Dynamic Behavior of Materials, Volume 1

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamic Behavior of Materials, Volume 1 Leslie E. Lamberson Proceedings of the 2019 Annual Conference on Experimental and Applied Mechanics River Publishers

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

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.

River Publishers Dynamic Behavior of Materials, Volume 1 Proceedings of the 2019 Annual Conference on Experimental and Applied Mechanics Leslie E. Lamberson Editor

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-990-0 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction 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, expressed 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.

Preface Dynamic Behavior of Materials represents one of six volumes of technical papers presented at the 2019 SEM Annual Conference and Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics and held in Reno, NV, June 3–6, 2019. The complete proceedings also include volumes onChallenges in Mechanics of TimeDependent Materials; Fracture, Fatigue, Failure and Damage Evolution; Advancement of Optical Methods & Digital Image Correlation in Experimental Mechanics; Mechanics of Biological Systems and Materials & Micro- and Nanomechanics; Mechanics of Composite, Hybrid and Multifunctional Materials; and Residual Stress, Thermomechanics & Infrared Imaging and Inverse Problems. 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 our efforts to bring together researchers interested in the dynamic behavior of materials and structures and to provide a forum to facilitate technical interaction and exchange. Over the years, this track has been representing the ever-growing interests in dynamic behavior to the SEM community, working toward expanding synergy with other tracks and topics, and improving diversity and inclusivity, as evidenced by the increasing number and diversity of papers and attendance. The contributed papers span numerous technical divisions within SEM, demonstrating its relevance not only in the dynamic behavior of materials community but also in the mechanics of materials community as a whole. The track organizers thank the authors, presenters, organizers, and session chairs for their participation, support, and contribution to this track. The SEM support staff is also acknowledged for their devoted efforts in accommodating the large number of paper submissions this year, making the 2019 Dynamic Behavior of Materials track a success. Philadelphia, PA, USA Leslie E. Lamberson Gaithersburg, MD, USA Steven Mates San Diego, CA, USA Veronica Eliasson v

Contents 1 Dynamic Tensile Behavior of Soft Ferromagnetic Alloy Fe-Co-2V................................................ 1 Brett Sanborn, Bo Song, Don Susan, Kyle Johnson, Jeff Dabling, Jay Carroll, Adam Brink, Scott Grutzik, and Andrew Kustas 2 Toward Paradoxical Inconsistency in Electrostatics of Metallic Conductors..................................... 5 Michael Grinfeld, Pavel Grinfeld, and Steven B. Segletes 3 Ballistic Response of Woven Kevlar Fabric as a Function of Projectile Sharpness .............................. 13 Julia Cline, Paul Moy, Doug Harris, Jian Yu, and Eric Wetzel 4 Effect of Thermomechanical Couplings on Viscoelastic Behaviour of Polystyrene .............................. 17 Pankaj Yadav, André Chrysochoos, Olivier Arnould, and Sandrine Bardet 5 Dynamic Response of Layered Functionally Graded Polyurethane Foam with Nonlinear Density Variation ........................................................................................................ 25 Dennis Miller, Vijendra Gupta, and Addis Kidane 6 Numerical and Experimental Investigation of Density Graded Foams Subjected to Impact Loading......... 31 Vijendra Gupta, Dennis Miller, and Addis Kidane 7 Method for Characterizing Electric Current Effects on the Deformation of Metals............................. 37 Christopher Rudolf, Wonmo Kang, and James Thomas 8 Mechanical Properties of Transparent Laminates Fabricated Using Multi-Material Photopolymer Jetting ................................................................................................... 47 Michael Harr, Paul Moy, and Jian Yu 9 Structural Intensity Assessment on Shells via the Projection of Experimental Data on a Finite-Element Mesh....................................................................................................................... 53 F. Pires, S. Avril, S. Vanlanduit, and J. Dirckx 10 Dynamic Compressive Response of Carbon Fibre Laminar Composite and Carbon Fibre Corrugated Sandwich Panel .......................................................................................................... 59 W. X. Huang and L. Tsai 11 Strain Rate Dependence of Stabilized, Nanocrystalline Cu Alloy.................................................. 63 S. A. Turnage, M. Rajagopalan, K. A. Darling, C. Kale, B. C. Hornbuckle, C. L. Williams, and K. N. Solanki 12 Designing Future Materials with Desired Properties Using Numerical Analysis................................. 69 Constantine (Costas) G. Fountzoulas and Jian H. Yu 13 Kolsky Bar Testing of Pressure Sensitive Adhesives ................................................................. 73 Evan L. Breedlove, David Lindeman, and Chaodi Li 14 Full-Field Mechanical and Thermal Strain-Rate Dependence of CFRP Laminates ............................ 85 Brian Smith, Amos Gilat, and Jeremy Seidt vii

viii Contents 15 Enhanced Energy Absorption Performance of Liquid Nanofoam-Filled Thin-Walled Tubes under Dynamic Impact ......................................................................................................... 89 Mingzhe Li, Saeed Barbat, Ridha Baccouche, Jamel Belwafa, and Weiyi Lu 16 Effect of Heat-Treatment on Rock Fragmentation Using Dynamic Ball Compression Test .................... 95 Ying Xu, Wei Yao, and Kaiwen Xia 17 Effect of Confining Pressure on the Dynamic Mode II Fracture Toughness of Rocks ........................... 99 Wei Yao, Tony Zhang, and Kaiwen Xia 18 A Viscoelastic-Viscoplastic Characterization with Time Temperature Superposition for Polymer Under Large Strain Rates .............................................................................................. 103 V. Dorleans, F. Lauro, R. Delille, D. Notta-Cuvier, and E. Michau 19 Tensile Hopkinson Bar Analysis of Additively Manufactured Maraging Steel ................................... 111 Nicholas E. Taylor, David M. Williamson, Christopher H. Braithwaite, and Sarah J. Ward 20 Large-Diameter Triaxial Kolsky Bar for Evaluating Very-High-Strength Concrete ............................ 115 Brett Williams, William Heard, Bradley Martin, Colin Loeffler, and Xu Nie 21 Dynamic Compressive Tests of Alumina Dumbbells Using a Spherical Joint .................................... 119 Steven Mates, Richard Rhorer, and George Quinn 22 Experimental Method for Mode I Dynamic Fracture Toughness of Composite Laminates Using Double Cantilever Beam Specimens................................................................................... 127 G. Portemont, T. Fourest, and R. De Coninck 23 Numerical Study of Ring Fragmentation ............................................................................. 131 Brady Aydelotte 24 Development of a New Testing Method to Capture Progressive Damage in Carbon Fiber Reinforced Polymers Subject to a Simulated Lightning Strike .................................................................. 137 Brandon Hearley, Kara Peters, and Mark Pankow 25 Overview of the First SHPB Experiments on Oriented Single Crystal Explosives............................... 143 Christopher Meredith, Daniel Casem, Cheng Liu, Benjamin Morrow, Carl Cady, and Kyle Ramos 26 Hydrodynamic Richtmyer-Meshkov Instability of Metallic Solids Used to Assess Material Deformation at High Strain-Rates..................................................................................... 149 Joseph D. Olles, Matthew Hudspeth, Christopher F. Tilger, Christopher Garasi, Nathaniel Sanchez, and Brian Jensen 27 Combined Compression and Shear Impact Response of Polycrystalline Metals at Elevated Temperatures............................................................................................... 157 Bryan Zuanetti, Tianxue Wang, and Vikas Prakash 28 Dynamic Failure of Pure Tungsten Carbide Under Simultaneous Compression and Shear Plate Impact Loading.......................................................................................................... 163 Bryan Zuanetti, Tianxue Wang, and Vikas Prakash 29 A Kolsky Bar with a 50 ns Rise-Time: Application to Rates Beyond 1 M/s ...................................... 171 Daniel T. Casem 30 Strain Stiffening Effects of Soft Viscoelastic Materials in Inertial Microcavitation ............................. 175 Jin Yang and Christian Franck 31 Assessment of Dynamic Fracture in Ultra-High Performance Concrete Using Synchrotron X-ray Source... 181 Nesredin Kedir, Shane Paulson, Cody Kirk, Tao Sun, Kamel Fezzaa, and Wayne Chen 32 High Rate Mechanical Characterization of Sensitized 5083-H131 Aluminum Alloy ............................ 185 Timothy Walter, Heather Murdoch, Paul Moy, Denise Yin, and Julia Cline 33 Application of High-Speed Digital Image Correlation to Taylor Impact Testing................................. 189 Phillip Jannotti, Nicholas Lorenzo, and Chris Meredith

Contents ix 34 Observation of Dynamic Adhesive Behavior Using High-Speed Phase Contrast Imaging...................... 197 Shane Paulson, Nesredin Kedir, Tao Sun, Kamel Fezzaa, and Wayne Chen 35 Strain-Rate Effect on the Deformation Mechanisms of Agglomerated Cork..................................... 201 Louise Le Barbenchon, Jean-Benoît Kopp, Jérémie Girardot, and Philippe Viot 36 Dynamic Mechanical Behavior of Reinforced Cork Agglomerate ................................................. 209 Louise Le Barbenchon, Jean-Benoît Kopp, Jérémie Girardot, and Philippe Viot 37 Use of Edge-on Impact Tests with Synchrotron-Based MHz Radioscopy to Investigate the Multiple Fragmentation Process in SiC Ceramics.............................................................................. 215 Pascal Forquin, Bratislav Lukic, Yannick Duplan, Dominique Saletti, Daniel Eakins, and Alexander Rack 38 Low Temperature Seawater Effects on the Mechanical, Fracture, and Dynamic Behavior of E-Glass and Carbon Fiber Laminates .......................................................................................... 219 James LeBlanc, Paul Cavallaro, Jahn Torres, Eric Warner, Andrew Hulton, Ryan Saenger, and David Ponte 39 Using the SURF Model to Simulate Fragment Impact on Energetic Materials .................................. 223 Xia Ma and Brad Clements

Chapter 1 Dynamic Tensile Behavior of Soft Ferromagnetic Alloy Fe-Co-2V Brett Sanborn, Bo Song, Don Susan, Kyle Johnson, Jeff Dabling, Jay Carroll, Adam Brink, Scott Grutzik, and Andrew Kustas Abstract Fe-Co-2V is a soft ferromagnetic alloy used in electromagnetic applications due to excellent magnetic properties. However, the discontinuous yielding (Luders bands), grain-size-dependent properties (Hall-Petch behavior), and the degree of order/disorder in the Fe-Co-2V alloy makes it difficult to predict the mechanical performance, particularly in abnormal environments such as elevated strain rates and high/low temperatures. Thus, experimental characterization of the high strain rate properties of the Fe-Co-2V alloy is desired, which are used for material model development in numerical simulations. In this study, the high rate tensile response of Fe-Co-2V is investigated with a pulse-shaped Kolsky tension bar over a wide range of strain rates and temperatures. Effects of temperature and strain rate on yield stress, ultimate stress, and ductility are discussed. Keywords Kolsky tension bar · Fe-Co-2V alloy · Temperature effects · Magnetic · Material properties Introduction Soft ferromagnetic alloys are used in electromagnetic applications where good magnetic properties are needed. Fe-Co-2V is an example of a soft ferromagnetic alloy used in magnetic bearings and electrical generators. While Fe-Co-2V was chosen for these applications due to its desirable magnetic properties, the material may undergo mechanical loading at different rates and temperatures. Quasi-static tension experiments have been conducted on Fe-Co-2V up to 800 ◦C [1]. The behavior was elastic, followed by a plastic plateau (Lüders banding) before significantly hardening. In general, the yield strength increased with increasing strain rate. When the temperature was below 300 ◦C, the yield strength decreased with increasing temperature. The yield strength of the material as highly dependent on grain size, which followed a Hall-Petch relationship. Despite being well-characterized quasi-statically at elevated temperatures, Fe-Co-2V used in applications may be subjected to impact loading at elevated or even low temperatures. Hence, the high rate tensile response over a range of temperatures is needed to improve material models used in numerical simulations for improved design. In this study, the dynamic properties of Fe-Co-2V were investigated over a wide range of strain rates and temperatures. Materials and Experiments In this study, Fe-Co-2V alloy was characterized over strain rates from 40 to 230 s−1 using two different experimental setups. Intermediate rate experiments (40–110 s−1) were conducted on a drop-Hopkinson bar setup wherein a drop table is used to provide a long loading pulse at a speed lower than what is usually conducted with a Kolsky/split Hopkinson bar. Experiments at 230 s−1 were conducted with a Kolsky tension bar. The Kolsky tension bar had a thermal chamber installed at the testing section which subjected the material to a temperature range of −100 to 100 ◦C. A recently developed specimen-strain correction method [2] was applied to the dynamic tensile tests of the Fe-Co-2V alloy such that the specimen stress-strain response over the gage section was obtained. B. Sanborn ( ) · B. Song · D. Susan · K. Johnson · J. Dabling · J. Carroll · A. Brink · S. Grutzik · A. Kustas Sandia National Laboratories, Albuquerque, NM, USA e-mail: bsanbor@sandia.gov © Society for Experimental Mechanics, Inc. 2020 L. E. Lamberson (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-30021-0_1 1

2 B. Sanborn et al. Strain Rate and Temperature Effects Stress-strain curves for the Fe-Co-2V alloy at 40, 110, and 230 s−1 at ambient temperature are shown in Fig. 1.1. A quasi-static stress-strain curve is also shown for comparison. As shown in Fig. 1.1, the upper yield strength increased with increasing strain rate. At low strain rates, the stress-strain curves showed a typical Lüders banding response prior to hardening. However, at high strain rates, i.e., 230 s−1, the material exhibited a significant upper and lower yield response. The effect of temperature on material response at a strain rate of 230 s−1 is shown in Fig. 1.2. While the overall shape of the stress-strain curve is similar at different temperatures, the yield and flow stresses increased with decreasing temperature. Fig. 1.1 Stress-strain behavior of Fe-Co-2V at various strain rates at 20 ◦C Fig. 1.2 Tensile stress-strain curves at various temperatures at a strain rate of 230 s−1

1 Dynamic Tensile Behavior of Soft Ferromagnetic Alloy Fe-Co-2V 3 The upper yield stress increased about 18% at −100◦C compared to ambient temperature. No necking was observed for any of the failed specimens at any of the temperatures. Conclusion The strain rate- and temperature-dependent tensile properties of Fe-Co-2V alloy were measured using a Kolsky bar and drop-Hopkinson bar. The material displayed an elastic, followed by either upper/lower yield behavior at high strain rates or a Lüders banding behavior at lower rates, prior to linearly hardening until failure. The specimens failed in a brittle manner. The yield stress increased with increasing strain rate, while the hardening rate was constant. The yield strength of the material increased with decreasing temperature. The hardening rate was independent of temperature. The material behavior measured in this study can be used for material model development. Acknowledgements 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 the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. References 1. Ren, L., Basu, S., Yu, R.-H., Xiao, J.Q., Parvizi-Majidi, A.: Mechanical properties of Fe-Co soft magnets. J. Mater. Sci. 36, 1451–1457 (2001) 2. Song, B., Sanborn, B., Susan, D., Johnson, K., Dabling, J., Carroll, J., Brink, A., Grutzik, S., Kustas, A.: Correction of specimen strain measurement in Kolsky tension bar experiments on linear work-hardening materials, 2019 SEM Annual Conference on Experimental and Applied Mechanics, June 3–6, 2019, Reno, NV (2019)

Chapter 2 Toward Paradoxical Inconsistency in Electrostatics of Metallic Conductors Michael Grinfeld, Pavel Grinfeld, and Steven B. Segletes Abstract In a recent report, we drew attention to the paradoxical inconsistency of classical electrostatics with the model of a crystalline conductor. There are different ways to avoid this inconsistency. Mostly, they rely on physical adhoc assumptions associated with the introduction of additional constants and models of charged liquids. In this report, we suggest another approach, which does not require the introduction of any additional physical constants. The proposed theory includes classical electrostatics as a special case. Keywords Electrostatics · Laplace and Poisson equations · Simple layers · Thermodynamics Introduction Electromagnetism is in the background of various military applications, including the problem of protection against shaped charges, which Russian adversaries call “cumulative” jets [1]. One of the many problems faced in the computer implementation of physical models is associated with the fact that certain physical fields experience discontinuities across boundaries and interfaces. Some fields (the models thereof) even become infinitely large when approaching such interfaces. Theorists have accumulated various sophisticated mathematical tools for handling these problems when treating the problems analytically. The problem of infinite charge density in the vicinity of boundaries in metals is further aggravated when using numerical modeling, since the singularities in the underlining physical theories are in violent discord with the inability of finite discretizations to handle sharp gradients. One of the methods for handling this violent discord is based on the replacement of the classical physical theories with modified ones that allow for the avoidance of infinite charge density at the boundaries. We suggested one approach of this sort in a prior report [2] and continue developing the approach in this one. Let us recall the paradoxical inconsistency of classical electrostatics first noted in our prior report [2]. According to classical electrostatics, all excess electric charges, positive or negative, concentrate on the conductor’s boundary with a finite 2-D density. In mathematical physics, these sort of boundaries are known as “simple layers” of charges [3, 4]. This surface density can be either positive or negative, depending on the total excess charge of the conductor. What is the physical meaning of this finite 2-D density of electric charge? Actually, this concept implies that the associated 3-D density of the electric charge is infinite. Of course, this infinity, on its own, is an essential inconsistency of classical electrostatics. However, in many cases, this particular limitation is not very important and classical electrostatics provides researchers with reasonable results [3, 4]. It is not, however, this inconsistency that we discussed in our prior work [2]. The paradoxical inconsistency that we introduced [2] is different. In fact, the density of the easily movable negative charges can grow and reach quite large values (for example, on the surface of a body), in accordance with classical electrostatics. If the positive charges are easily movable and compressible also, like in the case of ionized gas, the same can be said about the gas of positive ions. In this sense, classical electrostatics can be applied to a gaseous plasma. The situation, however, changes dramatically when the positive charges belong to a (near rigid) lattice. In this case, the maximum positive charge density is basically fixed, and we cannot expect that it might assume infinite values, even in an idealized sense. In other words, there cannot be surfaces (or interfaces) with finite positive charges—excess positive charge can only M. Grinfeld ( ) · S. B. Segletes Weapons and Materials Research Directorate, Army Research Laboratory, Aberdeen Proving Ground, MD, USA P. Grinfeld Drexel University, Philadelphia, PA, USA © Society for Experimental Mechanics, Inc. 2020 L. E. Lamberson (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-30021-0_2 5

6 M. Grinfeld et al. manifest in regions (i.e., domains). Thus, we deal in this case with the asymmetry of how to treat the situations of negative versus positive excess charge. This very asymmetry dictates the necessity to revise classical electrostatics. To address this inconsistency, we have to reformulate classical electrostatics in such a way that nowhere does the positive 3-D charge density exceeding the value q+ appear. A Simple Consistent Model of Electrostatics to Avoid the Paradoxical Inconsistency We propose the following model of a crystalline rigid conductor. The model is shown schematically in Fig. 2.1.1 The key suggestion is the following: the total domain of the conductor is split into two subdomains, n and a. Generally speaking, both subdomains may comprise several separate regions. In Fig. 2.1, the subdomains of n are shown in gray, whereas the subdomains of a are shown in red. We assume that, in n, the macroscopic density of both positive and negative charges (q+ and q−, respectively) are equal to each other. Thus, the net macroscopic charge densityq vanishes in n: q =q++q− =0 within n. (2.1) Within the subdomain a, our assumption is different. We just assume that the density of negative charges q− vanishes; in other words, we arrive at the following relationships: q− =0, q =q+ within a. (2.2) Thus, by the construction, the positive charge density exceeds the value q+ nowhere. Also, from Fig. 2.1, we point out that, between the two domains and between each domain and the vacuum surround, we postulate zero-thickness interface layers. These interfaces are denoted , Sa, and Sn and are further described and analyzed later in this report. Let us formulate the system of equations in the mathematical form of a boundary value problem. In view of Eqs. 2.1 and 2.2, the electrostatics bulk equations inside the domains n and a read ∇i∇iϕ =0 (2.3) and ∇i∇iϕ =−4πq+ , (2.4) respectively. Outside the external boundary S (the union of Sn and Sa), we also have to use the Poisson equation ∇i∇iϕ =−4πqext , (2.5) where qext is a given distribution of the external sources of the electrostatic field. Within the domain n (with the mobile negative charges potentially on the adjoining interface), the electrostatic potential is assumed constant everywhere: ϕ =ϕ0 =const. inside n , (2.6) as always in classical electrostatics. Equation 2.6 automatically satisfies the electrostatics Eq. 2.3; thus, there is no need to deal with the Laplace equation (Eq. 2.3) inside the domain n. 1In this and the figures to follow (though its justification remains to be assumed or derived later in this report), one may follow the heuristic ruleof-thumb that dark- and light-gray denote electrically neutral domains and interfaces, respectively, red denotes a domain void of negative charges, while blue denotes an interface comprised solely of mobile negative charges and characterized by a surface charge density.

2 Toward Paradoxical Inconsistency in Electrostatics of Metallic Conductors 7 Fig. 2.1 The cross-sectional (cutaway) geometry of a conductor Sn Ωn Ωa Sa ∑ Ωa Ωa Ωa Let us now discuss the boundary conditions for our bulk equations of electrostatics. Per Fig. 2.1, a priori, there are three sorts of zero-thickness interfaces. First, there are interfaces Sn between the domain n and vacuum. Secondly, there are interfaces Sa between the domain a and vacuum. Lastly, there are interfaces between the domains n and a. For all three types of interfaces, the boundary conditions are different: 1. We begin with the Sn interfaces. Those are the traditional interfaces of classical electrostatics, possessing the boundary conditions ϕ + − = 0 Ni∇iϕ Sn =−4πσSn (2.7) where Ni is the interface’s outward normal andσSn is the 2-D density of the surface charges. The former of the boundary conditions given as Eq. 2.7 reflects the continuity of the electrostatic potential; the latter is the consequence of the Gauss law applied to the interface with the finite 2-D density of electric charges (the so-called “simple layer of charges”). It is essential that this density is negative since only negative charges are mobile and are able to generate the unlimited 3-D density of the electric charge (as the boundary layer near Sn approaches zero thickness). 2. We proceed with the Sa interfaces. Since, by the postulated models, there are no surface charges at these interfaces, we suggest using the standard boundary conditions of continuity of electrostatic potential and its first derivatives: ϕ + − = 0 ∇iϕ + − Ni =0 . (2.8) 3. Finally, we consider the -interfaces, separating the n and a domains. Across those interfaces, we are still using the electrostatics boundary conditions ϕ + − = 0 ∇iϕ + − Ni =−4πσan (2.9) Thus, a priori, we assume that there can be 2-D accumulations of the negative charges. This is a quite plausible assumption for the -interfaces since there is a source of negative mobile charges from the domain n. There is, however, a significant difference between the Sn and Sa interfaces, on the one hand, and the interfaces, on the other hand. The difference is the following: the location of Sn and Sa are known up-front, from the conductor geometry, whereas the location of the interfaces are not. In order to determine the location of the interfaces, we need one more equation. This additional equation can be chosen based on various principles. We postulate this additional condition in the simplest form, rejecting the possibility of charge accumulation on interfaces: σan =0 . (2.10) Finally, we need a charge balance equation for the mobile negative charges. Let their total charge be equal to Q−. These charges are located within the domain n with the volumetric density q− = q+ and at the interface Sn with the surface

8 M. Grinfeld et al. density σSn. Thus, we arrive at the charge balance relationship: n d q−+ Sn dSσSn =Q− . (2.11) As a side note, the assumption given by Eq. 2.10 can be substantiated on the basis of a rigorous mathematical analysis if we accept the principle of minimum electrostatic energy [5, 6]. 1-D Solutions for a Flat Layer Consider a flat 1-D system, which carries the total number of Qmobile charges per unit cross section. We explore the distributions of charges that are symmetric with respect to the plane z =0. Since both Sa and interfaces accumulate zero charge and possess zero thickness, their visible presence is omitted in the figures that follow. Configuration 1 If the net bounded charge Q+, which includes the positive charges of the lattice ions and negative charge of the bounded negative charges, is less than |Q−|, we get the standard configuration of classical electrostatics, shown in Fig. 2.2. According to our terminology, there is no a domain, which is free of the mobile negative charges. The whole domain between the boundaries appears to be the neutral domain n, having the Sn-type interfaces with the negative surface charge density equal to σSn = Q+−|Q−| 2 . (2.12) Configuration 2 Consider a flat 1-D system presented in Fig. 2.3. Within the gray n domain, the 1-D Laplace equation reads d 2 ϕ dz2 = 0 for |z| ≤H . (2.13) Within the red a domain, the 1-D Laplace equation reads d 2 ϕ dz2 =−4πq+ for H <|z| ≤Ha . (2.14) Fig. 2.2 The “standard” configuration of classical electrostatics Ωn Sn z Hs Sn

2 Toward Paradoxical Inconsistency in Electrostatics of Metallic Conductors 9 Fig. 2.3 A configuration with finite a domains (red) at the external boundaries Ωn z H∑ Ωa Ha Ωa Outside the plate, in the absence of the outside charges, we arrive at the Laplace equation again: d 2 ϕ dz2 = 0 for |z| >Ha . (2.15) Boundary conditions at both interfaces H and Ha are the same; namely, ϕ + − = 0 dϕ dz + − =0 . (2.16) At z =0, we choose the following boundary conditions: ϕ(0) =0 dϕ dz 0 =0 . (2.17) The charge balance equation (Eq. 2.11) is 2H q− =Q− . (2.18) In view of the bulk Laplace equation (Eq. 2.13) and the boundary conditions (Eq. 2.17), the electrostatic potential ϕ vanishes everywhere inside the gray n domain: ϕ(z) =0 for |z| ≤H . (2.19) Now, combining Eqs. 2.1 and 2.18, we arrive at the following relationship for the thickness of the neutral domain: H = 1 2q+ |Q−| . (2.20) The general solution of the Poisson equation (Eq. 2.14) inside the red a domain reads ϕ(z) =−2πq+z 2 +C1z +C2 for H <|z| ≤Ha , (2.21) whereC1 andC2 are the constants that should be determined from the boundary conditions (Eq. 2.16) and solution (Eq. 2.19) within the gray n domain, which gives us two linear equations −2πq+H 2 +C1H +C2 =0 −4πq+H +C1 =0 (2.22)

10 M. Grinfeld et al. with the solution C1 =4πq+H C2 =−2πq+H 2 . (2.23) Using Eq. 2.23, we can rewrite the general solution Eq. 2.21 as follows ϕ(z) =−2πq+ z −H 2 for H <|z| ≤Ha . (2.24) The solution given by Eq. 2.24 implies ϕ(Ha) =−2πq+ Ha −H 2 dϕ dz Ha =−4πq+ Ha −H . (2.25) The general solution of the Laplace equation (Eq. 2.15) within the vacuum domain has the following general solution ϕ(z) =ϕ z=Ha + z −Ha dϕ dz z=Ha for |z| ≥Ha . (2.26) Using Eq. 2.25, we can rewrite the general solution (Eq. 2.26) as follows: ϕ(z) =−2πq+ Ha −H 2 − 4πq+ z −Ha Ha −H for |z| ≥Ha . (2.27) Configuration 3 Consider now the same flat 1-D system but with a different arrangement of the domains, presented in Fig. 2.4. For this configuration, we arrive at the system d 2 ϕ dz2 =−4πq+ for |z| ≤H , (2.28) ϕ =const for H <|z| ≤HS , (2.29) d 2 ϕ dz2 = 0 for |z| >HS . (2.30) Fig. 2.4 A configuration with a finite a domain (red) in the middle of the body Ωn z H∑ Ωn Hs Ωa Sn Sn

2 Toward Paradoxical Inconsistency in Electrostatics of Metallic Conductors 11 The system of bulk equations, Eqs. 2.28–2.30, should be considered with the following boundary conditions: 1. at z =0, we choose the following boundary conditions: ϕ(0) =0 dϕ dz 0 =0 (2.31) 2. at z =H , we use the following boundary conditions; namely, continuity of ϕ and its derivative across the interface: ϕ + − = 0 dϕ dz + − =0 (2.32) 3. at z =HS, the surface charges can accumulate and we choose the boundary conditions ϕ + − = 0 Ni∇iφ Sn =−4πσSn (2.33) The charge balance equation (Eq. 2.11) reads (HS −H )q−+σSn = Q− 2 . (2.34) We then obtain ϕ =−2πq+z 2 for | z| ≤H , (2.35) ϕ =−2πq+H 2 for H <|z| ≤HS . (2.36) According to the solutions given by Eq. 2.36, ϕ is constant in the domain n, and so dϕ/dz =0 when approaching the interface at z =H from above. In contrast, the parabolic form of Eq. 2.35 for the central a domain would lead us to the conclusion that dϕ/dz =−4πq+H when approaching the interface at z =H from below. These two violate the interface boundary condition for dϕ/dz at z =H , given by Eq. 2.32, which calls for continuity of the derivative of ϕ across the interface. Therefore, one must conclude that the hypothetical configuration 3 represents a nonphysical configuration of the electric charge distribution. Conclusion In an earlier report [2], we demonstrated the paradoxical inconsistency of classical electrostatics for the models permitting only finite density of positive charges. To handle this paradox, classical electrostatics should be significantly modified. There are several possibilities of reasonable modifications. Basically, they lead to the introduction of additional physical mechanisms and additional material constants. Introduction of additional material constants and effects is the precursor of significant changes in the technical complexity of the associated boundary value problems. The growth of the complexity can entail the disappearance of the keynote feature of classical electrostatics: this feature is the possibility of analytical exploration of the interesting physical problems. To minimize the complexification of classical electrostatics, we suggested another approach, which is not associated with the introduction of any additional material constants. After formulation of the novel boundary value problem, we demonstrated how it can be solved analytically in the simplest instructive 1-D cases of the flat plate.

12 M. Grinfeld et al. References 1. Fedorov, S.V.: Electrodynamic protection against shaped charge weapons: physics aspects of functioning. Vestnik of the Baumann’s Technical University: Mashinostroenie (3) (2014) 2. Grinfeld, M., Segletes, S.B.: Toward paradoxical inconsistency in electrostatics of metallic conductors. Technical Report ARL-TR-8365, Army Research Laboratory (US), Aberdeen Proving Ground (2018) 3. Stratton, J.A.: Electromagnetic Theory. McGraw Hill, New York (2008). Originally published 1941 4. Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media. Pergamon, Oxford (1960) 5. Grinfeld, P.A., Grinfeld, M.A.: Towards thermodynamics of elastic electric conductors. Phil Mag. A81(6), 1341–1354 (2001) 6. Grinfeld, P.: Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer, New York (2013)

Chapter 3 Ballistic Response of Woven Kevlar Fabric as a Function of Projectile Sharpness Julia Cline, Paul Moy, Doug Harris, Jian Yu, and Eric Wetzel Abstract Right circular cylinders (RCC) are a common fragment-simulating projectile used to simulate debris resulting from an improvised explosive device (IED) blast. The specifications for ballistic performance of soft body armor specify the geometry and mass of RCC projectiles used for testing. The edges on the impacting side of an RCC projectile have a fillet radius, and recent investigations indicate that the fillet radius (or “sharpness”) of the projectile may affect the ballistic limit velocity, V50. It is hypothesized that this happens because the sharpness of the projectile changes the fiber failure mechanism, so to test this, 4 gr RCC projectiles with varying fillet radii (0.64 mm, 1.27 mm and 1.70 mm) are precisely manufactured. Previously collected, unpublished data for 4 gr RCC projectiles with fillet radii of 0.10 mm, 0.18 mm and 0.25 mm are also included to assess a range of projectile sharpness values. Ballistic impact tests on single layer woven Kevlar K706 fabric are conducted using a laboratory gas gun to measure the limit velocity, and it is found that the limit velocity decreases with increasing fillet radii. Back surface images and impacted targets are examined post-mortem to identify failure mechanisms. Keywords Limit velocity · Kevlar · RCC · Ballistic response Introduction RCCs are non-deformable projectiles commonly used to simulate airborne fragments accelerated by an IED blast that pose a risk to soft body armor. Requirements for fragmentation protection against several masses of RCC projectiles are outlined in the Enhanced Combat Helmet (ECH) purchase description [1]. For 4 gr (259 mg) RCCs, the mass is specified within a tolerance of 0.15 gr (10 mg), an outer diameter of 3.40 ±0.03 mm and a length of 3.73 mm. The corners of each RCC are fillet machined with a radius of 0.18 ±0.08mm. The limit velocity (V50) is defined as the velocity at which 50% of impacting projectiles will penetrate a given target. A previous, unpublished study of the limit velocity for RCC projectiles with precisely machined fillet radii of 0.10 mm, 0.18 mm and 0.25 mm indicates that the fillet radii may have an effect on the limit velocity. The cause of this has not been studied extensively and merits further investigation. It is hypothesized that the “sharpness” of the projectile will govern the mechanism needed to penetrate the fabric, which will affect the limit velocity. For computational models aimed at predicting V0–V100 probability curves, realistic failure mode information is critical to obtaining accurate predictions. This work investigates sensitivity to the fillet radii within the current specified tolerance and characterizes the trends related to limit velocity and fillet radii in order to better inform fiber/yarn-level computational models of ballistic impact. Background For woven fabrics, there are several mechanisms by which a projectile can penetrate [2, 3]. If the yarns are stretched beyond the maximum fiber failure strain, they will fail in tension. Transverse shear failure of the yarns can occur when a blunt nose projectile (RCC) cuts or shears through the fabric yarns. For loosely woven fabrics or projectiles with smooth curvature (ex. hemisphere or spherical shape), it is possible for the projectile to push the yarns aside and “window” through the fabric without cutting or failing the yarns. J. Cline ( ) · P. Moy · D. Harris · J. Yu · E. Wetzel CCDC Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, USA e-mail: julia.e.cline@nasa.gov © Society for Experimental Mechanics, Inc. 2020 L. E. Lamberson (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-30021-0_3 13

14 J. Cline et al. Fig. 3.1 The dimensions and fillet radius of the 4 gr RCC projectiles tested in this study. Projectile length is adjusted for the rounder nosed projectiles to maintain projectile mass Fig. 3.2 Experimental setup for ballistic impact testing. Two cameras are set-up to record velocity and yaw data as well as images of the back surface of the target during impact. Cameras are triggered using a sound trigger. Ballistic gel is placed behind the target to catch penetrating projectiles. Inset is a schematic of the circular mount for Kevlar targets To evaluate the effect of projectile sharpness (fillet radii) on the limit velocity, we choose a single layer, scoured state Kevlar K706 fabric consisting of 60 denier KM2 Kevlar yarns woven at 13 ×13 yarns per centimeter. Targets are cut from fabric into 10 cm squares and mounted on a circular frame that is secured using a hose clamp and torqued to 14 N· m. To observe any pull-in from the frame edge, a thin, black sharpie line is drawn around the outer edge of the frame on the target to denote the original location. 4 gr RCC projectiles are precisely machined with fillet radii of progressively decreasing sharpness as shown in Fig. 3.1. Mass and fillet radius of a sample set of projectiles are verified prior to testing. A laboratory gas gun is used to accelerate projectiles toward the targets. A 3.40 mm diameter barrel is installed on the laboratory gas gun with the end of the barrel set at 7.11 mm from the target. A 10 cm slot at the end of the barrel allows for high speed imaging of the projectile before it leaves the barrel and impacts the target, enabling both projectile velocity and yaw angle measurements. Figure 3.2 shows a full schematic of the experimental setup. A second high speed camera is placed behind the target to capture images of the impact and penetration.

3 Ballistic Response of Woven Kevlar Fabric as a Function of Projectile Sharpness 15 SenTest [4] software is used to guide target velocity choices. SenTest is based on the Neyer D-Optimal test method and is a generic software designed for conducting and analyzing sensitivity tests. For our use, we specify the upper and lower bounds of the velocity measurements and SenTest selects target velocity values to test based on whether the previous velocity test was a success (complete penetration) or a failure (partial penetration). The limit velocity bound we use is 50m/s to 200 m/s with a standard deviation of 10 m/s. Analysis Approximately 40 shots are taken per projectile geometry and any shots with significant yaw prior to impact are disregarded in the limit velocity calculation. SenTest software is used to calculate the limit velocity and standard deviations for each projectile type. Figure 3.3 is a plot of the limit velocity for each RCC fillet radius tested in this study. It shows a clear linear, decreasing relationship indicating that penetration is more likely for the rounder nose projectiles at lower velocities. Further analysis of the back surface images of the impacts and examination of the impacted targets allows for understanding of the failure mechanisms occurring during penetration. For the hemispherical nosed projectiles, penetration is commonly achieved by windowing or sliding through gaps in the weave (Fig. 3.4a) or by a yarn spreading around the projectile nose (Fig. 3.4b). As the fillet radius is decreased, the edges of the projectile become sharper and more broken and cut fibers/yarns are observed which indicates that the failure mechanism is transverse shearing (Fig. 3.4c). Fig. 3.3 Limit Velocity (V50) decreases as the RCC projectile fillet radius increases Fig. 3.4 Penetration occurs by (a) a hemispherical projectile windowing through the Kevlar weave, (b) a yarn spreading and sliding out of the way of a hemispherical projectile, and (c) the cutting of some fibers/yarns around the nose of a sharp projectile

16 J. Cline et al. Conclusion A study of the effect of varying fillet radius of RCC projectiles on limit velocity of Kevlar K706 woven fabric shows that protective capabilities reduce as projectiles become more rounded and smooth. Round-nosed (hemispherical) projectiles can easily find gaps in the weave and slide through to defeat the armor material whereas sharper filleted projectiles are forced to expend more energy to cut through yarns/fibers to achieve penetration. To extend this study, newer materials such as ultra-high molecular weight polyethylene (UHMWPE) fiber cross-ply composite could be evaluated in the same manner. The matrix should prevent the projectile from sliding through the fibers and could allow study of other relevant failure mechanisms. 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. Department of the Army: “Purchase Description Enhanced Combat Helmet (ECH),” PD-ECH-ICE-PG16-0001. Project Manager – Infantry Combat Equipment, Quantico, VA (2009) 2. Nilakantan, G., Merrill, R., Keefe, M., Gillespie, J., Wetzel, E.: Experimental investigation of the role of frictional yarn pull-out and windowing on the probabilistic impact response of Kevlar fabrics. Compos. Part B. 68, 215–229 (2015) 3. Nilakantan, G., Wetzel, E., Bogetti, T., Gillespie, J.: A deterministic finite element analysis of the effects of projectile characteristics on the impact response of fully clamped flexible woven fabrics. Compos. Struct. 95, 191–201 (2013) 4. SenTest. Neyer Software LLC. http://neyersoftware.com/SensitivityTest/SensitivityTestFlyer.htm. Accessed Feb 2019

Chapter 4 Effect of Thermomechanical Couplings on Viscoelastic Behaviour of Polystyrene Pankaj Yadav, André Chrysochoos, Olivier Arnould, and Sandrine Bardet Abstract Analysis of the thermo-mechanical behaviour of the polymers has been and still is the subject of many rheological studies both experimentally and theoretically. For small deformations, the modelling framework retained by rheologists is often of linear viscoelasticity which led us to the definition of complex moduli and to the rules of the renowned timetemperature superposition principle (TTSP). In this context, the effect of time (i.e., rate dependence) is almost unanimously associated with viscous effects. It has however been observed that the dissipative effects associated with viscous effects may be superimposed with thermo-elastic coupling effects, indicating a high sensitivity of polymeric materials to temperature variations (thermodilatability). Indeed, because of heat diffusion, it was also noticed that these strong thermo-mechanical couplings may induce a time dependence of the material behaviour. Using traditional experimental methods of viscoanalysis i.e., dynamic mechanical thermal analysis (DMTA) and via an experimental energy analysis of the behaviour using quantitative infrared techniques, the relative importance of thermoelastic heat sources compared to viscous dissipation was analysed with the increasing frequency of monochromatic cyclic tensile tests made at different ambient temperatures. Keywords Time-temperature superposition (TTSP) · Viscoelasticity · Dissipation · Thermo-mechanical coupling · DMTA Introduction Polymeric materials are widely known for their high viscoelasticity, which signifies wide dependence of their mechanical responses on the applied strain rate which is very important in duration for numerous engineering applications. This is one of the reasons why a protocol capable of predicting the viscoelastic behaviour of polymeric materials over time scales and temperatures of use has been developed. This protocol gave rise to DMTA techniques which allow to gather viscoelastic behaviour characteristics at easily adaptable frequencies and temperature of use in laboratories and extrapolate them to very large ranges of strain rate and temperatures. During the DMTA tests, a monochromatic sinusoidal signal is applied on the specimen to observe the stress-strain response and derive the so called dynamic moduli i.e., E and E respectively named as storage and loss moduli. In general, the storage modulus is associated with the stored elastic energy which is recoverable during unloading while the loss modulus is supposed to be associated with the viscous dissipated energy. The viscous part of the behaviour is equivalently characterised by the loss tangent defined by tan δ =E /E [1]. The DMTA constitutive equations can then be summarized as follows: ⎧ ⎪ ⎨ ⎪ ⎩ ε =ε0 sinωt σ =σ0 sin(ωt +δ) =E sinωt +E cosωt E =σ0/ε0 cosδ E =σ0/ε0 sinδ (4.1) where ε0 is the loading amplitude, σ0 the stress amplitude and ω=2πf, the pulsation and f being the loading frequency. Mechanical tests, performed at constant temperature and carried at several frequencies (also known as frequency sweep), permit the investigation of viscoelastic characteristics on a small time scale equivalent to few decades. According to the literature, the obtained data, extracted from tests carried out at different frequencies and constant temperature, show the P. Yadav · A. Chrysochoos ( ) · O. Arnould · S. Bardet Laboratory of Mechanics and Civil Engineering (LMGC), University of Montpellier, CNRS, Montpellier, France e-mail: andre.chrysochoos@umontpellier.fr © Society for Experimental Mechanics, Inc. 2020 L. E. Lamberson (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-30021-0_4 17

18 P. Yadav et al. same evolutions of E and E as those obtained from isochronal tests conducted at different temperatures [2]. This implies that the variation of temperature corresponds to a shift in time scale and this relation between time (or loading frequency) and temperature is termed as time-temperature superposition principle [3] and is known to be valid only for “thermorheologically simple” materials. These are the materials which show that the variation of temperature corresponds to a shift in time scale [4]. For example, considering a given relaxation time τ, its changes with the temperature then introduce a shift factor aT such that: τ(T) = τ (T0) aT(T) (4.2) According to Eq. 4.1, the linear viscoelastic behaviour of materials leads to stabilized hysteretic responses (stress-strain loops) whose area corresponds to the mechanical energy lost in form of the dissipated mechanical energy over a cycle, often leading to a self-heating of the specimen. The intensity of the self-heating naturally depends on the material characteristics, on the loading frequency and on the thermal boundary conditions. However, Zener [5] observed that the stress-strain loop may not only be induced by the dissipation mechanisms but also by thermoelastic effects. For all purposes, a theoretical thermodynamic description of these “thermoelastic damping” can be found in [6]. This present work analysed the experimental results obtained from DMTA device equipped with an Infrared camera. The Infrared data were used to estimate the temperature variations of the specimen during the cyclic loading in order to detect the possible thermo-mechanical coupling and/or dissipative effects. The outputs of the different tests are discussed in terms of energy balance. This paper also presents an analytical way to derive the concept of thermorheologically simple materials used in the DMTA protocol within a thermomechanical viscoelastic framework where the status of the temperature is the one of a controlled parameter but not (yet) those of a thermodynamic variable. Experimental Methods Firstly, the standard DMTA setup used for polystyrene samples was shown followed by the experimental home-made arrangement developed to allow us to perform a thermographic analysis during the DMTA tests. Standard DMTA Polystyrene (PS) sample with a weight averaged molecular weight (Mw) of 111,500 and a polydispersity of 3, was used to carry out the DMTA tests. The samples of dimension 85 ×13 ×4 mm were made from the sheets (300 ×300 mm) of PS from Goodfellow. The tests were made on a classical DMTA (BOSE ELF 3230) equipped with strain control module (Fig. 4.1). The samples were tested in the temperature range from 40 ◦C to 90 ◦C and for 3 decades of frequency from 0.01 Hz to 10 Hz i.e., frequency sweep was made with 5 points per decade in the tension mode with a strain ratio of R=−1. A virgin sample was utilized for each new loading parameter to avoid possible damage or ageing [7]. The glass transition temperature of PS samples, measured using the Dynamic Scanning Calorimetry (DSC) was 108 ◦C. This glass transition temperature value was not the exact value of glass transition but was an adequate value for us to stay below the glass transition region. Thermographic Approach The thermography techniques were used during the DMTA measurements to observe the temperature variations of the sample during the cyclic loading. The pixel-to-pixel calibration [9] was performed for each temperature for the Infrared (IR) camera (CEDIP SC7000 series) before using it on the DMTA tests as the temperature variations induced by material deformation

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