterizing very soft polymeric foams with a Kolsky compression bar because radial-inertia-induced stress can overshadow the intrinsic material response. Currently existing analyses on radial inertia effect are based on the assumption of constant Poisson’s ratio [1–5], which is not applicable to foam materials. Foam materials usually possess very small Poisson’s ratio before densification and can approach a nearly rubbery state after densification. This drastic change in Poisson’s ratio may result in a sudden radial confinement in the specimen during densification process. Therefore, it is desirable to understand the radial inertia effect of polymeric foams that are subjected to large deformation (passing densification) at high strain rates. In this paper, we develop an analytical method to calculate the additional stress induced by radial inertia in a polymeric foam specimen. The effect of changing profile of Poisson’s ratio during deformation on radial inertia is investigated. The analytical results are also compared with experimental results obtained from Kolsky compression bar tests on a silicone foam.

13.2 Radial Inertia in a Compressible Cylindrical Solid In this study, a cylindrical specimen configuration, shown in Fig. 13.1, with initial dimensions of radius, a0 , and length, l0 , for radial inertia analysis. An axial compression at velocity Vx = Vx (t) generates a radial expansion with the velocity, Vr = Vr (r, t). From mass and momentum conservations, the additional axial stress induced by radial inertia is calculated as

B. Song () · B. Sanborn Sandia Neational Laboratories, Albuquerque, NM, USA e-mail: [email protected] W.-Y. Lu Sandia National Laboratories, Livermore, CA, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_13

85

86

B. Song et al.

Fig. 13.1 Cylindrical specimen configuration

 e˙x2 (t) a 2 (t) − r 2 (t) dν (ex ) 2 · ν(t)e¨x (t) + ν(t) · (ν(t) + 1) σr (r, t) = ρ(t) · e˙ (t) + 2 (1 − ex (t)) 1 − ex (t) dex x

(13.1)

where ρ and ν are instantaneous density and Poisson’s ratio of the specimen; ex is engineering axial strain subjected to the specimen. Equation (13.1) indicates that the radial inertia effect includes three parts induced by (1) strain acceleration (the first term in the bracket in Eq. (13.1)); (2) large strain and high strain rate (the second term in the bracket in Eq. (13.1)); and (3) change of Poisson’s ratio and high strain rate (the third term in the bracket in Eq. (13.1). The first two parts are also dependent on instantaneous Poisson’s ratio. For a hyperelastic foam, i.e. silicone foam, the Poisson’s ratio follows a Boltzmann sigmoidal function of engineering axial strain, ν (ex ) =

ν1 − ν2 + ν2  x0 1 + exp ex −e δ

(13.2)

where ν 1 , ν 2 , ex0 , and δ are constants. Applying Eq. (13.2) into Eq. (13.1) yields an average radial-inertia-induced stress along axial direction, ⎡





⎤

ν1 −ν2  e (t)−e 1+exp x δ x0

⎢ e¨x (t) · ν2 + ⎢ ⎢    ⎢ 2 ρ0 · a0 ⎢ ν1 −ν2  · 1 + ν2 + σ r (t) = · ⎢ + ν2 + e (t)−e 1+exp x δ x0 4(1 − ex (t))2 ⎢ ⎢   ex (t)−ex0 ⎢ ⎣ + e˙2 (t) · ν2 −ν1 ·  exp  δ 2 x δ e (t)−e 1+exp

x

δ

 ν1 −ν2  e (t)−e 1+exp x δ x0

⎥ ⎥ ⎥ ⎥ e˙x2 (t) ⎥ · 1−ex (t) ⎥ ⎥ ⎥ ⎥ ⎦

(13.3)

x0

For a silicone foam specimen with a density of ρ 0 = 600 kg/m3 , a porosity of 50%, a diameter of 15 mm (a0 = 7.5 mm), the constants of Poisson’s ratio are ν 1 = 0.21, ν 2 = 0.43, εx0 = 0.525, δ = 0.01. The average radial-inertia-induced stress of a silicone foam specimen subjected to a constant strain rate of 4000 s−1 with a linear ramping time of 40 μs was calculated with Eq. (13.3) and is plotted in Fig. 13.2.

13.3 Conclusions A general radial inertia analysis is conducted in this study to better understand the dynamic compressive response of compressible materials, particularly polymeric foam materials with drastic change of Poisson’s ratio before and after densification. This analysis is applicable to any compressible or incompressible material, with a constant or varied Poisson’s ratio, subjected to high-strain-rate compression. Strain acceleration, large strain, high strain rate, and the change of Poisson’s ratio all contribute, but not necessarily equally, to radial inertia, which results in additional axial stress in dynamic specimen stress measurements.

13 Radial Inertia Effect on Dynamic Compressive Response of Polymeric Foam Materials

87

Fig. 13.2 Additional axial stress induced by radial inertia

Acknowledgement Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in this article do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

References 1. Kolsky, H.: An investigation of the mechanical properties of materials at very high rates of loading. Proc. R. Soc. Lond. B62, 679–700 (1949) 2. Dharan, C.K.H., Hauser, F.E.: Determination of stress-strain characteristics at very high strain rates. Exp. Mech. 10, 370–376 (1970) 3. Forrestal, M.J., Wright, T.W., Chen, W.: The effect of radial inertia on brittle samples during the split Hopkinson pressure bar test. Int. J. Impact Eng. 34, 405–411 (2006) 4. Song, B., Ge, Y., Chen, W.W., Weerasooriya, T.: Radial inertia effects in Kolsky bar testing of extra-soft specimens. Exp. Mech. 47, 659–670 (2007) 5. Warren, T.L., Forrestal, M.J.: Comments on the effect of radial inertia in the Kolsky bar test for an incompressible material. Exp. Mech. 50, 1253–1255 (2010)

Chapter 14

Examining Material Response Using X-Ray Phase Contrast Imaging B. J. Jensen, B. Branch, F. J. Cherne, A. Mandal, D. S. Montgomery, A. J. Iverson, and C. Carlson

Abstract Propagation based X-ray phase contrast imaging (PCI) offers unique opportunities for ultrafast, high-resolution measurements to examine dynamic materials response at extreme conditions. Within the past decade, efforts on the IMPULSE system at the Advanced Photon Source included the development of a novel Multi-frame X-ray PCI (MPCI) system that was used to obtain the first shock-movies to examine material deformation with micron spatial resolution on nanosecond timescale. The MPCI system has been systematically developed over the years to improve optical efficiencies, spatial resolution, obtain more images per experiment, and to develop a dual-imaging, dual-zoom feature useful for many applications. With the MPCI system, X-ray PCI has been successfully used to study a wide range of phenomena including jet-formation in metals, crack nucleation and propagation, response of additively manufactured materials, and detonator dynamics to name a few. In this paper, a brief overview of the MPCI system development is provided along with its application to study shock propagation in materials. Keywords PCI · phase contrast imaging · x-ray imaging · shock compression · matter at extremes

14.1 Introduction Traditional shock wave experiments have proven successful over the years in relating the evolution of the shock wave propagation to the underlying mechanisms responsible for the material response [1–4]. Because these methods do not access the microscopic length scale directly, new diagnostics are needed that can provide real-time, in-situ, and spatially resolved measurements on the microscopic length scale with nanosecond time resolution. In recent years, there has been significant efforts in obtaining X-ray diffraction data in shock wave experiments [5–7] and in coupling dynamic loading platforms to the X-ray beam lines at Argonne National Laboratory’s Advanced Photon Source (APS, Argonne, IL) to take advantage of diagnostics such as X-ray diffraction (XRD) and X-ray Phase Contrast Imaging (PCI) [8–11]. In particular, X-ray PCI provides a unique opportunity for acquiring high-resolution, spatially resolved, in-situ images of materials response and shock wave propagation during dynamic compression experiments. In the current work, X-ray PCI images of shock wave propagation through a polycarbonate sample were obtained to determine the in-situ shock wave velocity. The results were compared with the known shock response of the material and data obtained using traditional shock wave methods.

14.2 Experiments and Methods Propagation based X-ray phase contrast imaging (PCI) is sensitive to differences in the index of refraction which is proportional to the density of the material. PCI requires a spatially coherent X-ray beam to achieve the enhanced contrast needed to bring out features within a material such as cracks, edges, and voids. The application to shock wave experiments was proposed by Montgomery [12] and first demonstrated experimentally at the Advanced Photon Source (APS; Argonne, IL) on the IMPULSE (IMPact System for ULtrafast Synchrotron Experiments) gas gun system [8–11, 13]. Experiments on IMPULSE at the APS have been performed to study a wide range of phenomena including crack propagation in materials B. J. Jensen () · B. Branch · F. J. Cherne · A. Mandal · D. S. Montgomery Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] A. J. Iverson · C. Carlson National Security Technologies, Los Alamos, NM, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_14

89

90

B. J. Jensen et al.

[14], jet formation in metals [15], dynamic extrusion tests on polymers [16, 17], response of additively manufactured (AM) materials to shock loading [18, 19], dynamic response of detonator initiator systems [20–22], compaction of powder materials [23], and response of explosives [24] to name a few. The multi-frame X-ray PCI (MPCI) system in routine use at the APS and at the new Dynamic Compression Sector (DCS) consists of individually gated ICCD (Intensified charge coupled device) PI-MAX-4 cameras (Princeton Instruments, Inc.) optically multiplexed together to obtain sequential images (or “shock movies”) during dynamic loading. A schematic of the system is shown in Fig. 14.1 and described in detail elsewhere [8–10, 15, 25, 26]. Briefly, X-rays from the synchrotron propagate through the sample and are incident upon a scintillator (typically LSO) which converts the X-rays to visible light. A series of optical components including mirrors (M1 and M2), beam splitters (BS), relay lenses (RL), and objective lenses (OB1 and OB2) image the light onto the ICCD cameras (ICCD 1–4). Because each ICCD is individually gated, the cameras can be synchronized with the desired X-ray bunch to capture a specific event during loading. Since its inception in 2011, the MPCI system has undergone significant improvement which includes the ability to take 8 images per experiment by interleaving in time four PI-MAX-4 ICCDs with the dual image feature (DIF). This feature allows for two images per camera with a minimum separation between images of at least 500 nanoseconds. The entire system is motorized for remote operation and optimization while the system is in the X-ray beam. Most recently, a dual imaging scheme for the scintillator (image both sides of the scintillator) has improved the light efficiency and allows the use of two different microscope objectives simultaneously (OB1 and OB2) for multiple magnification (5x, 7.5x, 10x) or fields-of-view (FOV) [21]. Note: to switch from the dual imaging mode (shown in Fig. 14.1) back to the standard single imaging mode, the pellicle has to be removed from the optical path and the mirror M1 must be replaced with a beam splitter. To show the utility of using MPCI to study the shock response of materials, the results for two similar experiments are shown: one that used the traditional velocimetry diagnostics to measure the shock velocity and another that used the MPCI system to measure the shock velocity in-situ. The traditional shock wave experiment [3, 27] is shown in Fig. 14.2 (top) and consists of a polycarbonate projectile and flyer plate impacting the polycarbonate sample. Multiple photonic Doppler velocimetry (PDV) probes [28] were positioned about the target to measure the particle velocity at the impact surface,

MPCI Experiment Confguration X ray Beam

Impacts of 0.5 mm borosilicate spheres (BS) stacked on a steel anvil

Sample/Target

Static Image Projectile Gun Barrel

Objective Lens (5x, 7.5x, 10x)

Pellicle

OB2

Scintillator OB1

Turning Mirror

RL

M2

M1 RL

BS

Target Anvil ICCD 1

RL

Compaction wave through a cylinder filled with 100 micron BS spheres Shock in plastic cylinder

ICCD 4

B S

ICCD 3

ICCD 2

Compacted Material

Compacted Material

Fig. 14.1 Experiment configuration for typical X-ray PCI impact experiments using the Los Alamos Laboratory MPCI system. (Top-Right) Example PCI data showing the compaction of glass spheres [26]. Complete absorption is indicated by black and complete transmission shown as yellow (fire colormap). The data show the sphere deforming and fracturing after impact. (Bottom-right) Example of using PCI to study shock wave propagation through a tube filled with 100 micrometer borosilicate spheres to study the compaction process. The shock wave in the confining material (PMMA) is visible along with the slower moving compaction front. In both examples the shock wave or impact occurs from left to right

14 Examining Material Response Using X-Ray Phase Contrast Imaging

Traditional Shock Wave Experiment Configuration

Polycarbonate Cylinder Mirror Polycarbonate Foam

PDV Data showing Particle velocity vs. Time

Particle Velocity (km/s)

PDV Probes

91

Peak Particle Velocity

Shock at back of sample

t = 0.668 s

Impact

X-ray PCI Experiment Configuration

Time ( s)

X-ray Beam Polycarbonate Cylinder

Sequential multi-frame X-ray PCI Images T =306.8 ns T = 0 ns T = 153.4 ns x = 852 pixels x = 236 pixels x = 543 pixels

FOV

6 mm

Scintillator Cu Foam

Shock Front Moving at Us Cu Impactor

Cu Impactor

To MPCI System

Fig. 14.2 Experiment configuration and example data for two experiments that use traditional and X-ray diagnostics. (Top) Experiment configuration for MPCI experiments along with three PCI images taken at 153.4 ns intervals to observe the shock wave propagation through polycarbonate. The shock front and copper impactor are indicated by the white arrows. The position of the shock wave in pixels is indicated by the parameter, x. (Bottom) Experiment configuration for a traditional symmetric impact (flyer and sample materials are the same) experiment to measure the particle velocity at the impact surface and the shock wave speed through the sample. Example data obtained using PDV are shown to the right. Dynamic images have been corrected using bright-field and dark-field images

back surface of the sample, and the projectile velocity up to impact simultaneously. Because the polycarbonate samples are transparent to the 1550 nm laser light used in the PDV, the shock wave speed can be measured directly using a single PDV probe located on the center of the sample. In contrast, the experimental configuration for the X-ray PCI experiments performed on the IMPULSE gas gun system at Sector 35 of the APS is shown in Fig. 14.2 (bottom). A polycarbonate projectile fitted with a copper flyer plate impacted a polycarbonate cylinder (sample) generating a steady shock wave the propagated through the material. X-rays were transmitted through the sample and were detected using the MPCI system to obtain images of the shock wave as a function of time. For all experiments, multiple images were taken including dark-field images to measure the camera background with no X-rays and bright-field images (X-ray beam with no sample) to calibrate 100% transmission, variations in the X-ray beam itself, and to account for phase contrast features along the beam line (e.g. dust, imperfections in window materials, etc.). Static images (pre-shot) were taken to position the object and visualize features prior to the experiment. A detailed description of the PCI analysis can be found elsewhere [26].

14.3 Results Example data for the traditional, symmetric impact experiment is shown in Fig. 14.2 (top) [27]. The measured projectile velocity was Vp = 0.674 ± 0.001 km/s. The PDV data is shown in Fig. 14.2 (top-right) where the particle velocity (km/s) is plotted versus time (μs). A velocity jump to a steady state particle velocity of up = 0.343 ± 0.005 km/s was observed at t = 1.27 μs. This steady state persists in the PDV data until the shock wave reaches the back surface of the sample at

92

B. J. Jensen et al.

t = 1.938 μs resulting in a release wave that propagates back through the sample. The difference between the shock jump and the shock arrival at the back surface is the transit time for the shock wave determined to be t = 0.668 μs. Given the known thickness of the polycarbonate sample (1.919 mm), the shock velocity was determined to be 2.873 ± 0.005 km/s in reasonable agreement with the calculated value of Us = 2.838 km/s obtained from the available Hugoniot data [29–31]. Example data obtained using the MPCI are shown in Fig. 14.2 (bottom-right) as three images taken 153.4 nanoseconds apart. The typical FOV was approximately 1.4 mm square (7.5× objective) with a pixel size of 1.687 μm/pix. The data show a well-defined shock front propagating through the sample from the left to the right (in successive images), and indicated by the white arrow that shows the direction of propagation. The copper flyer appears as the black region (100% X-ray absorption) coming into the FOV on the left of each image. The measured projectile velocity was Vp = 0.717 ± 0.001 km/s. To determine the shock wave velocity, horizontal lineouts were obtained from the images and averaged together to locate the position of the shock front at each time step. These are shown in the images as x = 236, 543, and 852 pixels (± 5 pixels uncertainty in locating the shock front) for X-ray bunch times of t = 0, 153.4, and 306.8 nanoseconds, respectively. The shock wave positions were plotted versus time and then fit to a line with the slope equal to the shock wave velocity. The resulting shock velocity was Us = 3.388 ± 0.055 km/s in reasonable agreement with the calculated value of Us = 3.323 km/s obtained from available Hugoniot data [29–31].

14.4 Conclusions Recent developments in coupling dynamic loading platforms to the X-ray beam lines at the APS are providing new methods for diagnosing shock physics experiments. In this work, two experiments were performed that used a traditional shock method to measure shock velocity and compared it to a similar experiment that used X-ray PCI to measure the shock wave velocity directly as it propagated through the material. Overall, the two methods resulted in shock wave speeds that are in reasonable agreement with predicted values. The traditional, optical technique does result in lower uncertainty in the shock velocity measurement, however it is a useful method only for materials that are optically transparent. The MPCI method will be essential for situations where the materials are opaque, when the shock wave profile exhibits a complex structure, and/or when the materials themselves are heterogeneous (e.g. additively manufactured materials, granular systems, powders, etc.). Efforts are underway to use the PCI images shown here to calculate the density behind the shock front to better characterize the high-pressure state of the material. The ability to measure shocked density states will pave the way for the analysis and understanding of more complicated materials such as powders, granular systems, foams, and other engineered materials. Acknowledgements This work was performed by Los Alamos National Laboratory (LANL) at Los Alamos and at the Dynamic Compression Sector (DCS) at the Advanced Photon Source (APS). All x-ray phase contrast images shown here were obtained using LANL’s multi-frame xray phase contrast imaging system (MPCI) developed on the IMPULSE capability at APS. Chuck Owens, Joe Rivera, and John Wright (LANL) are thanked for sample assembly, experiment preparation and execution. Nick Sinclair and Adam Schuman (DCS/WSU) are thanked for their technical support at the Sector 35 beamline setting up the X-ray beam. The authors gratefully acknowledge the financial support provided by Science Campaigns, Joint Munitions Program (JMP), and National Security Technologies (NSTec) Shock Wave Physics Related Diagnostic (SWRD) program. LANL is operated by Los Alamos National Security, LLC for the U.S. Department of Energy (DOE) under Contract No. DE-AC52-06NA25396. DCS is supported by the Department of Energy (DOE), National Nuclear Security Administration, under Award Number DE-NA0002442 and operated by Washington State University (WSU). This research used resources of APS, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC0206CH11357.

References 1. Asay, J.R., Fowles, G.R., Duvall, D.E., Miles, M.H., Tinder, R.F.: Effects of point defects on elastic precursor decay in LiF. J. Appl. Phys. 43, 2132 (1972) 2. Dolan, D.H., Johnson, J.N., Gupta, Y.M.: Nanonsecond freezing of water under multiple shock wave compression: continuum modeling and wave profile measurements. J. Chem. Phys. 123, 064702 (2005) 3. Jensen, B.J., Gray, G.T., Hixson, R.S.: Direct measurement of the α-ε transition stress and kinetics for shocked iron. J. Appl. Phys. 105, 013502 (2009) 4. Jensen, B.J., Cherne, F.J.: Dynamic compression of cerium in the low-pressure γ-α region of the phase diagram. J. Appl. Phys. 112, 013515 (2012) 5. Jensen, B.J., Gupta, Y.M.: Time-resolved x-ray diffraction experiments to examine the elastic-plastic transition in shocked magnesium-doped LiF. J. Appl. Phys. 104, 013510 (2008)

14 Examining Material Response Using X-Ray Phase Contrast Imaging

93

6. Jensen, B.J., Gupta, Y.M.: X-ray diffraction measurements in shock compressed magnesium doped LiF crystals. J. Appl. Phys. 100(5), 053512 (2006) 7. Kalantar, D.H., Belak, J.F., Collins, G.W., Colvin, J.D., Davies, H.M., et al.: Direct observation of the $\alpha-\epsilon$ transition in shockcompressed iron via nanosecond X-ray diffraction. Phys. Rev. Lett. 95, 075502 (2005) 8. Jensen, B.J., Lou, S.N., Hooks, D.E., Fezzaa, K., Ramos, K.J., Yeager, J.D., Kwiatkowski, K., Shimada, T., Dattelbaum, D.M.: Ultrafast, high resolution, phase contrast imaging of impact response with synchrotron radiation. AIP Adv. 2(1), 012170–012176 (2012) 9. Luo, S.N., Jensen, B.J., Hooks, D.E., Fezzaa, K., Ramos, K.J., Yeager, J.D., Kwiatkowski, K., Shimada, T.: Gas gun shock experiments with single-pulse x-ray phase contrast imaging and diffraction at the advanced photon source. Rev. Sci. Instrum. 83(7), 073903 (2012) 10. Jensen, B.J., Owens, C.T., Ramos, K.J., Yeager, J.D., Saavedra, R.A., Iverson, A.J., Luo, S.N., Fezzaa, K.: Hooks DE impact system for ultrafast synchrotron experiments. Rev. Sci. Instrum. 84(1), 013904–013905 (2013) 11. Gupta, Y.M., Turneaure, S.J., Perkins, K., Zimmerman, K., Arganbright, N., Shen, G., Chow, P.: Real-time, high-resolution x-ray diffraction measurements on shocked crystals at a synchrotron facility. Rev. Sci. Instrum. 83(12), 123905 (2012) 12. Montgomery, D.S., Nobile, A., Walsh, P.J.: Characterization of National Ignitition Facility cryogenic beryllium capsules using x-ray phase contrast imaging. Rev. Sci. Instrum. 75, 3986–3988 (2004) 13. Yeager, J.D., Luo, S.N., Jensen, B.J., Fezzaa, K., Montgomery, D.S., Hooks, D.E.: High-speed synchrotron X-ray phase contrast imaging for analysis of low-Z composite microstructure. Compos. A: Appl. Sci. Manuf. 43(6), 885–892 (2013). https://doi.org/10.1016/j.compositesa.2012.01.013 14. Ramos, K.J., Jensen, B.J., Yeager, J.D., Bolme, C.A., Iverson, A.J., Carlson, C.A., Fezzaa, K.: Investigation of dynamic material cracking with in situ synchrotron-based measurements. In: Song, B., Casem, D., Kimberley, J. (eds.) Dynamic Behavior of Materials, vol. 1, pp. 413–420. Springer International Publishing, Dordrecht (2014). ISBN 978–3–319-00770-0 15. Jensen, B.J., Cherne, F.J., Ramos, K.J., Iverson, A.J., Carlson, C.A., Yeager, J.D., Fezzaa, K., Dimonte, G., Hooks, D.E.: Multiphase material strength determined through shock generated Richtmyer-Meshkov instabilities. J. Appl. Phys. 118, 195903 (2015) 16. Brown, E.N., Ramos, K.J., Dattelbaum, D.M., Jensen, B.J., Gray III, G.T., Matterson, B.M., Trujillo, C.P., Martinez, D.T., Pierce, T.H., Iverson, A.J., Carlson, C.A., Fezzaa, K., Furmanski, J.: In situ and postmortem measures of damage in polymers at high strain-rate. Conference Proceedings of the Society for Experimental Mechanics Series. 65(1), 53–59 (2015) 17. Brown, E.N., Furmanski, J., Ramos, K.J., Dattelbaum, D.M., Jensen, B.J., Iverson, A.J., Carlson, C.A., Fezzaa, K., Trujillo, C.P., Martinez, D.T., Gray III, G.T., Patterson, B.M.: High-density polyethylene damage at extreme tensile conditions. J. Phys. Conf. Ser. 500, 112011 (2014) 18. Hawreliak, J., Lind, J., Maddox, B., Barham, M., Messner, M., Barton, N., Jensen, B.J., Kumar, M.: Dynamic behavior of engineered lattice materials. Nature Scientific Reports. 6, 28094 (2016) 19. Branch, B., Ionita, A., Clements, B., Montgomery, D., Jensen, B.J., Patterson, B., Mueller, A., Dattelbaum, D.M.: Controlling shockwave dynamics using architecture in periodic porous materials. J. Appl. Phys. 121, 135102 (2017) 20. Willey, T.M., Champley, K., Hodgin, R., Lauderbach, L., Bagge-Hansen, M., May, C., Sanchez, N.J., Jensen, B.J., Iverson, A.J., Van Buuren, T.: X-ray Imaging and 3D Reconstruction of In-Flight Exploding Foil Initiator Flyers. J. Appl. Phys. 119, 235901 (2016) 21. Sanchez, N.J., Neal, W.E., Jensen, B.J., Iverson, A.J., Carlson, C.A.: Dynamic Exploding Foil Initiator Imaging at the Advanced Photon Source. AIP Conf. Proc. 1979(160023), (2018) 22. Neal, W.E., Sanchez, N., Jensen, B.J., Gibson, J., Martinez, M., et al.: The effect of surface heterogeneity in exploding metal foils. AIP Conf. Proc. 1979(180007), (2018) 23. Mandal, A., Jensen, B.J., Aslam, T.D., Iverson, A.J.: Dynamic Compaction of Nickel Powder Examined by X-ray phase contrast imaging. AIP Conf. Proc. 1979(110010), (2018) 24. Ramos, K.J., Jensen, B.J., Hooks, D.E., Fezzaa, K., Yeager, J.D., Iverson, A.J., Carlson, C.A., Cherne, F.J.: In situ investigation of the dynamic response of energetic materials using IMPULSE at the advanced photon source. J. Phys. Conf. Ser. 500, 142028 (2014) 25. Jensen, B.J., Hooks, D.E., Fezzaa, K., Ramos, K.J., Yeager, J.D., Iverson, A.J., Carlson, C.A., Cherne, F.J., Kwiatkowski, K.: Dynamic experiments using IMPULSE at the advanced photon source. J. Phys. Conf. Ser. 500, 042001 (2014) 26. Jensen, B.J., Montgomery, D.S., Iverson, A.J., Carlson, C.A., Clements, B., Short, M., Fredenburg, D.A.: X-ray phase contrast imaging of granular systems. LA-UR-17-27104 Los Alamos Laboratory Report (2017) 27. Branch, B., Jensen, B.J.: Dynamic X-ray diffraction to study the shock-induced a-e phase transition in iron. AIP Conf. Proc. 1979(040001), (2018) 28. Jensen, B.J., Holtkamp, D.B., Rigg, P.A., Dolan, D.H.: Accuracy limits and window corrections for Photon Doppler velocimetry. J. Appl. Phys. 101, 013523 (2007) 29. LASL SHOCK HUGONIOT DATA (University of California Press, Berkley and Los Angeles, CA, 1980) 30. Carter WJ, Marsh SP. Hugoniot Equation of State of Polymers. LA-12006-MS Los Alamos National Laboratory (1995) 31. Millett, J.C.F., Bourne, N.K.: Shock and release of polycarbonate under one-dimensional strain. J. Mater. Sci. 41, 1683–1690 (2006)

Chapter 15

History Note: Machining, Strain Gages, and a Pulse-Heated Kolsky Bar R. Rhorer, S. Mates, E. Whitenton, and T. Burns

Abstract A special Kolsky bar apparatus with the capability to pulse heat the sample was built at the National Institute of Standards and Technology (NIST). This Kolsky bar laboratory’s initial purpose was to measure dynamic material properties in support of machining analysis research. Machining is a high-strain, high-strain-rate, high-temperature, high-heating-rate process. Developing mathematical models to analyze machining processes presents unique challenges, including appropriate material stress-strain relationships. The NIST system can heat a sample to over 1000 C in less than a second immediately prior to a Kolsky bar impact test. Although there are many traditional Kolsky bars in operation, the ability to achieve a very rapid temperature increase by electrical resistive heating is unique. The development and construction of this laboratory involved cooperative effort of several researchers from different NIST divisions. The work benefitted from a rich history of strain measuring research at the National Bureau of Standards (NBS), now named NIST, as well as drawing on a long history of interaction with the Society for Experimental Mechanics (SEM). . Keywords Experimental mechanics history · Kolsky bar · High-temperature dynamic testing · Machining analysis · Strain gages

15.1 Introduction This paper has been prepared for the Society for Experimental Mechanics (SEM) 2018 Annual Meeting honoring the 75th Anniversary of the Society. The call for papers included sessions on the industrial application of experimental mechanics. This paper reviews a project directly supporting machining research—machining is arguably the most fundamental of all industrial processes. This work relied on a rich history of experimental mechanics at the National Bureau of Standards (NBS), now named the National Institute of Standards and Technology (NIST), as well as extensive interactions with other researchers through SEM. The history of the scientific modeling of the mechanics of machining shows the traditional back-and-forth of analytical and experimental work. Highly skilled artisans developed machining systems that were capable of producing the ‘machines’ that undergirded the Industrial Revolution. Later, to guide the improvement of the machining processes, analytical models evolved to explain the basic ideas and concepts. The scientific/mathematical models allowed the analytical prediction of forces and temperatures associated with machining. As improved experimental measurement techniques were developed, the measuring of actual machining process parameters became possible. The experimental studies then highlighted the limitations of the models, thereby encouraging the development of improved models. With the advent of highly refined Finite Element Analysis (FEA) techniques being applied to machining analysis by the 1990s, it became clear that the available material properties models were often limiting the reliability of machining analysis. In the late 1990s a group of researchers at NIST initiated a project to improve the material-constitutive models—stress-strain relationships—that could be used in the FEA machining models.

R. Rhorer () · E. Whitenton Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, USA S. Mates Materials Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, USA T. Burns Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_15

95

96

R. Rhorer et al.

In general, machining is a very high strain, high strain-rate, high temperature, and high heating-rate process. Therefore, to produce useful constitutive models a dynamic testing process with high temperature and high heating rate was indicated. A traditional Kolsky bar (also often called a Split-Hopkinson Pressure bar) system was designed and built at NIST. This system incorporates a feedback-controlled electrical pulse capability that heats the sample prior to the traditional Kolsky bar test. The pulse heating system was already in existence at NIST having been developed and used for many years in a thermophysics program. Supporting the development of the NIST Pulse-heated Kolsky Bar laboratory included the cooperative effort of many different disciplines represented at NIST. Analytical mechanics, machining analysis, dynamic strain measurement, metallurgy, basic material science, high temperature thermophysics, infrared temperature measurements, high-speed infrared and visible imaging, and computer based digital data recording/analysis were all essential. The fabrication capability onsite at NIST was also important for the timely construction of the very specialized and precise experimental apparatus.

15.2 Machining Analysis The basic concept of machining is as old as human history. The need to remove material to shape a raw piece of stock into a useful shape is obvious. A sharp tool is used to assist in material removal. By the time of the Industrial Revolution— often defined as starting in the eighteenth Century—people began to develop machines that would hold a ‘cutting tool’ and mechanically move the tool to shape a workpiece. With the increasing need for metal objects, skilled workers continually improved the machines and associated cutting tools. By using the traditional practical methods, sometimes referred to as ‘trial-and-error’, there was a steady improvement in methods, machine tools, and cutting tools. By the early nineteenth Century skilled craftsman had developed techniques—often referred to as “secrets”—to produce amazing metal components. An interesting quote from a ‘trip report’ in about 1800 highlights the primitive level of technology and the challenges of machining: During his visit Cope watched the boring of one of the Philadelphia engines. The cylinder was 6 feet 6 inches long with an internal diameter, before boring, of 38-1/4 inches. The boring was to increase the internal diameter by three quarters of an inch. When Cope visited Soho, an increase of half an inch had been accomplished in 87 days! The drilling machine headpiece, slightly smaller than the bore of the cylinder, was fitted directly to the waterwheel shaft. Into this headpiece were fitted three steelings (cutters), each 3-1/2 inches wide. The boring continued night and day, stopping every ten minutes to change steelings, a process which itself took ten minutes. One workman “almost lives in the cylinder,” Cope reported. This mechanic had a hammer and it was his duty to “keep things in order” and to attend to the steelings [1].

It was clear to all that the basic idea machining worked, but improvements were essential. By the early twentieth Century scientific methods began to be employed to understand and improve the machining process. A famous engineer, F.W. Taylor, President of ASME, published an extensive work in 1907 that addressed in detail machining methods [2]. This work was based on empirical machining data and practical understanding of the processes, but did not use a scientific model. By the mid-twentieth Century, scientific modeling approaches were published, starting with the 1945 paper by Eugene Merchant, “Mechanics of the Cutting Process”, arguably the most significant paper in the history of scientific modeling of machining [3]. In the last half of the twentieth Century, many papers regarding the analysis of machining processes were published. In 1993 at the Oklahoma State University, Stillwater, Oklahoma, a symposium gathered together many of the top researchers in machining analysis. This symposium, titled U.S. Machining and Grinding Research in the twentieth Century, was documented with a special issue of Applied Mechanics Reviews [4]. This report includes a short comment by Professor Merchant, who attended the symposium, plus an extensive historical review with over 200 references prepared by Professor Ranga Komanduri [5]. Merchant’s approach included a simple material model with only one material property parameter: flow stress. In modern studies, constitutive models with five or more parameters are used to describe the material behavior in dynamic loading cases such as machining.

15.3 Strain Measurment The measurement of strain is fundamental to the discipline of Experimental Mechanics. Since the mid-twentieth Century, the history of the Society for Experimental Mechanics parallels the history of strain measurement developments. The SEM, which started as the Society for Experimental Stress Analysis (SESA) in 1938, was the technical society that led the

15 History Note: Machining, Strain Gages, and a Pulse-Heated Kolsky Bar

97

development of the metal-foil strain gages that made, by mid-twentieth Century, dynamic strain measurements possible. An excellent history of strain measurements can be found in the first SESA Handbook (Hettenyi, 1950) [6]. By the end of the twentieth Century, SEM had become a leader in the development and application of the Digital Image Correlation (DIC) strain measurement technology. In the U.S. from the time of the Revolutionary War until the twentieth Century, measurement standards and other scientific work was led by the U.S. Coast Survey and other federal agencies. In 1901, the U.S. Congress officially chartered the National Bureau of Standards (NBS). (In 1988, the name of NBS was changed to the National Institute of Standards and Technology (NIST)). From the early days of NBS, strain measurements were an important scientific endeavor. In the 1920s, NBS researchers produced special designs for quasi-static mechanical strain gages (we would now call these gages ‘extensometers’), such as the Whittemore and Tuckerman gages. NBS was often tasked with helping American industry, and these strain gage developments are an example: One device, made in the mechanics division for an investigation of riveted joints in the construction of Navy ships, was to have a wide application. This was the Tuckerman’s optical strain gage, devised in 1923, which gave consistent readings sensitive to two-millionths of an inch of deformation. It proved as reliable in measuring strains in the duralumin members in the framework of dirigibles, in concrete models of dams, or in steel and cement models of building structures, as in ship construction [7].

As electrical resistance strain gages were developed in the 1930s and 1940s, NBS pursued several related development and measurement efforts. In 1951, NBS sponsored a symposium “Characteristics and Applications of Resistance Strain Gages” as part of their activities celebrating the 50th anniversary of the founding of NBS. In the Foreword to the publication of the symposium papers (NBS Circular 528, 1954), NBS Director A. V. Astin, commented about work on strain measurements at NBS in the 1930s, stating: Work in this field was greatly accelerated with the introduction of strain sensitive fine wires a few years later. Extensive performance tests were made on wire strain gages of different manufacture. Studies were made of applications of strain gages to the measurement of mechanical quantities, such as acceleration, impact force, and dynamic pressure, and calibrations were carried out on a multitude of instruments employing wire strain gages as the sensing element. As in other laboratories, wire strain gages were used in large quantities to determine strain distributions in structures under load.

In addition, Director Astin further commented about current and future work related to strain gages at NBS: Work is in progress on strain gages consisting of a conducting coating applied by an evaporation technique, .. and on the application of strain gages to the determination of dynamic properties of materials . . . [8].

There was participation of NBS researchers during this era with the SESA conferences and publications. Using wire strain gages on a long copper bar subjected to impact loading, NBS engineer W.R. Campbell demonstrated a method of determining a dynamic stress-strain curve. He published this work in the Proceedings of the Society for Experimental Stress Analysis (SESA) in 1952 [9]. Campbell’s approach consisted of measuring the velocity of elastic-plastic waves and determining the tangent modulus to then calculate stresses as a function of strain. He demonstrated that the method was possible, but the uncertainty of the calculated stresses was high and the stress-strain data was not very useful. The challenges encountered were difficulties in accurately recording the strain gage signals and the effects of the relatively long gage length gages (he used the standard SR-4 type A-3 wire gage with a gage length of ¾ inch). This work was about the same time as the early Kolsky bar work, for example Kolsky used a “modified form of the Davies bar” in 1949 for dynamic testing [10]. The work performed and published by H. Kolsky became known as the Kolsky bar, or Split Hopkinson Pressure bar, and evolved into the primary method for dynamic material property determination by the later decades of the twentieth Century. Almost all Kolsky bars use the traditional metal foil strain gages, developed in the 1950s, but the strain gages are not used to measure strains in the sample material directly (as was done by Campbell), but the plastic stress-strain curve of the sample is determined indirectly from the dynamic elastic strain signals from gages on the bars. There is an extensive body of literature related to the design and application of the Kolsky bar to determine dynamic material properties. An excellent reference that gives not only the basic operation and background of the Kolsky bar, but also an extensive bibliography, is the article by K.T. Ramesh in the new “SEM Handbook” [11]. By the late decades of the twentieth Century, there were hundreds of Kolsky bars in university, industrial, and national laboratories. NBS had an extensive history of strain measurement work, but we find no records of follow on work of Campbell’s project, or any Kolsky bar work at NBS/NIST, until the development of the NIST Pulse-heated Kolsky Bar in about 2000.

98

R. Rhorer et al.

15.4 NIST Pulse-Heated Kolsky Bar A group of NIST researchers, starting in the late 1990s, initiated a project to “develop the capability to obtain and validate the material response data that are critical for accurate simulation of high-strain-rate manufacturing processes.” [12] The project was funded in 2001 as part of an internal NIST “Advanced Technology Intramural Program.” The project involved participation from five different NIST Divisions: Manufacturing Metrology Division, Mathematical and Computational Sciences Division, Metallurgy Division, Optical Technology Division, and the Fabrication Technology Division. Drawing on a rich NIST history of cooperation within many different scientific disciplines, researchers designed and built the new Pulse-Heated Kolsky Bar laboratory. This new Kolsky bar facility was directed at providing dynamic material properties of important industrial materials (for example carbon steels used in the automotive industry) at elevated temperatures in high-heating-rate environments like machining. The initial team of researchers involved Matthew Davies and Timothy Burns (analytical mechanics and mathematical modeling), Richard Fields and Lyle Levine (metallurgy and material property measurements), Debasis Basak (thermophysics and expertise with the existing pulse heating system at NIST), Howard W. Yoon (applied optics and expertise in infrared temperature measurements), Eric Whitenton (electronics, computer programing and high speed videography); Brian Dutterer (senior machinist, mechanical design, fabrication), and Michael Kennedy (senior mechanical technician, lab set up, strain gage application). Matthew Davies was the Principal Investigator until 2001, when he moved from NIST to the University of North Carolina at Charlotte and then Richard Rhorer took on the project leadership role. A project publication—“Temperature Control of Pulse Heated Specimens in a Kolsky Bar Apparatus Using Microsecond Time-Resolved Pyrometry” [13]—provides a description of the heating system. The development of the heating approach involved many experiments. The use of modern thermal-imaging high-speed video recording cameras verified that we were achieving uniform heating of the sample. Throughout the project results from material tests using the NIST Pulse Heated Kolsky Bar have been reported in several venues, including several papers presented at SEM annual meetings such as “Recent Results from the NIST Pulse Heated Kolsky Bar” in 2007 [14]. With a facility such as the NIST Kolsky bar laboratory with many different users, both internal as well as external, a valuable asset has been a computer based acquisition, storage, and data reduction program. This program, written by Eric Whitenton, was described in a 2005 SEM conference paper [15]. Laboratory users could have access to all test data from their own offices any time after a test. In 2005, Steven Mates joined the Kolsky lab project and in 2009 he became the project leader. Under Mates’ leadership, the laboratory has expanded to include a tension bar system (utilizing the same pulse heating system) as well as updating many of the hardware, control, and data acquisition systems. By early 2018, approximately 5000 Kolsky bar tests have been completed in the laboratory. These tests include many heated-metal tests plus numerous room temperature tests ranging from very soft materials like flesh simulants to hard materials like ceramics.

15.5 Conclusions and Future Work The NIST Pulse-Heated Kolsky Bar Laboratory met the original goal of providing constitutive models for assisting the analysis of machining processes. After 15 years of operation, this laboratory continues to provide dynamic material properties for research efforts. The laboratory now has additional capabilities to provide pulse-heated tension dynamic testing. The visible light and thermal high-speed imaging capabilities developed in the Kolsky laboratory have been applied to realistic high-speed machining tests in the NIST Shops. A rich history of strain measuring work at NBS/NIST, combined with the material science and thermophysics capabilities, allowed the rapid development of the new Kolsky bar apparatus. The relationship of this project with the Society for Experimental Mechanics has been beneficial, both from the standpoint of the history of strain gage applications and providing an excellent forum for discussing our results with leaders in the field of dynamic material properties. We see future experimental mechanics work at the NIST laboratory in working with other institutions to develop standards for Kolsky bar work. In addition, there are experimental mechanics challenges in the field of machining research such as the development of advanced cutting force transducers with much higher frequency capabilities than those commonly used today.

15 History Note: Machining, Strain Gages, and a Pulse-Heated Kolsky Bar

99

References 1. Pursell, Carroll W., Jr., Early Stationary Steam Engines in America: A Study in the Migration of a Technology, Smithsonian Institution Press, City of Washington, 1969, pg. 55 2. Taylor, F.W.: On the art of cutting metals. Trans ASME. 28, 31–248 (1907) 3. Merchant, M.E.: Mechanics of the Cutting Process. J. Appl. Phys. 216, 267–318 (1945) 4. Komanduri, R., Merchant, M.E., Shaw, M.C.: U.S. Machining and Grinding Research in the 20th Century. Special Issue, Appl. Mech. Rev. 46(3), 72–132 (1993) 5. Komanduri, R.: U.S. Machining and Grinding Research in the 20th Century. Special Issue, Appl. Mech. Rev. 46(3), 129–132 (1993) 6. Hetenyi, M.: Handbook of Experimental Stress Analysis. Wiley, NY (1950) 7. Cochrane, R.C.: Measures for Progress: a History of the National Bureau of Standards. National Bureau of Standards, U.S. Department of Commerce (1966) 8. Astin, A.V., Director, National Bureau of Standards, Characteristics and Applications of Resistance Strain Gages: Proceedings of the NBS Semicentennial Symposium of Resistance Strain Gages, Held at the NBS on November 8 and 9, 1951, National Bureau of Standards Circular 528 (1954) 9. Campbell, W.R.: Determination of dynamic stress-strain curves from strain waves in long bars. Proc. Soc. Exper. Stress Anal. X(1), 113–124 (1952) 10. Kolsky, H.: Stress Waves in Solids, Dover Publications, NY, 1963, p. 154. (The Dover edition, first published in 1963, is an unabridged and corrected republication of the work first published by the Clarendon Press, Oxford, in 1953) 11. Ramesh, K.T.: High rates and impact experiments. In: Sharpe Jr., W.N. (ed.) Springer Handbook of Experimental Solid Mechanics, pp. 929– 959. Springer, NY (2008) 12. Davies, M.: Internal NIST Memo: ATP Proposal 2000 13. Basak, D., Yoon, H.W., Rhorer, R., Burns, T.J., Matsumoto, T.: Temperature control of pulse heated specimens in a Kolsky bar apparatus using microsecond time-resolved Pyrometry. Int. J. Thermophys. 25(2), 561–574 (2004) 14. Burns, T.J., Mates, S.P., Rhorer, R.L., Whitenton, E.P., Basak, D.: Recent Results from the NIST Pulse-Heated Kolsky Bar. Proceedings of the Society for Experimental Mechanics Annual Conference (2007) 15. Whitenton, E.: The NIST Kolsky Bar Data Processing System. Proceedings of the Society for Experimental Mechanics Annual Conference, Portland, Oregon (2005)

Chapter 16

Improved Richtmyer-Meshkov Instability Experiments for Very-High-Rate Strength and Application to Tantalum Michael B. Prime, William T. Buttler, Saryu J. Fensin, David R. Jones, Ruben Manzanares, Daniel T. Martinez, John I. Martinez, Derek W. Schmidt, and Carl P. Trujillo

Abstract Recently, Richtmyer-Meshkov instabilities (RMI) have been used for studying metal strength at strain rates up to at least 10ˆ7/s. RMI experiments involve shocking a metal interface with geometrical perturbations that invert, grow, and possibly arrest subsequent to the shock. In experiments one measures the growth and arrest velocities to study the specimen’s flow (deviatoric) strength. In this paper, we describe experiments on tantalum at three shock pressure from 20 to 34 GPa, with six different perturbation sizes at each pressure, making this the most comprehensive set of RMI experiments on any material. In addition, these experiments were fielded using impact loading, as compared to high explosive loading in previous experiments, allowing for more precise modeling and more extensive interpretation of the data. Preliminary results are presented. Keywords Dynamic strength · Richtmyer-Meshkov instability · High-rate strength · Shock physics · Hydrocode

16.1 Introduction Recently, researchers have shown that Richtmyer-Meshkov Instabilities (RMI) are sensitive to strength at strain rates up to at least 107 /s [1–18]. Figure 16.1 illustrates a nominal RMI experiment for strength. The initial perturbations invert after shock, and the subsequent peaks are called spikes and the valleys are called bubbles. The initial perturbation size is characterized by the non-dimensional number η0 k (where k = 2π /λ). Previous RMI experiments to interrogate strength used high explosive loading to generate the shock [4, 19]. This work reports the first such measurements using impact loading as an improvement. Interpreting the data requires modeling the experiments in a hydrocodes or something similar [19]. Impact loading can be modelled both more simply and more accurately than high explosive, as illustrated by the significant previous effort to model explosive loading [19]. More accurate modeling should make for easier and more accurate strength estimation. Furthermore, impact loading makes it simpler to adjust impact pressure and interrogate strength over a wider range of conditions.

16.2 Experiments The experiment used the same batch of “Starck” tantalum that has been well characterized and used in other work [20–22]. The three tantalum targets were 40 mm diameter and 1.45 mm thick. On one face, diamond turning techniques were used to machine six regions containing multiple wavelengths of a sinusoidal surface perturbation, with η0 k ranging from 0.3 to 0.9 with a constant wavelength of 0.25 mm except for the η0 k = 0.9 region which had a wavelength of 0.3 mm. These targets were then pressed into a tantalum momentum ring, 55 mm diameter, 1.45 mm thick, the purpose of which was to avoid edge release waves reaching the central region where diagnostics were placed, ensuring a simple one-dimensional loading. Finally, a 0.25 mm thick tantalum foil was glued to the impact face, covering both the sample and the momentum ring, resulting in a final target thickness of 1.7 mm. Each target was bonded to the front of a Lexan plate. This plate served to both mount the target in the gas gun and position the PDV probes. A 50 mm hole in the Lexan plate allowed the probes to directly image the rear of the target. A total of 24

M. B. Prime () · W. T. Buttler · S. J. Fensin · D. R. Jones · R. Manzanares · D. T. Martinez · J. I. Martinez · D. W. Schmidt · C. P. Trujillo Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_16

101

102

M. B. Prime et al.

Fig. 16.1 A Richtmyer-Meshkov instability experiment. The perturbed surface of the sample is accelerated by a shock. At later time (upper figure), the perturbations have inverted

PDV probes were used, measuring the response of both the perturbed regions and flat regions, see Fig. 16.2. One probe was mounted in the Lexan plate, collimated down the gas-gun barrel, to measure the impact velocity. A single piezo-electric pin was also mounted in the Lexan plate to provide a trigger signal for the diagnostics at impact. A series of irises, mirrors, and a HeNe laser was used to align the Lexan plate normal to the end of the gas-gun barrel to provide a planar impact. Typical impact-tilts achieved with this technique are sub milli-radian. For all three experiments, the flyer plate was a 38 mm diameter, 2.5 mm thick, tantalum disc. This was mounted to the front of the projectile, supported by a glass micro-bead material to avoid bowing of the flyer plate during the launch. The 80 mm diameter gas-gun in MST-8 at LANL was used to accelerate the projectile to velocities from ∼630 m/s to ∼1000 m/s, generating stresses on the order of 20 GPa, 30 GPa and 34 GPa.

16.3 Results Figure 16.3 shows an exemplar PDV velocity spectrogram. For the 34 GPa experiment, velocity results for the largest initial perturbation size are shown. The highest velocities in the spectrogram come from the spike. In this case, the peak spike velocity is about 1900 m/s, which is well above the rest of the velocities thus indicating spike growth. Over about 1 μs, the velocity decays to the background level of about 1000 m/s indicating arrest: the spike growth has ceased. The lowest velocities after shock breakout are around 600 m/s, which come from the bubble region during the perturbation inversion process. As with the spike, those velocities return to the background level indicating cessation of the instability growth.

16.4 Discussion The experiments provided good velocity data for all perturbation sizes at all three shot pressures. Work is underway to estimate the strength for each shot following a previously reported procedure [19]. Following that, the strength will be calculationally assigned to the appropriate strain, strain, rate and temperature [14] so that the strength values can be used to help calibrate a constitutive (strength) model valid for wide ranges of conditions [23]. Acknowledgements Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the Los Alamos National Security, LLC for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396. By

16 Improved Richtmyer-Meshkov Instability Experiments for Very-High-Rate Strength and Application to Tantalum

103

Fig. 16.2 A view of the target free surface and the PDV probes prior to execution

Fig. 16.3 The PDV velocity spectrogram for the region with η0 k = 0.9 in the 34 GPa experiment shows a distinct signal for the spike with a maximum velocity near 1900 m/s during growth and a later return to the background velocity, indicating arrest

approving this article, the publisher recognizes that the U.S. Government retains nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos National Laboratory strongly supports academic freedom and a researcher’s right to publish; as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness.

104

M. B. Prime et al.

References 1. Piriz, A.R., Cela, J.J.L., Tahir, N.A., Hoffmann, D.H.H.: Richtmyer-Meshkov instability in elastic-plastic media. Phys. Rev. E. 78(5), 056401 (2008) 2. Piriz, A.R., Cela, J.J.L., Tahir, N.A.: Richtmyer–Meshkov instability as a tool for evaluating material strength under extreme conditions. Nucl Instrum Meth A. 606(1), 139–141 (2009) 3. Dimonte, G., Terrones, G., Cherne, F.J., Germann, T.C., Dupont, V., Kadau, K., Buttler, W.T., Oro, D.M., Morris, C., Preston, D.L.: Use of the Richtmyer-Meshkov instability to infer yield stress at high-energy densities. Phys. Rev. Lett. 107(26), 264502 (2011) 4. Buttler, W.T., Oró, D.M., Preston, D.L., Mikaelian, K.O., Cherne, F.J., Hixson, R.S., Mariam, F.G., Morris, C., Stone, J.B., Terrones, G., Tupa, D.: Unstable Richtmyer-Meshkov growth of solid and liquid metals in vacuum. J. Fluid Mech. 703, 60–84 (2012) 5. López Ortega, A., Lombardini, M., Pullin, D.I., Meiron, D.I.: Numerical simulations of the Richtmyer-Meshkov instability in solid-vacuum interfaces using calibrated plasticity laws. Phys. Rev. E. 89(3), 033018 (2014) 6. Mikaelian, K.O.: Shock-induced interface instability in viscous fluids and metals. Phys. Rev. E. 87(3), 031003 (2013) 7. Plohr, J.N., Plohr, B.J.: Linearized analysis of Richtmyer-Meshkov flow for elastic materials. J. Fluid Mech. 537, 55–89 (2005) 8. Prime, M.B., Vaughan, D.E., Preston, D.L., Buttler, W.T., Chen, S.R., Oró, D.M., Pack, C.: Using growth and arrest of Richtmyer-Meshkov instabilities and Lagrangian simulations to study high-rate material strength. J. Phys. Conf. Ser. 500(11), 112051 (2014) 9. Opie, S., Gautam, S., Fortin, E., Lynch, J., Peralta, P., Loomis, E.: Behaviour of rippled shocks from ablatively-driven Richtmyer-Meshkov in metals accounting for strength. J. Phys. Conf. Ser. 717(1), 012075 (2016) 10. John, K.K.: Strength of Tantalum at High Pressures through Richtmyer-Meshkov Laser Compression Experiments and Simulations. Ph.D. Dissertation, California Institute of Technology, Pasadena, CA (2014) 11. Buttler, W.T., GrayIII, G.T., Fensin, S.J., Grover, M., Prime, M.B., Stevens, G.D., Stone, J.B., Turley, W.D.: Yield strength of Cu and a CuPb alloy (1% Pb). AIP Conf. Proc. 1793(1), 110005 (2017). https://doi.org/10.1063/1.4971668 12. Sternberger, Z., Maddox, B.R., Opachich, Y.P., Wehrenberg, C.E., Kraus, R.G., Remington, B.A., Randall, G.C., Farrell, M., Ravichandran, G.: A comparative study of Rayleigh-Taylor and Richtmyer-Meshkov instabilities in 2D and 3D in tantalum. AIP Conf. Proc. 1793(1), 110006 (2017). https://doi.org/10.1063/1.4971669 13. Prime, M.B., Buttler, W.T., Buechler, M.A., Denissen, N.A., Kenamond, M.A., Mariam, F.G., Martinez, J.I., Oró, D.M., Schmidt, D.W., Stone, J.B., Tupa, D., Vogan-McNeil, W.: Estimation of metal strength at very high rates using free-surface Richtmyer–Meshkov instabilities. J. Dyn. Behav. Mater. 3(2), 189–202 (2017). https://doi.org/10.1007/s40870-017-0103-9 14. Prime, M.B.: Strain rate sensitivity of Richtmyer-Meshkov instability experiments for metal strength. In: Kimberley, J., Lamberson, L., Mates, S. (eds.) Dynamic Behavior of Materials, Volume 1: Proceedings of the 2017 Annual Conference on Experimental and Applied Mechanics, pp. 13–16. Springer International Publishing, Cham, Switzerland (2018). https://doi.org/10.1007/978-3-319-62956-8_3 15. Opie, S.: Effects of Phase Transformations and Dynamic Material Strength on Hydrodynamic Instability Evolution in Metals. Ph.D. thesis Arizona State University, Tempe. Arizona, USA (2017) 16. Sternberger, Z., Opachich, Y., Wehrenberg, C., Kraus, R., Remington, B., Alexander, N., Randall, G., Farrell, M., Ravichandran, G.: Investigation of hydrodynamic instability growth in copper. Int. J. Mech. Sci. (2017). in press, https://doi.org/10.1016/j.ijmecsci.2017.08.051 17. Zhou, Y.: Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720-722, 1–136 (2017). https://doi.org/10.1016/j.physrep.2017.07.005 18. Zhou, Y.: Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723–725, 1–160 (2017). https://doi.org/10.1016/j.physrep.2017.07.008 19. Prime, M.B., Buttler, W.T., Buechler, M.A., Denissen, N.A., Kenamond, M.A., Mariam, F.G., Martinez, J.I., Oró, D.M., Schmidt, D.W., Stone, J.B., Tupa, D., Vogan-McNeil, W.: Estimation of metal strength at very high rates using free-surface Richtmyer-Meshkov instabilities. J. Dyn. Behavior Mater. 3(2), 189–202 (2017). https://doi.org/10.1007/s40870-017-0103-9 20. Vachhani, S.J., Trujillo, C., Mara, N., Livescu, V., Bronkhorst, C., Gray, G.T., Cerreta, E.: Local mechanical property evolution during high strain-rate deformation of tantalum. J. Dyn. Behav. Mater. 2(4), 511–520 (2016). https://doi.org/10.1007/s40870-016-0085-z 21. Buchheit, T.E., Cerreta, E.K., Diebler, L., Chen, S.-R., Michael, J.R.: Characterization of Tri-lab Tantalum (Ta) Plate. Sandia National Laboratories Report SAND2014-17645 (2014) 22. Lim, H., Bong, H.J., Chen, S.-R., Rodgers, T.M., Battaile, C.C., Lane, J.M.D.: Developing anisotropic yield models of polycrystalline tantalum using crystal plasticity finite element simulations. Int. J Solids Struct. 730(11), 50–56 (2018) 23. Preston, D.L., Tonks, D.L., Wallace, D.C.: Model of plastic deformation for extreme loading conditions. J. Appl. Phys. 93(1), 211–220 (2003)

Chapter 17

Mechanical Characterization and Numerical Material Modeling of Polyurea James LeBlanc, Susan Bartyczak, and Lauren Edgerton

Abstract The mechanical behavior of four unique blends of polyurea materials has been investigated through a combined experimental and computational study. Mechanical characterization of each material was evaluated under both tensile and compressive loading at strain rates ranging from 0.01 to 100 strains per second (1/s). Planar blast wave experiments utilizing a 40 mm light gas gun were also conducted which imparted strain rates up to 104 strains per second (1/s). The material testing results showed that stress-strain response is a function of loading, strain level, and strain rate. These results were utilized to define a non-linear rubber material model in Ls-Dyna which was validated against the test data through a series of “block” type simulations for each material. Each material model was shown to replicate both the tensile and compressive behavior as well as the strain rate dependence. The material models were subsequently extended to the simulations of the blast wave experiments. The blast wave simulations were shown to accurately capture wave propagation resulting from a shock type pressure loading as well as the stress magnitudes of the transmitted waves after passing through the respective polyurea materials. The current study has resulted in the mechanical characterization of four polyurea materials under tensile/compressive loading at increasing strain rates, a suitably validated numerical material model, and suitable correlations between experimental and simulation results. Keywords Polyurea materials · Blast loading · Computational modeling · Material characterization · Strain rate effects

17.1 Material Characterization Polyurea materials in general exhibit non-linear responses which differ under tensile and compressive loadings, as well as exhibit both strain and strain rate response dependence. Baseline material characterization for each of the four unique polyurea blends was conducted for tensile and compressive loading at strain rates from 0.01-to-100 1/s. The results of the mechanical characterization show that the materials in the current study exhibit unique responses under compressive and tensile loading, as well as a stiffening effect under increasing strain rate. The typical stress-strain response is shown in Fig. 17.1. The response of the materials under strain rates up to 104 1/s was studied through a series of gas gun experiments which produce uni-axial stress wave loading through the material thickness. A schematic of the gas gun experimental setup is shown in Fig. 17.2.

17.2 Experimental / Numerical Correlation The finite element model of the gas gun target assembly is shown in Fig. 17.3 and includes the front aluminum cylinders, polyurea sample, and rear aluminum cylinders. Figure 17.4 presents the correlation between the experimental and numerical results for the front and back face PVDF gauges. From this figure it is seen that due to pressure loading on the front-face of the aluminum there is a dilatational stress wave that propagates through the aluminum cylinder and ultimately into the polyurea disk. The wave is nearly a stepwise shape characterized by a near instantaneous rise time, followed by a constant stress profile and a rapid decay as the wave passes through the gauge. It is shown that the model accurately captures both the J. LeBlanc () Naval Undersea Warfare Center (Division Newport), Newport, RI, USA e-mail: [email protected] S. Bartyczak · L. Edgerton Naval Surface Warfare Center (Dahlgren Division), Dahlgren, VA, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_17

105

106

Fig. 17.1 Material characterization

Fig. 17.2 Gas gun experimental schematic

Fig. 17.3 Computational model schematic

J. LeBlanc et al.

17 Mechanical Characterization and Numerical Material Modeling of Polyurea

107

Fig. 17.4 PVDF stress gauge correlation

magnitude as well as the time duration of the incident stress wave in the forward aluminum cylinders and into the polyurea samples. The correlation of the transmitted wave, as recorded by the back face PVDF gauge, is also shown in Fig. 17.4. It is seen that the magnitude and general shape of the transmitted stress wave from the simulation is in good agreement with the experimental result. It is noted that the simulation does predict the arrival of stress wave at the PVDF gauge approximately 2 micro-sec sooner than is observed in the experiments. Based on these results, it is determined that the material models suitably predict the stress wave propagation through the material in terms of dispersion; however, there is a slight difference in wave speed. Acknowledgements The authors acknowledge the financial support provided by the Naval Undersea Warfare Center (Division Newport) and the Naval Surface Warfare Center (Dahlgren Division) through their respective internal investment programs.

Chapter 18

Full-Scale Testing and Numerical Modeling of Adhesively Bonded Hot Stamped Ultra-High Strength Steel Hat Sections Y. B. Liu, D. Cronin, and M. Worswick

Abstract The implementation of structural adhesives to join multi-material lightweight vehicle structures requires advanced computer aided engineering (CAE) and therefore thorough material characterization and model validation at the component level. Hot stamped, 1.2 and 1.8 mm thick ultra-high strength steel hat section channels were joined to form closed tubular structures using a two-part toughened epoxy adhesive applied to the flanges, with a bondline thickness of 0.007 (0.178 mm). The joined tubes were tested under quasi-static loading in two configurations: three-point bending to load the adhesive in shear (Mode II) and axial crush resulting primarily in Mode I loading. Finite element models of the tests were developed using previously measured material properties for the adhesive implemented using cohesive zone elements. The three-point bending response included a linear loading regime followed by localized plastic deformation of the tube and finally abrupt failure of the adhesive joint between the hat sections at an average load of 34.0 kN for the 1.2 mm tubes and 78.8 kN for the 1.8 mm tubes. The axial crush response included an initial average peak force of 260 kN followed by a local folding or global deformation mode, leading to progressive separation of the adhesive joint and an average energy absorption of 8.45 kJ. Finite element models based on published adhesive and metal properties demonstrated good correlation with experimental results in predicted peak force and overall loading response. Keywords Structural adhesive · Material characterization · Cohesive elements · Finite element modeling · Ultra-high strength steel

18.1 Introduction and Background Automotive manufacturers face increasing challenges to improve the fuel efficiency of their fleets to reach a Corporate Average Fuel Economy (CAFE) Standard of 54.5 mpg by 2025 [1]. This need has led to the implementation of ultra-high strength materials and the consideration of multi-material lightweight vehicle (MMLV) structures where joining of dissimilar materials may be achieved with structural adhesives. For instance, through the use of advanced adhesive and multi-material designs, a new vehicle platform was able to shed 320 kg over the previous model and saw an increase of 28% in fuel efficiency [2]. In addition to weight reduction, adhesives also improve noise, vibration and hardness (NVH) and corrosion resistance due to the formation of a continuous and sealed joint [3, 4]. The growing popularity of adhesive bonding has also increased the need for material characterization to support CAE analysis. The mechanical properties that are needed to predict the adhesive behavior in the finite element (FE) environment can be obtained with tests such as the lap shear, rigid double cantilever beam, tapered double cantilever beam and bulk adhesive testing [5]. The focus of this study was to develop a method to reliably join hot-stamped hat sections into tubes with adhesive-only joints, test the tubes in three-point bending and axial crushing under quasi-static loading rates, and then finally to assess an adhesive model developed from bulk material and coupon-level test data.

18.2 Methodology Ultra-high strength steel (Usibor® 1500-AS, ArcelorMittal) blanks (1.2 mm thickness) were austenitized at 930 ◦ C for six minutes and then hot stamped into hat sections 590 mm in length. A fully martensitic microstructure was achieved through

Y. B. Liu () · D. Cronin · M. Worswick University of Waterloo, Waterloo, ON, Canada e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_18

109

110

Y. B. Liu et al.

in-die quenching. To maximize the adhesive joint strength and achieve consistent cohesive failure, the Al-Si intermetallic coating on the formed hat sections was removed by grit blasting. Immediately following the coating removal, the hat section flanges were cleaned with Methyl Ethyl Ketone to remove surface contaminates. This combination of surface preparation technique was found to optimize bond quality and reduce variability, increasing single lap shear failure strength by 4.3 MPa (19.8%) while reducing standard deviation from 1.75 MPa to 0.28 MPa. Finally, the hat sections were joined with a two-part epoxy structural adhesive (#07333 Impact Resistant Structural Adhesive, 3 M Corporation) using a bond line thickness of 0.178 mm, maintained with brass shims. A custom fixture was used to secure the hat sections while oven-curing the adhesive (80 degrees Celsius for 30 minutes). It was found that proper fixturing was important to achieve a consistent bond line thickness and to reduce variability in the component-level response of the tubes. In addition, minimizing the time between application of the adhesive and joining of the components further improved the consistency of the experimental results. Three-point bending was used to test the adhesive in a Mode II (shear) loading condition. The test setup comprised a hydraulic load frame and an indenter attached to the piston with an outer diameter of 100 mm. The supports were spaced 375 mm apart and had an outer diameter of 50 mm. The indenter and supports were lined with Teflon to reduce friction, allowing the tube to deform freely without binding to the metal surface. Although 1.2 mm thick hat sections were the focus of the study, three preliminary three-point bend tests used 1.8 mm gauge thickness to benchmark crush response. Mixed-mode loading was evaluated by loading the tubes in the axial direction. The tube length was reduced to 490 mm to mitigate global buckling response during the test. A fold initiator was introduced 70 mm from the top of the tube, to initiate local buckling and folding in the material. The experimental tests were simulated using a commercial explicit finite element code (LS-DYNA R7.1.2, Livermore Software Technology Corporation). Principles of Cohesive Zone Modelling (CZM) were applied, where cohesive elements with a defined traction-separation response were connected to the hat section shell elements accounting for the shell thickness. The fixtures were modeled as rigid materials with dimensions and masses corresponding to the experimental test setup. Current models did not use failure criteria since no failure was observed in three-point bend experiments and extensive failure in axial crush could result in instability.

18.3 Experimental Results The three-point bend test force-displacement response (Fig. 18.1) of the 1.8 mm thick specimens showed good repeatability in peak force (78.8 kN, standard deviation 0.9 kN) and displacement to ultimate load (28.2 mm, standard deviation 1.2 mm), where the adhesive joint had failed. Tests of the 1.2 mm thick specimens showed consistent peak force (34.0 kN, standard deviation 1.6 kN) but had variation in displacement to failure. This was due to an asymmetric (out-of-plane) deformation occurring in the second and third tests (3P-2 and 3P-3) (Fig. 18.1), which delayed the failure of adhesive joint considerably, thus increasing total displacement to failure. The axial crush tests (Fig. 18.2) demonstrated good repeatability in peak force (260 kN, standard deviation 6.9 kN); however, variations in deformation after the initial peak (folding or global buckling) led to differences in total energy absorbed. Folding behavior in the first test (AX-1) was associated with higher energy absorption (10.5 kJ) compared to global buckling in the second test (AX-2) (Fig. 18.2) and the third test (AX-3) (7.26 kJ and 7.56 kJ respectively).

Fig. 18.1 Force displacement of three point bend tests (left), with examples of symmetric failure (center) and asymmetric failure (right)

18 Full-Scale Testing and Numerical Modeling of Adhesively Bonded Hot Stamped Ultra-High Strength Steel Hat Sections

111

Fig. 18.2 Force displacement of axial crush tests (left), with examples of local folding mode (center) and global buckling mode (right)

Fig. 18.3 Comparison of experimental and numerical force-displacement plots (left) and predicted separation of flange (1.8 mm center, 1.2 mm right)

Fig. 18.4 Comparison of experimental and numerical force-displacement plots (left) and predicted global buckling deformation mode (center, right)

18.4 Numerical Results Numerical models of three-point bending for both the 1.8 mm preliminary tests and the 1.2 mm primary tests showed good correspondence to the experimental data in loading response and peak force (Fig. 18.3). Being the thicker and thus more dimensionally stable structure, the 1.8 mm model was able to predict the onset and propagation of the adhesive failure, similar to that observed in the experiment. The 1.2 mm model predicted a response in agreement with the experiment up to the peak force, but did not predict abrupt failure of the joint until a much higher displacement value (64 mm compared to 29 mm). The axial crush numerical model predicted a higher initial peak force (Fig. 18.4) and a stiffer response compared to the experimental data. This could be related to the need to include material thinning resulting from forming in the tube model.

112

Y. B. Liu et al.

Another cause could be boundary or contact conditions that did not accurately match that of the experiments. The model predicted a deformation mode (Fig. 18.4) similar to tests AX-2 and AX-3, where global buckling was prevalent over local folding.

18.5 Conclusions Current results demonstrated that hot stamped ultra-high strength steel hat sections can be successfully joined with structural adhesives into tubes with repeatable quasi-static loading response under three-point bending and axial crush loading. The average peak loads in three-point bending loading were 34.0 kN (standard deviation 1.6 kN) for the 1.2 mm thick material and 78.8 kN (standard deviation 0.9 kN) for the 1.8 mm thick material. The peak load for axial crushing was 260 kN (standard deviation 6.9 kN) for the 1.2 mm thick material. The consistency achieved in the experiments provided reliable data for assessing the finite element model. The finite element models, including a cohesive zone adhesive model derived from coupon level and bulk material testing, demonstrated good correlation with the experiments in predicted peak force, loading response, and deformation mode.

References 1. Del-Colle, A.: Obama Announces 54.5 mpg CAFE Standard by 2025. Retrieved from http://www.popularmechanics.com/cars/a7015/obamaannounces-54-6-mpg-cafe-standard-by-2025/ (2011, July 29) 2. Kumar, S.: Special issue on functionally graded adhesively bonded systems. Int. J. Adhes. Adhes. 76, 1–2 (2017) 3. Katakis, M.: Cadillac ATS, CTS Achieve High Refinement, Light Weight By Using Adhesives, Aluminum: Feature Spotlight. Retrieved from May 13, 2013 http://gmauthority.com/blog/2013/05/cadillac-ats-cts-achieve-high-refinement-low-weight-with-use-of-adhesives-aluminum/ (2013) 4. Galm Intelligence. The Right Material In The Right Place – Mercedes Lightweight Aluminum Body In White. Retrieved from https:// www.galmintelligence.com/p3081/the-right-material-in-the-right-place-mercedes-lightweight-aluminium-body-in-white/ 5. Trimino, L., Cronin, D.S.: Evaluation of numerical methods to model structural adhesive response and failure in tension and shear loading. J. Dyn. Behav. Mater. 2(1), 1–16 (2016). https://doi.org/10.1007/s40870-016-0045-7

Chapter 19

Mechanical Characterization of ZrO2 Rich Glass Ceramic Balamurugan M. Sundaram, Jamie T. Westbrook, Charlene M. Smith, and John P. Finkeldey

Abstract Glass-Ceramics (GCs) find wide applications in electronic packaging, kitchenwares, optics, acoustic systems, aerospace industry, as armor materials, and as aesthetic material for dental restoration due to their simple processing, easy machinability, low porosity and high strength.. However, they are inevitably subjected to tensile load resulting in catastrophic failure due to their sometimes low fracture toughness and high brittleness. In this context, a zirconia containing lithium disilicate glass ceramic is developed and mechanically characterized. Its fracture toughness and hardness are measured using Chevron Notch Short Bar (CNSB) method and Vickers indent respectively.. Further, the material was subjected to single edge notched bar (SENB) loading in 3-point bend configuration. The non-linearity in the load-deflection curve suggested the presence of R-curve behavior which was subsequently measured. The results of this technique are compared with those of Corning Ultra Low Expansion (ULE) glass, which was used as a standard for the measurement. In the glass-ceramic material a rising R-curve, a desirable attribute as it suppresses subcritical crack growth, was evident. With higher fracture toughness, rising R-curve and improved brittleness index, this GC has advantaged mechanical attributes. Further, the fracture surface exhibited significant roughness as compared to ULE glass. With multiple potential factors contributing to the improved fracture toughness, each of their contributions is yet to be fully understood. Keywords Fracture toughness · R-curve behavior · Glass ceramic · Zirconia · Material characterization

19.1 Introduction Since its discovery in 1953, glass ceramics have found wide applications as kitchenwares, optical components, electronic components and dental implants, among other applications. They are produced by controlled crystallization of certain glasses resulting in materials with residual glassy phase and one or more embedded crystalline phases. By controlling the ceramization, one can yield an array of materials with various combinations of properties. The main advantages of Glassceramics are that a) they can be produced by simple glass forming technique, b) their nano/micro structure can be altered based on its application c) they have little to no porosity, and d) they have the possibility of combining desired properties such as low thermal expansion and high chemical durability [1]. In spite of these favorable characteristics, their high brittleness may lead to catastrophic failures, especially under tensile stress. Further, any pre-existing cracks/flaw may also get activated by these tensile loads to the point of catastrophic failure. The stress intensity factor (SIF) at this point of failure is often referred to as fracture toughness (KIc ), a desirable property that can be enhanced in GC over glass. Enstatite [2] and mica-containing glass-ceramics [3], for example, have KIc of √ ∼3 MPa m. Fracture toughness, considered an intrinsic material property, describes a material’s ability to resist unstable crack growth. Hence KIc could be critical material parameters to compare various GCs. There are numerous methods to evaluate KIC that includes Chevron notch short bar (CSNB), single edge notched bar (SENB), double torsion, and indentation fracture. Some materials show an increase in SIF/fracture resistance with crack extension under stable crack growth. This phenomenon is called R-curve behavior. This behavior can be attributed to several factors including bridging effects, the frictional or elastic interaction between crack faces. It is this effect that influence R-curve behavior the most. Other factors include energy-consuming effects like development of micro-cracks, crack deflection and crack branching [4]. Another reason for R-curve behavior is phase transformation, which is a characteristic of zirconia ceramics. R-curve behavior of

B. M. Sundaram () · J. T. Westbrook · C. M. Smith · J. P. Finkeldey Corning Research and Development Corporation, Corning, NY, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_19

113

114

B. M. Sundaram et al. 250

S1 S2

Load (N)

200 150 100 50 0

0

0.02

0.04

0.06 0.08 0.1 0.12 Deflection (mm)

0.14

0.16

Fig. 19.1 Load deflection plot from CNSB test for two samples of GC

ceramic materials is a desirable mechanical effect as it means that an additional energy is necessary for a crack to keep propagating until failure. Besides the suppression of subcritical crack growth, R-curve behavior can reduce the scatter in measured material strength [5]. In this study, a newly synthesized lithium disilicate glass ceramic reinforced with Zirconia is mechanically characterized. The glass ceramic was prepared in the usual way, melting the appropriate precursor glass and subsequently subjecting it to a prescribed heat treatment. The heat treatment yields both the desired phase assemblage (lithium disilicate plus tetragonal ZrO2 ) and microstructure. The fracture toughness is measured using CNSB samples. The material is also characterized for its Vickers hardness. Further, a 3-point-bend based loading setup in conjunction with laser based noncontact extensometer is used to evaluate R-curve behavior in them, if any. A stiffness-based method is used to monitor the crack growth. This measurement methodology is followed on ULE glass as well as it is known not to exhibit an R-curve behavior. The roughness of the fracture surface was also measured using non-contact optical profiler. A digital image correlation (DIC) based method in conjunction with high-speed camera to evaluate R-curve is currently being explored to improve the temporal resolution. Further, the material also needs to be characterized for its dynamic properties.

19.2 Fracture Toughness Mode-I fracture toughness (KIc ) was evaluated using a CNSB based tensile setup. The specimen geometry and the loading setup were in accordance with ASTM E1304. Fifteen bar-shaped specimens were machined from bulk material using diamond impregnated saw which was later polished to eliminate machining flaws. The samples were subjected to tensile loading using universal testing machine (UTM) at 0.05 mm/min until failure. The characteristic load deflection plot for GC is shown in Fig. 19.1. Using the peak load, the mode-I fracture toughness was evaluated as [6], KI c =

Fmax Ym √ , B W

where, F is the peak load, B is the specimen thickness, W is the specimen width, and Ym is the dimensionless stress intensity factor coefficient based on specimen geometry.

19.3 Vickers Hardness Vickers hardness of the materials was measured using digital micro-hardness tester in accordance with ASTM C1327. The Vickers hardness of the material was evaluated as [7], H = 0.018186

p d2

Where, H is the hardness (GPa), p is the applied load (kg), and d is the mean indentation’s diagonal length (mm).

19 Mechanical Characterization of ZrO2 Rich Glass Ceramic

115

19.4 Brittleness Index The brittleness index (B) was evaluated as [7], B=

H KI c

where H is the hardness and KIc is the mode-I fracture toughness.

19.5 R-Curve A 3-point-bending test setup using single edge notched beam (SENB) samples was used to study R-curve behavior. UTM under displacement control was used to load the specimen at 0.05 mm/s rate. A non-contact laser extensometer was used to measure the flexural displacement. The advantage of this detection system is that it provided high sampling rate as compared to the machine crosshead. The schematic of the test setup is shown in Fig. 19.2. The specimen is a rectangular rod of size 90 mm × 8 mm × 6 mm. However, prior to testing the specimens, it was important to understand the effect of test setup on the measured stiffness. Hence, the system stiffness was determined by loading a rectangular steel bar of size 90 mm × 25 mm × 10 mm up to the estimated failure load of these specimens to obtain a load deflection (system stiffness) plot. This system stiffness was removed from the test data of the materials being tested while plotting their stiffness curves. The load deflection plot at various notch lengths for the GC is shown in Fig. 19.3. Using these stiffness curves, a relationship between stiffness and notch length can be established. With notch length being an alternate to crack length, an ability to monitor crack length through stiffness is developed. Consequently, a SENB specimen was loaded until failure to observe any non-linearity in load deflection curve. Load deflection curves for ULE glass and GC are shown in Fig. 19.4.

Load

8 mm

Specimen Initial notch-tip 6 mm

2 mm

Laser extensometer

80 mm

Fig. 19.2 3-point-bending based experiment setup for R-curve measurement 30

0 mm 2 mm 2.25 mm 2.5 mm

Load (N)

25 20 15 10 5 0 0

0.002

0.004 0.006 0.008 Deflection (mm)

Fig. 19.3 Stiffness curves at 0 mm, 2 mm, 2.25 mm, and 2.5 mm notch length for GC

0.01

0.012

116

B. M. Sundaram et al.

180 160

Glass-ceramic

140

ULE

Load (N)

120 100 80 60 40 20 0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Deflection (mm)

Load (N)

Fig. 19.4 Load deflection curve obtained from 3-point-bend test

Slope 1

Slope 2

0

0

Deflection (mm)

Fig. 19.5 Schematic describing the evaluation of stiffness slopes along the non-linearity using secant method

GC showed non-linearity suggesting the existence of R-curve behavior whereas ULE glass did not show any non-linearity as expected. A non-linear region in load displacement curve for an otherwise stiff material is attributed to crack propagation. With a stable crack initiated in GC, a change in stiffness is monitored. Using secant method at various points of non-linear region it is possible to obtain stiffness slopes as depicted in Fig. 19.5. As a relationship between stiffness and crack length has been developed, the crack length can be evaluated throughout the non-linear region. The corresponding instantaneous SIF is evaluated as [8],  !" F S 3(ξ )1/2 1.99 − ξ (1 − ξ ) 2.15 − 3.93 (ξ ) + 2.7(ξ )2 a KI = ,ξ= w Bw 3/2 2 (1 + 2ξ ) (1 − ξ )3/2 where, F is the applied load, S is the distance between the supports, B is the specimen thickness, w is the width of the specimen and a is the instantaneous crack length. The SIFs were plot against the crack length to obtain the R-curve.

19.6 Results √ The mean fracture toughness using CNSB tests for ULE glass and GC were measured to be 0.73 ± 0.05 MPa m and √ 4.35 ± 0.1 MPa m, respectively. The KIc of the GC is higher than other lithium disilicate or ZrO2 reinforced GCs (2.0– √ 3.0 MPa m) reported in literature [7, 9]. The Vickers hardness measurement for ULE glass and GC are 4.33 GPa and 5.95 GPa, respectively. Their brittleness indexes were evaluated as 5.93 μm-0.5 and 1.36 μm-0.5 respectively. The material is

19 Mechanical Characterization of ZrO2 Rich Glass Ceramic

117

4.8

KIc (MPa m0.5)

4.7

Slope = 1.16 GPa m-0.5

4.6 4.5 4.4 4.3 4.2 4.1 0

0.1

0.2 0.3 Crack length (mm)

0.4

0.5

Fig. 19.6 Stress intensity factor (KIc ) vs crack length plot showing R-Curve behavior of GC

Fig. 19.7 Fracture surface roughness measurements of ULE glass and (b) GC measured using optical profiler

considered machinable if the brittleness index is below 4.3 μm-0.5 [10] and hence the newly developed GC can be considered machineable. The R-curve measured for GC is shown in Fig. 19.6. It can be seen that the GC exhibits a rising R-curve, which is noteworthy. The R-curve follows a more linear path that usual power law within the region (crack length) of measurement. The slope of the curve was 1.16 GPam-0.5 . The crack needs to be monitored over a longer region to obtain the tail end of the R-curve behavior. The SIF at zero crack length, which is essentially the KIc , is comparable to that measured from CNSB. The toughening mechanism that results in rising R-curve behavior in this GC is yet to be fully understood. Transformation toughening of the Zirconia phase from tetragonal to monoclinic structure could be one of the mechanisms for the observed R-curve behavior similar to that seen in ZrO2 ceramics.

19.7 Surface Roughness Measurement Surface roughness of the fracture zone for GC along with ULE glass were measured using non-contact optical surface profiler and are shown in Fig. 19.7. The mean surface roughness for ULE glass and GC are 0.086 ± 0.02 μm and 12.13 ± 1.6 μm, respectively. It can be seen that the GC has significantly higher surface roughness than ULE glass which is consistent with higher KIc . It is suspected that the large microstructure of the lithium disilicate phase is resulting in the torturous path.

118

B. M. Sundaram et al.

19.8 Work-in-Progress Improving the temporal resolution for R-curve measurement by the use of Digital Image Correlation (DIC) method and high speed photography is currently underway. Further, characterization of dynamic properties of this GC is being carried out. As the surface roughness correlated with the increased fracture toughness, the effect of microstructure needs to be decoupled from that of the reinforcement from addition of Zirconia.

19.9 Summary Lithium disilicate Glass-ceramic (GC) reinforced with Zirconia was successfully synthesized. Mechanical characterization of GC was carried out and the results compared with that of ULE glass which acted as a standard. Hardness of 5.95 GPa and √ a fracture toughness of 4.35 ± 0.1 MPa m were observed for GC. The brittleness index of the GC is within the threshold (107 s−1 ) on 5 μm thick soda-lime glass samples. Normal and transverse particle velocities were measured using a newly developed diffracted beam photonic Doppler velocimeter (DPDV) and a complimentary PDV arrangement, which provided an independent measurement of the normal particle velocity. Normal and shear stress profiles were inferred using an experimentally calibrated strength model of the employed pure WC anvil plates [8], which sandwiched the thin glass

30 Pressure-Shear Plate Impact Experiments on Soda-Lime Glass at Pressures Beyond 20 GPa

165

Fig. 30.2 (a) Normal and transverse free surface velocity profiles measured by the DPDV diagnostic for an experiment with v0 = 736 m/s. (b) Inferred normal and shear stresses using a strength model for pure WC anvil plates [8]

specimen. The shear stress observed in the PSPI experiments quickly reach a maximum value above 1.2 GPa after which a sudden loss of shear strength occurs. Comparisons to previous experiments conducted at lower pressures by Jiao et al. [10] showed a similar shear strain level (1.5–2) at which this maximum shear stress is achieved. Acknowledgments The authors are grateful for support from the Office of Naval Research (Award No. N00014-16-1-2839) for the development of the PSPI capability at high pressures and the Air Force Office of Scientific Research (Award No. FA9550-12-1-0091) for development of the PDV-DPDV interferometer system.

References 1. Grunschel, S.E.: Pressure-shear plate impact experiments on high-purity aluminum at temperatures approaching melt, Ph.D. thesis, Brown University (2009) 2. Frutschy, K.J., Clifton, R.J.: High-temperature pressure-shear plate impact experiments on OFHC copper. J. Mech. Phys. Solids. 46(10), 1723–1744 (1998) 3. Spitzig, W.A., Richmond, O.: The effect of pressure on the flow stress of metals. Acta Metall. 32, 457–463 (1984) 4. Gleason, A.E., Bolme, C.A., Lee, H., Nagler, B., Galtier, E., Milathianaki, D., Hawreliak, J., Kraus, R.G., Eggert, J., Fratanduono, D., Collins, G.W., Sandberg, R., Yang, W., Mao, W.L.: Ultrafast visualization of crystallization and grain growth in shock-compressed SiO2 . Nat. Commun. 6, 8191 (2015) 5. Mello, M., Kettenbeil, C., Bischann, M., Ravichandran, G.: In: Chau, R., Germann, T., Lane, M. (eds.) Shock Compression of Condensed Matter – 2017. American Institute of Physics, Melville (2018) 6. Dolan, D.H.: Accuracy and precision in photonic Doppler velocimetry. Rev. Sci. Instrum. 81(5), 053905 (2010) 7. Kim, K., Clifton, R.J., Kumar, P.: A combined normal-and transverse-displacement interferometer with an application to impact of y-cut quartz. J. Appl. Phys. 48(10), 4132–4139 (1977) 8. Clifton, R.J., Jiao, T.: Pressure and strain-rate sensitivity of an elastomer: (1) pressure-shear plate impact experiments; (2) constitutive modeling, in elastomeric polymers with high rate sensitivity. In: Barsoum, R.G. (ed.) Elastomeric Polymers with High Rate Sensitivity, pp. 17–64. Oxford, Elsevier (2015) 9. Dandekar, D.P., Grady, D.E.: Shock equation of state and dynamic strength of tungsten carbide. AIP. Conf. Proc. 620(1), 783–786 (2002) 10. Jiao, T., Kettenbeil, C., Ravichandran, G., Clifton, R.J.: In: Chau, R., Germann, T., Lane, M. (eds.) Shock Compression of Condensed Matter – 2017. American Institute of Physics, Melville (2018)

Chapter 31

Dynamic Mechanical Response of T800/F3900 Composite Under Tensile and Compressive Loading Yogesh Deshpande, Peiyu Yang, Jeremy Seidt, and Amos Gilat

Abstract The effect of strain rate on the mechanical response of T800/F3900, a strengthened epoxy carbon-fiber reinforced polymer, is studied by conducting compression and tensile tests at different strain rates. Low strain rate tests (0.001 s−1 and 1 s−1 ) are done using a hydraulic frame and high strain rate tests (300 s−1– 600 s−1 ) are done with the SHB technique. Digital Image Correlation is used in all tests to obtain full-field strain measurements. Tension tests have been done on unidirectional laminates in the 90 ◦ direction. Compression tests have been done on unidirectional laminates in the 90 ◦ and through the thickness directions. No or small strain rate effect is observed between the low strain rate tests. The results from the high strain rate tests show significant strain rate effects. Keywords Composite material · Material testing · Strain rate effect · Digital image correlation · Split Hopkinson bar

31.1 Introduction Composite materials are made from two or more constituent materials with different material properties. The material used, T800/F3900 is a strengthened epoxy carbon-fiber reinforced polymer. Tension and compression tests were conducted to study the material response under quasi-static and dynamic loading. Split Hopkinson pressure bars are used for the dynamic tests and MTS hydraulic load frame is used for compression and tension loading at strain rates of 10−3 and 1 s−1 . DIC is used to obtain full-field measurement of strain during the tests at all strain rates. The testing is done in support of the development of a new material model (MAT_213) in LS-DYNA (Goldberg et al. 2014) [1].

31.2 Experimental Setup The specimens are machined from a 0.125 in. thick T800/F3900 unidirectional plate. The specimen used for the tensile tests at various strain rates has dog-bone geometry with fibers in the 90◦ direction. Figure 31.1 shows the dimensions of the specimen and the adapters used for connecting the specimen to the testing machines. The gage section is 0.125 in. wide and 0.125 in. thick. Its length is 0.38 in. For the high rate tests, the specimen is glued to aluminum adapters that are then glued to the incident and transmitted bars. In the quasi-static tests, the adapters are pinned into double universal joints that are then attached to the load frame using wedge grips. The universal joints help prevent bending and correct alignment issues with the specimen during testing. The compression specimen is a short rectangular prism, shown in Fig. 31.2. The specimen is machined in 90◦ and through-thickness fiber directions. To prevent transverse motion during the test, titanium platens with a shallow 0.02 in deep rectangular notch, Fig. 31.2, are placed at the specimen ends. The SHB setup for tension and compression tests is shown schematically in Fig. 31.3. Figure 31.4 shows photographs of the setups.

Y. Deshpande () · P. Yang · J. Seidt · A. Gilat Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_31

167

Y. Deshpande et al.

R0 .50 0

168

0.125

0.50

0.422

0.38

Fig. 31.1 Tensile specimen geometry (left), specimen with adapters (middle and right)

0.13

Unit: in

0.13

0.25

Fig. 31.2 Compression specimen geometry (left), Titanium platen (right) Pulley

Clamp

Gage B

Incident Bar

ei

Transmitter Bar

er

et

Gage A

Striker Bar

Specimen

Gas Gun

n

Gage C

Gage A

Specimen

ei

et

er Incident Bar

Transmitter Bar

Gage B

P

u1

u2

u1

u2

Fig. 31.3 Schematic drawing of the tension (left) and the compression (right) SHB apparatus

Fig. 31.4 Tension (left) and compression (right) SHB apparatus

31.3 Results 31.3.1 Tension Tests Stress-strain curves from 90 ◦ tensile tests at high and low strain rates are shown in Fig. 31.5. The strain in the figures is obtained by averaging the strain on the surface of the gage section. Images from the DIC measurements of strain from the low rate and high rate tensile tests are shown in Fig. 31.6a, b respectively. The image on the left shows the axial strain at one instant during the loading. As shown, the gage section has uniform strain distribution. The image on the right shows deformation at the instant of fracture. At high rates, the specimens fail at three separate locations. Comparison of stress strain curves shows almost no strain rate effect between the 1 s−1 and 10−3 s−1 tests. There is significant strain rate effect between 300 s−1 tests and the lower rate tests. At low rate, the curve is nearly linear, showing a constant modulus of elasticity. At high

31 Dynamic Mechanical Response of T800/F3900 Composite Under Tensile and Compressive Loading

169

16

Stress(ksi)

12

8 90 deg tension strain rate effect 1 per s_3 1 per s_2 1E-3 per s_3 1E-3 per s_2 1E-3 per s_1 300 per s_1 300 per s_2 300 per s_3

4

0 0

0.004

0.008

0.012

0.016

Strain Fig. 31.5 90 ◦ tension stress strain curves

rate, the curve has two distinct parts that are nearly straight lines. The first part shows a significantly higher modulus than the low rate modulus. The curve then transitions into a nearly parallel curve to the low strain rate curve. The maximum strain seen in high rate tests is about half the value in low rate tests, whereas maximum stress at all rates is about the same.

31.3.2 Compression Tests Stress-strain curves from 90 ◦ and through-thickness compression tests at different rates are shown in Fig. 31.7. The strain is obtained by averaging the strain on the surface of the specimen. Images from the DIC measurements of strain from the low rate and high rate compression tests are shown in Fig. 31.8a, b respectively. The image on the left shows the axial strain at one instant during the loading. It shows that the axial strain is uniform on the surface. The image on the right shows deformation at the instant of fracture. Comparison of stress-strain curves at different strain rates show almost no strain rate effect at low rates for both the 90 ◦ and through-thickness compression specimens. There is noticeable strain rate effect between the 600 s−1 tests and the low rate tests for the 90 ◦ specimens. The maximum stress seen in high rate tests is higher than in lower rate tests, whereas the maximum strain is about the same at all strain rates.

170

Y. Deshpande et al.

Fig. 31.6 DIC Images from 90 ◦ tensile testing (a) low rate test (b) high rate test 40 50

30

Stress(ksi)

Stress(ksi)

40

30

20

90 deg Compression rate effect

20

0.001 per s_3 0.001 per s_2

through thickness Compression rate effect

0.001 per s_1

1 per s_1 1 per s_2 1 per s_3 600 per s_1 600 per s_2

10

0.001 per s_3 0.001 per s_2 0.001 per s_1 1 per s_1 1 per s_2 1 per s_3

10

0

0 0

0.01

0.02

0.03

0.04

0.05

0

0.01

Strain

Fig. 31.7 90 ◦ compression (left) and through-thickness compression stress-strain curves

0.02

0.03 Strain

0.04

0.05

31 Dynamic Mechanical Response of T800/F3900 Composite Under Tensile and Compressive Loading

171

Fig. 31.8 DIC images from compression testing (a) low strain rate test (b) high strain rate test

31.4 Conclusions Tensile and compressive high strain rate tests have been conducted using split Hopkinson technique. Full field measurement of deformations in the specimen has been done using the Digital Image Correlation method. Low strain rates were conducted using a hydraulic load frame. Specimens machined from a unidirectional T800/F3900 composite plate were tested. No or small strain rate effect is observed between the low strain rate tests. The results from the high strain rate tests show significant strain rate effects. Acknowledgements The research presented was supported by the U.S.A Federal Aviation Administration, Grant No. 16-G007. The authors are grateful to Mr. Bill Emmerling, Dr. Chip Queitzch, and Mr. Danial Cordasco of the FAA.

References 1. Goldberg, R.K., Carney, K.S., Dubois, P., Hoffarth, C., Harrington, J., Rajan, S., Blankenhorn, G.: Development of an orthotropic elasto-plastic generalized composite material model suitable for impact problems. J. Aerosp. Eng. J. Aerosp. Eng. 30, 04015083 (2015). Web

Chapter 32

Experimental Investigation of Rate Sensitive Mechanical Response of Pure Polyurea K. Srinivas, C. Lakshmana Rao, and Venkitanarayanan Parameswaran

Abstract The application of layered composite structural systems to protect the main structure during vehicle collisions, blast loading and other high impact applications has been a major focus of interest in the automobile industry over the last few decades. Enhancing the energy absorption, improving blast resistance and improving the dynamic fracture resistance of metallic plate structures has been the main challenges in these studies. Polyurea is a good candidate which has excellent mechanical as well as chemical properties. In the present work, a detailed study has been made on the rate sensitive response of pure form of polyurea. The experiments were carried out by Split Hopkinson Pressure Bar(SHPB) over a wide range of strain rates(1000–4500 s−1 ) to measure the stress-strain response of the material. A brief analysis has been made on the stiffness calculation & to see how stress is varying with respect to strain rate at different strains. Keywords Pure Polyurea · High-strain rate loading · Rate sensitivity · SHPB · Pulse shaper · Impact resistance · Etc

32.1 Introduction Polyurea coatings have been found to enhance the blast protection of concrete wall structures. For these concrete walls, polyurea coating proved to be effective in mitigation of blast than any other polymers of 21 types which were actually tested by U.S Air Force Research Laboratory(AFRL) people [1]. Also, coating the metal plates with a polyurea layer has proven to significantly affect the energy absorption & reducing permanent deformation of such structures subjected to high-intensity impulsive loads [2, 3]. Polyurea is a coating material that has received extensive research interest in the recent past in the applications of armor fighting vehicles, automobile industries and many blasts & ballistic applications due to its attracting properties at high strain rates [4, 5]. Polyurea, which is cross-linked amorphous isocyanate monomer or prepolymer and polyamine curative, is a special type of polymer. It features a fast setting time, lightweight, abrasion resistance, fracture resistance, corrosion protection & impact resistance. For last few decades, it is one of the candidate coatings that is explored in armed vehicles and impact applications like crashworthiness and blast [9–11]. Due to so many attracted mechanical properties particularly at high strain rates, many authors showed great interest to find its mechanical properties subjected to high strain rates [14–18]. The main objectives in the present work are, • A detailed analysis of studying the rate sensitive behavior of Pure form of Polyurea which has gone under many laboratory level tests to check purity by considering all the requirements of SHPB. • Investigating the effect of pulse shaper on the stress-strain response of polyurea and how it will affect other parameters like strain rate, minimizing dispersion, etc. in SHPB tests.

K. Srinivas () · C. Lakshmana Rao Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India e-mail: [email protected] V. Parameswaran Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_32

173

174

K. Srinivas et al.

Fig. 32.1 Schematic representation of split Hopkinson pressure bar(SHPB) Table 32.1 Complete specifications of SHPB setup Bar Properties S.No Part 1 Striker bar 2 Incident bar 3 Transmitter bar Strain gauge properties S.No Part 1 Incident bar 2 Transmitter bar

Dia.(mm) 19.5 19.5 19.5 Circuit type Quarter bridge type Half bridge type

Length(mm) 400 2000 1800

E(MPa) 200,000 200,000 200,000

ρ(kg/m3 ) 8100 8100 8100

ν 0.33 0.33 0.33

Excitation voltage 4 V excitation 6 V excitation

32.1.1 General Description of Split Hopkinson Pressure Bar(SHPB) and Working The experiments are carried out by a special experimental technique called Split-Hopkinson Pressure Bar(SHPB). It is widely used for determining the mechanical properties of various materials at different strain rates ranging from 102 s−1 to 104 s−1 [6–13]. A general SHPB apparatus consists of three major components: a loading device, bar components, and a data acquisition and recording system, as schematically shown in the Fig. 32.1. Some authors are making necessary modifications to SHPB for achieving more accurate data in terms of constant strain rate, minimizing dispersion and wave attenuation effects [7, 8]. In the present study, a copper pulse shaper has been used to get all the results. Here the bars of the set up are chosen maraging steel type, which keeps enough hardness without undergoing any great deformations even at high strain rates. Complete specifications of the bars as well as strain gauges used for the SHPB development as shown in the below in Table 32.1.

32.1.2 Polyurea Sample Preparation Polyurea, as we have stated already, is a cross linked amaorphous polymer. The grade of polyurea used in the present study is a pure form of polyurea. The polyurea was made from the rapid chemical reaction between polyamine and diisocyante took like 4:1 by weight. The mixture has been sprayed on the appropriately prepared surface with the help of a special highpressure liquid spraying machine which mixes the amine and cyanate part stoichiometrically. After the spraying process, the large samples were obtained and those are machined according to Table 32.2 by Abrasive Water Jet machine to avoid thermal distortions. The dimensions of the specimen have chosen according to the equation given in section 2.1 and the Youngs modulus is calculated by simple Universal Testing Machine.

32 Experimental Investigation of Rate Sensitive Mechanical Response of Pure Polyurea

175

Table 32.2 Polyurea specimen specifications S.No 1

Part Cylindrical specimens

Dia.(mm) 12.5

Length(mm) 5.5

E(MPa) 85

ρ(kg/m3 ) 1111

Fig. 32.2 Strain gauge signals for SHPB calibration

32.1.3 Calibration of SHPB To check the accuracy of the setup and to validate the one-dimensional wave theory conditions, it is required to calibrate SHPB setup before doing an experiment. Both incident and transmitted bars are pressed together without placing specimen. The experiment has to be done in this condition and voltage signals were recorded on both incident and transmitted bars which were shown in the Fig. 32.2.

32.2 Requirements for Valid Split Hopkinson Pressure Bar(SHPB) Experiments There are certain requirements which should be satisfied by SHPB to get an exact constitutive response of the material to be tested [18]. All these requirements are satisfied by placing a copper pulse shaper of 0.6 mm thickness and choosing smooth surface bars. Some of the requirements include, 1. Equilibrated Stresses 2. Dispersion Effects 3. Inertia Effects and other requirements include friction, inertia,etc.

32.2.1 Effect of Pulse Shaper As described in the previous sections, pulse shaper is a necessary element for keeping the equilibrium of stresses, minimizing dispersion effects. The Fig. 32.3 shows the effect of pulse shaper on the reflected pulse, which in turn gives strain rate. Keeping strain rate constant is one of the main challenges in SHPB tests. As the intrinsic behavior of the material to be calculated, the stress on the specimen to be free from inertial effects. The specimen stress is calculated by taking average of stresses at both ends along with additional inertia terms [7, 8].

176

K. Srinivas et al.

Fig. 32.3 Reflected pulse with & without pulse shaper

Fig. 32.4 Stress strain plot by considering pulse shaper effect

(σ1 + σ2 ) σ= +ρ 2



   r2 r2 t2 t2 ε¨ + ρ ε˙ 2 + − 8t 12 16 12

here, ρ is the density of specimen and r,t are the radius and thickness of the specimen. So, the average stress is obtained by maintaining constant strain rate and proper specimen dimensions as given in Table 32.1. In Fig. 32.4 stress strain response plot, it is observed that the loading is sudden in case of no pulse shaper and it is gradual with copper pulse shaper. It is because the pulse shaper itself will absorb some energy while loading. Also, there is a slight decrease in the final stress value because of pulse shaper.

32.3 Results and Discussion In the present work, a detailed study has been made on the rate sensitive response of polyurea. The SHPB experimental data for Polyurea specimens obtained at different strain rates in the form of voltage vs time as shown in the Fig. 32.5a. This is recorded by the data acquisition system in the form of incident, reflected & transmitted wave voltage. By using proper conversion factors, this data has converted to corresponding stress, strain & strain rate values. Stress-strain response at different strain rates was calculated by substituting the above strain pulse values in analytical equations. To get good

32 Experimental Investigation of Rate Sensitive Mechanical Response of Pure Polyurea

177

Fig. 32.5 (a) Voltage vs time data & (b) Engg. stress vs true Engg. strain at different strain rates

Fig. 32.6 Stress as a function of strain rate

repeatability, three specimens were tested at each strain rate. The combined stress-strain plot is obtained by taking the average of all three repetitions. The results of Engineering Stress vs Engineering Stress at different strain rates as shown in the Fig. 32.5b. From the plots, it can be seen that initial modulus is high. After some initial deformation, the material started apparently yielding up to certain strain. In this region material experiences relatively large elongation with a small increase in the stress and beyond this yielding, material hardens which implies stress increases rapidly with strain. Plots in the Fig. 32.6 show that stress as a function of strain rate at different strain: 0.1, 0.15 and 0.25. As the strain rate increases, stress also increases, which means the effect of strain rate is more pronounced at high strain rates. Also, it can be seen that the curves are of diverging type, which means the increase in the stress is higher at large strains. A material should be rate sensitive to withstand higher loads under impact loading condition. Many materials show rate sensitive behavior at high strain rates. It is calculated by taking some quasistatic stress as a reference of same nature and taking the ratio of the difference to the quasistatic stress. The Fig. 32.7 shows the how rate sensitivity parameter(R) varies over the strain rate. Quasistatic stress is taken as 1.869 MPa at 0.001 s−1 . Rate Sensitive Parameter (R) =

σ − σstatic σstatic .

178

K. Srinivas et al.

Fig. 32.7 Variation of rate sensitivity with strain rate for two different strains

Fig. 32.8 Stiffness variation with strain rate

It shows that Polyurea material is highly rated sensitive with respect to strain rate. The sensitivity is more at high strains than at low strains. So, the strength of the material increases with loading which is one of the desired parameters for impact loading applications. The Fig. 32.8 shows the variation of stiffness with respect to strain rate. It is calculated by taking tangent modulus at a point on stress-strain plot for each strain rate. As stiffness is the property of the structure, i.e., intrinsic property, it depends on the dimensions of the specimen. The material showing much stiffer at high strain rates.

32.4 Conclusions The detailed experiments were conducted to investigate the mechanical properties of the pure form of polyurea at different strain rates ranging from 0.001/s to 4500/s in this paper, which is actually the domain of many crashworthiness cases. Before doing the experiments, all the conditions which are required for SHPB tests were carried out, so that the results obtained are free from many errors and assumptions. The material shows good rate sensitive behavior at a wide variety of strain rates, which means stress-strain properties of Polyurea are strongly dependent on strain rates. It has shown that pulse shaper has

32 Experimental Investigation of Rate Sensitive Mechanical Response of Pure Polyurea

179

a significant effect on the output results. By using copper pulse shaper, the strain rate is almost uniform throughout the experiment which is a major requirement for stress equilibrium. Acknowledgements All the experimental work was done at High-Speed Experimental Mechanics(HSEM) Laboratory, IIT Kanpur. The authors thank Mr.Manoj Kumar for experimental assistance and Mr.Ramesh Chandra of Fosroc Constructive Solutions for supplying pure form of Polyurea required for this work.

References 1. Ackland, K., Anderson, C., Ngo, T.D.: Deformation of polyurea-coated steel plates under localised blast loading. Int. J. Impact Eng. 51, 13–22 (2013) 2. Xue, L., Willis Jr., M., Belytschko, T.: Penetration of DH-36 steel plates with and without polyurea coating. Mech. Mater. 42(11), 981–1003 (2010) 3. Rijensky, O., Rittel, D.: Polyurea coated aluminum plates under hydrodynamic loading: does side matter? Int. J. Impact Eng. 98, 1–12 (2016) 4. Youssef, G.H.: Dynamic Properties of Polyurea. University of California, Los Angeles (2011) 5. Shim, J.: Finite Strain Behavior of Polyurea for a Wide Range of Strain Rates. Massachusetts Inst of Tech Cambridge Dept of Civil And Environmental Engineering, (2010) 6. Rao, C., Lakshmana, V., Narayanamurthy, K., Simha, R.Y.: Applied Impact Mechanics. Wiley, Chichester (2016) 7. Jiang, T.Z., Xue, P., Butt, H.S.U.: Pulse shaper design for dynamic testing of viscoelastic materials using polymeric SHPB. Int. J. Impact Eng. 79, 45–52 (2015) 8. Song, B., Chen, W.: Loading and unloading split Hopkinson pressure bar pulse-shaping techniques for dynamic hysteretic loops. Exp. Mech. 44(6), 622–627 (2004) 9. Amini, M.R., Isaacs, J.B., Nemat-Nasser, S.: Experimental investigation of response of monolithic and bilayer plates to impulsive loads. Int. J. Impact Eng. 37(1), 82–89 (2010) 10. Mohotti, D., et al.: Plastic deformation of polyurea coated composite aluminium plates subjected to low velocity impact. Mater. Des. 56(2014), 696–713 (1980-2015) 11. Gardner, N., et al.: Blast mitigation in a sandwich composite using graded core and polyurea interlayer. Exp. Mech. 52(2), 119–133 (2012) 12. Song, B., et al.: A long split Hopkinson pressure bar (LSHPB) for intermediate-rate characterization of soft materials. Exp. Mech. 48(6), 809–815 (2008) 13. Meyers, M.A.: Dynamic Behavior of Materials. Wiley, New York (1994) 14. Albrecht, A.B., Liechti, K.M., Ravi-Chandar, K.: Characterization of the transient response of polyurea. Exp. Mech. 53(1), 113–122 (2013) 15. Johnson, T.P.M., Sarva, S.S., Socrate, S.: Comparison of low impedance split-Hopkinson pressure bar techniques in the characterization of polyurea. Exp. Mech. 50.7, 931–940 (2010) 16. Sarva, S.S., et al.: Stress–strain behavior of a polyurea and a polyurethane from low to high strain rates. Polymer. 48(8), 2208–2213 (2007) 17. Shim, J., Mohr, D.: Using split Hopkinson pressure bars to perform large strain compression tests on polyurea at low, intermediate and high strain rates. Int. J. Impact Eng. 36(9), 1116–1127 (2009) 18. Chen, W.W., Song, B.: Split Hopkinson (Kolsky) Bar: Design, Testing and Applications. Springer Science & Business Media, Boston (2010)

Chapter 33

Experimental Study on Dynamic Fracture Response of Al6063-T6 Under High Rates of Loading Anoop Kumar Pandouria, Purnashis Chakraborty, Sanjay Kumar, and Vikrant Tiwari

Abstract The focus of the current article is to investigate the dynamic fracture toughness of aluminum alloy Al6063T6 under three-point transient loading conditions. Prior to the dynamic evaluation, three-point bend experiment were also conducted under the quasi-static conditions to evaluate the static fracture toughness using standard formulations available in the literature. Modified Hopkinson pressure bar and ultra-high speed 3D-Digital image correlation (DIC) procedure was utilized to identify the crack initiation time and the crack mouth opening displacement (CMOD) of the specimen under transient loading conditions. To develop the better understanding of the failure mechanism of the specimen, fracture toughness at different strain rates were evaluated. A good agreement between the both Strain gauge measurements and DIC results was observed. Keywords CMOD · DIC · Dynamic fracture toughness · MHPB · High strain rate

33.1 Introduction Fracture is a failure mechanism of materials that have a great importance to the performance of the structures. Failure from fracture can occur due to many reasons like uncertainties in the loading conditions, defects in the material, and inadequacies in design. As failure of load-bearing components can be catastrophic, fracture is a very important consideration for engineers in designing structures such as automobiles, airplanes, power plants, bridges etc. Thus the need to understanding the dynamic behavior of ductile metals at high strain rates of loading is of critical importance. Also it has been proved by early researchers that, when subjected to dynamic loading the behavior of metals is quite distinct from that observed under quasi-static conditions; generally, the flow stress has higher rate dependency. At high strain rate loading, a tiny fluctuation in the plastic flow field induces important acceleration to material particles. Thus, significant inertia effects are taking place at the macroscopic level and sometimes also at the level of microscopic deformation mechanisms. One of the common technique ustilized in understanding the fracture behaviors is three-point bend (TPB) evaluations, it is extensively used for evaluation of static (K1C) , and dynamic (K1d ) fracture toughness. Among other parameters stress intensity factor (SIF), K1 (t), and time, tf, (time required to initiate the crack) are most important in determining of fracture toughness. Several researchers [1–5] have used different technique to determine dynamic SIF but strain-gauge based measurement technique [1] using Modified Hopkinson Pressure Bar to perform three-point bend test under high strain rate is the most appreciated technique among them. Rubio et al. [2] used static SIF formula to determine the dynamic SIF from crack mouth opening displacement (CMOD).They evaluated dynamic SIF from load point displacement by modifying the earlier research efforts by Bacon et al. [1–8]. The same formula has been used here to evaluate the static SIF from CMOD. In this study experiments were conducted on three-point bend specimens of Al6063-T6 by using a modified Hopkinson pressure bar (MHPB). Strain gauge measurement technique and digital image correlation [7] were also used to evaluate static SIF, load point displacement and crack initiation time.

A. K. Pandouria () · P. Chakraborty · S. Kumar · V. Tiwari Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_33

181

182

A. K. Pandouria et al.

33.2 Experimental Procedure The TPB evaluates were performed using modified Hopkinson pressure bar (MHPB) that can be seen in Fig. 33.1. It was designed and assembled in-house. It consists of a Maraging steel rod of length 1510 mm and a diameter of 20 mm. The wave propagation speed (c0 =5091.75 m/s) and density (ρ = 8100 kg/m3 ) were experimentally measured. The material of the projectile and the bar are kept same to avoid noise at the contact during the impact. Two diametrically opposite 120  strain gauges were used to measure the surface strain signal on the bar. In a traditional three-point bend test, a well-designed three-point bend notched specimen with suitable geometrical tolerance is placed in front of input bar. Once the projectile is propelled with the help of single stage gas gun, it impacts one end of the input bar. This results in compressive stress wave of geometric wavelength two times of striker length getting induce in the input bars. The shape of this compressive wave is almost trapezoidal and the amplitude of the wave is proportional to the impact velocity of the striker bar. As compressive wave travels towards the input bar-specimen interface, the signal is recorded by strain gauge mounted on the incident bar, usually denoted by incident pulse εi (t), Due to the impedance mismatch between the incident bar and specimen a wave reflection from the incident bar- specimen takes place and strain gauge mounted on the bar records the strain signal, which is defined as reflected pulse εr (t). During strain pulse reflection from incident bar-specimen interface, a part of strain pulse propagate through the specimen. Specimen undergoes deformation during the period of stress wave propagation, until the specimen’s dynamic limit reached. From the 1-D elastic wave propagation theory, load point displacement of the specimen, u (t), can be expressed as [2]. t u(t) = c0

(εi (t) − εr (t)) dt

(33.1)

0

33.2.1 Material and Specimen The material that has been used in this study is Al6063-T6, which was procured from local vendors. The Chemical compositions and mechanical properties of the receiving material are shown in Tables 33.1 and 33.2. Three-point bend (TPB) were specimen design according to ASTM E399 standard criteria [6]. The Crack size is nominally varies between 0.45 and 0.55 times the width and width to thickness (W/B), ratio follows a 1 ≤ W/B ≤ 4 limits. One of the actual notched TPB specimen made of Al6063-T6 that were used in this study of evaluating fracture toughness can be seen shown in Fig. 33.2. The dimensions of the specimen are as followed: crack size a = 10 mm, thickness B = 10 mm, width W = 20 mm, span S = 80 mm and total length L = 100 mm.

Fig. 33.1 Experimental setup with High-Resolution Cameras Table 33.1 Chemical composition (Wt. %) of the Al6063-T6 Component Al6063-T6

Al Reminder

Cr Max 0.1

Cu Max 0.1

Fe Max 0.35

Mg 0.45–0.9

Mn Max 0.1

Si 0.2–0.6

Ti Max 0.1

Zn Max 0.1

33 Experimental Study on Dynamic Fracture Response of Al6063-T6 Under High Rates of Loading

183

Table 33.2 Mechanical properties of the Al6063-T6 Young’s modulus E (GPa) 68.9

Yield strength (MPa) 214

Ultimate strength (MPa) 241

Poisson’s ratio V 0.33

Elongation % 12

Mass density P (kg/m3) 2700

Fig. 33.2 Three-point bend specimen of Al6063-T6

33.2.2 Determination of Static Fracture Toughness Static three-point bend experiments were performed on Zwick/Roell testing machine coupled with high-resolution cameras to measure the crack mouth opening displacement (CMOD). The stress intensity factor (SIF), K1 (t), was evaluated from CMOD, using a expression by Rubio et al. [2] for TPB specimen geometry as shown below: K1 (t) =

E w(t) kβ (α) √ 4 aα vβ (α)

(33.2)

Where α is the ratio of crack size (a) to specimen width (W), β is the ratio of span length (S) to width (W) and kβ (α), vβ (α) are the non-dimensional function which depends on α and β parameters. It’s expression for the case β = 4 was found by Guinea et al. [9] and is given in Appendix 2. Time to fracture is sufficiently long in this case because test was performed at very low speed (1 mm/min). Thus, static fracture toughness, K1C, has obtained as SIF value at the crack initiation time (ti ), i.e. K1C = K1 (ti )

(33.3)

33.2.3 Determination of Dynamic Fracture Toughness Dynamic three-point bend experiments were performed using MHPB under different strain rate loading conditions. Two Photron SA5 high-speed digital camera in conjunction with 3D DIC were used to identify the crack imitation time and measure load point displacement. The load point displacement of the specimen under high strain rate deformation (u), was calculated from both DIC & Strain Gauge separately. Picture of the experimental setup that was used in the dynamic fracture toughness test is shown in Fig. 33.1. The dynamic stress intensity factor was found using the standard formulation also given by Rubio et al. [2]. K1 (t) =

3 β kβ (α) u(t) √ 2 B W C (α)

Where C(α) is the compliance of the cracked specimen, details are provided in Appendix 1.

(33.4)

184

A. K. Pandouria et al.

33.3 Results and Discussion 33.3.1 Quasi-Static Experiments Quasi-static experiments were performed (Fig. 33.3) at 1 mm/min loading speed on a 3-point bend specimen. This setup was coupled with high-resolution cameras to take meaning full measurements. 3D DIC was also utilized in determining the crack mouth opening displacement and crack initiation time. Figure 33.4 shows a typical response where CMOD exhibited the linear relation with loading time. The nature of SIF (Fig. 33.5) was also found to be linear conforming the relation √ between the SIF and CMOD as given in Eq. 33.1. The measured static fracture toughness, K1C, was found to be 17.31 MP a m (the value of SIF at the time to fracture, which is 35 second as measured using DIC).

33.3.2 Dynamic Experiments These experiments were performed at two different impact velocities (14.1 m/s and 15.95 m/s) on the notched Al6063-T6 specimen. Experimental details are given in Table 33.3. The Deformed and un-deformed 3-point bend specimen before and after dynamic experiment are shown in Fig. 33.6a, b. The load point displacement and SIF of the specimen at impact velocity 14.1 m/s and 15.95 m/s, are shown in Figs. 33.7 and 33.8 respectively. The image captured by high-speed cameras has shown that after 200 μs the paint get detached from the specimen surface. So the results are valid only 200 μs. This time was found to be sufficient to evaluate dynamic fracture toughness (as shown in Fig. 33.9) as crack initiation time varies between 41 or

Fig. 33.3 Quasi-static bending test detail Fig. 33.4 CMOD time history for Static test

CMOD 5

w (mm)

4 3 2 1 0 0

50

100

150

t (s)

200

250

300

350

33 Experimental Study on Dynamic Fracture Response of Al6063-T6 Under High Rates of Loading

185

Fig. 33.5 Static SIF time history

SIF from CMOD 300

K1 (MPa.m1/2)

250 200 150 100 50 0 0

50

100

150

200

250

300

350

t (s) Table 33.3 Dynamic test results of the Al6063-T6 Experiment no. 1 2

Velocity(m/s) 14.1 15.95

Strain rate (1/s) 1569 1960

Crack initiation time (ti ) (μs) 45 41

√ K1d (MPa m) 27.34 27.67

Fig. 33.6 Three point bend specimen (a) Un-deformed (b) Deformed DISP_14.1m/s

SIF_14.1m/s

2.6

220

2.4

200

2

180

1.8

160

1.6

140

K1(MPa.m1/2)

u (mm)

2.2

1.4 1.2 1 0.8

120 100 80 60

0.6 0.4

DIC

40

0.2

Strain Gauge

20

0

0 0

20 40 60 80 100 120 140 160 180 200 Time (μs)

DIC Strain Gauge 0

20 40 60 80 100 120 140 160 180 200 Time (μs)

Fig. 33.7 Load point displacement and SIF for experiment no. 1

45 μs (Table 33.3). The Figs. 33.7 and 33.8 clearly shows good agreement between results obtained DIC and Strain Gauge measurement technique, up to the first 100 μs. After 100 μs SIF evaluated from strain gauge was found to be lower than the SIF calculate from the DIC. The reason behind this mismatch may be attributed due to the loss of contact between specimen and input bar after 100 μs.

186

A. K. Pandouria et al. SIF_15.95m/s

DISP_15.95m/s 2.6

220

2.4

200

2.2

180

1.8

160

1.6

140

K1C (MPa.m1/2)

u (mm)

2

1.4 1.2 1 0.8

120 100 80 60

0.6 0.4

Strain Gauge

40

Strain Gauge

0.2

DIC

20

DIC

0

0

20

40

60

80 100 120 140 160 180 200

0

Time (μs)

0

20 40 60 80 100 120 140 160 180 200 Time (μs)

Fig. 33.8 Load point displacement and SIF for experiment no. 2

a

t=41μs

U (mm)

t=200μs

-1.7 -1.74063 -1.79125 -1.84188 -1.9025 -1.96313 -2.11375 -2.164838 -2.20625 -2.25687 -2.3075 -2.35725 -2.4112 -2.5612

-0.748 -0.764375 -0.78075 -0.797125 -8135 -0.84625 -0.879 -0.895375 -0.91175 -0.928125 -0.960875 -0.97725 -0.993625 -1.01

b

t=45μs

U (mm)

U (mm)

t=200μs

-0.589 -0.602937 -0.616875 -0.658687 -0.672625 -0.7005 -0.714437 -0.728375 -0.742312 -0.75625 -0.770187 -0.784125 -0.812

U (mm) -1.53 -1.57375 -1.6175 -1.66125 -1.71875 -1.86625 -1.91524 -1.9575 -2.0175 -2.08521 -2.17875 -2.2425 -2.3425 -2.4241

Fig. 33.9 Load point displacement profile of the specimen at velocity (a) 14.1 m/s and (b) 15.95 m/s

33.4 Summary and Conclusions In this study a static and dynamic fracture evaluation has been performed on TPB specimen of Al6063-T6, in conjunction with 3D digital image correlation technique (DIC). This provides a direct measurement of√CMOD and crack initiation time. Result has shown that the dynamic initiation √ fracture toughness (K1d = 27.5 MP a m) is higher than the static initiation fracture toughness (K1C = 17.31 MP a m). This study has also showed that the value of dynamic SIF increase with the velocity but dynamic fracture toughness does not change significantly with the impact velocity. The dynamic SIF was calculated using two different techniques (1) strain measurement technique and (2) DIC, results from both were found to be in good agreement with each other’s.

33 Experimental Study on Dynamic Fracture Response of Al6063-T6 Under High Rates of Loading

187

Appendix 1. Compliance of Cracked Specimen The compliance of cracked specimen is given by [2] C (α) = CO + Cc (α)

(33.5)

Where C0 is the compliance of any type of cracked specimen can be calculated as L3 CO = 48E  I

  24 (1 + v) I 1+ k A L2

(33.6)

Cc (α) take into the account the effect of crack in compliance of the specimen is expressed by [9]. Cc (α) =

1  h1 (α) + βh2 (α) + β 2 h3 (α) EB

(33.7)

0.29 + 1.39α − 1.62α 2 h1 (α) = −0.378α 3 ln (1 − α) + α 2  1 + 0.54α − 0.84α 2 h2 (α) = 1.1α 3 ln (1 − α) + α 2

(33.8)

−3.22 − 1.64α + 28.1α 2 − 11.4α 3  (1 − α) 1 + 4.7α − 4α 2

h3 (α) = −0.176α 3 ln (1 − α) + α 2

8.91 − 4.88α − 0.435α 2 + 0.26α 3  1 − α 2 (1 + 2.9α)

(33.9)

(33.10)

Appendix 2. The Expression of kβ (α) and vβ (α) The expression of kβ (α) & vβ (α), for the case β = 4, is given by √ kβ (α) =

3

α

(1 − α) 2 (1 + 3α)



1.9 + 0.41α + 0.51α 2 − 0.17α 3

0.66 vβ (α) = 0.76 − 2.28α + 3.87α 2 − 2.04α 3 +  1 − α2

 (33.11)

(33.12)

References 1. Bacon, C., Farm, J., Lataillade, J.L.: Dynamic fracture toughness determine from load point displacement. Exp. Mech. 34, 217–223 (1993) 2. Rubio, L., Fernandez-Saez, J., Navarro, C.: Determination of dynamic fracture-initiation toughness using three-point bending tests in a modified Hopkinson pressure bar. Exp. Mech. 43, 379–386 (2003) 3. Singh, R.P., Parameswaran, V.: An experimental investigation of dynamic crack propagation in a brittle material reinforced with a ductile material. Opt. Lasers Eng. 40, 289–306 (2003) 4. Galvez, F., Cendon, D., Garcia, N., Enfedaque, A., Sanchez Galvez, V.: Dynamic fracture toughness of a high strength armor steel. Eng. Fail. Anal. 16, 2567–2575 (2009) 5. Ibrahim, U., Irfan, M.A.: Dynamic crack propagation and arrested in rapid prototype material. Rapid Prototyp. J. 18, 154–160 (2012) 6. ASTM Standard E 399–90: Standard Test Method for Plain Strain Fracture Toughness of Metallic Materials. Annual Book of ASTM Standards, ASTM International, West Conshohocken (1994) 7. Sutton, M.A., Orteu, J.J., Schreier, H.: Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Springer Science & Business Media, Boston (2009)

188

A. K. Pandouria et al.

8. Nishioka, T., Atluri, S.: A method for determining dynamic stress intensity factors from COD measurement at the notch mouth in dynamic tear testing. Eng. Fracture Mech. 16(3), 333–339 (1982) 9. Guinea, G., Pastor, J., Planas, J., Elices, M.: Stress intensity factor compliance and CMOD for a general three-point bend beam. Int. J. Fracture. 89(3), 103–116 (1998)

Chapter 34

Ballistic and Material Tests and Simulations on Ultra-High Performance Concrete Sidney Chocron, Alexander Carpenter, Nikki Scott, Oren Spector, Alon Malka-Markovitz, Zev Lovinger, and Doron Havazelet

Abstract Ultra-high performance concretes (UHPC), meaning concretes with compressive strengths above 150 MPa (B150), introduce improved properties such as stiffness, compressive strength, and post-failure compliance as compared to standard concretes. Advantages are shown in standard applications of construction, yet, a large potential exists in applications of protective structures to withstand impulsive loadings of blast or direct impact. In this work an UHPC with a compression strength of 200 MPa was used to test and develop a material model to enable predictions for impact and penetration. The material was first tested to characterize the material behavior under quasistatic loading in torsion, compression and triaxial compression, up to confinement pressures of 500 MPa. Moreover, the UHPC was characterized under dynamic loading, using a Kolsky bar (Split Hopkinson Pressure Bar). Based on these lab-scale tests, a Johnson-Holmquist material model was calibrated for the numerical simulations. Finally, ballistic tests were performed with two projectile geometries, using two configurations: a standalone UHPC panel to obtain the ballistic limit, and depth of penetration (DOP) measurements, with aluminum backing, to better relate to the concrete strength during penetration conditions. Preliminary ballistic computations with the UHPC model, calibrated from the lab-scale tests for LS-DYNA, provided good predictions when compared to most of the tests. Keywords Ultra-high performance concrete · High strain-rate material properties · Ballistic impact · Computer simulations

34.1 Introduction Ultra-high performance concretes (UHPC) are an emerging class of material that exhibit improved stiffness, compressive strength, and post-peak strength as compared to the standard concretes used in construction applications [1]. Advantages are shown in standard applications of construction, yet, a large potential exists in applications of protective structures to withstand impulsive loadings of blast or direct impact. For this reason, there is significant interest in further researching their low and high-rate mechanical properties and behavior under ballistic impact. This paper summarizes the compression tests performed with different specimen sizes, strain-rates, and confinement pressures. A limited number of torsion tests were also conducted. The data were used to determine the Young’s modulus of the material and its strength at low and high pressures. These values were used to create a Johnson-Holmquist material model for the UHPC to predict the depth of penetration and ballistic limit of the target.

S. Chocron () · A. Carpenter · N. Scott Southwest Research Institute, San Antonio, TX, USA e-mail: [email protected] O. Spector · A. Malka-Markovitz · Z. Lovinger Rafael Advanced Defense Systems, Haifa, Israel D. Havazelet Israel Ministry of Defence, Tel Aviv, Israel © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_34

189

190

S. Chocron et al.

34.2 Materials and Methods 34.2.1 Ultra-High Performance Concrete Material The ultra-high performance concrete utilized during this study was provided by G.tecz (Angersbachstr. 12b, 34,127 Kassel, Germany, phone: +49,561 8617 555). Twenty-five concrete slabs, each measuring 500 mm × 500 mm × 50.8 mm, were manufactured. Twenty-three of the slabs were for ballistic tests while two were for material tests. The material consists of a fine cement matrix incorporating many small, randomly oriented steel fibers. Although no measurements of the cement aggregates were obtained from the material studied here, typical UHPC has a maximum particle size of 1 mm [1]. A simple analysis of the material suggests that the short steel fibers content in the concrete is approximately 1.9% in weight. The concrete also contains voids which can be as large as 3–4 mm in diameter. The UHPC material was designed to exhibit a nominal compressive strength of 200 MPa.

34.3 Test Performed on UHPC 34.3.1 Quasistatic and Medium-Rate Compression Tests A servo-hydraulic MTS loadframe was used to test the UHPC in compression at strain rates up to 0.2 s−1 . In order to obtain accurate measurements of the displacement within the specimen, a clip-on axial extensometer. Three different specimen sizes were tested in compression in order to determine if there is any effect of the loaded volume on the measured strength: (a) 27.9 mm diameter, 50.8 mm length, (b) 15.2 mm diameter, 30.5 mm length, (c) 12.7 mm diameter, 25.4 mm length. The strain at which fracture of the UHPC specimens began was typically small (∼0.5%), so the engineering stresses and strains are essentially equivalent to the corresponding true stresses and strains for all tests. The hydrostatic pressure within the material was calculated by dividing the compressive engineering stress by 3.

34.3.2 High-Rate SHPB Compression Tests A number of higher-rate compression tests were conducted using SwRI’s 1.5 inch diameter split Hopkinson pressure bar (SHPB). The dimensions of the UHPC cylindrical specimen were 27.9 mm for the diameter and 50.8 mm for the length. Some of the tests were performed with a pulse shaper to ensure dynamic equilibrium berfore failure of the specimens.

34.3.3 High-Pressure Compression Tests The behavior of the UHPC under increasing hydrostatic pressure was studied by performing compression tests in a pressure chamber. Before each test, a 12.7-mm-diameter, 25.4-mm-long cylindrical specimen was placed between two alumina anvils. A clip gage was used to measure displacements.

34.3.4 Torsion Tests A limited number of torsion tests were conducted on torsion specimens with square grips and a short circular gage section. Each specimen was placed between two collets containing square openings in the torsion tester. One collet was then rotated relative to the other using a low-mass, hydraulically-driven torsional actuator and shaft in order to impose shear stresses/strains within the gage region of the specimen. The average strain rate within the specimens during the tests was approximately 1 s−1 . The torques on the specimens were measured by a load transducer during the tests. The gage section of the specimens was 19.05-mm-diameter and 2.54-mm-height.

34 Ballistic and Material Tests and Simulations on Ultra-High Performance Concrete

191

34.3.5 Ballistic Tests Ballistic tests were performed at Rafael. Rigid projectiles (HRC 53) with hemispherical nose, 38.1-mm-long and 12.7-mmdiameter, were shot on single UHPC slabs 50 cm × 50 cm × 5.08 cm for ballistic limit experiments as well as on single slabs backed by aluminum 6061-T6 plates (5.08 cm-thick) for DOP experiments. A scaled version of the projectile with a diameter of 7.62-mm was also used to study scale effects. The set-up is shown in Fig. 34.1a where the mirror used to measure projectile pitch and yaw at impact is can be seen in front of the concrete target. The shots were recorded with a high speed video camera that provided impact velocity, residual velocity, and impact angle.

34.4 Test Results 34.4.1 Quasistatic and Medium-Rate Compression Results The strength measured for the different specimens results from the quasistatic and medium-rate compression tests is summarized in Table 34.1. For the larger specimens (27.9-mm-diameter and 50.8-mm-long) the average Young modulus was 41 GPa. A smaller series of medium-rate compression tests was conducted using the same specimen size. The average strain rate during these tests was 1.1 × 10−1 s−1 . The average from the six tests was 216 MPa, only marginally larger than that obtained from the slower rate tests, so only a marginal effect is seen at medium strain rates. The average compressive strength for smaller specimens was 170–175 MPa. The smaller strength representative of these specimens as compared to the larger specimens suggests that there is an effect of the specimen size on the strength of the material. This size effect may be attributable to the presence of voids within the cement matrix of the UHPC. As the crosssectional area of the specimen is reduced, the voids appear proportionally larger as compared to this area. Another possibility is that when specimen size is close to the characteristic size of the steel fibers, these are less effective at reinforcing the material.

Fig. 34.1 (a) Set-up for ballistic tests. (b) The effect of strain rate on the strength of the UHPC material is shown Table 34.1 Summary of slow- and medium-rate compression test results Diameter (mm) 27.9 27.9 15.2 12.7

Length (mm) 50.8 50.8 30.5 25.4

Avg. strain rate (s−1 ) 1.2 × 10−4 1.1 × 10−1 7.2 × 10−5 2.2 × 10−5

Min strength (MPa) 183 167 157 157

Max strength (MPa) 233 274 183 202

Avg. strength (MPa) 206 216 170 175

192

S. Chocron et al.

34.4.2 High-Rate Compression Results The maximum stresses exhibited by the specimens tested at high rates (>100 s−1 ) are significantly larger than those from the slower rate tests. The average strength measured from the first three tests is 403 MPa. There was some concern that the specimens were not in dynamic equilibrium during these first tests, so copper pulse shapers were added to the test setup during the last two tests. The addition of the pulse shapers did provide equilibrium in the specimen and reduced the measured strength by 13% to 357 MPa. This value is thought to be more representative of the strength of the UHPC at 400 s−1 . Other investigations into the high-rate behavior of standard and ultra-high performance concretes [2, 3] found similar large increases in the strength of the material above 1 s−1 . The strain-rate effect typically manifests itself as a small increase of strength with increasing strain rate below 1 s−1 , and the effect then becomes significantly larger for faster strain rates. This effect was also observed during the current study, as is illustrated by Fig. 34.1. The data suggest that the strain-rate dependence could be modeled by a log-linear relationship between strain rate and strength below 1 s−1 , but a different relationship (either linear with a different slope or exponential) will be needed for faster strain rates. Note that it has been observed that the strain rate sensitivity of concrete may be significantly different in tension as compared to compression [3]. That said, it might be expected that accurately modeling the strain rate dependence of the material in compression is more important, as strengths in tension are much weaker.

34.4.3 High-Pressure Triaxial Compression Results Test results had scatter in the data but it appears that the strength of this UHPC is approximately constant over most of the pressures investigated (0–500 MPa). The average strength obtained from the five tests is 222 MPa. If one of the tests that seems an outlier is removed from the average, the strength of the UHPC at high pressures increases to an average of 234 MPa. Interestingly, a purely hydrostatic test with a maximum pressure of 400 MPa resulted in an elastic unload of the specimen, which was recovered undeformed after the test.

34.4.4 Torsion Results There is some scatter present within the data: one of the specimens exhibited a peak torque approximately 50% larger than the other two. Elastic theory was used to convert the peak torques from the test into the maximum shear stress experienced by the specimens and the corresponding equivalent strengths. The average strength of the UHPC in shear is only 30 MPa. The data cannot be used to measure the shear modulus of the UHPC because of the machine compliance present in the rotation measurements. However, Young’s modulus was estimated from the compression tests as 41 GPa, and Poisson’s ratio for UHPC is reported in the literature as 0.2 [1]. It is expected that the UHPC is isotropic because of the random orientation of the steel fibers, so the shear modulus can be estimated using Young’s modulus and Poisson’s ratio as 17.1 GPa.

34.4.5 Ballistic Test Results Three depth of penetration (DOP) and nine standalone tests were performed. For some of the DOP tests the projectile stopped before reaching the aluminum plate. These tests are reported in Fig. 34.2b as “proj. in UHPC” and provide the measure depth of penetration into the UHPC. For the tests that fully perforated the UHPC and stopped in the aluminum plate, the legend is “proj. in Al”, and reports the depth of penetration in the aluminum. Figure 34.2a summarizes the residual velocity vs. impact velocity for the standalone UHPC targets. Note that the ballistic limit is between 313 and 432 m/s.

34 Ballistic and Material Tests and Simulations on Ultra-High Performance Concrete

193

Fig. 34.2 Summary of ballistic test results for (a) standalone UHPC targets, (b) DOP targets

Fig. 34.3 LS-DYNA simulation of penetration into UHPC. (a) Initial geometry, (b) final DOP and damage plot

34.5 Numerical Simulations Results Preliminary numerical simulations were performed using the Holmquist-Johnson [4] concrete model. Most of the constants for the constitutive model were derived from the material tests presented above while the equation of state constants were from Erzar [5]. Figure 34.3a shows the initial geometry of the impact on UHPC by the 12.7 mm projectile. Note that the mesh is refined around the area of impact. The projectile material model was purely elastic since it was apparent that it did not deform plastically during the penetration process. Figure 34.3b was taken at the last time-step of the computation, and the depth of penetration predicted was 22.6 mm, close to the 19 mm observed in the test. One of the material constants that could not be directly derived from the tests was the damage constant D1. The reason was that various hydrostatic load/unload cycles up to 400 MPa resulted in an elastic (zero damage) unload of the specimen. The D1 constant defines the failure of the material through an empirical relation: D2  f f εp + μp = D1 P ∗ + T ∗ f

f

where εp is the equivalent plastic strain at failure, μp the plastic volumetric strain at failure, P* the presure normalized by the quasitatic compression strength and T* is the normalized maximum tensile hydrostatic pressure the material can bear. A parametric study was performed on the constant D1 (D2 was assumed to be 1.) to see the sensitivity of the results. As expected, since D1 is the major damage constant, both the DOP and the residual velocities of the projectiles were found to be very sensitive to D1. For example residual velocities varied from 93 to 330 m/s for values of D1 from 0.6 to 0.04 respectively (for the test it was 303 m/s). Similarly, DOP in aluminum could change from 15.6 mm to 32.6 mm. The final value of D1 will probably be calibrated by comparing numerical simulations with the 12 ballistic tests performed and finding the optimum value.

194

S. Chocron et al.

34.6 Summary and Conclusions Ultra-high performance concrete was tested in compression and shear to obtain data used to construct a material constitutive model for ballistic impact. A split Hopkinson pressure bar was used to obtain data at rates faster than 100 s−1 , while a pressure chamber provided data at larger hydrostatic pressures. The Young’s modulus and strength values for the material were calculated from the test data. The observations to highlight from this study were: (a) there appears to be a small size effect on the UHPC strength, where smaller specimens appear to be slightly weaker, (b) he strain-rate dependence of strength is weak below 1 s−1 , but it appears to strengthen more rapidly at faster rates, (c) UHPC is strongly pressure-dependent for hydrostatic pressures below 70–80 MPa, but its strength is roughly pressure-indepedent above that point, (d) ballistic tests were performed to determine depth of penetration and ballistic limit for some configurations, (e) a preliminary HolmquistJohnson model provides good predictions for most of the test, (f) the constant D1 could not be determined directly from the material tests. A parametric study on constant D1 showed that results are very sensitive to this parameter and hence, the constant can probably be calibrated using the ballistic tests.

References 1. Fehling, E., Schmidt, M., Walraven, J., Leutbecher, T., Fröhlich, S.: Ultra-High Performance Concrete UHPC. Ernst & Sohn, Berlin/Germany (2014) 2. Nöldgen, M., Riedel, W., Thoma, K., Fehling, E.: Properties of ultra-high performance concrete (UHPC) in tension at high strain rates. Proceedings of the VIII International Conference on Fracture Mechanics of Concrete and Concrete Structures. Toledo/Spain (2013) 3. Pajak, M.: The influence of the strain rate on the strength of concrete taking into account the experimental techniques. Archit. Civ. Eng. Environ. 4(11), 77–86 (2011) 4. Holmquist, T.J., Johnson, G.R., Cook, W.H.: A computational constitutive model for concrete subjected to large strains, high strain rates, and high pressures. In Proceedings of the 14th International Symposium on Ballistics, Quebec City, Canada, September (1993) 5. Erzar, B., Pontiroli, C., Buzaud, E.: Shock characterization of an ultra-high strength concrete. Eur. Phys. J. Spec. Top. 225(2), 355–361 (2016). https://doi.org/10.1140/epjst/e2016-02637-4

Chapter 35

Mechanical Behavior of Ta at Extreme Strain-Rates Daniel Casem, Daniel Magagnosc, Jonathan Ligda, Brian Schuster, and Timothy Walter

Abstract The compressive stress-strain response of a commercially pure (99.98%) Ta was investigated at strain-rates ranging from 0.001/s to 500 k/s. Strain-rates up to 20 k/s were obtained using conventional load frames and Kolsky bar methods. The higher strain-rates were obtained using optically instrumented miniature Kolsky bars. Because these experiments require sample sizes as small as ∼30 um, a fine grain structure was desired. To achieve this, we study a Ta billet that was processed by ECAE to produce an ultrafine grain structure. The billets were subsequently annealed at 1203 K under high vacuum for 2 h to coarsen the grain size to approximately 2 um. The as-worked and annealed microstructures were investigated by electron backscatter diffraction to verify the grain structure. A strong rate dependence is observed over this range of strain-rates, although there is a discrepancy between data at similar strain-rates using different sample sizes. This discrepancy is the subject of on-going investigation. Keywords Tantalum · Split Hopkinson pressure bar · Kolsky bar · High strain-rate · Plasticity

35.1 Introduction The goal of this program is to investigate the mechanical behavior of tantalum (Ta) over a wide range of strain-rates (0.001/s to 500 k/s). Ta is a high density metal with a high melting temperature, and like other bcc metals, its flow strength has a strong dependence on strain-rate. All experiments are uniaxial stress compression, with samples initially at room temperature (∼293 K). Rates of 0.1/s and below were achieved using servo-hydraulic load frames, and the higher strain-rates were achieved using Kolsky bar, or split Hopkinson pressure bar, techniques [1]. Strain-rates beyond ∼20 k/s utilized specialized miniature Kolsky bars that require optical instrumentation, detailed in [2–6]. Indeed a major objective is to study the behavior in the range beyond that easily accessible with conventional Kolsky bars. As a general rule, it is necessary to use both smaller diameter Kolsky bars and smaller samples to obtain higher strain-rates. At the highest rates, specimens with dimensions of the order of 30 um were used. Because our goal is to study continuum level properties, a homogeneous material with a small grain size is required. For this reason, we selected as our subject material a billet that had been processed using Equal Channel Angular Extrusion (ECAE) to produce an ultrafine grained structure (∼200 nm). More details of this process and the resulting materials are discussed in [7–9]. For our work, specimens were annealed for 2 h at 930 ◦ C under high vacuum (∼5 × 10−6 Torr) in order to recrystallize the heavily deformed grain structure. After annealing, pieces were cut and ion polished in the extrusion direction and surface normal direction. After polishing, the grain structure was observed by EBSD in each direction, see Fig. 35.1. The grain structure shows little texture and predominantly equiaxed grains. The corresponding grain size was found to be 2.2 um using the line intercept method.

D. Casem () RDRL-WMP-C, US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA e-mail: [email protected] D. Magagnosc · J. Ligda · T. Walter RDRL-WMM-B, US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA B. Schuster RDRL-WML-H, US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_35

195

196

D. Casem et al.

Fig. 35.1 EBSD scans of the ECAE Ta discussed in this paper, after annealing. (a) extrusion direction, and (b) lateral direction

Fig. 35.2 Sketch (not to scale) showing how specimens were machined from billet

35.2 Experiments Our first objective was to determine if the material was isotropic and uniform in mechanical behavior throughout the billet. The original billet was 25 mm x 25 mm x 150 mm. A matrix of approximately cube shaped samples was machined by wire EDM (Electro Discharge Machining) from the cross-section of material at the center of the billet, as sketched in Fig. 35.2. Actual dimensions were nominally 1.4 mm × 1.5 mm × 1.6 mm, with the shorter dimension in the extrusion direction. Unequal dimensions were used to keep track of the orientation of each sample. In all cases, loading faces were polished to remove the EDM finish. In many cases, all six faces were polished; however, this did not make any noticeable difference in the measured stress-strain curves so to save time and effort this practice was not used for all samples (i.e., only the loading faces were polished).

35 Mechanical Behavior of Ta at Extreme Strain-Rates

197

1400 c

true stress (MPa)

1200 1000 800

b

600 a

400 200 0 0

0.05

0.1

0.15 true strain

0.2

0.25

0.3

Fig. 35.3 Stress-strain curves for Ta. (a) Rates at 0.001/s, at different locations throughout the cross-section, and at different orientations. (b) Rates of 5-25 k/s, all in the extrusion direction, using the 3.2 mm diameter Kolsky bar. (c) Rates of 25 k/s-500 k/s, using the L = 30 um, D = 60 um samples and the miniature Kolsky bars (extrusion direction)

Fig. 35.4 Typical sample of Ta used for the highest strain-rates. The slight striations are due to “curtaining,” a feature common to milling with a FIB. The top loading face is polished flat but is covered with debris in the image

Figure 35.3 shows stress-strain curves for 29 experiments at 0.001/s (group “a” in the figure). These curves include data from samples at different locations (center, edge, etc.) and in different orientations (the extrusion direction and the two natural lateral directions). No definitive trends were found with direction or original location within the billet, i.e., the scatter is apparently random. Additional experiments were conducted using a 3 mm diameter Kolsky bar with these same size samples, all in the extrusion direction. These data are plotted as group “b” in the figure, and range in strain-rates from 5 k/s to 25 k/s. The increase in strength due to strain-rate is apparent. Also noticeable is the increased softening due to adiabatic heating during the high-rate experiments. The final group of experiments (group “c”) were conducted using much smaller cylindrical specimens that were nominally 30 um tall and 60 um diameter. To fabricate these samples, a thin layer of material was EDM’d from the center of the original billet that was subsequently polished down to a 30 um thickness to create a thin foil without an EDM affected zone. Disk shaped samples were then machined from this foil with a fs laser mill and finished to final dimensions using a Focused Ion Beam (FIB). A typical sample is shown in Fig. 35.4. Two separate miniature Kolsky bars were used to test these samples. The first was a 305 um diameter steel bar, optically instrumented, and was used to obtain strain-rates from 80 k/s to 500 k/s. At these extreme rates, specimen equilibrium is a legitimate concern. However, due to the small sample size, the state of equilibrium is quite good, in spite of the rapid deformation. To confirm this, forces measured by each bar at the ends of the sample can be compared in the standard way.

198

D. Casem et al. 25 20

orange is F2 force blue is F1 force

force (N)

15 10 5 0 5.0

6.0

7.0

8.0

9.0

10.0

11.0

-5 time (μs)

Fig. 35.5 Force equilibrium of the sample at the highest strain-rate (500 k/s) 1200

1000

true stress (MPa)

800

600 1.6 mm micro samples

400

200

0 0.0001

0.001

0.01

0.1

1

10

100

1000

10000 100000 1000000

true strain-rate (1/s)

Fig. 35.6 True stress at 0.07 true strain for Ta. Note the discrepancy between the sample sizes at 25 k/s

This comparison is shown in Fig. 35.5, where F1 is the force between the sample and the input bar and F2 is the force between the sample and the output bar. The close agreement is a good indication that the sample is in a state of quasistatic equilibrium, and that the results are free from inertial effects. It was not practical to obtain strain-rates lower than 80 k/s for these samples using this bar, so an additional experiment was conducted using a specialized miniature bar designed to obtain lower strain-rates with samples of this size. This bar is discussed in [10] and was used to perform a test at 25 k/s, which overlaps the rate range obtained with the larger samples using the 3 mm diameter bar. The stress-strain curve for this experiment is also plotted in group “c” in the figure.

35.3 Discussion/Conclusion From the data in Fig. 35.3, strain at 7% true stress was measured and plotted as a function of true strain-rate in Fig. 35.6. Additional data from experiments at 0.01/s and 0.1/s, not shown in Fig. 35.3, are also plotted here. The data show an increase in strength with strain-rate over the complete range of strain-rates. Of particular concern, however, is the discrepancy of approximately 100 MPa between the data using the two different sample sizes. Even when tested at the same strain-rate (25 k/s), the data do not agree. There is also a change in slope, which may indicate a change in mechanism. This discrepancy is currently not understood and its cause is currently under investigation. Some disagreement is expected because of the

35 Mechanical Behavior of Ta at Extreme Strain-Rates

199

different sample geometries, the larger cube shape as compared to the smaller L/D = 0.5 cylinders, since the different shapes will be affected differently by friction as the loading faces. However, this seems to be a rather large discrepancy to be explained by friction. Another possibility is that the two sample geometries are affected differently by impurities (e.g., oxygen). Grain size effects are another obvious suspect, and has been credited to varying degrees of scatter in similar testing programs on pure aluminum [10] and Ti-6Al-4 V [11]. However, as mentioned above, the small grain material selected here was chosen specifically to avoid this problem, and even the smallest samples should contain thousands of grains. We are currently planning additional chemical and microstructural analyses along with additional mechanical testing with different sample geometries to resolve the inconsistencies in the data. Acknowledgements The authors would like to thank Mr. Kevin Taylor for his help conducting the low-rate experiments and for preparing the samples.

References 1. Chen, W., Song, B.: Split Hopkinson (Kolsky) Bar. Springer, New York (2011) 2. Casem, D.T.: A Small Diameter Kolsky bar for High-rate Compression. Proc. of the 2009 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Albuquerque, NM, 1–4 June 2009 3. Casem, D.T., Grunschel, S.E., Schuster, B.E.: Normal and transverse displacement interferometers applied to small diameter Kolsky bars. Exp. Mech. 52(2), 173–184 (2012) 4. Huskins, E.L., Casem, D.T.: Compensation of Bending Waves in an Optically Instrumented Miniature Kolsky Bar. J Dyn Behav Mater. 1, 65 (2015). https://doi.org/10.1007/s40870-015-0006-6 5. Casem, D.T., Zellner, M.: Kolsky bar wave separation using a Photon Doppler Velocimeter. Exp. Mech. 53, 1467–1473 (2013) 6. Casem, D.T., Huskins, E.L., Ligda, J., Schuster, B.E.: A Kolsky Bar for High-rate Micro-compression – Preliminary Results. In: Proc. 2015 SEM Annual Conference and Exposition, Springer, Costa Mesa, CA (2015) 7. Wei, Q., Jiao, T., Mathaudhu, S.N., Ma, E., Hartwig, K.T., Ramesh, K.T.: Microstructure and mechanical properties of tantalum after equal channel angular extrusion (ECAE). Mater. Sci. Eng. A. 358, 266–272 (2003) 8. Wei, Q., Ramesh, K.T., Kecskes, L., Mathaudhu, S.N., Hartwig, K.T.: Ultrafine and nanostructured refractory metals processed by SPD: microstructure and mechanical properties. Mater. Sci. Forum. 579, 75–90 (2008) 9. Wei, Q., Schuster, B.E., Mathaudhu, S.N., Hartwig, K.T., Kecskes, L.J., Dowding, R.J., Ramesh, K.T.: Dynamic behaviors of body-centered cubic metals with ultrafine grained and nanocrystalline microstructures. Mater. Sci. Eng. A. 493(1–2), 58–64 (2008) 10. Casem, D.T., Ligda, J.P., Schuster, B.E., Mims, S.: High-rate mechanical response of aluminum using miniature Kolsky bar techniques. In: Kimberley, J., Lamberson, L., Mates, S. (eds.) Dynamic Behavior of Materials, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham (2018) 11. Casem, D.T., Weerasooriya, T., Walter, T.R.: Mechanical behavior of a low-cost Ti–6Al–4V alloy. J. Dyn. Behav. Mater. (2018). https://doi.org/10.1007/s40870-018-0142-x

Chapter 36

Constitutive Modeling of Polyamide Split Hopkinson Pressure Bars for the Design of a Pre-stretched Apparatus A. Bracq, G. Haugou, and H. Morvan

Abstract This paper aims to model the constitutive behavior of polyamide material used in the Split Hopkinson Pressure bars (SHPB). The Hopkinson bars apparatus is employed for the mechanical characterization of many materials under high strain rates at large strains. Nevertheless, testing soft materials is a challenging task regarding their low impedance properties and the difficulty to achieve a dynamic equilibrium. To address that issue, polyamide (nylon) SHPB are employed. However, the application of the pre-stretched technique to tensile apparatus using polyamide bars may provide a flexible mechanical characterization device reaching moderate to high strain rates at large strains. It requires bars of several meters where wave attenuation and dispersion are dominant. Moreover, the design of such apparatus is extremely complex with respect to the sample shape and rigidity as well as connectors. While analytical techniques are proposed in the literature, they are not sufficient to provide guidance in the design and the optimization of a pre-stretched apparatus. Therefore, the aim of the present study is to develop a finite element model of polyamide SHPB. Various experimental tests are conducted using compressive polyamide SHPB. These tests are computationally modeled using the Radioss explicit FE code through an axisymmetric analysis. The generalized Maxwell model is chosen to consider the viscoelastic material properties. An optimization procedure by inverse method is applied using both experimental and numerical strain signals to identify the material coefficients. Experimental tests are repeatable for each test configuration. The viscoelastic model parameters of the bars are identified through one configuration and validated against three others. This model gives very satisfactory results and presents interesting predictive abilities. Keywords Hopkinson bars · Viscoelastic bars · Experimental testing · Constitutive modeling · Inverse technique

36.1 Introduction and Statement of Problem The last few decades have seen the development and processing of different soft materials in numerous industrial fields such as automotive, defense, biology. However, the mechanical characterization of highly compliant materials remains a difficult task using conventional techniques especially under high strain rates. To reach dynamic strain rates, the Split Hopkinson Pressure bars (SHPB) have been employed [1]. Testing low impedance materials still results in a low signal to noise ratio using regular Hopkinson bars. Therefore, the literature points out the use of polymeric SHPB with impedance values close to the material involved [2–4]. Indeed, polymeric SHPB as for instance polyamide bars have proved to be an essential choice to characterize gels, foams and adhesives [5–7]. Whereas, the viscoelastic material nature of polyamide is taken into account through analytical techniques with the consideration of wave attenuation and geometrical dispersion, no constitutive modeling of the viscoelastic material is proposed. Indeed, it is required for the Finite Element (FE) modeling of dynamic loading using SHPB apparatus. Actually, the enhancement of numerical tools using the FE method would benefit the prediction of the elastic waves and the test apparatus design by taking into account connectors and the sample shape. More precisely, the application of the pre-stretched technique to tensile apparatus using polyamide bars would lead to a flexible mechanical characterization device reaching moderate up to high strain rates at large strains. This technique enables to stretch up the specimen until fracture based on greater duration times against conventional SHPB [8, 9]. The duration time depends on the pre-stretched length and the material speed of sound. This duration time implies a specific strain range achievable following the targeted strain rate. For instance, a pre-stretched length of three meters for a polyamide

A. Bracq () · G. Haugou · H. Morvan Laboratory LAMIH UMR CNRS 8201, University of Valenciennes and Hainaut Cambrésis, Valenciennes, France e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_36

201

202

A. Bracq et al.

bar and a target strain rate of 250 and 500 s−1 would lead respectively to an ultimate engineering strain of 86% and 172%. Consequently, the requirement of bars of several meters to avoid signals superposition would lead to a significant wave attenuation. Hence, such a complex apparatus require precise calculations with first the constitutive modeling of the viscoelastic polyamide bars. To address this challenge, experimental testing configurations with standard SHPB are computationally modeled using the Radioss explicit FE code through an axisymmetric analysis. A similar study has been carried out but solely for polymethyl methacrylate (PMMA) [10]. Strain signals data have been exploited in the optimization procedure. In the same way, strain measurements are employed in different studies to determine the sample precise behavior during Hopkinson bars tests [11, 12]. Thus, an optimization process by inverse method is applied using both experimental and numerical strain signals to identify the material coefficients. The generalized Maxwell model is chosen to consider the viscoelastic material properties. The ability of the numerical model to predict wave attenuation and dispersion are ensured through comparisons with various experimental data and configurations.

36.2 Methods Experimental tests are performed using existing polyamide SHPB in compressive configurations. Indeed, common compressive impact experiments with a free-end incident bar are carried out in order to get insight into the wave attenuation and dispersion. Two configurations are chosen with specific bars length and striker characteristics (Figs. 36.1 and 36.2). The first corresponds to a striker bar (16.1 mm in diameter and 950 mm in length) impacting a free-end incident bar (20.3 mm in diameter and 3040 mm in length) including a glued strain gauge. The second one is a smaller striker bar (16.1 mm in diameter and 150 mm in length) impacting two incident bars placed end to end including each a strain gauge. Theses strain gauges, located to avoid waves superposition, enable to measure strain waves propagating through the bar by means of a conditioning amplifier system (VISHAY 2210B) and a fast acquisition system (DL 850 Yokogawa) with a sampling rate fixed at 5 MHz. Table 36.1 summarizes experimental tests conducted for each configuration with the averaged striker speed. Repeat tests are performed for each striker speed revealing consistent results but are not mentioned in Table 36.1 for clarity. Moreover, experiments are chosen for specific purposes: identify and validate the finite element described hereafter. An axisymmetric FE model for each set-up is created with HyperMesh software (Altair HyperWorks 14.0) and simulations are run with the explicit code Radioss (Altair HyperWorks 14.0). The viscous nature of polyamide is modeled using a generalized Maxwell model (Eq. 36.1) defined as follows: G(t) = G∞ +

N 

Gi exp (−βi t)

(36.1)

1 τi

(36.2)

i=1

βi =

G (t = 0) = G0 = G∞ +

0

N i=1

(36.3)

Gi

Strain gauge

Strain gauge

3040 mm

3020 mm

150m Fig. 36.1 Configuration N◦ 2: schematic of striker bar, incident bars and instrumentation Fig. 36.2 Configuration N◦ 1: schematic of striker bar, incident bar and instrumentation

0

950 mm

Strain gauge

3040 mm

36 Constitutive Modeling of Polyamide Split Hopkinson Pressure Bars for the Design of a Pre-stretched Apparatus

203

Where G(t) is the shear modulus, G0 the initial shear modulus, G∞ the long-term shear modulus. Gi and β i are respectively shear moduli and time decay parameters defining Maxwell elements. Furthermore, an ultra-sound device (EPOCH LT) is employed to determine the elastic properties of polyamide bars based on the wave speed measurement. It leads to the Young’s modulus value E and initial shear modulus G0 . Then, an optimization by inverse technique is applied using a global surface response methodology (HyperStudy – Altair HyperWorks 14.0). Experimental strain measurements are employed to calculate the objective function based on the mean square error (MSE): MSE =

N 2 1  i i Xexp − Xnum N

(36.4)

i=1

Where N = 26 is the amount of data points used for correlations. Xexp and Xnum correspond respectively to the experimental and numerical strain. Once the material model parameters identified, the model predictive ability is assessed through comparison with three independent set of data covering a wide range of impact conditions.

36.3 Results and Discussion First of all, the determination of the elastic properties of the material enables to highlight the significance of viscous properties during an impact experiment and its FE modeling. Indeed, the first experimental configuration (case 1 – Fig. 36.1) is modeled using first using a simple elastic law for polyamide bars (Table 36.1). Figure 36.3 (left) points out the error introduced by the model for several back and forth of the incident wave. It corresponds to a distance covered by the incident wave of about 7.5 meters. Such distance may coincide with the length of an input bar using the pre-stretched set-up. Therefore, the viscous material nature is preponderant and can’t be approximate by a simple elastic law. Figure 36.3 (right) displays the excellent agreement of numerical and experimental results. This time a viscoelastic material model is chosen to simulate the

Table 36.1 Characteristics of dynamic experiments using polyamide SHPB apparatus Case 1 2 3 4

Configurations 1 1 2 2

Averaged striker speed [m/s] 12.2 9.8 9.2 14.75

0,004

Purpose Model identification Model validation Model validation Model validation

0,004 Experimental

0,003 Numerical elastic

0,001 0 -0,001

0,001 0

-0,001

-0,002

-0,002

-0,003

-0,003 0

1

2

Numerical viscoelastic

0,002

Strain [-]

Strain [-]

0,002

-0,004

Experimental

0,003

3

Time [ms]

4

5

6

-0,004

0

1

2

3

4

5

6

Time [ms]

Fig. 36.3 Experimental strain signals and numerical ones based on an elastic model (left) and a viscoelastic model (right). This test corresponds to the case 1

204

A. Bracq et al.

Table 36.2 Mechanical properties of the polyamide material identified for the modeling of the elastic and viscoelastic behavior

Density [kg/m3] E [MPa] 1150 3500 Maxwell model parameters G∞ [MPa] G1 [MPa] 55.54 G2 [MPa] 67.51 G3 [MPa] 85.76 G4 [MPa] 64.99

Nu [−] 0.4 976.2 β1 [s−1 ] β2 [s−1 ] β3 [s−1 ] β4 [s−1 ]

5.9 29 710 20,560

4

5

0,003 Experimental 0,002

Numerical viscoelastic

Strain [-]

0,001

0 -0,001 -0,002 -0,003 0

1

2

3

4

5

6

Time [ms] Fig. 36.4 Experimental and predicted numerical strain signals corresponding to the case 2

0,002

Experimental 0,002

Numerical viscoelastic

0,001

Numerical viscoelastic

0,001

0

Strain [-]

Strain [-]

0,003

Experimental

-0,001

0

-0,001 -0,002

-0,002

-0,003 -0,004

-0,003 0

1

2

3

Time [ms]

4

5

0

1

2

3

Time [ms]

Fig. 36.5 Experimental and predicted numerical strain signals corresponding to the case 3 (left) and 4 (right)

polyamide behavior and the model parameters are identified by means of the optimization process. Four Maxwell elements are sufficient to accurately describe the viscous material properties. Their values along with the G∞ value are indicated in Table 36.2. Second of all, the viscoelastic model of the bars is assessed by modeling three other impact configurations and comparing with experimental results (cases 2–4). It allows to evaluate the model ability to predict the wave attenuation and dispersion for various impact conditions. Figure 36.4 presents the results obtained for the case 2 demonstrating an excellent correlation of the numerical model with experimental results. It is a first step towards the model validation. To go further, the cases 3 and 4 are replicated numerically. The short length of the striker leads to a reduced pulse duration. Yet again, the numerical model is able to precisely predict the strain wave signals (Fig. 36.5).

36 Constitutive Modeling of Polyamide Split Hopkinson Pressure Bars for the Design of a Pre-stretched Apparatus

205

36.4 Conclusion A viscoelastic material model based on the generalized Maxwell model is identified for polyamide Split Hopkinson Pressure Bars. Polymeric bars are useful for the characterization of highly compliant materials. Indeed, low impedance properties and high incident wave rise time are achieved with such bars. However, the use of the pre-stretched technique applied to tensile tests at moderate up to high strain rates requires bars of several meters. Therefore, the wave dispersion and attenuation is predominant and complicate the design of new SHPB configurations. In a preliminary step, the design and optimization of the pre-stretched apparatus may benefit from the advantages of the numerical tools. To that end, an inverse approach is adopted based on compressive dynamic experiments using polyamide bars and their respective finite element models. The viscoelastic model parameters of the bars are identified through one configuration and validated against three others based on comparison with experimental strain signals. This model gives very satisfactory results and presents interesting predictive abilities. It may benefit the optimization of pre-stretched bars by adjusting bars diameter, sample dimensions and the pre-stretched load following sample mechanical properties and targeted strain/strain rates.

References 1. Kolsky, H.: An investigation of the mechanical properties of materials at very high rates of loading. Proc. Phys. Soc. Sect. B. 62(11), 676–700 (1949) 2. Van Sligtenhorst, C., Cronin, D.S., Wayne Brodland, G.: High strain rate compressive properties of bovine muscle tissue determined using a split Hopkinson bar apparatus. J. Biomech. 39(10), 1852–1858 (2006) 3. Salisbury, C.P., Cronin, D.S.: Mechanical properties of ballistic gelatin at high deformation rates. Exp. Mech. 49(6), 829–840 (2009) 4. Cronin, D.S.: Ballistic gelatin characterization and constitutive modeling. In: Proulx, T. (ed.) Dynamic Behavior of Materials, Volume 1: Proceedings of the 2011 Annual Conference on Experimental and Applied Mechanics, pp. 51–55. Springer New York, New York (2011) 5. Morin, D., Haugou, G., Bennani, B., Lauro, F.: Experimental characterization of a toughened epoxy adhesive under a large range of strain rates. J. Adhes. Sci. Technol. 25(13), 1581–1602 (2011) 6. Morin, D., Haugou, G., Lauro, F., Bennani, B., Bourel, B.: Elasto-viscoplasticity Behaviour of a structural adhesive under compression loadings at low, moderate and high strain rates. J. Dyn. Behav. Mater. 1(2), 124–135 (2015) 7. Bracq, A., Haugou, G., Delille, R., Lauro, F., Roth, S., Mauzac, O.: Experimental study of the strain rate dependence of a synthetic gel for ballistic blunt trauma assessment. J. Mech. Behav. Biomed. Mater. 72, 138–147 (2017) 8. Haugou, G., Leconte, N., Morvan, H.: Design of a pre-stretched tension Hopkinson bar device: Configuration, tail corrections, and numerical validation. Int. J. Impact Eng. 97, 89–101 (2016) 9. Haugou, G., Morvan, H., Leconte, N.: Direct compression loading using the pre-stretched bar technique: Application to high strains under moderate strain rates. In: Kimberley, J., Lamberson, L., Mates, S. (eds.) Dynamic Behavior of Materials, Volume 1, pp. 169–173. Springer International Publishing, Cham (2018) 10. Bustamante, M., Cronin, D.S., Singh, D.: Experimental testing and computational analysis of viscoelastic wave propagation in polymeric split hopkinson pressure bar. In: Kimberley, J., Lamberson, L., Mates, S. (eds.) Dynamic Behavior of Materials, vol. Volume 1, pp. 67–72. Springer, Cham (2018) 11. Bracq, A., et al.: Characterization of a visco-hyperelastic synthetic gel for ballistic impacts assessment. In: Kimberley, J., Lamberson, L., Mates, S. (eds.) Dynamic Behavior of Materials, Volume 1: Proceedings of the 2017 Annual Conference on Experimental and Applied Mechanics, pp. 109–113. Springer, Cham (2018) 12. Oliveira, I., Teixeira, P., Ferreira, F., Reis, A.: Inverse characterization of material constitutive parameters for dynamic applications. Procedia Eng. 114, 784–791 (2015)

Chapter 37

Investigating the Mechanical and Thermal Relationship for Epoxy Blends Michael Harr, Paul Moy, Timothy Walter, and Kevin Masser

Abstract The mechanical response of epoxy networks was investigated under uniaxial compression at low, intermediate, and high strain rates. These epoxy blends are tailored to achieve a broad range of glass transition temperatures. Previous studies have shown a correlation of the epoxies’ Tg in relationship to its ballistic performance as well as its mechanical properties at quasi-static rates. To better understand these phenomena, a MWIR camera was used to directly measure the transient surface temperature and determine temperature change. The extent and rate of deformation highly influences the flow stress behavior which coincides with the rise in the adiabatic temperatures, hence the thermal softening response. One intriguing aspect for results at rates 0.01/s – 0.1/s and higher reveals that the surface temperature continues to increase despite pressure ceasing. This would indicate the core temperatures are still gradually transferring to the surface. The experimental setup and results are discussed. Keywords Digital image correlation · Epoxy · Glass transition temperature · Hopkinson bar · Thermal imaging

37.1 Introduction Polymer epoxies are increasingly being used in the aerospace, transportation, construction, and defense industries for a variety of applications. Within these applications, materials are subjected to a large variety of complex dynamic loads within a wide range of operational temperatures. In defense, ballistic performance is of particular interest. Previous work has shown the ballistic performance of epoxies to be largely dependent on the temperature difference the material’s glass transition temperature (Tg ) and the temperature upon impact [1–4]. Materials with Tg closer to the operational temperature perform better. The mechanical response of polymers has been shown in previous work to be dependent on both the temperature of the material and the strain rate of loading [1, 3, 5–7]. In addition to these dependencies, it is well established that adiabatic heating occurs at higher strain rates due to the deformation of the material [3, 5–7]. Given the combination of the influence of temperature on the mechanical response of these materials, the heat generation occurring due to deformation, and their necessary operational temperature proximity to Tg for better ballistic performance, it is necessary to understand heat generation due to deformation of the material and how it affects the mechanical response.

37.2 Materials The epoxy resin examined in this study is diglycidyl ether of bisphenol A (DGEBA), i.e. EPON 825, supplied by MillerStephenson. Four different curing agents were examined: three polypropylene oxide based Jeffamine diamines supplied by Hunstman, with nominal molecular weights of 230 g/mol, 400 g/mol, and 2000 g/mol (D230, D400, and D2000 respectively), and 4,4 -methylenebis(cyclohexylamine) (PACM) supplied by Sigma Aldrich [2]. Formulations of the epoxy blends were created with the intent to provide a varying range in Tg as follows: DGEBA/D230 (Tg = 100 ◦ C), DGEBA/D400 (Tg = 50 ◦ C), DGEBA/PACM (Tg = 160 ◦ C), and DGEBA/PACM/D2000 (Tg = 80 ◦ C). The materials were molded into 12.2 mm diameter cylinders and machined to 12.2 mm in length to provide a diameter to length ratio of one.

M. Harr () · P. Moy · T. Walter · K. Masser Weapons and Materials Research Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, Scotland © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_37

207

208

M. Harr et al.

37.3 Methodology Quasi-static and intermediate strain rate experiments (10−4 /s – 100 /s) were performed using a servo-hydraulic Instron 1331 axial load frame along with a stainless steel compression test fixture from Wyoming Test Fixtures. A FLIR SC6100 MWIR camera was used to measure the transient surface temperature of the specimens. High strain rate experiments (>102 /s) were performed using a 6061 aluminum modified split-Hopkinson pressure bar. The working principle behind the split-Hopkinson pressure bar and one-dimensional wave theory are well documented [8, 9]. Pulse shaping was used to achieve dynamic equilibrium in the specimen and a near-constant strain rate. Both the incident and transmission bar were 19 mm in diameter and 6.1 m in length. Full field strain was determined using stereo digital image correlation with two Specialised Imaging Kirana high speed video cameras and a high contrast speckle pattern on the specimens.

37.4 Results and Discussion Experiments were successfully performed on four formulations of epoxy with varying Tg at low, intermediate, and high strain rates. A stress-strain plot for DGEBA/PACM/D2000, shown in Fig. 37.1, clearly shows the increase in yield strength with the increase in strain rate as well as the strain hardening effect. This is representative of the trend displayed by all four materials. Both the yield strength and elastic modulus increase with increasing strain rate. For the high strain rate experiments, the specimens almost completely elastically recover after unloading. At strain rates 0.01/s and higher for DGEBA/D230 and DGEBA/D400 and strain rates 0.1/s and higher for DGEBA/PACM and DGEBA/PACM/D2000, the surface temperature of the specimens was observed to continue to rise immediately after unloading where the pressure was no longer present. This leads to the conclusion that the higher internal temperatures continue to transfer to the surface of the material at a faster rate than heat dissipates to the surroundings. Figure 37.2 shows representative curves for the change in temperature for each material at the 0.1/s strain rate. The unloading event is visible in the sharp drop in temperature between 6 and 8 s. An interesting phenomenon is observed to occur in the D230 and D400 formulations at strain rates of 0.1/s and higher. Thermal images for DGEBA/PACM and DGEBA/PACM/D2000 show uniform heating, whereas DGEBA/D230 and DGEBA/D400 show heating along a 45◦ shear band. Figure 37.3 shows representative still images from a DGEBA/D400 specimen and a DGEBA/PACM/D2000 specimen at a strain rate of 0.1/s where the differences between the two heating modes can be observed. 160 140

True Stress (MPa)

120 100 80 60 40 20 0 0

0.1

0.2

0.3

0.4 0.5 True Strain

0.6

Fig. 37.1 True stress as a function of true strain for DGEBA/PACM/D2000 across a range of strain rates

0.7

0.8

0.9

37 Investigating the Mechanical and Thermal Relationship for Epoxy Blends

209

Change in Temperature (°C)

25

20

15

10

5

0 0

2

4

6

8

10

12

10

Time (s) Fig. 37.2 Change in surface temperature as a function of time for the four materials at 0.1/s strain rate

Fig. 37.3 Thermal IR still frames for DGEBA/D400 (a–c) and DGEBA/PACM/D2000 (d–f) at a strain rate of 0.1/s

Acknowledgements This research was supported in part by an appointment to the Postgraduate Research Participation Program at the U.S. Army Research Laboratory administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and USARL.

References 1. Knorr, D., Yu, J., Richardson, A., Hindenlang, M., McAninch, I., La Scala, J., Lenhart, J.: Glass transition dependence of ultrahigh strain rate response in amine cured epoxy resins. Polymer. 53, 5917–5923 (2012) 2. Masser, K., Knorr, D., Hindenlang, M., Yu, J., Richardson, A., Strawhecker, K., Beyer, F., Lenhart, J.: Relating structure and chain dynamics to ballistic performance in transparent epoxy networks exhibiting nanometer scale heterogeneity. Polymer. 58, 96–106 (2015)

210

M. Harr et al.

3. Whittie, S., Moy, P., Schoch, A., Lenhart, J., Weerasooriya, T.: Strain Rate Response of Cross-Linked Polymer Epoxies under Uni-Axial Compression. Proceedings of the 2011 Annual Conference on Experimental and Applied Mechanics 4. Whittie, S., Moy, P., Gunnarsson, C.A., Knorr, D., Weerasooriya, T., Lenhart, J.: Fracture Response of Cross-Linked Epoxy Resins as a Function of Loading Rate. Proceedings of the 2012 Annual Conference on Experimental and Applied Mechanics 5. Aruda, E., Boyce, M., Jayachandran, R.: Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mech. Mater. 19, 193–212 (1995) 6. Rittel, D.: On the conversion of plastic work to heat during high strain rate deformation of glassy polymers. Mech. Mater. 31, 131–139 (1999) 7. Nasraoui, M., Forquin, P., Siad, L., Rusinek, A.: Influence of strain rate, temperature, and adiabatic heating on the mechanical behavior of Poly-methyl-methacrylate: experimental and modelling analyses. Mater. Desig. 37, 500–509 (2012) 8. Chen, W., Lu, F., Zhou, B.: A quartz-crystal-embedded split Hopkinson pressure bar for soft materials. Exp. Mech. 40(1), 1–6 (2000) 9. Chen, W., Song, B.: Split Hopkinson (Kolsky) Bar Design, Testing, and Applications. Springer, Boston (2011)

Chapter 38

A Novel Auxetic Structure with Enhanced Impact Performance by Means of Periodic Tessellation with Variable Poisson’s Ratio M. Taylor, L. Francesconi, A. Baldi, X. Liang, and F. Aymerich

Abstract This study proposes a new approach to designing impact resistant elastomeric structures using innovative bidimensional patterns composed of a combination of circular and elliptical voids with variable aspect ratios. Key to the design are discrete sections each with different effective Poisson’s ratios ranging from negative to positive. Cubic samples 80 × 80 × 80 cm in size with different void geometry and effective Poisson’s ratios were fabricated and successively tested under compressive and low-velocity impact loads as a proof-of-concept, showing good agreement with finite element simulations. The numerical comparisons for different porosity levels demonstrated that the variable Poisson’s ratio materials resulted in better impact responses compared to those characterized by a positive (constant) value of the effective Poisson’s ratio. The promising results also show that the variable shape of the voids can lead to a modular trigger of overall effective auxetic behavior, opening up new ways design and use auxetic macro-structures with variable porosity and variable Poisson’s ratio for a wide range of applications and, in particular, for impact and protecting devices. Keywords Auxetic structures · Impact · Variable Poisson’s ratio · Digital image correlation · Buckling

38.1 Introduction Most common engineering materials have a positive Poisson’s ratio (PR), which is defined as the ratio of lateral strain to an axially applied strain [1–3]. Positive PR materials expand laterally when compressed. While materials with negative PR are theoretically possible, they are uncommon engineering application. Such materials (i.e., auxetics, from the Greek “auxeticos” -which tends to increase) will contract laterally when compressed. Starting with the pioneering work of Lakes [4], so-called auxetic metamaterials have been discovered and analyzed for a range of applications [5–7]. Metamaterials derive their effective (i.e., homogenized) properties from a combination geometric design and bulk material properties. Recent research has included using auxetic structures to extend the fatigue life of ductile thin low-porosity samples [8] along with the design of auxetic foams [9–15], two-dimensional (2D) periodic [16–18] and 3D [16, 19] materials, metallic structures [20, 21], films [22], zeolites [23], composites [24], lattices [25], medical devices [26–28], cushions [10], filters [29], novel textiles [30], pressure vessels [31], perforated structures for structural applications [20, 32–34], and shock absorbers [35–37]. While auxetic materials have shown excellent dynamic response and good shock absorption capabilities in different applications [6, 11–14, 19, 35–37], very few investigations have been conducted focusing on compressive and impact load cases and, to the authors’ knowledge, no investigations have yet been dedicated considering metamaterials with variable Poisson’s ratio. Such structures are the focus of this work. The primary design goal of Variable Poisson Ratio (VPR) structures is to take advantage of the unique benefits both auxetic and non-auxetic materials provide during impact loading. An auxetic material tends to contract into the region of impact acting to arrest the penetration of that impact. This comes with a downside—this contraction acts to concentrate the impact forces over a decreasing area. On the other hand, non-auxetic materials spread those impact forces out and away from the impact region. By layering these different behaviors, we can have auxetic regions near the point of impact (where projectile arresting is important) and non-auxetic structures farther away (where decreasing stress is more important). M. Taylor · L. Francesconi () · X. Liang Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA, USA e-mail: [email protected] A. Baldi · F. Aymerich Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università degli Studi di Cagliari, Cagliari, Italy © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_38

211

212

M. Taylor et al.

The possibility of tuning the auxetic properties of perforated high and low porosity structures by varying the aspect ratio of an array of elliptical voids was suggested recently by Bertoldi et al. [16], Taylor et al. [20] and Francesconi et al. [32]. In this paper, we use this concept to modify the mechanical response of high porosity (15–40% of overall porosity) polymers by gradually modifying the PR from positive to negative values. We investigate the behavior of several VPR geometries via numerical simulation using the commercial finite element code Abaqus/Standard (Simulia, Rhode Island, USA). In addition to numerical simulation, we perform quasi-static compression and low-velocity impact experimental testing of silicone rubber samples to compare the behavior of VPR structures to those with uniform PR. We also use digital image correlation (DIC) to compare the experimental and the numerical results.

38.2 Material and Testing Methods 38.2.1 Material Design and Fabrication For this study, we consider an elastomeric material with simple three-section VPR structures made up of two-dimensional patterns of auxetic, neutral (i.e., zero Poisson ratio), and non-auxetic (i.e, circular void) layers. Figure 38.1 shows a typical VPR structure: cubic samples of 80 × 80 × 80 mm with varying periodic tessellations that go through the whole thickness. All of the structures analyzed have a fixed void center-to-center distance of 10 mm in both planar dimensions; although, we considered porosities ranging from 5% to 40% and different Poisson’s ratios in the elliptical void sections (i.e. Sects. 38.1 and 38.2 in Fig. 38.1) to analyze their performance in compression and impact loading. Different overall porosities were achieved by simply varying the geometrical dimensions of the pattern’s features. The modification in the Poisson’s ratios of these sections is obtained by changing the aspect ratio (i.e., ratio of the major to minor axis length) of the ellipses while keeping porosity fixed. The effect of the interaction of the three sections is visible in Fig. 38.2 where three consecutive frames were captured during a compression test (described in the next section): under the action of the impactor the first section to be solicited is the auxetic one (top) that immediately collapse causing a densification of the structure. This contracting auxetic region (Sect. 38.1) slows the impactor and transmits the load to Sects. 38.2 and 38.3 (with neutral and positive Poisson’s ratio, respectively), which distribute the load and dissipate its residual energy by an expansion of the bottom part of the structure. The role of the auxetic upper layer is, as visible in Fig. 38.2, to focus the deformation toward the loading axis and, by doing that, slowing down the motion of the impactor, avoiding the material expansion that is a common problem when indenting or impacting material with a positive value of the Poisson’s ratio. To test the VPR structures in compressive and impact loading, we fabricated samples using “TAP Platinum Silicone” a silicone rubber with excellent tear resistance, high flexibility, low shrinking after curing (below 0.01%) and very low viscosity, that vulcanizes at room temperature without the need of a de-air stage or additional procedures. The material properties are summarized in Table 38.1.

Fig. 38.1 An example of a three-section VPR geometry (overall dimension 80 × 80 mm) with superimposed finite element mesh

38 A Novel Auxetic Structure with Enhanced Impact Performance by Means of Periodic Tessellation with Variable Poisson’s Ratio

213

Fig. 38.2 Consecutive frames of a compression test of a silicone elastomeric variable Poisson’s ratio structure (overall porosity = 30%, maximum applied load = 2250 N) Table 38.1 Principal chemical and physical characteristics of the TAP Platinum Silicone

Mixing Ratio Tensile Strength Elongation at break Viscosity pre-catalisation Working time (at 23◦ ) Setting time (at 23◦ )

1:1 1.5 MPa 280% 900–1100 cP 15–18 min 2h

Fig. 38.3 Sample fabrication and impactor. The modular mold fabricated by 3D printing (a), the casting of the silicone VPR structure with 30% of porosity, the sample with circular holes (c) and the squared tip of the impactor (d) used to impact the samples

Figure 38.3a shows the 3D printed mold used to fabricate the samples and two coupons with an overall porosity of 30%; the one of Fig. 38.3c, characterized by circular voids, has a positive Poisson’s ratio while the second one (shown within the mold in Fig. 38.3b) presents a variable Poisson’s ratio (VPR) made by means of three cells with different tessellations.

38.2.2 Numerical Method VPR structures were numerically tested (in static compression) to determine their effective (i.e., homogenized) mechanical properties and the reciprocal interactions between the sections for different levels of porosity. These results were compared with a positive PR benchmark, composed of uniform circular voids with the same overall porosity. The structures were modeled using a single layer of four-node, quadratic two-dimensional solid elements with reduced integration (Abaqus element type CPE4R), and Abaqus’ built-in automatic adaptive mesh refinement algorithm was used to refine the mesh around the complex geometry of the samples and accurately predict the material response near the voids. The final meshes resulted in approximately 100,000 elements (Fig. 38.1 shows a typical mesh) and the experimental compression is approximated via a static displacement step. The finite element simulations were then used to estimate the effective Poisson’s ratio for all the samples considered. To do this, we focused on a squared portion of 80 mm by 80 mm along the perimeter of the samples, extracting the horizontal and the vertical displacement components (respectively, εxx  and εyy ) along the 4 edges using an arithmetic mean of all the considered nodes. Then, we calculated the 4 “averaged” displacements ux R , ux L , uy U , uy B (the superscripts

214

M. Taylor et al.

“L”, “R”, “U”, and “B” are intended for the left, right, upper, and bottom boundaries, respectively). Afterwards, we used the average displacements to obtain the local strain averages εxx , εyy  and computed the effective Poisson’s ratio according to the Eqs. (38.1, 38.2, and 38.3).

εxx  =

(

)

ux R − ux L L0 (

εyy =

uy

)U

( )B − uy L0

εxx  υ=− ( ) εyy

(38.1)

(38.2)

(38.3)

38.2.3 Experimental Testing Method The static tests were performed using a hydraulic MTS LandMark 370 universal testing machine equipped with a 100 kN load cell in load-control mode. During the compression tests, multiple images were taken at 8 bits and at regular intervals of 50 N (for a total deflection of 250 mm) in order to run DIC (Digital Image Correlation) analyses. The experimental setup used for DIC consists in a monochromatic Allied Vision F421-B Pike high-resolution camera equipped with a Schneider Kreuznach Macro camera lens and in an isotropic spot-free professional illuminator. After the compression tests, the samples were impacted at low-velocity using an instrumented drop-weight impact machine equipped with an instrumented square-ended tip of 80 × 80 mm and a total mass of 5.32 kg. During the impacts the samples were simply supported on a steel plate and the impact force between the impactor and the sample was measured by a semiconductor strain-gauge bridge bonded to the ended rod of the impactor, while the velocity of the impactor, immediately before the collision and at the rebound, was extracted by an infra-red sensor. The histories of the velocity and of the displacement of the impactor were finally calculated by double integration of the contact force signal.

38.3 Results Following the general concept presented in Fig. 38.1, an extensive finite element analysis was performed to characterize the mechanical response of several VPR samples and to better understand the complex interaction between the sections with different Poisson’s ratio. Figure 38.4 shows the U1 (horizontal) displacement generated by a uniaxial compression of three samples with porosities of respectively, 5% (Fig. 38.4a), 15% (Fig. 38.4b), and 30% (Fig. 38.4c). While the periodic patterns are similar, the ways that the structures deform are different. The samples with the 5% and the 15% of porosity effectively expand when compressed (considering the U1 displacements along the two lateral sides of Fig. 38.4a, b), although not uniformly along the vertical axis. On the other hand, the structure with the highest porosity (30%, Fig. 38.4c) shows a contraction of the top-mid section (aspect ratio of the ellipses = 5) while the mid-bottom section still responds, as expected, with an expansion. It is also clear from the samples presented in Fig. 38.4 that, by increasing only the porosity, it’s possible to control the effective Poisson’s ratio of the VPR structures without modifying the pattern of voids. Table 38.2 reports the value of the effective PR calculated at the different sections in the structures as well as the overall effective Poisson’s ratio for different porosities. When the porosity reaches a critical value of approximately 20%, the material changes behavior, transitioning from effectively non-auxetic to auxetic. This trend, that shows an overall reduction of the PR when increasing the porosity, was further confirmed by the tests performed on higher porosity structures. The zoomed plots of Fig. 38.5 show the two deformed configurations to better understand the changes in the response of the samples in the transition from the positive to the negative Poisson’s ratio, when the porosity increased from 15% (Fig. 38.5a) to 30% (Fig. 38.5b).

38 A Novel Auxetic Structure with Enhanced Impact Performance by Means of Periodic Tessellation with Variable Poisson’s Ratio

215

Fig. 38.4 Horizontal (direction 1) X displacement maps of the low-porosity samples under a compression load directed along Y for 3 different levels of overall porosity (starting from left to right: 5, 15 and 30%) Table 38.2 Poisson’s Ratio of the different sections for the different porosities considered in the study

Porosity 5% 10% 15% 20% 25% 30% 35%

PR – Sect. 38.3 (Bottom) Circle, AR =1

PR – Sect. 38.2 (Mid) Ellipsis, AR =3

PR – Sect. 38.1 (Top) Ellipsis, AR =5

Overall Effective PR (Displacement = 5 mm)

0.3 0.29 0.28 0.27 0.25 0.23 0.21

0.27 0.2 0.1 −0.03 −0.19 −0.37 −0.54

0.21 0.04 −0.18 −0.43 −0.64 −0.81 −0.92

0.785 0.624 0.376 0.07 −0.239 −0.61 −0.798

Fig. 38.5 Horizontal (direction 1) X displacement maps of two low-porosity samples under a tensile load directed along Y

After using the finite element simulations to identify the mechanical responses of several specimens, two VPR structures (with 15% and 30% of porosity) and one structure with constant (positive) PR and 30% of porosity (obtained cutting a pattern of circular voids), were manufactured and tested. The experimental testing consisted of uniaxial quasi-static compression tests and low velocity-impact tests performed at different impact energies. During the compression tests, several pictures were taken at increasing load levels to run a DIC analysis and calculate the strain contour map over the samples’ surfaces. The obtained experimental planar Lagrangian strains for different level of compression are presented in Fig. 38.6.

216

M. Taylor et al.

Fig. 38.6 Contour of the horizontal Lagrangian Strain (DIC) obtained during quasistatic compression testing for different displacements and different porosities

Figure 38.6 shows the DIC deformation maps for three of the considered structures at compressions ranging from 0 to 15 mm. The top row (Fig. 38.6a–c) shows the deformation of the sample with a uniform pattern of circular holes, clearly demonstrating its corresponding uniformly non-auxetic behavior by expanding laterally in response to the compressive loading. The second row (Fig. 38.6d–f) shows a VPR material with a positive effective Poisson’s ratio. Despite also being non-auxetic, its response is markedly different than that of the circular void sample, owing to the varying PR of its individual layers. The deformation is concentrated just in the top layer at the beginning of the compression test (which is characterized by a negative Poisson’s ratio of −0.18 (see Table 38.2) and, after increasing the load, the collapse of the top layers lets the deformation move to the central portion of the sample (displacement = 10 mm) and, finally, to the bottom layer (displacement =15 mm). The situation is completely different from the last case (Fig. 38.6g–i) where the VPR sample with the 30% of porosity is considered. Following the numerical prediction, the VPR sample has global effective auxetic behavior and, in particular, the top and the mid-section, both auxetic (PR = −0.37 and PR = −0.8, respectively) collapse immediately towards the loading axis for small compression and only at the end of the test (for an applied of displacement 15 mm) the load starts to affect the bottom section by distributing the strains and causing an expansion. The force deflection curves for the three porous samples together with a non-porous benchmark specimen are shown in Fig. 38.7. All samples experience a nonlinear response with stiffness generally increasing as a function of displacement. The notable exception is the circular void sample, which experiences a global buckling instability at a displacement of about 16 mm (Fig. 38.7b). In practical applications this could be a problem because the onset of buckling is sudden and dramatic, which could lead to catastrophic failure of components. The VPR structures introduced in this study seem, from our preliminary results, to be immune to this type of instability at the loadings considered. This suggests a possible side benefit to using VPR materials over non-auxetic porous structures in applications where the compression behavior is crucial. In addition to quasi-static compression testing, the samples were impacted at different energies with an instrumented impactor. The force-displacement response of the circular void sample and the VPR with the 30% of porosity are reported in Fig. 38.8. The experimental curves are similar especially for the lower energy regime. At an impact energy of 35.5 J, we see that the VPR sample has a slightly lower peak force and also a more regular increase of the force than the circular void sample. This fact can be attributed to the more gradual way the VPR structures absorb the loads owing to its varied section PRs. We also see that the VPR samples experience less displacement overall in response to impact, which support our initial hypothesis that VPR structures would be beneficial for arresting impacts.

38 A Novel Auxetic Structure with Enhanced Impact Performance by Means of Periodic Tessellation with Variable Poisson’s Ratio

217

Fig. 38.7 Force-Displacement curves obtained during the compression test. The two magnified boxes, immediately (a) before and (b) after the displacement of 15 mm, show the elastic instabilities of the sample with the circular voids

Fig. 38.8 Force-Displacement curves of the samples with variable Poisson’s ratio and the one with the pattern of circular voids impacted at 15.5 and 35.5 J

38.4 Summary and Conclusions In this study, we have proposed innovative material structures with variable Poisson’s ratio. The VPR materials are characterized by layers, or sections, of elliptical voids each with a unique PR. This design allows the material to densify in some regions while expanding in others when acted upon by compressive or impact loads, taking advantage of the positive aspects of both kinds of behavior. In order to analyze the behavior of VPR materials and prove the concept, we performed both experiments and numerical simulations on candidate three-section VPR geometries in a silicone elastomer. Finite element simulations show that the considered materials have an overall non-auxetic behavior at porosities less than 20%. Above this porosity, the materials are effectively auxetic, despite having a non-auxetic layer. Experimental results are promising, showing that the VPR materials experience a smoother deformation with stiffer response than corresponding non-VPR structures. Moreover, VPR structures appear to be resistant to the global buckling instability experienced by corresponding materials with uniform circular voids. DIC analysis of the strain contours compare favorably with finite element results. We anticipate that the proposed structures will provide benefits over uniformly porous materials where impact or compressive resistance is crucial or where global buckling must be avoided. Areas for future investigation include a more detailed parameter study including more layers and a wider range of porosities and void shapes in order to better understand VPR behavior and determine optimal designs. In addition, it would be interesting to analyze the response of VPR structures to ballistic impact to see how auxetic layers may assist in arresting projectiles.

218

M. Taylor et al.

References 1. Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, vol. 7. Pergamon Press, Oxford (1970) 2. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944) 3. Ashby, M.F., Jones, D.R.H.: Engineering Materials 1: an Introduction to their Properties and Applications. Butterworth Heinemann, Oxford (1996) 3. Lakes, R.: Advances in negative Poisson’s ratio materials. Adv. Mater. 5, 293–296 (1993) 4. Greaves, G.N., Greer, A.L., Lakes, R.S., Rouxel, T.: Poisson’s ratio and modern materials. Nat. Mater. 10, 823–837 (2011) 5. Prawoto, Y.: Seeing auxetic materials from the mechanics point of view: a structural review on the negative Poisson’s ratio. Comput. Mater. Sci. 58, 140–153 (2012) 6. Stavroulakis, G.E.: Auxetic behaviour: appearance and engineering applications. Phys. Status Solidi B. 242(3), 710–720 (2005) 7. Evans, K.E., Alderson, A.: Auxetic materials: functional materials and structures from lateral thinking. Adv. Mater. 12(9), 617–628 (2000) 8. Javid, F., Liu, J., Rafsanjani, A., Schaenzer, M., Pham, M.Q., Backman, D., Yandt, S., Innes, M.C., Booth-Morrison, C., Gerendas, M., Scarinci, T., Shanian, A., Bertoldi, K.: On the design of porous structures with enhanced fatigue life. Extreme Mech. Lett. 16, 13–17 (2017) 9. Wojciechowski, K.W., Branka, A.C.: Negative Poisson ratio in a two-dimensional “isotropic” solid. Phys. Rev. Lett. A. 40, 7222–7225 (1989) 10. Lakes, R.S., Lowe, A.: Negative Poisson’s ratio foam as seat cushion material. Cell. Polym. 19, 157–167 (2000) 11. Lakes, R.: Foam structures with a negative Poisson’s ratio. Science. 235, 1038–1040 (1987) 12. Scarpa, F., Giacomin, J., Zhang, Y., Pastorino, P.: Mechanical performance of auxetic polyurethane foam for antivibration glove applications. Cell. Polym. 24(5), 1–16 (2005) 13. Bianchi, M., Scarpa, F.L., Smith, C.W.: Stiffness and energy dissipation in polyurethane auxetic foams. J. Mater. Sci. 43(17), 5851–5860 (2008) 14. Chan, N., Evans, K.E.: Fabrication methods for auxetic foams. J Mater Sci. 32(22), 5945–5953 (1997) 15. Chan, N., Evans, K.E.: Microscopic examination of the microstructure and deformation of conventional and auxetic foams. J. Mater. Sci. 32(21), 5725–5736 (1997) 16. Bertoldi, K., Reis, P.M., Willshaw, S., Mullin, T.: Negative Poisson’s ratio behavior induced by an elastic instability. Adv. Mater. 22(3), 361–366 (2010) 17. Kureta, R., Kanno, Y.: A mixed integer programming approach to designing periodic frame structures with negative Poisson’s ratio. Optim. Eng. 15(3), 773–800 (2014) 18. Wojciechowski, K.W.: Two-dimensional isotropic system with a negative Poisson ratio. Phys. Lett. A. 137(1–2), 60–64 (1989) 19. Shim, J., Perdigou, C., Chen, E.R., Bertoldi, K., Reis, P.M.: Buckling-induced encapsulation of structured elastic shells under pressure. Proc. Natl. Acad. Sci. U. S. A. 109, 5978–5983 (2012) 20. Taylor, M., Francesconi, L., Gerendás, M., Shanian, A., Carson, C., Bertoldi, K.: Low porosity metallic periodic structures with negative poisson’s ratio. Adv. Mater. 26(15), 2365–2370 (2013) 21. Milstein, F., Huang, K.: Existence of a negative Poisson ratio in fcc crystals. Phys. Rev. B. 19, 2030–2033 (1979) 22. Ravirala, N., Alderson, A., Alderson, K.L., Davies, P.J.: Auxetic polypropylene films. Polym. Eng. Sci. 45(4), 517–528 (2005) 23. Grima, J.N., Jackson, R., Alderson, A., Evans, K.E.: Do zeolites have negative Poisson’s ratios? Adv. Mater. 12(24), 1912–1918 (2000) 24. Herakovich, C.T.: Composite laminates with negative through-the-thickness Poisson’s ratios. J. Compos. Mater. 18(5), 447–455 (1984) 25. Doyoyo, M., Wan Hu, J.: Plastic failure analysis of an auxetic foam or inverted strut lattice under longitudinal and shear loads. J. Mech. Phys. Solids. 54(7), 1479–1492 (2006) 26. Ali, M.N., Rehman, I.U.: An Auxetic structure configured as oesophageal stent with potential to be used for palliative treatment of oesophageal cancer; development and in vitro mechanical analysis. J. Mater. Sci. Mater. Med. 22(11), 2573–2581 (2011) 27. Caddock, B.D., Evans, K.E.: Negative Poisson ratios and strain-dependent mechanical properties in arterial prostheses. Biomaterials. 16, 1109–1115 (1995) 28. Dolla, W.J.S., Fricke, B.A., Becker, B.R.: Structural and drug diffusion models of conventional and Auxetic drug-eluting stents. J Med Devices. 1, 47–55 (2007) 29. Alderson, A., Rasburn, J., Evans, K.E., Grima, J.N.: Auxetic polymeric filters display enhanced de-fouling and pressure- compensation properties. Membr. Technol. 137, 6–8 (2001) 30. Alderson, A., Alderson, K.: Expanding materials and applications: exploiting auxetic textiles. Tech. Text. 14, 29–34 (2005) 31. Ellul, B., Muscat, M., Grima, J.N.: On the effect of the Poisson’s ratio (positive and negative) on the stability of pressure vessel heads. Phys. Status Solidi B. 246(9), 2025–2032 (2009) 32. Francesconi, L., Taylor, M., Bertoldi, K.: Baldi, static and modal analysis of low porosity thin metallic Auxetic structures using speckle interferometry and digital image correlation. Exp. Mech. 58(2), 283–300 (2018) 33. Grima, J.N., Evans, K.E.: Auxetic behavior from rotating squares. J. Mater. Sci. Lett. 19(17), 1563–1565 (2000) 34. Grima, J., Gatt, R.: Perforated sheets exhibiting negative poisson’s ratios. Adv. Eng. Mater. 12, 460–464 (2010) 35. Sanami, M., Ravirala, N., Alderson, K., Alderson, A.: Auxetic materials for sports applications. Procedia Eng. 72, 453–458 (2014) 36. Wan, H., Ohtaki, H., Kotosaka, S., Hu, G.: A study of negative Poisson’s ratios in auxetic honeycombs based on a large deflection model, European journal of mechanics. Eur. J. Mech. A. Solids. 23(1), 95–106 (2004) 37. Scarpa, F., Panayiotou, P., Tomlinson, G.: Numerical and experimental uniaxial loading on in-plane auxetic honeycombs. J. Strain Anal. Eng. Des. 35(5), 383–388 (2000)

Chapter 39

On the Response of Polymer Bonded Explosives at Different Impact Velocities Suraj Ravindran, Addis Tessema, and Addis Kidane

Abstract Multiscale experiments are performed to understand the deformation mechanisms in polymer bonded explosives at a range of impact velocities. The experiments are conducted in a direct impact configuration, where a polycarbonate projectile is directly shot onto polymer bonded sugar samples at different impact velocities. During the deformation process, the images are captured at five million frames/second using an ultrahigh speed camera. For the macro scale experiment, the images are captured at a resolution of 75 μm/pixel and for the mesoscale experiments, the magnification factor is 10 μm/pixel. The deformation field is obtained using digital image correlation technique. From the macroscale displacement field, the spatial stress distribution is calculated using a nonparametric method. The meso-scale experiments are used to explain the deformation mechanisms observed at the macroscale. Keywords Compaction · PBX · direct impact · shock

39.1 Introduction The accidental detonation of polymer bonded explosives (PBX) is an unresolved problem in energetic material research [1, 2]. It is shown that the accidental explosions are mainly due to the formation high-temperature regions called hot spots in the material. The spatial length scale of the hot spot formation is in the order of micrometers. It is believed that the local failure in the material is responsible for the formation of hotspots. However, the experiments that show the dynamics of the local deformation and failure is very limited due to the requirement of high spatial and temporal resolution for such experiments. High spatio-temporal resolution experiments based on digital image correlation (DIC) are recently developed to understand the mechanism of deformation in heterogeneous materials [3–5]. This experimental method has potential to study the stress wave propagation in material at different spatial length scales. In this study, results from the direct impact experiments performed in polymer bonded sugar, a mechanical simulant of PBX are presented. The main objective of the work is to understand compressive stress wave propagation in PBS at macro and meso-scale in order to understand the multiscale dynamics of the wave propagation. The spatial stress profile is estimated from the displacement field obtained using digital image correlation in macroscale experiment.

39.2 Materials and Methods 39.2.1 Material Preparation Polymer bonded sugar (PBS), a mechanical simulant of PBX was used in the present study. PBS has been extensively used as the simulant material of high melting explosive (HMX) based polymer bonded explosives. However, the failure of the HMX crystal is not confirmed to be same. The constituents of the PBS sample are shown in Table. 39.1. The specimen is prepared by pressing the molding powder in a cylindrical mold of 25.4 mm inside diameter. For the sake of brevity, a complete procedure of specimen preparation is repeated here, the detailed procedure of making the samples are given in [6]. The final samples are of cylindrical shape, therefore, rectangular samples were extracted and machined to final dimensions

S. Ravindran () · A. Tessema · A. Kidane Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_39

219

220

S. Ravindran et al.

Table 39.1 Material constituents and composition Material constituents Sugar Crystal Hydroxyl-terminated polybutadiene (HTPB) Di-octyl Sebacate (DOS) Toluene diisocyanate (TDI)

Composition (% of weight) 87.5 9.00 2.40 1.10

Fig. 39.1 Schematic and complete experimental setup

using a milling machine. Samples used in this study has a density of 1.34 gm/cm3 . While the theoretical density of the sample was close to 1.46 gm/cm3 . Therefore, the material contains approximately 9% of voids. To facilitate the specimen surface for the DIC measurements, appropriate speckles were applied on the sample. For the macroscale experiments, an average speckle size of 270 μm was used. These speckles were applied to the sample by using an airbrush. The speckles for the meso-scale experiments were applied on the specimen following the procedure, first, a thin layer of white paint was applied, before the complete drying of the paint, toner powder was sprayed. After the complete drying of the paint, compressed air was applied to the sample to remove the particles that do not adhere to the paint. A complete schematic of the experimental setup used in this study is shown in Fig. 39.1. The experiment was conducted by shooting a foam backed polycarbonate flyer projectile of 10 mm thickness and 25.4 mm diameter on to the specimen. The experimental setup consists of a compressed helium operated gas-gun for propelling the flyer, and an imaging setup to capture the images of the deforming sample. The velocity of the projectile is adjusted by changing the pressure of the helium in the gas-gun. Imaging setup comprises of a high framing rate camera (Shimadzu, HPV-X2), Flash light, far-field microscope, tikona lens and computer to control the camera. High speed camera used in this study is capable of capturing images at a varying framing rate with a maximum framing rate of five million/second. The resolution of the images is 400 × 250 pixel2 at all framing speeds. In this study, for the macroscale experiment, a framing rate of five million/second was used. Whereas, in the meso-scale experiment, the maximum framing rate of five million/second was used. The optics used for the macroscale experiment was a 100 mm Tikona lens. The file of view of macroscale experiment was 30 × 8.75 mm2 . While the mesoscale experiment utilizes a high magnification extension tube from Navitar. Field of view of the meso-scale experiment was 4.26 × 2.5 mm2 at a spatial resolution of 10 μm/pixel. The post-processing of the images was performed in Vic-2D software from correlated solutions. A subset size of 13 × 13 pixel2 with a step size 1 was used for the macroscale experiment. While the post-processing of the meso-scale experiment was performed by using a subset size of 9 × 9 pixel2 with a step size of 1.

39.2.2 Full-Field Stress Calculation The spatial stress wave profile across the specimen during compressive wave propagation is estimated by calculating the inertia component of the stress. A similar approach can be seen in [7–9]. Inertia stress is calculated by using the following procedure. Instantaneous density field is calculated by using the full-field displacement field obtained from the experiment. Consider, the initial density of the specimen as ρ0 , then the density of the material at any time is related to the initial density by the following equation,

39 On the Response of Polymer Bonded Explosives at Different Impact Velocities

221

a

b

d

c

Fixed end

Flyer

Fig. 39.2 Schematic of the specimen impacted at the left end of the sample

X1=0

ρ0 = Jρ

(39.1)

Where J is the Jacobian at a point at any time t and it is calculated by J = detF, F is the deformation gradient and it is calculated from the displacement field obtained from DIC. For small strain, the Jacobian is calculated as follows, J = det F ≈ (1 + ε11 ) (1 + ε22 ) (1 + ε33 )

(39.2)

Compression wave stress before the wave reaches the support end of the specimen can be calculated using the acceleration field obtained from the full-field displacement. A brief description of the method is outlined below. Figure 39.2 shows a specimen impacted by a projectile at the left end (X1 = l mm) of the specimen. The DIC measurement is performed on the face abcd. However, the data for calculating the stress is taken in the middle of the sample (strip highlighted) in order to reduce the effect of lateral relief wave. The linear momentum conservation in Lagrangian description neglecting the body force term can be written as * ∂u ** ∇ • P = ρ0 * ∂t X=cons tan t

(39.3)

Where P is the First Piola Kirchoff’s stress and is the Lagrangian acceleration. The expansion of the Eq. 39.3. gives,   A

 ∂P11 (X1 , t) ∂P12 (X1 , t) ∂P13 (X1 , t) ∂u1 dX 2 dX 3 = ρ0 dX 2 dX 3 + + ∂X1 ∂X1 ∂X1 ∂t

(39.4)

A

Neglecting the shear terms in Eq. 39.4, the axial First Piola stress can be calculated as, X1 P11 (X1 , t) =

ρ0

∂u1 dX 1 ∂t

(39.5)

0

In order to calculate the true stress (Cauchy’s stress), First Piola stress has to be transformed to the deformed coordinates incorporating change in the volume which can be done by the following general equation, σ = J −1 F T P

(39.6)

The true stress can be calculated using the following equation, σ11 (X1 , t) =

P11 (X1 , t) (1 + ε22 (X1 , t)) ((1 + ε33 (X1 , t))

(39.7)

Where ε22 (X1 , t) can be obtained from DIC. However, the DIC is based on surface deformation, therefore, out of plane strain ε33 (X1 , t) cannot be measured. Assuming isotropy, the strain ε22 (X1 , t) at the top edge of the sample is used as ε33 (X1 , t). In short, digital image correlation is used calculate the displacement at equal intervals. Once we obtain the displacement field, the acceleration field is calculated by finding second time derivative of the displacement by numerical differentiation.

222

S. Ravindran et al.

After that, the acceleration field is spatially averaged in a small strip at the center of the sample in X2 -direction assuming one-dimensional wave propagation in the material.

39.3 Results and Discussion 39.3.1 Macroscale Axial Strain Field The axial strain contour plots at different times for two impact velocities are shown in Fig. 39.3. It is clear that a stress front that separates an unstressed and stressed region in the material. It is presumable that pores in the specimen are closed which might be forming a compaction type wave that densifies the sample upon propagation. The stress front in the higher impact velocity loading is appeared to propagate faster compared to the lower impact velocity scenario. The wave propagating across the specimen is seen to be slightly non-planar and rough in nature. This may be either due to microstructural heterogeneity or due to lateral wave release that reaches the center of the sample.

39.3.2 Spatial Stress Profile Spatial stress profile is calculated using Eq. 39.7 and plotted in Fig. 39.4. Figure 39.4a shows the spatial stress profile at different times for the impact velocity of 95 m/s. Maximum stress at the impact end is close to 80 MPa and it is seen to be decreasing with time. It is possible due to the failure or the arrival of the lateral relief at the center of the sample. Figure 39.4b shows the comparison of the spatial stress profile at two impact velocities. At the higher impact velocity, the stress at the impact end is significantly higher compared to the stress for lower impact velocity experiment. Spatial stress profile two impact velocities are shown in Fig. 39.5. It is clear that the stress at the impact end is higher for the higher particle velocity compared to the lower particle velocity.

39.3.3 Mesoscale Deformation Mechanisms The mesoscale axial strain contour and the underlying microstructure is shown in Fig. 39.5 for the impact velocities of 68 and 90 m/s. It is apparent that the axial strain field is highly heterogeneous for both the impact velocities. Interestingly, the axial strain localization is predominantly concentrated in the polymer binders with very less amount of crystals deformation.

Fig. 39.3 Full-field axial strain with time

39 On the Response of Polymer Bonded Explosives at Different Impact Velocities

b t=12 µs

90

90

70

70

95 m/s

50

70 m/s

50

6µs

8µs

30

10µs 12µs

10 -10

Stress (Mpa)

Stress (MPa)

a

223

0

14µs

2.5 5 7.5 10 Axial Lagrangian coordinate (mm)

-30

30 10 -10

0

2.5

5

7.5

10

Axial Lagrangian coordinate (mm) -30

Fig. 39.4 Spatial stress profile, (a) Spatial stress profile at different times for 95 m/s impact, (b) Comparison of spatial stress profile for the impact velocities 95 and 70 m/s, respectively

Fig. 39.5 The meso-scale local strain field and corresponding microstructure

However, the crushing of the crystals is apparent in both the loading cases, therefore for the impact velocities considered, the mechanism of stress transmission is same. It can be expected to have a critical velocity at which the crystal crushing dominates the stress wave propagation in the material. In order to find that critical velocity, experiments at a wide range of impact velocities are required.

39.4 Summary An experimental setup was developed to perform the high spatio-temporal deformation measurements in heterogeneous material. In this study, the multiscale compressive wave propagation in polymer bonded sugar is studied at two impact velocities. It is seen that a compaction type wave propagates across the material. The mesoscale experiment shows that the main local deformation mechanisms in both the impact velocities are the binder deformation and crystals fracture. Acknowledgement The financial support of Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-14-1-0209 is gratefully acknowledged.

224

S. Ravindran et al.

References 1. Barua, A., Horie, Y., Zhou, M.: Microstructural level response of HMX–Estane polymer-bonded explosive under effects of transient stress waves. Proc. R. Soc. London A Math. Phys. Eng. Sci. 468, 3725–3744 (2012) 2. Baer, M.R.: Modeling heterogeneous energetic materials at the mesoscale. Thermochim. Acta. 384, 351–367 (2002) 3. Bodelot, L., Escobedo-Diaz, J.P., Trujillo, C.P., et al.: Microstructural changes and in-situ observation of localization in OFHC copper under dynamic loading. Int. J. Plast. 74, 58–74 (2015) 4. Ravindran, S., Tessema, A., Kidane, A.: Note: Dynamic meso-scale full field surface deformation measurement of heterogeneous materials. Rev. Sci. Instrum. 87, 36108 (2016) 5. Ravindran, S., Tessema, A., Kidane, A.: Local deformation and failure mechanisms of polymer bonded energetic materials subjected to high strain rate loading. J. Dyn. Behav. Mater. 2, 146–156 (2016) 6. Ravindran, S., Tessema, A., Kidane, A.: Multiscale damage evolution in polymer bonded sugar under dynamic loading. Mech. Mater. 114, 97–106 (2017) 7. Othman, R., Aloui, S., Poitou, A.: Identification of non-homogeneous stress fields in dynamic experiments with a non-parametric method. Polym. Test. 29, 616–623 (2010) 8. Koohbor, B., Kidane, A., Lu, W.-Y., Sutton, M.A.: Investigation of the dynamic stress–strain response of compressible polymeric foam using a non-parametric analysis. Int J Impact Eng. 91, 170–182 (2016) 9. Pierron, F., Zhu, H., Siviour, C.: Beyond Hopkinson’s bar. Philos. Trans. R. Soc. London A Math. Phys. Eng. Sci. 372, 20130195 (2014)

Chapter 40

Localized Microstructural Deformation Behavior of Dynamically Deformed Pure Magnesium Peter Malchow, Suraj Ravindran, and Addis Kidane

Abstract Dynamic grain boundary region deformation of pure magnesium was investigated and verified by utilizing insitu full field strain measurements obtained from Digital Image Correlation (DIC) techniques. This method was confirmed to effectively characterize the microstructural response of an area of interest in the vicinity of multiple grain boundaries and triple junctions. The highly heterogeneous evolution of the material strain patterns was quantified, and the highest concentrations of the localized response were seen to occur primarily at the interfaces between grains, while the amount of in-grain deformation, specifically in the larger grains was minimal by comparison. Locations of suspected active slip and twinning regions were identified and conclusions about other possible modes of deformation are discussed. Keywords Metals and alloys · Microstructure · Deformation and fracture · Magnesium · High-strain rate · Digital image correlation

40.1 Introduction Magnesium and its alloys have been implemented in many engineering branches, such as aerospace and automotive applications [1]. A primary reason for its increased use is that it possesses a high specific strength at room temperature along with an inherent resistance to corrosion [2]. However, magnesium is a low-ductility metal that exhibits weak grain boundaries under loading as a direct result of the limited number of slip systems at room temperature [3]. This gives rise to a significant degree of the local deformation occurring in the regions of grain boundaries. Magnesium features a hexagonal closed packed (hcp) crystal structure which influences increased formation of mechanical twins under plastic deformation [4]. The dominant deformation mechanisms in magnesium are prismatic and pyramidal slip modes, as well as {0112} twinning systems [5]; with the onset of twinning being positively correlated with grain size [6]. Under high strain rates, the number of twins in magnesium is greater and occurs at higher levels of global strain when compared to the quasi-static case [7]. In general, material strain rate sensitivity is related to the grain size and overall grain coarseness [8]. Additionally, polycrystalline metals exhibit different amounts of deformation characteristics depending on the applied strain rate [9]; as twinning, basal slip, grain boundary sliding, creep and work hardening are all rate sensitive parameters [10]. Grain Boundary Sliding (GBS) is known to cause cracks along the boundaries particularly in the presence of triple junctions [11]. Triple junctions are associated with concentrated boundary action; contrary to a single boundary they experience additional rotations and different modes of grain motion as the dislocation slip tends to change direction when reaching a triple point [12, 13]. GBS is also correlated with the amount of triple junction migration that occurs; which is a variant of grain boundary motion [14]. In light of this, the ability to quantify the local dynamic deformation behavior and modes in pure magnesium becomes critically important for effectively including it into designs and other practices. Recently, an experimental technique based on digital image correlation is developed to understand the local deformation mechanisms in materials under high rate loading [15–17]. This work documents the full-field in-situ dynamic deformation response of polycrystalline pure magnesium. Conclusions on the suspected modes of deformation are discussed based on the localized strain data.

P. Malchow · S. Ravindran () · A. Kidane Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_40

225

226

P. Malchow et al.

40.2 Material and Methods 40.2.1 Material Preparation Nominally pure (99.9%) as-cast magnesium was used in the present work. This cast magnesium exhibits sufficiently large grain sizes (∼500–1000 μm) to enable meso-scale measurements. Additionally, the as-cast condition shows relatively weak grain boundaries; increasing the probability of observing grain boundary deformation modes. The weak boundary regions are a product of impurities which tend to collect in these areas. A cubic sample (12 × 12 × 12 mm3 ) was cut from a cast ingot using a band saw and machined to its final dimensions using a milling instrument. The surface of the sample was polished using incremental grits of sandpaper and further detailed utilizing a range of aqueous alumina powders thereby eliminating scratches and defects. The purpose of polishing was to prepare the surface such that the microstructure would be clearly visible following chemical etching. The specimen was etched for 10 min in a solution consisting of 6 g picric acid, 5 mL acetic acid, 10 mL water and 100 mL ethanol. To enable DIC measurements, a black and white uniform speckle pattern was then applied to the polished surface using white spray paint and black carbon powder. Note that for our purposes, the powder offers a finer speckle configuration at higher magnifications than traditional black paint. The powder particle size was approximately 10 μm; sufficiently small to choose optimal subset sizes during image correlation, which in turn will yield more detailed data.

40.2.2 Experimental Setup The schematic of the experimental setup used in this study is shown in Fig. 40.1. A strain rate of 1000/s was applied using a Split Hopkinson Pressure Bar, which consisted of 25.4 mm diameter and 1.82 m long stainless-steel incident, transmitted and striker bars. The striker bar was housed inside a barrel connected to an inert helium gas-gun. The loading was initiated when the striker bar was propelled into the incident bar. The sample was held in between the incident and transmitted bars while illuminated by a Photogenic flash lamp. Strain gauges were attached to the incident and transmitted bars which is used to detect the impact signal as it traveled through the bars. The strain gauges were connected to a signal amplifier which outputted to an oscilloscope that captured the impact waveform and also initiated the triggering of the camera. Imaging was performed using an ultra-high-speed CMOS camera (Shimadzu Hypervision HPV-X2, Hadland Imaging) 128 images were recorded at 500,000 frames per second with an exposure time of 1700 ns. The oscilloscope trigger and delay of the camera were selected to enable precise timing for accurately capturing the event. To verify the grain boundary activity with sufficient spatial resolution, a high magnification Navitar extension tube was utilized. The field of view was approximately 4.2 × 2.3 mm2 with a spatial resolution of 10.66 μm/pixel. The images were post-processed using subset and step sizes of 9 × 9 pixel2 and 1 pixel respectively, with a filter size of 11.

Fig. 40.1 Schematic of the experimental setup

40 Localized Microstructural Deformation Behavior of Dynamically Deformed Pure Magnesium

227

40.3 Results and Discussion 40.3.1 Macroscale Strain Field Figure 40.1 displays the local full-field strain map and the microstructure is overlaid to provide context on the location of varying strain domains. The images are captured at approximately 1, 2, 3 and 4% global axial strain. Here, the strain localization along the grain boundaries is displayed by the large contrasts appearing close to the boundary lines. This phenomenon is particularly evident when examining the εxx and εyy components and this large variability in the normal components of strain is accompanied by the formation of shear deformations in the material. From Fig. 40.1, the localized strain values demonstrate a highly heterogeneous deformation pattern as certain region exhibit higher magnitudes than the global strain. For instance, the local values in select regions of εxx and εyy are ≥10% in the images of only 2% global strain. Again, the highest values of εxx and εyy occur at the boundaries and in the presence of triple junctions. Conversely, the amount of deformation inside the larger grains remains minimal. This concentration at the triple junctions is expected since damage nucleation is known to be more probable here [18] (Fig. 40.2). The high strain concentration around the boundaries coupled with a relatively low degree of intragranular deformation implies the main failure mechanism in this material is grain boundary cracking or intergranular fracture; (i.e. the grain boundaries are weak in comparison to the granular strength). This concentration of strain is most likely due to compatibility stresses which accumulate at interfaces between grains.

40.4 Summary The deformation of nominally pure magnesium was captured in-situ under high strain rate loading and analysis of the heterogeneous strain patterns across the microstructure was conducted using a DIC-based technique. Based on the observation of highly localized strain occurring at the grain boundaries particularly at triple junctions, the boundaries in

Fig. 40.2 Contour maps of the sample illustrating the evolution of the in-plane components εxx , εyy and εxy . The sample was loaded in the x-direction and the grain boundary lines are overlaid on the plots

228

P. Malchow et al.

this material appear mechanically weak, while the larger grains are more resilient to in-grain deformation. Therefore, we conclude that the main mode of failure featured here is grain boundary fracture. Ultimately, this demonstrates a unique methodology of applying DIC for effectively characterizing the microstructural response of a material at high strain rates.

References 1. Hussein, R.O., Northwood, D.O.: Improving the performance of magnesium alloys for automotive applications. High Performance and Optimum Design of Structures and Materials, vol. 137, pp. 1743–3509. ASM International, Ohio (2014) 2. Shaw, B.A.: Corrosion resistance of magnesium alloys. ASM Handbook. 13A, 692–696 (2003) 3. Mordike, B.L., Ebert, T.: Magnesium: properties, applications, potential. Mater. Sci. Eng. A. 302, 37–45 (2001) 4. Yoo, M.H., Morris, J.R., Ho, K.M., Agnew, S.R.: Nonbasal deformation modes of HCP metals and alloys: role of dislocation source and mobility. Metall. Mater. Trans. A. 33A, 813–822 (2002) 5. Styczynski, A., Hartig, C., Bohlen, J., Letzg, D.: Cold rolling texture in AZ31 wrought magnesium alloy. Scr. Mater. 50, 943–947 (2004) 6. Meyers, M.A., Vohringer, O., Lubarda, V.A.: The onset of twinning in metals: a constitutive description. Acta Mater. 49, 4025–4039 (2001) 7. Hazeli, K., Kingstedt, O.T., Kannan, V., Ravichandran, G., Ramesh, K.T.: Strain evolution and twinning modes in magnesium single crystal. Preprint submitted to Elseiver. (2016) 8. Wang, Y.M., Ma, E.: Strain hardening, strain rate sensitivity, and ductility. Mater. Sci. Eng. A. 375–377, 46–52 (2004) 9. Dixit, N., Xie, K.Y., Hemker, K.J., Ramesh, K.T.: Microstructural evolution of pure magnesium under high strain rate loading. Acta Mater. 87, 56–67 (2015) 10. Dixit, N., Hazeli, K., Ramesh, K.T.: Twinning in magnesium under dynamic loading. EPJ Web Conf. 94, 02018 (2015) 11. Shimokawa, T., Nakatani, A., Kitagawa, H.: Grain size dependence of the relationship between inter and intra granular deformation of nanocrystalline al by molecular dynamics. Phys. Rev. B. 71, 224110 (2005) 12. Gutkin, M.Y., Ovid’Ko, I.A., Skiba, N.V.: Crossover from grain boundary sliding to rotational deformation. Acta Mater. 51, 4059–4071 (2003) 13. Ravindran, S., Koohbor, B., Kidane, A.: Experimental characterization of meso-scale deformation mechanisms and the RVE size in plastically deformed carbon steel. Strain. 53(1), 224105-1–224105-9 (2017) 14. Van Swygenhoven, H., Derlet, P.M.: Grain boundary sliding in nanocrystalline fcc metals. Phys. Rev. B. 64, 224105 (2001) 15. Ravindran, S., Tessema, A., Kidane, A.: Note: dynamic meso-scale full field surface deformation measurement of heterogeneous materials. Rev. Sci. Instrum. 87(3), 036108 (2016) 16. Ravindran, S., Tessema, A., Kidane, A.: Local deformation and failure mechanisms of polymer bonded energetic materials subjected to high strain rate loading. J Dyn Behav Mater. 2(1), 146–156 (2016) 17. Ravindran, S., Tessema, A., Kidane, A.: Multiscale damage evolution in polymer bonded sugar under dynamic loading. Mech. Mater. 114, 97–106 (2017) 18. Bieler, T.R., et al.: The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals. Int. J. Plast. 25, 1655–1683 (2009)

Chapter 41

Energy Absorption Characteristics of Graded Foams Subjected to High Velocity Loading Abigail Wohlford, Suraj Ravindran, and Addis Kidane

Abstract In this study the effect of layer stacking arrangement on the energy absorption characteristics of density-graded cellular polymers subjected to high velocity impact is investigated experimentally. Dynamic loading is performed using Split Hopkinson Pressure Bar (SHPB) which is also modified for a direct impact experiment. Different bulk density polymeric foam layers are bonded together in different stacking arrangements and subjected to impact loading. Ultra-high speed imaging is implemented to measure the deformation and observe the formation and propagation of compaction waves during direct impact. The effect of the orientation of the discrete layers on the dynamic stress-strain response is analyzed using digital image correlation (DIC). The effects of material compressibility are implemented to the analysis. The approach uses DIC to calculate the full-field acceleration and material density, later used to determine the stress gradients developed in the material. The best arrangement of layer structure is chosen by the highest energy absorption characteristics measured. Failure mechanisms associated with energy dissipation in graded materials are discussed. Keywords Dynamic loading · Polymeric foam · Digital image correlation · Energy absorption · Graded materials

41.1 Introduction Polymeric foams are cellular materials that in many industries such as aerospace, automotive, and military have become of great deal of interest. These cellular structures are ultra-light solids which absorb substantial energy in compression. Its many applications include absorbing impact and shock mitigation through energy dissipation by progressive local crushing. It is well known that energy absorption is strongly related to the foam density. Functionally Graded Materials (FGM) are advanced engineering materials that enable a material to have the best properties of multiples materials. The concept of Functionally Graded Foam Material (FGFM) has been introduced to improve upon certain properties as compared to a homogenous foam. The significant advantage of functionally graded foam materials is the optimization of strength to weight ratio. It is well established that higher density results in higher strength. Graded foam has the appeal of higher strength (higher density) combined with lighter weight (lower densities). The advent of graded foam has mostly been analyzed through simulations and analytical works. Many researchers have experimentally investigated the dynamic response of homogeneous foams, as well as there has been some theoretical and numerical work done on density-graded cellular materials. Ciu [1] tested the variation of gradation of the foam characteristics with finite element simulations and found improved performance over single-phase foams. Kiernan [2] developed a finite element model of the SHPB to study the wave propagation through FGFM’s. The impact response of density graded cellular polymers has been analyzed by observing the propagation of compaction waves using DIC [3]. The objective of this present study is to experimentally analyze the energy absorption and strength properties of FGFM’s subjected to high velocity loadings. The stacking sequence of discretized layers of different bulk densities is varied to determine the ideal gradation for load bearing performance. FGFM samples are constructed and subjected to dynamic loading via SHPB to generate stress in the discretized foam samples. Based on the experimental results the effects of orientation on the response of FGM’s is analyzed by observing the stress-strain response. The yield point of the tests was calculated to determine which of the four different stacking arrangements was stronger. To demonstrate the SHPB results are admissible, DIC is utilized to characterize the full field deformation.

A. Wohlford () · S. Ravindran · A. Kidane Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_41

229

230

A. Wohlford et al.

41.2 Materials and Methods The polyurethane foam was purchased from General Plastics, specifically the FR-6700 Aerospace grade series. This rigid, closed-cell foam is ideal because of its high strength-to-weight ratio. The five polymeric foams used to fabricate the layered specimens had densities of 160, 240, 290, 320, and 400 kg/m3 . Each layer was machined to 17 × 17 × 5 mm3 and then polished using silicon carbide papers to have a smooth surface, suitable for speckling. The layers were bonded together with a thin layer of highly-flexible polyurethane adhesive. It is vital that the adhesive is flexible to account for small relative lateral deformation of the layers and to minimize shear stress developed within the interface. The final length of the specimens is 25 mm. The different sequences studied took on either a stepwise or sandwich configuration as the following stacking arrangements: 160/240/290/320/400, 400/320/290/240/160/, 160/290/400/290/160, and 400/290/160/290/400. A schematic and picture of one of the stacking sequences is shown in Fig. 41.1. Table 41.1 reports the manufacturer mechanical properties given. Before speckling, a marker was used to draw a line between the layers to indicate the interfaces. A small piece of tape was used to cover a narrow strip of the front surface before speckling to help distinguish the different layers during deformation. The front surface of the specimens was painted white and then black speckling was applied using conventional spray painting method. A schematic of the experimental setup is shown in Fig. 41.2. The elements of the operation include a dynamic loading device, bar components, data acquisition and recording device, and high speed imaging system. The sample was loaded dynamically using a SHPB. Polycarbonate bars were used because testing low impedance materials such as foams requires low impedance bars to ensure a clear transmission wave is recorded and is easily discernible from any noise. To measure the full-field deformation during loading a single ultra-high speed HPV-X2 camera (Shimadzu) was used. The camera is capable of acquiring 128 images at the full-field resolution, and for this particular work and imaging rate of 200,000 frames per second was utilized. The camera was equipped with a 100 mm macro lens (Tokina) providing an optical resolution of 100 μm/pixel. A flash monolight (Photogenic) was used to illuminate the area if interest on the sample after trigger from the strain gages.

41.3 Results and Discussion The results presented are just from the preliminary tests and much of this work is still ongoing. The stepwise increasing density and stepwise decreasing density results are shown in this current study. The primary means by which a foam absorbs energy is by plastic deformation [4]. The deformation response is analyzed using the digital image correlation code, VIC-2D. Figures 41.3 and 41.4 shows the contour plots of the axial strain in the two converse specimens during the collapse of the

Fig. 41.1 Specimen stacking arrangement schematic and photograph Table 41.1 Manufacturer elastic modulus and compressive strength Density (kg/m3) 160 240 290 320 400

Elastic modulus (MPa) 60.7 161.0 196.9 244.1 344.2

Compressive strength (MPa) 2.3 5.8 7.2 9.6 14.0

41 Energy Absorption Characteristics of Graded Foams Subjected to High Velocity Loading

231

Fig. 41.2 Experimental setup schematic

a

exx [1] Lagrange 0 -0.00625 -0.0125 -0.01875 -0.025 -0.03125 -0.0375 -0.04375 -0.05

b

-0.05625 -0.0625 -0.06875 -0.075 -0.08125 -0.0875 -0.09375 -0.1

Fig. 41.3 Axial strain of (a) low to high density (b) high to low density

first layer. It is obvious that the first picture, the high-to-low density specimen, sees the highest strain at the far end of the sample, where the 160 kg/m3 layer is. The low-to-high density specimen sees the highest strain at the impacted end, where the layer with the lowest compressive strength is located. A point was extracted from the center of each layer during the post-processing using VIC-2D and the axial strain versus time is shown in the graphs below. It is apparent from the plots that the strain along the specimen is heterogeneous, varying spatially along the axis of the length. In both arrangements it can be observed that all the layers had begun to all be compressed together, but then the lowest density layer started to see much higher deformation. The failure progresses sequentially along the length of the specimen. The lowest density deforms plastically and then is fully densified (final failure of that layer) and then the strain propagates to the next lowest density layer. During the first incident wave only the two lowest density layers saw total failure. The two crushed layers started to compress the next lowest density layer, but the two high density layers still saw little effect. It was not possible to capture the final strain of the three highest density layers due to fragments of foam that were detached from the first two layers when they were being crushed. There does appear to be a slight variation in the time of failure of the different layers based on their orientation to the loading direction.

232

A. Wohlford et al.

time (µs) 0

100

200

300

400

500

0

strain

-0.05 -0.1

160 240 290 320 400

-0.15

high

-0.2

low

-0.25

time (µs) 0

100

200

300

400

500

0

strain

-0.05 160

-0.1

240 290

-0.15

320 400

-0.2

low

high

-0.25

Fig. 41.4 Axial strain points from each layer

41.4 Summary Functionally graded materials have the advantage of achieving tailored morphologies for desirable structural properties. The effects of stacking sequence on the energy dissipation of functionally graded foams under dynamic deformation was studied. The goal is to optimize the arrangement of layers of different bulk densities for superior energy absorption. High strain rate experiments were conducted on specimens that were made in-house and deformed using a standard SHPB. The compressive behavior was observed using a high speed camera and image correlation software. The full-field strain maps extracted from DIC were used to evaluate the mechanical response of the graded sample. By rearranging the configuration of the density gradation compared to the loading direction, a variation in the densification and strain progression occurs. These deformation characteristics will be used to tailor the material to a specific load and timeframe. Acknowledgment The financial support of US Army Research Office with grant number W911NF-17-S-0002 is greatly acknowledged.

References 1. Ciu, L., Kiernan, S., Gilchrist, M.: Designing the energy absorption capacity of functionally graded foam materials. Mater Sci Eng A. 507, 215–225 (2009) 2. Kiernan, S., Cui, L., Gilchrist, M.: Propagation of a stress wave through a virtual functionally graded foam. Int J Non Linear Mech. 44, 456–468 (2009) 3. Koohbor B, Ravindran S, Kidane A. Impact response of density graded cellular polymers 4. Kiernan, S., Cui, L., Gilchrist, M.: A numerical investigation of the dynamic behavior of functionally graded foams. In: IUTAM Symposium on Multi-Functional Material Structures and Systems. Springer, Dordrecht (2010) 5. Koohbor, B., Kidane, A., Lu, W.: Effect of specimen size, compressibility and inertia on the response of rigid polymer foams subjected to high velocity direct impact loading. Int J Impact Eng. 98, 62–74 (2016)

Chapter 42

Residual Structural Capacity of a High-Performance Concrete George Vankirk, William Heard, Andreas Frank, Mike Hammons, and Jason Roth

Abstract In this study, the residual unconfined compressive strength of a high-performance concrete (f’c ∼ 140 MPa) was investigated using samples that were pre-loaded to specific states of triaxial confinement. The residual unconfined compressive strengths of the samples were then compared to the unconfined compressive strength of pristine samples not subjected to the pre-load triaxial conditions. To accomplish the pre-load triaxial conditions, the samples were first subjected to specified stress-strain paths corresponding to pure hydrostatic compression and uniaxial strain in compression. Both the hydrostatic compression and uniaxial strain (in compression) tests were performed at low- and high-pressure levels under controlled conditions to prevent reaching the material failure limit. Once the samples were tested through either hydrostatic compression or uniaxial strain, they were recovered and subjected to unconfined compression until failure. Data from these samples were compared to the unconfined compressive strength of pristine samples from the same concrete batch. Residual structural capacity was determined through a comparison of these values and as a means to quantify damage induced (both with and without shear) by the specified stress-strain paths. Applications of these data are discussed for future improvements to concrete constitutive models commonly used at the U.S. Army Engineer Research and Development Center to simulate dynamic events. Keywords Residual structural capacity · High-strength concrete · Constitutive model · Damage · Failure surface

42.1 Introduction Modeling of extreme dynamic events, such as impact and blast effects in geomaterials, is an area of interest for the U.S. Department of Defense. Events of specific interest include projectile penetration, shaped-charge penetration, and close-in explosive detonation. These events provide a wide range of challenges with various types of material response and failure modes. These events occur at high strain rates, create large deformations, and are highly impulsive. Understanding and quantifying material damage, or residual structural capacity, is essential for advancing and improving constitutive models for these extreme dynamic events. Constitutive models must capture a spectrum of multiple, complex stress-strain paths; however, most current phenomenological models rely mainly on laboratory experiments to calibrate the model for simulating dynamic events. Fully describing a material’s response through laboratory data is a challenge. There is a current shortage of datasets available that provide a comparative measure of compressive strength of pristine untested concrete specimens to the residual compressive strength of concrete specimens that have undergone a specified stress path, but recovered prior to reaching the failure limit. Investigating this type of material behavior can lead to a more fundamental understanding of material damage, and help to improve constitutive models to more accurately predict material damage, failure, and residual strength.

42.2 Experiments In this study, a high-strength self-consolidating concrete (commonly referred to as BBR9) was investigated for residual structural capacity. This concrete was developed for laboratory use by the U.S. Army Engineer Research and Development Center with the following constituent materials: crushed limestone sand, type I/II cement, grade 100 slag, microsilica,

G. Vankirk () · W. Heard · A. Frank · M. Hammons · J. Roth U.S. Army Engineer Research and Development Center, Vicksburg, MS, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_42

233

234

G. Vankirk et al.

and high-range water-reducing admixtures. A set of 30 unconfined compression tests were performed on 50- by 114-mm cylindrical specimens yielding an average compressive strength of 140 MPa for a high-resolution statistical baseline. Triaxial experiments were conducted on 50- by 114-mm cylindrical specimens at confining pressures of 0, 10, 20, 50, 100, 200, and 300 MPa in order to determine the failure limit state for shear stress at these increasing levels of confinement. From the triaxial data, the maximum principal stress difference (calculated as σ1 - σ2 , or the difference between the total axial pressure and the hydrostatic pressure) and the mean normal stress (calculated as (σ1 +σ2 +σ3 ) / 3) were used to generate a failure surface. A failure surface is generated to describe the bounds of the material’s shear capacity under increasing levels of confinement. In order to investigate the effects of residual strength, BBR9 specimens were subjected to four different controlled stress paths under quasi-static conditions, but each specimen was unloaded and recovered prior to the point of failure. Each load path included 5 repeated laboratory tests of 50- by 114-mm cylindrical specimens in order to provide a statistical grouping of test results. The specimens were tested in hydrostatic compression and uniaxial strain at low and high levels of confinement (mean normal stress levels of 133 MPa and 400 MPa) in order to observe material damage below the failure surface. Once the samples had undergone their initial prescribed stress path, they were then tested in unconfined compression for comparison to the baseline compressive strength of 140 MPa.

42.3 Results A calibrated failure surface for BBR9 was generated using the parameters described by the Advanced Fundamental Concrete (AFC) model [1] and is presented in Fig. 42.1. This failure surface was fit to a full suite of quasi-static laboratory triaxial compression data [2]. Irreversible and complete material damage occurred when the material was stressed up to the failure surface shown in Fig. 42.1. Hydrostatic compression (HC) tests were performed on pristine 50- by 114-mm cylindrical specimens at low and high levels of confinement (mean normal stress levels of 133 MPa and 400 MPa) in order to volumetrically compress the samples without inducing shear. The stress history for each of the five HC samples at 133 MPa and the five HC samples at 400 MPa are shown in Fig. 42.1. Likewise, uniaxial strain in compression (UX) tests were also performed on pristine 50- by 114-mm cylindrical specimens at low and high levels of confinement (mean normal stress levels of 133 and 400 MPa). The stress history for each of the five UX samples at 133 MPa and five UX samples at 400 MPa are

400

Principal Stress Diference (MPa)

350

300

Triax. data

250

Failure Surface HC MNS-400

200

HC MNS-133 UX MNS-400

150

UX MNS-133

100

50

0

0

50

100

150

200 250 300 350 Mean Normal Stress (MPa)

400

450

500

Fig. 42.1 BBR9 triaxial compression test results with failure surface along with loading stress path plots for HC and UX laboratory experiments at low and high levels of confinement

42 Residual Structural Capacity of a High-Performance Concrete

235

Fig. 42.2 Comparison of unconfined compressive strengths for the baseline samples to the pre-loaded HC and UX for BBR9 material

shown in Fig. 42.1. The uniaxial strain tests induced shear for comparison to the samples tested hydrostatically, which did not induce shear. Figure 42.1 helps to illustrate the amount of pre-loading stress history of the 20 specimens relative to the known failure limit of BBR9. After these twenty specimens were subjected to an initial load path, they were unloaded and recovered to test in unconfined compression. The results of the unconfined compression tests are shown in Fig. 42.2.

42.4 Conclusion Triaxial compression data were used to fit a failure surface as defined by AFC parameters of BBR9 high-strength concrete material. A failure surface can be used to describe the shear state of the material, and material damage. The failure surface generated for BBR9 material using AFC parameters is only as accurate as the resolution of the laboratory characterization data that was used to generate it, therefore it may only be accurate for pressures below 375 MPa. The AFC material model dictates that the failure surface describes the limit state for which irreversible material damage occurs. Tests performed on BBR9 material followed stress paths that were below the failure surface as defined by AFC parameters and experimental data indicating that irreversible material damage should not have occurred. BBR9 material tested in uniaxial strain exhibited a higher reduction in structural capacity than material tested in hydrostatic compression. UX MNS-400 and UX MNS 133 exhibited a 29% and 10% loss in structural capacity, respectively. Samples previously tested in hydrostatic compression with a confining pressure of 400 MPa exhibited a 13% loss in structural capacity. The average compressive strength of HC MNS-133 was higher than baseline samples by 3%. This increase is not out of the immediate vicinity of the baseline mean and should not be interpreted as an increase in residual structural capacity. It has been shown that there is a loss of structural capacity when the BBR9 material has been stressed below the failure surface as defined by AFC parameters. The physical mechanisms for this capacity loss will need to be further investigated. Additionally, there are plans to incorporate the findings from this study into the failure surface definition of the AFC constitutive model to improve model accuracy. Permission to publish was granted by Director, Geotechnical & Structures Laboratory.

236

G. Vankirk et al.

References 1. Adley, M.D., Danielson, K.T., Frank, A.O.: Virtual Material Laboratory (VML), Version 1.0: Applications to Advanced Fundamental Concrete (AFC) Model. U.S. Army Engineer Research and Development Center (U.S.), Vicksburg (2013) 2. Williams, E.M., Graham, S.S., Reed, P.A., Rushing, T.S.: Laboratory Characterization of Cor-Tuf Concrete with and Without Steel Fibers. U.S. Army Engineer Research and Development Center, Vicksburg (2009)

Chapter 43

Dynamic Mode II Fracture Response of PMMA Within an Aquatic Environment Vivianna Gomez, Ian Delaney, Rodrigo Chavez, and Veronica Eliasson

Abstract Sheets of Poly Methyl Methacrylate (PMMA) 3 mm thick are lasercut into rectangles, and then subjected to a high-energy impact while submerged in distilled water, to investigate the effect that submersion may have on mode II fracture. The experiment has been carried out using a compressed air gun, which launches a delrin projectile towards the PMMA test specimen. A unique apparatus was constructed, which contains and allows the test specimen to be suspended in distilled water, while the projectile impacts a buffer adhered to the test specimen. The buffer transfers the impact onto the edge of the test specimen. Transparent walls in the apparatus allow the use of ultra-high-speed photography (up to 10Mfps), and digital image correlation (DIC) to measure dynamic response within the specimen throughout fracture. Data collected was used to calculate critical stress intensity factor. Results are compared to those from experiments in which test samples are not submerged. Keywords Fracture Mechanics · Mode 2 Fracture · Submerged Fracture · Polymeric Materials · Dynamic Loading · Extreme Environment · Boundary Conditions · Water · Aquatic Environment · DIC · Digital Image Correlation · CSIF · Critical Stress Intensity Factor · PMMA · Polymers

43.1 Extended Abstract Dynamic mode fracture is a specific type of fracture that investigates a kinetic approach to fracture behavior and crack propagations [1]. In this experiment, physical fracture behavior of submerged Poly Methyl Methacrylate (PMMA) samples will be evaluated with consideration to crack propagation rate, critical stress intensity factor, and kinetic energy behavior. The sample will act as a rigid, deformable body that will be subjected to dynamic loading. A projectile, later defined, fired from connected pressurized gas gun will impact submerged sample, projectile velocity and strain response will be recorded. The projectile’s velocity is dependent on pressure present in gas-gun chamber at time of trigger launch. The impact will induce rapid crack propagation that will emit stress waves. The stress waves will travel transversely along the width of the sample, radiating along the region of the crack [1]. The kinetic energy associated with the propagating crack must be sufficient enough to overcome the surface energy [2]. Considering such a phenomenon, the projectile’s velocity will be relevant when considering energy behavior and crack propagation rates. The expected fracture pattern will be a continuous fracture along the width of the sample, produced by in-plane shear forces. Experiments will be conducted following Mode II sample and impact configurations, where sample is subjected to in-plane shear loads (Fig. 43.1). To be specific, the experiment defines a wet sample as one that is fully submerged in distilled water, where fluid is idealized as thin layer around the sample. Hydrostatic pressure created by water is ignored. The sample will be submerged into wet environment housed in manufactured aquarium. The aquarium is designed to restrict any vertical displacements, allow translational movement, and transfer dynamic load onto the sample without any direct contact between projectile and submerged sample. While fully submerged, the sample will experience dynamic loading that will emit stress waves. PMMA was the chosen material due to isotropic material properties, defined solid shape, and relevance as a structural material in engineering fields [4]. The experimental sample will have a thin static notch located midway of the height on opposite edge of impact location. Sample and notch configuration allow the experiment to focus on the area of interest: the tip of crack. Stress wave activity will be recorded with strain gauges and an ultra-high-speed camera, the Shimadzu HPV-X2. All information recorded, including projectile velocity, camera footage, and strain response, will be used, with support of

V. Gomez · I. Delaney () · R. Chavez · V. Eliasson University of California: San Diego, Department of Structural Engineering, San Diego, CA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_43

237

238

V. Gomez et al.

Fig. 43.1 Modes of Fracture [3]

Fig. 43.2 Expected Fracture [5]

MATLAB and digital image correlation (DIC) software, to quantify critical stress intensity factor and crack propagation rate during Mode II fracture. The critical stress intensity factor will be determined by considering fracture occurring under shear loads (Fig. 43.2). The wet specimens to be tested are sheets of 3.175 mm thick PMMA which have been laser-cut to 200 mm in height by 100 mm in width. These samples have a horizontal notch 25.4 mm long and 0.3 mm wide cut into them in order to provide a point of fracture initiation, which is placed on the trailing edge of the sample exactly halfway from the bottom. This geometry is intended to provide ample distance between the point of fracture and any boundaries of the sample which would reflect stress waves. The sample will have a speckle pattern painted onto one side, which will then be viewed by an ultra-high-speed camera and analyzed using digital image correlation (DIC) software (Fig. 43.3). A projectile will be launched from a compressed air gun at approximately 10 (m/s), which will cause the fracture event. The projectile is a 76.2 mm long cylinder of Delrin, which has a diameter of 50 mm. This projectile strikes a 2.175 mm thick steel buffer, which is is bonded to the edge opposite of the notched edge of the sample (Fig. 43.4). This buffer evenly distributes the dynamic loading and transfers the energy from the impact. The impact between buffer and projectile will applying uniform, pseudo-instantaneous stress to the upper half of the sample, creating a shear in the plane direction of the added notch. The sample sits in an enclosed volume of water referred to as the aquarium. This aquarium is constructed of transparent acrylic, which will be filled with water to create the submerged environment key to the experiment. The internal cavity is 240 mm tall, 178 mm deep, and 28 mm wide. The most critical measurement being the width, which accommodates the thickness of the sample and allows for 12.7 mm of water on either side of the sample. The sample is free to move in the horizontal direction. After impact there exists 76 mm of clearance for the sample to recoil into before colliding with the far end of the aquarium, at which point the fracture event will have long been complete. The transparency of the acrylic walls will allow the fracture event to be viewed by an ultra-high-speed camera, set perpendicular to the aquarium.

43 Dynamic Mode II Fracture Response of PMMA Within an Aquatic Environment

239

Fig. 43.3 PMMA Fracture Sample [6]

25.40 mm 0.30 mm 203.20 mm

101.45 mm

101.60 mm

Fig. 43.4 Aquarium that will hold PMMA sample submerged (Projectile shown)

Exterior to this aquarium is a protective structure assembled from aluminum extrusions. These extrusions are placed in between the end of the gun barrel and the aquarium, with space enough for the end of the buffer to protrude through. This will allow the projectile to make contact with the buffer, however then be stopped by the extrusions before it could otherwise continue into the aquarium (Fig. 43.5). The camera in question is a Shimadzu HPV-X2, which will be operated at anticipated frame rate of five Million frames per seconds (Mfps), with the capability of viewing the event at up to 10 Mfps. 128 frames captured from the camera per fracture event will be analyzed using VIC-2D image correlation software. This software will recognize the speckle pattern applied to the sample and track the displacement of each point between frames, which is then translated into a strain field mapped onto the sample. Displacements from the software will be used to calculate the critical stress intensity factor of the sample, and the speed of crack propagation.

240

V. Gomez et al.

Fig. 43.5 Gas Gun Experimental Apparatus

Data will be recorded for PMMA samples inside the aquarium housing with and without water. Information from trials will indicate if any sort of calibration is needed, and whether the proximity of the acrylic window panels affect any type of stress wave behavior in samples. Initial predictions of that the complete submersion of the samples in a new medium, water, are that the water will significantly alter the crack propagation and overall dynamic fracture behavior. It is suspected that being submerged in water will create a unique boundary condition along the exterior of the fracture sample, which may remove energy from the system. This possibility has been tested for the case of mode I fracture, to which no significant effect was observed [5]. Mode II cracks expand at much higher velocities than mode I, and hence require a much greater amount of energy for the same material [5]. Because of this energy necessity, there is reason to believe that the underwater environment may affect this form of fracture by acting as an energy sink. It has been shown that mode II fracture is challenging to recreate in the laboratory environment, and that even if it is achieved, the event often transitions to the type of mode I in enough time [5]. It is predicted water will remove energy, thus decreasing the crack propagation rate, and/or increasing critical stress intensity factor. The removal of energy may be a product of increased dampening occurring along the surface of the sample as stress waves propagate. An additional significance of the submerged environment could be that this transition to mode I occurs much earlier than in a dry environment. The velocity can be manually defined so the lowest possible velocity is ideal for isolating strain conditions in the sample. Strain rate conditions caused by impact conditions will be considered when quantifying underwater crack propagation. Recent research on underwater dynamic fracture initiation of PMMA [4] reveal that water media has no significant effect on dynamic fracture when subjected to impact resulting in high strain rates. Results from this experiment will be compared to previous underwater dynamic fractures [4]. If crack tip speed and stress intensity factor results are similar, strain rates and surface energy behaviors underwater will be further explored. Once submerged sample experiments produce consistent data, with acceptable variations in values, data will be used to calculate critical stress intensity factor and crack propagation rate. The primary purpose of subjecting PMMA samples to submerged environments is to assess physical differences in fracture behavior, if any. The conclusions made about Mode II dynamic underwater fractures will lead to the experimentation of other materials in similar environment; ultimately, creating a database for composite materials. The recording and expression of dynamic underwater fracture will prove beneficial and efficient in the selection of materials subject to submerged environments and potential impacts.

43 Dynamic Mode II Fracture Response of PMMA Within an Aquatic Environment

241

References 1. Freud, L.B.: The Mechanics of Dynamic Fracture, pp. 1–15. Brown University, Providence (1986) 2. Anderson, T.L.: Linear elastic fracture mechanics. In: Fracture Mechanics: Fundamentals and Applications, 4th edn, pp. 25–26. CRC press, New York (2017) 3. Mach, K.J., Nelson, D.V., Denny, M.W.: Techniques for predicting the lifetimes of wave-swept macroalgae: a primer on fracture mechanics and crack growth. J. Exp. Biol. 210(13), 2213–2230 (2007) 4. Delpino, O.: On the dynamic fracture behavior of polymeric materials subjected to extreme conditions, PhD. Thesis, University of Southern California, (2016) 5. Broberg, K.B.: Differences between mode I and mode II crack propagation. Pure Appl. Geophys. 163, 1867 (2006) 6. Kalthoff, J.F.: On the Measurement of Dynamic Fracture Toughnesses–A Review of Recent Work. Dynamic Fracture, pp. 151–172. Springer, Dordrecht (1985)

Chapter 44

An Image-Based Inertial Impact Test for the High Strain Rate Properties of Brittle Materials Lloyd Fletcher and Fabrice Pierron

Abstract Testing ceramics at high strain rates presents many experimental difficulties due to the brittle nature of the material being tested. When using a split Hopkinson pressure bar (SHPB) for high strain rate testing, adequate time is required for stress wave effects to damp out. For brittle materials, with small strains to failure, it is difficult to satisfy this constraint. Thus, most available high strain rate data for ceramics focuses on using the SHPB for strength testing in compression. Due to the limitations of the SHPB technique, there is minimal data on the stiffness and tensile strength of ceramics at high strain rates. Recently, a new image-based inertial impact (IBII) test method has shown promise for analysing the high strain rate behaviour of brittle materials. This test method uses a reflected compressive stress wave to generate tensile stress and failure in an impacted specimen. Throughout the propagation of the stress wave, full-field displacement measurements are taken. Strain fields and acceleration fields are derived from the displacement fields. The acceleration fields are then used to reconstruct stress information and identify the material properties. The aim of this study is to apply IBII test methodology to analyse the stiffness and strength of ceramics at high strain rates. Preliminary results have shown that it was possible to use the IBII test method to identify the elastic modulus and strength of tungsten carbide at strain rates on the order of 1000/s. For a tungsten carbide with 13% cobalt binder the elastic modulus was identified as 520 GPa and the tensile strength was 1400 MPa at nominal strain rate of 1000/s. Further tests are planned on several different grades of tungsten carbide and other ceramics including boron carbide and sapphire. Keywords High strain rate testing · Full-field measurement · Ultra-high speed imaging · Image-based methods · Ceramics

44.1 Introduction Engineering ceramics and ceramic composites are used in a variety of applications that require them to withstand high strain rate loading. Monolithic ceramics such as boron carbide are typically used in ballistic armour applications. Ceramic metal composites (often referred to as cermets) such as tungsten carbide are used in high speed mill and drill bits for machining purposes. For both of these cases it is essential that the high strain rate response of these materials is well characterised to ensure that these ceramic components function adequately under dynamic loading. Current high strain rate data for ceramics has been obtained using the split-Hopkinson pressure bar technique. In order to infer the material properties of the test sample from strain gauge measurements on the bars it is assumed that the force on both ends of the sample is equal (i.e. the specimen is in a state of quasi-static equilibrium). For brittle materials that have characteristically small strains to failure it can be difficult to satisfy the assumption of quasi-static equilibrium before fracture occurs, especially for testing in tension. Therefore, current test data has focused on testing ceramics in compression [1, 2]. To the authors’ knowledge there is only a single study investigating the dynamic properties of tungsten carbide cermets [1]. This study only investigated the effect of strain rate on the compressive strength at strain rates up to 500/s. Unfortunately, data is not currently available for the tensile strength of this material at high strain rates. Recently, the Image-Based Inertial Impact (IBII) test has emerged as a viable alternative to the SHPB technique. This technique uses a ‘spall’ configuration in which a reflected compressive stress wave is used to generate tensile stress and cause failure of the test sample. The IBII test was first applied to study the dynamic tensile strength of concrete in [3]. This technique has also been extended to study the transverse properties of unidirectional composites at strain rates that

L. Fletcher () · F. Pierron Engineering Materials Research Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_44

243

244

L. Fletcher and F. Pierron

y

10 mm

50 mm

60 mm

thickness, t = 4 mm 45 mm

30 mm

x WC Specimen

Wave Guide

45 m/s Projectile

Fig. 44.1 Schematic of the test components

are not achievable with a traditional SHPB [4]. Therefore, the aim of this study is to apply the IBII test to investigate the tensile properties of ceramics. This paper will focus on the application of the IBII test to a tungsten carbide cermet as a demonstration of the methodology.

44.2 Experimental Methodology The material tested in this study was a ‘fine’ grain tungsten carbide cermet with a cobalt binder. The density of the material was measured to be 14,000 kg/m3 with a binder volume fraction of ∼12.5%. The specimen was impact tested using a gasgun at an impact speed of 45 m/s. A schematic of the main experimental components is shown in Fig. 44.1. The specimen was bonded to a cylindrical waveguide which was mounted on a wedge shaped foam stand in front of the gas gun barrel. The purpose of the waveguide is to minimise the effects of misalignment. The waveguide and projectile were made from high strength steel (alloy 15CDV6) so that the impact did not cause the projectile to yield and clip the input pulse. A grid pattern was applied to the samples, the grid had a pitch of 0.7 mm and was sampled at 5 px/period. The test was imaged using a Shimadzu HPV-X camera at 5 Mfps. The camera was triggered using copper contacts on the front of the waveguide. A Bowens Gemini 1000 photographic flash was used for lighting. As the flash has a rise time of ∼110 μs it was triggered from a custom Arduino system connected to the light gates at the gas gun barrel exit.

44.3 Data Processing The grid images of the impacted sample were processed using an open source code [5]. The displacement fields were then post-processed using a custom Matlab program. The strain and accelerations fields were calculated from the displacement fields using a centred finite difference method. As the displacement fields contain noise spatial and temporal smoothing were applied prior to differentiation. The spatial filter used here was Gaussian with a kernel length of 41 pixels and the temporal filter was a third order Savitsky-Golay filter applied over 13 frames. The acceleration fields were then used to reconstruct stress averages along the specimen length. As in [4], the stress-gauge equation can be used for the reconstruction of the average axial stress in the specimen ‘σxx y ’ at an arbitrary position ‘x’ from the free edge at time ‘t’: y

σxx (x, t) = ρxa (x, t)S

(44.1)

where ‘ρ’ is the material density and ‘a (x, t)S ’ is the average acceleration over the specimen surface from the free edge to the point ‘x’. A stress-strain curve can then be obtained by plotting the average stress (σxx y ) against the average strain y (εxx + νεyy ). The stress strain curve can then be linearly fitted to obtain the stiffness component, Q11 . Furthermore, as described in [4] three rigid body virtual fields can be used to identify a linear distribution of stress. This ‘linear’ stress-gauge equation can be used to identify the tensile strength at the failure location of the test sample.

44 An Image-Based Inertial Impact Test for the High Strain Rate Properties of Brittle Materials

245

Fig. 44.2 Stress-strain curves at several sections along the specimen length (free edge x = 0). Note that the tensile fracture occurs at x = 36 mm

44.4 Results and Discussion The stress-strain curves reconstructed at several sections along the specimen length are shown in Fig. 44.2. The Poisson’s ratio was identified using the virtual fields method. The identified value was ν = 0.22. This was used to calculate the axial strain for constructing the stress-strain curves. The response is linear elastic and extremely consistent along the length of the sample. The peak axial averaged strain rate observed during the test was 1100/s. The elastic modulus was identified over all axial sections from the stress-strain curves. Taking the average modulus over the middle portion of the specimen gives an identified value of E = 520 GPa. It is not expected that the elastic modulus of this tungsten carbide cermet will be strain rate dependent. In [6] the quasi-static modulus of a similar tungsten carbide cermet was found to be 540 GPa with a Poisson’s ratio of 0.22 (Kennametal KF312, 12% Co binder, density of 14,300 kg/m3 ). This compares extremely well to the results obtained in this study. The strength was also identified using the linear stress-gauge equation as in [4]. The stress field reconstructed from the strain and the stress field calculated with the linear stress-gauge equation are shown in Fig. 44.3a, b at the fracture time. In Fig. 44.3 the local gauge area for strength identification is shown as a black rectangle. A comparison of these two stress measures is shown in Fig. 44.3c and the local stress strain curve over the gauge area is shown in Fig. 44.3d. The tensile strength was identified as 1400 MPa with a maximum average strain rate in the failed section of 930/s. It was not possible to find a quasi-static reference value for the tensile strength of a similar tungsten carbide cermet so any possible strain rate effects cannot be commented on at this stage.

44.5 Conclusion and Future Work The image-based inertial impact (IBII) test was successfully used to identify the elastic modulus and tensile strength of a tungsten carbide cermet at strain rates on the order of 1000/s. To the authors’ knowledge, this data does not currently exist in the literature for this material at strain rates this high. This demonstrates the considerable potential for using the IBII test to investigate the properties of other engineering ceramics at high strain rates. In the future, the test method will be applied to other ceramic materials including silicon carbide, boron carbide, glass and sapphire. Acknowledgements The authors want to thank Dr. Leslie Lamberson from Drexel University for providing the samples and for useful discussions about the material. Funding from EPSRC, grant EP/L026910/1, is also gratefully acknowledged.

246

L. Fletcher and F. Pierron

Fig. 44.3 Strength identification diagnostics at the fracture time t = 17 μs. (a) Stress field from the strains. (b) Stress field from the linear stress-gauge equation. (c) Comparison of stress measures over time. (d) Local stress strain curve at the fracture location

References 1. Mandel, K., Radajewski, M., Krüger, L.: Strain-rate dependence of the compressive strength of WC–Co hard metals. Mater. Sci. Eng. A. 612, 115–122 (2014) 2. Swab, J.J., Meredith, C.S., Casem, D.T., Gamble, W.R.: Static and dynamic compression strength of hot-pressed boron carbide using a dumbbellshaped specimen. J. Mater. Sci. 52, 10073–10084 (2017) 3. Pierron, F., Forquin, P.: Ultra-High-Speed full-field deformation measurements on concrete spalling specimens and stiffness identification with the virtual fields method. Strain. 48, 388–405 (2012) 4. Fletcher, L., Van-Blitterswyk, J., Pierron, F.: A novel image-based inertial impact (IBII) test for the transverse properties of composites at high strain rates. J. Dyna. Behav. Mat. (2018). Under Review 5. Grédiac, M., Sur, F., Blaysat, B.: The grid method for in-plane displacement and strain measurement: a review and analysis. Strain. 52, 205–243 (2016) 6. Getting, I.C., Chen, G., Brown, J.A.: The strength and rheology of commercial tungsten carbide cermets used in high-pressure apparatus. Pure Appl. Geophys. 141, 545–577 (1993)

Chapter 45

An Image-Based Approach for Measuring Dynamic Fracture Toughness Lloyd Fletcher, Leslie Lamberson, and Fabrice Pierron

Abstract In order to model the dynamic failure of engineering structures it is necessary to have a thorough understanding of dynamic fracture processes. Dynamic fracture toughness has been experimentally analysed by fitting the K-dominant solution to the displacement field measured with a local or full-field technique, such as caustics, photoelasticity or digital image correlation. For highly dynamic crack propagation the stress state at the crack tip is influenced by stress waves. A dynamic propagating crack emits stress waves which can be reflected and/or scattered away. These waves are felt by the evolving crack front, as well as through the sample configuration, and hence material inertia may lead to effects more subtle (yet still present) than those associated with load transfer. The K-dominant solution only indirectly accounts for these inertial effects by including the crack velocity as an input or by using higher order terms in the series expansion. The aim of this work is to develop a new image-based method for measuring dynamic fracture toughness. This method uses full-field measurements to perform an energy balance on a fracture specimen and calculate the energy consumed by crack growth. Using full-field data the impact energy, strain energy and kinetic energy can be measured. When the material cracks the fracture energy is the difference between the impact energy and the sum of the strain and kinetic energy. Explicit dynamics simulations using cohesive elements were used to validate the methodology. The finite element data was used for simulated image deformation experiments. These virtual experiments were used to analyse measurement error propagation from camera spatial and temporal resolution. Future work will include additional image deformation simulations and a first experimental validation of the test method. Keywords Dynamic fracture · Full-field measurement · Ultra-high speed imaging · Image-based methods · Energy balance

45.1 Introduction The efficient design of impact resistant structures requires accurate models of material fracture processes under dynamic loading. For example: the design of ballistic armour or the design of composite panels to resist hail strikes. There has been significant work undertaken to measure dynamic fracture properties using a variety of experimental and numerical modelling techniques. More recently, dynamic fracture toughness has been determined utilising full-field displacement measurements taken using digital image correlation [1]. This methodology utilises the solution for the displacement field around the crack tip given by a series expansion with a typical square root singularity. However, for the case of highly dynamic crack propagation, stress wave interactions lead to a complex and rapidly oscillating stress state at the crack tip. The effects of inertia are generally assumed to be contained within the ‘higher order terms’ of the series expansion [2]. In [2] it is noted that as a fracture event becomes more influenced by inertia the singular term of the series expansion becomes less significant. Furthermore, the derivations in [2] neglect the effects of reflected stress waves which will occur for dynamic fracture events in finite specimens. Therefore, a more general approach is required to characterise dynamic fracture toughness for a variety cases were inertial effects are significant.

L. Fletcher () · F. Pierron Engineering Materials Research Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK e-mail: [email protected] L. Lamberson Department of Mechanical Engineering & Mechanics, Drexel University, Philadelphia, PA, USA Materials Science and Engineering, Drexel University, Philadelphia, PA, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_45

247

248

L. Fletcher et al.

P

Displacement Field ux, uy Cohesive Zone 12μs

45 mm

75 MPa

dv/dt Acceleration ax, ay

du/dt Velocity vx, vy

du/dx du/dy Strain εxx, εyy, εxy

t

E F = E I - EK - ES 70 mm Fig. 45.1 Schematic representation of the image-based energy balance method

The aim of this work is to develop an image-based dynamic fracture analysis and explore how it could alleviate some of the limitations of current methods. The proposed method utilises ultra-high speed imaging to perform a transient energy balance on a fracture specimen and calculate the energy consumed during crack growth. In order to develop this new methodology the following objectives are proposed: (1) numerical validation using finite element simulations; (2) image deformation simulations to account for measurement errors coming from camera noise spatial resolution and temporal resolution; and (3) experimental validation of the test concept using a model brittle material, acrylic (PMMA). In this paper the formulation of the test concept is presented along with numerical validation on finite element and image deformation simulations. Future work will include the first experimental validation of the new method. It should also be noted that the transient full-field energy balance described herein is not limited to the study of energy consumed due to fracture processes. The methodology could also be used to study energy consumed in phase transformations, damage or plastic work. To the authors’ knowledge, this paper describes the first use of a full-field energy balance for transient dynamics using fracture as a first example.

45.2 Theory and Concept The proposed test concept is based on an energy balance applied to a dynamically impacted sample (shown in Fig. 45.1). For the case of an un-cracked sample that does not fracture the energy balance takes the form: EI = EK + ES where EI is the input energy from the impact force, EK the kinetic energy and ES the strain energy in the sample. When the energy balance is applied to a cracked specimen an additional term is introduced to account for the energy consumed by fracture processes ‘EF ’. For this case the energy balance can be written as: EI = EK + ES + EF . If full-field displacement data is available, with sufficient temporal and spatial resolution, then impact energy EI , kinetic EK and strain energy ES can be calculated. The impact energy can be calculated by using the average acceleration over the field of view to reconstruct the impact force; the kinetic energy is obtained from the velocity field; and the strain energy can be obtained from the strain field by assuming a suitable material model to reconstruct the stress field. The fracture energy is then the energy remaining after subtracting the kinetic and strain energy from the impact energy.

45.3 Numerical Modelling The specimen configuration shown in Fig. 45.1 was modelled in Abaqus explicit (v 6.14). The bulk of the model was meshed using three dimensional 8-node reduced integration elements (C3D8R). The mesh size, time step and damping were selected to minimise the error on the energy balance for an un-cracked specimen. The mesh size was set to 0.25 mm, the simulation time step was allowed to float and the beta damping was set to 0.5 × 10−8 s. The specimen was modelled as PMMA with nominal material properties of ρ = 1160 kg/m3 , E = 5.5 GPa, ν = 0.35. The input loading was modelled as a trapezoidal pulse 12 μs in duration with a peak of 75 MPa. Cohesive surface elements were used to model the crack path with a bilinear traction separation law. The cohesive law was specified using the input maximum surface traction Tmax = 50 MPa and the critical strain energy release rate Gc = 2200 J/m2 . The displacement fields were extracted from the finite element model and used to synthetically deform grid images using a procedure similar to [3]. The images were processed using a freely available grid processing code to extract the displacement fields, see [4]. A custom Matlab code was then used to calculate the strain and velocity fields using a centred finite difference method. Two different camera setups were simulated: (1) 400 × 250 pixels, 0.9 mm pitch grid sampled at 5 pixels/period, with a frame rate of 2 Mfps (Shimadzu HPV-X camera)

45 An Image-Based Approach for Measuring Dynamic Fracture Toughness

249

and (2) 800 × 500 pixels, 0.45 mm pitch grid sampled at 5 pixels/period, with a frame rate of 2 Mfps. These two cases were selected to explore the effects of spatial resolution on performance. Additionally, case (1) was analysed with a frame rate of 4 Mfps hereafter referred to as case (3), this was used to explore the effects of temporal resolution.

45.4 Results and Discussion The energy balance for the finite element model is shown in Fig. 45.2 for the case of ‘no fracture’ without a cohesive zone and for the ‘fracture’ case including the cohesive zone. The error on the energy balance is extremely low for the case of perfect simulation data. For the ‘fracture’ case the total energy consumed was 0.08 J, giving a ratio of fracture energy on impact energy of EF /EI = 12%. In order to measure this energy, the error with a real camera system must be much less than 12%. The energy balance calculated from the image deformation simulations is shown in Fig. 45.3. At this stage this has only been calculated for the ‘no fracture’ case with no noise in order to assess the underlying systematic errors. For the 400 × 250 pixels simulation the bulk of the error is attributed to the measurement of the impact energy. The errors for this case are: EI,err = 14%, Ek,err = 4%, Es,err = 3%. Comparison of the error for both of these cases show that for 800 × 500 pixels the error in the energy balance is drastically reduced. Figure 45.4 (left) shows the energy balance for the 400 × 250 pixel simulated camera and Fig. 45.4 (right) shows the energy balance as applied to the experimental data from [5]. For both the simulated and experimental case the error on the energy balance is similar. This preliminary analysis does not include the effects of noise or the effects of high strain gradients that will occur at the crack tip for the ‘fracture’ case. For an experiment including fracture it is likely that a much higher spatial resolution will be required to account for the high strain gradients at

Fig. 45.2 Energy balance calculated directly from the finite element model. Left: energy balance for the ‘no fracture’ case and right: energy balance for the ‘fracture’ case

Fig. 45.3 Energy balance calculation for image deformation simulations, ‘no fracture’ case. Left: camera setup 1 (400 × 250 pixels) and right: camera setup 2 (800 × 500 pixels). Both simulated cameras had a frame rate of 2Mfps

250

L. Fletcher et al.

Fig. 45.4 Right: energy balance for simulated camera setup 3 (400 × 250 pixels, 4Mfps) and Left: energy balance applied to experimental data from [5] (400 × 250 pixels, 5Mfps)

the crack tip. Future image deformation simulations will be conducted at a higher resolution for the ‘fracture’ case including the effects of noise on the measurement. The best spatial resolution for an ultra-high speed camera is currently the Cordin 580 rotating mirror camera (3200 × 2400 pixels), so this methodology has practical potential.

45.5 Conclusion and Future Work This work presents a new methodology for determining the dynamic fracture toughness of a material using a full-field transient energy balance. The new test concept has been validated using explicit dynamics simulations coupled with image deformation simulations. The methodology has also been validated using experimental data for the ‘no fracture’ case. Continuing work is underway to perform additional image deformation simulations and to conduct a first experimental validation including the fracture energy. The results of this study show that for experimental validation a high resolution camera (e.g. rotating mirror camera at >1 Mpixels) is required to adequately resolve the energy components. Acknowledgments LF and FP gratefully acknowledge funding from EPSRC through grant EP/L026910/1.

References 1. Shannahan, L., Weerasooriya, T., Gunnarsson, A., Sanborn, B., Lamberson, L.: Rate-dependent fracture modes in human femoral cortical bone. Int. J. Fract. 194, 81–92 (2015) 2. Freund, L.B., Rosakis, A.J., Mech, J.: The structure of the near-tip field during transient elastodynamic crack growth. J Mech Phys. Solids. 40, 699–719 (1992) 3. Rossi, M., Lava, P., Pierron, F., Debruyne, D., Sasso, M.: Effect of DIC spatial resolution, noise and interpolation error on identification results with the VFM. Strain. 51, 206–222 (2015) 4. Grédiac, M., Sur, F., Blaysat, B.: The grid method for in-plane displacement and strain measurement: a review and analysis. Strain. 52, 205–243 (2016) 5. Pierron, F., Zhu, H., Siviour, C.: Beyond Hopkinson’s bar. Philos. Trans. A. Math. Phys. Eng. Sci. 372, 20130195 (2014)

Chapter 46

The Effect of in-Plane Properties on the Ballistic Response of Polyethylene Composites Julia Cline

Abstract Using developed experimental and analytical methods for in-plane shear characterization of quasi-statically loaded polyethylene laminates, this work seeks to evaluate the effect the in-plane shear behavior has on ballistic performance (resistance to penetration and back face deflection). In-plane shear is a matrix-dominated phenomena and processing pressure is known to influence noticeable changes in the shear properties of polyethylene composites, so by varying matrix materials and processing conditions it is possible to probe an array of configurations. Quasi-static tensile tests of laminates with [±45◦ ] orientation are performed to obtain the in-plane shear properties. To evaluate the ballistic response, a high pressure helium laboratory gas gun is used to accelerate 0.22 caliber spherical steel projectiles toward specimen panels to characterize the V50 ballistic limit velocity and back-face deflection. Keywords In-plane shear characterization · UHMWPE · Ballistic performance · Back face deflection

46.1 Extended Abstract Soft ballistic composites that are used for soldier body and head protection consist of high strength, high stiffness fibers embedded in a compliant matrix whose strength is several orders of magnitude lower than the fibers. Increasing the resistance to penetration (or V50 ) of a material is simply an energy absorption problem. The material must absorb and disperse the kinetic energy from the inbound projectile. For non-penetrating shots, typically the energy is absorbed by deforming the material perpendicular to the projectile’s velocity vector i.e. back face deformation (BFD). Energy dissipation is primarily via fiber extension but recent studies link low shear strength with improved ballistic performance [1]. Stiff fibers in a compliant matrix rotate toward the loading direction, which engages fibers in non-principal directions [2, 3]. It is hypothesized that this fiber rotation results in a higher BFD because it engages more material, which in turn raises the V50 . In order to investigate how in-plane shear properties and resulting fiber rotation, as a deformation mechanism, affect the ballistic response of soft ballistic composites, we first quantify the quasi-static, in-plane shear response. Then, ballistic testing of composite panels is performed to quantify the V50 and BFD, and the resulting data is compared back with the quasi-static results to draw conclusions. Two materials are selected for this work: DSM Dyneema’s most recent grades of hard ballistic (HB) fiber-based material systems HB 210 and HB 212 [4]. Both material systems are based on the high performance SK 99 gel-spun UHMWPE fiber with the same volume fraction and yarn size. HB 210 has a polyurethane matrix and HB212 has a more compliant Kraton matrix (rubber). Both material grades are commercially available on a continuous roll as a pre-consolidated sheet of four plies in a [0◦ /90◦ ]2 cross-ply configuration. By selecting material systems that have the same fiber and different matrix material, we eliminate the effect of the fiber properties on the ballistic response and can focus on shear (matrix)-dominated deformation mechanisms. Test specimens are cut from 45.7 cm by 45.7 cm thin panels that are manufactured according to Dyneema’s specified processing cycle (20 MPa; 132 ◦ C; 1 h) in a confined mold. The HB 212 panels are vacuum bagged prior to hot-pressing. Silicone rubber pads are placed between the mold and UHMWPE sheets to distribute pressure more uniformly across the panel during processing. To probe the effect of processing pressure on the in-plane shear response [5], additional panels of HB 212 processed at 7 MPa and 34 MPa are also produced. 7 panels for each configuration are manufactured. Quasi-static testing specimens measuring 30 cm long and 2.54 cm wide, are cut from a single panel on the bias, so the resulting layup is [±45◦ ]. The remainder of the panels are cut into 23 cm by 23 cm square specimens for ballistic testing.

J. Cline () Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_46

251

252

J. Cline

Fig. 46.1 Plots of the (a) engineering stress-strain curves for HB 210 and 212 specimens and fiber rotation angle as a function of (b) stress and (c) longitudinal stretch

The in-plane shear tests are performed by quasi-statically applying tensile load to the [±45◦ ] laminates using an electromechanical load frame. Stereo-vision Digital Image Correlation (DIC) is used to measure the large deformation surface strains as a function of applied load until failure. A contrasting speckle pattern is applied to the surface of the specimens using black and white spray paint to provide trackable surface features. Krylon dulling spray is applied prior to testing to reduce reflections. DIC images are analyzed using VIC3D [6] to obtain surface deformation and strain measurements. A continuum mechanics approach is taken to analyze the resulting data and determine the in-plane shear behavior of the material as described in [2]. The engineering stress-strain curve is plotted in Fig. 46.1a; the calculated fiber rotation angle, θ , as a function of stress is plotted in Fig. 46.1b and the fiber rotation angle as a function of deformation is plotted in Fig. 46.1c. The difference in shear response between the HB 210 and HB 212 materials is evident, whereas the effect of pressure on the HB 212 response is minimal. This is consistent with the work of Hazzard [3]. The fibers do not deform significantly but fiber rotation is observed to be a primary mechanism of deformation. To evaluate the ballistic response, a high pressure helium laboratory gas gun is used to accelerate 0.22 caliber spherical steel projectiles toward specimen panels to characterize the V50 ballistic limit velocity and back-face deflection. The 23 cm by 23 cm panels are inserted into a window frame fixture, with a 15 cm by 15 cm opening, to stabilize the panel during testing. High grit sand paper is affixed to the inside surfaces of the window frame to increase friction between the frame and panel. Clamps are placed around the edges of the panel frame (see Fig. 46.2) to secure the panel in place and minimize the edge pull-in during impact. 15–20 panels are tested per material configuration with one center shot per panel. A variety of shots are performed of both partial (V0 ) and complete (V100 ) penetrations to yield a data set by which to calculate V50 . The gas gun is pressurized at different pressures to accelerate the projectile at chosen impact velocities. The velocity is measured with a Doppler radar (BR-4502, Infinition Inc.). The gas gun pressure, resulting velocity and penetration type are recorded. V50 results and representative impacted panels for HB 210 and 212 are shown in Fig. 46.2.

46 The Effect of in-Plane Properties on the Ballistic Response of Polyethylene Composites

253

Fig. 46.2 (a) V50 data calculated for each material system and partially impacted panels for (b) HB 210 and (c) HB 212 showing clear deformation differences

Two panels from each set are used to characterize the BFD response. High-speed, stereo-vision DIC is used to capture the BFD. The back side of each panel is speckled using a sharpie marker and dulling spray is used to reduce reflections. The panels are shot at velocities corresponding to 90% of the lowest V50 for the set of material configurations (this corresponds to the V50 for the HB 210 system). VIC3D is used to extract the 3D displacements on the back surface of the panel. The 3D deformation data is used to calculate the stretches along principal and non-principal directions and the results are used to draw conclusions on the influence of in-plane shear properties on the ballistic response of PE composite materials. Acknowledgements This research was supported in part by an appointment to the Postgraduate Research Participation Program at the U.S. Army Research Laboratory administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and USARL.

References 1. Karthikeyan, K., et al.: The effect of shear strength on the ballistic response of laminated composite plates. Eur. J. Mech. A. Solids. 42, 35–53 (2013) 2. Cline, J., Bogetti, T., Love, B.: Comparison of the in-plane shear behavior of UHMWPE fiber and highly oriented film composites, Proceedings of the 32nd American Society for Composites Technical Conference, West Lafayette, IN, 23–25 Oct 2017 3. Hazzard, M. et al.: An investigation of in-plane performance of ultra-high molecular weight polyethylene composites, Proceedings of the 20th International Conference on Composite Materials, Copenhagen, 19–24 July 2018 4. DSM Dyneema® Industries, http://www.dsm.com/products/dyneema/en_US/home.html, Accessed Feb 2018 5. Greenhalgh, E.S., et al.: Fractographic observations on Dyneema® composites under ballistic impact. Compos. Part A. 44, 51–62 (2013) 6. Correlated Solutions, Inc. http://www.correlatedsolutions.com, Accessed Feb 2018

Chapter 47

Storage and Loss Moduli of Low-Impedance Materials at kHz Frequencies Wiroj Nantasetphong, Zhanzhan Jia, M. Arif Hasan, Alireza V. Amirkhizi, and Sia Nmeat-Nasser

Abstract Standard Dynamic Mechanical Analysis (DMA) is generally used to measure the mechanical properties of polymers at frequencies around and below 100 Hz. Ultrasonic (US) techniques measure wave speeds and impedances at higher frequencies. However, both approaches run into issues between the two regimes. DMA systems become less reliable due to the dynamic response of the frames and load path as one tries increasing the frequency. On the other hand, the internal multiple reflections in the wave propagation techniques introduce challenges in clean measurements and require careful analysis. In this presentation, we introduce a robust procedure for determining the storage and loss moduli of low-impedance materials, where a cylindrical sample is placed between two long metal bars, similar to SHPB technique. However, unlike SHPB, the incidence signal is created by a very light impact, to ensure that the sample does not experience permanent or large deformation. Furthermore, due to the length of the specimen, dynamic equilibrium is neither guaranteed nor intended. The reflected and transmitted pulses are measured using semi-conductor strain gages. The wave speed may be determined using a phase spectral analysis of the time-resolved signals. Determination of the material loss requires a more thorough transfer matrix analysis. The method was applied to a soft polyurea elastomer that was tested in a temperature-control chamber and results were compared with DMA and US data using time-temperature superposition (TTS). While the predictions of the storage modulus using DMA and TTS matched very well with the direct measurements, the DMA/TTS predictions generally underestimate the material loss at higher frequencies. We expect that this method may be applied successfully to other low impedance materials including foams and metamaterials. Keywords Low impedance materials · Polyurea · Low frequency wave propagation · Modified Hopkinson bar · Impact

47.1 Introduction The frequency-dependent mechanical properties of rubbery polymers have been traditionally studied using dynamic mechanical analysis (DMA). While modern DMA machines have higher frequency capabilities (in some cases quoted up to 1000 Hz), the measurements can easily be affected by a number of non-constitutive factors. For example, the sample size and mass could be affecting the response in a dynamic sense that will be interpreted, naively, as material response. Gripping and other frictional issues may also become more tricky. Finally, at such frequency levels running a steady state mechanical oscillation tends to heat up the sample and there may not be an easy way of controlling and maintaining uniform temperature throughout the sample. On the other end of the spectrum, ultrasonic (US) testing is a robust way of characterizing the elastic and loss properties materials at moderate and high frequencies (100 kHz and above). The temperature rise is not so much an issue in this case as the pulse lengths could be limited (short duty cycle). The challenge, on the other hand, is the potentially high loss factors

W. Nantasetphong SCG Chemicals Co., Ltd., Bangsue, Bangkok, Thailand Z. Jia Department of Mechanical Engineering, University of Connecticut, Storrs, CT, USA M. Arif Hasan · S. Nmeat-Nasser Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA, USA A. V. Amirkhizi () Department of Mechanical Engineering, University of Massachusetts, Lowell, Lowell, MA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_47

255

256

W. Nantasetphong et al.

in shear waves and low signal to noise ratio in low impedance materials under test with piezo-based transducers. In previous work, we have shown the signal to noise issue and loss factor may be overcome [1–4] even for rubbery low impedance media. Frequency of acoustic applications lies in between the two ranges and introduces a different set of challenges. In particular, the wavelengths generally become larger, not only in the sample but also in the test apparatus media. This leads to internal reflections. The complexity of resolving multiple internal reflections in transmission/reflection measurements may be avoided by expanding the test setup sizes. But this approach becomes expensive and particularly challenging around 10 kHz range. The low impedance of elastomers confounds the measurement using standard wave propagation techniques as well since the transmitted signals become too small when the test medium has a high impedance. Time-temperature superposition (TTS) may be used to establish estimates of polymer properties at such experimentally difficult ranges by making measurements at lower or higher frequencies and assuming simple thermo-rheology. In the past this method has provided very good estimates of storage moduli but has fallen short in predicting the loss response of complex elastomers such as polyurea [3]. Therefore, when accurate estimates of both storage and loss moduli are required for design purposes at such frequency ranges, one would need a direct measurement. This has motivated the development of a low frequency setup to measure transmission and reflection of low intensity stress waves from a sample of low impedance material.

47.2 Experimental Setup and Procedure The split Hopkinson pressure bar (SPHB) setup is used for testing of materials under very high strain rates and in general is ideal for establishing flow stress and hardening response beyond small linearly elastic strain ranges. The same setup, however, cannot be used for elastic and viscoelastic measurements for two reasons. First, the dynamic equilibrium is not easily attainable for the linear part of the stress-strain curve with any reasonable modulus. Second, in a normal SHPB test, the strains are immediately pushed beyond proportional limits leading to plastic deformation. To avoid both these challenges, but to utilize the long wavelength capabilities of SHPB system, the pulse generation was changed from a high intensity impact (e.g. striker accelerated in a gas gun) to a ball-drop impact. Furthermore, the standard strain gages were replaced by semiconductor strain gages that could be used for much lower levels of strain in dynamic testing. The sample is enclosed in an environmental chamber to make measurements at multiple temperatures, enabling estimates of thermal effects at mid-range frequencies and secondary check of the TTS procedure. At each temperature level, cylindrical samples of multiple lengths and the same diameter as the bars are placed in between the two bars and an input pulse is sent from the far end of the incident bar. The incident and reflected signals are measured at the same location, yet, due to the length of the bar they are separated in time, therefore eliminating the need to resolve internal reflections (as is usually the case in SHPB technique). The transmission signal is measured in the second bar.

47.3 Analysis Method and Discussion The collected signals are first treated using phase spectral analysis method [5]. In this method, the phase of the transmitted signal is determined as a function of frequency and its ambiguity is removed by enforcing its continuity. It is assumed that the measurement starts at low enough frequency that at least two samples of different length are close enough to provide an unwrapped phase difference as the basis. The loss is estimated based on the comparison of the amplitudes of the transmitted signals at two sample lengths. It is clear that this approach, particularly with regards to the loss estimate, does not consider the internal reflections in the sample. While this may not be a significant issue when the wavelength is significantly higher or lower than the sample length, it becomes a major source of error when the two lengths are comparable. Very short sample sizes limit the accuracy of phase and wave speed measurements, while very long ones could be quite lossy and deteriorate the accuracy of amplitude measurements. To overcome this challenge, the results may be improved using a transfer matrix (TM) method to correct for the loss calculations [6]. Thus, the corrected wave speed and loss factor values may be converted to storage and loss moduli. These moduli were compared with the DMA-based master-curves for polyurea samples. The results have shown marked correction when compared with pure US data treated with TTS shifts. In particular the storage moduli from DMA master-curves determined

47 Storage and Loss Moduli of Low-Impedance Materials at kHz Frequencies

257

from low frequency and low temperature tests match very closely with the results of the present method at kHz frequency and room temperature. Furthermore, although not seen as the perfect match as in the storage modulus case, the agreement in the comparison of the loss moduli was closer than observed previously with US data and certainly supports use of DMA/TTS master-curves as basis for estimates. Acknowledgment This work has been conducted at the Department of Mechanical and Aerospace Engineering at University of California, San Diego and Department of Mechanical Engineering, University of Massachusetts, Lowell, and has been partially supported through DARPA and ONR grants to the two universities.

References 1. Nantasetphong, W., Jia, Z., Amirkhizi, A.V., Nemat-Nasser, S.: Dynamic properties of polyurea-milled glass composites Part I: Experimental characterization. Mech. Mater. 98, 142–153 (2016) 2. Nantasetphong, W., Amirkhizi, A.: V and Nemat-Nasser S, Constitutive modeling and experimental calibration of pressure effect for polyurea based on free volume concept. Polymer. 99, 771–781 (2016) 3. Qiao, J., Amirkhizi, A.V., Schaaf, K., Nemat-Nasser, S., Wu, G.: Dynamic mechanical and ultrasonic properties of polyurea. Mech. Mater. 43, 598–607 (2011) 4. Qiao, J., Nantasetphong, W., Amirkhizi, A.V., Nemat-Nasser, S.: Ultrasonic properties of fly ash/polyurea composites. Mater. Des. 89, 264–272 (2016) 5. Sachse, W., Pao, Y.H.: On the determination of phase and group velocities of dispersive waves in solids. J. Appl. Phys. 49, 4320 (1978) 6. Nemat-Nasser, S., Sadeghi, H., Amirkhizi, A.V., Srivastava, A.: Phononic layered composites for stress-wave attenuation. Mech. Res. Commun. 68, 65–69 (2015)

Chapter 48

Effects of Pressure and Strain Rate on the Mechanical Behavior of Glassy Polymers Abigail Wohlford, Timothy Walter, Daniel Casem, Paul Moy, and Addis Kidane

Abstract In this study the mechanical response of transparent polymers under varying strain rates and hydrostatic pressures is investigated. Quasi-static and dynamic tests are performed under uniaxial and multi-axial loadings and the effect of hydrostatic pressure on the response of the material is explored. In both quasi-static and dynamic experiments, the confinement pressure was increased to a predetermined level and kept constant during the test. This test subjects specimens to confining pressure (up to 200 MPa) prior to loading. Loading was either applied quasi-statically using a servo-hydraulic load frame or at dynamic rates using a modified SHPB. The strain rate dependency of the Polycarbonate material is studied using an Instron UTM and split Hopkinson pressure bar (SHPB) technique. Digital image correlation is used to record full field deformation and calculate the strain. Ultra-high strain rate tests are also performed using a micro-Kolsky bar and small (50–100 um length) specimens. The confinement experiments show that the yield stress is linearly proportional to the confinement pressure but the elastic modulus is insensitive to confinement pressure. Significant rate sensitivity is observed at moderate strain rates and becomes insensitive to strain rate at the highest strain rates measured. Keywords Hydrostatic pressure · Multi-axial loading · Polycarbonate · Ultra-high strain rate

48.1 Introduction Glassy polymers are thermoplastic materials widely used in many industries and everyday products. An important application for these polymers is transparent armor for impact protection, such as blast shields. Polycarbonate is a prime material commonly used due to its high transparency, high impact resistance, pliability, and its light weight. It can undergo large plastic deformation without fracturing, unlike many other thermoplastics. It is well known that yield stress depends on strain rate, temperature, and pressure. The mechanical response due to increasing strain rate has been studied using a variety of experimental techniques up to a peak strain rate. Dwiveldi [1] conducted tensile and compressive experiments at a wide range of strain rates to verify the JC and ZA polymer constitutive models. Millet and Bourne [2] were among the first to report the pressure dependency of PC using plate impact experiments. The characterization of polymers under multiaxial loading using a confining cylinder was examined by Ravi-chandar [3]. More experimental data is needed to increase our knowledge of deformation mechanisms for polymers and their response to loading rate and external pressure. The understanding of the impact response of PC is of great importance, therefore the events taking place during this reaction are needed to be understood. During impact it is well understood that high strain rates are occurring and this has been extensively studied. Confinement testing ideally reflects the stress states to that of real-life scenarios. During dynamic loading out in the field, it is often found that stress state is multiaxial, but the behavior under such conditions is not well understood. Therefore it is essential to have reliable experimental work done to attain a rudimentary definition of these polymer’s dynamic properties and nonlinear mechanical response. Previous testing methods for confinement used a metal sleeve around the sample. This technique was not ideal because the pressure on the sample

A. Wohlford () Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Army Research Lab, Aberdeen Proving Ground, Aberdeen, MD, USA e-mail: [email protected] T. Walter · D. Casem · P. Moy US Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA A. Kidane Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_48

259

260

A. Wohlford et al.

wasn’t constant and many assumptions had to be made about the strain in the metal ring. There are experimental difficulties that cause there to be little to no data of PC reported above strain rates of 104 s−1 . In this work, the mechanical behavior of polycarbonate and its sensitivity to strain rate and hydrostatic pressure is investigated. Results from these efforts will lead to the development of accurate material models for thermoplastic polymers. The compressive strength of the material is evaluated to study these external effects. The uniaxial quasi-static and high rate tests are done using traditional experimental techniques but the ultra-high rate and confinement tests are executed using recent techniques.

48.2 Materials and Methods For the current work commercial Polycarbonate was purchased from McMaster Carr. The translucent thermoplastic polymer was extruded into cylindrical rods. Then the 0.5 bars were machined into specimens of 0.5 length, and the 1 diameter bar into 2 long samples. Also 0.25 length sample were cut from the 0.5 bar in order to get higher strain rates for the quasi-static testing. A femtosecond laser integrated into a Scanning Electron Microscope was used to cut 300 μm diameter samples from PC sheet of thickness 150 μm for ultra-high strain rate experiment. The quasi-static compression tests were performed with a servo-hydraulic Instron machine. Strain rates of 0.001/s up to 10/s were achieved under displacement control. During the tests the load cell measured the force applied to the specimen and stereoscopic DIC was implemented to measure the full-field deformation. Point Gray 2.3 MP cameras were used to capture the deformation during loading. The captured images were then post-processed in the digital image correlation code VIC-3D (Correlated Solutions) to get the strain. The quasi-static confinement tests were carried out with a servo-hydraulic load frame with high pressure confinement chamber designed by Dynamic Systems and Research. The pressure intensifier can pressurize the confining chamber up to 200 MPa. The axial displacement as well the lateral deformation of the sample are measured with LVDT sensors. The internal load cell and a pressure transducer connected to the intensifier measured the applied load and confining pressure. First the PC sample was prepared by attaching extensometer feet and then enclosing it with a rubber seal membrane along with two steel platens put on the top and bottom of the sample. The entire thing was covered with a nitrile rubber coating to ensure that no fluid could seep into the membrane. The LVDT’s were then connected to the sample and it was placed inside the chamber. The pressure chamber was sealed shut and the compressible fluid was pumped inside to radially confine the sample to a predetermined pressure. The axial shear load was applied with displacement control while maintaining a constant confining pressure. Von Mises yield criterion is applied since it has been found to govern the deformation of materials under complex loading situations (Fig. 48.1).

Axial Load Rod Load Cell

High Pressure Supply Line

Top Cap Upper Pressure Chamber Triaxial Loader

Specimen

Bottom Seal Plate

Bottom Cap Control Station Pressure Intensifier

Lowered

Fig. 48.1 Quasi-static confinement system schematic and photograph

48 Effects of Pressure and Strain Rate on the Mechanical Behavior of Glassy Polymers

261

Fig. 48.2 Triaxial split Hopkinson pressure bar set-up Fig. 48.3 Strain rate vs. yield stress

120 Yield Stress (MPa)

100 80 60 40 20 0 0.001 0.01

0.1

1

10

100

1000 10000

Strain Rate (/s)

Intermediate to high strain-rate compression tests at 102 –103 s−1 were carried out using a Split Hopkinson Pressure Bar (SHPB) with 20 ft. long Aluminum bars. Each of the ¾” bars were instrumented with strain gages to measure the elastic waves. Applying 1D wave analysis we can infer the stress-strain history of the sample from the gauges. Pulse shapers were utilized to help obtain a constant axial strain rate. The uniformity of the deformation during loading was monitored with a high speed camera. In order to perform compression tests at even higher strain rates, beyond 10,000/s, a mini-Kolsky bar set-up is implemented. Since it is unfeasible to use strain gages on an 800 μm bar, laser interferometer-based techniques are employed. A Transverse Displacement Interferometer (TDI) measured the incident and reflected pulses in the incident bar, and a Normal Displacement Interferometer (NDI) measured the transmitted pulse in the transmitter bar. From these displacement measurements, basic wave analysis can be used to analyze the response. High rate confinement compression tests were executed using a modified SHPB. The experimental setup consists of a SHPB integrated with a hydrostatic pressure system. As seen in Fig. 48.2 there are two confining vessels, one for the test sample and the other for the free end of the transmitter bar. A tie-rod assembly is incorporated to eliminate a reaction moment created in the bars. The pressure intensifier supplies hydrostatic confining pressure to the sample and the SHPB applies the dynamic axial load.

48.3 Results and Discussion The results presented are just from the preliminary tests and much of this work is still ongoing. First the effects of strain rate on the deformation behavior are investigated. The force and displacement are converted into stress and strain, and the maximum stress is plotted as the yield stress. The quasi-static experiments were conducted at three strain rates: 0.001/s, 1/s, and 10/s. The dynamic compression tests were done at two different rates: 250/s and 1200/s. The ultra-high rate data is not currently shown due to defects in the small samples from machining. The results are shown to be consistent with the literature, the yield stress increases with increasing strain rate. The strength of the polymers has a positive correlation to strain rate. At some strain rate between the quasi-static and dynamic data there appears to be a dramatic increase in the effect of strain rate on the apparent yield stress. All of the tests were found to be consistent and repeatable (Fig. 48.3). The quasi-static and dynamic yield stresses as functions of pressure are shown in Fig. 48.4. The zero pressure data are the yield stresses found during the uniaxial tests. As would be predicted, as the pressure went up so did the yield stress. It can be observed that the yield stress exhibits strong pressure dependence. The high rate tests appear to have a stronger correlation

262 140 Yield Stress (MPa)

Fig. 48.4 Pressure vs. yield stress

A. Wohlford et al.

Quasi-static (1/sec) Dynamic (1000 /sec)

120

100

80

0

50

100

150

200

Pressure (Mpa)

to pressure. More tests will be conducted to further study this trend. It is very apparent that strain rate and pressure are important factors that influence the mechanical response of polymers. These elements will need to be taken into account when considered this material for certain problems.

48.4 Summary The effects of strain rate and hydrostatic pressure on the mechanical response of Polycarbonate was studied by performing a series of experimental tests. Multiple test setups were used to perform this study using quasi-static test machine and split Hopkinson bar along with modified setups to include confinement. The compression tests covered a strain-rate range of 0.001/s to 20 k/s and a hydrostatic pressure range up to 200 MPa. Digital Image correlation was used to analyze the slow to high rate uniaxial tests, and laser interferometry was implemented to evaluate the ultra-high rate tests. It was clearly observed that the strain rate and pressure are very influential factors on the mechanical response of the material. Flow stress for the PC shows significant rate sensitivity but at rates 250/s and above there appears to be a saturation point most likely due to adiabatic heating affects. Acknowledgement This research was supported in part by an appointment to the Postgraduate Research Participation Program at the U.S. Army Research Laboratory administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and USARL.

References 1. Dwivedi, A., Bradley, J., Casem, D.: Mechanical response of polycarbonate with strength model fits 2. Millet, J., Bourne, N.: Shock and release of polycarbonate under one–dimensional strain 3. Ravi-Chandar, K., Ma, Z.: Inelastic deformation in polymers under multiaxial compression 4. Van Melick, H.G.H.: The influence of intrinsic strain softening on the macroscopic deformation behaviour of amorphous polymers

Chapter 49

The Role of Texture on the Strain-Rate Sensitivity of Mg and Mg Alloy AZ31B Nathan Briggs, Moriah Bischann, and Owen T. Kingstedt

Abstract In this study, the role of texture on the quasi-static and dynamic response of pure magnesium and magnesium alloy AZ31B is investigated. Texture is imparted through thermo-mechanical processes of hot-rolling or equal channel angular extrusion. Constant strain-rate, both dynamic and quasi-static, and strain-rate jump experiments, dynamic-quasistatic, quasi-static-dynamic and dynamic-dynamic, are used to examine plastic flow anisotropy and strain hardening response. Observations of the macroscopic material behavior, specifically the stress-trajectory, are supported by electron back-scatter diffraction analysis to gain insights of texture evolution and predominant deformation processes taking place during deformation increments. Keywords Magnesium · AZ31B · Dynamic response · Strain-rate sensitivity · Strain hardening

49.1 Introduction Magnesium (Mg) and Mg alloys have received significant research attention because their high specific strength potentially makes them ideal candidates for use as structural members in light-weight applications such as automotive and aerospace vehicles. Recent work has been directed towards investigating Mg and Mg alloy microstructure texture development as a result of thermomechanical processing routes (i.e., extrusion, rolling and equal channel angular extrusion (ECAE) [1]). The limited number of textures that can be obtained in Mg through conventional processing techniques are well understood, particularly in the case of rolling. To summarize results relevant to the current study, when rolled material loaded is loaded in compression along the rolling direction (RD) or transverse direction (TD), the material behavior is largely strain-rate insensitive [2]. The strain-rate insensitivity is due to deformation primarily being accommodated via deformation twinning which is generally accepted as both temperature and strain-rate insensitive [3]. Loading along the normal direction (ND) results in a rate-sensitive response as deformation occurs primarily through pyramidal slip [4]. The microstructure texture obtained through ECAE is processing route, alloying component and temperature dependent. Thus there is a rich space of processing variables to explore which could potentially lead to positive increases in mechanical strength and strain to failure. Mukai et al. [5] were the first to demonstrate increased properties in Mg alloy AZ31B after ECAE processing, showing a two to threefold increase in elongation under quasi-static tensile loading. Following this initial study, a body of work was conducted which examined the texture evolution and mechanical properties of Mg [6] and Mg alloys (e.g., AZ31 [5, 7, 8], AZ61 [9], ZK60 [7]). Mechanical results are difficult to interpret because there have been conflicting reports in literature. For instance, it has not been definitively established if ECAE processing of Mg and Mg alloys increases yield strength [10–12], decreases yield strength [13–15], increases strain to failure [13, 15], or decreases strain to failure [10]. Yet, from the available collection of ECAE studies on Mg and Mg alloys, it can be concluded that texture and loading orientation are the dominant features in determining mechanical response. With the goal for Mg and Mg alloys to supplant currently used structural materials, greater understanding of the mechanical response under high strain-rate conditions are necessary. The focus of this study is to provide insight into the macroscopic mechanical response of hot rolled AZ31B, ECAE AZ31B and ECAE pure Mg, over the quasi-static and dynamic range through the use of constant strain-rate, strain-rate jump, and interrupted experiments.

N. Briggs · O. T. Kingstedt () Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA e-mail: [email protected] M. Bischann Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_49

263

264

N. Briggs et al.

49.2 Experimental Methodology Three thermomechanically processed materials were examined in this study (see Fig. 49.1). The first, a hot rolled AZ31B Mg alloy comprised of 3% Al, 1% Zinc, and other trace elements, with the basal plane normal aligned to the processing ND of the rolled material. The elevated temperature of the rolling process allows for a relatively large grain size of 32 μm to be maintained. The second material examined was ECAE AZ31B (grain size 4.2 μm), and the third was ECAE Mg (grain size 8.6 μm). To prevent a strong texture from developing in the ECAE material, the 4Bc route was selected. The 4Bc processing route consists of four processing passes with a ninety-degree rotation about the extrusion direction of each pass. To prevent shear localization and failure during ECAE processing, elevated temperatures of 250 ◦ C and 200 ◦ C were used for AZ31B and pure Mg, respectively [7]. Using electrical discharge machining (EDM), cylinders of standard dimensions for Kolsky (split-Hopkinson) pressure bar experiments were cut from hot rolled plate such that their axial direction was parallel to the normal direction (ND), rolling direction (RD), or at 45◦ to the normal direction (45◦ ). Specimens of the ECAE material were machined to examine the material response with loading directions matching the principle processing directions: the extrusion direction (ED), longitudinal direction (LD), and transverse direction (TD). High strain-rate experiments were conducted using a Kolsky (split-Hopkinson) pressure bar [16]. Dynamic constant strainrate experiments were conducted using striker bars with the same diameter (19.05 mm) as the incident and transmitted bars. A subset of experiments were conducted to predetermined levels of strain using stop rings. Dynamic-dynamic strain-rate jump experiments were conducted using striker bars with a discontinuous cross-section in line with previous studies [17–19]. For this study, the striker bars used for strain-rate jump experiments had a 50% diameter reduction (19.05–9.52 mm) halfway along their length. To promote a strain-rate decrement experiment (ε˙1 > ε˙2 ), the 19.05 mm diameter section of the striker bar impacts the incident bar. Conversely, to promote a strain-rate increment experiment (ε˙1 < ε˙2 ), the reduced diameter section (9.52 mm) impacts the incident bar. Strain gage signals of the wave reflected from the incident bar / specimen interface, which is used to calculate the nominal strain-rate experienced by a specimen during loading, are shown in Fig. 49.2 with the orientation of the striker bar above each wave profile.

Fig. 49.1 Representative EBSD color maps (a,d,g) and inverse pole figures (b,c,e,f,h,i) of the three materials examined

49 The Role of Texture on the Strain-Rate Sensitivity of Mg and Mg Alloy AZ31B

265

Fig. 49.2 Loading rate profiles for constant, increment and decrement experiments with striker bar orientation

Fig. 49.3 Mechanical behavior of (a) Rolled AZ31B, (b) ECAE AZ31B, and (c) ECAE Mg

Quasi-static strain rate experiments were accomplished using an Instron servo-hydraulic loadframe. Constant rate experiments were conducted at 10−2 , 10−3 and 10−4 s−1 . Strain rate increment and decrement experiments were carried out between these strain rates to achieve a detailed picture of material behavior of deformation behavior over the quasi-static regime.

49.3 Results and Discussion The strongly textured rolled AZ31B material response showed the greatest anisotropy in plastic flow behavior between the selected loading orientations. Loading along the ND results in the characteristic concave down stress-strain curve (green trace with downward pointing triangle in Fig. 49.3). The initial slight sigmoidal shape occurring over the first 0.015 strain, is attributed to the early stages of deformation occurring through basal slip, due to slight misalignment of the basal plane normal direction and the loading direction. Loading along the RD places ! ( the) c-axis of the hcp crystal in tension, resulting in deformation primarily occurring through extension twinning 1012 1011 . Extension twinning causes an 86.3◦ rotation of the crystal with a shear corresponding to a strain of ∼0.065 along the c-axis [20]. The 86.3◦ rotation reorients the c-axis to be nearly parallel to the RD loading direction. Further deformation is accommodated through deformation slip which corresponds to a dramatic increase in strain hardening seen at strains greater than ∼0.065 [21]. As was seen in previous studies, loading along the RD results in a strain-rate insensitive response [2]. Specimens cut at 45 ◦ to the ND and RD have basal planes aligned to the plane of maximum shear, and thus deformation is accommodated through basal slip. As a result, the yield stress of the 45◦ orientation was the lowest of the examined orientations and demonstrated nearly constant strain hardening over all strain-rates examined point to a lack of strain-rate sensitivity of basal slip accommodated deformation. The texture imparted by ECAE softens the anisotropic deformation response. As is shown in !Fig. ( 49.3, ) ECAE AZ31B loading along the TD results in a sigmoidal response characteristic of extension twinning 1012 1011 occurring during the first portion of deformation. Unlike loading along the RD in the rolled AZ31B, there is an apparent strain-rate sensitivity

266

N. Briggs et al.

indicating the likelihood of both deformation slip and twinning mechanisms accommodating deformation as it is well known that deformation twinning in Mg is rate insensitive. Loading along the LD and ED results in a similar deformation response of constant hardening before localization and failure along a plane of maximum shear inclined at 45 ◦ to the loading direction. As has been seen in previous studies of Mg and Mg alloys, there is an increasing strain to failure with increasing strain-rate [2]. The mechanical response of the comparatively weak textured ECAE Mg is shown in Fig. 49.3 for loading along the ED, LD, and TD directions. Curves marked with upwards or downwards triangles are for specimens loaded to failure. Each of the loading orientations failed at similar levels of strain when subjected to the highest strain rates (curves marked by downward pointing triangles). The ED specimens were both loaded to failure showing an increase in the strain to failure with increasing strain rate. The TD loading orientation possessed the greatest strength followed by the LD and ED directions. Of important note is the complete absence of a sigmoidal stress-strain response pointing to the ability of ECAE processing to reduce plastic flow anisotropy between loading directions through texture modification. In other words, the softened texture prevents single mechanisms such as pyramidal or extension twinning from dominating the material response. From the stress-strain response an apparent strain-rate sensitivity is present when loading along the LD.

49.4 Conclusions In the present work, the mechanical response of two ECAE processed hcp materials, pure Mg and Mg alloy AZ31B, have been examined over in the dynamic regime. Constant strain-rate and dynamic-dynamic strain-rate jump experiments were utilized to investigate the effect of imparted texture on the strain-rate sensitivity with respect to loading along principle processing directions. From this study the following conclusions can be made: 1) ECAE 4Bc processing is an appropriate technique to reduce the microstructural texture strength in Mg and Mg alloy AZ31B. 2) The smeared texture obtained in the ECAE Mg reduced the anisotropy of the material response over the loading orientations examined. 3) Strain-rate sensitivity was observed in the ECAE processed material systems in the hardening response. In ECAE AZ31B rate sensitivity was observed when loading along the ED but not definitively along the LD. In ECAE Mg rate sensitivity was observed in constant strain-rate experiments along the LD and TD, but not along the ED. Strain-rate sensitivity was also apparent during the second strain-rate regime (ε˙2 ) when loading along the LD and TD.

References 1. Furukawa, M., Horita, Z., Nemoto, M., Langdon, T.G.: Review: processing of metals by equal-channel angular pressing. J. Mater. Sci. 36(12), 2835–2843 (2001) 2. Tucker, M.T., Horstemeyer, M.F., Gullett, P.M., El Kadiri, H., Whittington, W.R.: Anisotropic effects on the strain rate dependence of a wrought magnesium alloy. Scr. Mater. 60(3), 182–185 (2009) 3. Christian, J.W., Mahajan, S.: Deformation twinning. Prog. Mater. Sci. 39(1–2), 1–157 (1995) 4. Knezevic, M., Levinson, A., Harris, R., Mishra, R.K., Doherty, R.D., Kalidindi, S.R.: Deformation twinning in AZ31: influence on strain hardening and texture evolution. Acta Mater. 58(19), 6230–6242 (2010) 5. Mukai, T., Yamanoi, M., Watanabe, H., Higashi, K.: Ductility enhancement in AZ31 magnesium alloy by controlling its grain structure. Scr. Mater. 45(1), 89–94 (2001) 6. Suwas, S., Gottstein, G., Kumar, R.: Evolution of crystallographic texture during equal channel angular extrusion (ECAE) and its effects on secondary processing of magnesium. Mater. Sci. Eng. A. 471(1–2), 1–14 (2007) 7. Agnew, S.R., Mehrotra, P., Lillo, T.M., Stoica, G.M., Liaw, P.K.: Texture evolution of five wrought magnesium alloys during route A equal channel angular extrusion: experiments and simulations. Acta Mater. 53(11), 3135–3146 (2005) 8. Xia, K., Wang, J.T., Wu, X., Chen, G., Gurvan, M.: Equal channel angular pressing of magnesium alloy AZ31. Mater. Sci. Eng. A. 410, 324–327 (2005) 9. Kim, W.J., Hong, S.I., Kim, Y.S., Min, S.H., Jeong, H.T., Lee, J.D.: Texture development and its effect on mechanical properties of an AZ61 Mg alloy fabricated by equal channel angular pressing. Acta Mater. 51(11), 3293–3307 (2003) 10. Su, C.W., Chua, B.W., Lu, L., Lai, M.O.: Properties of severe plastically deformed Mg alloys. Mater. Sci. Eng. A. 402(1–2), 163–169 (2005) 11. Su, C.W., Lu, L., Lai, M.O.: Mechanical behaviour and texture of annealed AZ31 Mg alloy deformed by ECAP. Mater. Sci. Technol. 23(3), 290–296 (2007) 12. Janecek, M., Popov, M., Krieger, M.G., Hellmig, R.J., Estrin, Y.: Mechanical properties and microstructure of a Mg alloy AZ31 prepared by equal-channel angular pressing. Mater. Sci. Eng. A. 462(1–2), 116–120 (2007) 13. Agnew, S.R., Horton, J.A., Lillo, T.M., Brown, D.W.: Enhanced ductility in strongly textured magnesium produced by equal channel angular processing. Scr. Mater. 50(3), 377–381 (2004)

49 The Role of Texture on the Strain-Rate Sensitivity of Mg and Mg Alloy AZ31B

267

14. Kim, H.K., Kim, W.J.: Microstructural instability and strength of an AZ31 Mg alloy after severe plastic deformation. Mater. Sci. Eng. A. 385(1–2), 300–308 (2004) 15. Kim, W.J., Jeong, H.T.: Grain-size strengthening in equal-channel-angular-pressing processed AZ31 Mg alloys with a constant texture. Mater. Trans. 46(2), 251–258 (2005) 16. Gama, B.A., Lopatnikov, S.L., Gillespie, J.J.W.: Hopkinson bar experimental technique: a critical review. Appl. Mech. Rev. 57(4), 223–250 (2004) 17. Nicolazo, C., Leroy, M.: Dynamic behaviour of alpha-iron under decremental step pulses. Mech. Mater. 34(4), 231–224 (2002) 18. Rittel, D., Ravichandran, G., Venkert, A.: The mechanical response of pure iron at high strain rates under dominant shear. Mater. Sci. Eng. A. 432(1–2), 191–201 (2006) 19. Leroy, M., Raad, MK., Nkule, L., Cheron, R.: Influence of instantaneous dynamic decremental incremental strain rate tests on the mechanicalbehavior of metals – Application to high-purity polycrystalline. Institute of Physics Conference Series, vol. 70, pp. 31–38. (1984) 20. Wonsiewicz, B.C., Backofen, W.A.: Plasticity of magnesium crystals. Trans. Metall. Soc. AIME. 239(9), 1422 (1967) 21. Lou, X.Y., Li, M., Boger, R.K., Agnew, S.R., Wagoner, R.H.: Hardening evolution of AZ31B Mg sheet. Int. J. Plast. 23(1), 44–86 (2007)

Chapter 50

Shock Compaction of Al Powder Examined by X-Ray Phase Contrast Imaging A. Mandal, M. Hudspeth, B. J. Jensen, and S. Root Abstract Shock compaction response of ∼50% porous aluminum powder, encapsulated in PMMA cylinders and impacted at 0.3–1.7 km/s using 6061-T6 Al impactors, was examined in situ and in real time using a propagation-based X-ray phase contract imaging (PCI) technique capable of providing micron spatial resolution at the Advanced Photon Source. Numerical simulations of the PCI data accurately captured the propagating compaction shock wave in the powder and the deformation of the powder column. Keywords Shock compaction · Aluminum powder · X-ray phase contrast imaging · Shock-polar method · CTH simulation

50.1 Introduction Shock response of porous materials has been examined extensively to explore thermodynamic states not reachable using fully dense solids [1–3], to determine Grüneisen parameter [2, 4], and to gain insight into a wide range of phenomena that are important to industrial and military applications [5–8], as well as planetary science [9, 10]. Shock response of porous bodies is significantly more complex and far less understood compared to that of typical solids, because it involves processes operative at multiple length- and time-scales that are influenced by intrinsic (elastic moduli, strength etc.) and extrinsic properties (e.g. porosity, particle morphology etc.) of the powder [6]. Traditionally, stress gauges, laser velocimetry (VISAR/PDV) [2, 4], and post-shock analysis of recovered specimens [5, 6] have been used to examine shock response of porous materials. While useful, these techniques provide only limited and indirect information about the compaction process. In this work, the coupling of propagation-based X-ray phase contrast imaging (PCI) technique with dynamic loading platforms at the Dynamic Compression Sector (DCS) of the Advanced Photon Source (APS) [11, 12] was utilized to examine shock response of ∼50% porous aluminum (Al) powder (−325 mesh) in situ and in real time. The PCI data were simulated to gain insight into the observed material response and to evaluate an Al powder material model.

50.2 Experimental Details As shown in Fig. 50.1, PMMA cylinders (8 mm dia. × 6 mm long), encapsulating an 800-μm diameter Al powder column, were impacted by 6061-T6 Al impactors at 0.3–1.7 km/s using Los Alamos National Laboratory’s (LANL) IMPULSE gun [13, 14] and DCS powder gun. Upon impact, a shock wave propagated into the PMMA base plate and subsequently interacted with the PMMA/powder interface resulting in an initially planar transmitted shock wave into the powder. Propagating shock waves and the subsequent deformation of the Al powder column were recorded through a 1.726 mm-wide field of view (FOV) using LANL multi-frame X-ray PCI (MPCI) system. This was achieved as follows: 23.5 keV white X-ray beam propagating through the PMMA cylinder impinged onto a LSO scintillator 700 mm away (Fig. 50.1). The X-ray photons, converted to visible photons at the scintillator, were then imaged via four PI-MAX intensified charge-coupled device (ICCD) detectors

A. Mandal () · B. J. Jensen Shock and Detonation Physics Group, Los Alamos National Laboratory, Los Alamos, NM, USA M. Hudspeth · S. Root Dynamic Material Properties, Sandia National Laboratories, Albuquerque, NM, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_50

269

270

A. Mandal et al.

Fig. 50.1 Schematic of the experimental setup. The Multi-frame X-ray Phase Contrast Imaging (MPCI) system developed on the IMPULSE (IMPact System for ULtrafast Synchrotron Experiments) by LANL [13, 14] is also shown

(Princeton Instruments) at 7.5x magnification using an objective microscope. The ICCDs were triggered in sequence to capture images from a series of 80 ps-width X-ray bunches arriving every 153.4 ns. Synchronization of the detectors with the dynamic event and the desired X-ray bunches was achieved using a radio frequency (RF) signal that is coincident with each X-ray bunch (bunch clock), a delay generator (DG) and a PZT impact pin. For further details, see Refs. [11–15].

50.3 Results and Discussion The PCI images and the corresponding CTH simulations [16], presented in the top and bottom rows of Fig. 50.2a, respectively, show the evolution of the propagating compaction shock front in the powder and the powder column shape inside the FOV. The CTH simulations used the SESAME 3700 equation of state (EOS) for Al, which treats it as an elasticperfectly plastic solid. Additionally, a P-α model [17] was used to describe the crush-up behavior of Al powder. Figure 50.2a suggests that the compaction shock velocity was accurately captured in the numerical simulations. The propagating shock was vertical near the center and curved/oblique toward the horizontal powder/PMMA interfaces. The compacted powder density depended on both axial and radial locations, and it was the largest along the horizontal powder/PMMA interfaces. Figure 50.2b suggests that the shape of the compaction shock front depended on the impact velocity.

50.4 Conclusions Propagation of the compaction shock wave in Al powder and subsequent deformation of the compacted powder were visualized directly using a propagation-based X-ray PCI technique. Numerical simulations accurately captured the compaction shock velocity and the observed deformation of the Al powder column, suggesting that the Al powder material model used in our simulations can sufficiently describe its compaction response over the wide range of pressure/density states

50 Shock Compaction of Al Powder Examined by X-Ray Phase Contrast Imaging

271

Fig. 50.2 Phase contrast images (top, shown in false color) and corresponding CTH simulations (bottom) showing (a) the evolution of the propagating compaction shock front in the Al powder and the deformation of the powder column (impact velocity: 1.7 km/s), and (b) the differences in compaction shock structure (impact velocity: 0.3 – 1.7 km/s). Shock wave propagation direction is indicated with the black arrows. Time is relative to impact

realized in our work. To further confirm of the accuracy of this material model, density of the shock compacted powder will be determined directly from the PCI images [15, 18] and will be compared against the density calculated from the CTH simulations. Acknowledgements This publication is based upon work performed by Los Alamos National Laboratory (LANL) and Sandia National Laboratories (SNL) at the Dynamic Compression Sector (DCS) of the Advanced Photon Source (APS). All PCI data shown in this work were obtained using LANL’s novel multi-frame X-ray phase contrast imaging (MPCI) system developed on the IMPULSE capability at APS. A.M. and B.J.J. acknowledge the financial support provided by LANL Science Campaigns, Joint Munitions Program (JMP), and MaRIE concept, and National Security Technologies (NSTec) Shock Wave Physics Related Diagnostic (SWRD) program. M.H and S.R. acknowledge financial support provided by the Truman fellowship (LDRD) and Science Campaigns within SNL. Paulo Rigg and the DCS team is thanked for their assistance with the experiments; Chuck Owens, Joe Rivera (LANL) and Jim Williams (SNL) are thanked for target assembly; Adam Iverson, Carl Carlson and Matt Teel (NSTec) are thanked for their assistance with the PCI system. LANL is operated by Los Alamos National Security, LLC for the U.S. Department of Energy (DOE) under Contract No. DE-AC52-06NA25396. SNL is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. DOE, NNSA under contract DE-NA0003525. DCS is supported by the U. S. DOE, NNSA, under Award Number DE-NA0002442 and operated by Washington State University (WSU). This research used resources of APS, a U.S. DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

References 1. Anderson, G.D., Doran, D.G., Fahrenbruch, A.L.: Equation of State of Solids: Aluminum and Teflon, Tech. Rep. AFWL-TR-65-147 (Air Force Weapons Laboratory), 1965 2. Ahrens, T.J.: Equation of state. In: Asay, J.R., Shahinpoor, M. (eds.) High Pressure Shock Compression of Solids, pp. 75–114. Springer, New York (1993.) Chap. 4) 3. Bonnan, S., Hereil, P.-L., Collombet, F.: Experimental characterization of quasi static and shock wave behavior of porous aluminum. J. Appl. Phys. 83(11), 5741–5749 (1998) 4. Kraus, R.G., Chapman, D.J., Proud, W.G., Swift, D.C.: Hugoniot and spall strength measurements of porous aluminum. J. Appl. Phys. 105, 114914 (2009) 5. Gourdin, W.H.: Dynamic consolidation of metal powders. Prog. Mater. Sci. 30, 39–80 (1986) 6. Eakins, D.E., Thadhani, N.N.: Shock compression of reactive powder mixtures. Int. Mater. Rev. 54(4), 181–213 (2009) 7. Perry, J.I., Braithwaite, C.H., Taylor, N.E., Jardine, A.P.: Behavior of moist and saturated sand during shock and release. Appl. Phys. Lett. 107, 174102 (2015) 8. Sheffield, S.A., Gustavsen, R.L., Anderson, M.U.: Shock loading of porous high explosives. In: Davison, L., Horie, Y., Shahinpoor, M. (eds.) High-Pressure Shock Compression of Solids IV: Response of Highly Porous Solids to Shock Loading, pp. 23–62. Springer, New York (1997.), Chap. 2) 9. Stöfller, D., Gault, D.E., Wedekind, J., Polkowski, G.: Experimental hypervelocity impact into quartz sand: distribution and shock metamorphism of ejecta. J. Geophys. Res. 80(29), 4062–4077 (1975)

272

A. Mandal et al.

10. Housen, K.R., Holsapple, K.A., Voss, M.E.: Compaction as the origin of the unusual craters in the asteroid Mathilde. Nature. 402, 155–157 (1999) 11. Jensen, B.J., et al.: Ultrafast, high resolution, phase contrast imaging of impact response with synchrotron radiation. AIP Adv. 2, 012170 (2012) 12. Luo, S.N., et al.: Gas gun shock experiments with single-pulse x-ray phase contrast imaging and diffraction at the advanced photon source. Rev. Sci. Instrum. 83, 073903 (2012) 13. Jensen, B.J., et al.: Impact system for ultrafast synchrotron experiments. Rev. Sci. Instrum. 84, 013904 (2013) 14. Jensen, B.J., et al.: Dynamic experiment using IMPULSE at the advanced photon source. J. Phys. Conf. Ser. 500, 042001 (2014) 15. Jensen, B.J., et al.: X-ray phase contrast imaging of granular systems, submitted to Springer (LA-UR-17-27104) 16. McGlaun, J.M., Thompson, S.L., Elrick, M.G.: CTH: a three-dimensional shock wave physics code. Int. J. Impact Eng. 10(1), 351–360 (1990) 17. Hermann, W.: Constitutive equation for the dynamic compaction of ductile porous materials. J. Appl. Phys. 10(6), 2490–2499 (1969) 18. Paganin, D., et al.: Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J. Microsc. 206(1), 33–40 (2002)

Chapter 51

Shock Compression Response of Model Polymer/Metal Composites David Bober, Yoshi Toyoda, Brian Maddox, Eric Herbold, Yogendra Gupta, and Mukul Kumar

Abstract Heterogeneous materials do not respond mechanically to an impulse in the manner of homogeneous metals and alloys. Wave propagation in a microstructure with chemically distinct identities, that are only in incidental contact with each other, is a complex process and not well understood. Here we report on a series of plate-impact experiments on a polymer-metal composite, where the volume fraction of the metallic phase was systematically varied from 0% to 40%, while other parameters like the sample thickness were kept constant. The velocity histories at the sample/window interfaces were measured to examine the continuum response corresponding to the internal materials processes. The unfilled polymer demonstrated a steady shock wave response; whereas the wave profiles obtained from mixture samples showed structured waves that depended on the volume fraction of the fill. The shock wave profiles were quantified using parameters strongly correlated to the material composition. The likely physical basis of these correlations is discussed. Keywords Shock response · Heterogeneous material · Particulate composite · Plate impact · Stress wave

51.1 Introduction Shock waves propagating in matrix-particle composites exhibit complex structures that are distinct from their homogenous counterparts. This class of materials has relevance to many shock wave related applications, such as polymer bonded explosives, electronics packaging, and saturated soils. Thus, a good understanding of shock wave structures in heterogeneous materials is important for developing improved material models, which are needed for engineering hydrocode calculations. In the simple case of a matrix filled lightly with perfectly elastic spheres, shock loading results in a multiphase flow characterized by unsteady stress/velocity fields around each particle. In principle, point mass models with appropriate equations for momentum and heat transfer between the phases could be used to provide a macroscale understanding of these flows [1]. However, particle deformation, fracture, and particle-particle interaction constitute significant complications. In the most densely filled composites, like wet sand, the response is further complicated because large stresses are transmitted over long distances through particle-particle contact [2]. To amplify heterogeneity effects, we chose to study the case of stiff/dense tungsten particles embedded in a soft/light polymer matrix. The acoustic impedance mismatch between these materials represents an extreme difference achievable between two fully dense solids. Our objective was to use this pronounced heterogeneity to search for trends in the shock wave profiles and identify potentially relevant scaling parameters. To that end, a series of composites having different matrix-fill ratios were prepared and subjected to shock loading via gas-gun driven flyer plates. Analysis of the shock wave profiles is presented that quantifies their dependence on the composite’s phase fraction. We close with a discussion of the potential physical mechanisms.

D. Bober () · B. Maddox · E. Herbold · M. Kumar Lawrence Livermore National Laboratory, Livermore, CA, USA e-mail: [email protected] Y. Toyoda · Y. Gupta Washington State University, Pullman, WA, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_51

273

274

D. Bober et al.

51.2 Materials and Methods A proprietary thermosetting polymer resin/catalyst mixture was loaded with varying volume fractions of tungsten particles. The tungsten particles were spheroidal, with a mean diameter of 16.5 μm. 1 mm sheets of the composite were made by pressing the uncured mixture between two flat steel plates, which were then heated to cure the polymer. The target assembly consisted of a polymer-tungsten sample (12 mm diameter, 1 mm thick) sandwiched between an aluminum buffer (25.4 mm diameter, 1-2 mm thick) and a lithium fluoride (LiF) optical window (8 mm diameter, 4 mm thick). These were bonded using thin layers of Epo-Tek 301 epoxy (Epoxy Technology, Billerica MA). The LiF face adjoining the polymer-tungsten sample was prepared with a diffuse finish and coated with a thin layer of aluminum using physical vapor deposition. The velocity of this interface was monitored with a velocity interferometer system for any reflector (VISAR) [3]. Three additional VISAR probes directed at the release side of the polished aluminum buffer were used to determine shock arrive times and tilt. The VISAR velocity resolution was estimated at ±1%. Each probe had a diameter of 200 μm. The impactor was an aluminum disc 5.5 mm thick, which was driven by a powder gun or two-stage light gas gun. The measured tilt in every experiment was below 3.6 mrad.

51.3 Results The velocity history at the polymer-LiF interface for the neat polymer sample showed a typical sharp shock profile, as shown in Fig. 51.1. The material reached greater than 95% of its final velocity in less than 0.8 ns. The rounded top or ‘lazy S’ profiles often observed in polymers were not evident. The wave profile was dramatically altered by the addition of only 5% tungsten, as shown Fig. 51.1. In this material, the rapid initial acceleration continued only briefly, followed by a protracted period of an ever-decreasing acceleration. In contrast to the neat polymer, the 5% tungsten material did not reach 95% of its final velocity for more than 50 ns after the initial rise; a difference of approximately three orders of magnitude by the addition of only 5% filler material. Higher tungsten loading produced qualitatively similar wave profiles, and in every case the shock rise times in the W-polymer composites were far longer than in the pure polymer. These profiles match the rounded top response observed for some neat polymers and most particle-polymer composites [4–8]. In addition, steady wave behavior was confirmed for the 10% tungsten composite by the self-similar behavior observed for 1 mm and 2 mm samples. Additional experiments were performed at 2300, 3500 and 5000 m/s impact velocities, with results qualitatively similar to those in Fig. 51.1. The principal difference observed at higher impact velocities was the tendency for the shock front to become somewhat less extended in time. These higher velocity results will be presented in a subsequent publication. For this report, the 1800 m/s experiments (Fig. 51.1) were selected because they showed the most pronounced compositional effects. We decided that a numerical fit for the wave profiles might be helpful in organizing the subtler differences in the wave profile and in revealing possible trends. A piecewise linear and exponential forms were chosen because they appeared likely to fit the data using a small number of parameters. The slope of the initial linear acceleration was fit via a least squares algorithm. The asymptotic velocity was taken to be steady particle velocity. The transition velocity at which the piecewise function switches from linear to exponential was found by using a non-linear optimization routine in Matlab (Mathworks, Natick MA), which simultaneously fit the exponential time constant. The transition velocity was then nondimensionalized Fig. 51.1 The shock profiles for the five composites are much more extended in time than that of the neat polymer (0%W). There are also noticeable differences between the composites, including a curvature which sharpens with tungsten content and the velocities at which they deviate from the neat polymer’s response

51 Shock Compression Response of Model Polymer/Metal Composites

275

Fig. 51.2 The fast rise fraction shown in part (a) quantifies how much of the acceleration follows the rapid rise seen in the neat polymer versus how much occurs in the slow exponential portion. The exponential time constants plotted in part (b) measure how long it takes the slower second phase of acceleration to bring the material to its steady state velocity. Both are plotted as functions of the tungsten volume fraction

by dividing by the peak particle velocity. The result is a quantity that represents the fraction of the total acceleration that occurred in the fast linear portion of the profile, a parameter that we term the fast rise fraction. In contrast, the exponential time constant provides a measure of time required to reach the steady state velocity after the fast rise fraction had ended. This parameter was left in units of μs, although possible normalizing time scales will be discussed later. As expected from the introductory observations above, the value of the fast rise fraction for the neat polymer was approximately 1, and dropped to 0.51 for the 5% tungsten sample. Increasing the tungsten content caused this value to drop further until a minimum of 0.29 was reached at 20–30% tungsten. At the highest tungsten loading examined (40%), the fast rise fraction rebounded to 0.35. This trend is shown in Fig. 51.2a. Tungsten content also had a significant effect on the time constant of the shock profile. As Fig. 51.2b shows, the time constant dropped monotonically as the amount of tungsten fill was increased. No time constant was determined for the neat polymer because the dominant fast linear rise left an insignificant exponential portion. Organizing the polymer-tungsten materials in terms of tungsten volume fraction is intuitive from a material fabrication standpoint, but may not be the best approach to understanding the shock response. For this, some notion of the microstructural length scale may be helpful. One approach is to calculate the mean interparticle spacing (IPS), which is a measure of the distance between particles surfaces [9]. To better explain the use of IPS, consider a set of monodisperse spheres of radius r’ packed together at an arbitrary ‘maximum’ density, for example the tapped density. The IPS of this packing is defined to be zero because most particles surfaces would be in contact. If a smaller set of spheres of radius r is inscribed concentrically within them, their IPS is given by 2(r’ − r). Of course, the relative packing density of the rsize spheres is easily calculated. More formally, the IPS is given by,  I P S = 2r

m 

1/3

 −1

(51.1)

where r is the particle radius, φ m will be taken as the tapped relative density (0.60), and φ is the actual packing density [9]. Dividing the IPS by the particle radius produces a non-dimensional microstructural length scale. The fast linear rise fraction and the time constant are plotted in terms of this normalized IPS in Fig. 51.3a, b. It is striking that the fast rise portion shows a nearly linear relationship with IPS in cases where the particles are widely spaces, i.e. values above about 0.9r. As the IPS drops to about 0.9–0.5r, an abrupt transition occurs in the fast rise fraction. Figure 51.3b also shows a distinct downward curvature in the time constant that was not evident from the volume fraction plot.

51.4 Discussion We first consider the time constant for the case of large IPS and use a working hypothesis that the particles do not interact with one another. This is to say that we will assume for the present that their local flow fields decay over short distances and no particle-particle contact occurs. Sufficiently far from any particle, the shock front will resemble a planar shock in the

276

D. Bober et al.

Fig. 51.3 The fast rise fraction (a) and time constant (b) shown in Fig. 51.2 are replotted here in terms of the normalized interparticle spacing, which gives a more direct measure of the microstructural length scale

neat material. In those places, the sharp discontinuity separates regions of high stress/high velocity from the ambient state. Closer to a particle, the flow field becomes more complex. As the shock front passes over an isolated particle, there will be a pressure differential between the leading and trailing sides of the particle. This is the so-called inviscid unsteady force [10]. During this same period, the viscous unsteady force will also be accelerating the particle [10]. Together, these forces will act on a time scale (τ s )set by the particle diameter (d) and the shock speed (us ), which is given by [1], τs =

d us

(51.2)

The time scale over which these forces act is shorter than our measured time constants, meaning that a longer lasting force must be sought. For a particle moving slower than a surrounding fluid, this longer lasting acceleration can be supplied by the quasi-steady drag, which will persist so long as the velocity gradient lasts [10]. The time scale is given by, τd =

ρd 2 18μ

(51.3)

where ρ is the particle density and μ is the polymer’s dynamic shear viscosity [1]. Given that drag scales linearly with the particle radius and inertia with the radius cubed, larger particles should produce longer time constants. This agrees with the earlier observation by Herbold et al. [11] that larger tungsten particles embedded in a polymer matrix led to a slower shock response than did small particles. Setchell, Anderson and Montgomery [4] likewise found that small spherical alumina particles led to less extended shock fronts than coarse angular ones. The importance of viscosity in an epoxy-alumina composite was illustrated by Setchell et al. [12], who tested the effect of initial temperature. It was found that colder initial temperatures (−55 ◦ C) lead to a faster shock response, while warming to 75 ◦ C lead to more extended wave profiles [12]. The inference is that more a more viscous matrix leads to faster momentum transfer and a more compact shock front [12]. The time constants measured here would imply a viscosity of ∼10 Pa·S, which is line with our estimate from ongoing numerical simulations. This drag mechanism would partially justify the chosen exponential fit because it implies an accelerating force that decays as the particles gain velocity. Shock waves reflected from other particle-polymer interfaces may also contribute to a second long-time accelerating force. The relative momentum transferred via drag versus shock reverberation is difficult to estimate, though the discussion above and ongoing numerical simulations both suggest that the former dominates. While the fast-moving polymer slowly accelerates the dense tungsten particles, the particles would in turn act to slow the mean velocity of the nearby fluid. With only a few particles in the system, the total volume of polymer that experiences a drag is minimal, and most polymer will accelerate just as in the neat case. When more particles are added, their collective volume of influence grows and so more of the polymer is subjected to the slow acceleration. Measuring a spatially averaged velocity in such a system would produce a piecewise linear-exponential behavior, with a greater portion for lower particle loadings. This matches the observed trend for the fast rise fraction, which drops for increasing particle loading, at least up to 20%. This also matches Setchell’s [12] observation that increased alumina content in an epoxy composite led to longer rise times. Of course, the preceding analysis assumes non-interacting particles and is expected to break down as the IPS becomes small.

51 Shock Compression Response of Model Polymer/Metal Composites

277

The critical value of IPS at which particle interactions become significant is unknown a priori, but the wave profiles provide some valuable clues. As a start, the conclusions of the preceding paragraph break down rapidly as the IPS drops below about 0.9r. Beyond this point, increasing the particle loading leads to the opposite trend from that predicted for noninteracting particles. The local minimum in Fig. 51.3a can therefore be interpreted as a sign of the transition from isolated to interacting tungsten particles. At this transition, it is expected that the flow fields around neighboring particles will interact. At the small IPS values, the apparent viscosity will probably begin to depend on particle volume fraction. As is well-known in colloidal systems, the addition of rigid particles concentrates the shear flow through smaller interparticle fluid volumes [13]. An analogous concentration of shear in a shocked particle-polymer matrix has been seen in simulations, including interparticle jetting [11]. The expected effect is to amplify the drag force, even though the intrinsic matrix properties are essentially unchanged. This provides a plausible explanation for the downward slope and curvature of the time constant when plotted with respect to IPS. Increased drag would shorten the shock width by causing faster momentum transfer between the phases. This is analogous to previous observations in epoxy particle composites in which colder initial temperatures led to more extended shock profiles. While Setchell [12] observed a change due to intrinsic viscosity, here the hypothesis is that the apparent viscosity is being modified by the changing characteristic microstructural length scale. As the IPS is decreased, the particles themselves will eventually encounter one another. At very high packing densities, this could lead to stress transmission over long distances via force chains, as is well-known to occur in granular materials [2]. Indeed, at a sufficient packing density the distinction between a matrix-particle composite and a saturated granular material disappears. For randomly packed equal sized spheres, the critical volume fraction for the emergence of continuous percolating chains is 18.3% [14]. Of course, that does not necessarily imply that these chains have sufficient lateral confinement to transmit high stresses over long distances [15]. Indeed, there is evidence in the literature that the properties of the particles remain relatively unimportant up to a density of at least 45% [16]. It has been observed that epoxy composites with alumina, aluminum, tungsten carbide, and a mixture of aluminum and tungsten particles all had similar shock responses [16]. The main difference was a shift due to the overall density, which is contrary to what would be expected if particle force chains had a dominant role [16, 17]. This conclusion agrees with direct numerical simulations which did not show evidence of force chains at 45% particle fraction [8]. Even in the case where particles are certainly close enough to interact, thin intervening layers of polymer can have a dramatic effect on wave propagation [18]. This effect can also be seen in the dependence of storage and loss moduli on composition [19]. Additional evidence for the dominance of the matrix material at packing a fraction of 43% comes from the viscous/temperature dependence [12]. White et al. [20] found that increasing the metal fill of an epoxy composite from 30% to 40% led to a percolation-like jump in the dynamic strength. In the context of these previous studies, the maximum particle fraction studied here (40%) is approaching, but may not reach, the regime in which granular force chains dominate the response.

51.5 Conclusion It was found that the shock profiles of a set of polymer-metal composites depend strongly on composition. These trends can be quantified in terms of a single time constant and a transition velocity, both of which can be related to physical phenomena. The governing mechanism is believed to be a particle-matrix drag effect, in which the small dense tungsten spheres retard the acceleration of the surrounding polymer. For dilute composites, increasing the tungsten content led to more extended shock profiles, probably because the drag effect scales with the polymer volume that interacts with the filler. While we see evidence that neighboring particles interact at small interparticle spacings, long range particle to particle stress transfer is unlikely. Even at 40% tungsten, neighboring particle interaction are most likely still strongly mediated by the intervening polymer. The current hypothesis is that this leads to a rise in the apparent viscosity, which in turn causes a drop in the measured time constant. The relative role of shock reverberation requires more study, but appears to be modest. Three dimensional direct numerical simulations are planned to further refine these conclusions and better quantify the particle-matrix interaction. It is desirable to collect shock profiles for different size tungsten particles to provide valuable scaling information. The relationship between the time constant and the steady wave development time is also an area of active pursuit. Acknowledgments This work was partly performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The work at Washington State University was supported in part by DOE/NNSA (DENA0002007).

278

D. Bober et al.

References 1. Ling, Y., Haselbacher, A., Balachandar, S., Najjar, F.M., Stewart, D.S.: Shock interaction with a deformable particle: direct numerical simulation and point-particle modeling. J. Appl. Phys. 113, 013504 (2013) 2. Nesterenko, V.: Dynamics of Heterogeneous Materials. Springer, New York (2013) 3. Barker, L.M., Hollenbach, R.E.: Laser interferometer for measuring high velocities of any reflecting surface. J. Appl. Phys. 43, 4669–4675 (1972) 4. Setchell, R.E., Anderson, M.U., Montgomery, S.T.: Compositional effects on the shock-compression response of alumina-filled epoxy. J. Appl. Phys. 101, 083527 (2007) 5. Rauls, M.B., Ravichandran, G.: Shock wave structure in particulate composites. Procedia Eng. 103, 515–521 (2015) 6. Munson, D.E., Boade, R.R., Schuler, K.W.: Stress-wave propagation in Al2O3-epoxy mixtures. J. Appl. Phys. 49, 4797–4807 (1978) 7. Setchell, R.E., Anderson, M.U.: Shock-compression response of an alumina-filled epoxy. J. Appl. Phys. 97, 083518 (2005) 8. Vogler, T.J., Alexander, C.S., Wise, J.L., Montgomery, S.T.: Dynamic behavior of tungsten carbide and alumina filled epoxy composites. J. Appl. Phys. 107, 043520 (2010) 9. Hao, T., Riman, R.E.: Calculation of interparticle spacing in colloidal systems. J. Colloid Interface Sci. 297, 374–377 (2006) 10. Parmar, M., Haselbacher, A., Balachandar, S.: Modeling of the unsteady force for shock-particle interaction. Shock Waves. 19, 317–329 (2009) 11. Herbold, E.B., Nesterenko, V.F., Benson, D.J., Cai, J., Vecchio, K.S., Jiang, F., et al.: Particle size effect on strength, failure, and shock behavior in polytetrafluoroethylene-Al-W granular composite materials. J. Appl. Phys. 104, 103903 (2008) 12. Anderson, M.U., Cox, D.E., Montgomery, S.T., Setchell, R.E.: Initial temperature effects on the shock compression and release properties of different alumina-filled epoxy compositions. In: Elert, M., Furnish, M.D., Chau, R., Holmes, N., Nguyen, J. (eds.) Shock Compression of Condensed Matter–2007, Pts 1 and 2, pp. 683–686. American Institute of Physics, Melville (2009) 13. Mooney, M.: The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 6, 162–170 (1951) 14. Powell, M.J.: Site percolation in randomly packed spheres. Phys. Rev. B. 20, 4194–4198 (1979) 15. Tordesillas, A., Muthuswamy, M.: On the modeling of confined buckling of force chains. J. Mech. Phys. Solids. 57, 706–727 (2009) 16. Jordan, J.L., Herbold, E.B., Sutherland, G., Fraser, A., Borg, J., Richards, D.W.: Shock equation of state of multi-constituent epoxy-metal particulate composites. J. Appl. Phys. 109, 013531 (2011) 17. Millett, J.C.F., Bourne, N.K., Deas, D.: The equation of state of two alumina-filled epoxy resins. J. Phys. D. Appl. Phys. 38, 930–934 (2005) 18. Daraio, C., Nesterenko, V.F.: Strongly nonlinear wave dynamics in a chain of polymer coated beads. Phys. Rev. E. 73, 026612 (2006) 19. Veazie, D., Jordan, J.L., Spowart, J.E., White, B.W., Thadhani, N.N.: Model for elastic Modulus of multi-constituent particulate composites. Exp. Mech. 53, 1213–1222 (2013) 20. White, B.W., Thadhani, N.N., Jordan, J.L., Spowart, J.E.: The effect of particle reinforcement on the dynamic deformation of epoxy-matrix composites. In: Elert, M.L., Buttler, W.T., Furnish, M.D., Anderson, W.W., Proud, W.G. (eds.) Shock Compression of Condensed Matter – 2009, Pts 1 and 2, pp. 1245–1248. American Institute of Physics, Melville (2009)

Chapter 52

High-Strain Rate Interlaminar Shear Testing of Fibre-Reinforced Composites Using an Image-Based Inertial Impact Test J. Van Blitterswyk, L. Fletcher, and F. Pierron

Abstract In this work a novel image-based inertial impact test is proposed to measure the interlaminar shear modulus of fibre-reinforced polymer composite materials at high strain rates. The principle is to combine ultra-high-speed imaging and full-field measurements to capture the dynamic kinematic fields, exploiting the inertial effects generated under high strain rate loading. The kinematic fields are processed using the virtual fields method to reconstruct stress averages from maps of acceleration. In this way, the specimen acts like a dynamic load cell, with no gripping or external force measurement required. This paper focusses on validation of the test principle using explicit dynamic simulations in ABAQUS. Simulations demonstrate the potential for the proposed method to identify the shear modulus at strain rates where current test methods become unreliable (500 s−1 on average, and on the order of 2000 s−1 locally). Access to spatial maps of stress averages provides opportunity to estimate the shear strength in the future. Further design work is required to amplify shear stress and strain in the specimen, after which the test will be validated experimentally. Eventually, the objective is to tailor the test to begin populating regions of a tension-shear failure envelope. Keywords High strain rate · Composite materials · Full-field measurements · Virtual fields method · Ultra-high-speed imaging

52.1 Introduction The mechanical response of fibre-reinforced polymer (FRP) composite materials subjected to high rates of deformation (crash, blast, etc.), is dependent on the interlaminar properties. As the interlaminar properties are matrix-dominated, literature suggests that strain rate has a significant effect on stiffness and strength [1]. Testing FRP composites in shear is commonly done using thin-walled tubular specimens or lap-shear specimens, loaded in torsion or compression, respectively with a splitHopkinson pressure bar (SHPB) system [1]. The effect of strain rate on the interlaminar shear modulus and shear strength has not been well characterised, with unacceptably high scatter shown across available studies [1]. The uncertainty can be primarily attributed to limitations of the SHPB system. Under high strain rate loading, inertial effects induce heterogeneous kinematic fields. This violates the assumption of quasi-static equilibrium required to infer the response of the material in SHPB test. Due to low wave speeds in the interlaminar directions of FRP composites, strain rate is limited to a few 100 s−1 in shear with the standard SHPB method. The prolonged time required for inertial effects to dissipate at higher strain rates in low wave speed materials, has resulted in a consensus that the SHPB system provides unreliable measurements of the material stiffness [2]. In addition, the lap-shear specimen generates a non-uniform shear stress state and significant normal stresses, resulting in high scatter in strength measurements. The thin-walled tubular specimens are subjected to a state of pure shear but suffer from a high sensitivity to specimen geometry and machining defects. Therefore, there is a need to consider alternative methods for high strain rate testing of FRP composites subjected to interlaminar shear. Recently, the image-based inertial impact (IBII) test has emerged as a viable alternative to obtain in-plane properties [3, 4]. With this approach, ultra-high-speed imaging is combined with the grid method [5] to measure kinematic fields at submicrosecond resolution. These fields are processed using the virtual fields method (VFM) to identify stiffness and strength, using the specimen as a dynamic load cell. This alleviates many of the assumptions associated with existing techniques, which hinder reliable characterisation of these properties. This work describes the extension of the IBII test to attempt to measure the interlaminar shear modulus at strain rates where existing techniques are unreliable. Estimating shear strength will be considered in the future with a view to characterise a shear/tension failure envelope.

J. Van Blitterswyk () · L. Fletcher · F. Pierron Mechanical Engineering, Faculty of Engineering and the Environment, University of Southampton, Southampton, BJ, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_52

279

280

J. Van Blitterswyk et al.

52.2 Test Concept A preliminary configuration for the proposed IBII interlaminar shear test is shown in Fig. 52.1. The specimen is subjected to high strain rate loading using a stress pulse from an impact with a projectile. The projectile is embedded in a nylon sabot, with a diameter matching that of the gas gun that will be used for later experimental validation (see [4] for further detail about the experimental implementation of the IBII test). The projectile impacts the waveguide, which reduces effects from slight misalignments at impact. The pulse propagates through the waveguide and is transferred into the specimen. Shear is generated in the region of the specimen overhanging the waveguide. The dynamic response of this region is measured using ultra-high-speed imaging, coupled with the grid method. The idea of the tapered waveguide is to minimise the shear stress concentration at the interface with the specimen. From equilibrium, the average shear stress, σxy x , at any horizontal plane, y, and time, t, can be expressed as a function of the measured surface accelerations (‘stress-gauge approach’ Eq. 52.1). σxy (y, t)x = ρyax (y, t)S

(52.1)

In Eq. 52.1, ρ is the material density, the superscripts x and s, coupled with the overline, respectively denote the horizontal line average at a slice located at y, and the average surface acceleration between the bottom free edge and y. Using reconstructed stress averages, stress-strain curves can be generated at any horizontal slice in the specimen and used to identify the shear modulus. Although not considered in this work, the stress-gauge approach can be used to estimate the failure strength and can be similarly applied to vertical slices within the specimen to extract the interlaminar Young’s modulus. Knowing that the shear stress at the free edges is zero, shear stress averages can be used to fit a parabolic stress distribution. This is likely to give a more accurate identification of the shear strength.

52.3 Numerical Modelling The material used in this study is a unidirectional carbon/epoxy pre-preg (AS4–145/MTM45–1). The plate has a nominal thickness of 18 mm, which sets the width of the specimen. The specimen overhang of 27 mm is selected to closely match the specimen aspect ratio with the camera spatial resolution (Shimadzu HPV-X, 400 × 250 pixels). The 2–3 interlaminar plane is considered (E33 = E22 = 7.6 GPa, ν 23 = 0.225, Gxy = 3.65 GPa, ρ = 1605 kg·m−3 ), but the same method can be easily applied for testing in the 1–3 interlaminar plane. Explicit dynamics simulations in ABAQUS were used to explore the feasibility of the proposed configuration. Plane stress CPS4R elements (2D, 4 node, reduced integration) were used with a mesh size of 0.1 mm. For all materials, β damping (7 x106 ms) was applied to eliminate high frequency ‘ringing’ in the acceleration fields. The time step was allowed to float; however, field outputs were fixed to 0.2 μs intervals to match the frame rate of the HPV-X camera. The projectile and waveguide are modelled as 6061-T6 aluminium (E = 70 GPa, ν = 0.3, ρ = 2700 kg·m−3 ), and the sabot is modelled as Nylon 6–6 (E = 3.45 GPa, ν = 0.4, ρ = 1140 kg·m−3 ), The trial test configuration has a total waveguide length, LWG , of 75 mm, projectile length, LP , of 10 mm, and impact speed, VP , of 50 m·s−1 . The tapered section of the waveguide is 25 mm long with a 45◦ angle to allow for lateral wave expansion.

Fig. 52.1 Schematic of the simulated IBII interlaminar shear experiment

52 High-Strain Rate Interlaminar Shear Testing of Fibre-Reinforced Composites Using an Image-Based Inertial Impact Test

281

Fig. 52.2 (a) Map of acceleration (m·s−2 ) at 14 μs, and (b) stress-strain curve at y = 17 mm

52.4 Results and Discussion Figure 52.2a shows the map of acceleration 14 μs after the wave enters the specimen for a 2–3 interlaminar plane specimen. One can see that the acceleration fields generated by the impact pulse are highly heterogeneous and exceed 3x105 g. Simulated acceleration and strain fields are used to validate the stress-gauge approach. Reconstructed stress averages, from the acceleration fields, and average shear strain along a horizontal slice at y = 17 mm, are used to generate the shear stressshear strain curve shown in Fig. 52.2b. This validates the use of the stress-gauge approach for identifying the reference shear modulus. In the future, image deformation simulations will be considered to account for the effects of grid sampling, and spatial and temporal noise on the identification of the shear modulus. Defining a strain rate for the measured properties is challenging as the inertial effects create highly heterogeneous strain and strain rate maps. For the given configuration, strain rates reach 500 s−1 on average, and on the order of 2000 s−1 locally.

52.5 Conclusions and Future Work This paper demonstrates how full-field measurements, coupled with the virtual fields method, can be used to exploit inertial effects under high strain rate loading to identify the interlaminar shear modulus. Having access to the full-field, dynamic response of the material removes many of the limitations associated with existing test methods. Experimental validation of the proposed test will be the primary focus of future work. Before doing so, additional design work is required to increase the amount of shear strain generated within the specimen. The virtual fields method provides localised measurements of stress-averages, which will enable estimates of shear strength to be made. More advanced formulations of the virtual fields method will also be considered for simultaneous identification of multiple constitutive properties from a single test.

References 1. Van Blitterswyk, J., Fletcher, L., Pierron, F.: Characterisation of the Interlaminar properties of composites at high strain rates: a review. Adv. Exp. Mech. 2, 3–28 (2017) 2. Gama, B.A., Lopatnikov, S.L., Gillespie, J.W.: Hopkinson bar experimental technique: A critical re- view. Appl. Mech. Rev. 57(4), 223 (2004) 3. Pierron, F., Zhu, H., Siviour, C.R.: Beyond Hopkinson’s bar. Phil. Trans. R. Soc. A. 372, 20130195 (2014) 4. Fletcher, L., Van Blitterswyk, J., Pierron, F.: A novel Image-Based Impact (IBI) test for the transverse properties of unidirectional composites at high strain rates, Submitted to Comp. A (2017) 5. Grédiac, M., Sur, F., Blaysat, B.: The grid method for in-plane displacement and strain measurement: a review and analysis. Strain. 52, 205–243 (2016)

Chapter 53

Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys in Shear Using In-Situ Synchrotron Radiation X-Ray Diffraction Christopher S. Meredith, Zachary Herl, and Marcus L. Young

Abstract A fundamental understanding of magnesium-based alloys during high rate, large deformation processes that occur during impact and penetration are not well-known. This metal possesses a limited number of deformation mechanisms, each with their own disparate strengths, strain hardening rates, and strain rate sensitivities. Consequently, these alloys exhibit severe tension-compression asymmetry and anisotropy dictated by their processing history and the applied deformation. Thus, an understanding of material behavior undergoing large shears at dynamic rates is required. Experiments have been performed on a specimen geometry that induces shear localization in “pure” simple shear, called the compact forced simple shear (CFSS) specimen. The deformation occurs on a 2D plane in the specimen, which is oriented with respect to directional aspects of the material’s microstructure and deformation modes. Experiments at dynamic strain rates have been performed to determine how the mechanical behavior in shear evolves and correlates to the microstructural deformation mechanisms. The experiments were performed at the Dynamic Compression Sector of the Advanced Photon Source at Argonne National Laboratories using in-situ synchrotron x-ray diffraction aimed to probe the microstructural evolution during shear-induced localization. By correlating the propensity for shear localization to occur with the mechanical response of various orientations, we have built a data set to compare existing models to identify key deformation mechanisms responsible for localization. Keywords Magnesium alloys · In-situ x-ray diffraction · Shear deformation · Dynamic loading · Texture evolution

53.1 Introduction Magnesium is the lightest structural metal and can have specific strengths equal to or greater than other typical structural metals [1]. However, this metal has a limited number of deformation mechanisms, and each possesses its own critical resolved shear strengths, strain hardening rates, and strain rate sensitivities [2–5]. Thus, magnesium alloys often have a large tension-compression asymmetry, strong anisotropy, and limited ductility depending on the processing history and the applied deformation. The most often observed deformation mechanisms are basal, pyramidal, and prismatic slip and extension twinning. Basal slip is the easiest to activate due to its low critical resolved shear stress; however, prismatic and pyramidal slip have been observed when the activation of basal slip is unfavorable. These three mechanisms are confined to the basal plane so other mechanisms are required to accommodate deformation along the c-axis. Pyramidal slip ! can cause deformation parallel to the c-axis; however, it is difficult to activate. As a result, the 1012 twin system typically initially accommodates an extension of the c-axis. Contraction twin systems are possible, but they are not or infrequently observed experimentally. Basal slip and extension twinning are generally considered to be strain rate insensitive, while pyramidal and prismatic are observed to be rate sensitive [2–5]. To take advantage of magnesium’s light weight for situations where high-rate, large deformations occur, such as during impact and penetration events, a better understanding of the response to shear deformation at these conditions needs to be explored. Most of the existing studies focus on either uniaxial tension or compression, and a reasonably good understanding of the active deformation mechanisms has been obtained under these conditions [6–8]. However, shear

C. S. Meredith () Impact Physics Branch, Army Research Lab, Aberdeen Proving Ground, MD, USA e-mail: [email protected] Z. Herl · M. L. Young Department of Materials Science & Engineering, University of North Texas, Denton, TX, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_53

283

284

C. S. Meredith et al.

Fig. 53.1 Schematic of the geometries of the (a) “top-hat” specimen, (b) shear-compression specimen, (c) torsional specimen with hexagonal flanges, and (d) compact forced simple shear specimen. The red part in each of the specimens indicates the location of the shear zones

dominated deformation has only been minimally studied, and usually at quasi-static strain rates. Although not trivial, several methods exist for applying shear deformation at high rates. Unfortunately, the most commonly used methods have some serious disadvantages for isolating shear behavior. Figure 53.1 shows a schematic of four of these testing geometries for shearing: (a) the “top-hat” specimen, (b) the shear-compression specimen (SCS), and (c) a torsional specimen with hexagonal mounting flanges, as well as, (d) the compact forced simple shear (CFSS) specimen used in this study. As illustrated in Fig. 53.1a, the “top-hat” geometry has been most widely used and consists of two halves; one is a circular disk or “crown” (the top of the “hat”) and the other is a ring (the “brim” of the hat) [9]. For shear testing using a Kolsky bar, the two halves are compressed together which induces a thin circumference of material at the interface between each half to experience shear dominated deformation. The disadvantages of this method are that shear strain cannot be measured directly and anisotropy cannot be readily analyzed due to the loading occurring around a circumference within the geometry. As illustrated in Fig. 53.1c, a torsional specimen with hexagonal flanges can be utilized using a torsional Kolsky bar [10], which can measure the shear strain; however; due to the sample being a thin-walled tubular geometry, it is not possible to isolate the anisotropic behavior. Furthermore, the usual method of loading in the torsional Kolsky bar apparatus is complex and temperamental; thus, obtaining high-quality data is challenging. The shear-compression specimen (SCS) is another promising geometry (Fig. 53.1b) [11], but a large compressive component to the loading is generated, which was not desirable in this study. A recent geometry developed at Los Alamos National Laboratory is called the compact forced simple shear (CFSS) specimen, as illustrated in Fig. 53.1d [12]. The deformation is confined to a plane so anisotropy can easily be probed, but the major disadvantage of this method is that shear strain cannot be measured. A major advantage of the geometry, however, is that the shear location is thin and easily isolated. Thus, the combination of a localized thin region and a low Z material like Mg allows for in-situ synchrotron X-ray diffraction (SR-XRD) measurements using a synchrotron source, like the Dynamic Compression Sector (DCS) at the Advanced Photon Source at Argonne National Laboratories. The texture evolution can be directly observed by passing x-rays through the shear zone during high strain rate deformation. This SR-XRD data can then be correlated to the deformation mechanisms using crystal plasticity models Fig. 53.1. In this study, we present in-situ SR-XRD data on CFSS samples from two Mg-based alloys, AZ31B and AMX602, during high strain rate deformation using a Kolsky bar. The AZ31B samples came from a typical rolled plate where strong basal texture is aligned with the normal direction of the plate, while the AMX602 samples came from a powder-compacted, finegrained alloy with a relatively weak rolling texture. The SR-XRD patterns were collected at various microsecond times during loading of the samples for direct time-dependent correlation of the mechanical behavior to the texture evolution from the diffraction patterns. The observed textural evolution is related to the likely deformation mechanisms active, i.e. slip versus twinning.

53 Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys. . .

285

53.2 Experimental Methods Two Mg-based alloys were examined in this study. The first alloy was a rolled plate of AZ31B-H24 (Mg-3 at.%Al-1 at.%Zn), which was annealed to remove twins present after the initial rolling processing. The second alloy was AMX602 (Mg6 at.%Al-0.5 at.%Mn-2 at.%Ca) which possessed a fine-grained microstructure. The bulk plate was produced using the spinning water atomization process (SWAP) and then extruded. SWAP consists of dropping molten material under gravity from a crucible into a large container where water is rapidly spinning around the interior surface. The molten material is atomized using high pressure argon gas and the droplets rapidly solidify and cool upon contacting the water, which locks in a fine-grained microstructure. The powder is collected and compacted into a “green” cylinder. The cylinder is then extruded at an elevated temperature into a plate-like geometry with a thickness of 25.4 mm [13]. These two alloys will be referred to throughout the paper as either AZ31B or AMX602. CFSS samples were machined from the bulk of each material with dimensions shown in Figs. 53.1d and 53.2a. The rolling (RD) or extrusion (ED), transverse (TD), and normal (ND) directions were tracked during machining and marked on each sample. Figure 53.2b highlights the area of shear of the sample with an example orientation shown. This example is interpreted as follows: ND-RD refers to the area of shear (TD is perpendicular to this plane), where the first direction, RD, refers to the direction of loading (or displacement) by the Kolsky bar, and the second, ND, refers to the direction of the x-ray beam and is also the second direction that forms the area of shear. Note the area of shear is not the same as the shear plane, which is ND-TD in this example. In this work, the different orientations are defined by their shear plane. Table 53.1 shows the orientations tested for each alloy. Samples were cut such that the deformation would occur on different shear planes at dynamic strain rates to determine the influence of anisotropy on the mechanical response. Since each alloy exhibits texture typical for a rolled Mg-based alloy plate, the shear planes of the samples were chosen strategically to load parallel or perpendicular to the strong basal texture.

90.0°

.188

.050

SECTION A-A

.094

B

.083

.050 .050

A

.083

a

SECTION B-B

.440

.215

B

A

b

Fig. 53.2 Detailed schematic of the (a) CFSS sample geometry and dimensions in inches and (b) three dimensional view of the CFSS sample and shear plane, where an example ND-RD orientation is shown by a blue hatch pattern Table 53.1 Sample orientations which were tested for each alloy

Alloy AMX602 AMX602 AMX602 AZ31B AZ31B AZ31B

Loading axis ND ED ED RD RD ND

Beam axis ED TD ND ND TD RD

Shear plane ND-TD ED-ND ED-TD RD-TD RD-ND ND-TD

286 Striker

C. S. Meredith et al. Pulse Shaper

Incident Bar

Strain Gauge

Specimen

Strain Gauge

Transmission Bar

Damper

Fig. 53.3 Schematic of the Kolsky Bar used in these experiments

Kolsky Bar

Beam Splitting Prisms Camera 1 Camera 2 Camera 3 Camera 4

Sample Vert. & Hor. KB Mirrors

Defining Slits

Guard Slit

Phosphor Plate

Fig. 53.4 Schematic of the experimental setup used during synchrotron radiation x-ray diffraction measurements at beam line 35-ID-B at the Advanced Photon Source (APS) in Argonne National Laboratories (ANL)

The samples were loaded using a Kolsky bar which was shipped to DCS for our allotted beam time. Figure 53.3 shows a schematic of the Kolsky bar set up for compression loading. The typical design consists of a pair of long, coaxially aligned bars that are made of the same material and have the same diameter. For compression testing, a cuboid or cylindrical sample is typically placed between the bars before loading. A shorter bar, called the striker or hitter, which is also coaxially aligned with the others and made of the same material and diameter, is fired out of a gas gun and impacts one end of one long bar, which is called the incident bar. At impact, an elastic compression wave travels down the incident bar and when it reaches the sample, part of the wave reflects back into the incident bar and the other part transmits into the second long bar, called the transmitted bar. Strain gages at the center of the incident and transmitted bars record the incident, reflected, and transmitted waves, which are used to calculate the stress, strain, and strain rate within the sample. The only difference with compression experiments and these shear experiments presented in this study is the atypical sample geometry, which means strain and strain rate cannot be calculated. In these experiments, only the net sample displacement and velocity can be accurately determined. The shear stress can be calculated, although the values become dubious as the sample fractures. The Kolsky bar set up consisted of Al 7075-T6 4.8 mm diameter incident and transmitted bars which were 762 mm long, and a striker which was 100 mm long. The experimental setup for Kolsky bar testing at DCS (beam line 35-ID-B) is shown in Fig. 53.4. A 22.83 keV (λ = 0.0543 nm) x-ray beam was shaped into a thin, rectangular beam with a cross-section of 100 × 800 μm2 by a pair of vertical and horizontal K.B. mirrors. This rectangular beam passes orthogonally through the sample which is positioned in the Kolsky bar which is rest on a platform inside the experimental hutch. SR-XRD patterns diffracted from the sample were collected before and during loading of the CFSS specimens by a phosphor plate, which had a diameter of 155 mm and was positioned approximately 90 mm from the sample. By use of beam-splitting prisms, a series of four cameras captured two frames each for a sequence of eight 1024 × 1024 pixel images as snapshots during the experiment. In this paper, Frame 0 refers to the diffraction pattern of the samples at ambient conditions, i.e. prior to the sample being loaded, and Frames 1–8 are those collected during the experiment. Sometimes due to poor timing the first frame or two capture the diffraction data prior to the sample being loaded. To maintain safety and facilitate time-resolved measurements, two shutters are used that prevent the x-ray beam from coming into the experimental hutch: a “slow” and “fast” shutter, where the name indicates their relative opening time. A major issue with the cameras is they have large non-linearities above a certain number of counts, which we did not know about during the experiments. This means quantitative comparisons between orientations are not possible, only qualitative conclusions can be drawn. Prior to testing, the incident and transmission bars are placed by using brass spacers and the sample is sandwiched between the bars using a block to assist in placing it squarely in the beam path. A photo diode was placed behind the sample and the shutters were opened to expose the sample and diode to the x-ray beam. The sample was then centered by measuring the

53 Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys. . .

287

Fig. 53.5 One quarter of an x-ray diffraction pattern from an AMX602 sample before shear testing. The diffraction peaks have been labeled and correspond to the α-Mg phase. Brighter pixels indicate higher diffracted intensity

light intensity transmitted through the sample to the diode, as the Kolsky Bar platform was moved. Finally, the shutters were closed and the diode removed in preparation to fire the striker. Before the striker contacts the incident bar, it passes a laser beam which triggers the “slow” shutter to open. When the incident wave reaches the incident strain gage, it takes 75.4 μs for the wave to reach the sample and initiate sample loading. As the compression wave passes the strain gage, the measured voltage triggers the “fast” shutter to expose the sample to the x-ray beam, which initiates recording of both strain gages by the oscilloscopes as well as the imaging feedback from the cameras that record the x-ray diffraction pattern from the detector. Thus, all measuring components start at the same time, i.e. share a universal t = 0. By setting an appropriate delay time for each camera, each camera records an X-ray diffraction pattern from the phosphor plate at a different time during loading. The camera draws a voltage through the oscilloscopes when they are recording an image, which can be matched to loading by comparison to the transmitted wave shifted by the (known) travel time from sample to strain gage. The experimental setup and measurements are very similar as used by Hustedt et al. [14]. The captured images show a cross-section of the Debye-Scherrer cones where they intersect the phosphor plate. As a representative example, one quarter of a x-ray diffraction pattern from an AMX602 sample before shear testing is shown in Fig. 53.5, with the diffraction rings identified. Although full diffraction rings were collected, only one quarter of the pattern is shown so that peak identification is clear. This pattern is characteristic of the experimental setup, with the radial spacing of the peaks dependent on the wavelength of light, as well as the orientation and locations of the phosphor plate. To evaluate the diffraction patterns, a set of calibration samples were placed in the beam at the same position as the shearing samples. The LaB6 and polycrystalline Si patterns were compared to known standards by using the software FIT2D in order to more precisely determine the beam center, detector tilt, and sample-to-detector distances for each camera. By knowing the location and orientation of the detector in space, the diffraction pattern can be converted from polar coordinates to Cartesian coordinates. Furthermore, with the wavelength/energy of the beam, the pattern can be integrated into line profiles comparing the diffracted intensity against the d-spacing, which measures the interplanar spacing of the sample.

53.3 Results The shear stresses as a function of displacement of the samples is shown in Fig. 53.6. All experiments resulted in the macroscopic failure of the sample, which is reflected in the stress-displacement curves as a quick or sudden drop in the flow stress at large displacements. Multiple experiments were run for AMX602 at the three sample orientations, but not for AZ31B due to time limitations. The behavior of AMX602 is not very consistent for samples with the same orientation, and does not show clear differences between the different orientations, either. It is unclear why this is the case–it could be due to the relatively weak texture of the material, variations in the machining of the samples, the samples bending during the loading, or combinations thereof. The behavior of AZ31B shows clear differences due to the orientation. Note, we have

288

C. S. Meredith et al.

a

b ND-TD ED-ND ED-TD

300

180 160 Shear Stress (MPa)

Shear Stress (MPa)

250

ND-TD RD-ND RD-TD

200

200 150 100

140 120 100 80 60 40

50

20 0

0

0.025

0.05 0.075 0.1 Displacement (mm)

0

0.15

0.125

0

0.05

0.15

0.1

0.2

0.25

0.3

Displacement (mm)

Fig. 53.6 Shear stress vs. displacement curves for (a) AMX602 and (b) AZ31B samples in the various orientations

a

b 2

Frame 1

3

4

5

6

7

8

150 100 50 0

0

5

10

15

20 25 Time (µs)

30

35

40

250 2 3 4 5

Frame 1 200

67 8

Shear Stress (MPa)

200

c 250

Shear Stress (MPa)

Shear Stress (MPa)

250

150 100 50 0

0

5

10

15

20 25 Time (µs)

30

35

40

200

2 3 4 5 6 7

Frame 1

8

150 100 50 0

0

5

10

15

20 25 Time (µs)

30

35

40

Fig. 53.7 Shear stress vs. time curves from AMX602 CFSS samples with shear planes of (a) ND-TD, (b) ED-ND, and (c) ED-TD, with the SR-XRD camera framing overlaid

Fig. 53.8 SR-XRD patterns from! AMX602 with ! the ND-TD shear plane at approximately (a) 0 μs, (b) 15 μs, and (c) 30 μs, respectively. Three diffraction planes, (0002), 1010 , and 1120 , have been labeled and correspond to the α-Mg phase

performed subsequent experiments on the AZ31B (not shown) that confirm the behavior is consistent for multiple samples at the same orientation. The stronger texture likely aids in improving consistency. Overall, the AZ31B is weaker due to it possessing larger grains, but is more resistant to shear induced failure. The RD-ND orientation has a low yield that is half the other two orientations. The ND-TD orientation has a slightly lower yield than the RD-TD, but it has a slightly higher work hardening rate and lower resistance to failure. Figure 53.7 shows the shear stress as a function of time for one experiment at each orientation of the AMX602. The timing for each camera measurement, i.e. frame number, for collecting the SR-XRD patterns is also shown in Fig. 53.7, where the arbitrary unit signal increases from zero. The images show that we captured diffraction data covering the full response of the material at each orientation–from still within the elastic range to just prior to failure. Since the samples are only loaded for 25–30 μs, correct timing of the frames was tricky and was one reason we conducted multiple experiments at each orientation.

53 Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys. . .

289

Fig. 53.9 Integrated SR-XRD line profiles of AMX602 with the ND-TD shear plane, which are offset vertically for clarity. Marked in blue are the d-spacings of the unstrained peaks in Mg for reference

Figure 53.8 shows frames 0, 3, and 8 of the SR-XRD data collected for the sample with the ND-TD shear plane. These snapshots were selected to show the ambient state, within the early part of plastic loading, and the state at or near failure. Starting with 8a, the second inner-most ring is brightest on the left and right sides of the pattern. This ring is diffracted from the (0002) plane and, thus, the peaks are parallel to the ND which confirms what we already know. During loading, the (0002) ring rotates clockwise toward the TD, with the greatest intensity centered at approximately 143 and − 37◦ with respect to the ND (CCW is positive) in the last frame. This rotation is apparent in the first frames of loading, and continues to develop further until the sample fractures. In Fig. 53.9, the diffraction lineouts of the eight frames, as well as a “zeroth” frame from before the experiment are shown. These patterns have been integrated into d-spacing to show how interplanar distances change with time and strain, and have vertical lines marking the d-spacing of non-strained Mg for reference. There isn’t any significant shift of the peaks, but there is some broadening. Due to the nonlinearity of the cameras at peak intensity, as well as residual gain in later frames, the relative height and width of the peaks is dubious. Figure 53.10 shows SR-XRD frames 0, 2 and 8 of a CFSS sample with the ED-ND shear plane. The (0002) planes rotates several degrees away from the ND and toward the displacement direction, however, a second peak develops with an orientation rotated ! approximately ! 60◦ from the initial peak (frame 2). This trend is reflected by the orthogonal families of planes, 1010 and 1120 . In Fig. 53.11, the integrated diffraction line profiles are presented and compared to the ! ambient diffraction pattern (Frame 0). Due to the non-linearity of the detectors used, the intensity relative to the 1011 is suspect, but what is clear is the peak shifts to a lower d-spacing over time. This suggests a compressive strain/stress in the crystal that seems to occur from the beginning of load being applied. A compressive stress component is certainly possible because the loading type with this sample geometry is not pure shear. This compressive loading could be an artifact of the split peaks with the basal planes, however, every orientation appears to show this peak shifting, or it’s an outcome of the anisotropic response of the material. Figure 53.12 shows the snapshots of frame 0, 2 and 8 of AMX602 with the shear plane of ED-TD. Here, the (0002) ring is essentially non-existent prior to loading because the strong basal texture is along the normal direction, so most (0002) planes do not diffract. The orthogonal planes are better situated to diffract, and appear as complete rings with a slight orientation

290

C. S. Meredith et al.

Fig. 53.10 SR-XRD patterns of AMX602 with the ED-ND shear plane with snapshots at approximately (a) 0 μs, (b) 13 μs, and (c) 28 μs

Fig. 53.11 Integrated SR-XRD line profiles of AMX602 with the ED-ND shear plane, which are offset vertically for clarity. Marked in blue are the d-spacings of the unstrained peaks in Mg for reference

preference along the displacement direction. During loading the (0002) plane rotates from its initial state facing the cameras to rotated 143 and −37◦ with respect to the ED, where CCW is positive. The relative orientation of the peaks is similar to the ND-TD sample, but with a different orientation of each sample, so it is not clear if this is a coincidence. As loading continues, the intensity of the (0002) grows until the sample fractures. The line profiles for this sample are shown in Fig. 53.13, where the peak for (0002) is shown to grow in intensity as the ring becomes stronger in the diffraction patterns. In this orientation no peak shifting is observed.

53 Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys. . .

291

Fig. 53.12 SR-XRD patterns of AMX602 with the ED-TD shear plane with snapshots at approximately (a) 0 μs, (b) 15 μs, and (c) 28 μs

Fig. 53.13 Integrated SR-XRD line profiles of AMX602 with the ED-TD shear plane, which are offset vertically for clarity. Marked in blue are the d-spacings of the unstrained peaks in Mg for reference

In Fig. 53.14, one of each orientation of the AZ31B CFSS samples is shown in the shear stress vs. time plot, with the camera timings for the SR-XRD snapshots overlaid. The timing of the frames capture the entire loading well, including one or two frames during fracture. The samples are loaded for 45–50 μs, which is almost twice that of AMX602, but the shear flow stresses are lower. Figure 53.15 shows the patterns of frame 0, 2 and 7 of AZ31B with the RD-TD shear plane. The majority of the (0002) planes do not diffract because the basal texture is parallel to the normal direction, while the orthogonal planes diffract in all 360◦ . During loading, the (0002) plane rotates into view at 45 and 225◦ with respect to the RD, with the intensity of

292

C. S. Meredith et al.

a

b 2

3

4

5

6

7

8

150 100 50 0

10

20

30 40 Time (µs)

50

60

250 Frame 1

200

2

3

4

5

6

7

8

Shear Stress (MPa)

Frame 1

200

0

c 250

Shear Stress (MPa)

Shear Stress (MPa)

250

150 100 50 0

0

10

20

30 40 Time (µs)

50

60

200

Frame 1

2

3

4

5

6

7

8

150 100 50 0

0

10

20

30 40 Time (µs)

50

60

Fig. 53.14 Shear stress vs. time curves for select AZ31B CFSS samples with shear planes of (a) RD-TD, (b) RD-ND, and (c) ND-TD, with the SR-XRD camera framings overlaid

Fig. 53.15 SR-XRD patterns of AZ31B with the RD-TD shear plane with snapshots at approximately (a) 0 μs, (b) 20 μs, and (c) 50 μs

this peak increasing over time, which can be indicative of extension twinning. In Fig. 53.16, the integrated line profiles show how the (0002) plane re-orientates with respect to the strongly present orthogonal planes. The (0002) plane shows an increase in intensity, and the orthogonal planes become less intense. As with the other alloy at this orientation, no peak shifting is observed. Figure 53.17 shows SR-XRD frames 0, 2, and 7 of the RD-ND shear plane orientation. The (0002) planes split into a second peak which are rotated about 50◦ from each other, with the initial peak further rotating about 10◦ total. In Fig. 53.18, the line profiles of this sample are shown, and the intensity of the (0002) planes is decreasing as shearing increases. Additionally, the peak shifting here is not nearly as significant as was seen in the AMX602 sample, and very little changes other than the broadening of the peaks as the sample near fracture. Lastly, the patterns for the sample with the NDTD shear plane in AZ31B are shown in Fig. 53.19 with the integrated lineouts present in Fig. 53.20. The crystal re-orientates itself by about 27◦ clockwise to center the (0002) peaks at 146 and −34◦ . This is consistent with behavior shown by the AMX602 samples of the same orientation. In the line profiles, there is a compressive strain from the initial loading in the (0002) plane, which is oriented such that the compressive wave of the Kolsky bar compresses the basal planes. As the crystal is re-orientated, this compressive strain on the (0002) is shown to relax, becoming tensile near failure. Comparing the two Mg alloys studied here, there are some similarities to their microstructural evolution because they share a similar texture, where the c-axis is preferentially orientated parallel, or nearly parallel, to the normal direction of the plate. However, since the texture in AZ31B is stronger, the microstructural changes observed in the diffraction patterns and line profiles are more obvious. The ND-TD shear plane orientation show incremental rotation of the (0002) planes away from the ND and toward the TD, which does not seem to be indicative of twinning. The observed peak broadening, observed in the line profiles, is further evidence for slip activity. The second orientation, ED/RD-ND, shows peak splitting of the basal planes, where similar reorientation behavior has been observed at low strain rates when extension twins form to orientate the basal plane for subsequent slip [15]. But, the 60◦ and 50◦ angles between the (0002) peaks in AMX602 and AZ31B, respectively, is similar to the rotation of contraction twinning in Mg, which is approximately 56◦ . Contraction twinning is very difficult to active so it is rarely observed except under extreme conditions, such as during shock loading. More investigation is required to determine if it is truly activated. Extension twinning is easy to activate when the basal planes elongate, is a rotation of 86◦ . The peak splitting occurs early in the loading which could be indicative of twinning at yield, whereas the reduction of the

53 Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys. . .

293

Fig. 53.16 Integrated SR-XRD line profiles of AZ31B with the RD-TD shear plane, which are offset vertically for clarity. Marked in blue are the d-spacings of the unstrained peaks in Mg for reference

Fig. 53.17 SR-XRD patterns of AZ31B with the RD-ND shear plane with snapshots at approximately (a) 0 μs, (b) 20 μs, and (c) 50 μs

basal plane peak in the corresponding line profiles could indicate slip deformation becomes dominant at later loading times. Finally, the low shear stress at yield for this orientation versus the others is also evidence for extension twinning, because low yield is associated with this twin mode in uniaxial compression perpendicular or tension parallel to the c-axis. The last orientation is the ED/RD-TD shows approximately 90◦ rotations that are indicative of extension twinning. Further study is required to determine whether twinning is occurring, and what mode if it is.

294

C. S. Meredith et al.

Fig. 53.18 Integrated SR-XRD line profiles of AZ31B with the RD-ND shear plane, which are offset vertically for clarity. Marked in blue are the d-spacings of the unstrained peaks in Mg for reference

Fig. 53.19 SR-XRD patterns of AZ31B with the ND-TD shear plane with snapshots at approximately (a) 0 μs, (b) 20 μs, and (c) 51 μs

53 Mechanical Behavior and Deformation Mechanisms of Mg-based Alloys. . .

295

Fig. 53.20 Integrated SR-XRD line profiles of AZ31B with the ND-TD shear plane, which are offset vertically for clarity. Marked in blue are the d-spacings of the unstrained peaks in Mg for reference

53.4 Conclusions Experiments on two Mg alloys (AMX602 and AZ31B) have been performed on a specimen geometry that induces shear localization in “pure” simple shear, called the compact forced simple shear (CFSS) specimen. The deformation occurs on a 2D plane in the specimen, which is oriented with respect to directional aspects of the material’s microstructure and deformation modes. Experiments at dynamic strain rates have been performed at the Dynamic Compression Sector of the Advanced Photon Source at Argonne National Laboratories using in-situ synchrotron x-ray diffraction, which aimed to probe the microstructural evolution during shear-induced localization. Both alloys showed similar textural evolution for the same orientation, however, the changes were more pronounced with AZ31B owing to its stronger initial texture. Samples whose shear plane corresponded to the ND-TD likely showed slip dominated deformation, while the RD/ED-ND and RD/ED-TD orientations showed evidence of contraction and extension twinning, respectively. Acknowledgements This publication is based upon work performed at the Dynamic Compression Sector, which is operated by Washington State University under the U.S. Department of Energy (DOE)/National Nuclear Security Administration award no. DE-NA0002442. This research used resources of the Advanced Photon Source, a DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract no. DE-AC02-06CH11357. The work was performed under a cooperative agreement between the Army Research Laboratory and the University of North Texas (W911NF-16-2-0189). Thanks to Nick Lorenzo (ARL) for helping to conduct the experiments at DCS.

296

C. S. Meredith et al.

References 1. Mordike, B.L., Ebert, T.: Magnesium properties–applications–potential. Mat. Sci. Eng. A. 302, 37–45 (2001) 2. Avedesian, M., Baker, H. (eds.): Magnesium and Magnesium Alloys, ASM Specialty Handbook. ASM International, Metals Park (1999) 3. Jain, A., Agnew, S.R.: Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet. Mat. Sci. Eng. A. 462, 29–36 (2007) 4. Agnew, S.R., Duygulu, Ö.: Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B. Int. J. Plast. 21, 1161–1193 (2005) 5. Chang, Y., Kochmann, D.M.: A variational constitutive model for slip-twinning interactions in hcp metals: application to single- and polycrystalline magnesium. Int. J. Plast. 73, 39–61 (2015) 6. Agnew, S.R., Tomé, C.N., Brown, D.W., Holden, T.M., Vogel, S.C.: Study of slip mechanisms in a magnesium alloy by neutron diffraction and modeling. Scripta Mat. 48, 1003–1008 (2003) 7. Proust, G., Tomé, C.N., Jain, A., Agnew, S.R.: Modeling the effect of twinning and detwinning during strain-path changes of magnesium alloy AZ31. Int. J. Plast. 25, 861–880 (2009) ! ! 8. Lentz, M., Risse, M., Schaefer, N., Reimers, W., Beyerlein, I.J.: Strength and ductility with 1011 — 1012 double twinning in a magnesium alloy. Nat. Commun. 7, 11068 (2016) 9. Bronkhorst, C.A., Cerreta, E.K., Xue, Q., Maudlin, P.J., Mason, T.A., Gray III, G.T.: An experimental and numerical study of the localization behavior of tantalum and stainless steel. Int. J. Plast. 22, 1304–1335 (2006) 10. Marchand, A., Duffy, J.: An experimental study of the formation process of adiabatic shear bands in a structural steel. J. Mech. Phys. Solids. 36, 251–283 (1988) 11. Rittel, D., Lee, S., Ravichandran, G.: A Shear-compression specimen for large strain testing. Exp. Mech. 42, 58–64 (2002) 12. Gray III, G.T., Vecchio, K.S., Livescu, V.: Compact forced simple-shear sample for studying shear localization in materials. Acta Mater. 103, 12–22 (2016) 13. Meredith, C.S., Lloyd, J.T., Sano, T.: The quasi-static and dynamic response of fine-grained Mg alloy AMX602: an experimental and computational study. Mat. Sci. Eng. A. 673, 73–82 (2016) 14. Hustedt, C.J., Lambert, P.K., Huskins-Retzlaff, E.L., Casem, D.T., Tate, M.W., Philipp, H.T., Woll, A.R., Purohit, P., Weiss, J.T., Gruner, S.M., Ramesh, K.T., Hufnagel, T.C.: In situ time-resolved measurements of extension twinning during dynamic compression of polycrystalline magnesium. J. Dyn. Behav. Mat. 4, 222–230 (2018) 15. Brown, D.W., Agnew, S.R., Bourke, M.A.M., Holden, T.M., Vogel, S.C., Tomé, C.N.: Internal strain and texture evolution during deformation twinning in magnesium. Mat. Sci. Eng. A. 399, 1–12 (2005)

Chapter 54

Developing an Alternative to Roma Plastilina #1 as a Ballistic Backing Material for the Ballistic Testing of Body Armor Randy Mrozek, Tara Edwards, Erich Bain, Shawn Cole, Eugene Napadensky, and Reygan Freeney

Abstract Ballistic clay (Roma Plastilina #1; RP1) is currently used as a backing material that is meant to simulate the penetration resistance of the human body during the ballistic testing of body armor. RP1 is a modeling clay with a primary market in the artistic community. Over time, RP1’s formulation and performance have changed to meet the demands of the artistic community. As a result, RP1 must now be heated to 100 ◦ F to obtain the desired response and exhibits a strong temperature-dependent performance such that the backing material is considered out of calibration after 45 min. This presentation will focus on our efforts to develop a replacement for RP1 that exhibits the desired backing material response at room temperature with minimal temperature-dependence. Specifically, the challenges of designing a viscoplastic material with a controlled response that exhibits dimensional stability while providing minimal elastic recovery from deformation even at high strain rates and linking the quasistatic mechanical response with the ballistic performance. Keywords RP1 · Backing material · Clay · Ballistic · Backface deformation

54.1 Introduction Ballistic clay (Roma Plastilina #1; RP1) was demonstrated in 1977 to provide a penetration response consistent with an averaged tissue response obtained from animal testing [1]. RP1 is a useful test media because it is believed to exhibit minimal elastic recovery which means that the indentation can be measured post-test, rather than in-situ, to determine the maximum backface deformation (BFD) behind the armor (Fig. 54.1a). The depth of this deformation is what would be felt by the warfighter and can be linked to the probability of soldier mortality. Although RP1 is important for the military, the primary market for RP1 is the artistic community and over time the material formulation has changed due to changing feedstocks and the demands of the artistic community. These formulation changes have resulted in the current RP1 needing to be heated up to 100 ◦ F to obtain a similar response to 1977. More importantly, RP1 exhibits a significant temperaturedependence such that the material is considered out of calibration after 45 min of use. These inconsistencies complicate the armor effectiveness assessment and may increase the tendency of vendors to over-engineer their armor leading to increased weight for the warfighter. In addition, RP1 can exhibit performance variations depending on the working conditions prior to testing, age, and shear history. Partially due to concerns with RP1, Dr. J. Michael Gilmore from Director, Operations, Test, and Evaluation (DOT&E) requested the National Research Council (NRC) of the National Academies to establish a study committee to evaluate the methodologies used by the US Army to evaluate body armor from 2009 to 2010 [2–4]. In that series of three reports, a significant finding of the NRC was the need to develop a replacement for RP1 that provides a similar response without heating, exhibits minimal changes with time and temperature, and is composed of a minimum number of well-controlled components amongst other criteria. It is critical that the RP1 replacement provides dimensional stability while exhibiting minimal elastic recovery from deformation to obtain an accurate measurement of the BFD. Polymer-based materials are a good choice for an RP1 replacement due to a similarity in material consistency however; the design of the material is non-trivial. The mechanical response of polymers is typically described as viscoelastic referring to the combination of liquid-like viscous flow and the ability to elastically recover from deformation. Both of these attributes will lead to inaccuracies in the BFD measurement. Viscous flow will distort the impression made upon impact if the flow timescale is faster than the time between the shot and R. Mrozek () · T. Edwards · E. Bain · S. Cole · E. Napadensky US Army Research Laboratory, APG, Aberdeen, MD, USA e-mail: [email protected] R. Freeney Aberdeen Test Center, APG, Aberdeen, MD, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_54

297

298

R. Mrozek et al.

Fig. 54.1 Force as a function of penetration depth from spherical indentation measurements at 8 mm/s and a temperature of 100 ◦ F (red) and 72 ◦ F (black), respectively for (a) ARTIC and (b) RP1

the measurement. While elastic recovery of the material will result in the measured indentation after the test being less than the maximum indentation during the impact event. An RP1 replacement material must provide a viscoplastic response where it exhibits dimensional stability, flows in response to deformation and provides minimal elastic recovery after deformation. RP1 accomplishes this balance of properties by using more than 20 components however, the focus of the new material development is to accomplish this same performance using three components or less and at a cost around $21.16 per liter (the current price of RP1). This paper describes initial temperature-dependent characterization of an RP1 candidate material, ARTIC (ARL’s Reusable, Temperature-Insensitive Clay) as compared to RP1 at room temperature and 100 ◦ F.

54.2 Experimental Penetration Testing Penetration testing was performed to evaluate the normal force relative to the displacement into both the ARTIC and RP1 at 72 ◦ F and 100 ◦ F. Testing was conducted using the same 6.35 mm diameter spherical indenter fixed to the end of an 8 mm diameter top platen on a Physica MCR501 rheometer (Anton Paar, Graz, Austria) to enable the use of its CTD 450 environmental chamber for the accurate control of temperature. A stationary 50 mm diameter plate was used as the base platform. Penetration samples were produced by shaping a piece of the material to be tested into a cube with dimensions of approximately 25 mm × 25 mm × 25 mm. Samples sat within the environmental chamber at temperature for a minimum of 10 min to reach thermal equilibrium prior to measurements at 100 ◦ F. Measurements were performed at a rate of 8 mm/s with the probe initiating 30 mm above the base platform (i.e., approximately 5 mm above the top of the 25 mm tall material sample) to allow the probe to accelerate to full speed before contacting the sample. The probe was driven through the material to within 5 mm of the bottom plate to avoid any influence of the boundary conditions. Tests were repeated on a minimum of five material samples at each temperature to examine the reproducibility and range of the penetration force. Gas Gun Testing Samples were formed into stainless steel rectangular molds covered with release paper to form blocks with dimensions of 125 mm × 175 mm × 46 mm. Materials for ballistic penetration depth testing were stored at their respective test temperature (i.e., 72 ◦ F and 100 ◦ F) for a minimum of 24 h before being packed into the molds and for a successive 24 h after the blocks were formed and the molds were released. A total of 3 blocks of a given material were produced for testing at each temperature. Just before shooting, the sample block to be tested was transferred into an environmentally controlled chamber with a front opening to allow for projectile entry. Ballistic penetration testing was performed using a 0.22 caliber gas gun. A 5.56 mm diameter stainless steel ball bearing (Type 302, 0.69 g) was used as the projectile to impact the sample. Projectile speeds between 50–600 m/s were determined with a Doppler radar system (BR-3502, Infinition, Inc., Trois-Rivieres, Quebec, Canada). The projectile’s penetration depth was determined by inserting a stainless steel needle down the length of the channel made by the projectile to the trailing end of the projectile lodged within the material at the conclusion of testing. The distance on the needle at the entry channel hole was marked, the needle was removed from the cavity, and the distance on the needle was measured using a standardized precision metal ruler. The reported penetration depth is this measured penetration depth plus the diameter of the projectile (i.e., distance to the leading edge of the projectile at the conclusion of testing).

54 Developing an Alternative to Roma Plastilina #1 as a Ballistic Backing Material for the Ballistic Testing of Body Armor

299

Fig. 54.2 5.56 mm spherical projectile penetration as a function of velocity for (a) ARTIC at 72 ◦ F compared to RP1 at 100 ◦ F and 72 ◦ F, respectively, and (b) ARTIC at 72 and 100 ◦ F

54.3 Results/Discussion The materials were initially evaluated with a quasi-static penetration testing due to previous work performed at the Aberdeen Test Center to evaluate RP1 and the relative ease of the test. Figure 54.1 provides plots of the force as a function of penetration depth for RP1 and ARTIC tested at 72 ◦ F and 100 ◦ F, respectively. Figure 54.1a demonstrates the significant temperaturedependence of the RP1 response where a test temperature decrease from 100 ◦ F to 72 ◦ F results in a two-fold increase in the force required for penetration. In contrast, ARTIC provides a nearly identical response at both test temperatures and the penetration response is similar to RP1 at 100 ◦ F for both (Fig. 54.1b). In addition, the measured force for both materials quickly drops to zero after the test (not shown) indicating minimal residual stress that would provide a mechanism for elastic recovery. While these results are positive, the test is performed at low rates for a high rate application and may not capture a difference in the strain rate dependent mechanical response at ballistic rates between the two materials. Spherical projectile penetration was used to gain insight into the higher rate response of both ARTIC and RP1. The materials were impacted with 5.56 mm spherical projectile at velocities of 50 to 600 m/s using a gas gun to determine the depth of projectile penetration. A sample chamber was used to provide temperature control at 72 ◦ F and 100 ◦ F, respectively. The RP1 demonstrated noticeably reduced penetration depths at 100 ◦ F compared to 72 ◦ F consistent with the stiffer response observed in the penetration tests (Fig. 54.2a). ARTIC exhibited a very similar penetration response at both test temperatures (Fig. 54.2b). In addition, the depth of penetration as a function of velocity for ARTIC at 72 ◦ F also provided a good match to RP1 at 100 ◦ F in the entire velocity range tested. This data supports that the ARTIC 6.5 may provide a suitable match to heated RP1 even at high strain rates however, full scale ballistic testing is required before that determination can be made.

54.4 Conclusions This initial work indicates that it may be feasible to replace RP1 with a new material that exhibits the desired response without the need for heating. Significant addition evaluation is needed prior to making any determination. Most importantly, a material like ARTIC needs to be evaluated side-by-side with heated RP1 under ballistic test conditions. In addition, ARTIC must be evaluated to determine whether there are any time-dependent changes, the performance at reduced temperatures, and the reproducibility of the performance between lots.

References 1. Prather, R.N., Swann, C.L., Hawkins, C.E.: Backface Signatures of Soft Body Armors and the Associated Trauma Effects; ARCSL-TR-77055; DTIC Document: (1977) 2. Lehowicz, L.G.: Phase I Report on Review of the Testing of Body Armor Materials for Use by the U.S. Army: Letter Report. National Academies Press, (2009) 3. Lehowicz, L.G.: Sciences, DEP Testing of Body Armor Materials for Use by the U.S. Army – Phase II: Letter Report. National Academies Press, (2010) 4. Lehowicz, L.G.: Testing of Body Armor Materials: Phase III. National Academies Press (2012)

Chapter 55

IBII Test for High Strain Rate Tensile Testing of Adhesives A. Guigue, L. Fletcher, R. Seghir, and F. Pierron

Abstract This paper presents the application of the new Image-Based Inertial Impact (IBII) test methodology to study the high strain rate response of adhesives. It relies on an inertial impact (spalling-like) test configuration and the use of ultrahigh speed imaging to record the deformation of the test specimen in the MHz range. The underlying novelty is to use the acceleration obtained from the time-resolved displacement maps to derive stress information leading to the identification of the material constitutive parameters. Here, an epoxy adhesive is tested at strain rates up to 1000 s−1 and its modulus and tensile strength are successfully derived from just the deformed images. Keywords High strain rate · Adhesive · High speed imaging · Virtual fields method · Impact

55.1 Introduction Structural adhesive bonding is a lightweight joining alternative to riveting and bolting. Many polymers can be used as adhesives depending on the application, and polymers are known to be strain rate dependent. Since structures can undergo a wide range of impact loadings, it is essential to understand the behaviour of such adhesives at high rates of strain. The gold standard for high strain rate testing is the Kolsky (or split Hokinson) bar apparatus. However, this technology however suffers from intrinsic limitations, in particular that related to the assumption of stress uniformity. This implies that the test cannot provide reliable data in the transient wave dominated regime, which is of particular significance when testing materials with low wave speeds and low stresses to fracture, which is the case for adhesives submitted to high rate tension. This problem is well detailed in [1] where it is reported that measuring the strain rate with the standard Kolsky bar analysis could result in an error of a factor of 10 on the initial stiffness of the adhesive, while using direct measurement on the sample (in that case, digital image correlation) led to more accurate results. Nevertheless, the impact speeds used in this article were still low enough so that the assumption of quasi-static force equilibrium could be made. This strongly limits the maximum achievable strain rates that the specimens could be tested at. Using a small hat-shaped specimen, Yokoyama and Nakai [2] managed to reach higher speeds without losing the static equilibrium assumption, but their stress rate (estimated at 106 MPa.s−1 ) was still an order of magnitude lower than what was achieved here. It is also possible to test bulk adhesive specimens, and by employing very small specimens [3], strain rates up to 3000 s−1 could be achieved. However, the behaviour of the adhesive in-situ is likely to be somewhat different from that of the bulk adhesive. The present paper explores the possibility to use the new Image-Based Inertial Impact (IBII) test methodology [4] to provide the dynamic response of adhesives at strains rates on the order of 1000 s−1 .

55.2 Methodology The adhesive tested here is an EPOTECNY 504 white coloured epoxy marketed by Epotecny France. This is the adhesive used in [5] to transfer grids for the grid method. 2D butt joint specimens were prepared using an in-house built rig to control the alignment of the two bonded plates and to produce a bondline of 2 mm thickness. The substrate material used here is Aluminium 6061-T6. A schematic of the specimen is provided in Fig. 55.1 and a picture can be seen in Fig. 55.2 (right). The idea of the IBII test is to spall the test specimen using the reflection of the compressive wave off the specimen free edge.

A. Guigue () · L. Fletcher · R. Seghir · F. Pierron Engineering and the Environment, University of Southampton, Southampton, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_55

301

302

A. Guigue et al.

Fig. 55.1 Specimen dimensions and loading configuration

Fig. 55.2 Gas gun (left), projectile and sabot (centre) and test set-up (right)

The dimensions of the test specimen, impactor and waveguide were obtained after a parametric study using explicit finite element modelling. The intent was to ensure that the adhesive would fail in tension and not in compression in the first stages of the test. The specimen was equipped with a bidirectional grid of period 0.9 mm using a photolithography technique and a physical mask. Unfortunately, the photolithographic film was only available in blue colour, which resulted in limited contrast. In the future, the grids will be printed directly onto the specimen using a flat table printer. A Shimadzu HPV-X camera with a 400 × 250 pixel sensor was used to record images of the deforming specimen at 2 MHz. A sampling of 5 pixels per period was used, which is customary for the grid method [6]. The specimen was bonded onto the waveguide using cyanoacrylate glue, and the waveguide was rested onto a foam stand which was carefully aligned using a long rod protruding from the gas gun barrel. A projectile was mounted into an oil filled nylon (oilon) sabot and fired from the compressed gas gun at a speed of approximately 10 m.s−1 . Two thin pieces of copper tape were attached to the impact face of the waveguide so that contact with the projectile resulted in the closure of an electric circuit providing the trigger for the camera. A Bowens Gemini 1000 flash was used to illuminate the event. Because of its relatively long rise time (100 μs) compared to the test duration, it was triggered off the light gates at barrel exit using an in-house developed Arduino system that automatically calculates the required triggering delay as a function of the projectile velocity and target distance. The different parts of the setup are illustrated in Fig. 55.2.

55 IBII Test for High Strain Rate Tensile Testing of Adhesives

303

55.3 Data Processing The raw data consists of 128 grid images at 500 ns time intervals, covering a total test time of 64 μs. The grey level images were processed with spatial phase shifting, using the iterative algorithm proposed by Grédiac et al. (procedure 2) in [6]. This yields maps of the two components of the displacement. Gaussian spatial smoothing over a kernel of 41 data points (8 grid lines) was then used before deriving strains using centred finite differences, while a Savitsky-Golay temporal filter (3rd order over 21 frames) was used before deriving accelerations by double centred finite differences. The underpinning idea of the IBII tests is to use acceleration to reconstruct stress averages across the specimen width. Considering the part of the test specimen between a section of coordinate x and the free edge, the following equilibrium equation can be derived [4] for a homogeneous material: σxx y = ρxax S

(55.1)

where σxx y is the average of the longitudinal stress over a cross-section of coordinate x, ρ is the density of the tested material and ax S is the surface average of the longitudinal acceleration between the considered section and the free edge. This equation holds for any transverse cross-section. The only refinement here is that the material density is different between the glue and the aluminium substrate. Therefore, the following formulae have to be employed instead: σxx y = ρs xax Ssub for x < Ls σxx y = ρs Ls ax Ssub + ρa (x − Ls ) ax Sadh for Ls < x < Ls + e σxx y = ρs Ls ax Ssub + ρa eax Sadh + ρs (x − e − Ls ) ax Ssub−adh for Ls + e < x < L

(55.2)

where ρs and ρa represent the densities of the substrate and adhesive, respectively, and Ssub-adh is the surface between the considered section in the upstream substrate and the upstream adhesive bondline. All other parameters are defined in Fig. 55.1. From the strain maps, the average longitudinal strain can also be computed in each transverse section. Since the stress is predominantly uniaxial, stress-strain curves can be reconstructed in each section. The specific difficulty here lies in the fact that the bondline is wide enough to influence the kinematic fields but too small to provide well-resolved strain information. In particular, spatial smoothing over the bondline is prohibited because of the large difference in stiffness. Therefore, two strain metrics were used to provide stress-strain curves. First, the strain was calculated in the adhesive by the width-averaged displacement difference between the two edges of the bondline. In parallel, it was also evaluated from the width-averaged displacement in the substrate just outside the bondline. This is a crude way of processing the data but was thought reasonable for a first attempt.

55.4 Results and Analysis Figure 55.3 shows strain and strain rate maps at three instances in the test. Here, for representation purposes, strain smoothing was performed and the strain ‘leakage’ between substrate and adhesive is clear. However, one can see that the maximum compressive strain reaches just below 1%, while the maximum tensile strain is only about 0.5% just before fracture. The strain rate maps are also very heterogeneous and strain rates vary between ±1000 s−1 . Figure 55.4 shows the adhesive stress-strain curves, calculated from the two different strain metrics as explained previously. A slight difference is observed in stiffness, the value from the adhesive data is deemed to be more representative as the value from the substrate is likely to integrate some much stiffer substrate data leading to a slightly larger modulus, as observed. The value of this modulus is compatible with typical data for epoxy resins. Another experiment (not detailed here) using the image-based DMTA methodology detailed in [7] led to a modulus for 7 GPa for strain rates an order of magnitude lower than here, which is consistent again for such data. The values found here are consistent with the results in [1]. In Fig. 55.4, the fracture stress can also be observed to be approximately 27 MPa. This is rather low for an epoxy but unfortunately, no reference data is available for this system. It should be mentioned however that this adhesive is not a structural one and its low strength is therefore less of a surprise. Finally, it is clear from Fig. 55.5 that the failure is indeed cohesive. Other tests conducted on carbon-epoxy substrates with the image-based DMTA methodology exhibited adhesive failure.

304

A. Guigue et al.

Fig. 55.3 Strain and strain rate at different times during the test

Fig. 55.4 Stress-strain curves for the two strain calculation methods

Fig. 55.5 Fractured specimen

55.5 Conclusion and Future Work This paper reports work in progress. The current results are promising but future work is needed to refine the test procedure and the data processing. The current data processing is very simple and in the future, the full Virtual Fields Method (VFM, [8]) will be implemented. One of the main difficulties lies in the strong contrast between substrate and adhesive and it is essential to derive more adapted ways to produce the strains. An excellent tool to assess the impact of the data processing on the results is the synthetic image deformation procedure as reported in [9, 10] for instance. This is currently underway.

55 IBII Test for High Strain Rate Tensile Testing of Adhesives

305

Another point to be addressed is the strong heterogeneity in strain rate. In fact, this can be turned into an advantage as there is potential to extract a strain rate dependence from a single test using the VFM. This is currently under study for visco-plastic models [11] and will be extended to elastic stiffness identification in the near future. Acknowledgments The authors gratefully acknowledge funding from EPSRC, grant EP/L026910/1.

References 1. Neumayer, J., et al.: Experimental determination of the tensile and shear behaviour of adhesives under impact loading. J. Adhes. 92(7–9), 503–516 (2016) 2. Yokoyama, T., Nakai, K.: Determination of the impact tensile strength of structural adhesive butt joints with a modified split Hopkinson pressure bar. Int. J. Adhes. Adhes. 56, 13–23 (2015) 3. Goglio, L., et al.: High strain-rate compression and tension behaviour of an epoxy bi-component adhesive. Int. J. Adhes. Adhes. 28(7), 329–339 (2008) 4. Pierron, F., Zhu, H., Siviour, C.: Beyond Hopkinson’s bar. Philosophical Transactions of the Royal Society A: Mathematical. Phys. Eng. Sci. 372(2023), 20130195 (2014) 5. Piro, J.L., Grediac, M.: Producing and transferring low-spatial-frequency grids for measuring displacement fields with moire and grid methods. Exp. Tech. 28(4), 23–26 (2004) 6. Grédiac, M., Sur, F., Blaysat, B.: The grid method for in-plane displacement and strain measurement: a review and analysis. Strain. 52(3), 205–243 (2016) 7. Seghir, R., Pierron, F.: A novel image-based ultrasonic test to map material mechanical properties at high strain-rates. Exp. Mech. 58(2), 183–206 (2018) 8. Pierron, F., Grédiac, M.: The virtual fields method. In: Extracting Constitutive Mechanical Parameters from Full-Field Deformation Measurements, p. 517. Springer, New York (2012) 9. Rossi, M., et al.: Effect of DIC spatial resolution, noise and interpolation error on identification results with the VFM. Strain. 51(3), 206–222 (2015) 10. Rossi, M., Pierron, F.: On the use of simulated experiments in designing tests for material characterization from full-field measurements. Int. J. Solids Struct. 49(3–4), 420–435 (2012) 11. Bouda, P., et al.: Optimization of an image-based experimental setup for the dynamic behaviour characterization of materials. In: Annual SEM Conference. Springer, Greenville (2018)

Chapter 56

Two Modified Digital Gradient Sensing with Higher Measurement Sensitivity for Evaluating Stress Gradients in Transparent Solids Chengyun Miao and Hareesh V. Tippur

Abstract Two modified full-field Digital Gradient Sensing (DGS) methods with higher measurement sensitivity are presented for quantifying small angular deflections of light rays caused by a non-uniform state-of-stress in a transparent solid. These methods are devised by combining or altering previously proposed methods, reflection-mode DGS (r-DGS) (Periasamy and Tippur, Meas Sci Technol 24:025202, 2013) and transmission-mode DGS (t-DGS) (Periasamy and Tippur, Appl Opt 51:2088–2097, 2012). In this presentation, the working principles of r-DGS and t-DGS are introduced first. Then, the socalled t2-DGS method is proposed with the aid of a separate reflective planar surface located behind the transparent solid. The sensitivity of t2-DGS is shown to be twice that of t-DGS. Next, an even higher sensitivity method called the transmissionreflection DGS or simply tr-DGS is developed by making the back surface of a transparent planar solid specularly reflective. The governing equations of tr-DGS are proposed followed by a comparative demonstration of t2-DGS and tr-DGS methods by measuring stress gradients in the crack tip region during a dynamic fracture experiment. The tr-DGS is ∼1.5 times more sensitive than t2-DGS, and at least three times more sensitive than t-DGS approach. Keywords Digital gradient sensing · Quantitative visualization · Angular deflections · Full-field measurements · Photomechanics

56.1 Introduction Transparent materials are widely used in engineering applications. Among them, transparent ceramics and glasses are popular due to favorable characteristics such as, high stiffness and scratch resistance, and very high compression strength, to name a few. Hence, understanding their failure mechanisms is critical for assuring mechanical integrity of such transparent structures. In recent years, full-field methods such as Digital Image Correlation (DIC) have become attractive for measuring surface deformations, due to the advantages of simplicity of surface preparation, ordinary white light illumination, and feasibility of 2D or 3D measurements [3, 4]. In its wake, a new full-field optical method called Digital Gradient Sensing (DGS) was proposed by Periasamy and Tippur for measuring two orthogonal angular deflections of light rays caused by stresses in transparent solids [2, 5]. Subsequently, DGS was modified to study opaque, optically reflective objects by measuring orthogonal surface slopes [1]. The simplicity of the experimental setup and its high measurement sensitivity make DGS popular for experimental mechanics investigations. Furthermore, it has been shown that these measured quantities can be numerically integrated to evaluate surface profiles or stress fields with high accuracy. Miao et al. investigated the feasibility of reflection-mode DGS (r-DGS) in conjunction with a robust Higher-order Finite-difference-based Least-squares Integration (or simply HFLI) scheme to measure the surface topography of thin structures [6, 7]. Then, the authors also extended the application of r-DGS with HFLI to measure sub-micron deformations and stresses in back-side impacted composite plates [8]. Sundaram and Tippur have applied transmission-mode DGS (t-DGS) to study fracture mechanics of ductile materials [9–11]. Recently, they extended the research to study fracture mechanics of very brittle material such as soda-lime glass [12–14]. As noted in [14], due to low fracture toughness and high stiffness, the crack-tip deformations in soda-lime glass are extremely challenging to evaluate. Hence, the ‘optical lever’ of t-DGS was increased significantly to increase the measurement sensitivity of t-DGS [12–14]. This however results in insufficient light and increased need for vibration isolation (to study quasi-static events). Hence, DGS methods with higher measurement sensitivity without significant changes to the

C. Miao · H. V. Tippur () Department of Mechanical Engineering, Auburn University, Auburn, AL, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_56

307

308

C. Miao and H. V. Tippur

‘optical lever’ are attractive. In this context, two modified DGS methods are presented here. These methods are devised by combining or altering r-DGS and t-DGS methods resulting in higher measurement sensitivity.

56.2 Experimental Setup and Working Principles 56.2.1 Reflection-Mode Digital Gradient Sensing (r-DGS) A schematic of the experimental setup for reflection-mode Digital Gradient Sensing (or r-DGS) to measure surface slopes is shown in Fig. 56.1. A digital camera is used to record random speckles on a target plane via the reflective specimen surface. To achieve this, the specimen and the target plate are placed perpendicular to each other, and the beam splitter is placed at 45◦ relative to the specimen and target plate, respectively. The target plate is coated with random black and white speckles, and is illuminated uniformly using ordinary white light. For simplicity, the angular deflections of light rays only in the y-z plane are shown in Fig. 56.2. When the specimen is in the undeformed state, a point P on the target plate comes into focus through point O on the specimen surface. An image is recorded at this time as the reference image. After the specimen suffers out-of-plane deformation due to an impact load, a neighboring point Q on the target plate comes into focus through the same point O on the specimen surface. The corresponding deformed image is recorded next. The local displacements δ y : x can be obtained by performing a 2D image correlation of the reference and deformed images. OP makes an angle φ y with OQ and φ y = θ i + θ r where and θ r (=θ i ) are incident and reflected angles relative to the normal to the specimen. The two orthogonal surface slopes can be expressed as ∂w 1 = tan φ y:x . The governing equations for r-DGS are [1]: ∂y:x 2 ∂w ∂y:x

=

1 2

 tan φy:x ≈

1 2

 φy:x ≈

1 2



δy:x 

 (56.1)

where Δ is the distance between the specimen and the target. The details of the specimen surface for r-DGS are shown in Fig. 56.3a. The specimen surface is made reflective using vapor deposition of aluminum film. (The front face of the specimens illustrated in Fig. 56.3 is towards to the camera and the target.)

Beam splitter

Lamps

Camera

Specimen

x

Target plate z

Crack Fig. 56.1 Schematic of experimental setup for r-DGS, t2-DGS and tr-DGS methods [15]

y

56 Two Modified Digital Gradient Sensing with Higher Measurement Sensitivity for Evaluating Stress Gradients in Transparent. . .

309

y O

Specimen

θi

Δ

θr

Load

n w

n O

φy

Beam splitter

δy

P

Q

y0

Target plate

z Fig. 56.2 Working principle of r-DGS [6]

a

x

Ref lective front surface

y z

b

c

Crack

y

Coated ref lective surface on the back side

x

z

y

x

z

Separate/detached planar ref lective surface behind the specimen

Transparent specimen

Transparent specimen Fig. 56.3 Specimen configurations for: (a) r-DGS; (b) t2-DGS; (c) tr-DGS

56.2.2 Transmission-Mode Digital Gr adient Sensing (t-DGS) The details of experimental setup and working principles for transmission-mode Digital Gradient Sensing (t-DGS) method can be found in [2, 5]. In t-DGS, the refractive index and thickness of the specimen change after imposing a load on the specimen. As a result, light rays deviate from the initial path, which is referred to as the elasto-optical effect. The optical path change, δS, caused by the deformation of the specimen can be expressed as [16]:  1/2  1/2 δS (x, y) = 2B (n − 1) εzz d (z/B) + 2B δn d (z/B) (56.2) 0

0

310

C. Miao and H. V. Tippur

The two terms in the above equation represent the contribution of normal strain in the thickness direction, εzz , and the change in the refractive index, δn, to the overall optical path change, respectively. Equation (56.2) can be reduced to [2] under plane stress conditions as,  δS (x, y) = Cσ B σxx + σyy ,

(56.3)

where Cσ = D1 − (υ/E)(n − 1) is the elasto-optic constant of the specimen. The two angular deflections of light rays φ x : y , which are related to in-plane stress gradients, can be expressed as [2]: φx:y

 ∂ σxx + σyy δx:y . = Cσ B ≈  ∂ (x : y)

(56.4)

56.2.3 Double Transmission Digital Gradient Sensing (t2-DGS) A schematic of experimental setup for double transmission Digital Gradient Sensing or simply t2-DGS method is shown in Fig. 56.1. It is similar to the setup for r-DGS (Fig. 56.3a) except a separate reflective planar surface is placed on the backside of the transparent specimen, as shown in Fig. 56.3b. (The gap between the additional reflective surface and the rear surface of the specimen is nominally zero.) The light rays, initiated from the target plate, pass through the transparent specimen and reach the reflective surface. Then, they are reflected back by the reflective surface and pass through the transparent specimen again. Hence, the speckles on the target are recorded by the camera via the additional reflective surface. As in r-DGS, a reference image is recorded first and then the deformed images as the specimen is subjected to load. The local displacements δ x : y can be measured by correlating the reference image and the deformed images. Thus, light rays experience elasto-optical effects over twice the specimen thickness as a result of reflection from the rear face. Hence, the optical path change here is twice that of t-DGS, or, δSt2 − DGS = 2(δSt − DGS ). The two angular deflection of light rays of t2-DGS, (φ x : y )t2 − DGS , which are related to the in-plane stress gradients, can be then expressed as [15],    ∂ σxx + σyy . φx:y t2−DGS = 2 φx:y t−DGS = 2Cσ B ∂ (x : y)

(56.5)

From the above, it is evident that the sensitivity of t2-DGS is twice that of t-DGS.

56.2.4 Transmission-Reflection Digital Gradient Sensing (tr-DGS) The schematic of the experimental setup for transmission-reflection Digital Gradient Sensing (tr-DGS) method is again similar to the one shown in Fig. 56.1. Furthermore, the experimental setup for tr-DGS is similar to the one for t2-DGS. However, in tr-DGS, the specimen is transparent but its rear face is made reflective by a reflective film deposition, as shown in Fig. 56.3(c). The light rays, initiated from the target plate, pass through the transparent specimen and reach the reflective surface. Then, they are reflected back by the reflective surface and pass through the transparent specimen again. Hence, the speckles on the target are recorded via reflective rear surface. As in r-DGS, a reference image is recorded first and then the deformed images as the specimen is subjected to load. The local displacements can be measured by correlating the deformed images with the reference image. In r-DGS, the reflective surface deforms when the specimen suffers impact load. In t2-DGS, the refractive index and thickness of the specimen change when the specimen subjected to impact load, while the separate reflective surface doesn’t deform. The tr-DGS method combines r-DGS and t2-DGS. Hence, the angular deflections of light rays of tr-DGS (φ x : y )a − DGS is a combination of r-DGS (φ x : y )r − DGS and t2-DGS (φ x : y ) t2-DGS, 

φx:y

tr−DGS

=

  δx:y = φx:y r−DGS + φx:y t2−DGS . 

(56.6)

  ∂w As noted earlier, φx:y r−DGS = 2 ∂(x:y) and for plane stress, εzz ≈ 2w = − Eυ σxx + σyy . Hence, w ≈ B  − υB 2E σxx + σyy , where υ is the Poisson’s ratio, B is the undeformed thickness, and E is the elastic modulus of the specimen.

56 Two Modified Digital Gradient Sensing with Higher Measurement Sensitivity for Evaluating Stress Gradients in Transparent. . .

311

As a result, Eq. (56.7) can be expressed as [15], 

φx:y

tr−DGS

   δx:y υB ∂ σxx + σyy . = 2Cσ B − =  E ∂ (x : y)

(56.7)

56.3 Experimental Verification The working principle for t2-DGS is relatively straightforward; hence, the verification for t2-DGS is not demonstrated here. To verify the tr-DGS method, a comparative demonstration of t2-DGS and tr-DGS methods is carried out on a PMMA plate (152 mm × 76 mm × 5.8 mm) by measuring the stress gradients in the crack tip region during a dynamic fracture experiment. The specimen configuration with its front surface facing to the camera is shown in Fig. 56.4. A 10 mm long horizontal pre-notch was cut along the edge in the mid-span of the specimen. The top-half surface of the backside of the specimen was deposited with a thin aluminum film to make it reflective to implement tr-DGS. The bottom-half surface was placed flush with a reflective planar surface for studying t2-DGS. The specimen was subjected to symmetric mode-I loading by impacting it on the uncracked long-edge. Hence, during the experiment, stress gradients can be measured by tr-DGS and t2-DGS simultaneously in the upper- and lower-halves on the same specimen. The schematic of the overall experimental setup used for the dynamic impact experiment is shown in Fig. 56.5. A modified Hopkinson pressure bar was used for loading the specimen. It consists of a 1.83 m aluminum long rod of 25.4 mm diameter, and a 305 mm long, 25.4 mm diameter aluminum striker placed in the gas gun barrel. The striker was launched towards the long-bar at a velocity of ∼ 4 m/s during tests. A Kirana-05 M ultrahigh-speed camera was used to record the speckle images at 400 K frames per second (inter frame period 2.5 μs). The distance between the specimen and the camera lens plane (L) was ∼795 mm and the one between the specimen mid-plane and the target plane () was 72 mm. The time-resolved angular deflection contour plots φ x : y in the PMMA plate are shown in Fig. 56.6. In these plots, t = 0 μs corresponds to crack initiation at the original notch. In each contour plot, the top-half are measured by tr-DGS, and the bottom-half by t2-DGS. The interface of these two methods is along the growing crack path. It can be observed that the top-half measured by tr-DGS contours are denser and larger than the bottom-half counterparts measured by t2-DGS, which indicates tr-DGS is more sensitive than t2-DGS. It should be noted here that, although the φ x : y contours are different in the

152 mm

Top-half made ref lective for tr-DGS.

Crack

Bottom-half with a detached reflective surface for t2-DGS.

Fig. 56.4 Back-side of the specimen with and without reflective coating in the upper-half and lower-half, respectively. (Front-side of the specimen faces the camera)

312

C. Miao and H. V. Tippur

Fig. 56.5 Schematic of the experimental setup for dynamic plate impact study

Fig. 56.6 Angular deflection contour plots proportional to stress gradients of (σxx + σyy ) in the x- and y-directions. Contour increments = 3 × 10−4 rad

top- and bottom-halves measured by the different methods, they are due to the same stress field. The stress conditions are symmetrical about the crack path. Hence, the values of φ x : y along any symmetric angular paths with respect to the crack-tip polar coordinates (±θ ), measured by tr-DGS and t2-DGS, can be extracted to compare these two methods as a verification of tr-DGS approach.

56 Two Modified Digital Gradient Sensing with Higher Measurement Sensitivity for Evaluating Stress Gradients in Transparent. . .

0.4

0.6

0.8

1

1.2

1.4

t = 2.5 μs

t = 15 μs

313

1.6

0

Ratio

-0.5 -1 -1.5 -2 Theoretical band

-2.5

t = -2.5 μs

-3 Fig. 56.7 Ratio of

t = 0 μs

r/B

(φy )tr−DGS within 0.5 ≤ r/B ≤ 1.5 along ±45◦ (φy )t2−DGS

The contours of φ y are used to make the comparison in this study. Theoretically, the ratio of values of φ x : y measured by tr-DGS and t2-DGS, based on the Eq. (56.5) and (56.7), can be written as:    φx:y tr−DGS υ  . = 1− 2Cσ E φx:y t2−DGS

(56.8)

It is evident that this ratio is dependent only on the properties of the specimen material. For PMMA in [2, 5], the result of Eq. (56.8) in the dynamic loading case should be ∼1.32 whereas ∼1.57 for quasi-static loading conditions due to the value of Cσ reported from different sources in the literature [17]. In the region around the crack-tip, discrete angular deflection values of φ y along 45◦ paths in the range 0.5 ≤ r/B ≤ 1.5 were considered. That is, (φ y )tr − DGS along −45◦ , and (φ y )t2 − DGS along +45◦ , were extracted for comparison. The details of the data paths are shown in Fig. 56.6b. The red dot represents the crack-tip, r is the radial distance from the origin and the heavy black line represents the crack. It can be observed in the φ y plot, the sign of the angular deflection values are opposite with respect to the growing crack path and hence, the values of (φy )tr−DGS is negative. (φy )t2−DGS (φy ) The experimental results of φ tr−DGS in 0.5 ≤ r/B ≤ 1.5 range along ±45◦ are shown in Fig. 56.7 at a few select time ( y )t2−DGS instants. The theoretical band represents the range −1.32 to −1.57. Ratios at t = ±2.5 μs indicate the experimental results just before and after crack initiation at t = 0 μs. It can be observed from Fig. 56.7 that the data points at these time instants are nearly constant and close to the prediction. It is worth noting that, during analysis, sufficient care was exercised to locate the crack-tip.

56.4 Conclusion In this paper, the concept and the feasibility of two new experimental configurations of Digital Gradient Sensing (DGS) methods with enhanced measurement sensitivity are described. These configurations could be particularly valuable for studying high stiffness and low toughness substrates such as transparent ceramics. The concept identified as tr-DGS has measurement sensitivity in excess of 3 relative to t-DGS. It exploits doubling of the optical path within the specimen besides reflection from the deformed back surface achieved by a reflective coating. When the deposition of a reflective coating is not feasible, a simple doubling of the optical path and hence the sensitivity over t-DGS can be achieved in t2-DGS configuration by using a detached reflector kept flush with the back surface of the specimen. Acknowledgement Support for this research through Army Research Office grants W911NF-16-1-0093 and W911NF-15-1-0357 (DURIP) are gratefully acknowledged.

314

C. Miao and H. V. Tippur

References 1. Periasamy, C., Tippur, H.V.: A full-field reflection-mode digital gradient sensing method for measuring orthogonal slopes and curvatures of thin structures. Meas. Sci. Technol. 24, 025202 (2013) 2. Periasamy, C., Tippur, H.V.: Full-field digital gradient sensing method for evaluating stress gradients in transparent solids. Appl. Opt. 51(12), 2088–2097 (2012) 3. Sutton, M.A., Orteu, J.J.: Image Correlation for Shape, Motion and Deformation Measurements. Springer, New York (2009) 4. Pankow, M., Justusson, B., Waas, A.M.: Three-dimensional digital image correlation technique using single high-speed camera for measuring large out-of-plane displacements at high framing rates. Appl. Opt. 49, 3418–3427 (2010) 5. Periasamy, C., Tippur, H.V.: Measurement of orthogonal stress gradients due to impact load on a transparent sheet using digital gradient sensing method. Exp. Mech. 53, 97–111 (2013) 6. Miao, C., Sundaram, B.M., Huang, L., Tippur, H.V.: Surface profile and stress field evaluation using digital gradient sensing method. Meas. Sci. Technol. 27, 095203 (2016) 7. Miao, C., Tippur, H.V.: Measurement of orthogonal surface gradients and reconstruction of surface topography from digital gradient sensing method. In: Advancement of Optical Methods in Experimental Mechanics, pp. 203–206. Springer, Cham (2017) 8. Miao, C., Tippur, H.V.: Measurement of Sub-micron Deformations and Stresses at Microsecond Intervals in Laterally Impacted Composite Plates Using Digital Gradient Sensing. J. Dynamic Behav. Mat. 1–23 (2018). https://doi.org/10.1007/s40870-018-0156-4 9. Sundaram, B.M., Tippur, H.V.: Dynamic crack growth normal to an interface in Bi-Layered materials: an experimental study using digital gradient sensing technique. Exp. Mech. 56, 37–57 (2015) 10. Sundaram, B.M., Tippur, H.V.: Dynamics of crack penetration vs. branching at a weak interface: an experimental study. J. Mech. Phy. Solids. 96, 312–332 (2016) 11. Sundaram, B.M., Tippur, H.V.: Dynamic mixed-mode fracture behaviors of PMMA and polycarbonate. Eng. Fract. Mech. 176, 186–212 (2017) 12. Sundaram, B.M., Tippur, H.V.: Dynamic fracture of soda-lime glass: a full-field optical investigation of crack initiation, propagation and branching. J, Mech. Phy Solids. (2018). https://doi.org/10.1016/j.jmps.2018.04.010 13. Sundaram, B.M., Tippur, H.V.: Full-field measurement of contact-point and crack-tip deformations in soda-lime glass. Part-I: Quasi-static Loading. Int. J. Appl. Glas. Sci. 9, 114–122 (2018) 14. Sundaram, B.M., Tippur, H.V.: Full-field measurement of contact-point and crack-tip deformations in soda-lime glass. Part-II: Stress wave loading. Int. J. Appl. Glas. Sci. 9, 123–136 (2018) 15. Miao, C., Tippur, H.V.: Higher sensitivity DigitalGradient Sensing configurations for quantitative visualization of stress gradients in transparent solids. Opt. Lasers Eng. 108, 54–67 (2018) 16. Tippur, H.V., Krishnaswamy, S., Rosakis, A.J.: Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip K-dominance. Int. J. Fract. 52, 91–117 (1991) 17. Xu, L., Tippur, H., Rousseau, C.-E.: Measurement of contact stresses using real-time shearing interferometry. Opt. Eng. 38, 1932–1937 (1999)

Chapter 57

Quantitative Visualization of Sub-Micron Deformations and Stresses at Sub-Microsecond Intervals in Soda-Lime Glass Plates Chengyun Miao and Hareesh V. Tippur

Abstract Full-field optical measurement of deformations and stresses on transparent brittle ceramics such as soda-lime glass is rather challenging due to the low toughness and high stiffness characteristics. Particularly, the surface topography and stress field evaluation from measured orthogonal surface slopes and stress gradients could be of considerable significance for visualizing and quantifying deformation of glass plates under dynamic impact loading. In this work, two full-field optical techniques, reflection Digital Gradient Sensing (or r-DGS) and a new DGS method, called transmission-reflection Digital Gradient Sensing (or tr-DGS) are employed to quantify surface slopes and stress gradients, respectively, as glass specimens are subjected dynamic impact loading using a modified Hopkinson pressure bar. These two methods can measure extremely small angular deflections of light rays caused by surface deformations and local stresses in specimens. The trDGS methodology is especially more sensitive than r-DGS. Using such optical methods, sub-micron surface deflections and the corresponding stress field, (σxx + σyy ), can be quantified using a Higher-order Finite-difference-based Least-squares Integration (HFLI) scheme. When used in conjunction with ultrahigh-speed photography, microsecond or sub-microsecond temporal resolution is possible. Keywords Digital gradient sensing · Dynamic impact loading · Transparent brittle ceramics · Deflections and stresses · Ultrahigh-speed photography

57.1 Introduction Transparent ceramics such as soda-lime glass is popular engineering material as it offers benefits such as high stiffness and hardness, and very high compression strength besides low cost and high sustainability. Further, polymer-laminated glasses are used as transparent armor in military applications. The ability to bear load under dynamic impact is critical as well. Hence, quantitative visualization of deformations under dynamic impact loading is important. A few years ago Periasamy and Tippur [1] proposed a full-field optical method called Digital Gradient Sensing (DGS) for measuring two orthogonal in-plane stress gradients in transparent solids. It employs 2D DIC for quantifying elastooptic effects in transparent materials. Subsequently, they [2] extended DGS to study reflective objects by measuring two orthogonal surface slopes. The simplicity of the experimental setup, good accuracy, and easy access to image correlation algorithms make DGS very attractive for experimental mechanics investigations. Furthermore, these measured quantities can be numerically integrated readily to evaluate surface topography or stress fields. Miao et al. investigated the feasibility of rDGS in conjunction with a robust Higher-order Finite-difference-based Least-squares Integration (or simply HFLI) scheme to measure the surface topography of thin structures [3, 4]. Later on, the authors showed that r-DGS can be used with HFLI to measure sub-micron deformations at microsecond resolution on composite plates under dynamic loading [5]. Very recently, the authors proposed two modified DGS variants with even higher measurement sensitivity [6], which are valuable for studying high stiffness and low toughness materials such as glass. Sundaram and Tippur have applied transmission-mode DGS (t-DGS) to study dynamic fracture mechanics of transparent and brittle materials including glass [7–12]. Considering their work, the feasibility of other DGS methods being applied to study glass needs to be examined. In this context, submicron deformations in glass plates are measured at sub-microsecond intervals and stresses are evaluated in glass plate.

C. Miao · H. V. Tippur () Department of Mechanical Engineering, Auburn University, Auburn, AL, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_57

315

316

C. Miao and H. V. Tippur

57.2 Experimental Details 57.2.1 Reflection-Mode Digital Gradient Sensing (r-DGS) A schematic of the experimental setup for reflection-mode Digital Gradient Sensing (or r-DGS) to measure surface slopes is shown in Fig. 57.1. A digital camera is used to record random speckles on a target plane via the reflective specimen surface. To achieve this, the specimen and the target plate are placed perpendicular to each other, and the beam splitter is placed at 45◦ relative to the specimen and target plate, respectively. The target plate is coated with random black and white spray painted speckles, and is illuminated uniformly using ordinary white light. For simplicity, the angular deflections of light rays only in the y-z plane are shown in Fig. 57.2. When the specimen is in the undeformed state, a point P on the target plate comes into focus through point O on the specimen surface. The speckle image is recorded at this instant as the reference image. After the specimen suffers out-of-plane deformation due to the impact load, a neighboring point Q on the target plate comes into focus through the same point on the specimen surface O. The corresponding deformed image is recorded next. The local displacements δ y : x can be obtained by performing a 2D image correlation of the reference and deformed images. OP makes an angle φ y with OQ and φ y = θ i + θ r where and θ r (=θ i ) are incident and reflected angles relative to the normal to the specimen. The two orthogonal surface slopes can be  ∂w expressed as ∂y:x = 12 tan φy:x . The governing equations for r-DGS are [2]: ∂w ∂y:x

=

1 2

 tan φy:x ≈

1 2

 φy:x ≈

1 2



δy:x 

 ,

(57.1)

where Δ is the distance between the specimen and target plate. The details of the specimen surface for r-DGS are shown in Fig. 57.3a. The specimen surface is made reflective using vapor deposition of aluminum film. (The front face of the specimens illustrated in Fig. 57.3 is towards to the camera and the target plate.)

57.2.2 Transmission-Reflection Digital Gradient Sensing (tr-DGS) The schematic of the experimental setup for transmission-reflection Digital Gradient Sensing (tr-DGS) method is similar to the one shown in Fig. 57.1. However, in tr-DGS, the specimen is transparent but its rear face is made reflective by a reflective

Beam splitter

Lamps

Camera

Specimen

x

Target plate z

Fig. 57.1 Schematic of experimental setup for r-DGS and tr-DGS methods [6]

y

57 Quantitative Visualization of Sub-Micron Deformations and Stresses at Sub-Microsecond Intervals in Soda-Lime Glass Plates

317

y O

Specimen

θi

Δ

θr

Load

n w

n O

φy

Beam splitter

δy

P

Q

y0

Target plate

z Fig. 57.2 Working principle of r-DGS [3]

a

b Coated reflective surface on the back side y

y x

x z

z

Reflective front surface

Transparent specimen

Fig. 57.3 Specimen configurations for: (a) r-DGS; (b) tr-DGS [5]

film deposition, as shown in Fig. 57.3b. The light rays from the target plate pass through the transparent specimen and reach the reflective surface. Then, they are reflected back by the reflective surface and pass through the transparent specimen again. As in r-DGS, a reference image is recorded first and then the deformed images as the specimen is subjected to load. The local displacements can be measured by correlating the reference image with the deformed images. In r-DGS, the reflective surface deforms when the specimen suffers impact load. In tr-DGS, when the specimen is impacted, the refractive index and thickness of the specimen change; furthermore, the reflective rear surface also deforms. Hence, tr-DGS is much more sensitive than r-DGS. The two angular deflections of light rays of tr-DGS (φ x : y )tr − DGS , which are related to the in-plane stress gradients, can be expressed as [5], 

φx:y

tr−DGS

=

   δx:y υB ∂ σxx + σyy = 2Cσ B − .  E ∂ (x : y)

(57.2)

318

C. Miao and H. V. Tippur

57.3 Glass Plate Subjected to Dynamic Impact 57.3.1 Quantitative Evaluation of Sub-Micron Magnitude Deformations The dynamic backside/lateral impact of soda-lime glass plate was studied using r-DGS in conjunction with ultrahigh-speed digital photography. The specimen was 152.4 mm × 101.6 mm × 4.6 mm. One of the two 152.4 × 101.6 mm2 faces of the specimen was made reflective by depositing with a thin aluminum film. The reflective surface faces the target plate and the camera. The schematic of the experimental setup employed is shown in Fig. 57.4. A modified Hopkinson pressure bar (or simply a ‘long-bar’) was used for loading the uncoated backside of the specimen. It included a 1.83 m steel rod of 25.4 mm diameter with a tapered end of 3 mm diameter impacting the backside of the specimen. A 305 mm long, 25.4 mm diameter steel striker was placed in the gas-gun barrel to impact the long-bar. The striker was launched towards the long-bar at a velocity of ∼ 2.5 m/s during the tests. At the same time, a Kirana-05 M ultrahigh-speed camera was used to record speckles on the target plane at 1.25 million frames per second (inter-frame period of 0.8 μs) in this experiment. The distance between the specimen and the camera lens plane (L) was ∼950 mm and the one between the specimen mid-plane and the target plane (Δ) was 102 mm. ∂w The time-resolved orthogonal surface slope contours ∂w ∂x and ∂y due to transient stress wave propagation in the glass plate are shown in the first two rows of Fig. 57.5 at a few select time instants. The start of the impact is at t = 0 μs. In the early stages of impact, deformations are concentrated close to the center of the plate resulting in sparse surface slope contours. With the passage of time, the contours get denser and larger with a higher concentration of contours near the contact point. The reconstructed 3D surfaces computed through 2D integration by using surface slope data from r-DGS in conjunction with HFLI are plotted in row 3 of Fig. 57.5. The out-of-plane deformation is around 0.63 μm at t = 4 μs, which shows that this method is able to detect sub-micron deformations at sub-microsecond intervals.

57.3.2 Quantification of Stress Fields Quantitative evaluation of stress fields in soda-lime glass plate subjected to dynamic lateral impact was studied using tr-DGS in conjunction with ultrahigh-speed digital photography. The specimen was 76 mm × 24 mm × 5.6 mm strip with one of the two 76 × 5.6 mm2 faces made reflective by depositing a thin aluminum film. The non-reflective surface of the glass strip faces the camera. The schematic of the experimental setup for this experiment is similar to the previous one. A long-bar was used for impact loading the 24 mm × 5.6 mm face of the specimen using a 1.83 m steel rod of 25.4 mm diameter with a tapered

Fig. 57.4 Schematic of the experimental setup for dynamic plate impact study. Inset shows close-up view of the optical arrangement

57 Quantitative Visualization of Sub-Micron Deformations and Stresses at Sub-Microsecond Intervals in Soda-Lime Glass Plates

319

Fig. 57.5 Evolution of w,x (row 1) and w,y (row 2) contours and surface topography (row 3) for a clamped glass plate subjected to central impact. Note: Location (0, 0) is made to coincide with the loading point. Contour increments = 8 × 10–5 rad

25.4 × 6.7 mm2 rectangular tip. A close-up view of the optical arrangement is shown in Fig. 57.6. As in the previous experiment, a striker was launched towards the long-bar at a velocity of ∼ 5 m/s during tests. Simultaneously a Kirana-05 M ultrahigh-speed camera was used to record the speckle on the target plane at 500 K frames per second. The distance between the specimen and the camera lens plane (L) was ∼1500 mm and the one between the specimen mid-plane and the target plane (Δ) was 177 mm.

320

C. Miao and H. V. Tippur

Fig. 57.6 Close-up view of the optical arrangement for tr-DGS

Fig. 57.7 Evolution of φ x (row 1) and φ y (row 2) contours and stress fields (row 3) for a glass plate subjected to flank side impact. Contour increments = 2.5 × 10−5 rad

The time-resolved orthogonal stress contours φ (x : y) due to transient stress wave propagation in the glass plate are shown in the first two rows of Fig. 57.7 at a few select time instants. The start of the impact event is at t = 0 μs. It can be observed that all the contours are symmetric about the center of the sample, confirming a symmetric surface-to-surface contact. The stress concentrations can be observed at the two corners of the specimen. The reconstructed 3D stress fields (σxx + σyy ) were obtained from the measured stress gradients by using HFLI and are plotted in the 3rd row of Fig. 57.7. The stress concentrations are also very clear in the 3D depiction of the stress fields.

57 Quantitative Visualization of Sub-Micron Deformations and Stresses at Sub-Microsecond Intervals in Soda-Lime Glass Plates

321

57.4 Conclusions In this paper, two full-field optical methods have been applied to quantitatively visualize out-of-plane deformations and (σxx + σyy ) stress fields in thin soda-lime glass subjected to dynamic impact. First, the feasibility of evaluating sub-micron deformations at sub-microsecond temporal resolutions over relatively large regions-of-interest has been demonstrated using r-DGS and HFLI along with ultrahigh-speed photography. Subsequently, another much higher sensitivity method, tr-DGS, has been developed and applied using HFLI to visualize the evolution of (σxx + σyy ) stress fields in glass. Thus, both out-ofplane deformations and (σxx + σyy ) stress fields are successfully evaluated using variants of DGS methodology. Acknowledgement Support for this research through Army Research Office grants W911NF-16-1-0093 and W911NF-15-1-0357 (DURIP) are gratefully acknowledged.

References 1. Periasamy, C., Tippur, H.V.: Full-field digital gradient sensing method for evaluating stress gradients in transparent solids. Appl. Opt. 51(12), 2088–2097 (2012) 2. Periasamy, C., Tippur, H.V.: A full-field reflection-mode digital gradient sensing method for measuring orthogonal slopes and curvatures of thin structures. Meas. Sci. Technol. 24, 025202 (2013) 3. Miao, C., Sundaram, B.M., Huang, L., Tippur, H.V.: Surface profile and stress field evaluation using digital gradient sensing method. Meas. Sci. Technol. 27, 095203 (2016) 4. Miao, C., Tippur, H.: Measurement of orthogonal surface gradients and reconstruction of surface topography from digital gradient sensing method. In: Advancement of Optical Methods in Experimental Mechanics, pp. 203–206. Springer, Cham (2017) 5. Miao, C., Tippur, H.V.: Measurement of sub-micron deformations and stresses at microsecond intervals in laterally impacted composite plates using digital gradient sensing. J. Dyn. Behav. Mater. 1–23 (2018). https://doi.org/10.1007/s40870-018-0156-4 6. Miao, C., Tippur, H.V.: Higher sensitivity digital gradient sensing configurations for quantitative visualization of stress gradients in transparent solids. Opt. Lasers Eng. 108, 54–67 (2018) 7. Sundaram, B.M., Tippur, H.V.: Dynamic crack growth Normal to an Interface in bi-layered materials: an experimental study using digital gradient sensing technique. Exp. Mech. 56, 37–57 (2015) 8. Sundaram, B.M., Tippur, H.V.: Dynamics of crack penetration vs. branching at a weak interface: an experimental study. J. Mech. Phys. Solids. 96, 312–332 (2016) 9. Sundaram, B.M., Tippur, H.V.: Dynamic mixed-mode fracture behaviors of PMMA and polycarbonate. Eng. Fract. Mech. 176, 186–212 (2017) 10. Sundaram, B.M., Tippur, H.V.: Dynamic fracture of soda-lime glass: a full-field optical investigation of crack initiation, propagation and branching. J. Mech. Phys. Solids. (2018). https://doi.org/10.1016/j.jmps.2018.04.010 11. Sundaram, B.M., Tippur, H.V.: Full-field measurement of contact-point and crack-tip deformations in soda-lime glass. Part-I: quasi-static loading. Int. J. Appl. Glas. Sci. 9, 114–122 (2018) 12. Sundaram, B.M., Tippur, H.V.: Full-field measurement of contact-point and crack-tip deformations in soda-lime glass. Part-II: stress wave loading. Int. J. Appl. Glas. Sci. 9, 123–136 (2018)

Chapter 58

Microstructural Effects in the High Strain Rate Ring Fragmentation of Copper Sarah Ward, Christopher Braithwaite, and Andrew Jardine

Abstract Understanding the failure of rings and shells at high strain rates is a longstanding challenge (Mott, N.F. (1947), Fragmentation of rings and shells, Proc. Royal Soc., A189, 300–308, January.). Predicting the distribution of fragment sizes, shapes and velocities has been an important objective for the many modelling techniques applied to the problem; a detailed history of this research can be found in the work of Grady (Grady, Fragmentation of rings and shells. Springer, Berlin Heidelberg, 2006). Physical testing is particularly important for model development and validation. More recently, ring fragmentation has been used to study the relationship between material microstructure and dynamic fracture. A series of explosively loaded fragmentation experiments were conducted to investigate microstructural effects in the dynamic tensile behaviour of high purity copper. Rings of high purity copper were expanded at strain rates of approximately 104 s−1 . Diagnostics include PDV, high speed photography and soft capture. The fracture mechanism is studied through the detailed analysis of fracture surfaces and fragments using scanning electron microscopy. Microstructural changes induced by the plastic deformation developed during the applied loading are also examined. Keywords Copper · Ring · Fracture · Fragmentation · Microstructure

Ring fragmentation experiments were conducted using C110 OFHC copper rings of a 2 mm square cross-section and a 22 mm inner diameter. The rings were fitted to an EN3B mild steel driver cylinder containing an explosive charge. Upon the detonation of the explosive charge, the rings were radially expanded. The explosive mass contained within the driver cylinder was varied to allow for control of the strain rate of the experiment. Figure 58.1 shows a schematic of the driver cylinder and ring assembly. Diagnostics included PDV, high speed photography and soft capture within paraffin wax. PDV probes were positioned in alignment with the direction of radial expansion, and velocity traces were recorded for each experiment in a variety of circumferential positions (between 12◦ and 90◦ ) to study any variation of expansion velocity in the later stages of fragmentation. The results of six experiments are presented here. They were conducted at peak strain rates of approximately 1 × 104 s−1 . The number of fragments produced in each experiment was found to increase with increasing strain rate, as was the failure strain of the ring (in agreement with results from other authors [1]). Figure 58.2 shows velocity-time traces for all experiments. Experiments conducted with a small explosive charge showed a sharp increase to a peak velocity, followed by a linear decrease in velocity until the ring became stationary. The ring was then re-accelerated to a much lower second peak velocity. The ring is initially accelerated into free flight from the detonation shockwave. The linear nature of the deceleration of the ring is due to the flow stress in the deforming copper slowing the ring [2], while the reacceleration phase was caused by the driver cylinder catching up with the ring. For experiments where a larger explosive charge was used, the ring was initially accelerated to a much higher peak velocity. Some traces showed a short section of linear deceleration at the same rate as those seen on experiments conducted with a smaller explosive charge. Higher strain rates also correlated with increased variation in the expansion velocity around the circumference of the ring.

S. Ward () · C. Braithwaite · A. Jardine Univeristy of Cambridge, SMF Fracture and Shock Physics Group, Cavendish Laboratory, Cambridge, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_58

323

324

S. Ward et al.

Fig. 58.1 Cross-section of driver cylinder and ring assembly

Fig. 58.2 Velocity time traces for varying explosive charges. The legend states the mass of explosives used followed by the channel number if multiple traces were recorded for a single experiment. Traces from the same experiment are shown in the same colour group

Soft captured fragments were examined under a scanning electron microscope. There was strong evidence of a slip based failure mechanism. The fracture surfaces themselves (Fig. 58.3a) were smooth and highly reflective, indicative of the new surfaces formed during a slip based failure [3]. There was no evidence of classic ductile fracture voids. The outer surfaces of the fragments were heavily textured, exhibiting an orange-peel effect (Fig. 58.3b), suggested to be the result of the interaction of slip bands with the outer surface [3]. Acknowledgments We acknowledge support of this research by EPSRC and AWE. The PDV data analysis was performed using software written by N. Taylor of the SMF group at the Cavendish Laboratory. © British Crown Owned Copyright 2018/AWE.

58 Microstructural Effects in the High Strain Rate Ring Fragmentation of Copper

325

Fig. 58.3 (a) Fracture surface from experiment (b) Orange-peel effect on outside of fragment

References 1. Altynova, M., Hu, X., Daehn, G.S.: Increased ductility in high velocity electromagnetic ring expansion. Mertallurgical Mater. Trans. A. 27(7), 1837–1844 (1996) 2. Gourdin, W.H., Lassila, D.H.: The mechanical behavior of pre-shocked copper at strain rates of 10− 3− 10 4 s− 1 and temperatures of 25–400◦ C. Mater. Sci. Eng. A. 151(1), 11–18 (1992) 3. Ghahremaninezhad, A., Ravi-Chandar, K.: Ductile failure in polycrystalline OFHC copper. Int. J. Solids Struct. 48(24), 3299–3311 (2011)

Chapter 59

Uncertainties in Low-Pressure Shock Experiments on Heterogeneous Materials Tracy J. Vogler, Matthew Hudspeth, and Seth Root

Abstract Understanding and quantifying the uncertainties in experimental results are crucial to properly interpreting simulations based on those results. While methods are reasonably well established for estimating those uncertainties in highpressure shock experiments on homogeneous materials, it is much more difficult to treat relatively low-pressure experiments where shock rise times are significant and material strength is not negligible. Sample heterogeneity further complicates the issue, especially when that heterogeneity is not characterized in each sample. Here, we extend the Monte Carlo impedance matching approach used in high-pressure Z experiments to low-pressure experiments on heterogeneous porous materials. The approach incorporates uncertainties not only in the equation of state of the impedance matching standard but also those associated with its strength. In addition, we also examine approaches for determining material heterogeneity and evaluate its effect on the experimental results. Keywords Shock wave · Porosity · Uncertainty analysis · Granular materials · Heterogeneity

59.1 Introduction It is necessary to understand the uncertainties and systematic errors associated with experimental data in order to make appropriate decisions based on simulations based on or calibrated to those data. Mitchell and Nellis [1] applied standard uncertainty analyses to shock physics experiments, and their methodology has been widely used since. Their high-pressure shock experiments involved only a few experimental variables (impact velocity, sample thickness, shock arrival times, etc.), and the governing equations were relatively simple. Thus, they could utilize analytic expressions for partial derivatives in calculating the uncertainties. An alternate has been proposed by Root et al. [2] in which forward Monte Carlo (MC) simulations are performed based on appropriate distributions of experimental parameters. Thus, if there are k experimental parameters ξ = (ξ 1 , ξ 2 , . . . , ξ k ), then one generates N realizations of ξ with each element of it chosen from an appropriate distribution. Typically, an individual parameter ξ i is described by a normal distribution with a mean value and a standard deviation. If there are correlations between the experimental parameters, then those can be accounted for in generating the realizations. For example, when impedance matching utilizing a standard material (e.g. Al, Cu, Ta), the values of co and s are correlated rather than independent. Besides the ability to incorporate correlations, MC uncertainty analysis allows more complex behaviors that are not amenable to closed form solutions. In general, MC uncertainty analysis should provide the same results as the approach of Mitchell & Nellis when the two methods are applied in a consistent manner. While the MC approach has been applied in high-pressure experiments on the Z machine, to date it has not been applied to the low-pressure regime that is relevant to the crush response of porous materials such as foams and granular materials. Fredenburg et al. [3] and Vogler et al. [4] have utilized the approach of Mitchell & Nellis. Here, we extend the MC approach to the low-pressure regime by incorporating strength of standard material. We also attempt to address issues related to sample heterogeneity that are of particular concern for these materials.

T. J. Vogler () Sandia National Laboratories, Livermore, CA, USA e-mail: [email protected] M. Hudspeth · S. Root Dynamic Material Properties, Sandia National Laboratories, Albuquerque, NM, USA © The Society for Experimental Mechanics, Inc. 2019 J. Kimberley et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-95089-1_59

327

328

T. J. Vogler et al.

59.2 Overview of Experiments The approach outlined here is applied to the experiments similar to those described by Vogler et al. [4] and Brown et al. [5]. Briefly, a loose powder such as sand is poured into a cavity created in a PMMA block (the cell) and covered with an aluminum plate (the driver). The average density of the sample ρ oo is calculated from the sample’s mass and the volume it occupies in the cell. A second, thick plate (the impactor) is launched at a few 100’s of meters per second using a smooth bore gas gun so that it impacts the driver at velocity Vimp , transmitting a planar shock into the driver and subsequently into the sample. The velocity of the impactor is measured using shorting pins, and the tilt of the impactor is also characterized by shorting pins around the circumference of the target. Laser interferometry is used to measure the shock arrival time tj at different sample thicknesses, xj . Impactor tilt is accounted for in the arrival times, and the shock velocity Us is found as a best fit to the arrival data (xj , tj ).

59.3 Monte Carlo Approach Each of the experimental parameters given above is assigned a statistical distribution, typically a normal distribution characterized by a mean and standard deviation. In general, these statistical parameters are based on studies of those measurement techniques. Additional uncertainties in timing associated with bowing of the impactor, δtbow , and tilt, δttilt , are also include for each of the arrival times. A large number of realizations of the experimental parameters is then generated, and the final shock state calculated. While this is relatively straightforward, a few issues merit more discussion. For low pressures, the velocity histories measured are not abrupt step functions but rather gradual ramps [4, 5] as shown in Fig. 59.1. While the velocity histories from different positions are typically quite similar, some variations are typically seen. The variation between two velocity histories is characterized by calculating values for the time difference for different values of up , Δtshape ( up ). This is done over the range of 10–90% of the peak velocity (approximately 0.5 km/s in the figure) to generate a statistical distribution for the timing uncertainty due to shape, δtshape that is sampled as part of the MC analysis. Impedance matching utilizes the known behavior of the aluminum impactor/driver to determine the stress in the sample. This is shown schematically in Fig. 59.2a. The red curve is referred to as the Rayleigh line and has a slope given by the product of the initial density and shock velocity ρ oo Us . In this version, the unloading curve for aluminum is assumed to be identical to the shock response (it is reflected). There is uncertainty associated with the shock response of aluminum, which is indicated by the dashed lines in the unloading curve. This is incorporated through the variance and covariance matrices associated with the parameters co and s, which come from analysis of high pressure shock data for aluminum. These uncertainties are assumed to hold for the low-pressure regime as well. Impedance matching typically does not account for the strength of the standard material, but for the relatively low stress levels of these experiments that effect should not be neglected. In an effort to account for the strength, the response of aluminum is simulated in CTH utilizing the Steinberg-Guinan rate-independent strength model. The release path

Fig. 59.1 Velocity histories for different sample thicknesses illustrating the arrival time difference for two histories at a specific value of velocity

59 Uncertainties in Low-Pressure Shock Experiments on Heterogeneous Materials

329

Fig. 59.2 Impedance matching for the porous sample to the aluminum impactor and driver (a) utilizing a reflected Hugoniot response with uncertainty in the Al shock response and (b) showing the effect of strength on the aluminum release path

incorporating strength is shown with the purple solid curve in Fig. 59.2b. It lies below and to the left of the reflected Hugoniot. A correction ustr p for strength is tabulated from simulations for various values of impact velocity and sample impedance (the slope of the dashed lines). The correction is then found for a given experiment by interpolation. In general, the correction increases with impact velocity and decreases as the sample impedance increases. Corrections of 10–20 m/s are typical. Finally, sample density and uniformity present a difficult issue. Sample mass can be measured very accurately as can, at least for regular shapes, the volume of the sample. However, samples of heterogeneous materials typically have local variations in the sample density at different scales. At present, each realization incorporates an uncertainty in the initial density δρ oo that is relatively large. However, a more appropriate approach is to characterize the variation in local density using conventional radiography or tomography. Once the heterogeneity is characterized, appropriate methods for incorporating it into the MC analysis will be developed. Work in this area is ongoing.

59.4 Results The uncertainty analysis outlined above was applied to experimental data for a porous material as shown in Fig. 59.3. The lower stress data point corresponds to about 30% residual porosity, while the higher stress one has minimal (