SHOCK W A V E S Measuring the Dynamic Response of Materials
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SHOCK W A V E S Measuring the Dynamic Response of Materials
SHOCK IAIAV=S Measuring the Dynamic Response of Materials
William M. Isbell ATA Associates, USA
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Imperial College Press
Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
SHOCK WAVES: MEASURING THE DYNAMIC RESPONSE OF MATERIALS Copyright © 2005 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 1-86094-471-X
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Dedication To my wife, Virginia, who spent countless hours preparing the draft of the original manuscript, then spent as many additional hours as a computer-widow, while I finalized the work.
Preface The 1950s and 1960s are sometimes referred to as the "Golden Age of Shock Wave Physics," characterized by ample private and government funding and with new insights and discoveries arriving on a regular basis. . . an exciting period, indeed. After a period of relative quiet during the 1970s, shock wave research resumed its expansion in the 1980s, with substantial support shown for the science worldwide. To some degree, however, the emphasis had changed from basic to applied studies. While numerous studies of a basic nature were still being pursued, the application of shock wave research to practical matters such as spacecraft shielding, fragmentation of spacecraft by explosions and by impact with space debris, and the development of advanced kinetic energy weapons, has provided a major support for the research efforts. Perhaps the most significant advance during this period was in the incorporation of physical models of shock wave behavior into finite difference and finite element routines, run both on supercomputers and on desktop personal computers. The capabilities of these routines to provide three-dimensional representations of complex target and projectile geometries have produced quantitative answers in areas previously not studied. Support for these computer techniques and the models they employ has grown in proportion to their utility. New launchers and diagnostics have become available to the experimenter, substantially expanding measurement capability. Nanograms are launched at 100 km/s by Van de Graaff generators, milligrams are launched at 15-30 km/s by exploding foils and by laser acceleration, and grams are launched at 10-13 km/s by three-stage guns. As with their predecessors at lower velocities, new insights will be obtained into material behavior, pushing back the frontiers.
vii
Contents Dedication Preface Acknowledgments
v vii xv
1 Introduction 1.1 1.2 1.3 1.4 1.5
Foreword Motivation for Research Chapter Organization Convention for Units References
1 2 3 4 5
2 Characteristics of High-Intensity Waves 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
2.9
Foreword Description of Shock Wave Formation Rankine-Hugoniot Relations Attenuation Waves Constitutive Relations High Pressure Region: Mie-Gruneisen Equation of State Elastic-Plastic Flow in Uniaxial Strain Time-Dependent Effects 2.8.1 Strain Rate 2.8.2 Spallation References
ix
7 7 8 11 12 13 15 17 17 18 20
x
Shock Waves: Measuring the Dynamic Response of Materials
3 Experimental Techniques for Measurement of the Dynamic Properties of Materials 3.1 3.2 3.3
Introduction Stress-Strain-Strain-Rate Studies Uniaxial Stress Tests 3.3.1 Low-Rate Testing 3.3.2 Medium-Rate Testing 3.3.3 High-Rate Testing 3.4 Multi-Axial Stress Tests, Biaxial Machine 3.4.1 Confined Pressure Device 3.5 High Heating Rate Tests 3.5.1 Heating and Testing Machine 3.5.2 Temperature Measurement 3.5.3 Strain Measurement 3.6 Ultrasonics Measurements 3.6.1 Elastic Wave Velocities 3.7 Equation of State and Wave Profile Studies 3.8 Gas Guns 3.8.1 Compressed-Gas Gun, 102 mm 3.8.2 Compressed-Gas Gun, 63.5 mm 3.8.3 Light-Gas Gun 3.9 Instrumentation 3.9.1 X-Cut Quartz Gages 3.9.2 ManganinGage 3.9.3 Streak Camera Techniques 3.9.4 Laser Velocity Interferometer 3.9.5 Slanted Resistance Wire 3.9.6 Magnetic Wire Gage 3.10 Spall Tests 3.10.1 Recovery Tests 3.10.2 Room Temperature Testing 3.10.3 Elevated Temperature Testing 3.10.4 Metallographic Examination 3.11 References
23 23 24 24 24 26 28 29 29 29 31 32 34 35 39 40 40 42 44 47 47 51 52 55 57 58 60 61 62 63 64 66
xi
Contents
4 Dynamic Response of Materials at Low and Moderate Stresses (<20 GPa) 4.1
4.2
4.3
Summary of Results 4.1.1 Introduction 4.1.2 Compressive Wave Behavior 4.1.3 Elastic Behavior 4.1.4 Release Wave Behavior 4.1.5 Fracture 4.1.6 Degraded Properties 4.1.7 Check Data 4.1.8 Summary Measurements of the Dynamic Properties of Alpha Phase Tantalum 4.2.1 Introduction 4.2.2 Material Properties 4.2.3 Stress-Strain Studies 4.2.4 Elastic Behavior 4.2.5 Low-Pressure Shock Wave Equation of State for Tantalum 4.2.6 Ultrasonic Equation of State 4.2.7 Yield Behavior 4.2.8 Wave Propagation 4.2.9 Spall Fracture 4.2.10 Summary References
69 69 72 76 77 79 80 82 84 85 85 86 88 90 96 99 102 104 109 117 120
5 Hugoniot Equations of State to 0.5 TPa 5.1 5.2 5.3
Foreword Theoretical Considerations Experimental Techniques 5.3.1 Instrumentation 5.3.2 Target Design and Construction 5.3.3 Data Analysis 5.3.4 Shock Wave Velocity 5.3.5 Impact Velocity 5.3.6 Hugoniot of the Impactor
125 127 130 130 132 137 137 138 139
xii
5.4
5.5 5.6 5.7
Shock Waves: Measuring the Dynamic Response of Materials
Experimental Results Fansteel-77 OFHC Copper 2024-T4 Aluminum Depleted Uranium Nickel -. Type 304 Stainless Steel Titanium Beryllium AZ3 IB Magnesium Plexiglas (PMMA) Quartz Phenolic HugoniotData 5.5.1 Tables of HugoniotData 5.5.2 Graphs of Hugoniot Data Summary References
139 140 142 144 145 147 148 149 150 151 152 152 153 153 158 180 184
6 Attenuation of Shock Waves from High Pressure 6.1 6.2 6.3
6.4
6.5 6.6
Foreword Introduction Description of the Impact Experiments 6.3.1 Design of Experiments 6.3.2 Laser Velocity Interferometer 6.3.3 Suppression of Spall 6.3.4 Manganin Wire Gage 6.3.5 Stepped Targets with Shims 6.3.6 Wedge Targets Experimental Results 6.4.1 OFHC Copper 6.4.2 6061-T6 Aluminum 6.4.3 Titanium 6.4.4 Beryllium 6.4.5 Uranium Alloy Conclusions References
187 188 190 191 192 194 194 196 202 206 207 214 220 223 225 226 230
xiii
Contents
7 Response of Porous Materials to Static and Dynamic Loading 7.1 7.2 7.3 7.4 7.5 7.6
Introduction Equation of State Surface Dynamic Behavior The Porous Constitutive Model Description of the Porous Berylliums Experimental Data 7.6.1 Static Compression Data 7.6.2 Shock Wave Tests 7.7 Comparison of Predictions with Data 7.8 Electron-Beam Tests 7.8.1 Experiment Design 7.8.2 Experimental Results 7.8.3 Applicability of Experimental Techniques to Other Materials 7.9 Summary 7.10 References
233 233 235 235 238 239 240 241 245 248 248 250 255 256 256
8 Interferometric Methods 8.1 8.2 8.3 8.4
8.5 8.6 8.7
Foreword The Displacement Interferometer The Velocity Interferometer The VISAR Interferometer 8.4.1 Glass Etalon Delay Legs 8.4.2 Air Delay Legs 8.4.3 Extended High Velocity Differential Etalon 8.4.4 Fringe Recording Systems 8.4.5 Increased Stability, Reduced Size Laser Illumination Photodetectors and Recording Systems Light Collection Systems 8.7.1 Direct Beam Technique 8.7.2 Fiber Optic Cable Technique 8.7.3 Multiple Fiber Technique
257 257 258 261 263 264 265 267 267 269 269 270 270 270 270
xiv
Shock Waves: Measuring the Dynamic Response of Materials
8.8
VISAR Configurations 8.8.1 "Line" VISAR 8.8.2 Imaging "White Light" VISARs 8.9 Summary 8.10 References
Appendix A.I
Index
271 271 271 272 273
Compendium of Wave Profiles
Summary Profiles Profiles Profiles Profiles
in Aluminum in Titanium in Tantalum in Fused Quartz
275 276 302 307 317 319
Acknowledgments Much of the work reported herein is the result of collaborations between the author and talented researchers from Stanford Research Institute, General Motors Technical Center, Lawrence Livermore National Laboratory, and General Research Corporation. In several instances, these colleagues were co-authors of papers and reports which formed the basis of chapters in this book. In particular, I wish to acknowledge the contributions of Mr. Douglas Christman, Dr. Arfon Jones, Sir Colin Maiden, Mr. Nilson Froula, Mr. Sidney Green, Mr. Alan McMillan, Dr. Francis Ree, Mr. Herbert Shipman, and Mr. Robert Warnica. I am very conscious of and grateful for their participation in the research efforts [1-8]. Also, I greatly appreciate the early training provided me in the area of shock wave physics by previous colleagues and mentors, including Drs. George Abrahamson, Alex Charters, George Duvall, G. Richard Fowles, Robert McQueen, and Mark Wilkens. These pioneering researchers, who originated many of the early efforts to describe the response of materials and structures to shock wave loading, have provided inspiration and knowledge helpful throughout my career. Finally, the new generation of shock wave researchers is acknowledged. An encouraging example is the rapidly expanding research efforts by the author's Japanese colleagues, such as Prof. Kazuyoshi Takayama of Tohoku University in Sendai, Japan, who is utilizing shock waves for peaceful and commercial purposes, portending an excellent future for a discipline which has been based primarily upon military applications. Dr. William M. Isbell Santa Barbara, California June 2003 XV
xvi
Acknowledgments
References 1. Christman, D. R., W. M. Isbell, S. G. Babcock, A. R. McMillan, and S. J. Green. "Measurements of Dynamic Properties of Materials, Vol. Ill: 6061T6 Aluminum." DAS A 2501-3, Manufacturing Development, General Motors Corporation, 1971. 2. Jones, A. H., W. M. Isbell, and C. Maiden. J.A.P., vol. 37, 3493 (1966). 3. Maiden, C. J., and S. J. Green. "Compressive Strain-Rate Tests on Six Selected Materials at Strain Rates from 10'3 to 104 in/in/sec." ASME Trans. J. Appl. Mech., Ser. E, vol. 33, 496-504 (1966). 4. McMillan, A. R., W. M. Isbell, and A. H. Jones. "High Pressure Shock Wave Attenuation." DASA-2425, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971. 5. Ree, F. H., W. M. Isbell, and R. R. Horning. Lawrence Livermore Laboratory Rept. UCRL-51682, Part 4 (1974). 6. Isbell, W. M., F. H. Shipman, and A. H. Jones. "Use of a Light-Gas Gun in Studying Material Behavior at Megabar Pressures." Symposium High Dynamic Pressure, Paris, France, September 1967. 7. Isbell, W. M., and D. R. Christman. "Shock Propagation and Fracture in 6061-T6 Aluminum from Wave Profile Measurements." DASA-2419, General Motors Materials and Structures Laboratory (AD705536), April 1970. 8. Isbell, W. M., F. H. Shipman, and A. H., Jones. "Hugoniot Equation of State Measurements for Eleven Materials to Five Megabars." MSL-68-13, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD721920), 1968.
1. Introduction 1.1
Foreword
In the investigation of shock waves in solids, the experimenter is placed in the role of indirect observer of the phenomena involved [1]. The duration of the event is far too brief for the eye to observe, and the release of energy which creates the shock waves tends to overwhelm the senses. Thus, much of the progress in understanding shock processes has awaited the development of adequate diagnostic tools to record short-lived, transient phenomena. To a large extent, the development of theoretical models has paralleled and followed the development of shock wave instrumentation and the production of experimental data. The 1950s and 1960s produced a wealth of experimental techniques and data which rapidly advanced the understanding of the mechanical, thermodynamic, and electromagnetic response of materials to high-intensity waves and provided the framework of modern shock wave physics. In a seminal 1955 paper, J. M. Walsh and R. H. Christian [2] described measurements of the equations of state of aluminum, copper, and zinc to pressures between 10 and 50 GPa (100 and 500 kilobars). Explosive lenses were used to generate high pressures in specimens, and free surface and shock wave velocities were measured by argon flash gaps and rotating mirror streak cameras. Other work and experimental techniques quickly followed. Measurements of shock wave profiles by use of shorting pins provided definition of elastic waves preceding the shock wave and led to the discovery by Minshall, Bancroft, and Peterson of Los Alamos of the phase transition in iron [3]. The major developments of the early period were summarized in a 1986 paper by G. E. Duvall [1]. Each discovery
1
2
Shock Waves: Measuring the Dynamic Response of Materials
was occasioned, to some degree, by the introduction of new experimental techniques, as shown in Table 1.1, adapted from Reference 1. In addition to developing new methods for measuring shock wave phenomena, new techniques were sought to create well-controlled shock waves in materials. Explosive techniques were augmented by single- and two-stage gas guns [4-5], and the range of pressures achievable was extended both higher and lower, from less than 0.1 GPa to nearly 1 TPa. Table 1.1. Techniques Utilized to Study Shock Wave Phenomena (Circa 1965)
PHENOMENON
TECHNIQUE
Equation of state
Flash gaps, shorting pins, streak camera
Phase transitions
Shorting pins
Electrical conduction
Ionization pins, quartz gage
Charge release-polarization
Piezoresistive gage
Shock demagnetization
Electromagnetic velocity gage
Shock-induced opacity
Laser interferometer
Elastic precursor and decay
Pins and laser interferometer
Residual stress
Recovery
Extremes of pressure have now been achieved by pulsed laser loading of materials and by the deposition of radiation energy in underground nuclear tests. In this manner, the available pressure regime has been raised to tens of TPa.
1.2
Motivation for Research
Following the discovery of new phenomena, the general trend has been to accumulate data sufficient to construct physical models. The process of discovery, model formulation, data accumulation, model refinement, and inclusion in computer routines has been repeated many times over the past fifty years. An important aspect of this cycle has been the measurement of shock wave parameters under the carefully controlled conditions provided by the impact of flat plates launched by guns. Beginning in the mid-1960s,
1. Introduction
3
two types of guns were developed for shock wave studies. Single-stage guns provided measurements to approximately 20 GPa, while two-stage light-gas guns increased this range to over 0.5 TPa. Of particular significance was the original utilization of a light-gas gun to extend equation of state measurements to above 0.5 TPa [4-5]. Innovations in instrumentation included development of pin techniques with sub-nanosecond accuracy for measurement of high-pressure equations of state and improvements to interferometers capable of measuring shock wave profiles with high accuracy [6-11]. The precision afforded by these launchers and the relative ease (compared to explosive techniques) of applying sophisticated sensors to measure shock wave parameters have led to detailed descriptions of material behavior under shock wave loading [12-13]. Combined with other techniques (static high pressure, material yield under uniaxial and biaxial stress, elastic wave velocities, etc.), these measurements furnish modelers with sufficient information to formulate highly accurate predictive routines. The studies presented in this book have provided experimental data for a wide variety of materials upon which models of material behavior have been formulated. In general, the emphasis has been on the procurement of data, rather than on its utilization. Experimental techniques were developed or improved upon to measure shock wave parameters. New launchers were built which extended the range of measurements beyond previous limits. Close coordination with theoretical shock wave physicists provided guidance on the parameters needed for models of material behavior. An iterative approach was utilized that made effective use of available equipment and manpower.
1.3
Chapter Organization
The book is arranged so as first to familiarize the reader with the theoretical and experimental foundations upon which the work is based. A description of the theoretical basis of shock wave formation and propagation is presented in Chapter 2 as a framework for the remainder
4
Shock Waves: Measuring the Dynamic Response of Materials
of the chapters. Chapter 3 describes the experimental techniques employed for the study, with details of launchers, sensors, and auxiliary equipment. Results of the experimental investigations of material response to dynamic loading in the regime below 20 GPa, where the effects of material strength must be incorporated in models, are presented in Chapter 4, beginning with a summary of the results from the four materials tested and continuing with a detailed description of the response of tantalum. Chapter 5 describes Hugoniot measurements of eleven materials to 0.5 TPa, using the technique of launching flat plates to 8 km/s against specimens suspended near the gun muzzle. Chapter 6 continues the description of impact tests where the objective is to describe the release behavior of materials from very high pressures. A qualitatively different class of material response is described in Chapter 7 — that of porous materials. While the equation of state surface of a solid material is independent of the path used to arrive at that point, no such unique relationship exists for porous materials. The mathematical model describing shock wave propagation in this class of materials necessarily contains the thermodynamic path by which the material arrives at its final state. The behavior of porous materials under shock wave loading is described in the chapter, using porous beryllium as the example. In Chapter 8, the use of laser interferometry is discussed as a method for obtaining highly accurate wave profiles, from which a wealth of shock wave parameters is obtained. The chapter is followed by Appendix A, which catalogs a large number of wave profiles, obtained with interferometry and stress gages.
1.4
Convention for Units
In general, International System (SI) units are used in the text throughout the book. Where tests were conducted before the widespread adoption of SI standards, however, the data are frequently presented using the
5
1. Introduction
original units. This convention is especially followed in tables and figures. The table below allows conversion of values. Quoted Units
1.5 1.
SI Units
1 kilobar (kb)
0.1 Gigapascal (GPa)
1 Megabar (Mb) 1g lg/cm 3
0.1 Terapascal (TPa) 0.001 kg 1,000 kg/m3
References
Duvall, G. E. "Shock Wave Research: Yesterday, Today, and Tomorrow." Shock Waves in Condensed Matter, 1-12. New York and London: Plenum Press, 1986. 2. Walsh, J. M., and R. H. Christian. "Equation of State of Metals from Shock Wave Measurements." Phys. Rev. 97:1544 (1955). 3. Bancroft, D., E. L. Peterson, and S. Minshall. "Polymorphism of Iron at High Pressure." J. Appl. Phys. 27:291 (1956). 4. Jones, A. H., W. M. Isbell, and C. J. Maiden. "Measurement of Very High Pressure Properties of Materials Using a Light-Gas Gun." J. Appl. Phys., vol. 37 (August 1966). 5. Isbell, W. M., A. H. Jones, and F. H. Shipman. "Use of a Light Gas Gun in Studying Material Behavior at Megabar Pressures." Proceedings of the Symposium on High Dynamic Pressure, Paris, September 1967. 6. Barker, L. M., and R. E. Hollenbach. Rev. Sci. Instr. 36:4208 (1965). 7. Barker, L. M., and R. E. Hollenbach. "Laser Interferometer for Measuring High Velocities of Any Reflecting Surface." /. Appl. Phys., vol. 43, no. 11 (November 1972). 8. Isbell, W. M. "Laser Interferometric Techniques for Precision Measurements of Shock Waves in Materials and Structures." Photomethods magazine (May 1982) and High Speed Photography and Photonics newsletter, vol. 2, no. 1 (winter 1982). 9. Hemsing, W. F. "Velocity Sensing Interferometer (VISAR) Modification." Rev. Sci. Instr. 50 (January 1979). 10. Isbell, W. M. "Advances in Laser Interferometer Techniques for Measurement of Dynamic Material Properties." Proceedings of International Congress on Instrumentation in Aerospace Simulation Facilities, Dayton, Ohio, September 30, 1981.
6
Shock Waves: Measuring the Dynamic Response of Materials
11. Isbell, W. M. "Modern Instrumentation for Measurements of Shock Waves in Solids." Proceedings, Japanese Shock Wave Symposium, Tokyo, Japan, 1999. 12. Isbell, W. M., and W. J. Tedeschi. "Hypervelocity Research and the Growing Problem of Space Debris." Proceedings, Hypervelocity Impact Symposium, 1992. 13. Cunningham, T. M., and W. M. Isbell. "Results from the Satellite Orbital Debris Characterization Test Series." Proceedings, Hypervelocity Impact Symposium, 1992.
2. Characteristics of High-Intensity Waves
2.1
Foreword
A brief description of shock wave theory is presented to provide a framework for the discussions of experimental techniques and results which follow. Several excellent texts are referenced which describe theoretical considerations in much greater detail. Much of the work quoted is obtained from studies conducted before 1970 [1-6] and summarizes the understanding of shock processes to that date. While later studies have elucidated certain aspects of the theories, the fundamental framework remains unchanged.
2.2
Description of Shock Wave Formation
As a general rule, liquids and solids have pressure-volume curves which are convex downwards, d2a7dV2 > 0 {a is the pressure, V is specific volume), resulting in an increase of sound velocity with pressure. Thus, in a simple compressional pulse, as illustrated in Figure 2.1, the waveform is continually changing. The wavelets carrying the greatest stress travel faster than any lower stress wavelets. The compressive region becomes shorter (and steeper) and the expansive region becomes longer. Eventually, the wave front becomes infinitely steep, theoretically forming a mathematical discontinuity. Dissipative mechanisms, on the other hand, tend to smooth out the discontinuities at the shock surface. However, in most solids and liquids the thickness of a high-intensity shock is so small that it is virtually a mathematical discontinuity. In the discontinuity region, the equations of 7
8
Shock Waves: Measuring the Dynamic Response of Materials
thermodynamically reversible flow are no longer applicable. Instead, the Rankine-Hugoniot relations apply, as discussed below.
2.3
Rankine-Hugoniot Relations
Planar shocks can be introduced in solids by the impact of two flat plates. Instrumentation in such experiments can measure the basic parameters of shock wave formation and propagation, including: • • • • • • •
Shock and release wave velocities in the target Free surface velocity at the rear of the target Particle velocity behind the shock wave Stress and strain behind the shock wave Density of the material compressed by the shock wave Temperature of the shocked surface Impact velocity of the impactor plate.
To obtain the thermodynamical quantities stress, density, and specific internal energy, which describe the state of the material behind the shock in the target from the measured velocities, it is necessary to invoke the celebrated Rankine-Hugoniot relations. These express the conservation of mass, momentum, and energy across a shock, assuming that equilibrium exists on either side of the shock front zone. Using the notation denoted in Figure 2.2, where all velocities to the right are measured with respect to a stationary observer, the conservation relations for the material passing through the shock are
A ) (u o
P o K - U s ) = p(u-Us)
(2.1)
<7O+/0OK-UB)2=(U-UB)2
(2.2)
- U s ) E o + K P o (u o - U s ) 3 + a o ( u s - U s )
= p(u-Us)E + >/2p(u-Us)3+a(u-Us)
(2.3)
2. Characteristics of High-Intensity Waves
9
Conservation of momentum and energy relations simplify to CT
- ^o = A) (U s - u 0 ) ( u - u 0 )
E-E o 4(. +
CTo)[i-i]
(2.4)
(2.5,
It is interesting to note that Equation 2.5 can be interpreted graphically. Specific internal energy increase across the shock is given by the area of the crosshatched triangle in Figure 2.3. For material initially at o — 0, u0 — 0, Equation 2.5 reduces to E
"Eo=^u2
(2-6)
In other words, the energy input into the target is divided equally into an increase in the internal and kinetic energy. The locus of states in the pressure-density plane obtained from experimental data using Equations 2.1 and 2.4 (or from the equation of state E = f (a, p) and Equation 2.5) from a known initial state (for solids it would be Go = 0, u 0 = 0) is called the Hugoniot curve. Sometimes it is referred to as the Hugoniot equation of state or simply as the Hugoniot. Early Russian work termed it the dynamic adiabat or shock adiabat. Historically, the development of the Rankine-Hugoniot relations stems from understanding the thermodynamical process at the discontinuity in the shock wave. Rankine [7] first realized that a region in which irreversible processes occurred was necessary, so that there would be an entropy gain on passing through the discontinuity. This increase in entropy is of the third order in the shock strength [8, 9], i.e., p - p o , c - O b . o r u - u 0 . Consequently, the rise of pressure, density, and temperature across a shock front differs from reversible adiabatic changes. The difference is again of the third order in shock strength. Expressed in another way, the Hugoniot has the same slope and curvature as the adiabat at the initial state. It follows that, for lowamplitude shock waves, the difference between the Hugoniot and adiabat is small. This difference increases with increasing shock strength, the Hugoniot lying above the adiabat.
10
Shock Waves: Measuring the Dynamic Response of Materials
Figure 2.1. Shock Wave Formation
Figure 2.2. Propagating Shock Wave. Subscript refers to quantities ahead of the shock.
Figure 2.3. Stress-volume schematic, showing relative position of Hugoniot and isentropes
2. Characteristics of High-Intensity Waves
11
Other properties of shock waves are (1) Shocks, as a rule, are compressive [10]. (Conditions for the existence of rarefaction shocks are discussed later.) (2) The flow velocity relative to the shock front is supersonic at the front side and subsonic at the back side. Proof of these is found in most textbooks on fluid mechanics [8, 9].
2.4
Attenuation Waves
Since, on the back side of the shock, the flow velocity relative to the shock wave is subsonic, it follows that any disturbance originating there will overtake the shock wave. Such disturbances can be either a superimposed shock wave or an attenuating rarefaction wave. Referring again to the impact of two plates, experimentally superimposed shock or rarefaction waves can be generated with a two-layer plate impactor. For a second layer of higher shock impedance, pU s , superimposed shock waves are generated which will eventually overtake the shock wave in the target. Rarefaction waves originate at the interface within the projectile for a lower shock impedance second plate or for a free rear surface. Equations 2.3 through 2.5 are applicable to the superimposed shock wave. However, across the rarefaction wave the state of the material is isentropically changed to lower stress levels. Since rarefaction waves are dispersive, each element travels at local sound speed. (For this reason, a rarefaction wave cannot be overtaken by another rarefaction wave.) Each element may be considered equivalent to a weak shock wave. Equations 2.1 through 2.5, put in differential form, describe the flow where all processes are at constant entropy.
c
{da %
<2 - 7)
12
Shock Waves: Measuring the Dynamic Response of Materials
Jdu = f^dp
(2.8a)
and
JdE = j 4 d / 5
(2.8b)
Equation 2.8b is recognized as the Riemann integral. Particle velocity increment across the release wave, calculated by means of Equation 2.8, differs from that across the Hugoniot. For actual materials there is a greater increase across the isentrope. Differences are small at low pressure and to even quite high pressures for incompressible materials. Al'tshuler et al. [11] indicated that differences are about 1-2% for iron shocked to 350 GPa. Porous materials, on the other hand, give large differences in velocity, even at low pressure. The stress-volume history for materials subjected to a onedimensional loading is shown in Figure 2.3. Material initially at (V0,0) is shocked to (V,, ax) and released to (V 2 ,0). Since the release isentrope, from state (Vj, c^) to state (V 2 ,0), is at a higher entropy level than the Hugoniot from state (Vo,0) to state (V^cri), it lies above the Hugoniot. Zero stress level is therefore reached at larger specific volume, with the material being at higher temperature than ambient. For a sufficiently intense shock, it is possible to melt or vaporize during unloading or possibly upon loading [12, 13].
2.5
Constitutive Relations
Stress-strain-time history at a given point in the target subjected to specified loading conditions may be calculated if the equation of state or constitutive equations for the material is formulated. In general, analytical solutions cannot be found for non-trivial problems, therefore reliance must be placed on numerical techniques. Analyses of experimental results based on such techniques are given in Chapter 3 of this book; however, a description of the numerical scheme is not
2. Characteristics of High-Intensity Waves
13
included. Interested readers may find details on the numerical approach in References 5 and 14. In the beginning of shock and acoustic wave research, the response of solids to shock loading was studied theoretically in regions of high pressure where the effect of material strength could be neglected, and in the lower pressure regime where thermodynamic effects could be neglected but strength effects were important. At high pressure, therefore, the solid was treated as a fluid (stress and pressure are used synonymously in text for this regime), and only for lower intensity waves was shear strength included. However, contributions to elasticplastic theory by E. H. Lee [15] and others led to the inclusion of thermodynamic influence on the strain deformation. Material descriptions used in these regimes are discussed in the next sections. The sections terminate with a brief description of the dynamic fracturing process known as spalling.
2.6
High Pressure Region: Mie-Gruneisen Equation of State
Information regarding the behavior of materials at high pressure can be obtained from Hugoniots with different initial densities or different temperatures and isentropes. However, a more complete thermodynamic description of high pressure states requires assuming the form for an equation of state and using available experimental data to adjust the constants [16, 17, 18]. A simple formulation which has found wide usage is the MieGruneisen equation of state. It is based on the assumption that the thermal energy of a crystal can be described adequately by the sum of the energies of a set of independent harmonic oscillators, and that the frequencies of all normal modes change proportionally with volume. With these assumptions, the equation of state can be reduced to
P-PH = ^ p ( E - E H )
(2.9)
14
Shock Waves: Measuring the Dynamic Response of Materials
where 7 (V) is the Gruneisen ratio, p the pressure, E the internal energy/unit mass, and V the specific volume, and subscript "H" refers to the Hugoniot state. Hugoniot data, obtained from flat plate experiments, are frequently represented in one of two forms: n
PH=Sai^
(2-10)
D = c+J 8 i /f i
(2.11)
i=l
or i=l
where
(2.12)
r o
Us is the shock velocity, u the particle velocity change across the shock, and &i, &\, and c are constants determined from data fitting. Two expressions relating the Gruneisen ratio to the 0°K isotherm are usually studied, (i) the Slater relation [19]
Vd2pJdV2
2
(2.13)
Pk being the pressure along the, 0°K isotherm, and (ii) the DugdaleMacDonald relation
vs'(pyV'")/3y'
7=1
1
a( ft v-)/3v ' 5
(2J4)
as derived by Rice et al. [6]. In order to obtain theoretically y(V), the 0°K isotherm must be evaluated.
2. Characteristics of High-Intensity Waves
15
It is possible to solve for pk in
700 =
V(Pk ~ PvH)
EH+EO+JpkdV
(2.15)
V0-K
with suitable initial conditions. Here / ( V ) is given by Equation 2.13 and Eo is a correction to allow for arbitrarily chosen zero energy at room conditions V = V o , and T = 298°K. E o is obtained by integrating the Debye specific heat curve. Isentropes, in principal, may be computed by setting (dE)
tavl="Ps
(2-16)
in Equation 2.9. If the volume dependence of the Gruneisen ratio is not available, assuming Y / V to be a constant leads to reasonable results. The reason is the insensitivity of the p - V loci to large changes in y [5, 6]. At pressures of several hundred GPa, the free electrons in metal contribute measurably to the increase in stiffness of the material. In References 16-19, a semi-empirical correction is added to allow for this effect and to permit the Mie-Gruneisen equation of state to be faired in with the Thomas-Fermi-Dirac [20] equation of state at higher pressures. With these equations of state, material shear strength is neglected; in other words, the media are treated as fluids. This approximation is adequate for many applications where only gross features are required. However, in many problems, strength significantly alters the results. For such problems more detailed constitutive equations are necessary, such as for the elastic-plastic behavior of a solid considered in the following sections.
2.7
Elastic-Plastic Flow in Uniaxial Strain
For appreciable plastic strain in the plate impact experiments, high pressures are needed. This results in high temperatures which necessitate
16
Shock Waves: Measuring the Dynamic Response of Materials
the inclusion of thermodynamic influences on strain deformation. To treat this problem analytically it is necessary to consider finite deformation thermoelastic-plastic theory. Physically, on loading, a stress-strain path lying 2/3 Y above the hydrostatic curve is followed (to point A in Figure 2.4). The initial release wave of intensity Y(4/3 + K///) follows path AC. The slope of AC is (K + 4/3/0, while the slope of the hydrostat is K. Thus, the relative slope depends on the shear modulus //. Once the elastic release wave has reached the loading shock, the shock amplitude is reduced. As the reduced amplitude shock propagates, the material is now loaded to a lower stress D. The influence of the work-hardening and Bauschinger effect on the stress-strain diagram for material loading is shown in Figure 2.5. Increase of yield with plastic working will result in the loading stress-strain path which has a greater offset from the hydrostat, but still 2/3 Y. Butcher and Cannon [21] observed that dynamic work-hardening of fully hard 4340 steel is in agreement with its static work-hardening properties. The dynamic (uniaxial strain) stress-strain relation wave obtained from the static relation (one-dimensional stress) by equating plastic work. This results in the relation in the plastic range S' =
3 Y 2a'-«K
(2'17)
where ax is the equivalent strain in uniaxial stress.
Figure 2.4. Stress-Strain Schematic for Elastic-Perfectly Plastic Material in Uniaxial Strain
2. Characteristics of High-Intensity Waves
17
Figure 2.5. Stress-volume schematic for loading and unloading in uniaxial strain, showing the influence of strain hardening and Bauschinger effect
2.8
Time-Dependent Effects
2.8.1
Strain Rate
High degree of resolution both in time (10~9 seconds) and position (300 nm) in the motion of the rear surface is possible, using laser interferometry. In flat plate experiments such measurements have demonstrated the time-dependent amplitude of the elastic precursor in iron and aluminum. In real media, the passage of material through a shock front is not an instantaneous process, but rather it involves a rapid transition from the front state to the back state through a narrow region, the shock layer. In the case of the elastic precursor, the wave front thickness seems to be in the region where a continuum mechanics approach is applicable. Many investigators, Taylor and Rice [22], Barker et al. [23], and Jones et al. [24], and this work have observed that the elastic wave does not have a sharp front. In the work of Barker et al. [23] on 1060 aluminum the thickness of the elastic wave fronts (defined by the time between the first-motion of
18
Shock Waves: Measuring the Dynamic Response of Materials
the free surface and the first maximum in the free surface velocity) is seen to vary from 10 to 40 ns. Stress magnitude at the first peak is above the elastic limit but decays as the shock moves into the material. As the stress decays towards the static yield point the elastic wave front attenuation is less rapid [25].
2.8.2
Spallation
Fracture, which is either partial or complete, results from the tension induced by the interaction of two rarefaction waves and is known as spalling. Such conditions are produced in the laboratory by the impact of two flat plates. Simplified representation of the process is given in Figure 2.6 in which the waves are elastic. Spallation occurs at FF when the tensile stress, which is greater than the stress required to fracture, has been maintained for sufficient time for cracking to nucleate. Maintenance of the stress for a longer time will allow the crack to propagate. Since spalling is time dependent in most metals tested (aluminum alloys, titanium alloys, beryllium alloys, copper, etc.), it is not possible to quote a single value for the spallation stress. For this reason it is important that the loading time-history be known, i.e., the wave profile that causes spalling, and each author's definition of spalling be clearly stated, i.e., whether it is incipient spall (nucleation) or complete separation. For example, Charest [26] sections and carries out a micrographic (500X) examination of each target in order to determine the condition for incipient spall. Consider the stress wave propagation shown in Figure 2.6. After impact a shock wave is propagated into both the impactor (driver) and target. When these shocks reach the free surfaces, they are reflected as rarefaction waves, as shown in Figures 2.6c and d, resulting in regions at zero stress level. These two rarefaction waves cross shortly after the condition shown in Figure 2.6d. The location at which this occurs is FF. Subsequently, this section will be the crack nucleation location, as it is subjected to damaging tensile stresses for the longest time.
2. Characteristics of High-Intensity Waves
19
For an impactor half as thick as the target, FF will be at the center of the target. The time for which material at section FF is under tension depends on the impactor thickness. The time-dependency of spallation typically is studied by conducting a series of tests in which the impact velocity for a set of impactor/target pairs is varied from below the velocity necessary to cause internal cracks to above the velocity needed to produce complete separation of the target. To change the time duration of the pulse, the impactor thickness is changed, and in order that only the time duration of the pulse be changed, the thickness of the target is changed proportionally such that a constant ratio (usually 1:2) of impactor to target thickness is maintained.
Figure 2.6. Stress wave propagation through target-impactor (driver) of similar material, showing the position at which spallation is initiated. In the schematic, impactor is onehalf of the target thickness.
20
2.9
Shock Waves: Measuring the Dynamic Response of Materials
References
1. Al'tshuler, L. V. Soviet Physics Uspekhi 8:52 (1965). 2. Deal, W. E., Jr. In R. H. Wentorf, Jr. (ed.), Modern Very High Pressure Techniques, 200. Washington: Butterworthlnc, 1962. 3. Doran, D. G. In A. A. Giardini and E. C. Lloyd (eds.), High Pressure Measurements, 59. Washington: Butterworthlnc, 1963. 4. Duvall, G. E., and G. R. Fowles. In R. S. Bradley (ed.), High Pressure Physics and Chemistry, vol. 2, 209. London and New York: Academic Press, 1963. 5. McQueen, R. G. In K. A. Gschneider, Jr., M. T. Hepworth, and N. A. D. Parlee (eds.), Metallurgy at High Pressures and High Temperatures, Metallurgical Society Conferences, vol. 22, 44. New York and London: Gordon and Breach Science Publishers, 1964. 6. Rice, M. H., R. G. McQueen, and J. M. Walsh. In F. Seitz and D. Tumbull (eds.), Solid State Physics, vol. 6, 1. New York and London: Academic Press, 1958. 7. Rankine, W. J. M. Phil. Trans. Roy. Soc. (London) 160 (1870) 277. 8. Serrin, J. In C. Truesdell (ed.), Handbuch der Physik, vol. 3/1, 224. Springer Verlag, 1959. 9. Courant, R., and K. O. Friedrich. Supersonic Flow and Shock Waves. New York: Interscience, 1956. 10. Drummond, W. E. J. Appl. Phys. 28:988 (1957). 11. Al'tshuler, L. V., K. K. Krupnikov, B. D. Ledenov, V. J. Zhuchikhin, and M. I. Brazhnik, Soviet Phys. JETP 34:606 (1958). 12. McQueen, R. G., and S. P. Marsh. /. Appl. Phys. 31:1253 (1960). 13. Urlin, V. D., Soviet Phys. JETP 22:341 (1966). 14. Wilkins, M. L. In B. Alder, S. Fernbach, and M. Rotenberg (eds), Methods in Computational Physics, vol. 3, 211. New York and London: Academic Press, 1964. 15. Lee, E. H. Proc. 5th U.S. National Congress of Applied Mechanics, 405 (1966). 16. Tillotson, J. General Atomic Division of General Dynamics Report No. GA3216 (1962). 17. Al'tshuler, L. V., A. A. Bakanova, and R. F. Trunin. Soviet Phys. JETP 15:65 (1962). 18. McCloskey, D. J. The Rand Corporation Memorandum RM-3905-PR (1964). 19. Slater, J. C. Introduction to Chemical Physics. New York and London: McGraw-Hill Book Company Inc., 1939. 20. Feynam, R. P., N. Metropolis, and E. Teller. Phys. Rev. 75:1561 (1947). 21. Butcher, B. M., and J. R. Cannon. AIAA Journal 2:2174 (1964). 22. Taylor, J. W., and M. H. Rice. J. Appl. Phys. 34:364 (1963). 23. Barker, L. M., B. M. Butcher, and C. H. Karnes. J. Appl. Phys. 37:1989 (1966).
2. Characteristics of High-Intensity Waves
21
24. Jones, O. E., F. Neilson, and W. B. Benedict. /. Appl. Phys. 33:3223 (1964). 25. Taylor, J. W. /. Appl. Phys. 36:3146 (1965). 26. Charest, J. A., and R. L. Warnica. GM Defense Res. Labs. Rept. TR66-34 (1966).
3. Experimental Techniques for Measurement of the Dynamic Properties of Materials 3.1
Introduction
The measurement of dynamic properties of materials requires use of specialized and often sophisticated equipment and techniques to obtain satisfactory data under extreme ranges of test conditions. In particular, laboratory studies of material response to impulsive loads presented in this document include pressures up to 0.5 TPa, temperatures up to 4000°C and loading times as low as 50 nanoseconds. These studies involve material characterization in four areas: (1) stress-strain-strain rate; (2) elastic constants; (3) equation of state and wave profiles; and (4) dynamic fracture. The methods described have been used in support of measurements of the dynamic properties of materials, experimental results of which are summarized in Reference 1 and given in detail in References 2 to 5. These reports include additional discussion on the application of the experimental techniques to specific materials.
3.2
Stress-Strain-Strain-Rate Studies
Shock wave and structural response codes require information on the behavior of materials under uniaxial and multi-axial stress conditions. Hydrocode inputs include constitutive equations relating stress to strain and to strain rate. Measurements are made of uniaxial compressive stress vs. strain at various strain rates, giving yield and flow stress behavior, strain-rate sensitivity, and work-hardening characteristics. The measurements are sometimes extended to uniaxial tension, and testing in orthogonal directions is necessary if material anisotropy is 23
24
Shock Waves: Measuring the Dynamic Response of Materials
significant. Bauschinger effect tests provide information on unloading and subsequent yield. Because of the temperature increase normally associated with energy-deposition loading, it is important to establish the influence of material temperature as well as heating rate and time-attemperature on strength properties and stiffness. Finally, extension of calculations to include long-time structural response requires information on yield and fracture surfaces for multi-axial stress states.
3.3
Uniaxial Stress Tests
3.3.1 Low-Rate Testing Uniaxial stress testing at low rates (<0.1/s) was performed on an Instron Model TT-D Universal Testing Instrument with load capacity of 20,000 lbs. Tension and compression tests were carried out at temperatures from -195°Cto+500°C.
3.3.2 Medium-Rate Testing [6-8] Medium-strain-rate testing was carried out at 103/s to 102/s, using the gas-operated, medium-strain-rate machine shown in Figure 3.1. The machine is capable of either upward or downward piston motion for compression or tension testing, respectively. The maximum strain rate depends on the accuracy required and on the specimen material and geometry, since stress waves generated by high velocity motion can impede accurate time resolution of stress and strain in the specimen. The upper limit is generally between 30/s and 100/s. The load applied to the specimen is measured by strain gages mounted on an elastic load bar directly above the specimen. Specimen strain is obtained by measuring piston displacement, by use of an optical extensometer, or by using strain gages.
3. Experimental Techniques
25
Figure 3.1. Medium-Strain-Rate Machine Schematic
The optical extensometer used was equipped with dual tracking units. This device detects the motion of a discontinuity in light intensity, such as the edge of a black and white band on a specimen, and gives a voltage output proportional to the position of the edge as the specimen is strained. To operate the extensometer, the test specimen is marked with paint, as shown in Figure 3.2. One tracker is focused on the upper portion of the gage length and the other on the lower portion. The extensometer is set so that the difference between movement of the upper and lower portions of the test specimen is recorded as output. This difference, divided by the original distance between the two marks, is the engineering strain. For testing at temperatures up to 500°C, a Research Incorporated Model ZH3A furnace was used. This furnace is divided into three independently controlled zones containing tungsten-filament quartz heating elements. A water-cooled heat exchanger is attached to the load
26
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.2. Tension Specimen for Medium-Strain-Rate Machine
bar to isolate the strain gages from the furnace heat. Specimen temperature was monitored with thermocouples placed in contact with each end of the specimen. These thermocouples are used to control the heating elements through the furnace controller. Specimens can be heated at a selected constant heating rate and held at a given temperature for as long as desired.
3.3.3 High-Rate Testing [8-11] High-strain-rate tests from about 2 x 102/s to 5 x 103/s were conducted on a Hopkinson bar device. The operation of the Hopkinson bar is based upon the theory of one-dimensional elastic wave propagation. Using this theory, deformation of a specimen sandwiched between two elastic bars and subjected to a stress wave may be calculated. Compression or tension tests can be carried out, as shown in Figures 3.3 and 3.4. The specimen behavior is obtained by measuring the output from strain gages mounted on the load bars on both sides of the specimen and applying one-dimensional elastic wave theory.
3. Experimental Techniques
27
Figure 3.3 Compression Device for Hopkinson Bar
Figure 3.4. Tension Device for Hopkinson Bar
The problem of non-uniformity of axial stress (axial inertia) in the early stages of testing (typically 1% to 2% strain at 103/s), where the specimen is being loaded by wave reverberations, makes interpretation of yield stress difficult. Also, the major assumption of uniaxial stress in the early stages will not be valid if radial inertia effects have not been eliminated. These effects are present immediately behind the wave front where radial, or transverse, stress waves propagate to the specimen sides and reflect back. Until these waves have reflected several times, a non-uniform stress distribution exists. This limits the upper strain rates achievable to a few thousand per second, since for higher strain rates the entire test (10-15% strain) will occur in the non-uniform stress condition. The calculation of the stresses beyond yield is estimated to be accurate to about 10%, although reproducibility is much better. Elevated temperature testing with the Hopkinson bar is done in compression only, and the technique is shown in Figure 3.5. The specimen is heated in a radiant-heat furnace similar to that used on the medium-rate machine. The specimen is brought up to temperature with
28
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.5. Elevated Temperature Technique for Hopkinson Bar
the load bars isolated outside the furnace. At test temperature, a pneumatically operated device quickly moves the bars into contact with the specimen and triggers the acceleration of the driver bar down the launch tube to impact the weigh-bar. This procedure eliminates any significant temperature increase in the bars at a distance of 1 mm from the specimen/bar interface for about 3 seconds, much longer than the time for the stress wave to be initiated and the 160 us stress pulse to propagate through the specimen.
3.4
Multi-Axial Stress Tests, Biaxial Machine [12,13]
This machine, shown in Figures 3.6 and 3.7, develops a biaxial state of stress by applying an axial load and a circumferential load simultaneously. The loads are applied on a tubular specimen through two independent gas-operated cylinders. The axial stress can be tension or compression, while application of internal or external pressure gives circumferential tension to compression. Axial and/or circumferential strain rates from 10"3/s to about 10/s are possible.
3. Experimental Techniques
29
Three-dimensional yield and/or fracture surfaces can be mapped at various strain rates, and constitutive equations relating stress, strain, and strain rate can be derived for multi-axial stress deformation states.
3.4.1
Confined Pressure Device [14]
For confined pressure testing, a pressure chamber replaces the standard compression specimen package and can accommodate both tubular and solid cylindrical specimens. The machine frame equalizes the hydrostatic component of the axial load so that the driving cylinder must overcome only the material reaction to deformation. The axial load is measured by a strain-gaged load cell mounted inside the pressure vessel to eliminate friction load from the seals. Pressure is measured with diaphragm-type pressure transducers. Axial and tangent specimen strains are obtained from strain gages mounted on the specimen.
3.5
High Heating Rate Tests [16,17]
5.5.7
Heating and Testing Machine
Electrically conductive materials can be tested at high heating rates and medium strain rates on a modified medium-strain-rate machine. Direct resistance (I2R) heating at rates up to 104 °C/s and temperatures up to 4000°C were used to study the high heating rate properties of materials. Figures 3.8 and 3.9 show the device. Load is measured with semiconductor strain gages mounted on the water-cooled elastic load bar directly above the specimen. A plug-in strain gage monitor is used to record load and specimen displacement as a function of time. Through an electronic switching panel, either uniaxial load or bending load in two orthogonal directions can be measured. Heating is accomplished with a direct resistance method. An 800 amp, 50 volt regulated direct-current power supply, capable of 1200 amp outputs for periods up to one second, is connected to the test sample through copper electrodes. Specimen grip design allows temperature extremes up to 4000°C on graphite materials while maintaining
30
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.6. Biaxial Medium-Strain-Rate Machine Schematic
moderate temperatures (below 800°C) in the grip jaws and load bars. At temperatures above 2800°C, specimen radiation losses are minimized by surrounding the specimen grip area with a cast alumina and boron nitride shield into which a graphite felt and pyrolytic graphite liner is placed. Two small holes accommodate material outgassing and optical viewing of the specimen. To achieve precise and repeatable heating profiles from test to test, the power supply is coupled to a highspeed temperature controller, and predetermined heating rates and timeat-temperature are programmed.
3. Experimental Techniques
31
Figure 3.7. Biaxial Medium-Strain-Rate Machine
3.5.2
Temperature Measurement
Inherent in heating and testing materials at high temperature is the uncertainty in the determination of the material temperature. The technique used with the high-temperature medium-strain-rate machine utilizes pyrometry, temperature measurement of hot bodies using selfemitted radiation. Two optical pyrometers were used to measure temperature in the range of 20°C to 4000°C: one for temperatures from 20°C to 1000°C, the other in the range 800°C to 4000°C. Both units have response times fast enough to measure temperature changes near 5 x 103/s/°C and are calibrated for emissivity corrections.
32
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.8. High-Temperature Medium-Strain-Rate Machine Schematic
3.5.3
Strain Measurement
Conventional strain measuring techniques such as strain gages or mechanical extensometers suffer severe drawbacks when used at high temperatures or high loading rates. Even the most sophisticated commercially available optical tracking systems are inadequate at temperatures where gage marks will not adhere to the sample or when radiant glow from the specimen interferes with the tracker response. To circumvent these problems, an optic strain measuring technique was developed to measure small dynamic strains at temperatures up to 4000°C.
3. Experimental Techniques
33
Figure 3.9. High-Temperature Medium-Strain-Rate Machine Test Chamber
An argon CW laser (488 nm) scans the sample surface through a quartz window and is reflected onto a photomultiplier tube (PMT), as shown in Figure 3.10. By operating at this wavelength, radiant energy from the surface of the hot sample does not interfere with the laser signal. A telescope then transfers the image of the laser-illuminated gage section to the narrow-band PMT. Electronic processing of this signal results in determination of gage section displacement (and hence specimen strain) as a function of time. Laser scanning of the surface at approximately 40 KHz is provided through an acoustic-optic diffraction cell. Strains as low as 10"4 occurring in times of one millisecond or longer can be measured. Specimen gage sections consist of RF-sputtered thin films of refractory metals such as tantalum, molybdenum, or hafnium, deposited in thicknesses of approximately one micron.
34
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.10. Scanning Laser Extensometer Schematic
3.6
Ultrasonics Measurements
Measurement of elastic constants of materials provides necessary inputs to the study of material response. Three basic measurements are initial density (p0), longitudinal wave velocity (CO, and shear wave velocity (Cs). Ultrasonics measurements of wave velocities at standard temperature and pressure (i.e., 20°C and atmospheric pressure) establish the parameters used in first-order (temperature- and pressureindependent) calculations of dynamic material behavior. Measurement of the temperature and pressure dependence of the wave velocities permits more exact calculations to be made and, in particular, leads to prediction of pressure-compression isotherms, adiabats, and hydrostats. If the temperature and pressure dependence is measured with sufficient accuracy (0.1%), the calculated pressurecompression behavior can frequently be extrapolated with reasonable confidence to higher pressures than those covered in the actual measurements.
3. Experimental Techniques
35
3.6.1 Elastic Wave Velocities Pulse Superposition [18, 19] Pulse superposition techniques give high accuracy (-0.02%) for velocity measurements and offer ease of measuring a change in velocity due to a change in temperature or pressure. The technique is shown schematically in Figure 3.11. A transducer, of either longitudinal or shear mode, is bonded to one end of a specimen with parallel, flat faces. A short radio frequency (RF) burst, matched in frequency to the resonant frequency of the transducer, is applied. The transducer, which is both a transmitter and receiver, transmits an ultrasonic wave into the specimen. This wave reflects off the back surface of the specimen and is received as an echo by the transducer. If a second RF burst is applied at the exact time this first echo is received, etc., the echo amplitudes will superpose. In order to determine when exact superposition occurs, the RF bursts are interrupted and the decaying echoes are observed. The repetition rate of the RF bursts, which determines the time between pulses, is adjusted to give maximum echo amplitudes, i.e., exact superposition. The wave transit time through the specimen is then one-half the reciprocal of the oscillator frequency controlling the RF repetition rate.
Figure 3.11. Pulse Superposition Technique Schematic
36
Shock Waves: Measuring the Dynamic Response of Materials
Pulse Transmission [19, 20] The pulse transmission technique is used when attenuation of RF pulses is too high to give sufficient measurable echoes in the pulse superposition mode. The accuracy is somewhat less (-0.05%) than with pulse superposition. The technique is shown schematically in Figure 3.12. A pulse is applied to the transmitting transducer, passed through the specimen, and received at the transducer on the opposite face, delayed by the transit time through the specimen. A comparison pulse is matched to the shape of the received pulse and is in turn used to adjust the shape of the initial pulse. Once the initial, delayed comparison pulses have the same shape, the oscillator frequency is adjusted to obtain an exact multiple of comparison pulses between the initial and delayed pulses. The reciprocal of oscillator frequency times the number of comparison pulses equals the transit time through the specimen. Temperature and Pressure Dependence [21, 22] The temperature dependence of elastic wave velocities is determined by immersing the specimen and transducer in a constant temperature bath. Generally, the upper temperature limit is determined by the
Figure 3.12. Pulse Transmission Technique Schematic
3. Experimental Techniques
37
Figure 3.13. Static High-Pressure Apparatus Schematic
transducer/specimen bond material and is about 300°C for longitudinal waves and 100°C for shear waves. Both pulse superposition and pulse transmission methods can be used. Specimen length must be corrected for thermal expansion to obtain correct velocities. The static high-pressure apparatus is shown in Figure 3.13 and employs a Harwood Engineering Co. pressure intensifier system. The hydraulic system consists of two parts: a low-pressure portion which transmits pressure to the large piston in the intensifier, and a highpressure portion which transmits pressure to the pressure vessel containing the specimen. The usable portion of the pressure vessel is 2.5 cm diameter by 15 cm long, and the pressure range is 0-0.9 GPa. Since pressure dependence is determined under isothermal conditions, temperature of the working fluid is monitored. Both pulse superposition and pulse transmission methods are used, with the specimen length corrected for hydrostatic compression to obtain correct velocities.
38
Shock Waves: Measuring the Dynamic Response of Materials
Elastic Constants [23-25] Once the longitudinal and shear wave velocities have been determined, the elastic constants can be calculated. Assuming the material is isotropic, the adiabatic elastic constants at 20°C and zero pressure are obtained as follows: Poisson's Ratio
z/ = [0.5-(C s /C L ) 2 ]/[l.0-(C s /C L ) 2 ]
(3.1)
Bulk Wave Velocity CB
= VCL2-%CS2
(3.2)
C E = V2(1 + ^)C S2
(3.3)
Sound Wave Velocity
Rayleigh Wave or Surface Wave Velocity CR = k' - 8kj + (24 - 16a/)k/ + (16a/ - 16) = 0 where k 2 = ( C R / C s ) 2 , a* = (Cs / CLf
(3.4)
Bulk Modulus
K = pCl=p(c2L~%Cl)
(3.5)
Shear or Rigidity Modulus G = pCl
(3.6)
E = 2p(l + i/)C2
(3.7)
A = 2 / O T /C 2 /(l-2^)
(3.8)
Young's or Elastic Modulus
Lame's Parameter
39
3. Experimental Techniques
After CL and C s are measured as functions of temperature (T) and pressure (P), the data are corrected for thermal expansion and hydrostatic compression. The adiabatic bulk and shear moduli (K and G) can then be determined as functions of T and P, after correcting for change in density. Finally, for use in compressibility calculations, the isothermal values of K and G and the adiabatic and isothermal pressure derivatives are evaluated at ambient conditions (i.e., 20°C and zero pressure).
3.7
Equation of State and Wave Profile Studies
The development of models of dynamic material behavior and the calculational codes used in predicting material response requires extensive data on material properties under uniaxial strain conditions, including Hugoniot equation of state, wave propagation and spall fracture. Most data are used directly in developing models of material behavior, although independent check data (e.g., attenuated wave shapes and spall fracture profiles) are necessary to determine accuracy of the calculation. The Hugoniot is the locus of equilibrium states reached after shocking of a material. Data are usually obtained either as stress-particle velocity points from x-cut quartz or piezoresistive gages or as shock velocity-particle velocity points from optical techniques and may be expressed in several forms. A convenient form for experimental work is the Hugoniot centered about zero stress-particle velocity, as established by a least-squares fit to data in the stress-particle velocity {a - up) plane. Transformation of the Hugoniot into various planes, such as shock velocity-particle velocity (Us - up) and stress-volume (cr - V), is performed by the assumption of a material model. Frequently, an ideal elastic-plastic wave structure with equilibrium initial and final states is applied to the mass and momentum conservation equations. This leads to: ^H = ^HEL + Pe( U s ~
Ue)(Up
~
U c)
(3-9)
(3.10)
40
Shock Waves: Measuring the Dynamic Response of Materials
where CTHEL is the Hugoniot elastic limit, V is specific volume, and pe and ue are density and particle velocity at the elastic limit. Therefore, when either an - up or Us - up relations are established experimentally and OHEL, °t, ue, and CL are known, then
3.8
Gas Guns [26, 27]
Gun-launched, flat-plate impact techniques are used for generating uniaxial strain conditions required for equations of state and wave propagation studies. Gun techniques give close control over the stresstime history of the input pulse, permit change in the shape of the pulse, are compatible with the use of high-resolution diagnostic techniques, and facilitate recovery of specimens for post-impact analysis.
3.8.1
Compressed-Gas Gun, 102 mm
A 102 mm, single-stage, compressed-gas gun was used to launch stress-free flat plates at velocities up to 0.6 mm/us. The gun is shown in Figures 3.14 and 3.15. The piston head is seated by filling the rear chamber with about 0.5 MPa air, and the high pressure chamber is filled to the desired firing pressure. The gun is fired by releasing the air from the rear chamber and then rapidly introducing air into the firing chamber. The piston is forced to the rear, and the high-pressure gas drives the projectile down the barrel. Air is supplied from a three-stage, high-pressure
3. Experimental Techniques
Figure 3.14. Schematic of 102 mm Compressed-Gas Gun
Figure 3.15. 102 mm Compressed-Gas Gun
41
42
Shock Waves: Measuring the Dynamic Response of Materials
air compressor for velocities less than 0.4 mm/us, while bottled helium is used for higher velocities. The projectile used in the 102 mm gun has three basic parts: a Micarta sabot; a flat plate (impactor) bonded to the front of the sabot; and a flexible, polyurethane seal bonded to the rear of the sabot to prevent blow-by of the compressed gas. The effect of air buildup between the impactor and target is minimized by evacuating the barrel and target chamber prior to firing, and by providing an expansion chamber between the end of the barrel and the target assembly. The target is shock mounted to prevent premature motion before impact, and additional shock isolation is provided by mounting the high pressure chamber and the impact chamber on separate concrete pads. Special attention is given to achieving a planar impact. Impacting surfaces are generally lapped flat to less than 0.5 um over the region of interest. Before firing, the target is aligned with the projectile in the angular position it will have at impact. Planarity of impact is measured with tilt pins and is generally less than 5 x 10~4 radians. Impact velocity is measured to better than 0.2% with a shorting pin system in which charged wires are placed between the end of the launch tube and the target. These pins are at accurately measured locations along the flight line of the projectile. Upon being struck by the impactor support plate (which is grounded), each pin is shorted and the resultant signal stops a counter with 10 ns time resolution.
3.8.2
Compressed-Gas Gun, 63.5 mm
The 63.5 mm gun shown in Figures 3.16 and 3.17 is used for launching smaller projectiles at velocities up to 0.5 mm/us. The test chamber instrumentation consists of a velocity measurement system and a single-flash photography station. The velocity system includes a light source, a collimating lens, and five sets of equally spaced slits in the launch tube. These slits create five narrow, parallel light beams crossing the launch tube and emerging from holes on the opposite side, where they are focused by another lens onto the sensitive element of a photomultiplier tube. An opaque projectile passing through the launch tube successively interrupts these light beams, so that when the output of
3. Experimental Techniques
43
the photomultiplier is recorded with an oscilloscope, projectile velocity can be measured to within 1%. This gun was used primarily for spall testing. Projectile target alignment methods as well as elevated temperature techniques are described below in the section on spall tests.
Figure 3.16. 63.5 mm Compressed-Gas Gun Schematic
Figure 3.17. 63.5 mm Compressed-Gas Gun
44
Shock Waves: Measuring the Dynamic Response of Materials
3.8.3 Light-Gas Gun An accelerated-reservoir light-gas (ARLG) gun is used for velocities in the range of 0.6 to 8.1 mm/us. Launch tube diameters are 29 mm and 64 mm (3 mm/us maximum velocity for the larger launch tube). An ARLG gun maintains a reasonably constant pressure on the base of the projectile during launch, allowing a relatively gentle acceleration with negligible heating. The gun is shown in Figures 3.18 and 3.19 and operates as follows: 1.
A weighted, plastic-nosed piston is placed in the pump tube breech and the projectile is placed in the launch tube breech. 2. Gunpowder is loaded into the powder chamber and the pump tube is filled with hydrogen. 3. The gunpowder is ignited, propelling the piston down the pump tube, which compresses the hydrogen. 4. A high-pressure, burst diaphragm ruptures, letting the expanding hydrogen accelerate the projectile down the launch tube. The piston is stopped in a tapered section between the pump tube and the launch tube. Prior to firing, the launch tube and target chamber are evacuated and then flushed with helium to ~10~2 torr to eliminate effects of gas buildup and ionization between projectile and target. A plastic sabot is used to hold the projectile, and the rear of the sabot has sealing lips which are pressed against the inside of the launch tube by an interference fit to prevent blow-by of high-pressure gas. The ARLG gun can be modified by putting the powder chamber at the breech end of the launch tube, giving a single-stage gun. This permits launch velocities of 0.3 to 1.5
Figure 3.18. Accelerated-Reservoir Light-Gas Gun Schematic
3. Experimental Techniques
45
Figure 3.19. Accelerated-Reservoir Light-Gas Gun
mm/u.s for projectiles up to 64 mm diameter. The instrumentation chamber is shown in Figures 3.20 and 3.21. The launch tube is inserted into the chamber through an O-ring seal, and the chamber is shock-mounted to prevent target displacement before impact. Several ports are available for optical, X-ray, and instrumentation access. The impact velocity measuring system consists of a two-channel laser triggering system and short-duration (30 ns) flash X-ray pulsers. This system gives accuracy in velocity measurement of better than 0.05%. Pictures taken with the orthogonal flash X-rays also show projectile tilt and integrity. The control room for remote loading and firing of the ARLG and 102 mm guns is shown in Figure 3.22. The equipment shown includes an X-Y recorder for monitoring temperature vs. time; the control console for the flash X-ray system; 1 ns and 10 ns time-interval counters; gas loading and gun fire-control panel; and streak camera remote control unit.
46
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.20. Target Chamber Detail, ARLG Gun
Figure 3.21. Target Chamber, ARLG Gun
3, Experimental Techniques
47
Figure 3.22. Control Room
3.9
Instrumentation
3.9.1 X- Cut Quartz Gages X-cut quartz crystals generate a current when opposing faces are at different stress levels [29]. In shock wave applications, one face remains at zero stress while the face in contact with the specimen is subjected to a time-varying stress pulse. This condition lasts until the shock wave reaches the back face of the gage, at which time the useful recording ends. Measurement of the generated current as a function of time permits circulation of stress-time history. The quartz gage is generally used in two configurations: direct impact or transmitted wave, as discussed below. Direct Impact The direct impact technique is shown in Figure 3.23. At impact, one stress wave propagates into the quartz crystal and one into the impactor
48
Shock Waves: Measuring the Dynamic Response of Materials
or specimen. The current generated by the stress difference between the impact face and the rear face of the quartz is collected from an electrode in the center of the rear face of the crystal, with the impact face at ground potential. The current is passed through low-loss, foam-dielectric cable to digitizing oscilloscpes, and the piezoelectric current vs. time signal is recorded. Since quartz calibration data become non-linear above about 3 GPa, a variation in the above technique is used to obtain higher stress data. An elastic, high impedance material (usually tungsten carbide) is bonded to the front of the quartz gage and is impacted by the specimen. Because of the large difference in impedance of the buffer and the quartz, high stresses are generated in the specimen and the buffer without exceeding the well-calibrated stress range of the quartz. Tungsten carbide buffers, for example, reduce the stress induced in the quartz to about one-quarter that in the specimen, allowing measurements to -12 GPa. Transmitted Wave The technique for measuring the profile of a shock wave transmitted through a specimen is shown in Figure 3.24. An x-cut quartz crystal is placed on the back face of the specimen, and the assembly is impacted with an impactor either of the same material as the specimen (where, because of symmetry, up = vi/2) or an impactor of different material with known Hugoniot. A stress wave is generated at impact and propagates through the specimen to the interface with the quartz gage. Here the wave is partially reflected back into the specimen and partially transmitted into the quartz, as determined by stress and particle velocity continuity requirements. Data recording is essentially the same as with direct impact. Elevated Temperature For testing at elevated temperatures, the target is radiantly heated by a nichrome-wire heater element placed between the target and the launch
3. Experimental Techniques
49
Figure 3.23. Quartz Direct Impact Technique
Figure 3.24. Quartz Transmitted Wave Technique
tube, as shown in Figure 3.25. The system is shown both with the heater in position before firing and retracted for firing (with the target removed to show heater and projectile detail). The target is heated from the front face only to permit instrumentation or optical access to the rear, and
50
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.25. Target Heating System (Target Chamber Removed)
temperature is monitored by a thermocouple. Heater element voltage is adjusted to bring the target to temperature in -10 minutes, and the target is then allowed to stabilize at test temperature for 5 to 10 minutes. The target and impactor are aligned at room temperature for planar impact. The target alignment is monitored during heat-up by an optical-lever system in which a laser beam is reflected off the rear face of the target and displayed, through a series of mirrors, on a screen in the control room. If the target position changes during heating, a remote control system using DC-drive motors allows realignment by rotating the target assembly about either of two axes. Tilt at impact is comparable to that achieved at room temperature. Although designed primarily for quartz gage testing, the heating system can be used with any type of instrumentation, provided the transducer, if any, can withstand exposure to elevated temperature, or if the rear surface remains reflective if optical techniques are being used. Velocity interferometer measurements at elevated temperature were easily accomplished with this technique.
3. Experimental Techniques
51
3.9.2 Manganin Gage [30] The manganin gage is one of a series of gages which utilize the piezoresistive properties of certain materials to give stress-time histories in impacted specimens. The gage has been used from below 0.1 GPa to over 40 GPa and can serve either as an in-material gage or as an interface stress gage. The gage can be used in a conducting material, but the active element must be electrically isolated, making the response time proportional to the gage plus insulation thickness. The manganin gage scheme and record from a high-pressure Plexiglas target test is given in Figure 3.26. The element can be wire or foil in various configurations with resistance ranging from 1 to 50 ohms. A constant-current source is discharged through the gage prior to impact, and voltage and voltage across the gage are monitored. A stress wave will change the gage resistance, which is recorded as a voltage-time trace. The record is reduced using the known piezoresistive coefficient for manganin. As with other "area-averaging" transducers, the effect of tilt must be considered in analyzing the results.
Figure 3.26. Manganin Gage Technique
52
3.9.3
Shock Waves: Measuring the Dynamic Response of Materials
Streak Camera Techniques [31]
Streak camera techniques provide equation of state data in the form of shock velocity-particle velocity points and wave profile data in the form of free surface or interface motion as a function of time. A Beckman and Whitley Model 339B streak camera was used for the three basic applications: shock arrival measurements (hat target), free surface velocity (shim target), and free surface motion (wedge target). "Hat" Target To measure shock velocity in an impacted specimen, targets were constructed in the form of a two-step cylinder ("hat" configuration) with shock arrival mirrors placed at two levels, as show in Figure 3.27. (If the surface of the target material can be polished to give good reflectivity, the mirrors can be eliminated.) The impactor and target are generally of the same material to give Up = Vi/2. At impact a shock is generated and traverses the target to the rear surfaces. The two surfaces are observed through the slit of a streak camera with a xenon flash tube used for lighting. The shock arrival at each level is recorded on film as a change in reflectance. The time required for shock wave travel from the first level to the second is calculated from the film record, knowing the camera sweep rate. This time and the known target dimensions give average shock wave velocity. The film record also provides a measure of shock tilt, which is used for transit time correction. Multiple shock waves can frequently be
Figure 3.27. "Hat" Target Technique
3. Experimental Techniques
53
observed by slight changes in mirror reflectivity as the waves arrive at different stress levels. "Shim" Target The shim target technique was used to measure velocity of a shim (~0.01 mm thick), initially weakly bonded to the rear surface of the target, as it crosses a gap of known width, as shown in Figure 3.28. Shims were used rather than the specimen surface itself to insure that, if the surface decelerated due to shock attenuation, the shim continued to move with maximum free surface velocity. The strength of the bond between the shim and target does not measurably influence shim motion. Attenuation targets were constructed with several steps to provide measurements of several propagation distances from one experiment. Upon arrival of the impact-generated shock wave at the rear surface, the reflectance of the shim surface changes. The shim then crosses the gap and strikes the mirror, giving an abrupt change in mirror reflectance. The gap transit time and the known gap size yield free surface velocity. Shock velocity and shock tilt were also obtained by including data from the side mirrors. Measurements of maximum free surface velocity attained by the specimen surface are estimated to be accurate to 2%.
Figure 3.28. Shim Target Technique
"Wedge" Target The wedge target technique illustrated in Figure 3.29 is used primarily for studying shock wave attenuation. The impactor and target dimensions were selected such that in the thinner portion of the wedge the shock
54
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.29. Wedge Target Technique
wave has not attenuated and constant shock velocity is obtained, while in the thicker part of the wedge the release wave from the rear of the thin impactor has started to attenuate the wave, leading to a decrease in average shock velocity. The shock velocity is proportional to the slope of the streak camera record and the arrival of the release wave and the subsequent attenuation can be deduced from changes in this slope. Tilt data were obtained by use of side mirrors and were used to correct the measured shock velocities. Since the record provides a continuous measure of shock velocity vs. x/xo, rather than at discrete x/x0 points, as provided by the shim technique, the wedge system is frequently used to measure the exact distance at which the rarefaction wave overtakes the shock wave.
3. Experimental Techniques
55
3.9.4 Laser Velocity Interferometer [32] The laser velocity interferometer measures surface velocity as a function of time. Either free surface motion or motion of the interface between the specimen and a transparent "window" material can be measured with high spatial and time resolution. Three major types of interferometers have been developed for use with shock wave studies. In order of their development, they are: • • •
Displacement interferometer for specular surfaces Velocity interferometer for specular surfaces Velocity interferometer for diffuse surfaces (VISAR)
The VISAR interferometer is the most versatile of the three, having the widest velocity range and the capability of measuring not only the motion of surfaces under one-dimensional motion but also both longitudinal and shear wave motions. Also, this type of interferometer is used to measure large surface displacements, such as the motion of projectiles down barrels. A detailed description of the development of VISARs is presented in Chapter 8. The laser velocity interferometer system shown in Figure 3.30 is of the second type and was used with specimens with polished rear surfaces. A sample record and the interferometer details are shown in Figure 3.31. The system consists of a single-frequency (632.8 nm) He-Ne laser, a series of mirrors directing the beam to the target, a lens to focus the beam onto the surface being observed, a second series of mirrors to direct the reflected beam to a beam splitter (which sends half the light around an adjustable delay leg), and a photomultiplier tube (PMT) with 0.8/ns risetime and 300 MHz frequency response. A fringe pattern is developed after the two beams recombine at the beam splitter and is focused on the PM tube. The fringe pattern will change as the target surface moves as a result of the Doppler shifting of the frequency of the beam reflected from the moving surface of the specimen. The number of fringes seen by the PM tube per unit time is proportional to the change in velocity of the surface.
56
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.30. Laser Velocity Interferometer Technique
Figure 3.31. Laser Velocity Interferometer Schematic
3. Experimental Techniques
57
The proportionality constant or sensitivity of the interferometer is controlled by the length of the delay leg, i.e., the longer the delay leg, the more fringes are obtained for a given change in velocity. The PMT outputs typically are recorded on high speed (1 ns/point minimum) digitizers. The voltage-time trace is then reduced to yield the velocity-time history of the moving surface, using appropriate interferometer constants (laser wavelength, delay leg length, etc.). A projectile and velocity interferometer target are shown in Figure 3.32. The sabot and target construction are typical for the 102 mm compressed-gas gun system.
3.9.5
Slanted Resistance Wire [33]
The slanted resistance wire (SRW) technique was used in obtaining the distance-time history of the motion of a conducting free surface. The technique can be used over a wide stress range (<0.1 GPa to >10 GPa) and, since it gives an analog record with good spatial resolution, it can provide data on material anisotropy parallel to the wave front.
Figure 3.32. Projectile and Velocity Interferometer Target
58
Shock Waves: Measuring the Dynamic Response of Materials
Figure 3.33. Slanted Resistance Wire Technique
Operation of the SRW transducer is shown in Figure 3.33. A thin (12um) platinum ribbon is placed at a small angle to the rear surface of the specimen, with one end of the ribbon touching or slightly above the surface and the other end held on the order of 0.1 mm further from the surface. A constant-current power supply is discharged through the wire just prior to impact. As the rear surface moves under the action of the incident stress wave, the ungrounded length decreases. The corresponding decrease in resistance of the ribbon is recorded by a digitizing oscilloscope as a decrease in voltage. Data reduction includes corrections for non-linearity in voltage vs. distance and for tilt. Velocity-time is then obtained by differentiation of the distance-time record.
3.9.6 Magnetic Wire Gage [34] The magnetic wire gage provides a direct measure of particle velocity in a non-conducting material as a function of time. The range of velocities over which the gage can be used is essentially limited only by the noise level and response time of the electronics used to monitor the gage,
3. Experimental Techniques
59
provided gage integrity is maintained. The gage is normally used in the "sandwich" configuration, with risetime proportional to gage thickness. The technique is shown schematically in Figure 3.34 with a representative record from a fused quartz target (which has a dispersive or ramped compressive wave). A thin (-10 (Am) copper foil is placed in the target, which is then placed in a magnetic field such that motion of the wire will cut the lines of force. A stress wave in the target will cause the wire to move at the material particle velocity, giving an induced voltage of: V = B • Up • d where B is the magnetic flux density, up is particle velocity, and d is gage length. Typical values of B and d are 0.1 Tesla and 6 mm, respectively. The digitized record is reduced to give a particle velocity-time history. Since the gage measures over a length, the data must be corrected for shock wave tilt.
Figure 3.34. Magnetic Wire Technique
60
Shock Waves: Measuring the Dynamic Response of Materials
3.10 Spall Tests Spall fracture by plate impact results from reflection of compressive stress pulses from a relatively low impedance interface (normally a free surface) and their subsequent interaction. Spall fracture studies can be carried out using both active and passive techniques. Active techniques provide quantitative, time-resolved data on the influence of internal fractures or spall surfaces on shock wave profiles [35]. The primary active instrumentation used is the laser velocity interferometer, which provides a time-history of surface motion. The relation between rear surface motion and spall plane growth is shown in Figure 3.35, and representative results from a spall test are given in Figure 3.36. Passive techniques involve the recovery and examination of shockloaded specimens. Subsequent metallographic examination establishes the type and degree of damage, which can be correlated with impact parameters such as velocity, impactor thickness, and target thickness.
Figure 3.35. Wave Interactions in Spall Test
3. Experimental Techniques
61
Figure 3.36. Velocity Interferometer Spall Test Results
3.10.1 Recovery Tests [26, 36, 37] The spall behavior of metals is generally studied by carrying out a series of impact and recovery tests, where the target specimen is sectioned across a diameter, polished, etched, and examined optically at a magnification of 50 to 100X. The specimen is then graded or classified according to the degree of fracture that has occurred, which can range from no observable fractures to complete material separation. The incipient spall threshold is defined as the impact velocity (for a given set of test parameters) corresponding to the onset of microfracture. One could also establish a complete fracture or separation, but this is generally not recommended as a primary spall criterion, since it is more likely to be influenced by edge effects (loss of one-dimensional strain conditions) and by recovery-induced damage.
62
Shock Waves: Measuring the Dynamic Response of Materials
3.10.2 Room Temperature Testing Spall recovery tests were conducted using the test setup shown in Figure 3.37. The impactor is bonded to a plastic sabot with epoxy resin. The back of the sabot is closed with a plastic disc, and the sabot is vented to allow pressure equalization when the barrel is evacuated. The target is mounted in a urethane alignment ring and placed in the mouth of the sabot stripper. To position the target, the sabot assembly is forced against the target until it slips, bringing the impactor and target into alignment. Verification of this scheme with the alignment fixture described below indicates that the two methods are comparable. Non-planarity of impact of approximately 10~3 radians is maintained, sufficient to regard tilt as unimportant in the interpretation of results. The front lip of the sabot stripper is slightly smaller in diameter than the sabot shell but larger than the target diameter. This allows the target and impactor to enter the recovery area within the tube but prevents any pieces of the sabot shell from entering and causing post-impact damage. The recovery tube is filled with an energy-absorbing material.
Figure 3.37. Room Temperature Spall Test Schematic
3. Experimental Techniques
63
3.10.3 Elevated Temperature Testing Elevated temperature spall tests utilized the furnace shown in Figure 3.38 (with one heater element removed for clarity), which provided radiant heating in the vacuum environment of the target chamber. The heating elements are spiral-wound resistance wires mounted in retractable doors. The targets are supported in the furnace on four ceramic pins and are heated uniformly from both sides. The specimen temperature is monitored at the nominal spall plane near the periphery, and the thermocouple output is recorded digitally. The voltage to the heating elements is adjusted to bring the target to test temperature in approximately 10 minutes, and the target is then allowed to stabilize at this temperature for an additional 5 to 10 minutes. Caution must be exercised in the choice of heating and stabilization times. Studies of several materials heated in times from milliseconds to hundreds of hours have shown marked decreases in yield strength occurring on the order of seconds to minutes, due to diffusion mechanisms [38]. Misleading results can be obtained if tests are conducted after softening has occurred and the test results are then applied to material failure where the heating occurs on a shorter time scale, for instance, under the influence of energy deposition.
Figure 3.38. Spall Test Furnace (One Door Removed)
64
Shock Waves: Measuring the Dynamic Response of Materials
Alignment of the impactor and target is made using an electrical fixture with three pins equally spaced on a 32 mm diameter circle, each pin consisting of two hardened steel dowels with a piece of 0.025 mm Mylar between them for electrical insulation. The outer ends of each pin are radiused and their lengths have been adjusted to define planes parallel within 2 um. Each pin segment is connected to an indicating light through a high impedance circuit which prevents arcing and heating between the contact point and the target. The actual alignment is accomplished by inserting the fixture between the target and impactor, with a slight pressure applied to the rear of the sabot. The furnace, which is mounted on a spherical bearing and is free to move with two degrees of freedom (sufficient to fix a plane in space), is then moved with micrometer screws until all six indicating lights (three for the impactor and three for the target) are lit.
3.10.4 Metallographic Examination Conventional light microscopy is the most common way of examining recovered spall specimens. A Zeiss Ultraphot II research metallograph was used which provided 8 to 2500X magnification, with bright field, dark field, polarized, or phase contrast illumination. A typical series of photomicrographs obtained for aluminum is shown in Figure 3.39. The level of damage for each impact velocity is indicated by the symbols at the left of each picture. These symbols are defined as follows: • Complete Separation © Above Incipient ® Incipient Spall Q Below Incipient O No Cracks at 50X Scanning electron microscopy (SEM) is also used in the study of spall fractures and fracture surfaces. SEMs with 20X to 105X magnification provide much greater depth of field and resolution than is possible with optical microscopy and permits detailed examination of fracture surfaces. SEM micrographs of sectioned spall specimens (similar to those in Figure 3.39) are shown in Figure 3.40, and micrographs of fracture surfaces from completely separated specimens are shown in Figure 3.41.
3. Experimental Techniques
Figure 3.39. Spall Fractures in 6061-T6 Aluminum. Optical Micrographs
Figure 3.40. Spall Fractures in 6061-T6 Aluminum. Scanning Electron Micrographs
Figure 3.41. Spall Fracture Surfaces. Scanning Electron Micrographs
65
66
3.11
Shock Waves: Measuring the Dynamic Response of Materials
References
1. Isbell, W. M., D. R. Christman, S. G. Babcock, T. E. Michaels, and S. J. Green. "Measurements of Dynamic Properties of Materials, Vol. I: Summary of Results." DASA-2501-1, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD712847), 1970. 2. Christman, D. R., W. M. Isbell, S. G. Babcock, A. R. McMillan, and S. J. Green. "Measurements of Dynamic Properties of Materials, Vol. Ill: 6061-T6 Aluminum." DASA-2501-3, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971. 3. Christman, D. R., T. E. Michaels, W. M. Isbell, and S. G. Babcock. "Measurements of Dynamic Properties of Materials, Vol. IV: Alpha Titanium." DASA-2501-4, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971. 4. Christman, D. R., W. M. Isbell, and S. G. Babcock. "Measurements of Dynamic Properties of Materials, Vol. V: OFHC Copper." DASA-2501-5, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971. 5. Isbell, W. M., D. R. Christman, and S. G. Babcock. "Measurements of Dynamic Properties of Materials, Vol. VI: Tantalum." DASA-2501-6, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971. 6. Clark, D. S., and D. S. Wood. "The Time Delay for the Initiation of Plastic Deformation at Rapidly Applied Constant Stress." ASTM Proc, vol. 49, 717-737 (1949). 7. Campbell, J. D., and K. J. Marsh. "The Effect of Grain Size on the Delayed Yielding of Mild Steel." Phil. Mag., vol. 7, 933-952 (1962). 8. Maiden, C. J., and S. J. Green. "Compressive Strain-Rate Tests on Six Selected Materials at Strain Rates from 10"3 to 104 in/in/sec." ASME Trans. J. Appl. Mech., Ser. E, vol. 33, 496-504 (1966). 9. Kolsky, H. "An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading." Phys. Soc. Proc. B, vol. 62, 676-700 (1949). 10. Davies, E. D. H., and S. C. Hunter. "The Dynamic Compression Testing of Solids by the Method of the Split Hopkinson Bar." J. Mech. Phys. Solids, vol. 11,155-179,196. 11. Babcock, S. G., and R. D. Perkins. "High Strain-Rate Response of Three Heat-Shield Materials at Elevated Temperatures." SAMSO-TR-68-71, vol. II, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD842604L), 1968. 12. Green, S. J., J. D. Leasia, R. D. Perkins, and C. J. Maiden. "Development of Multi-axial Stress High Strain-Rate Techniques." SAMSO-TR-68-71, vol. III, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD847163L), 1968.
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13. Perkins, R. D., A. H Jones, S. J. Green, and J. Leasia. "Multi-axial Loading Behavior of Four Materials Including ATJ-S Graphite and RAD-6300 Carbon Phenolic." SAMSO-TR-69, vol. I, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD874866L), 1969. 14. Perkins, R. D., A. H. Jones, and S. J. Green. "Determination of Multi-axial Stress Behavior of Solenhofen Limestone and Westerly Granite." DASA-2438, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1970. 15. Schierloh, F. L., R. D. Chaney, and S. J. Green. "An Application of Acquisition and Reduction of Data of Variable Short Time Tests." Exp. Meek, vol. 10, 23N-28N (1970). 16. Babcock, S. G., P. A. Hochstein, and L. J. Jacobs. "High Heating Rate Response of Two Materials from 72° to 6000°F." SAMSO-TR-69-393, vol. II, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD867427L), 1969. 17. Babcock, S. G., and P. A. Hochstein. "High Strain-Rate Testing of Rapidly Heating Conductive Materials to 7000°F." Exp. Mech., vol. 10, 78-83 (1970). 18. McSkimin, H. J. "Pulse Superposition Method for Measuring Ultrasonic Wave Velocities in Solids." Acoust. Soc. Am. J., vol. 33, 12-16 (1961). 19. Lingle, R., J. R. Havens, and A. H. Jones. "Ultrasonic Velocity Measurements in Nodular Iron Castings." MSL-69-49, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1969. 20. Mattaboni, P., and E. Schreiber. "Method of Pulse Transmission Measurements for Determining Sound Velocities." /. Geophys. Res., vol. 72, 5160-5163 (1967). 21. McSkimin, H., and P. Andreatch. "Analysis of Pulse Superposition Method for Measuring Ultrasonic Wave Velocities as a Function of Temperature and Pressure." Acoust. Soc. Am. J., vol. 34, 609-615 (1962). 22. Asay, J. R., D. L. Lamberson, and A. H. Guenther. "Pressure and Temperature Dependence of the Acoustic Velocities in Polymethylmethacrylate." J. Appl. Phys., vol. 40, 1768-1783 (1969). 23. Kolsky, H. Stress Waves in Solids, S1098. New York: Dover Publications, Inc., 1963. 24. Mason, W. P. "Acoustic Properties of Solids." American Institute of Physics Handbook, 2d ed., 3.22-3.97 (1963). 25. Barsch, G. R., and Z. P. Chang. "Adiabatic, Isothermal, and Intermediate Pressure Derivatives of the Elastic Constants for Cubic Symmetry." Phys. Stat. Sol, vol. 19, 129-151 (1967). 26. Christman, D. R., N. H. Froula, and S. G. Babcock. "Dynamic Properties of Three Materials, Vol. I: Beryllium." MSL-68-33, vol. I, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1968.
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27. Isbell, W. M., F. H. Shipman, and A. H. Jones. "Hugoniot Equation of State Measurements for Eleven Materials to Five Megabars." MSL-68-13, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD721920), 1968. 28. Ingram, G. E., and R. A. Graham. "Quartz Gauge Technique for Impact Experiments." SC-DC-70-4932, Sandia Laboratories, 1970. 29. Chin, H. C. "QZ: A Computer Program to Analyze X-Cut Quartz Data Obtained from Shock Loading." MSL-70-15, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1970. 30. Keough, D. D. "Procedure for Fabrication and Operation of Manganin Shock Pressure Gages." AFWL-TR-68-57, Stanford Research Institute (AD839983), 1968. 31. McMillan, A. R., W. M. Isbell, and A. H. Jones. "High Pressure Shock Wave Attenuation." DASA-2425, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971. 32. Barker, L. M. "Fine Structure of Compressive and Release Wave Shapes in Aluminum Measured by the Velocity Interferometer Technique." Behavior of Dense Media under High Dynamic Pressures, 483-505. New York: Gordon and Breach, 1968. 33. Barker, L. M., and R. E. Hollenbach. "System for Measuring the Dynamic Properties of Materials." Rev. Sci. Instr., vol. 35, 742-746 (1964). 34. Dremin, A. N., and G. A. Adadurov. "The Behavior of Glass Under Dynamic Loading." Sov. Phys. Solid State, vol. 6,1379-1384 (1964). 35. Isbell, W. M., and D. R. Christman. "Shock Propagation and Fracture in 6061-T6 Aluminum from Wave Profile Measurement." DASA-2419, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation (AD705536). 36. Lundergan, C. D. "Spall Fracture." AST-TDR-63-140, Proc. Symposium on Structural Dynamics Under High Impulse Loading, 357-358 (AD408777), May 1963; also Sandia Corporation Report SCDC-2845. 37. Warnica, R. L. "Spallation Thresholds of S-200 Beryllium, ATJ-S Graphite and Isotropic Boron Nitride at 75 °F, 500 °F and 1000 °F." MSL-68-18, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1968. 38. Babcock, S. G., J. J. Langan, D. B. Norvey, T. Michaels, and F. L. Schierloh. "Characterization of Three Aluminum Alloys." AMMRC-CR-71-3, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, 1971.
4. Dynamic Response of Materials at Low and Moderate Stresses (<20 GPa) 4.1
Summary of Results
4.1.1
Introduction
Detailed physical models form the basis of modern capabilities to describe the propagation of shock waves and their effects on materials in the important regime where strength effects dominate material behavior. While it is the ultimate goal to develop models based on physical first principles, constitutive relations are based, to a large degree, on insights gained from measurements. An experimental study was conducted to provide measurements upon which several models could be developed. The principal goals of the research program were: •
• •
To provide measurements of dynamic properties of materials to be used as inputs to model development for shock wave propagation and dynamic fracture. To provide data to establish accuracy of computer code predictions. To provide physical interpretation of experimentally observed material response to dynamic test conditions.
The materials chosen for study were four metals with widely varying response to shock waves. Selection criteria included:
69
70
Shock Waves: Measuring the Dynamic Response of Materials
• • • • •
Strain-rate sensitivity Lattice structure Density Strength and fracture characteristics Possible use as a structural material.
The materials selected were 6061-T6 aluminum, commercially pure tantalum, alpha phase titanium, and OFHC copper. Photomicrographs of the materials are shown in Figure 4.1.
Figure 4.1. Metals, As-Received Condition
The first two materials, aluminum and tantalum, were extensively characterized, while measurements on copper and alpha titanium were directed towards obtaining only that data necessary for models under development. A literature review was made in support of the experimental effort, and a compilation of references, including abstracts applicable to this work, is given in Reference 1. In this chapter, the results of the tests on the four materials are summarized. Section 4.2 details the measurements performed on tantalum and describes the experimental and analytical techniques utilized. Detailed descriptions of tests conducted on the remaining three materials are presented in References 2-4. The program plan for experimental work is shown in Figure 4.2. The plan is divided into areas which furnish necessary data for a particular
4. Dynamic Response of Materials at Low and Moderate Stresses
71
phase of the theoretical efforts. For instance, modeling of compressive wave behavior utilizes information from stress-strain behavior under uniaxial stress, from measurement of precursor decay, from the Hugoniot, and from the determination of conditions under which steadystate waves may be propagated. Release wave behavior is modeled from measurements of wave profiles, the material's unloading path, and strainand time-dependent measurements of the Bauschinger effect.
Figure 4.2. Outline of the Experimental Program
In addition to shock wave measurements, the equations of state were calculated from the measured elastic behavior of the materials. In this study, elastic constants were measured as functions of pressure and temperature. Hydrostats, adiabats, and isotherms were then calculated. Dynamic fracture was studied with measurements of spallation thresholds as functions of loading time and wave shape. The materials selected exhibited different fracture modes and were studied with optical microscopy and scanning electron microscopy. Finally, a study was made of degradation of material properties by elevated temperature, and a brief study was conducted on aluminum of the effects of pre-shocking on yield strength and spall threshold.
72
Shock Waves: Measuring the Dynamic Response of Materials
To obtain definitive checks on the ability of material models to predict accurately wave propagation and fracture, check data was obtained independently and consisted primarily of shock attenuation and wave forms of arbitrary shape against which the models could be exercised.
4.1.2
Compressive Wave Behavior
The behavior of materials under uniaxial and multi-axial stress conditions is the basis for development of constitutive equations relating stress to strain and strain rate. Primary measurements are of uniaxial compressive stress vs. strain at various strain rates. These tests give yield and flow stress behavior, strain-rate sensitivity, and work-hardening characteristics. The measurements are sometimes extended to uniaxial tension, and testing in orthogonal directions is necessary if material anisotropy is of importance. Stress-strain behavior of the metals is shown in Figure 4.3. At moderate rates, 6061-T6 aluminum is a relatively rate-insensitive alloy. The other materials show varying sensitivity to rate of loading, including yield drop in titanium and tantalum. Copper (1/2-Hard) shows a welldefined yield in uniaxial stress, but has a dispersive elastic wave structure in uniaxial strain.
Figure 4.3. Compressive Behavior in Uniaxial Stress
4. Dynamic Response of Materials at Low and Moderate Stresses
73
The development of models of dynamic material behavior and the calculational codes used in predicting material response requires extensive data on material properties under uniaxial strain conditions. This includes Hugoniot equations of state, compressive wave structure, elastic precursor behavior, release wave characteristics and spall fracture data. Compressive wave development and decay of the elastic precursor are used as basic input data for models of material behavior. In Figure 4.4, the dispersive nature of the plastic wave in tantalum is seen as the wave propagates from 2 to 15 mm. A decrease in amplitude of the elastic precursor with increasing distance of propagation is evident. On the right side of the figure, the precursor decay behavior for the four metals is summarized.
Figure 4.4. Elastic Precursor Behavior
The behavior of tantalum, titanium, and aluminum is somewhat similar, where the precursor amplitude decays rapidly during the first 5 mm, before relaxing to a stable value. Behavior of the precursor in copper is complicated by the dispersive nature of both the elastic and plastic waves. Because of this, it is difficult to assign single values to the precursor decay. Primary equation of state data was obtained as stress-particle velocity points using x-cut quartz gages, or as shock velocity-particle velocity points using streak-camera techniques. The Hugoniots thus determined are shown in Figure 4.5. Shock wave measurements were
74
Shock Waves: Measuring the Dynamic Response of Materials
made and analyzed, assuming steady-state wave behavior. There is some uncertainty in describing the stress region up to two or three times the elastic limit, where time-dependent processes cause non-steady-state wave behavior. The dashed portion of the titanium curve is based on evidence of a phase transition occurring at about 10 GPa.
Figure 4.5. Stress-Particle Velocity Hugoniots
Figure 4.6 summarizes the results of a study of compressive waves in aluminum and augments a detailed study of the formation of steady-state waves conducted at Sandia Corporation [2]. In an elastic-plastic wave system, the effects of rate dependency and non-linearity initially cause the plastic portion of the wave to spread. Eventually, the plastic portion can achieve a stable condition and the wave propagates relatively unchanged. The series of tests at 2.6 GPa shows increasing risetime in the plastic wave between 3 mm and 12 mm, with wave stability achieved by 20 mm. The series at 3 mm thickness shows a continually steepening plastic
4. Dynamic Response of Materials at Low and Moderate Stresses
75
Figure 4.6. Compressive Waves in 6061-T6 Aluminum
Figure 4.7. Steady-State Wave Characteristics in 6061-T6 Aluminum
wave with increasing stress. A more complete mapping of the formation of steady-state waves in aluminum is shown in Figure 4.7. Tests made at a given stress which exhibited steady-state behavior in the plastic portion of the wave are indicated by the solid symbols. The broad path shows the indistinct border between non-steady-state and steady-state behavior. A
76
Shock Waves: Measuring the Dynamic Response of Materials
study such as this has important implications when one is applying various forms of analyses to experimental data.
4.1.3
Elastic Behavior
Measurement of elastic constants of materials provides necessary inputs to the study of material response. The three basic parameters are density, longitudinal (dilatational) wave velocity and shear (transverse) wave velocity. Wave velocities were determined by measuring wave transit times through specimens of known thickness, using pulse superposition and pulse transmission techniques. Ultrasonics measurements of wave velocities at standard temperature and pressure (i.e., 20°C and atmospheric pressure) establish the inputs used in first-order (temperature- and pressure-independent) calculations of dynamic material behavior. Most important are elastic wave velocities, Poisson's ratio, bulk modulus, and the modulus. Measurement of the temperature and pressure dependence of the three basic parameters permits more exact calculations to be made and, in addition, leads to prediction of pressure-compression isotherms, adiabats, and Hugoniots. Results from the elastic wave velocity measurements are given in Table 4.1. If the temperature and pressure dependence is measured with sufficient accuracy (1 part in 103 or better), the calculated pressurecompression behavior can be extrapolated with reasonable confidence to pressures 1 to 2 orders of magnitude higher than that actually used for the measurements. Table 4.1. Elastic Wave Velocities MATERIAL
TEMPERATURE DEPENDENCE o-ioo"c, p = o
~L
I
PRESSURE DEPENDENCE o-io m>. T = an°c
FB
\
1
S
Allialmm (6O61-T6)
6 387-9.3x10 ~4T
3.214-8.4xlO~\
6.368)0.01BOP -1.0xl0~ 4 P 2
3. 10710 010BP -O.SxlO"4P2
Titanium (Alplia)
6.135-8.3xlO~ 4 T
3.261-7.6xl0~ 4 T
6. HBlO. 0056P
3.246|O. 00104P
Copper (OFIIC)
4.766-4.3x10"^
2.253-3.Oxlo" 4 T
4.757IO.OO685P
2.217)0 0024.11'
TuntalliK (CDMH. Pure)
4.150-1.46x10"^
2.036-1 .73x10
4. 14610.OO291P
2.03240.00137P
4T
4. Dynamic Response of Materials at Low and Moderate Stresses
11
4.1.4 Release Wave Behavior A complete understanding of wave propagation requires study of release wave behavior. Most previous studies have been conducted using impactors of the same material as the target. Thus, the measured release wave behavior is irretrievably linked with the compressive wave behavior in the impactor. An alternate and better technique for the study of release waves is shown in Figure 4.8, in which a square input wave has been induced in the material by the impact of a fused quartz disc. Fused quartz exhibits shock compressive behavior in the stress regime of -0-3 GPa in which the Hugoniot is concave downward, in contrast to most materials where the derivative of the stress-particle velocity curve is positive. The result is that, upon reflection of a shock wave from a free surface, a rarefaction shock is formed, rather than the customary rarefaction fan. Impact of a fused quartz disc on a sample thus produces a nearly instantaneous release wave at the impactor-specimen interface, allowing formation of release waves that are not related to the shape of the compressive wave. In the compressive portion of the wave in the sample, the elastic precursor decay is evident as the shock wave progresses into the material. On the release portion of the wave, the behavior of the elastic-plastic wave system can be followed in detail. The waves are dispersive in nature and decay of the elastic release wave is indicated. The profiles in Figure 4.8 have been adjusted to show the same maximum interface velocity and to coincide at release wave arrival time. Modeling the unloading wave requires knowledge of subsequent yield behavior of the material. Shown in Figure 4.9 are results of a study of the Bauschinger effect and its dependence upon strain and aging time. The Bauschinger effect can be evaluated in terms of the ratio of yield in tension on reload over the yield in compression on preload. For 6061-T6 aluminum, this ratio changes at low strains (< 2%), showing a decrease with increasing strain. The Bauschinger effect is also time dependent, and aluminum shows reload yield recovery as time between preloading in tension and reloading in compression is increased.
78
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.8. Wave Profiles in 6061-T6 Aluminum
Figure 4.9. Bauschinger Effect in 6061-T6 Aluminum
4. Dynamic Response of Materials at Low and Moderate Stresses
4.1.5
79
Fracture
Spall fracture usually results from the reflection of compressive stress waves from relatively low-impedance interfaces (normally a free surface) and their subsequent interaction. Analysis of spall fracture in metals requires consideration of a number of factors, most important of which are material properties, stress-time history, and test technique (i.e., stress pulse generation method). The spall fracture data were obtained by flat-plate impact, where uniaxial strain conditions exist. These studies were conducted using both active and passive methods. Active techniques provide quantitative, time-resolved data on the influence of internal fracture or spall surfaces on stress wave profiles [3]. The primary instrumentation used is the laser velocity interferometer, which provides a time history of free surface motion. Passive techniques involve the recovery and examination of shock-loaded specimens. Metallographic examination establishes the type and degree of damage, which can be correlated with impact parameters such as velocity, impactor thickness, and target thickness. A qualitative assessment of the spall behavior of the four metals is given in Figure 4.10. Each material exhibited ductile failure, although the nature of the plastic flow accompanying void formation differs. The aluminum and titanium specimens were taken from plate stock, and void growth and coalescence were influenced by the grain structuring resulting from the rolling process. There was very little dispersion of voids around the spall plane in the copper, while the tantalum showed sizable voids at appreciable distances from the nominal spall plane.
Figure 4.10. Spall Fractures
80
Shock Waves: Measuring the Dynamic Response of Materials
Spall tests were conducted using quasi-rectangular waves arising from symmetric impact. To study the dependence of the spall thresholds on time duration, tests were conducted using different impactor thicknesses to provide different pulse widths. For these tests, the ratio of impactor to target thickness was kept constant. A summary of the impact velocity necessary to create incipient spall is shown in Figure 4.11.
Figure 4.11. Spall Threshold Results
4.1.6
Degraded Properties
For the formulation of complete constitutive relations, it is important to establish the influence of material temperature as well as time-attemperature on strength, stiffness, and other material properties. The effect of heating rate on yield strength of 6061-T6 aluminum is shown in Figure 4.12. The yield at 20°C is compared with yield at 260°C and at 370°C. The data indicate that degradation of yield at elevated temperature has three distinct regimes. At high heating rates (>l°C/s for 260°C), the yield shows little dependence on heating rate, suggesting an almost instantaneous decrease in yield as temperature increases. At intermediate
4. Dynamic Response of Materials at Low and Moderate Stresses
81
rates (10~4 to l°C/s), a time-dependent softening occurs and the yield decreases with decreasing heating rate due to diffusion processes. At low heating rates (<10~4 °C/s), the yield reaches a constant value. Although the data in Figure 4.12 are presented in terms of heating rate, the conclusions would be qualitatively the same if discussed as a function of time at temperature. All tests were performed at constant strain rate and the observed response may be rate dependent.
Figure 4.12. Yield Dependence on Heating Rate.
Figure 4.13 shows a series of wave profile measurements made on 6061-T6 aluminum, in which initial temperature of the material was varied and the material was impacted above its spall threshold. Photomicrographs of recovered specimens are shown above the profiles. Evidence of fracture at the spall plane is shown by reversal in the velocity of the surface at approximately 0.7 us. Signals from internal fracture surfaces reach the rear surface with components of velocity opposite in sign to the rarefaction waves. As the fracturing progresses, the surface is no longer "pulled back" by the release wave system and accelerates, only to be decelerated again as the entrapped wave reverberates back and forth between the spall surface and the rear surface.
82
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.13. Wave Profiles at Various Temperatures. For purposes of comparison, the profiles have been slightly adjusted to show the same maximum free surface velocities and to coincide at release wave arrival time.
Loss of strength is evident in the elevated temperature profiles. As the temperature increases, the HEL decreases, as does the spall signal pullback.
4.1.7
Check Data
In general, the inputs to models of wave propagation and of fracture have been obtained from measurements of quasi-rectangular shock waves, along with a variety of other material properties which have been previously discussed. Once the model has been formulated and coefficients assigned to the parameters, definitive checks must be made on accuracy of the model. These checks must not employ the circular reasoning of using the input data as check data. Two types of data have been used to provide checks on the wave propagation calculations.
4. Dynamic Response of Materials at Low and Moderate Stresses
1.
2.
83
A stringent check on the model is its ability to calculate correctly the attenuation of a shock wave as it propagates. This requires that the accuracy with which the model can compute release behavior be high over the entire range of stresses in the wave. Non-rectangular waves can be induced in specimens and allowed to propagate. The model and coefficients developed for rectangular wave propagation must be able to predict behavior of waves of arbitrary shape for the system to be useful over a wide range of loadings.
Figures 4.14 and 4.15 show attenuation measurements made in three of the metals. In Figure 4.14, a 6 GPa shock wave is shown as it propagates through tantalum. The dotted line is based on calculations of peak pressure attenuation using the profile data. The plateau reached at the rear of the profile arises from the low-impedance backing material used on the rear of the thin impactor to maintain surface flatness during launch. Figure 4.15 shows attenuation of very thin pulses, approximately 80 nanoseconds in duration and with initial amplitudes greater than 10 GPa, as they attenuate in aluminum and titanium. These and other attenuation
Figure 4.14. Shock Wave Attenuation in Tantalum
84
Shock Waves: Measuring the Dynamic Response of Materials
series have been used as some of the principal checks to evaluate accuracy of wave propagation models.
Figure 4.15. Attenuated Wave Profiles
4.1.8
Summary
In summary, extensive tests have been performed on four metals in support of the development of constitutive relations and computer codes for wave propagation and dynamic fracture. The four metals (6061-T6 aluminum, alpha titanium, OFHC copper, and tantalum) have been characterized for shock wave calculations. Elastic constants, stress-strain-strain-rate behavior, equations of state, fracture characteristics, and wave propagation behavior have been quantitatively determined. In some cases, temperature dependence of these properties has been studied.
4. Dynamic Response of Materials at Low and Moderate Stresses
85
4.2
Measurements of the Dynamic Properties of Alpha Phase Tantalum
4.2.1
Introduction
The previous section summarized the results of tests on four materials. In this section, results are presented of an experimental study on the dynamic properties of tantalum. Areas studied included: • • • • •
Stress-strain relationships, including strain-rate and reverse loading behavior Elastic constants Equation of state Compressive and release wave characteristics Spall fracture.
The material showed approximately elastic-plastic behavior with some strain hardening under uniaxial stress compression, and exhibited substantial strain-rate sensitivity, with yield under uniaxial stress, increasing from 0.15 to 0.54 GPa for a strain rate increase of 0.001/s to 800/s. Longitudinal and shear wave velocities at 20°C were 4.146 and 2.032 mm/us, respectively. Temperature and pressure dependence was also measured, and various elastic constants were calculated. The shock wave equation of state (EOS) up to 20 GPa was determined as: crH = 4.0 + 550 up + 215 up2 (kb) The EOS was also calculated from the elastic constants, and comparisons were made with the shock wave EOS. Compressive wave tests showed a well-defined elastic precursor with a steady-state value of ~ 1.5 GPa. The impact velocity required for spall fracture was found to increase with decreasing pulse width.
86
Shock Waves: Measuring the Dynamic Response of Materials
4.2.2 Material Properties The tantalum on which tests were conducted was a commercially pure grade of 99.5% purity. This material was purchased as 50.8 mm diameter bar stock, which was preferable to plate or sheet stock, since it provided specimens of various thicknesses and gave little scrap, while ensuring that all specimens were from the same parent material. Chemical analysis performed by the supplier gave: Columbium Tungsten Iron Carbon Oxygen Nitrogen Hydrogen
210 ppm 120 20 <10 38 22 <5
Tantalum is not heat-treatable, and different strength levels are achieved by slight changes in impurity content or by cold-working. Although not specified, the material apparently had been annealed. Certifications gave 0.18 GPa tensile yield strength, 0.235 GPa ultimate tensile strength, and 52% elongation at 4D (measured along the bar axis). Measured hardness was 52 RK. A photomicrograph of a representative portion of the bar is shown in Figure 4.16. The bar axis or center line is vertical and the width shown is ~5 mm out of a total of 50.8 mm. There was a slight grain elongation
Figure 4.16. Tantalum Bar Stock, Grain Structure
4. Dynamic Response of Materials at Low and Moderate Stresses
87
along the axis and about a factor of 10 variation in grain size, ranging from 45 to 500 urn. For results presented in this book, the test or wave propagation direction was axial (e.g., vertical in Figure 4.16). The average measured density was 16.66 g/cm3. For use in equation of state calculations, several physical constants were compiled from the literature [5-9] and are listed below: Volume coefficient of thermal expansion: fi: 19.8 x 1(T6 + 0.001 x 10"6 T/°C Specific heat: Cp: 0.034 + 10"5T cal/g/°C Melting temperature:
2996°C
Boiling temperature:
5430°C
Latent heat of fusion:
35 cal/g
Latent heat of vaporization:
1000 cal/g
A portion of the tantalum bar was annealed for 1 hour at 1200°C in vacuum. The structure of this material is compared to the as-received material in Figure 4.17. There was no significant change in structure or hardness.
Figure 4.17. As-Received and Annealed Tantalum
88
4.2.3
Shock Waves: Measuring the Dynamic Response of Materials
Stress-Strain Studies
Shock wave and structural response code input requirements include constitutive equations relating stress to strain and strain rate. Uniaxial stress tests at various strain rates give yield and flow stress behavior, strain-rate sensitivity and work-hardening characteristics. Bauschinger effect tests were conducted to provide data on unloading and subsequent yield behavior. Uniaxial Stress Tests The compressive stress-strain-rate behavior of tantalum at 20°C is shown in Figure 4.18 for strain rates of 10'3/s to 600/s, with each curve the average of three tests. The material is very strain-rate sensitive, showing an almost 300% increase in yield level over this range of strain rates. A slight yield drop is evident above a rate of 40/s. Strain-rate data are cross-plotted in Figure 4.19 to give true stress vs. log true strain rate. Both yield and flow stress (6% strain) increase with strain rate, with sensitivity (slope) becoming very high above 100/s.
Figure 4.18. Compressive Stress vs Strain, Tantalum
F i g u r e 4.19.
Compressive Stress vs Log Strain Rate, Tantaium
Deformation mechanisms of tantalum (primarily under uniaxial stress conditions) have been studied by a number of other investigators. The reader is referred to the literature for details in such areas as strainrate effects [10-14], temperature effects [12-18], impurity content [10,
4. Dynamic Response of Materials at Low and Moderate Stresses
89
17-20], stress relaxation [13, 21], grain size effects [18, 21], dislocation behavior [15, 19-22], anisotropy [23], and fracture [16]. Reverse Loading Tests The reverse loading behavior or Bauschinger effect was studied by performing uniaxial tension tests after the material had been pre-strained in the opposite (compressive) direction. Results from low-strain-rate (0.001/s) tests are given in Figure 4.20 for maximum strains of 0.5, 1, and 3%. The Bauschinger strain (defined here as the plastic strain in the reverse loading path at 3/4 of the initial yield stress) is insensitive to prestrain, at least up to 3% strain. Flow stress in tension has increased with increasing compressive pre-strain.
Figure 4.20. Reverse Loading Behavior, Tantalum
Cyclic loading behavior is shown in Figure 4.21, where a single specimen was put in compression and then tension through a total of 5 cycles. The observed yield and flow stress behavior is indicative of an isotropic hardening mechanism.
90
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.21. Cyclic Loading Behavior, Tantalum
4.2.4
Elastic Behavior
Measurement of elastic constants provides additional inputs to the study of material response. Three basic measurements are density, longitudinal (dilatational) wave velocity, and shear (transverse) wave velocity. Measurements of temperature and pressure dependence of the wave velocities lead to prediction of pressure-compression isotherms, adiabats, and hydrostats. If these measurements are made with sufficient accuracy (-0.1%), the calculated pressure-compression behavior can frequently be extrapolated with reasonable confidence to higher pressures than covered in the actual measurements. Wave Velocity Measurements Initial measurements of longitudinal and shear wave velocities were made at 20°C and atmospheric pressure (i.e., P = 0). Values obtained parallel to the bar axis were: CL = 4.146 + 0.005 mm/us C s = 2.032 + 0.003 mm/us Although the accuracy of these measurements is estimated to be about ±0.1%, it should be noted that CL and Cs may vary depending on
4. Dynamic Response of Materials at Low and Moderate Stresses
91
the structural form of the material (i.e., plate, bar, etc.), its thermal and mechanical history, and the measurement direction relative to any structural anisotropy. Measurements were made of temperature and hydrostatic pressure dependence of wave velocities and the data corrected for thermal expansion and hydrostatic compression to give: Temperature Dependence, 0-100°C, P = 0 CL = 4.149 - 0.000146*T mm/us C s = 2.036 - 0.000173*T mm/us (T in °C) Pressure Dependence, 0-9 kb, T = 20°C CL = 4.146 + 0.00291 *P mm/us C s = 2.032 + 0.001378*P mm/us (P in kb) Temperature and pressure dependence of elastic constants for singlecrystal tantalum have been reported by Soga [24], Chechile [25], and Palmieri [26]. Armstrong and Brown [27] have published data on temperature dependence of elastic constants in polycrystalline tantalum. Palmieri also found CL (sound wave velocity) to vary from 3.245 to 3.366 mm/us and C s from 2.035 to 2.085 mm/us (at 0°C), depending on specimen orientation and heat treatment (i.e., annealing). Lamberson [28] studied both temperature and pressure dependence of CL and C s in polycrystalline tantalum with a density of 16.64 g/cm3 and 16 to 30 um grain size. CL was given as 4.156 - 0.0001573*T and -4.153 + 0.0025*P, and C s as 2.037 - 0.0001418*T and 2.034 + 0.0010*P (T in °C and P in kb). Elastic Constants Assuming polycrystalline tantalum is isotropic (CL and C s are independent of direction of measurement in a polycrystalline specimen), wave velocity data can be used to calculate various elastic constants. At
92
Shock Waves: Measuring the Dynamic Response of Materials
20°C and zero pressure, the following adiabatic constants were obtained: Bulk wave velocity,
CB = 3.42 mm/us
Sound wave velocity,
CE = 3.33 mm/us
Rayleigh wave velocity,
CR = 1.90 mm/us
Poisson's ratio,
r\ = 0.342
Bulk modulus,
K = 1946 kb
Shear modulus,
G = 688 kb
Elastic modulus,
E = 1846 kb
Lame's parameter,
X = 1488 kb
The adiabatic bulk and shear moduli were obtained as functions of T and P from the following:
K K
- Ac —P
UL V
G = pC2
2
4 p 21
- T
&
US
)
This gave: Temperature dependence, P - 0 K s =1948-0.084Tkb Gs = 691-0.131Tkb Pressure dependence, T = 20°C Ks = 1946 + 3.79P kb Gs = 688 + 1.28Pkb where the superscript S indicates adiabatic conditions. Isothermal values of K and G, as well as adiabatic and isothermal pressure and temperature derivatives at 20°C and zero pressure, were calculated and results are given in Table 4.2. Use of these constants in calculating the isotherm, adiabat, and hydrostat is discussed in the section on Equation of State.
4. Dynamic Response of Materials at Low and Moderate Stresses
93
Table 4.2. Elastic Constants for Tantalum (P « 0 , T - 20*C) PARAMETER
VALUE
KS
1 9 4 6 Waar
^
(|£-) p
- 0.084 kbar/'C
'#'s - *os
12
15.1 lcb«r/«C
s
KT a if T
1928 kbar T '
•
K0T
^ 0
f|f-lp
- 0.146 kb«r/*C
' I T ' S * K0S
^77
(^-)s
15.1 kbac/'C
GS - GT
688 Kbar
( lg»T
!-_»
" G0T
(||)p
- 0.131 kbar/-C
'If's * Gos
]^»
(|^)s
5.0 2 Itbar/'C
Graneisen Parameter The Graneisen ratio y is a parameter in the solid equation of state relating pressure to volume and energy, y = V (3P/5E)V (see, e.g.,
94
Shock Waves: Measuring the Dynamic Response of Materials
Reference 31). This parameter can be expressed thermodynamically as:
^CpldTJpldVJs
1
(4.1)
P%
For an isotropic elastic solid, this gives:
(4.2) where CL and C s are measured under adiabatic conditions. At 20°C and zero pressure: 7o=1.63 The zero-pressure Gruneisen parameter can also be estimated by several other methods, including those of Slater [32], Dugdale and MacDonald [33], Schreiber and Anderson [34], Anderson and Dienes [35], and Schuele and Smith [36]. An estimate of the temperature dependence of the Gruneisen parameter can be obtained by differentiating Equation 4.1 with respect to temperature. At constant pressure (P = 0, T = 20°C):
idTjp
°MdTP
KSUTJP
pidTk
cPldTJp](4.3)
=-0.00043/°C The pressure dependence can be estimated by differentiating with respect to pressure. At constant temperature (T = 20°C, P = 0):
apjx
Ks
dP JT
/3KTIOTP
=-0.00077/kb
UTJP) (4.4)
4. Dynamic Response of Materials at Low and Moderate Stresses
95
Another approach is to assume yfV constant, which permits (9y/9P)T to be estimated directly from the bulk modulus data. At constant temperature (20°C):
(
zio
=
[ KJ ] =
[
° + °T j
(4.5)
~ 1 1 9 3 19r , = -0.00085 / kb @ P=0 (1928 + 3.8P)
Debye Temperature The Debye temperature 6 is important in thermal energy calculations and indicates the temperature above which variations due to temperature for some thermodynamic parameters such as specific heat, thermal expansion, and Gruneisen ratio become small. There are a number of methods for calculating the Debye temperature [37]. One approximation suitable for use with elastic wave velocity data for poly crystalline metals is:
.
h ( 9N V / 3 f 1
2 f1/3
where h is Planck's constant, k is Boltzmann's constant, N is the number of mass points, and V is the sample volume. At 20°C, the elastically determined Debye temperature for tantalum is: 0 = 4.3xlO- 5 -
\J-
Vj [C L3
+ ^-\
CS3J
(4.6)
£ = 260°K for N = 2 (atoms per unit cell), V = (3.306 x 10"7 mm)3 (lattice constant) (34), CL = 4.146 mm/us, and C s = 2.032 mm/us.
96
4.2.5
Shock Waves: Measuring the Dynamic Response of Materials
Low-Pressure Shock Wave Equation of State for Tantalum
The Hugoniot equation of state is the locus of equilibrium states reached after shocking of a material. Data are usually presented either as stressparticle velocity points or as shock velocity-particle velocity points. The Hugoniot data presented in this book were obtained with x-cut quartz gages, and representative records for direct impact and transmitted wave tests are shown in Figure 4.22. The buffered direct impact method (tungsten carbide buffer plate on front of the quartz) permitted stresses up to 8 GPa in tantalum while keeping the stress in quartz at an acceptable level. The use of quartz gages for transmitted wave tests was primarily for the study of compressive wave development and elastic precursor decay. The results are discussed in the Wave Propagation section [38]. The direct impact records showed very fast risetimes (<10 ns) and then a straight peak stress region. The slight rounding at the front of the direct impact records in Figure 4.22 is due to impact tilt, and the ramped peak stress region results from finite-strain effects in the quartz which are corrected in the data analysis [39]. The Hugoniot may be expressed in several forms. A convenient form for experimental work is that established by a least-squares fit to data in the stress-particle velocity (crH - up) plane [40]. Transformation of the Hugoniot into other planes, such as shock velocity-particle velocity (Us - Up) or stress-volume (erH - v), is performed by assuming a material model. This was done by assuming an ideal elastic-plastic wave structure with equilibrium initial and final states and applying the mass and momentum conservation equations: ^H = ^
+ Pe ( U S " Ue ) (U p - Ufi )
(4.7)
and
v = vfl-^)[to where p e and ue are density and particle velocity at the elastic limit ae.
(4.8)
4. Dynamic Response of Materials at Low and Moderate Stresses
97
Figure 4.22. Quartz Gage Records, Tantalum
Therefore, when either aH - up or U s - up relations are established and at, pe, ue, and CL are known,
98
Shock Waves: Measuring the Dynamic Response of Materials
The resulting Hugoniots for tantalum at 20°C are given in Figures 4.23, 4.24, and 4.25 and are listed below: crH = 4.0 + 550up + 215up2 (Std. error 0.3 kb) CTH = 1.7 + 1820// + 3120/i2 (Where/u = v o / v - l )
(4.9)
PH = 1820/x + 3120/x2 (Hydrostat) Us =3.36 + 1.239up These equations are based on the data points indicated by the symbols in Figure 4.23 and on
Figure 4.23. Stress-Particle Velocity Hugoniot, Tantalum
4. Dynamic Response of Materials at Low and Moderate Stresses
99
Figure 4.24. Stress or Mean Pressure-Compression Hugoniot, Tantalum
4.2.6
Ultrasonic Equation of State
The equation of state can also be determined from ultrasonic measurements. Accurate measurement of elastic wave velocities (and, therefore, bulk modulus) as a function of hydrostatic pressure makes it possible to estimate directly the shock wave compression behavior. Numerous analytical and empirical relations have been developed for relating pressure, volume, and bulk modulus [42, 43, 44], including those of Birch, Murnaghan, and Keane [45, 46]. The use of these relations is summarized below: Birch EOS
PT = 289l[(l + /x)% - (1 + /i)^3][l - 0.15{(l + fif3 - l}] (4.10) Murnaghan EOS, Isotherm PT =507.l[(l + ^ ) 3 8 0 - l ]
(4.11)
100
Shock Waves: Measuring the Dynamic Response of Materials
Murnaghan EOS, Adiabat
P s = 516.2[(1 +/x) 3 ' 77 - l]
(4.12)
KeaneEOS, K ^ = 3 . 6 Ps = 566.l[(l + / i ) 3 ' 6 - l ] - 9 1 . 9 In (1 + fi)
(4.13)
KeaneEOS, K8'^ =1.10 Ps = 6063 [(1 + Ai)110 - l] - 4723 In (1 + /z)
(4.14)
In the above equations, subscripts T and S indicate isothermal and isentropic conditions, respectively. Duvall gives a method of calculating the increase in entropy across a shock which can be applied to the Murnaghan equation to give [28, 47]: PH=Ps+!oK!(Kos; 12
+
P °= P s + 1 2 6 1 (l^f
i)LiM3
"l + H
(4.15)
Zel'dovich gives an expression relating Ps and PT, assuming y/v is constant [28,48]:
Ps = PT + /?TX exp ( 7o M U 1 - (1 + fj.) 1
ll + 4
1
(4,6)
Ps = PT +10.4 expf 1.63f-^-)l-(l + /i) I U + MJJ The above equations of state are compared to the shock wave EOS in Table 4.3. The entropy correction based on the Murnaghan equation has been applied to the Birch and Keane equations as well. The ultrasonic
101
4. Dynamic Response of Materials at Low and Moderate Stresses
equations of state show good agreement with each other but are 3 to 6% above the shock wave hydrostat. Table 4.3. Equations of State Comparison-Hydrostat, Tantalum
P , IIUGONIOT MEAN PRESSURE (kb) V
V = — -1
BIRCH EQ. 10
MURNAGHAN EQ. 11 I EO. 12
KEANE EQ. 1} I EQ. 14
SHOCK WAVE EO. 9
0.02
19.7
39.7
40.0
40.0
40.0
37.6
0.04
81.6
91.9
82.4
32.3
62.2
77.8
0.06
120.0
126.}
127.0
127.0
126.3
120.4
0.08
172.7
173.3
174.3
174.2
172.6
165.6
0.10
221.6
222.8
224.1
223.9
220.9
213.2
The shock velocity-particle velocity relationship can also be determined from ultrasonic data. Following the method of Ruoff [49], one obtains: US=CB+Su1)+Aup2 C
B
=
^ -
(4.17)
where S = l/4«+l)
(4.18) 24U B
For tantalum U s = 3.42 + 1.193up + 0.14up2
(4.19)
This is compared with the linear Us - up relation obtained from lowpressure shock wave data in Figure 4.25.
102
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.25. Shock Velocity-Particle Velocity Hugoniot for Tantalum
4.2.7
Yield Behavior
A complete description of the low-pressure equation of state requires consideration of yield behavior or the Hugoniot elastic limit. Values of compressive yield were obtained by three independent methods. A.
Uniaxial Stress
The yield level was determined in uniaxial stress as a function of strain rate and then converted to uniaxial strain using:
'• = "'(rf£)
(420)
Strain Rate (s"1)
gJGPa)
600 10"3
1.12 0.29
4. Dynamic Response of Materials at Low and Moderate Stresses
B.
103
Wave Profiles
The elastic precursor level was measured as a function of propagation distance using quartz gages. These profiles are discussed in the section on Wave Propagation.
C.
Propagation Distance (mm)
gP (GPa)
2 12
1.9 1.6
Hugoniots
The yield level in uniaxial strain was inferred by comparison of the elastic (poCL) and plastic (erH vs. up) Hugoniots, i.e., by defining the elastic limit as the intersection of the elastic response line and the curve fit to the stress-particle velocity data. 10 /?0CLUe = A + Bue + Cue2 Solving for ue gives ue = 0.029 mm/us .-. Oe=2.0GPa For an isotropic, polycrystalline metal with elastic-perfectly plastic behavior and no strain-rate or time-dependent effects, one would expect the same yield to be obtained by each method. The yield obtained from wave profile tests and the Hugoniot measurements are in fair agreement, but both are much higher than the yield determined from uniaxial stress data for rates less than 103/s. For this particular tantalum, the equilibrium yield in uniaxial strain was assumed to be 2.0 GPa for use in the EOS transformations made in obtaining the a^- /u and Us - up relations given above (Equation 4.9).
104
4.2.8
Shock Waves: Measuring the Dynamic Response of Materials
Wave Propagation
Profiles of shock waves propagated through a specimen were recorded as stress-time or velocity-time histories. The stress-time data were obtained with quartz gages, and the results were transformed to material stress by application of an impedance matching technique [39], assuming timeindependent behavior. The velocity-time data were obtained with a velocity interferometer and are presented as measured. All wave profiles obtained are given in the Appendix for reference. Compressive Wave Behavior Structure in the compressive wave is shown in the quartz gage data in Figure 4.26. (Note that in this and subsequent figures containing quartz
Figure 4.26. Compressive Waves: Propagation Distance Dependence (6061-T6 Al Impactors)
gage data, tilt refers to the time required for a step-input to sweep across the gage electrode diameter.) The wave front is characterized by an elastic portion and a transition to a spreading elastic wave (nominal final stress of 3.4 GPa). (The elastic wave would be overdriven by the plastic wave at -44 GPa.) Also shown in Figure 4.26 are shock velocities as calculated from the shock wave Hugoniot (Equation 4.9). Although the calculated plastic wave velocity lies within the risetime of the measured wave, the plastic wave is not a step pulse and shows spreading with propagation distance
4. Dynamic Response of Materials at Low and Moderate Stresses
105
at this stress level. This means that transformation of the Hugoniot from the |
Figure 4.27. Compressive Waves, Material Difference (6061-T6AlImpactors)
Figure 4.28. Compressive Waves, Material Differences (6061-T6 Al Impactors)
Two tests with quartz gages were conducted using specimens machined from a piece of 76.2 mm diameter bar and are compared to 50.8 mm bar data in Figure 4.27 and Figure 4.28. Although the 76.2 mm bar was ordered to the same nominal specifications as the 50.8 mm bar, the compressive wave development was significantly different. The
106
Shock Waves: Measuring the Dynamic Response of Materials
larger diameter bar showed a less well-defined elastic limit but a higher velocity plastic wave. These differences can probably be attributed to changes in metallurgical characteristics of the material. Impurity content for the 76.2 mm bar was about the same as the 50.8 mm bar, but the yield and flow stress at low strain rate was about twice as high and hardness was 86 RK. Precursor decay data for tantalum, as determined from quartz gage data, is summarized in Figure 4.29. The calculated initial elastic impact stress was -3.7 GPa for these tests, and the precursor level has dropped -50% in 2 mm of travel. The equilibrium or steady-state elastic limit appears to be -1.5 GPa. The uncertainty bars in Figure 4.29 reflect dispersion and rounding at the elastic front, which is due to at least three factors.
Figure 4.29. Elastic Precursor Decay in Tantalum
First, since quartz gages average stress over the electrode area, small differences in wave front arrival times at the specimen/gage interface would give a finite risetime rather than an instantaneous stress jump. Second, the finite thickness of the epoxy between the specimen and the gage will increase the apparent risetime in the wave front.
4. Dynamic Response of Materials at Low and Moderate Stresses
107
Third, the influence of shock wave tilt on a finite-area gage is to smooth out abrupt changes in stress level as well as to increase recording risetime of the wave front. Gillis et al. [51] reported elastic precursor decay data for annealed, high-purity (99.9%) tantalum. Direct comparison with the present work is not possible because of material differences and variations in impact stress and propagation distance. However, their data tend toward the same steady-state yield value, 1.5 GPa. Gillis used quartz gages to record the stress profiles and indicated elastic wave risetimes on the order of 100 ns, after correction for tilt. The quartz gage data shown in Figure 4.26 (not corrected for tilt) show risetimes generally less than 50 ns, and velocity interferometer data, discussed below, showed risetimes to be less than 20 ns. To determine if risetime of the input pulse influenced precursor development, a series of velocity interferometer tests were performed with a fused quartz buffer on the front of the tantalum target. (Test conditions for all velocity interferometer tests discussed in this book are listed in Table 4.4.) Because of the nature of the fused quartz Hugoniot at stresses below -4.0 GPa, the compressive wave spreads as it propagates [52]. The risetime of the pulse at the fused quartz/tantalum interface was -300 ns, and the resulting compressive wave profiles after propagation through several thicknesses of tantalum are shown in Figure 4.30. The calculated elastic limits at 6 and 10 mm (1.8 and 1.6 GPa, respectively) are approximately the same as those determined from the quartz gage tests. This indicates that elastic wave development is insensitive to input pulse risetime (i.e., strain rate), at least for risetimes of 300 ns or less. Note that Test 225 in Figure 4.30 shows evidence of a slight yield drop or stress relaxation. Although this behavior might be expected on the basis of the relaxation observed in the uniaxial stress tests (see Figure 4.18), it was seen on only two velocity interferometer tests and on none of the quartz gage tests.
108
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.30. Compressive Waves, Long Risetime Input Pulse
Figure 4.31. Complete Wave Profiles, Tantalum
Release Waves and Wave Attenuation Release waves were studied by using relatively thin impactors and measuring the complete wave profile with the laser velocity interferometer. Unattenuated wave profiles are shown in Figure 4.31 for unbacked (free rear surface) and for Lexan-backed impactors. The
4. Dynamic Response of Materials at Low and Moderate Stresses
109
unloading wave in tantalum is dispersive and does not show a distinct elastic-plastic structure. Although the final interface velocity is higher for the Lexan-backed impactor, due to the incomplete unloading, the slopes of the rarefaction waves are essentially the same. This result is significant, and the attenuation results discussed below are for Lexanbacked impactor s. Lexan is a polycarbonate sheet made by General Electric Co. and has a density of 1.20 g/cm3. An approximate equation of state determined for use in wave propagation calculations is crH ~ 31 up +15 u p2 . If target thickness is large enough compared to impactor thickness, the release wave will overtake the compressive wave and attenuate the peak stress, as shown in Figure 4.32. The initial stress was -6.9 Gpa, and the impactors were backed with Lexan which gave only -80% unloading. The overtaking point can be calculated assuming ideal elasticplastic compression and release waves, which gives: 1 , l~up/Us X CL C[- u — ~ -^ _L— P 1 l-up/Us Xo Us C^+up
(4.21)
where CLis the elastic release wave velocity (assumed to be 4.3 mm/us at 6.9 GPa). At 6.9 GPa, Up = 0.11 mm/us and Us = 3.5 mm/us from Equation 9. This gives the release distance X/Xo = 7.9, which is larger than actually measured. This can be attributed to the ramped plastic compressive wave at this stress. If the velocity (-3.1 mm/us) of the trailing portion of the plastic wave is used for Us rather than the value obtained from Equation 4.9, the calculated overtaking point is X/Xo = 4.8. This is in better agreement with the experimental results shown in Figure 4.32. 4.2.9
Spall Fracture
Spall fracture by plate impact results from reflection of compressive waves from a relatively low-impedance interface (normally a free surface) and subsequent wave interaction. Spall studies were carried out
110
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.32. Wave Attenuation in Tantalum
with both passive and active techniques. Passive methods involved the recovery and examination of shock-loaded specimens. Metallographic examination established the type and degree of damage, which was correlated with test parameters such as velocity, impactor thickness, and target thickness. Active methods utilized the laser velocity interferometer and provided time-resolved data on the influence of spall fractures on shock wave profiles, as measured at the rear surface. Recovery Tests Spall behavior of tantalum was studied by carrying out a series of impact and recovery tests, where the target was sectioned across a diameter, polished, etched, and examined optically at a magnification of 50X. The specimens were then classified according to the degree of fracture that was observed, which ranged from no visible damage to complete material separation. The onset of significant fracture is generally referred to as incipient spall and is a critical parameter, since it can be used to deduce dynamic fracture strength and corresponds to the generation of sufficient free surface area within the material to reflect a portion of the interacting release waves as a compressive wave.
111
4. Dynamic Response of Materials at Low and Moderate Stresses
The incipient spall threshold was defined as the impact velocity (for a given set of impact parameters) corresponding to cracking over at least 50% of the width of the section estimated to be under a condition of plane strain during the time of loading. Hanneman [53] studied the thermal spall behavior of tantalum using a capacitor discharge technique. He reported a "spall threshold temperature" of 1900 to 2140°C for tantalum wire. Table 4.4. Velocity Interferometer Test Data, Tantalum vt Test
Impact
Ho.
Max.
Velocity (iws/us I
Stresa IKbar)
vt
vt
Impactor
Target
Thickness (imil
Thickness (mini
x/x °
Conficj.
154
0.269
92
1.488
4.552
3.06
155
0.218
66
1.48)
4.417
2.98
Ta-Ta
156
0.125
19
1.473
4.562
3.10
Ta-Ta
,
Ta-Ta
16B 2
0.228
69
0.599
1.519
2.54
Ta-Ta/ru
169 2
0.216
66
0.594
3.022
5.09
Ta-Ta/KU
170 2
0.225
69
0.617
5.057
8.20
Ta-Ta/Ft)
171 2
0.215
66
0.615
10.11
16.44
Ta-Tn/rt)
172
0.205
63
0.632
1.534
2.43
Ta-Ta/FQ
130
0.132
101
0.645
4.514
7.03
Ta-Ta
224
0.293
13
12.7
12.72/1.946
~
225
0.281
11
12.7
12.71 / f t . 121
—
M-FU/Ta/FO
226
0.286
13
12.7
12.72/10.15
—
Al-FU/Ta/FQ
l*
fc
* y • # T*I—TA (FQ i s
2.
is
tdittdluni
fused q u a r t !
1 futility t o r
into
t d n t A l uin
t a JTMC t
wit_h
IITGC
ircfiir
Al-FO/Ta/FO
sui^fcicru
window).
Lexan-backe<) impactor.
Incipient spall velocity results are summarized in Table 4.4 and the data are given in Figure 4.33. As is typically the case with metals, the impact velocity for incipient spall increases for decreasing impactor thickness. This implies that the peak stress (and therefore strain) required to create spall fractures increases with decreasing pulse width or time of loading. Optical photomicrographs of recovered specimens are shown in Figures 4.34 and 4.35. The specimens were polished and etched to provide maximum contrast between cracks and sound material. The procedure followed was:
112
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.33. Spall Data for Tantalum
Figure 4.34. Spall Fractures in Tantalum (1.5 mm - 4.5 mm)
1. Wet grind, 320/400/600 grit silicon carbide. 2. Rough polish, 6 um diamond with 6 parts alcohol and 1 part Gama-Medica Surgical Soap (Huntington Labs).
4. Dynamic Response of Materials at Low and Moderate Stresses
113
Figure 4.35. Spall Fractures in Tantalum
3. Final polish, 1 um diamond with alcohol and soap. 4. Swab etch (20°C), equal parts nitric acid, hydrofluoric acid, and water. All spall photomicrographs in this report are oriented such that initial shock wave propagation was from bottom to top. As noted above, spall cannot be rigorously defined in terms of a unique impact velocity or stress, but requires consideration of the degree of fracture. Fracture in tantalum is of a ductile nature with the development of spherical voids. As velocity increases (see Figure 4.34), the voids begin to coalesce and a complete fracture surface is formed. Note that there is dispersion of the voids around a nominal "spall plane." This was also found for 6061-T6 aluminum [54] and titanium [55], which have well-defined elastic wave structures, while copper [56] has a poorly defined, low amplitude elastic wave and showed very little void dispersion around the spall plane. Scanning microfractographs of fracture surfaces for completely spalled specimens are given in Figures 4.36 and 4.37. Fracture was by normal rupture with evidence of extensive plastic flow. At 195°C,
114
Shock Waves: Measuring the Dynamic Response of Materials
Figure 4.36. Spall Fracture Surfaces in Tantalum (1.5 m m - 4.5 mm, 0.32 mm/us)
Figure 4.37. Spall Fracture Surfaces in Tantalum, Temperature Effects
fracture is still ductile, with no evidence of a ductile-brittle transition, but void size is much smaller. Spall Profiles Complete wave profiles were obtained under spall-producing conditions using a tantalum-into-tantalum configuration, with a free rear surface on the impactor to give complete release and a free rear surface on the target for reflection of the compressive wave. The wave interactions resulting from such a test are discussed in Reference 57.
4. Dynamic Response of Materials at Low and Moderate Stresses
115
Figure 4.38. Spall Wave Profiles in Tantalum
Measured profiles for unattenuated pulses are given in Figure 4.38, along with photomicrographs of the recovered and sectioned targets. Also shown for reference are the incipient and complete spall levels for this impactor/target combination as determined from recovery tests. Evidence of fracture is shown as reversal of the velocity of the release wave. The amount of "pullback" (decrease in free surface velocity to the point of first reversal) can be related to spall strength of the material for a given impact geometry. An empirical relation for this has been given by Taylor [58] as: as = PC{Au{s/2)
(4.22)
where as is the spall strength, p and C are local density and longitudinal wave speed, and Aufs is pullback.
116
Shock Waves: Measuring the Dynamic Response of Materials
The data in Figure 4.38 indicate the spall stress to be 6.8 GPa. This calculated spall strength is not constant for a given material since it is proportional to Aufs. This factor has been found to change with target temperature, pulse width, and pulse shape for 6061-T6 aluminum [57], and other metals show similar behavior. A spall profile for poly crystalline tantalum reported by Taylor [58] showed a pullback of 0.12 mm/us for a pulse width of-1.5 us, which is less than the pullback of 0.19 mm/us shown in Figure 4.38 for an 0.8 us pulse width. However, for a given set of impact conditions (X, Xo, temperature), the pullback is relatively insensitive to maximum compressive stress as well as to degree of fracture. The spall profile from an attenuated pulse test, where the impact velocity was above that required for spall, is given in Figure 4.39.
Figure 4.39. Attenuated Spall Profile in Tantalum
4. Dynamic Response of Materials at Low and Moderate Stresses
117
4.2.10 Summary The dynamic properties of tantalum were measured under uniaxial stress and uniaxial strain conditions. The material tested was commercially pure bar stock with 45 to 500 urn grain size and 52 RK hardness. Material response to compressive uniaxial strain deformation approximated elastic-perfectly plastic behavior and exhibited strain-rate sensitivity with -0.14 GPa yield at a strain rate of 0.001/s and -0.54 GPa yield at 800/s. Longitudinal and shear wave velocities were measured as functions of temperature and pressure, giving the following (in mm/us): CL = 4.149 - 0.000146*T, C s = 2.036 - 0.000173*T (T in °C) CL = 4.146 + 0.00291*P, C s = 2.032 + 0.00137*P (P in kb) Principal elastic constants were evaluated at T = 20°C and P = 0, including: Bulk wave velocity,
CB = 3.42 mm/us
Sound wave velocity,
CE = 3.33 mm/us
Rayleigh wave velocity,
CR = 1.90 mm/us
Poisson' s ratio,
v = 0.342
Bulk modulus,
K = 1946 kb
Shear modulus,
G = 688 kb
Elastic modulus,
E = 1846 kb
Lame's parameter,
X - 1488 kb
Elastic wave velocity data were also used in determining the Gruneisen parameter y and the Debye temperature 0: 7=1.63 @T = 20°CandP = 0 dy/dT = - 0.00043/°C @ T = 20°C dy/dP = — 0.00077/kb @ P = 0 e= 260°K
118
Shock Waves: Measuring the Dynamic Response of Materials
The shock wave Hugoniot was determined experimentally to be: an = 4.0 + 550 Up + 215 up2 (CTH < 200 kb) Assumption of an ideal elastic-plastic wave structure with a 2.0 GPa Hugoniot elastic limit and equilibrium initial and final states gave: <7H = V.2+ 1820 JU + 3120 //2 P H =1820 / u + 3120J«2 U s = 3.36+1.239 Up The hydrostat as determined from elastic constants data was: PH = 6063 [(1 + [if10 - ll - 4723 ln(l + fi) + 1261 —^— ]
i + n)
Measurements of compressive wave development showed the wave front to be characterized by a well-defined elastic portion and a transition to a spreading plastic wave. Elastic precursor decay was -50% to 1.9 GPa after 2 mm of travel. Spall fracture behavior was determined for elastic pulse widths of 0.20 and 0.72 us. The incipient spall threshold expressed as impact velocity increased with decreasing pulse width. Tantalum showed a poorly defined spall plane with appreciable dispersion of almost spherical voids. The fracture surface for completely spalled tantalum showed ductile failure with void growth and coalescence accompanied by a large amount of plastic flow.
4. Dynamic Response of Materials at Low and Moderate Stresses
LIST OF SYMBOLS CB
Bulk Wave Velocity, mm/us
CL CL cp G K PH Ps PT T Us Up v Vi X Xo Xs XT
Longitudinal Wave Velocity, mm/us Shear Wave Velocity, mm/us Specific Heat, cal/g Shear Modulus, kb Bulk Modulus, kb Mean Pressure, Hydrostat, kb Mean Pressure, Adiabat, kb Mean Pressure, Isotherm, kb Temperature, °C Shock Wave Velocity, mm/us Particle Velocity, mm/us Specific Volume, cm3/g Impact Velocity, mm/us Target Thickness, mm Impactor Thickness, mm Adiabatic Modulus, kb Isothermal Modulus, kb
XOTS , XOTT
Adiabatic Pressure Derivative @ Zero Pressure = —
XOT T , XOT S
SI P= 0
Isothermal Pressure Derivative @ Zero Pressure = p ^ - T I P = 0
ft
Volume Coefficient of Expansion, /°C =
_Udp]
pidTJP = 0
y p, v p OH
Gruneisen Parameter Compression (= vjv - 1) Poisson's Ratio Density, g/cm3 Stress, Hugoniot, kb
119
120
4.3
Shock Waves: Measuring the Dynamic Response of Materials
References
1. Christman, D. R. "A Selected Bibliography on Dynamic Properties of Materials." General Motors Materials and Structures Laboratory, DASA2511, July 1970. 2. Johnson, J. N., and L. M. Barker. "Dislocation Dynamics and Steady Plastic Wave Profiles in 6061-T6 Aluminum," J. Appl. Phys., vol. 40, 4321-4334 (1969). 3. Isbell, W. M., and D. R. Christman. "Shock Propagation and Fracture in 6061-T6 Aluminum from Wave Profile Measurements." General Motors Materials and Structures Laboratory, DASA-2419, April 1970 (AD705536). 4. Babcock, S. G., J. J. Langan, D. B. Norvey, T. E. Michaels, and F. L. Schierloh. "Characterization of Three Aluminum Alloys for Use in Antiballistic Missile Vulnerability Analyses." General Motors Materials and Structures Laboratory, MSL-70-12, May 1970. 5. "A Selected Bibliography on Dynamic Properties of Materials," compiled by Christman, D. R. General Motors Corporation, Manufacturing Development, DAS A 2511, June, 1970 (AD 710823). 6. Metals Handbook, 8th ed., vol. 1: Properties and Selection of Metals, Lyman, T., ed. American Society for Metals, Metals Park, Ohio, 1961. 7. Handbook of Chemistry and Physics, 45th ed., Weast, R. C , ed. The Chemical Rubber Co., Cleveland, Ohio, 1964. 8. Thermophysical Properties of High Temperature Solid Materials, vol. 1: Elements, Touloukian, Y. S., ed. New York: The McMillan Co., 1967. 9. Aerospace Structural Metals Handbook, Chapter Code 5401 (Tantalum), Mechanical Properties Data Center, Belfour Stulen, Inc., Traverse City, Michigan, 1966. 10. Pink, E. "Zum Einfluss der Dehngeschwindigkeit auf Streckgrenzen Phanomene bei Tantalum." J. Less-Common Metals, vol. 16, 119-128 (1969). 11. Hoge, K. G. "Influence of Strain Rate on Flow Stress of Tantalum." Proc. Second Int. Conf. Strength of Metals and Alloys, vol. Ill, American Society for Metals, pp. 996-1000 (1970). 12. Arsenault, R. J. "Effects of Strain Rate and Temperature on Yield Points." AIME Trans., vol. 230, 1570-1576 (1964). 13. Hoge, K. G. "The Effect of Strain Rate on Mechanical Properties of Some Widely Used Engineering Metals." University of California Radiation Laboratory, UCRL-14599, 14 Dec. 1965. 14. Kossowsky, R. "Temperature and Strain Rate Dependence of Microyield Points in Tantalum." Refractory Metals and Alloys IV: Vol. I, vol. 41, Met. Soc. Conf., pp. 47-68. New York: Gordon and Breach, 1967. 15. Barbee, T. W., and R. A. Huggins. "Dislocation Structures in Deformed and Recovered Tantalum." Stanford University, ONRDMS Report No. 64-25, Nov. 1964.
4. Dynamic Response of Materials at Low and Moderate Stresses 16. Nunes, J., A. A. Anctil, and E. B. Kula. "Low Temperature Flow and Fracture Behavior of Tantalum." U.S. Army Materials Research Agency, AMRA-TR-64-22, Aug. 1964 (AD 448479). 17. Arsenault, R. J. "An Investigation of the Mechanism of Thermally Activated Deformation in Tantalum and Tantalum-Base Alloys." Acta Met., vol. 14, 831-838(1966). 18. Gazza, G. E. "Petch Analysis of Hydrogenated Tantalum Sheet." Refractory Metals and Alloys IV: Vol. I, vol. 41, Met. Soc. Conf., pp. 69-80. New York: Gordon and Breach, 1967 (also, AD 635596). 19. Eliasz, W., and Z. C. Szkopiak. "Strain Aging of Refractory Metals (Tantalum)." Air Force Materials Laboratory, AFML-TR-65-437, July 1965 (AD 808649). 20. Owen, W. S., D. C. Hull, J. Bryson, and C. L. Formby. "Plastic Deformation of Body-Centered Cubic Metals, Vol. I: Plastic Deformation of Tantalum." University of Liverpool, AFML-TR-66-369, vol. I, Feb. 1967 (AD 813542). 21. Barbee, T. W. "Some Aspects of Dislocation Dynamics in Metals," Stanford University, ONR SU-DMS Report No. 6533, Nov. 1965 (AD 625023). 22. Jewett, R. P., and E. D. Weisert. "Dislocation Morphology of Tantalum Deformed in Tension." High Temperature Refractory Metals, vol. 34, Met. Soc. Conf., pp. 160-172. New York: Gordon and Breach, 1966. 23. Hoddinott, D. S., and G. J. Davies. "The Anisotropy of Young's Modulus and Texture in Sheets of B.C.C. Metals." J. Inst. Met., vol. 97, 155-159 (1969). 24. Soga, N. "Comparison of Measured and Predicted Bulk Moduli of Tantalum and Tungsten at High Temperatures." J. Appl. Phys., vol. 37, 3416-3420 (1966). 25. Chechile, R. A. "Ultrasonic Equation of State of Tantalum." Case Institute of Technology, May 1967 (AD 655640). 26. Palmieri, L. "Ultrasonic Measurements of Elastic Moduli of Polycrystalline Tantalum and Niobium." Appl. Mat. Res., vol. 3, 139-143 (1964). 27. Armstrong, P. E., and H. L. Brown. "Dynamic Young's Modulus Measurements Above 1000 °C on Some Pure Polycrystalline Metals and Commercial Graphites." AIME Trans., vol. 230, 962-966 (1964). 28. Lamberson, D. L. "The High Pressure Equation of State of Tantalum, Polystyrene and Carbon Phenolic Determined from Ultrasonic Velocities." Doctoral Dissertation, Air Force Institute of Technology (AFIT-SE), Wright-Patterson AFB, Ohio, March 1969. 29. Bolef, D. I. "Elastic Constants of Single Crystals of the bcc Transition Elements V, Nb, and Ta.'V. Appl. Phys., vol. 32, 100-105 (1961). 30. Featherston, F. H., and J. R. Neighbours. "Elastic Constants of Tantalum, Tungsten, and Molybdenum." Phys. Rev., vol. 130, 1324-1333 (1963). 31. Rice, M. H., R. G. McQueen, and J. M. Walsh. "Compression of Solids by Strong Shock Waves." Solid State Physics, vol. 6, 1-63 (1958).
121
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Shock Waves: Measuring the Dynamic Response of Materials
32. Slater, J. C. Introduction to Chemical Physics. New York: McGraw-Hill Book Co., 1939. 33. Dugdale, J. S., and D. K. C. MacDonald. "The Thermal Expansion of Solids." Phys. Rev., vol. 89, 832-834 (1953). 34. Schreiber, E., and O. L. Anderson. "Pressure Derivatives of the Sound Velocities of Polycrystalline Alumina." Am. Ceram. Soc. J., vol. 49, 184— 190 (1966). 35. Anderson, O. L., and G. J. Dienes. Non-Crystalline Solids (Chapter 18), Frechette, V. D., ed. New York: John Wiley and Sons, 1960. 36. Schuele, D. E., and C. S. Smith. "Low Temperature Thermal Expansion of Rbl." J. Phys. Chem. Solids, vol. 25, 801-814 (1964). 37. Alers, G. A. "Use of Sound Velocity Measurements in Determining the Debye Temperature of Solids." Physical Acoustics, Vol. Ill, Part B (Lattice Dynamics), 1-42. New York: Academic Press, 1965. 38. Taylor, A., and B. J. Kagle. Crystallographic Data on Metal and Alloy Structures. New York: Dover Publications, Inc., 1963. 39. Chin, H. C. "A Computer Program to Analyze X-Cut Quartz Data Obtained from Shock Loading." MSL-70-15, Manufacturing Development, Materials and Structures Laboratory, General Motors Corporation, June 1970. 40. Group GMX-6, "Selected Hugoniots." Los Alamos Scientific Laboratory, LA-4167-MS, 1 May 1969. 41. Rohde, R. W., and T. L. Towne. "Shock-Compression Behavior of Tantalum at 250 and 900 °C." J. Appl. Phys., vol. 42, 878-880 (1971). 42. Birch, F. "The Effect of Pressure Upon the Elastic Parameters of Isotropic Solids, According to Murnaghan's Theory of Finite Strain." /. Appl. Phys., vol. 9, 279-288 (1938). 43. Birch, F. "Elasticity and Constitution of the Earth's Interior." J. Geophys. Res., vol. 57, 227-286 (1952). 44. Murnaghan, F. D. "The Compressibility of Media Under Extreme Pressures." Proc. Nat. Ac. Set, vol. 30, 244-247 (1944). 45. Keane, A. "An Investigation of Finite Strain in an Isotropic Material Subjected to Hydrostatic Pressure and Its Seismological Applications." Australian J. Phys., vol. 7, 323-333 (1954). 46. Anderson, O. L. "On the Use of Ultrasonic and Shock-Wave Data to Estimate Compressions at Extremely High Pressures." Phys. Earth Planet. Interiors, vol. 1, 169-176 (1968). 47. Duvall, G. E., and B. J. Zwolinski. "Entropic Equations of State and Their Applications to Shock Wave Phenomena in Solids." Acoust. Soc. Am., vol. 27, 1054-1058 (1955). 48. Zel'dovich, Y. B., and Y. P. Raizer. Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena, vol. II, 688-709. New York: Academic Press, 1967. 49. Ruoff, A. L. "Linear Shock-Velocity-Particle-Velocity Relationship." /. Appl. Phys., vol. 38, 4976^980 (1967).
4. Dynamic Response of Materials at Low and Moderate Stresses
123
50. Vaidya, S. N., and G. C. Kennedy. "Compressibility of 18 Metals to 45 kbar."/. Phys. Chem. Solids, vol. 31, 2329-2345 (1970). 51. Gillis, P.-P., K. G. Hoge, and R. J. Wasley. "Elastic Precursor Decay in Tantalum." J. Appl. Phys., vol. 42, 2145-2146 (1971). 52. Barker, L. M., and R. E. Hollenbach. "Shock-Wave Studies of PMMA, Fused Silica and Sapphire." J. Appl. Phys., vol. 41,4208-4226 (1970). 53. Hanneman, G. P. "Spallation Threshold Tests on Niobium, Tantalum and Zirconium." Battelle Northwest Laboratories, BNWL-756, Sept. 1968. 54. Christman, D. R., W. M. Isbell, S. G. Babcock, A. R. McMillan, and S. J. Green. "Measurements of Dynamic Properties of Materials, Vol. Ill: 6061T6 Aluminum." General Motors Corporation, Manufacturing Development, DASA 2501-3, 1971. 55. Christman, D. R., T. E. Michaels, W. M. Isbell, and S. G. Babcock. "Measurements of Dynamic Properties of Materials, Vol. IV: Alpha Titanium." General Motors Corporation, Manufacturing Development, DASA 2501-4, 1971. 56. Christman, D. R., W. M. Isbell, and S. G. Babcock. "Measurements of Dynamic Properties of Materials, Vol. V: OFHC Copper." General Motors Corporation, Manufacturing Development, DASA 2501-5, July 1971 (AD 728846). 57. Isbell, W. M., and D. R. Christman. "Shock Propagation and Fracture in 6061-T6 Aluminum from Wave Profile Measurements." General Motors Corporation, Manufacturing Development, DASA 2419, April 1970 (AD 705536). 58. Taylor, J. W. "Stress Wave Profiles in Several Metals." Dislocation Dynamics, 573-589. New York: McGraw-Hill Book Co., 1968.
5. Hugoniot Equations of State to 0.5 TPa 5.1
Foreword
Since the early 1950s, high explosives have been used to produce compressive waves with amplitudes from a few GPa to several hundred GPa in many materials [1]. High explosive plane wave generators placed either in direct contact with the material or in contact with a "standard" material upon which the specimens were placed, were used to measure Hugoniot states to approximately 60 GPa. Considerably higher pressures were obtained from explosive systems in which the high explosive was used to accelerate a thin flier plate across a gap and then impact the specimen surface. In this manner, pressures to approximately 200 GPa were generated in materials of high density by McQueen and Marsh [2] of Los Alamos Scientific Laboratory (LASL), now Los Alamos National Laboratory (LANL). Hart and Skidmore [3] increased the pressure range of these measurements to over 500 GPa, using a radially converging explosive system which accelerated plates to high velocities at some expense in precision; the converging shock wave system added complexity to the analysis. Al'tshuler et al. [4] extended the range of measurements to above 1000 GPa, accelerating his flier plates in a fashion undisclosed at that time. Until this study, which provided highly accurate extensions of lower pressure data to over 500 GPa, work in the United States had not progressed above the 200 GPa reported by McQueen. For this study, an "accelerated reservoir" light-gas gun was used to accelerate flier plates to velocities extending above 8 kilometers per second. This method of experimentation offers significant advantages over the explosive techniques previously used.
125
126
Shock Waves: Measuring the Dynamic Response of Materials
Using the light-gas gun, unshocked, stress-free impactors of similar or dissimilar material to the specimen were impacted over a wide and continuous pressure range. Of significance is the simplicity and accuracy of the calculation of the shock state in the specimen, using either symmetric impact assumptions or the measured Hugoniot of the gunlaunched impactor. hi contrast, for experiments in which the shock is created in a standard by either direct contact with explosives or on being struck by an explosively accelerated plate, the Hugoniot point is less readily calculable. In this case, it is necessary to assume a form of the equation of state for the standard in order to get the Hugoniot point for the specimen. A total of eighteen materials was tested using the gun-launched impactor technique, eleven of which are reported in this book. In each case, the pressures attained represented the highest pressures for accurate measurements in the Western world to that time. Table 5.1 lists the materials tested. Table 5.1. Summary of Materials Tested and Pressure Ranges Examined
MATERIAL
PRESSURE RANGE (Mb)
Fansteel-77
0.3-5.0
Copper (OFHC, 99.99%)
1.0-4.5
Aluminum (2024-T4)
0.5-2.2
Nickel (99.95%)
0.8^.7
Stainless Steel (Type 304)
0.8^.2
Titanium (99.95%)
0.4-2.7
Beryllium (1-400)
0.5
Beryllium (S-200)
0.3-1.6
Quartz Phenolic
1.3
Magnesium (AZ3 IB)
0.2-1.4
Plexiglas (Rohm and Haas, IIUVA)
0.7-1.0
Uranium (depleted)
0.8^1.6
5. Hugoniot Equations of State to 0.5 TPa
5.2
127
Theoretical Considerations
The application of the principles of conservation of mass, momentum, and energy across a discontinuity have led to the well-known RankineHugoniot equations. The equations were derived originally for fluids but may be applied to solids when the pressure P is understood to represent the one-dimensional stress normal to the wave front. These equations may be used to represent the discontinuous change of pressure P, density p, specific volume V, and internal energy E, across a shock front as they are related to the shock wave velocity Us and the particle velocity behind the shock front up (Figure 5.1). Material at Shock Conditions Pi
Material at Initial Conditions - ^
Po
Uj
Uo
Pi
Po
Ei
_ ^ -
Eo
Shock Front Velocity, U s Figure 5.1. Schematic of Shock Wave Parameters
P-Po Po U s
= /0o U s u p
= p(Us-up)
E - E O = 1 / 2 ( P + P O )(V-V O )
(5.1) (5.2) (5.3)
Thus, a measurement of the shock wave velocity and the particle velocity associated with the shock wave provides sufficient information to calculate the change in the thermodynamic state of the specimen (assuming that steady state conditions prevail behind the shock front).
128
Shock Waves: Measuring the Dynamic Response of Materials
Although measurement of shock velocity is relatively straightforward, the measurement of particle velocity is more difficult experimentally. For this study, two techniques were used to calculate the particle velocity associated with a measured shock velocity. The first method applies by symmetry. For an impact of a specimen launched at velocity v onto a specimen of the same material, a shock wave is induced with particle velocity equal to exactly one-half the impact velocity, or up=l/2v
(5.4)
To apply this condition rigorously, it is necessary that the impactor and specimen be in the same thermodynamic state, i.e., that the impactor has not been shock heated nor subjected to irreversible changes due to shock loading during acceleration. These conditions have been satisfied for this study. Since the impact velocity is measured with high precision (customarily -0.05%), the particle velocity also can be calculated with similar precision. A series of tests are conducted over a range of different impact velocities, the highest pressure being obtained at the highest impact velocity (about 8 km/s). Each test furnishes a point on the locus of final compressed states known as the Hugoniot. Figure 5.2 shows, in the pressure-particle velocity plane, a graphical description of the conditions of symmetrical impact. To obtain pressures higher than those created by symmetrical impact at the maximum launch velocity, it is possible to impact the specimen with a material of higher shock impedance (defined as the product of the initial density and the shock velocity) and to calculate the resultant particle velocity and pressure by an adaptation of the impedance matching technique developed by Walsh et al. [5]. With this technique, a measurement of the velocity of the impactor material (the Hugoniot of which has been previously measured) is sufficient, when combined with a measurement of the shock velocity in the specimen material, to determine a point on the Hugoniot of the specimen. Figure 5.3 shows an impactor of known Hugoniot striking a specimen with velocity v. A single shock wave of velocity Us is induced
5. Hugoniot Equations of State to 0.5 TPa
Target Hugoniot
Slope •I
/ I / i
\
I
/'/ \ ,/ \
/ /
\
\
\ /
\ ' /
a.
. / /' / / / / / > / / / / /
/— Reflected Impactor HtJ 9° niot
\
/ /
\ \
Particle Velocity UP = 1/2V i \ \
\
\
\ \ \
\
/
i— Impact Velocity /
Particle Velocity
Figure 5.2. Schematic of Symmetrical Impact Analysis Impedance Match Schematic (Dissimilar Materials)
Target Hugoniot^
^
| S °-
Slopes UsPn-v \
\ / V'
\A \ y/ \
\
\ // y/ A/ /' A
\ \ \y/ \
/-Reflected Impactor Hugoniot
Particle Velocity
Figure 5.3. Schematic of Dissimilar Materials Impact Analysis
129
130
Shock Waves: Measuring the Dynamic Response of Materials
in the specimen. The intersection of the line p\Js and the Hugoniot of the impactor, centered at velocity v, determines the shock pressure and the particle velocity in the specimen.
5.3
Experimental Techniques
The experimental determination of Hugoniot equations of state using the impedance match technique is based on the measurement of the shock velocity in the specimen and of the velocity of the impactor. With the techniques described below, these two measurements can be made with precisions of approximately 0.5 and 0.05 percent respectively, resulting in Hugoniots of good accuracy, considering the limited number of tests conducted on several of the materials. The light-gas gun range and its basic instrumentation have been described in several other papers [1, 6, 7, 8] and in Chapter 3 of this work. The next sections describe the instrumentation for shock wave measurements and the design of the equation of state targets.
5.3.1
Instrumentation
The impactor velocity measuring system consists of a laser triggering system and two short duration flash X-rays [8]. With this system, impactor velocities are measured accurate to 0.05%. The triggering system consists of a helium-neon gas laser aimed at a photodetector across the impact chamber orthogonal to and intersecting the line of flight of the projectile. A photomultiplier monitors the laser light output through a set of masks and a narrow band optical filter. When light interruption occurs due to projectile passage, a sharp change of voltage level is converted into a signal of sufficient amplitude to trigger a Field Emission Corporation, 30 nanosecond duration, dual flash X-ray unit. The X-ray flash exposes a Polaroid film plate on the opposite side of the chamber by means of a fluorescing intensifier screen. The trigger and X-ray flash system is then duplicated to record the passage of the projectile in the second field of view 30.5 cm further down range.
5. Hugoniot Equations of State to 0.5 TPa
131
The spacing between the two X-ray field center lines is indicated in the radiograph by fiducial wires, which are measured by an optical comparator to within 0.2 mm. Measurements of the impactor face position relative to the window fiducials allow calculation of actual projectile position and travel over the time interval measured between flash exposures. Figure 5.4 is an example of the shadowgraphs of the two Xray stations, showing the projectile in free flight before impact.
Figure 5.4. X-ray Shadowgraphs of Projectile Before Impact
A second method is also employed to measure impactor velocity. The time interval between the first X-ray flash and the impact of the projectile on the target is recorded electronically. The impactor and target positions are measured from the X-ray shadowgraphs and a velocity is calculated. Variations in measurements between the two techniques are usually less than 0.05%. The target is located approximately 60 cm from the launch tube muzzle and is included in the no. 2 X-ray field of view. Measurements of the shock wave transit time in the target are made using four coaxial selfshorting pins as sensors. The shorting of a pin results in a sharply rising current to ground which produces a signal across the input termination resistors of a time interval meter. The circuit is so designed that each pin signal can be seen on three output lines and is free of any reflections or ringing for several hundred nanoseconds. The individual circuits are "tuned" by the use of trimmer capacitors so that the risetime of each signal is 1.0 ± 0.1 ns to 12 volts. Thus it is possible for the combined mechanical-electronic signal system to make use of the ±1/2 ns resolution of the time recording instruments.
132
Shock Waves: Measuring the Dynamic Response of Materials
The shock wave transit time interval meters were Eldorado Model 793 counters. These counters have a specified time resolution of ±1/2 ns and may be read digitally to the nearest nanosecond. They require an input signal of 1 volt with a risetime of approximately 1 ns. Although instrument stability is specified to be one part in 104 for long term and five parts in 106 for short term, in actual practice, the instruments were calibrated prior to each shot over a period of about 10 minutes. The shot was then fired within five minutes of completion of the calibration procedure. The planarity of the shock wave induced in the target is dependent on the tilt and curvature of the impactor relative to the specimen and on the impactor flatness. The impactor surface is machine lapped and then hand polished flat to 0.5 x 10"3 inches rms. Tests performed with impactors of Fansteel-77 and OFHC copper indicate the surface curvature after launch to be less than the 5 nanosecond time resolution of the optical recording system at a launch velocity of 7 km/s. The impactor tilt relative to the target specimen front surface is sensitive to launch tube linearity and sabot alignment, as well as to target alignment. The capability to adjust the target position and perpendicularity relative to the launch tube center line by an optical technique brings the average tilt at impact to approximately 0.005 radians (approximately 15 nanoseconds of tilt at an impact velocity of 7 km/s). Because of the comparatively gentle acceleration of the projectile to its terminal velocity, the impactor plate is not shock heated. In addition, free flight in an evacuated range precludes aerodynamic heating. This accounts for the flatness of the impactor after launch and significantly reduces the complexity of the experiment. The estimated temperature rise during launch of the order of 1°C.
5.3.2
Target Design and Construction
The 29 mm diameter of the launch tube places restrictions on the diameter of the impactor plate and on the size of the specimen which are severe enough to require a thorough study of the optimization of target dimensions. In general, it is desirable to operate over as long a time base (specimen thickness) as possible for transit time measurements.
J. Hugoniot Equations of State to 0.5 TPa
133
However, the launch tube diameter controls the allowable specimen thickness since, for larger specimens, rarefactions arising from both the unconfined edges and the free rear surface of the impactor plate can overtake the head of the shock wave before measurements have been taken. To determine the maximum specimen thickness which would still maintain an un-rarefacted area on the rear surface of the specimen on which sensors could be placed to record wave arrival, the following analysis was used. A typical estimate of the angle of intrusion of plastic rarefaction waves from the specimen edges is to assume that (Xi ~ 45° (see Figure 5.5, below) and to ignore the elastic rarefaction wave system. Although for many materials this assumption is justified, for some materials this criterion is inadequate; in particular, materials with a low Poisson's ratio must be calculated more carefully. The elastic wave velocity, Ce , is given by
c -- t Wri I = 0 - K
<5-5)
where v is Poisson's ratio and Cp is the plastic wave velocity, which for strong shocks is a function of up, the particle velocity. Existing experimental results indicate that v is a weak function of the shock strength [6, 9, 10]. To a good approximation, the elastic rarefaction angle, a2,
(U - u
tan a2 = K2 tan 2 ax + (K2 - l) - ? — ^
2Ji
(5.6)
Figure 5.5 is a plot of a2 versus Poisson's ratio for the metals listed in Table 5.2. The calculation assumes a value of tan a2 = 0.7, taken from the work of AFtshuler, who notes that for compressions greater than -1.3, tan <xi becomes essentially constant at 0.70 ± .03, and (Us-up)/Us is taken as unity — its maximum value. Included in Figure 5.5 is a comparison of values of tan a2 measured in this test series [10] for
134
Shock Waves: Measuring the Dynamic Response of Materials
Figure 5.5. Intrusion Angle of Elastic Edge Rarefaction vs. Poisson's Ratio
copper, aluminum, titanium, and beryllium. As these measurements fall below the calculated line of the minimum allowable design angle, it is felt that Equation 5.6 provides a reasonably conservative design criterion. In practice, the design angle was chosen to be several degrees larger than the values listed. Impactor thicknesses were chosen to keep rarefactions originating at the impactor free rear surface from overtaking the shock wave until shock transit time measurements were complete. A single impactor thickness was calculated which was adequate for all materials and was used in all tests. The target specimen was a machined and ground disk with the impact and rear surfaces machine lapped and hand polished to a surface finish of 1 microinch rms or better. The lapping procedure employed produced surface flatness to within ~10~3 mm. Parallelism was maintained to ~10"3 radians. The thicknesses of all specimens were measured to an accuracy of ±0.5 x 10~3 mm.
137
5. a Hugoniot Equations of State to 0.5 TPa
Table 5.2. Minimum Design Angles (Maximum Edge Rarefaction Angles), Calculated from Equation 5.6
Material
Poisson's Ratio
Minimum Design Angle, a 2
Aluminum
.332
47°
Beryllium
.055
58°
Copper
.356
45°
Tantalum
.342
46°
Tungsten
.280
50°
Uranium
.402
43°
Titanium
.304
48°
Lead
.430
41°
Magnesium
.306
49°
Nickel
.300
48°
Steel (Mild)
.290
49°
Plexiglas
.327
47°
The target specimen thicknesses at the pin stations were measured with a Zeiss light-section microscope employed as a comparator. The use of the light-section microscope avoids the problem of an indicator marring the specimen surface, since no physical contact is made with the surface. Rather, the vertical position of a thin beam of light projected onto the specimen is compared with the position of the beam projected onto a laboratory-grade gage block, and the specimen thickness is calculated. The basic features of the target design employed in this work are illustrated in Figures 5.6 and 5.7. Two coaxial shorting pins were passed by the edge of the specimen disk and their cap faces aligned exactly in the plane of the specimen impact surface (within 10~3 mm). These pins were used to initiate the timing for the shock wave transit time measurement and to measure impactor tilt in terms of the time interval difference between their respective closures. Two rear surface pins (or one, for small diameter targets) were mounted in line with the tilt pins to record the shock wave arrival at the rear surface. All four pins were mounted in a guide fixture which assured the proper geometrical spacing. The tilt pins were fixed in position with a
136
Shock Waves: Measuring the Dynamic Response of Materials
dimensionally stable epoxy, while the rear surface pins were springloaded in place in the pin guide. The pin retainer and cable bracket lent rigidity to the assembly to prevent accidental damage to the pin shafts. The four-pin targets, in conjunction with the four Eldorado one-nanosecond time interval meters, produced four values of the shock wave transit time and a correction for impact tilt. From these four values a single shock Figure 5.6. Schematic of Target and wave velocity in the specimen was Impactor calculated and, by a system of cross-checking, an indication of the precision of the measurement was available. The degree of non-planarity of impact between target and impactor was calculated by comparing shock transit times recorded by the counters started by each of the two front surface pins. The time difference between the shorting of the front surface pins, At (which, when combined with the impact velocity, yields the tilt angle, 0), is calculated from Att = ti_c - tA-c = ti_B - tA-B, where ti_c is the time recorded on the counter started by pin no. 1 and stopped by pin C (Figure 5.6). For the highest velocity tests (7-8 km/s), it was necessary to lighten the projectile by reducing the diameter and the thickness of the impactor plate. The target designed for these highest pressure shots was thinner, had a slightly smaller diameter, and was Figure 5.7. Photograph of Target
5. Hugoniot Equations of State to 0.5 TPa
137
provided with only one coaxial shorting pin on the rear surface. The coaxial self-shorting pins employed in this work as sensors consisted of a one millimeter diameter tube of brass surrounding a Teflon sleeve and a copper inner conductor. The pins were connected to RG174 50 ohm cable by soldered joints and were made self-shorting by the placement of a brass cap over the sensing end. A shoulder on the cap left a small gap (on the order of 0.050 ± .002 mm) between its inner face and the flat end of the inner Figure 5.8. X-ray of Coaxial Shorting Pin conductor. When a large amplitude stress wave reaches the cap face, the cap is set into motion and the gap is closed at the free surface velocity of the cap material. The pin gaps were measured by X-ray shadowgraphy, of which Figure 5.8 is an example, and the measurements were employed in the shock velocity calculations for corrections to pin closure times.
5.5.5
Data Analysis
The analysis of the experimental data obtained in the research program is based upon the impedance match solution for the determination of particle velocity, pressure, energy, and volume of the shock state in the specimen. The requisite information in the analysis is the shock wave velocity, the impact velocity, and the Hugoniot of the impactor. A general description of the method of data analysis is given here, with the detailed equations presented in Reference 1.
5.3.4
Shock Wave Velocity
The measurements relevant to shock wave velocity are the target thickness and the shock wave transit time interval. In order to calculate the shock wave transit time, it is necessary to make refinements upon the recorded time interval.
138
Shock Waves: Measuring the Dynamic Response of Materials
Two sources of refinement to the measurement are: (1) Inclusion of the effects of shock wave tilt resulting from nonplanar projectile impact on the target. (2) Correction for the differences in closing times of coaxial pins with (slightly) different gaps, which involves: (a) Calculation of the interaction of the impactor with the two front surface pins. (b) Calculation of the interaction of the specimen material with the two rear surface pins. To calculate the closing velocity of the cap for the direct impact of the projectile material on the front surface pins, an impedance match solution was applied, using the impact velocity, the Hugoniot of the impactor, and the Hugoniot of the cap material (brass). The pin gap correction for the interaction of the rear surface pins and the specimen is based on the impedance match solution of the shock wave in the target being transmitted into the pin material. To begin the procedure, the Hugoniot of the specimen was first estimated to provide the necessary constants. A preliminary calculation was made with no pin gap corrections and a preliminary Hugoniot point was determined for the specimen. This Hugoniot point was then used to calculate pin interactions and, through a series of iterations (usually two), the preliminary Hugoniot point was modified until satisfactory convergence was reached (differences < 0.01%).
5.3.5 Impact Velocity Measurement of impact velocity has been discussed earlier under "Instrumentation." The flash X-ray system used furnishes a high precision velocity determination, providing the projectile maintains a constant velocity during the time of measurement. A check on this premise was provided by a second system which measured velocity over a longer baseline. No evidence of projectile acceleration or deceleration during its free flight was observed.
5. Hugoniot Equations of State to 0.5 TPa
5.3.6
139
Hugoniot of the Impactor
Measurement of impactor Hugoniots for tests in which specimens are impacted with dissimilar materials is discussed in the following section, "Experimental Results."
5.4
Experimental Results
Results of measuring Hugoniots over a period of several years are presented in this section, organized in three subsections: (1) A written description of test results for each material (eleven in all), including fits to the data and tables with chemical and physical analyses. (2) Tables of the data obtained. (3) Graphs of the data. The experimental work effort was first concentrated on the measurement of the Hugoniots of the three materials to be used as impactor plates for dissimilar material impacts. The materials were chosen: (1) To cover a range of shock impedances. (2) To be easily obtainable and consistent in their material properties from batch to batch. (3) To coincide with standards chosen by other laboratories. The materials investigated were Fansteel-77 (a tungsten alloy), OFHC copper, and 2024-T4 aluminum. The ratio of shock impedances of the copper and Fansteel-77 with respect to the aluminum is approximately 1:2:4. These three materials were more thoroughly investigated than the remaining materials, so that errors in the impedance match solution for materials impacted by these standards would be minimized. A typical test series for the determination of the Hugoniot of a specimen material, for instance nickel, began with a series of shots using nickel for both impactor and target over the full velocity range of the
140
Shock Waves: Measuring the Dynamic Response of Materials
gun. To obtain pressures higher than those created by like-like impact at the highest velocity, the series continued with impacts using a material of higher impedance than the specimen; in the case of nickel, Fansteel-77 was used. Impact velocities were adjusted to space the shots over the pressure range to be investigated. In the following section, the experimental data are presented in tabulated and graphical form. Fits to the shock velocity vs. particle velocity data have been made by the method of least squares and are listed for each material. The tabulations include measured and calculated parameters and an indication of the "weighting factor" used in the least squares fits. In general, test results were weighted according to whether the target had single or double pins, the double pin targets generally having a higher weighting factor, due to the redundant measurements of shock wave velocity. The tabulated data also include the impactor material and the impact velocity. Additional figures are included to illustrate comparison with other researchers. Fansteel-77 Fansteel-77, a tungsten alloy composed nominally of 90% tungsten, 6% nickel, and 4% copper, was employed as the standard for the highest pressure tests. Fansteel-77 was chosen over pure tungsten because the metallurgical structure reduces the brittleness of the material (which can cause plate fracture during launch) while maintaining high strength and density. In the initial work, the quality control of the Fansteel-77 stock material presented several problems. The material is produced by powder metallurgical techniques which include pressing and sintering of a billet of the material. The porosity of the surface of the billet was found to be somewhat dependent upon the compacting pressure prior to sintering. The core of the billet, however, was found upon metallographic examination to exhibit essentially no porosity, and sample to sample variations in density were less than 0.5% when the outside 1 mm was removed from a 50 mm diameter bar. The chemical and physical properties of Fansteel-77 are presented in Table 5.3.
5. Hugoniot Equations of State to 0.5 TPa
141
The test series on Fansteel-77 resulted in a Hugoniot over the pressure range 0.35 Mb to 5.0 Mb. Symmetric impacts were used on all tests, so that particle velocity may be assumed to be one-half the impact velocity. Both second- and third-order fits were made to the data; however, the linear relationship Us = C + Sup describes the data with a root mean square (RMS) deviation not significantly larger than the higher order fits. The relationship is given by: U s = 4.008+ 1.262 up km/s RMS deviation of Us was ±0.021 km/s for the seventeen data points. The results are presented in Table 5.11. The data points taken are displayed in the Us vs. up plane in Figure 5.9(a). hi one shot the velocity was not measured directly with the X-ray system but instead was calculated from the gun firing parameters. Estimates made in this manner are quoted with standard errors in velocity of ±2%. In Figures 5.9(b-d) are displayed the data for Fansteel-77 compared with the data obtained by researchers of the Ballistics Research Laboratory of the United States Army, Aberdeen Proving Grounds, and the data of Hart and Skidmore [3] for a tungsten alloy similar to Fansteel-77. The Ballistics Research Laboratory data are in excellent agreement with the results obtained here. The Hugoniot measured by Hart and Skidmore has the quadratic form: Us = 2.95 + 2.47 up - 0.342 up2 (km/s) and falls below the present data at lower values of up. This behavior is not readily explainable, since the intercept of the linear fit to the present work, Co = 4.008 km/s, is in reasonable agreement with the bulk sound velocity measured ultrasonically, Co = 3.912 km/s. It is interesting to note that the Hugoniots for pure tungsten (as measured by LASL) and for Fansteel-77 are quite similar, the LASL
142
Shock Waves: Measuring the Dynamic Response of Materials
linear fit for tungsten being given by: Us = 4.029+1.237 Up km/s although the density of the tungsten is 13% higher than that of Fansteel-77. Table 5.3. Chemical and Physical Properties of Fansteel-77 Chemical Properties Element Tungsten Copper Nickel
wt % 90 ±.1 3.8 ±.6 6.1 ±.2
Physical Properties Yield Strength (0.2% elongation) Ultimate Tensile Strength Density Poisson's Ratio Acoustic Velocities:
85,000 psi (min) 98,000 psi (min) 17.01 ± .01 g/cm3 0.286
Longitudinal Shear Bulk
CL = 5.049 km/s C s = 2.765 km/s Co = 3.912 km/s
OFHC Copper OFHC copper of 99.99% purity is employed as a standard for the intermediate and high pressure ranges in the Hugoniot experiments of a number of laboratories. Physical characteristics of the copper are described in Table 5.4. The results of twelve tests are presented here in Table 5.12. The linear fit to the data in the Us - up plane is given by: Us = 3.964+1.463 Up km/s Root mean square deviation of Us = 0.009 km/s for 12 data points. The (lower pressure) Hugoniot equation published by LASL [12] is given by: Us = 3.940+1.489 Up km/s in close agreement with the present results.
5. Hugoniot Equations of State to 0.5 TPa
143
The data are presented in Figure 5.10(a), which includes a legend to indicate the impactor material used. The use of different impactors provides a good method for cross-checking the Hugoniots [11] of the standards as suggested by McQueen. The Hugoniot established by Fansteel-77 impacting copper is indistinguishable from the copper on copper tests. This both checks the accuracy with which the Fansteel-77 Hugoniot was determined and provides additional confidence in the use of impedance matching techniques as used in this experimental system. A single test was conducted using 2024 aluminum as an impactor. All tests were used in the calculation of the fit given above. The resulting Hugoniot displays a very small RMS deviation (-0.1% in shock velocity). Figures 5.10(b-d) compare the present data with the data of Al'tshuler [9, 13], Walsh [5], and McQueen [2]. Agreement within - 2 % in the linear fit was obtained (i.e., the reported shock velocities are within - 2 % of the values predicted by the present linear fit). A close examination of the data indicates a small (< 1 %) deviation from a linear fit, beginning at up = 2.4 km/s. It is possible that this deviation is associated with melting in the shock front (see the following discussion on 2024-T4 aluminum). Table 5.4. Chemical and Physical Properties of OFHC Copper Chemical Composition Element
wt %
Iron Sulfur Silver Nickel Antimony Lead Copper
0.0005 0.0025 0.0010 0.0006 0.0005 0.0006 Remainder
Physical Properties Density: Poisson's Ratio Acoustic Velocities:
8.930 g/cm3 0.332 Longitudinal Shear Bulk
CL = 4.757 km/s C s = 2.247 km/s Co = 3.99 km/s
144
Shock Waves: Measuring the Dynamic Response of Materials
2024-T4 Aluminum The Hugoniot experiments performed on 2024-T4 aluminum extend over a pressure range of 0.45 to 2.2 Mb. Fourteen tests were conducted, employing the following impactors: Fansteel-77 (5 tests); OFHC copper (5 tests); and 2024-T4 aluminum (4 tests). In addition, Shot No. 98 from the series on copper is included as a cross-check by reversing the impactor and specimen materials and impacting an aluminum plate into a copper target. If the Hugoniot of copper is assumed to be known, the state in the aluminum may be calculated. The chemical and physical properties of 2024-T4 aluminum are tabulated in Table 5.5, and the data are presented in tabular form in Table 5.13. The measured initial density of the aluminum was 2.783 ± .001 g/cm3. Figure 5.11 (a) is a plot in the Us - up plane of the experimental results. The linear fit is from LASL [22]. A departure from linear behavior is seen, beginning at up = 3.5 km/s (P = 100 GPa), where the shock velocity falls below the line representing a linear fit to the data. Although the data show scatter, it was felt that the trend of the data in this region was beyond experimental error. The linear fit to the lower pressure data of LASL, Us = 5.328 + 1.338 up, was compared to the present data by calculating deviations of the data from this fit. These deviations are plotted in Figure 5.12, along with other high pressure data from Russian researchers. Only the ten tests showing least internal scatter and tilt are plotted. Urlin [15] has proposed a model for melting in the front of a shock wave and predicts an observable effect on the linear Us - up relation. For aluminum, the melting is calculated to begin at approximately 100 GPa. The present data follow the trend predicted by Urlin, although whether melting, experimental inaccuracy, or other phenomena is the explanation for the large deviations found between Us = 3.5 to 4.5 km/s remains to be verified. Evidence counter to this hypothesis is contained in Chapter 6, Attenuation of Shock Waves from High Pressure. Subsequent examination of the data indicated that some of the deviations could have been caused by curvature of the aluminum impactor plates when launched at the highest velocities. Unfortunately, scheduling difficulties precluded conducting additional tests to detemine the validity of the original measurements.
5. Hugoniot Equations of State to 0.5 TPa
145
Comparisons of Hugoniot data from other workers [9, 13, 14, 16] are shown in Figures 5.11(b-d). If all the present data points are combined, the data are represented by the equation: Us = 5.471+ 1.310 Up Root mean square deviation = .022 km/s for 11 data points. Table 5.5. Chemical and Physical Properties of 2024-T4 Aluminum Chemical Composition Element Silicon Iron Copper Manganese Magnesium Chromium Zinc Aluminum
wt % 0.5 0.5 3.8-4.9 0.3 - 0.9 1.2- 1.8 0.10 0.25 Remainder
Physical Properties Yield Strength Ultimate Tensile Strength Hardness (Brinell No.) Density (measured) Poisson's Ratio Acoustic Velocities:
47,000 psi 68,000 psi 120 2.783 g/cm3 0.332 Longitudinal Shear Bulk
CL = 6.38 km/s C s = 3.20 km/s Co = 5.20 km/s
(Based on Alcoa Aluminum Handbook and specimen certification)
Depleted Uranium Five tests were performed for the determination of the Hugoniot of depleted uranium over the pressure range 90 to 460 GPa. The impactors were either OFHC copper or Fansteel-77. No similar material impact tests were performed. The data are shown in Figures 5.13(a-c). Excellent agreement was obtained with the high pressure (above 200 GPa) data of Skidmore and Morris [17]. The density of the samples was 18.951 g/cm3. The chemical and physical properties are summarized in Table 5.6, and the Hugoniot data are presented in Table 5.14. It was noted that the freshly lapped surfaces
146
Shock Waves: Measuring the Dynamic Response of Materials
oxidized rapidly, turning from a light tan to a dark blue-brown color. That this oxide layer is very thin may be demonstrated by the fact that several swipes of the surface on a wet lapping plate removed the darkened layer. The linear fit to the present data is given by the equation: Us = 2.443+ 1.582 Up km/s Root mean square deviation = ±.029 km/s for 5 data points. Skidmore's data for 5 data points are included in the figures and are fit by the quadratic equation: Us = 2.55 + 1.504 up + .0901 up2 for Up > 2.5 km/s. The linear fit from LASL [12] is Us = 2.487 + 1.539 up. Table 5.6. Chemical and Physical Properties of Depleted Uranium Chemical Composition Element
wt (ppm)
Aluminum Boron Cadmium Chromium Copper Iron Magnesium Manganese Molybdenum Nickel Lead Silicon Samarium Uranium (wt %)
9 0.3 0.3 3 8 10 2 15 2 20 1 40 1 99.8%
Physical Properties: Yield (0.1% elongation) Tensile Strength Density (measured Poisson's Ratio Acoustic Velocities:
47,250 psi 124,000 psi 18.951 g/cm3 0.402 Longitudinal Shear Bulk
CL = 2.97 km/s C s = 1.20 km/s Co = 2.63 km/s
5. Hugoniot Equations of State to 0.5 TPa
147
Nickel Five tests were performed to obtain Hugoniot data for nickel. Three of the tests employed nickel impactors, and two impactors were of Fansteel77. The nickel was purchased under specification ASTM-B-160-61 as 99.5% purity. The chemical and mechanical properties of the material are listed in Table 5.7. The experimental data are tabulated in Table 5.15 and displayed in Figure 5.14(a). The least squares linear fit to the data is: Us = 4.456+1.555 Up km/s Root mean square deviation = 0.012 km/s for 5 data points. The data of McQueen and Marsh [2], Walsh et al. [5], and Al'tshuler et al. [18] are displayed in Figures 5.15(b-c) and show reasonable agreement with the present data. The data of Al'tshuler extend over a wider pressure range (110 to 920 GPa) than the present data. The Al'tshuler data are best fit by a quadratic curve: Us = 4.370 + 1.775 up - 0.047 up2 km/s Sigma Us = 0.008 km/s for 4 data points. Table 5.7. Chemical and Physical Properties of Nickel Chemical Composition Element
Weight %
Carbon Manganese Iron Sulfur Silicon Copper Nickel
0.11 0.26 0.09 0.005 0.02 0.02 99.47
Physical Properties Yield Strength (at 0.2% Elongation) Tensile Strength Density Poisson's Ratio Acoustic Velocities:
Longitudinal Shear Bulk
82,500 psi 89,000 psi 8.864 g/cm3 0.300 CL = 5.76 km/s C s = 3.08 km/s Co = 4.53 km/s
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Shock Waves: Measuring the Dynamic Response of Materials
Type 304 Stainless Steel The composition and mechanical properties of Type 304 stainless steel are presented in Table 5.8. The measured density is 7.905 g/cm3. The Hugoniot data over a pressure range of 80 to 430 GPa are listed in Table 5.16 and are presented graphically in Figures 5.15(a-c). The linear Hugoniot fit is given by: Us = 4.722+1.441 Up km/s Root mean square deviation of Us = 0.023 km/s for 4 data points. Also displayed are the data from the U. S. Army Ballistics Research Laboratory [19] for comparison with the present results. Although the pressure ranges tested are barely overlapping, the extrapolation of the present data to lower pressures is in reasonable agreement with the Ballistics Research Laboratory results. Table 5.8. Chemical and Physical Properties of Type 304 Stainless Steel Chemical Composition Element
Weight %
Carbon Manganese Phosphorous Sulfur Silicon Nickel Chromium Molybdenum Copper Iron Other (CO)
0.065 1.62 0.029 0.028 0.49 8.80 18.73 0.14 0.17 69.86 0.070
Physical Properties Yield Strength Tensile Strength Hardness Density Poisson's Ratio Acoustic Velocities:
55,000 psi 90,500 psi BHN192 7.905 g/cm3 0.290 Longitudinal Shear Bulk
CL = 5.74 km/s C s = 3.12 km/s Co = 4.47 km/s
5. Hugoniot Equations of State to 0.5 TPa
149
Titanium Three Hugoniot tests were performed on titanium over a pressure range of 40 to 270 GPa. The chemical and physical properties of the titanium samples are listed in Table 5.9. The Hugoniot data are presented in Table 5.17 and in Figure 5.16(a). The linear fit to the Hugoniot is given by: Us = 4.692+1.126 Up km/s Root mean square deviation of Us = 0.014 km/s for 3 data points. In Figures 5.16(b-d) are graphical comparisons of the data of Krupnikov [20], Walsh et al. [5], and LASL [12] with the present work. Krupnikov: Us = 4.89 + 1.11 up km/s (0.8 to 2.8 Mb). Walsh et al: Us = 4.590 + 1.259 up km/s (0.7 to 1.4 Mb). LASL: Us = 4.877 + 1.049 up km/s (0.8 to. 1.1 Mb). A surprisingly large spread in slope and intercept is seen. Consolidation of all the data above yields a linear fit: Us = 4.695+ 1.146 Up Root mean square deviation of Us = 0.045 km/s for 13 data points, deviating but very little from the present data alone. Table 5.9. Chemical and Physical Properties of Titanium Chemical Composition Element
Weight %
Carbon Nitrogen Iron Oxygen Titanium
0.02 0.010 0.25 0.115-0.123 99.60
Physical Properties Yield Strength (at 0.2% elongation) Tensile Strength Density Poisson's Ratio: Acoustic Velocities:
50,000-55,500 psi 75,600-76,300 psi 4.508 g/cm 3 0.304
Longitudinal Shear Bulk
Q,= 6.118 km/s Cs = 3.246 km/s Co = 4.83 km/s
150
Shock Waves: Measuring the Dynamic Response of Materials
Beryllium Hugoniot experiments were performed on two types of beryllium designated as S-200 and 1-400, having nominal compositions of 2% and 4% BeO respectively. The chemical and physical properties of the beryllium specimens as furnished by the supplier, Brush Beryllium Co., are listed in Table 5.10. The specimen preparation presented somewhat of a problem in that dust and small fragments of the material are considered toxic. All polishing and lapping of samples was performed in closed and vented work areas. The beryllium presented the most difficult case for accurate Hugoniot measurements with a restricted target size, because the elastic release wave velocity is relatively high and the target thickness is thus necessarily less than that for other materials. (See Figure 5.5 for the maximum angle of intrusion of the side rarefactions.) The Hugoniot data presented in Table 5.18 include seven tests on S200, the material of major interest, and one comparison test on 1-400. The impactors for the test series were OFHC copper (3 tests) and Fansteel-77 (5 tests). The data are presented graphically in Figures 5.17(a-d). A linear fit was made to the data for the S-200 beryllium: Us = 8.390 + 0.975 up km/s Root mean square deviation of Us = 0.017 km/s for 7 data points. The 1-400 beryllium is slightly more dense (-1.3%) than the S-200 beryllium, due to a greater percentage of the heavier beryllium oxide compound. The data point is slightly above the fit of the S-200. A comparison of the data with that of LASL [12] on a beryllium of similar BeO content shows a significant difference in the slopes of two linear fits, the slope of LASL's data indicating a somewhat "stiffer" Hugoniot than the slope of the present data (Us = 7.998 + 1.124 up ). The reason for this difference is difficult to determine. The possibility of edge rarefactions affecting the measurements on the present specimens was checked by firing several tests with specimens substantially thinner than the value dictated by the analysis in Section 5.1.5. No significant differences in the measured values were noted.
5. Hugoniot Equations of State to 0.5 TPa
151
Table 5.10. Chemical And Physical Properties Of S-200 And 1-400 Berylliums Chemical Composition Material
1-400 Wt. %
Beryllium Oxide Carbon Iron Aluminum Magnesium Silicon Manganese Beryllium Other
3.96 0.18 0.16 0.04 0.02 0.04 0.01 95.77 0.10
S-200 Wt. % 2.00 0.126 0.163 0.054 0.035 0.080 — 98.18 0.04
max
Physical Properties Yield Strength Ultimate Tensile Strength Density Poisson's Ratio
56,000 66,700 1.881 0.055
37,500 57,700 1.857 0.055
psi psi g/cm3
Acoustic Velocities: Longitudinal Shear Bulk
12.83 8.80 7.83
12.916 8.86 7.87
km/s km/s km/s
CL= Cs = Co=
AZ31B Magnesium Four Hugoniot tests were performed on AZ31B magnesium tooling plate material (p0 = 1.773 g/cm3). The nominal composition of the material is Mg 96.0%, Al 3.0%, and Zn 1.0%. The data show more scatter than was found in tests with other metals. The Hugoniot data are presented in Table 5.19 and in Figures 5.18(a-c). The Hugoniot is described by the linear equation: U, = 4.551+ 1.209 Up km/s Root mean square deviation of Us = 0.083 km/s for 4 data points. The figure compares graphically the measured values with the AZ31B magnesium data of the LRL Compiler [21] and the data on 99.5% pure magnesium by LASL [12]. Linear fits to the other data are:
a154
Shock Waves: Measuring the Dynamic Response of Materials
Us = 4.516 + 1.256 Up km/s (LASL, po = 1.745 g/cm3) Us = 4.648 +1.198 Up km/s (LRL Compiler, p0 = 1.78 g/cm3) Plexiglas (PMMA) The Plexiglas tested was identified as Rohm and Haas Type II UVA. Only two tests were performed, which served to provide a check on the extrapolation of the data of Hauver [23], taken above the phase transition at -20 GPa (line A of Figure 5.19(a)), into the higher pressure region investigated by Bakanova [24] (line B). The data are listed in Table 5.20. Fits to the above data are: Us = 3.51 + 1.25 Up km/s (Hauver) (2.8 < up < 3.5 km/s) Us = 3.10 + 1.32 up km/s (Bakanova) (3.0 < up < 8.0 km/s) Figure 5.19(a) displays the present data as compared to that of Hauver and Bakanova, with the line representing Bakanova's data between 3.0 < up < 8.0 km/s. The two data points determined in the present work are in good agreement with the data of Bakanova (difference in shock velocity less than 0.4%) and also are in agreement with the extrapolation of Hauver's data (difference in shock velocity of less than 2% at a pressure of 100 GPa). Quartz Phenolic A single test was performed to establish a very high pressure point on the Hugoniot of quartz phenolic. Lower pressure data were also obtained in another study and listed in Table 5.21 and displayed in Figures 5.20(a-c). A phase change is clearly evident, beginning at a pressure of -20 GPa. The data point obtained is on the higher pressure phase of the Hugoniot. The linear fit to the upper portion of the Hugoniot is given by: U s = 1.949+1.364 up km/s Root mean square deviation of Us = 0.016 km/s for 5 data points. The initial density of the quartz phenolic is 1.80 g/cm3. The properties of the high pressure phase suggest that this may be related to the properties of quartz under high pressure, i.e., the high pressure
5. Hugoniot Equations of State to 0.5 TPa
153
polymorph Stishovite [7, 26]. The tests were made with lay-up directions of the quartz cloth in the phenolic matrix both parallel and perpendicular to the shock front, but orientation effects on the wave velocity are apparently negligible at these pressures. Hugoniot data in the form of tables and graphs is presented next, followed by a summary of the data taken.
5.5
Hugoniot Data
5.5.1 Tables of Hugoniot Data Table 5.11. Hugoniot Data for Fansteel-77 V o = 0.0588 cm 3 /g; p 0 = 17.01 ± .01 g/cm 3
Shot
Impactor
S-23
FS
C-922*
FS
V,
Us
up
P
(km/s)
(km/s)
(Mb)
0.906
4.599
0.453
0.354
1.478*
4.978
Material (km/s)
VREL
VO
p0
(cm'/g)
(g/cm3)
.9015
.0530
18.87
3
.0501
19.98
1
0.740
0.627
.8514
±.019
±.016
±.0039
WF
C-1161
FS
1.952
5.194
0.976
0.862
.8121
.0477
20.95
3
C-939
FS
2.152
5.336
1.076
0.977
.7984
.0469
21.31
1
C-1218
FS
2.689
5.661
1.344
1.294
.7626
.0448
22.31
3
C-921
FS
2.689
5.756
1.344
1.316
.7665
.0451
22.19
3
C-933
FS
2.774
5.786
1.387
1.365
.7603
.0447
22.37
2
S-21
FS
2.849
5.855
1.425
1.419
.7566
.0445
22.48
3
C-1155
FS
3.413
6.163
1.706
1.788
.7232
.0425
23.52
3
C-914
FS
3.548
6.221
1.774
1.877
.7148
.0420
23.80
1
C-940
FS
3.618
6.365
1.809
1.959
.7158
.0421
23.76
3
S-18
FS
3.648
6.412
1.824
1.989
.7155
.0421
23.77
1
C-931
FS
3.994
6.490
1.997
2.205
.6923
.0407
24.57
2
C-923
FS
4.657
6.939
2.329
2.749
.6644
.0391
25.60
1
C-1220
FS
4.809
7.082
2.405
2.897
.6604
.0388
25.76
3
C-1232
FS
6.646
8.239
3.323
4.529
.6077
.0357
27.99
1
C-962
FS
6.936
8.390
3.468
4.949
.5867
.0345
29.00
1
* Impact velocity determined from measured gun firing parameters
154
Shock Waves: Measuring the Dynamic Response of Materials Table 5.12. Hugoniot Data for OFHC Copper Vo = 0.1120 cnvVg; p 0 = 8.930 g/cm3
„, Shot
Impactor V; Material (km/s)
Us (km/s)
up (km/s)
P (Mb)
VREL
VO (cmVg)
p0 (g/cm 3 )
WF
1081
Cu
3.426
6.463
1.713
0.989
.7350
.0023
12.15
3
1240
FS
3.875
7.414
2.343
1.551
.6840
.0766
13.06
3
S-96
Al
7.929
7.815
2.631
1.83G
.6633
.0743
13.46
1
S-15
FS
4.673
8.062
2.606
2.022
.6517
.0730
13.70
3
1082
Cu
6.434
6.669
3.217
2.490
.6209
.0704
14.20
3
S-76
Cu
7.010
9.076
3.509
2.943
.6134
.0687
14.56
3
963
Cu
7.097
9.149
3.549
2.900
.6121
.0685
14.59
1
1090
Cu
7.121
9.196
3.561
2.924
.6126
.0686
14.57
1
1249
FS
6.728
9.785
3.988
3.485
.5924
.0663
15.07
1
S-82
Cu
7.937
9.002
3.971
3.476
.5949
.06G6
15.01
1
S-19
FS
7.405
10.390
4.368
4.053
.57%
.0649
15.41
1
S-161
FS
7.932
10.785
4.674
4.502
.5666
.0635
15.76
1
VO
p0
WF
(cmVg)
(g/cm3)
Table 5.13. Hugoniot Data for 2024-T4 A l u m i n u m V o = 0.3593 cnrVg; p 0 = 2.783 g/cm 3
Shot
Impactor
Vi
Material (km/s)
Us
up
P
(km/s)
(km/s)
(Mb)
VREL
C-1238
FS
2.583
8.099
2.017
0.455
.7510
.2698
3.706
1
S-65
Al
6.161
9.504
3.081
0.815
.6758
.2428
4.118
1
S-74
Cu
4.883
9.793
3.289
0.896
.6642
.2387
4.190
1
S-198
Al
6.876
9.839
3.439
0.942
.6505
.2337
4.278
1
S-17
FS
4.775
10.047
3.663
1.024
.6354
.2283
4.380
1
S-197
Al
7.526
10.432
3.763
1.092
.6393
.2297
4.353
1 1
S-185
Cu
5.621
10.313
3.769
1.088
.6326
.2273
4.399
S-199
Cu
5.654
10.159
3 428
1.082
.6232
.2239
4.466
1
S-68
Al
8.067
10.594
4:034
1.189
.6192
.2225
4.494
1
C-1246
FS
5.400
10.585
4.127
1.216
.6101
.2192
4.562
1
S-183
Cu
6.567
10.884
4.438
1.344
.5922
.2128
4.699
1
S-66
Cu
7.833
12.351
5.237
1.800
.5760
.2070
4.832
1
S-98+
Cu
7.929
12.460
5.297
1.837
.5749
.2066
4.841
1
S-69
FS
7.947
13.228
5.962
2.196
.5491
.1973
5.068
1
+Aluminum impacted into OFHC copper standard
5. Hugoniot Equations of State to 0.5 TPa
155
Table 5.14. Hugoniot Data for Depleted Uranium V o = 0.0528 cnrVg; p 0 = 18.95 g/cm 3
Shot 1273
Impactor
Vi
Material (km/s) FS
2.044
Us
up
P
(km/s)
(km/s)
(Mb)
4.206
1.077
0.859
VREL
VO
p0
(cm3/g)
(g/cm3)
.0393
25.47
.7439
WF
3
S-125
Cu
3.885
4.902
1.602
1.488
.6732
.0355
28.15
3
S-124
Cu
3.918
4.940
1.613
1.510
.6735
.0355
28.14
3
1274
FS
4.134
5.806
2.086
2.295
.6407
.0338
29.58
1
S-128
Cu
7.967
7.596
3.214
4.627
.5769
.0304
32.85
1
VO (cnrVg)
p0 (g/cm3)
WF
3
Table 5.15. Hugoniot Data for Nickel V o = 0.1128 cm 3 /g; p 0 = 8.864 g/cm 3
Shot
Impactor Vf Material (km/s) 2.662
Us (km/s)
up (km/s)
P (Mb)
VRH.
6.546
1.331
0.772
.7967
.0899
11.13
S-85
Ni
S-87
Ni
3.839
7.407
1.919
1.260
.7409
.0836
11.96
3
S-86
Ni
7.184
10.070
3.592
3.206
.6433
.0726
13.78
3
S-120
FS
7.934
11.572
4.581
4.699
.6041
.0682
14.67
1
S-95
FS
7.993
11.607
4.616
4.749
.6023
.0680
14.72
1
WF
Table 5.16. Hugoniot Data for Type 304 Stainless Steel
Vo = 0.1265 cnrVg; p0 = 7.905 g/cm3 Us
up
P
Material (km/s)
(km/s)
(km/s)
(Mb)
S-93
St. Steel 2.993
6.877
1.497
0.814
S-77
St. Steel 5.648
8.760
2.824
S-88
St. Steel 7.214
9.980
3.607
11.459 ±.128
4.730
4.285 ±.046
Shot
S-119
Impactor
FS
Vi
7.883
VREL
VO
p0
(cnrVg)
(g/cm3)
.7823
.0990
10.11
3
1.956
.6776
.0857
11.67
3
2.846
.6386
.0808
12.38
3
.587 ±.004
.074
13.46
1
156
Shock Waves: Measuring the Dynamic Response of Materials Table 5.17. Hugoniot Data for Titanium V o = 0.2218 cnrVg; p 0 = 4.508 g/cm 3
Shot
Impactor
Vt
Material (km/s)
Us
up
P
(km/s)
(km/s)
(Mb)
VREL
VO
p0
(cnrVg)
(g/cm3)
WF
S-81
Ti
2.845
6.321
1.422
0.405
.7750
.1719
5.817
1
S-83
Ti
5.242
7.604
2.621
0.899
.6553
.1454
6.879
1
S-92
FS
7.849
10.934
5.534
2.728
.4939
.1096
9.128
1
VO
p0
WF
(cnrVg)
(g/cm3)
Table 5.18. Hugoniot Data for Beryllium S-200: V o = 0.5385 cnrVg; p 0 = 1 . 8 5 1 g/cm 3 1-400: V o = 0.5316 cm 3 /g; p 0 = 1.881 g/cm 3
Shot
Impactor
Vi
Material (km/s)
Us
up
P
(km/s)
(km/s)
(Mb)
VREL
1271
FS
2.205
9.965
1.631
0.302
.8363
.4504
2.220
3
S-122
Cu
2.695
10.266
1.903
0.363
.8147
.4387
2.280
3
1272
FS
4.330
11.779
3.459
0.757
.7063
.3804
2.629
3
S-123
Cu
5.596
12.218
3.990
0.905
.6734
.3627
2.758
1
1276
FS
7.734
13.034
4.792
1.160
.6324
.3405
2.937
3
S-126
Cu
7.945
13.970
5.678
1.473
.5936
.3196
3.129
3
1277
FS
7.734
14.324
6.139
1.633
.5714
.3077
3.250
1
1278 (1-400)
FS
3.191
10.981
2.549
0.526
.7679
.4082
2.450
0
WF
Table 5.19. Hugoniot Data for AZ31B Magnesium
Vo = 0.564 cm3/g; p 0 = 1.773 g/cm3
Shot
Impactor
V,
Material (km/s)
Us
up
P
(km/s)
(km/s)
(Mb)
VREL
VO
p0
(cm3/g)
(g/cm3)
S-79
Mg
3.539
6.731
1.769
0.211
.7372
.4158
2.405
3
S-80
Mg
5.911
8.020
2.956
0.420
.6314
.3561
2.808
3
S-89
Cu
6.631
10.884
4.959
0.957
.5444
.3070
3.257
1
S-91
FS
7.795
12.221
6.388
1.384
.4773
.2692
3.715
3
157
5. Hugoniot Equations of State to 0.5 TPa Table 5.20. Hugoniot Data for Plexiglas Vo = 0.8475 cm3/g; p0 = 1.180 g/cm3 Shot
Impactor
V;
Us
up
Material
(km/s)
(km/s)
(km/s)
(Mb)
P
S-96
Cu
6.907
10.482
5.592
0.692
S-97
FS
7.902
12.075
6.833
0.981
VREL
VO
p0
(cnvVg)
(g/cm 3 )
.4666
.3954
2.529
.4300
.3644
2.744
Table 5.21. Hugoniot Data for Quartz Phenolic Vo = 0.556 cnrVg; p 0 = 1.80 g/cm3 ,
Shot
Impactor
Vi
Us
up
P
Material
(km/s)
(km/s)
(km/s)
(Mb)
VREL
VO
p0
(cnrVg)
(g/cm 3 )
1159
Cu
1.619
4.479
1.341
.108
.7006
.3894
2.568
1175
Cu
2.654
5.248
2.164
.204
.5875
.3267
3.061
1157
Cu
3.088
5.343
2.522
.243
.5281
.2933
3.409
1173
Cu
4.798
7.160
3.793
.489
.4703
.2613
3.827
S-99
FS
7.985*
10.978
6.635
1.311
.3956
.2198
4.550
±.13
±.026
±.0118
1174
Cu
1.412
4.357
1.172
.092
.7310
.4061
2.462
1158
Cu
2.444
5.211
1.991
.187
.6179
.3433
2.913
1171
Cu
2.475
5.234
2.015
.190
.6149
.3416
2.927
1176
Cu
3.397
5.735
2.750
.284
.5205
.2892
3.458
1160
Cu
3.423
5.733
2.772
.286
.5165
.2869
3.486
1172
Cu
4.718
7.032
3.738
.473
.4685
.2603
3.842
L a y - u p with respect to the p l a n e of the shock front.
*Impact velocity determined from measured gun firing parameters.
158
Shock Waves: Measuring the Dynamic Response of Materials
(a)
Figure 5.9. Hugoniot Plots for Fansteel-77
(b)
5.5.2 Graphs of Hugoniot Data
(c)
Figure 5.9. Hugoniot Plots for Fansteel-77 {continued)
(d)
5. Hugoniot Equations of State to 0.5 TPa 159
Figure 5.10. Hugoniot Plots for Copper
(b)
Shock Waves: Measuring the Dynamic Response of Materials
(a)
160
(c)
Figure 5.10. Hugoniot Plots for Copper {continued)
(d)
5. Hugoniot Equations of State to 0.5 TPa 161
Figure 5.11. Hugoniot Plots for Aluminum
(b)
Shock Waves: Measuring the Dynamic Response of Materials
(a)
162
(c)
Figure 5.11. Hugoniot Plots for Aluminum (continued)
(d)
5. Hugoniot Equations of State to 0.5 TPa 163
Shock Waves: Measuring the Dynamic Response of Materials
Figure 5.12. Deviation of Aluminum Data from Linear Fit
164
5. Hugoniot Equations of State to 0.5 TPa
165
e
3
Figure 5.13. Hugoniot Plots for Uranium
2
w
Figure 5.14. Hugoniot Plots for Nickel
CM
Shock Waves: Measuring the Dynamic Response of Materials
(a)
166
(c) Figure 5.14. Hugoniot Plots for Nickel (continued)
(d)
5. Hugoniot Equations of State to 0.5 TPa 167
Shock Waves: Measuring the Dynamic Response of Materials
(a) Figure 5.15. Hugoniot Plots for Stainless Steel
168
^
Figure 5.15. Hugoniot Plots for Stainless Steel (continued)
(c)
5. Hugoniot Equations of State to 0.5 TPa 169
Figure 5.16. Hugoniot Plots for Titanium
(b)
Shock Waves: Measuring the Dynamic Response of Materials
(a)
170
(c)
Figure 5.16. Hugoniot Plots for Titanium (continued)
(d)
5. Hugoniot Equations of State to 0.5 TPa 171
Figure 5.17. Hugoniot Plots for Beryllium
Shock Waves: Measuring the Dynamic Response of Materials
(a)
172
(b)
Figure 5.17. Hugoniot Plots for Beryllium (continued)
(c)
5. Hugoniot Equations of State to 0.5 TPa 173
Shock Waves: Measuring the Dynamic Response of Materials
(a) Figure 5.18. Hugoniot Plots for Magnesium
174
5. Hugoniot Equations of State to 0.5 TPa
175
2 !•S s o
I & o o
I 00
E
Shock Waves: Measuring the Dynamic Response of Materials
(a) Figure 5.19. Hugoniot Plots for Plexiglas
176
(b)
Figure 5.19. Hugoniot Plots for Plexiglas (continued)
(c)
5. Hugoniot Equations of State to 0.5 TPa 111
Shock Waves: Measuring the Dynamic Response of Materials
(a) Figure 5.20. Hugoniot Plots for Quartz Phenolic
178
(b)
Figure 5.20. Hugoniot Plots for Quartz Phenolic (continued)
(c)
5. Hugoniot Equations of State to 0.5 TPa 179
180
5.6
Shock Waves: Measuring the Dynamic Response of Materials
Summary
The research program has resulted in the relatively high precision determination of the high pressure Hugoniot equations of state of eleven materials. The estimated error for shock velocity measurements with the experimental system was 0.2% to 0.6%. The estimated error in calculation of particle velocity was 0.05% for like-like impact but somewhat higher for dissimilar materials impact. Table 5.22 summarizes the results of the experimental program in terms of the constants of the linear Hugoniot equation Us = Co + S up. Also included are the pressure ranges measured and the RMS deviations of the measurements. Figures 5.21 (a) and 5.21(b) show the data points obtained, displayed in the P vs. up plane. Comparisons have been made of the data with that of other researchers. Space does not permit the inclusion of all data available for comparison, and the data quoted are selected principally on the basis of pressure ranges studied and similarity of the materials in terms of density and composition. The onset of melting in the shock front is evidenced by only very small changes in the measured parameters, shock velocity and particle velocity. Accordingly, very accurate data are required to substantiate theoretical predictions of the melting phenomenon. Although the linear fits to the high pressure data reported here are useful for calculating shock wave propagation, it is likely that several of the materials experience melting at the higher pressures, and the linear approximations become less accurate.
181
5. Hugoniot Equations of State to 0.5 TPa Table 5.22. Hugoniot Data for Materials Tested Material
Density Particle
Co
(g/cm3) Velocity Range (km/s)
(km/s)
S
RMS
No. of
Deviation from Linear Fit
Shots Used For Fit
Fansteel-77
17.01
0.4-3.9
4.008
1.262
0.021
17
OFHC Copper
8.930
1.7-4.7
3.964
1.463
0.009
12
2024-T4 Aluminum
2.783
2.0-6.0
5.471
1.310
0.022
11
Depleted Uranium
18.951
1.1-3.9
2.443
1.582
0.029
5
Nickel, Type 304
8.864
1.3-4.6
4.456
1.555
0.012
5
Stainless Steel
7.905
1.5-4.7
4.722
1.441
0.023
4
Titanium
4.508
1.4-5.5
4.692
1.126
0.014
3
Beryllium, 1-400 ( * )
1.881
2.5
—
—
—
1
Beryllium, S-200
1.851
0.9-6.1
8.390
0.975
0.017
7
Magnesium, AZ3IB
1.773
1.8-6.4
4.551
1.209
0.083
4
Plexiglas, Rohm-Haas IIUVA Quartz Phenolic (**)
1.18
5.6-6.8
—
—
—
2
1.80
2.5-6.6
—
1.364
0.016
5
( * ) Single test (excluded from least squares fit for S-200) ( **) High pressure phase (2nd wave)
182
Shock Waves: Measuring the Dynamic Response of Materials 5.01
]
"|
I
4.5 -
[
/Fansteel
4.0 -
/
-|
/
/Nickel
3.5 -
I
3.0 -
I
S3
%
|
2
-
I
If
2.5 -
j
/Titanium
II
t /
/
! F
0.01 0.0
/
y*
/ /
/
/*
I **~^ i 1.0 2.0
/ s"20c!
>*^ ^s'
Berylliurn^
^/Quartz ^ s ^ Phenolic
I I | 3.0 4.0 5.0 Particle Velocity, km/s (a)
Figure 5.21. Compilations of Hugoniots
I 6,0
7.0
5. Hugoniot Equations of State to 0.5 TPa
5.0 |
1
,
,
~
4 0
"
/
3.5 -
I
^
"
2.0 ~
I
1-5 ~
if
/
1.0 -
0 01 0,0
/
I 1.0
/
T
1
/
Tj
1
/OFHC Copper f
If
/
If
2024-T4
Z'
Aluminum/
/
X
i 3.0
-
/AZ31B
v*
i 2.0
"
1
f
'
,
7/304 Stainless Steel
1
J3
25
,
7 /Uranium
4b
3.0 I
1
183
Magnesium^'
^s
I 4.0
I 5.0
Plexiglass'
I 6.0
Particle Velocity, km/s (b) Figure 5.21. Compilations of Hugoniots {continued)
7.0
184
5.7 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
Shock Waves: Measuring the Dynamic Response of Materials
References Jones, A. H., W. M. Isbell, and C. J. Maiden. J.A.P., vol. 37, 3493 (1966). McQueen, R. G., and S. Marsh. J.A.P., vol. 31, 1253 (1960). Hart and Skidmore, I. C. (LRL Compendium) 1965. Al'tshuler, L. V., K. K. Krupnikov, and M. I. Brazhnik. Soviet Phys. JETP, vol. 34, 614 (1958). Walsh, J. M., M. H. Rice, R. G. McQueen, and F. L. Yarger. Phys. Rev., vol. 108, 196 (1957). Isbell, W. M., F. H. Shipman, and A. H. Jones. "Use of a Light-Gas Gun in Studying Material Behavior at Megabar Pressures." Symposium High Dynamic Pressure, Paris, France, September, 1967. Jones, A. H., W. M. Isbell, and F. H. Shipman. "Material Properties Measurements for Selected Materials." Contract NASA-3427, National Aeronautics and Space Administration, Ames Research Center, Moffett Field, California, April 1968. Lingle, R., and A. H. Jones. "Triggering System at the Muzzle of an Accelerating Light-Gas Gun." TR65-65, GM Defense Research Laboratories, Santa Barbara, California, September 1965. Al'tshuler, L. V., S. B. Kormer, and M. I. Brazhnik. Soviet Phys. JETP, vol. II, 766 (1960). Curran, D. R. J.A.P., 34 (9), 2677 (1963). McQueen, R. G., S. P. Marsh, and W. J. Carter. 'The Determination of New Standards-for Shock Wave Equation-of-State Work." Symposium High Dynamic Pressure, Paris, France, September 1967. McQueen, R. G., S. P. Marsh, W. J. Carter, J. N. Fritz, and J. W. Taylor. High Velocity Impact Phenomena, E. V. Colen, ed. New York: Academic Press Inc. (in press). Al'tshuler, L. V., S. B. Kormer, and A. A. Bakanova. Soviet Phys. JETP, vol. 11,573(1960). Kormer, S. B., A. I. Funtikov, V. D. Urlin, and A. N. Kolesnikova. Soviet Phys. JETP, vol. 15, 477 (1962). Urlin, V. D. Soviet Phys. JETP, vol. 22, no. 2, 341. Marsh, S. P. Private communication with LRL. Skidmore, I. C , and E. Morris. Thermodynamics of Nuclear Materials, 173-216. Vienna, 1962. Al'tshuler, L. V., A. A. Bakanova, and R. F. Trunin. Soviet Phys. JETP, vol. 16, 65 (1962). Minor, E. E. Private communication, BRL, 1968. Krupnikov, K. K., A. A. Bakanova, M. I. Brazhnik, and R. F. Trunin. Soviet Phys. JETP, vol. 8, 205 (1963). Van Theil, M. "Compendium of Shock Wave Data." UCRL50108 (2 vols.) ed., Lawrence Radiation Laboratory, University of California, Livermore, California, 1966.
5. Hugoniot Equations of State to 0.5 TPa
185
22. McQueen, R. G., and S. P. Marsh. Report No. GMX-6-566, 51-62 (1964), Los Alamos Scientific Laboratory, GMX-6. 23. Hauver, G. E., and A. Melani. B.R.L. Report No. 1259 (1964), Ballistics Research Laboratory, Aberdeen Proving Grounds, Aberdeen, Maryland. 24. Bakanova, A. A., I. P. Dudoladov, and R. F. Trunin. Fizika, Tverdogo Tela, vol. 7, 1616 (1965). 25. Isbell, W. M., N. H. Froula, and F. H. Shipman. Fundamental Material Study, vol. Ill, "Shock Wave Propagation and Equation of State Measurements of Quartz Phenolic," Technical Report TR67-07, vol. Ill, February 1967, prepared on Contract No. AF04-(694)-807 for Ballistic Systems Division, Norton AFB, California 92409. 26. Wacklerle, J. J.A.P., vol. 33, 922 (1962).
6. Attenuation of Shock Waves from High Pressure 6.1
Foreword
The attenuation of high-pressure shock waves has been investigated to provide experimental evidence upon which to construct models of shock wave propagation in the pressure regime of hundreds of kilobars to Megabars. High purity copper, 6061-T6 aluminum, 6A1-4V alpha phase titanium, an uranium alloy and two alloys of beryllium were examined experimentally at pressures as high as 0.3 TPa (3 Megabars). Three pressure regimes were investigated: (1) Low-pressure experiments (tens of kilobars) were instrumented with a velocity laser interferometer. (2) Intermediate pressures (to several hundred kilobars) were instrumented with Manganin wire piezoresistive gages. (3) High-pressure measurements (to several Megabars) were made through streak camera observation of free surface velocities. The results show that evidence of strength, as exhibited by elastic release wave behavior, persists to pressures of hundreds of kilobars and that the release wave structure is not elastic-perfectly plastic. For the materials examined, the release wave structures can be described as exhibiting a Bauschinger effect. At pressures beyond melting behind the shock front, fluid behavior is indicated by the lack of an elastic release wave.
187
188
6.2
Shock Waves: Measuring the Dynamic Response of Materials
Introduction
It is well known that most solids exhibit an elastic-plastic form of behavior when subjected to high-speed loading by moderate strength shock waves, i.e. from the Hugoniot elastic limit up to the order of 100 kilobars (1 kilobar = 109 dynes/cm2 =0.1 GPa). At higher pressures, the plastic wave velocity exceeds that of the elastic wave and only a plastic loading wave is observed. It is also expected that materials exhibiting elastic-plastic behavior on loading would show elastic-plastic behavior upon unloading. This is illustrated in Figure 6.1, which shows the stressstrain behavior for the theoretical model of an elastic-perfectly plastic material. For such a material, the loading path lies 2/3 Y (Y is the yield strength) above the hydrostatic curve and is followed to the peak stress point, A. The initial release path is AC and an elastic wave of that amplitude is generated. The slope of AC is (K + 4/3 G), where K is the bulk modulus and G is the shear modulus, while the slope of the hydrostat is K. The release wave velocity is then
I(K + %G)
C = J± V
/-A-L
P
(6.1)
where p is the density of the material. The unloading from the point C is then plastic. The question arises as to what degree this model matches experimental evidence. Several phenomena may be expected to modify this behavior. Many materials show strain-rate sensitivity, i.e., increased resistance to deformation with increased rate of deformation. This could be exhibited by increased yield strength upon loading and release. The Bauschinger effect on release (an initially small yield strength followed by a rapid, then a slow increase) would be to spread the elastic rarefaction. If the pressure is great enough the material behind the shock front will be molten and no elastic rarefaction would be expected. Therefore, the purpose of this work has been to investigate the release wave behavior of several metals from high pressures in an effort
6. Attenuation of Shock Waves from High Pressure
A_
y
2/3 "]// y / j /
/
j / /I
189
_
Release Wave Amplitude
1 2/3 Y
IP/Po-1) or STRAIN Figure 6.1. Loading and Unloading for an Elastic-Perfectly Plastic Material
to provide experimental information on the basic behavior of metals at high pressure. Attenuation of high-pressure shock waves has been studied previously by several investigators. Al'tshuler et al. [1] measured elastic release wave velocities at low pressures and plastic release wave velocities at high pressures by various methods. Curran [2] explained the results of his aluminum free surface velocity experiments in terms of an elastic release with the elastic yield increasing with pressure. Erkman and Christensen [3] described similar experiments with alloys of aluminum and concluded that the shear modulus may reach a maximum value at pressures between 100 and 340 kb. Although mention is made, no conclusion is reached of the Bauschinger effect. Barker [4] utilized the velocity interferometer to observe release waves in aluminum from pressures up to 90 kb. He concluded that the indistinct elastic-plastic structures of the observed release waves could be best described through the Bauschinger effect.
190
Shock Waves: Measuring the Dynamic Response of Materials
Other observations were made by Kusubov and Van Theil [5] and Fuller and Price [6], utilizing Manganin wire gages. These gages are capable of recording pressure as a function of time at one location within the material being examined. From both these studies it can be concluded that elastic unloading occurs in alloys of aluminum and magnesium up to pressures of ~200 kb. The magnitude of the elastic release wave indicates an increase of effective yield strength with pressure in this range. With the exception of the work of Al'tshuler et al. [1] and Fuller and Price [6], all work to date had been performed with alloys of aluminum. The work described herein details the investigation of shock wave attenuation of five metals and their alloys over a large range of pressures. Thus an alloy of aluminum (6061-T6), pure copper, titanium (6A1-4V and pure, in the alpha phase), two types of beryllium alloy (S-200 and a wrought alloy), and an uranium alloy were examined at pressures up to 3 Mb.
6.3
Description of the Impact Experiments
Four methods were used to examine the arrival of rarefaction waves at the shock front: (1) Measurement of the motion of the surface as a function of time (laser velocity interferometer). (2) Measurement of pressure as a function of time (Manganin gage). (3) Measurement of free surface velocity as a function of target thickness (step targets). (4) Measurement of shock velocity as a function of distance (optical wedge). The majority of the experiments were conducted at the higher pressure levels using techniques (3) and (4). Design considerations of all four techniques are described below.
6. Attenuation of Shock Waves from High Pressure
191
6.3.1 Design of Experiments The impactor used to produce an attenuated shock wave in a target was a thin (order 1 mm) plate with a free rear surface. Such a configuration, however, poses severe problems in the design of projectiles capable of carrying undeformed thin plates to extremely high velocities. The initial design of the plates for this study included a free rear surface with a reinforcing ring about the outside rear of the plate to provide stiffness. The plate thickness was one millimeter and the inside of the rear of the plate was supported by a thin plate of a strong polycarbonate, Zelux. It was found that this type of plate was very difficult to manufacture so that the impactor front and rear surfaces were flat and parallel. Also, the launch stresses caused severe plate curvature. The plate was then redesigned such that it could be machined with uniform thickness, but the plate was supported on a thick disk of Zelux. This is acceptable if the shock impedance of the plate material is sufficiently greater than that of the Zelux support that substantial attenuation would be achieved (as discussed later). Such a plate and sabot was manufactured and launched to over 7 km/s. It was determined by impacting a mirror with such a plate and observing the mirror with the streak camera that the flyer plate was flat to within 0.05 mm, causing less than 10 nanoseconds deviation. An additional restriction on the experimental design was that the diameter of the target disk was sufficient that the shock wave emerging from the rear surface was recorded before release waves from the corners and edges of the target arrived at the measurement position. Approximate shock relationships were established to provide reasonable design criteria. The velocity of the head of the rarefactions was estimated to be: ce = 1.2U. + u p
(6.2)
where ce is the velocity of the elastic rarefactions and Us is the shock velocity associated with the particle velocity up. The accuracy of this assumption is demonstrated in Figure 6.2, where release wave and shock velocities are compared for a variety of metals. This figure includes data
192
Shock Waves: Measuring the Dynamic Response of Materials
from the present study as well as from other sources. The tail of the rarefaction (return to zero pressure) was taken to travel at c2 = dP / dp taken at P = 0
(6.3)
where P is pressure and p is material density at pressure P.
Figure 6.2. Elastic Release Wave Velocity vs. Shock Velocity for Several Materials
6.3.2 Laser Velocity Interferometer Experiments were performed at low stress levels using the laser velocity interferometer. As described in more detail in Chapter 8, this technique employs a HeNe laser to provide a monochromatic coherent light source. The beam from the laser is reflected from the rear surface of the target and returned to the interferometer. The reflected beam is then split by a beam splitter into two beams, one of which goes directly to a second
6. Attenuation of Shock Waves from High Pressure
193
beam splitter, while the second beam is delayed by reflection through a delay leg before recombination with the direct beam. Because part of the beam is delayed by the length of time required to traverse the delay leg, the beams upon recombination will show a phase relationship dependent upon the length of the direct and delay legs and the frequency of the two beams. For dynamic measurements, the direct and delay leg path lengths will not change and changes in interference patterns are interpreted as changes in the frequency of the reflected laser beam. These frequency changes are due to changes in the velocity of the motion of the target rear surface. The laser velocity interferometer is therefore a Doppler system. The signal recorded shows no voltage change for constant velocities and (for uniform acceleration) sinusoidal variation of voltage for changes in velocity. The data resulting from interferometer experiments consist of oscilloscope traces showing non-periodic voltage oscillations. These oscillations correspond to changes in the interference fringe pattern upon the photomultiplier sensor. Each full oscillation corresponds to one full fringe shift. From the fringe shifts, the velocity of the surface can be found from AV = A / 2 T * A N
(6.4)
where v is the velocity of the surface, X is the wavelength of the laser light, r is the time delay of the delay leg and AN is the total number of fringes recorded. Small changes in this calculation of velocity are necessary during portions of the wave in which changes in velocity take place in periods of time comparable to r. In general, the adjustment is made to the time at which a given velocity is recorded to account for the time the signal spends traveling through the delay leg. (In its simplest form, this adjustment shifts the calculated times by ~r/2). The laser velocity interferometer provides very accurate velocitytime profiles to be obtained over a wide range of velocities. Profiles such as these contain a wealth of information on elastic-plastic compressive and release wave behavior.
194
Shock Waves: Measuring the Dynamic Response of Materials
6.3.3 Suppression of Spall Because the stress wave being observed is reflected at the target rear surface, the wave interactions taking place may include spall. To record as much of the wave profile as possible, it is desirable to suppress spall. This may be accomplished by placing a transparent material of approximately the same impedance as the target on the rear surface and recording the motion of the interface. In this way, a large part of the stress wave is passed through the target-window interface, thereby reducing the portion of the wave reflected as a tension wave and suppressing the tendency to spall. The amplitude of the reflected tension wave depends upon the impedance mismatch and on the initial amplitude of the compressive wave. Unfortunately, only a few transparent materials of limited impedance variation are available. This, and the fact that reflecting surfaces tend to degrade at higher pressure levels, generally limits the maximum pressure range for the interferometer technique to 100 to 200 kb. (Note: These tests were performed before the development of the VISAR interferometer which, while based on the same general principals as the velocity interferometer described above, allows measurements to be performed at substantially higher pressures.) Using the previously discussed criteria, disk-shaped targets were designed such that the area on the rear surface of the target used for reflection of the laser beam was unaffected by side rarefactions during the time of measurement. This design procedure was employed both for laser velocity interferometer and for Manganin gage experiments, where the entire wave was recorded. Examination of records obtained with these instrumentation techniques have demonstrated that the target designs used were quite conservative.
6.3.4 Manganin Wire Gage Experiments were performed with Manganin wire piezoresistive gages in aluminum, copper and titanium targets. These gages were developed and built by personnel of the Lawrence Radiation Laboratory (now, Lawrence Livermore National Laboratory). A description of the gage and
6. Attenuation of Shock Waves from High Pressure
195
its calibration can be found in Reference 8. The gages were constructed by plating copper on 0.025 mm diameter Manganin wire, leaving about 7 mm of unplated wire as the active gage element. The gage element resistance is about 7 ohms. The plated leads were then flattened and the assembly placed between two thin (.025 mm) sheets of mica for electrical insulation. Total thickness of the assembled gage is about 0.15 mm. A sketch of such a gage assembly is shown in Figure 6.3. In the lower portion of the figure is shown the installation of the gage in a target. Each target was equipped with two gages to provide redundant measurements.
Figure 6.3. Manganin Wire Gage Construction
The Manganin gage provides measurement of pressure as a function of time by means of the piezoresistive effect. The resistance of the gage is measured during the time the shock pressure pulse passes by the gage position. This is done by monitoring the voltage across the gage supplied by a constant current power supply. This supply provides a moderate voltage (tens of volts), high current (several amps) pulse to the gage for a short period of time (~100 us). By recording the voltage change and
196
Shock Waves: Measuring the Dynamic Response of Materials
through time and resistance calibration, the pressure-time history at the gage position may be determined. The Manganin wire gages were used at moderate pressures — from about 76 kb in aluminum and titanium to 400 kb in copper. Attempts to use these gages at pressures higher than 400 kb resulted in premature breakdown of the gages. The approach, however, provided the first measurements using a Manganin gage to such high pressures. The advantages of using the Manganin wire technique are: (1) Ability to measure pressure-time histories within the material being studied. (2) Capability to make measurements at higher pressure levels than are possible with quartz crystal techniques. The principal disadvantage is that the thickness of the gage and its different shock impedance than the surrounding material cause finite risetimes which can obscure details of the stress wave profile. Also, the question of gage hysteresis (different, and possibly variable, piezoresistive coefficients on compression and release) had not been resolved with sufficient accuracy to determine the effect on experimental error.
6.3.5
Stepped Targets with Shims
Because gages were not available to record stress-time or particle velocity-time at higher stress levels (>400 kb), it was necessary to employ an optical technique to observe free surface velocity at various depths in the target materials and from these measurements to infer release wave behavior. This was done by recording the time required for the surface to cross a known gap. The difficulty with this method is that if the shock wave that arrives at the free surface is attenuating, the surface may decelerate while crossing the gap. To avoid the timing error produced by this effect, a thin shim, preferably of the same material as the target, is applied to the rear surface. The situation of an attenuating shock in such a system is illustrated in Figure 6.4. A rarefaction wave is shown propagating with finite, discrete
6. Attenuation of Shack Waves from High Pressure
197
steps, represented by the characteristics AC, AE, etc., in the figure. After the attenuated compressive shock (OEB) reaches the rear surface of the shim and reflects as a rarefaction, the shim and target surface start to move together. When the shim-target interface is overtaken by the rarefaction wave (AC), the interface separates. The shim continues with the original free surface velocity while the surface itself is slowed to some lower velocity. Thus the shim indicates the free surface velocity associated with the peak particle velocity of the wave that struck the shim.
Figure 6.4. X-T Diagram of Shock-Rarefaction System
If the shock wave was attenuated during its passage through the shim, the shim velocity will not correspond to the peak particle velocity but rather some lower average. In general, free surface velocity error with this technique will depend upon the shape of the wave. For a triangular wave of duration corresponding to the impactor thickness (1.0 mm) and a shim of thickness 0.0125 mm, the error in the shim velocity would be 1.25%. This would seem to be representative of the worst case that would be encountered.
198
Shock Waves: Measuring the Dynamic Response of Materials
The basic target configuration was established as blocks with shims on the surface and measurement gaps for the shims to cross. The further side of the gap consisted of a mirror constructed by attaching aluminized Mylar to thick pieces of plate glass. Only one side of the Mylar was aluminized, and this surface was on the side opposite the side struck by the shim. This was done to avoid early extinction of the reflecting surface due to microjetting by the shim surface. Because of this thickness of film, the velocity measured was a combination of the shim velocity and the shock velocity in the Mylar due to the shim impact. A correction was made to determine the shim velocity using an estimate of the shock state in the Mylar. The magnitude of this correction can be estimated by calculating the shock velocity induced in the Mylar by assuming an impact of the shim material upon Plexiglas. Using this approximation, it is calculated that the maximum correction on the measured velocity ranges from 0.03% at an impact velocity of 2.5 mm/us to 0.7% at a velocity of 7.3 mm/us. The targets were designed with two or three different thicknesses in order to obtain as much information from each experiment as possible. The target design is shown in Figure 6.5. The target design provided the same unaffected area on all steps for any target thickness.
Figure 6.5. Step Target Design
6. Attenuation of Shock Waves from High Pressure
199
The reflectance of the shim is decreased when struck by the shock wave, indicating the start of the shim motion. The shim then crosses the gap and strikes the Mylar mirror. The mirror reflectance is also altered and the two changes in reflectance are measured on the streak camera record to provide the time of flight of the shim across the gap. A representative streak camera record is shown in Figure 6.6. Accuracies on the order of 0.5% to 2% in shim velocity are obtained when all errors are considered.
Figure 6.6. Step Target Result
For attenuating shock waves, the particle velocity associated with the peak pressure was calculated from the measured surface velocity. In several cases, the measured free surface velocity for unattenuated waves exceeded twice the known particle velocity. This effect is expected at high pressures as the irreversible entropy associated with the shock
200
Shock Waves: Measuring the Dynamic Response of Materials
becomes appreciable. For copper this effect was noted at moderate pressures and became more evident at higher pressures. In aluminum, the effect was appreciable only for high velocity impacts. Figures 6.7 and 6.8 present the initial free surface velocity as a function of particle velocity (half the impact velocity for similar materials impacts) for the copper and aluminum respectively. Also included in the figures are the computed free surface velocities. These were calculated from:
ufs=u0+J
-—
dV
(6.5)
assuming a Mie-Gruneisen equation of state and taking (F/V) to be a constant and where: ufs is the free surface velocity u0 is the particle velocity P is the pressure V is the volume H refers to the Hugoniot S refers to the entropy. As can be seen, the data are in good agreement with the computed values. From this it is possible to determine the particle velocity associated with a measured free surface velocity and to calculate the peak pressure in the attenuated shock as a function of distance of propagation. The free surface shim technique allows determination of the particle velocity as a function of distance, at stress levels at which pressure or particle velocity gages will not function. The principle disadvantage to the technique is the fact that data are taken in increments and several experiments are required to characterize a particular condition. Also, at high pressures the errors in measurement may be sufficient to obscure the detail of the release waves. Thus the technique is primarily applicable to measurement of the elastic release wave velocity and of the attenuation due to the release wave.
6. Attenuation of Shock Waves from High Pressure
Figure 6.7. Free Surface Velocity vs. Particle Velocity for Copper
Figure 6.8. Free Surface Velocity vs. Particle Velocity for Aluminum
201
202
6.3.6
Shock Waves: Measuring the Dynamic Response of Materials
Wedge Targets
A few experiments at moderate pressures were performed with targets in the form of wedges. The purpose of these experiments was to examine continuously shock propagation versus distance in the material. The basic wedge configuration is shown in Figure 6.9. The angle of the wedge was selected to insure inclusion of the overtaking point. The apparent velocity of the shock arrival along the wedge rear surface was greater than the velocity of the shock front in the material so that disturbances could not be communicated up the face of the wedge. The angles used ranged from 10° to 15°.
Figure 6.9. Wedge Target Design
The first attempt to use the wedge was made using rear surface shims and a free surface gap to a mirror. This method was discarded because of unevenness of the record caused by small imperfections in the shim surface and the mirror. Subsequent attempts to use wedges utilized direct observation of the target rear surface to record shock wave arrival.
6. Attenuation of Shock Waves from High Pressure
203
This is less desirable than observing free surface velocity because shock wave velocity is less sensitive to changes in the pressure of the attenuating wave than are the free surface or particle velocity. Figure 6.10 provides an example of a wedge record. The changes in shock velocity are evidenced by changes in the slope of the shock breakout.
Figure 6.10. Wedge Streak Camera Record for Copper: Impact Velocity 1.192 mm/us. Pulse Length 0.20 us.
Because the attenuation information is in the form of small changes in slope, extraction of data from records of this type is a formidable task. Two methods were used to process the data. The record was processed by converting the film record into a digital representation with an optical comparator (see Figure 6.11). The digital information was then processed in two ways. The entire shock arrival x-t path was fitted by a linear least squares fit for time as a function of distance. Each point was then compared to this line and the difference plotted and recorded. This type of result is shown in Figure 6.12, the results from a copper record. The irregularity of this display is caused by point to point dispersion in reading the film record on the optical comparator and by the expanded time scale of the plot. The slope of the plot changes at between 22.5 and 25 mm. This indicates the shock velocity decreased at this point, a
204
Shock Waves: Measuring the Dynamic Response of Materials
normalized depth of X/Xo (target thickness/impactor thickness) of 5.2 to 5.6. This decrease in shock velocity is interpreted as attenuation of the shock front by the release wave from the rear surface of the impactor.
Figure 6.11. Distance-Time Record from Copper Wedge Record
Figure 6.12. Delta Time vs. Distance from Copper Wedge Record
6. Attenuation of Shock Waves from High Pressure
205
The data were also analyzed by means of an "N" point differentiation scheme which calculates the local value of the derivative of distance along the face of the wedge with respect to time. The result of this processing is shown in Figure 6.13. As can be seen, there is again a decrease in the derivative starting between 22.5 and 25 mm along the face of the wedge, in agreement with the least squares fit treatment. The derivative given on the plot is the velocity of the shock breakout across the face of the target, not the shock velocity in the material. Correcting for the wedge angle produces a shock velocity of 4.86 mm/us for this experiment. This value is in good agreement with other copper measurements for the particle velocity of 0.596 mm/us.
Figure 6.13. Velocity vs. Distance from Copper Wedge Record
Use of the wedge allows continuous observation of the shock wave front as it moves through the target material. In principle, this allows a very accurate determination of the point at which the shock front is overtaken by the first release wave. In practice the observed quantity — the shock velocity — changes so little with the small changes in particle velocity which are associated with the initial portion of the release wave, that the overtaking point is very difficult to determine.
206
Shock Waves: Measuring the Dynamic Response of Materials
6.4
Experimental Results
During the course of the program the following materials were examined experimentally: (1) OFHC copper (2) 6061-T6 aluminum (3) 6A1-4V titanium (4) Pure titanium (in the alpha phase) (5) S-200 beryllium (6) A beryllium supplied by Lawrence Livermore National Laboratory, similar to Brush Beryllium QMV (7) A uranium alloy Table 6.1 presents the yield strengths, elastic properties and shock properties of these materials. Table 6.1 COMPRESSIVE
po g/cm3 Aluminum 6061-T6 Copper OFHC Titanium 6A1-4V Titanium q-Phase Beryllium S-200 Beryllium QMV Uranium Alloy
*
2.70
I
HUGONIOT
I I I Yield (kb) I (Refs. 10CL Cs Co 13) 10"3 103 mm/ mm/ mm/ /s /s |is jxs ps 3.2
I
3.2 3.2
I
6.368
3.197
5.19
0.331
2.00
I l+s I 5.326
1.338
1.99
3.94
1.489
1.09
4.695
1.146
9.6*
4.50
5.1
~9
6.118
3.246
4.83
0.304
1.09
4.695
1.146
1.85
2.3
3.3
12.20
8.86
7.87
.005
1.16
7.998
1.124
1.85
2.71
---
---
---
1.16
7.998
1.124
16.41
---
(for u <0.5 mm/us)
2.92
0.63
(for u>0.5 mm/us)
2.57
1.50
Manufacturer's Specification
0.356
I
4.50
---
3.99
I
S
2.7
---
2.247
I
y0
Linear Fits (Ref. 14) C mm/
8.92
---
4.757
[
v
6. Attenuation of Shock Waves from High Pressure
6.4.1
207
OFHC Copper
This pure (.999) form of copper has a relatively low yield (2.7 kb compressive yield strength), is ductile and strain-rate sensitive. Experiments performed at low pressures (52 kb) were instrumented with the laser velocity interferometer. These profiles are presented in Figure 6.14 in terms of interface velocity versus time. In order to suppress spall such that the unloading portion of the wave could be observed, the experiments were performed with fused quartz windows on the target rear surfaces. The laser beam was reflected from the quartz-copper interface. Due to the impedance mismatch between the target and window, the interface velocity is greater than the particle velocity but is less than the free surface velocity. Profiles were obtained for three values of X/Xo: 2.55, 5.11, and 11.01.
Figure 6.14. Copper: Interferometer Record
In all the profiles, a ramped elastic precursor is evident with a velocity of 4.75 mm/us, followed by a ramped plastic wave with a velocity of ~4.5 mm/us. There is no definite elastic-plastic release wave structure evident in these profiles. The indicated release wave velocity of greater than 5.3 mm/us exceeds the expected plastic release wave velocity. Although there is no clear-cut elastic-plastic release wave structure, the first release wave, because of its velocity, evidently is an elastic wave.
208
Shock Waves: Measuring the Dynamic Response of Materials
A number of intermediate pressure experiments were performed at about 400 kb with Manganin wire gages. Typical of the results of these experiments is the pressure-time profile shown in Figure 6.15. The elastic precursor is not observed at this pressure level as the shock is overdriven (the shock velocity is greater than the elastic precursor velocity). There is a finite risetime on the front of the wave that is attributable to impact tilt and to the finite gage thickness. There is also some unevenness on the top of the record due to system noise. The smeared elastic-plastic nature of the release wave can be seen in this record with an initial drop in pressure followed by a change in slope and then another pressure drop. The double inflection is taken to be the demarcation between elastic and plastic behavior. The stress does not return to zero because the copper impactor plate was backed with a plastic plate to insure impactor flatness. Again, examination of the velocity of the first release wave shows it to be much higher than expected for the plastic release wave.
Figure 6.15. Copper: Manganin Wire Record
209
6. Attenuation of Shock Waves from High Pressure
In order to obtain comparisons of the pressure-time records with other forms of data, the Manganin records were transformed through the conservation equations: d/9 = d P / c 2
(6.6)
dup=dP/pc
(6.7)
and
where p is the local material density, P is the pressure, c is the local sound speed, and up is the particle velocity. The results are then be transformed into the particle velocity or free surface velocity versus distance plane through the method of characteristics, as shown in Figure 6.16. The elastic-plastic nature of the release wave is no longer evident. 3 1
1
1
1
Copper
" a E e
2 —
o
^^-^^_^
>
^"--^^^
I/)
UJ
|
0
5
|
10
|
15
JO
Figure 6.16. Free Surface Velocity vs. Distance for Copper from Manganin Wire Gage Record
210
Shock Waves: Measuring the Dynamic Response of Materials
The same procedure was used to project the pressure-time results into the stress-strain plane. In Figure 6.17 the result of this transformation is shown, along with the Hugoniot of the material. The loading path here is taken along a Rayleigh line. The unloading path drops below the Hugoniot, indicating elastic release. The release path cannot be followed to zero stress because of the plastic backing of the impactor plate. This experimentally determined stress-strain release path shows no definite change from elastic to plastic behavior. If the maximum difference in stress between the Hugoniot and the release path is taken to be 4/3 of the yield, the value of the yield strength is then about 10 kb at 400 kb peak pressure from these experiments. This is to be compared with the quasi-static yield of 2.7 kb and the value of 3.2 kb yield at 103/s strain rate. Attempts to obtain pressure-time profiles at higher stress levels were not successful, due to premature gage failures. Data at higher pressures was obtained through the free surface technique. Experiments were i
'
i
Copper
/L.—• //
£ 2
oLi—
1
5
1
STRAIN (*]
10
Unloading Path
1
15
Figure 6.17. Copper Stress-Strain Release Paths from Manganin Wire Gage Records
6. Attenuation of Shock Waves from High Pressure
211
performed at impact velocities of 2.5, 5.6, and 7.3 mm/us, corresponding to pressures of 0.65, 2.01, and 3.03 Mb. Data from experiments at an impact velocity of 2.5 mm/us are shown in Figure 6.18. The overtaking point is clearly between values of X/Xo of 4.5 and 5.5. Also shown in this figure is the overtaking point determined from a wedge experiment. The overtaking point from the two types of experiments agree quite well.
Figure 6.18. Free Surface Velocity vs. Distance for Copper: Impact Velocity, 2.5 mm/us. Normalized to Uft = 2.5 mm/us at X/Xo = 2. Pulse Length, 0.17 us.
Results from experiments performed at an impact velocity of 7.3 mm/us are shown in Figure 6.19. Here, because of irreversible heating effects, the free surface velocity was significantly higher than the impact velocity. The overtaking point lies between X/Xo of 4 and 4.4, indicating a release wave velocity between 8.91 and 9.48 mm/us. Because of the large amount of heating at the peak pressure level, the free surface velocities cannot be taken to be twice the particle velocities. The values must instead be corrected for heating effects through Equation 6.5. The normalized attenuation for initially unattenuated pressures of 0.4, 0.65, and 3.03 Mb are presented in Figure 6.20. With increasing pressure, the distance before initial attenuation takes place becomes less and the initial rate of attenuation increases.
212
Shock Waves: Measuring the Dynamic Response of Materials
Figure 6.19. Free Surface Velocity vs. Distance for Copper: Impact Velocity, 7.3 mm/us. Normalized to Us = 8.2 mm/us at X/Xo =2. Pulse Length 0.16 us.
Figure 6.20. Normalized Attenuation for Copper
6. Attenuation of Shock Waves from High Pressure
213
The initial release wave velocities inferred from the experimental results are presented in Figure 6.21 as a function of p/p0 or relative compression. The plastic release wave velocities as determined by APtshuler et al. [1] are also shown along the calculated hydrodynamic release wave velocities. The calculated release wave velocities are obtained using the Mie-Gruneisen equation of state as the linear fit to the Hugoniot, resulting in:
c»=V»:£fl—±-) VI
1-SnJ
+ Po^
1 + Sn3
° ° (l-3n) 3
(6.8)
where V is the specific volume behind the shock, P is the shock pressure, Co and S are the constants of the linear fit of shock velocity, Us, to the particle velocity, uP, Us = Co + SuP, y is the Gruneisen parameter and 7=1-V/VO The sound velocities were calculated assuming F/V is a constant. The value shown on the ordinate of Figure 6.21 is the longitudinal wave velocity determined from ultrasonics measurements.
Figure 6.21. Release Wave Velocity vs. Compression for Copper
214
Shock Waves: Measuring the Dynamic Response of Materials
The initial release wave velocities determined from the lower pressure experiments agree well with the elastic release wave velocity for copper given by Al'tshuler et al. [1]. The two highest pressure points agree, within the error shown, with the Al'tshuler plastic wave data points and the calculated plastic release wave velocity. These results are interpreted to show an approach to purely hydrodynamic behavior.
6.4.2
6061-T6 Aluminum
The 6061-T6 alloy of aluminum is fairly strong, with a compressive yield of about 3.2 kb. The metal is relatively ductile and is strain-rate insensitive at room temperature. A series of intermediate pressure experiments was performed with Manganin wire gages to examine the unloading of the aluminum alloy. The experiments resulted in the series of pressure-time profiles shown in Figure 6.22. The peak pressures here range from 76 kb to 237 kb. An elastic precursor is not observed at these pressure levels and the finite risetime is attributed to shock tilt across the gage and the finite gage thickness.
Figure 6.22. Aluminum: Manganin Wire Records
215
6. Attenuation of Shock Waves from High Pressure
As higher pressures are produced in the material, the elastic-plastic structure of the unloading wave becomes more obvious. Although there is no elastic stress plateau, the waves are reasonably clear. An increase in the magnitude of the elastic release wave can be observed with increasing pressure. In these records, the release wave does not go to zero pressure because the impactor is backed with a plastic plate to insure flatness during launch. Again, in order to obtain comparisons of the pressure-time records with other forms of data, the Manganin records were transformed through the conservation Equations 6.6 and 6.7. When the Manganin records of the aluminum experiments are projected onto the free surface velocity versus normalized target thickness plane, the effect of increased release wave velocity with increased initial particle velocity may be seen (Figure 6.23). The overtaking points become progressively smaller for increasing free surface velocity, indicating higher release wave velocities. The initial rate of decrease is greater for the initially higher stress states. 3 I
J "
1
I
S
1
I
10
1
I
15
20
Figure 6.23. Free Surface Velocity vs. Distance for Aluminum from Manganin Wire Gage Records
Although the elastic-plastic nature of the release wave is fairly clear in the stress-time record of the Manganin wire record, it is not discernable in the particle velocity or free surface velocity versus target thickness presentation. Based upon these results, it would be very difficult to detect elastic-plastic behavior from experiments measuring free surface velocity as a function of target thickness.
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Shock Waves: Measuring the Dynamic Response of Materials
In Figure 6.24 the Manganin records have been translated into stressstrain release paths. Also shown is the Hugoniot for 6061-T6 aluminum. The release paths start from points on the Hugoniot and in all cases drop below the Hugoniot. This indicates an elastic-plastic release, as a hydrodynamic release would be placed above the Hugoniot. Once again, because of the plastic backing plate, the paths cannot be followed to zero stress. The offset from the Hugoniot of the release path increases with increasing pressure. If the difference between the Hugoniot and the release path is taken to be 4/3Y, then the following values of the yield are computed as a function of pressure. PEAK PRESSURE (kb)
Y(kb)
0
3
76
3
111
5
163
6
237
8
Figure 6.24. Aluminum Stress-Strain Release Path from Manganin Wire Gage Records
6. Attenuation of Shock Waves from High Pressure
217
Measurements made at higher pressures utilized the optical method for measuring free surface velocities. Experiments were performed at nominal impact pressures of 0.24, 0.69, and 1.01 Mb. Only the release wave overtaking points were measured at the higher pressures. The results of the lower pressure experiments are shown in Figure 6.25. Also shown in this figure is the hydrodynamic prediction of attenuation. A comparison of the free surface velocity data and the Manganin wire results is also shown in Figure 6.25. The Manganin wire results have been projected by the method of characteristics into the particle velocity-distance plane for this comparison. The Manganin wire results agree with the free surface measurements on the location of the overtaking but not upon the rate of attenuation. The free surface velocity measurements indicate a faster rate of attenuation. The normalized attenuation versus distance for aluminum is shown in Figure 6.26. Again, as with copper the overtaking distance decreases with increasing pressure. For the range of pressures shown here, 0.076 to 0.24 Mb, the initial rate of attenuation appears to be the same. The attenuation curves seem to be offset vertically in relative particle velocity by the same amount for the range of distances shown.
Figure 6.25. Free Surface Velocity vs. Distance for Aluminum: Impact Velocity, 2.5 mm/us. Normalized to UfS = 2.5 mm/us at X/Xo = 2. Pulse Length, 0.14 us.
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Shock Waves: Measuring the Dynamic Response of Materials
Figure 6.26. Normalized Attenuation for Aluminum
A number of high-pressure experiments were performed with 6061T6 aluminum. From these experiments, only approximate values of the release wave velocity could be obtained. The values obtained indicate that the initial release from pressure levels of 0.69 and 1.01 Mb is elastic. This may be seen in Figure 6.27, where the release wave velocities measured in this work and others are compared. The high-pressure points fall above the release wave velocities determined by Al'tshuler et al. [1], indicating an initial elastic release. Two calculated plastic release wave velocity curves are shown in Figure 6.27. Both were calculated from the Mie-Gruneisen equation of state and the linear fit to the Hugoniot, as described by Equation 6.8. The upper curve was calculated using the linear fit of Al'tshuler et al. [17]. Us = 5.341 + 1.353up (km/s)
(6.9)
where U s is the shock velocity and up is the particle velocity. This graph is shown to demonstrate the agreement between the Russian experimental release wave points and the calculation based on their linear fit. The velocity of the first release wave observed at the highest pressure level in this work is significantly higher than the
6. Attenuation of Shock Waves from High Pressure
219
Figure 6.27. Release Wave Velocity vs. Compression for Aluminum
expected plastic release wave velocity. This result is taken to imply that the first release wave at this pressure level is elastic. The evidence from the experiments with 6061-T6 aluminum at stress levels up to 0.25 Mb indicates that the initial release wave is elastic and that the magnitude of the elastic wave increases with increasing pressure, although the elastic release wave is not well separated from the plastic release wave. At pressures between 250 kb and 1 Mb, the initial release is believed to be elastic on the basis of the comparison with plastic release wave velocity. The elastic release wave velocity, initially about 20% higher than the plastic release wave velocity predicted by hydrodynamics, continues to
220
Shock Waves: Measuring the Dynamic Response of Materials
increase at that rate. The final two points, however, tend back toward plastic release wave velocity. Note that the various alloys of aluminum represented in Figure 6.27 cannot be distinguished (within experimental error) by their initial release wave velocities.
6.4.3
Titanium
Two types of titanium were examined, an alloy (6A1-4V annealed) and 0.995 pure titanium in the alpha phase. Only free surface velocity and wedge experiments were performed with the 6A1-4V alloy, and only Manganin wire gage experiments were performed with alpha titanium. Figure 6.28 presents a Manganin wire record for the alpha titanium. Examination of the front of the wave shows a slow rise (due in part to the gage response time) and some unevenness on the top of the record, due to system noise. The release wave shows a smooth transition from elastic to plastic behavior. This record was typical of the records obtained for a titanium at pressures of 80 and 134 kb. In Figure 6.29, the Manganin wire records have again been projected into the stress-strain plane. Here the Hugoniot has been extrapolated downward from the high-pressure Hugoniot and does not reflect a possible phase transition in this material at approximately 100 to 120 kb. The release path shows a large offset from the Hugoniot and, if taken to be 4/3 Y, the value of the yield is approximately 8.5 kb. This can be compared to a quasi-static compressive yield strength of about 5 kb. However, this material is very strain-rate sensitive and shows a compressive yield of about 8 kb when loaded at a rate of 600/s. Stressstrain tests have also shown that the yield is significantly reduced when the temperature is increased.
6. Attenuation of Shock Waves from High Pressure
Figure 6.28. Titanium: Manganin Wire Record
221
Figure 6.29. Titanium: Stress-Strain Release Path from Manganin Wire Gage Record
The Manganin wire gage results probably represent a mixture of the two effects, i.e., increase of the yield point due to rate of loading and pressure and reduction of the yield point due to shock heating. Additional data were obtained for an alloy of titanium, 6A1-4V. This is a high strength material with a compressive yield strength of about 9.6 kb, exhibiting moderate ductility (13% elongation) and strain-rate sensitivity. Free surface velocity measurements were made at a nominal impact velocity of 2.6 mm/fis. The results of these experiments are shown in Figure 6.30. The results suggest a gradual decrease in free surface velocity up to X/Xo of about 7, followed by a sharp decrease. A subsequent experiment performed with a wedge gave no clear indication of this behavior. Further, since the Manganin wire records gave no indication of this behavior with the pure titanium, it is concluded that it was caused by data scatter.
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Shock Waves: Measuring the Dynamic Response of Materials
Figure 6.30. Free Surface Velocity vs. Distance for Titanium: Impact Velocity = 2.6 mm/us. Normalized to Ufs = 2.6 at X/Xo = 2. Pulse Length 0.17 us.
Figure 6.31. Release Wave Velocity vs. Compression for Titanium
The overtaking point, as determined by the wedge experiment, agrees very well with the free surface velocity data. Figure 6.31 presents the initial release wave velocities observed in the titanium versus relative
6. Attenuation of Shock Waves from High Pressure
223
compression. Also shown here is the calculated plastic release wave velocity. As in the case of the aluminum alloys, the data from the 6A1-4V alloy and the pure alpha titanium are, within experimental error, the same. The initial difference between the calculated plastic release wave velocity is quite large but appears to have decreased at a compression of 1.2
6.4.4
Beryllium
Two types of beryllium were examined under the program: S-200 beryllium and a wrought beryllium (similar to Brush QMV) supplied by the Lawrence Livermore National Laboratory. Because of the high angle of intrusion of the lateral release waves, the targets were constructed with only two steps, rather than three as employed with targets made of the other materials. The data from the experiments are presented in Figure 6.32 for both materials. Within experimental error, there appears to be no difference in the results for the two materials. These data show a gradual decrease in. free surface velocity at X/Xo values of 1 to 3.5, then a sharp decrease at
Figure 6.32. Free Surface Velocity vs. Distance for Beryllium: Impact Velocity, 2.5 mm/us. Normalized to Uf, = 2.5 mm/us at X/Xo = 1. Pulse Length, 0.1 us.
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Shock Waves: Measuring the Dynamic Response of Materials
X/Xo of about 4.5. The sharp decrease is interpreted to be the arrival of the elastic release wave from the rear surface of the impactor plate. By using an overtaking distance of X/Xo of 4.2 with a shock velocity of 9.6 mm/us, an elastic release wave velocity of 13.6 mm/us is inferred. It is not clear why the gradual decrease in free surface velocity occurs between values of X/Xo of 1 to 3.5. Close examination of the experiments and the results show that this decrease is a real effect, since it is regular and was observed in all experiments. The accuracy of the individual experiments is ±2%. The data show decreases outside this range. The shims used on the target surface in all experiments were of aluminum foil. These aluminum shims were observed to fly off the beryllium surface with a velocity approximately three to four percent lower than the beryllium impactor velocity. This is the expected difference, due to the impedance mismatch between the beryllium and the aluminum at 225 kb. The decrease in the free surface velocity may be due to shock dispersion, a rate effect, or twinning. Two sources of possible error were examined for their effect on measured velocities. The data were examined to determine if the elastic precursor was affecting the results. The free surface velocity was measured by measuring the time required for the thin surface shim to cross a small gap of known width. The initial motion of the shims was detected by their loss of reflectivity. If the elastic precursor was causing the change in reflectivity (with resulting low free surface velocities) the measured shock velocities in the specimens would be the elastic precursor velocity. Because the shock velocity observed in all experiments corresponded to the plastic wave velocity, it was concluded that the signal of initial motion of the shim as recorded in the experiments was caused by the plastic wave, rather than the elastic wave. The possibility that the elastic precursor, although undetected, caused the surface to move prior to the arrival of the plastic wave was also examined. It was found that the motion due to the elastic precursor would cause a reduction of about 0.2% in the gap width for the experiments performed and an increase in the free surface velocity of the
6. Attenuation of Shock Waves from High Pressure
225
same amount. Therefore, the observed effect is associated with the plastic portion of the compressive wave.
6.4.5
Uranium Alloy
A limited group of experiments was performed using impactor plates and targets of a uranium alloy supplied by the Lawrence Livermore National Laboratory. The density of this alloy was 16.41 g/cm3 [16]. Copper shims were used in these experiments. The results are shown in Figure 6.33. These data indicate an overtaking distance of X/Xo of about 4.8. This, taken with a particle velocity of 1.26 mm/us and a shock velocity of 4.46 mm/us, gives a release wave velocity of 4.88 mm/u.s. Lack of ductility of the uranium alloy makes it difficult to launch unbroken and undeformed impactor plates. Examination of the streak camera records of the experiments showed unexplained discontinuities in the traces, making the data suspect. An experiment was performed in which a plate of the uranium alloy was impacted directly upon the reflecting surface of a mirror. The mirror was observed with a streak camera.
Figure 6.33. Free Surface Velocity vs. Distance for Uranium Alloy: Impact Velocity, 2.5 mm/us. Normalized to Ufs = 2.5 mm/us at X/Xo = 21. Pulse Length, 0.23 us.
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Shock Waves: Measuring the Dynamic Response of Materials
The record from this experiment showed that the impactor plate was not flat upon impact and indicated that it was broken in the center. Subsequent experiments of the same type were attempted using stronger backing plates behind the uranium plate and lower launch velocities without success.
6.5
Conclusions
From the preceding experimental results several conclusions may be drawn: (1) Evidence of elastic release waves at high pressures. This work has shown that elastic release waves persist to very high pressures. For all materials examined, elastic release wave behavior was observed. For those materials for which pressuretime profiles were recorded, increases in strength were observed with increases in pressure. The presence of significant shear strengths was observed through comparison of observed sound velocities with the observed or calculated plastic release wave velocities, as in Figure 6.34, where the velocity of the first observed disturbance is compared to plastic release wave velocity. (2) Evidence of melting during compression for copper. Most of the data taken on copper cluster about a line of slope 1.2. Data significantly below the 1.2 slope line are expected when the temperature behind the shock front becomes sufficient to melt the material in the compressed state and the shear strength becomes zero. Elastic waves will then no longer be observed. McQueen et al. [18] calculate the onset of melting for copper to occur at a pressure of 1.33 Mb. The upper two copper points shown in Figure 6.34 are above this pressure and fall about a line of slope equal to one. Since this represents plastic release wave velocity, we take this to indicate melting behind the shock front.
6. Attenuation of Shock Waves from High Pressure
227
Figure 6.34. Initial Release Wave Velocity vs. Plastic Release Wave Velocity
(3) Evidence for lack of melting in aluminum at 1 Mb. The highest pressure point for aluminum (1.01 Mb) indicates elastic release. Urlin [19] has predicted melting to occur in aluminum at approximately 1 Mb. These experimental results indicate that melting has not yet occurred at this pressure. The difference in melting characteristics of the alloys of aluminum may account for this behavior, however. With the exception of beryllium (see below) and the two copper points, all the data taken fall about the 1.2 slope line, indicating no melting behind the shock front. Because shock wave velocity is approximately equal to plastic release wave velocity, the fact that data cluster about a line of slope 1.2 provides the basis of approximating elastic release wave velocities for materials whose release wave properties are not known.
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Shock Waves: Measuring the Dynamic Response of Materials
(4) Evidence for loss of strength during melting. The observed increase of strength followed by an eventual loss of strength when melting is approached suggests that the two effects, heating and increasing strength with increasing pressure, are in competition. For pressures where there is little shock heating, the material strength increases. As the pressure is increased, heating produces a slower rate of increase in elastic strength and finally a decrease to essentially no strength when melting occurs. For this behavior there would be a pressure at which the material would show a maximum strength. (5) Lack of clearly defined elastic-plastic waves indicates a Bauschinger behavior. In none of the materials examined was elastic-perfectly plastic behavior clearly observed. From the stress-strain release paths it is concluded that the Bauschinger effect was observed for copper, a titanium, and 6061-T6 aluminum. From the free surface velocity experiments with other materials it was not possible to separate elastic and plastic release waves. (6) Changes in Poisson's ratio. Examination of the results shown in Figure 6.34 leads to the conclusion that Poisson's ratio (v) is not a strong function of pressure and temperature up to the onset of melting. Further, the elastic release wave velocity is related to plastic release wave velocity by
cp
I 1+ u )
(6.10)
For the materials shown in Figure 6.31, vis approximately 0.3, which leads to a ratio or slope of about 1.24. Beryllium is a significant departure from this with v = .055. The beryllium ratio of elastic to plastic wave speeds is predicted to be 1.64. Since a value of 1.41 is observed, a decrease in Poisson's ratio is indicated for this material.
6. Attenuation of Shock Waves from High Pressure
229
(7) Effects of alloying. From the results of the experiments and other data such as shown in Figure 6.31, alloys of metals show little difference in their elastic release wave velocities, at the high pressures tested. This does not imply that the details of the release waves are the same. (8) Wave profiles are required to provide accurate measurements of elastic-plastic behavior. Through the course of the experimental program it became clear that, without pressuretime or particle velocity-time measurements, it is very difficult, if not impossible, to identify elastic-plastic behavior. Examination of the pressure-time plots and the resulting projections into free surface velocity-distance plots demonstrates that the elastic-plastic release waves observed in the pressuretime plots are not clearly shown in the free surface velocitydistance plane. Experiments with wedges were useful only to identify the overtaking of the shock front by the first release wave. Extraction of attenuation data from wedge results requires differentiation of the data, which results in errors that can obscure the rate of attenuation. Examination of this problem revealed also that wedge experiments can be an extremely delicate measure of wave curvature produced by impactor curvature. Extremely small curvature of the impactor plate can obscure the rate of attenuation. Elastic release waves have been shown in this study to persist even at megabar pressures. The details of these waves depend upon complex factors, including the competing forces of 1) decreases in strength with increasing temperature behind the shock front and 2) increases in strength with increasing shock pressure. Models of material behavior at high pressures must account for these effects.
230
6.6
Shock Waves: Measuring the Dynamic Response of Materials
References
1. Al'tshuler, L. V., S. B. Kormer, M. I. Brazhnik, L. A. Vladimirov, M. P. Speranskaya, and A. I. Funtikov. Soviet Physics JETP, vol. 11, no. 4, 766775, October 1960. 2. Curran, D. R. J. Appl. Phys., vol. 34, no. 9,2677-2685, September 1963. 3. Erkman, J. O., and A. B. Christensen. J. Appl. Phys., vol. 38, no. 13, 53955403, December 1967. 4. Barker, L. M. "Fine Structure of Compressive and Release Wave Shapes in Aluminum Measured by the Velocity Interferometer Technique." Behavior of Dense Media Under High Dynamic Pressures, 483-505. New York: Gordon and Breach, 1968. 5. Kusubov, A. S., and M. Van Theil. J. Appl. Phys., vol. 40, no. 9, 3776-3780, August 1969. 6. Fuller, P. J. A., and J. H. Price, Brit. J. Appl. Phys., series 2, vol. 2, 275-286 (1969). 7. Lingle, R., and A. H. Jones. TR65-65, GM Defense Research Laboratories, Santa Barbara, California, September 1965. 8. Lyle, J. W., R. L. Schriever, and A. R. McMillan. J. Appl. Phys., vol. 40, no. 10,4663^1664 (1969). 9. Fowles, G. R., and W. M. Isbell. J. Appl. Phys., vol. 36, no. 4, 1377-1379 (1965). 10. Isbell, W. M., and D. R. Christman. "Shock Propagation and Fracture in 6061T6 Aluminum from Wave Profile Measurements." Materials and Structures Laboratory, Manufacturing Development, General Motors Corporation, Warren, Michigan, MSL-69-70,1970. 11. Isbell, W. M., D. R. Christman, S. G. Babcock, T. E. Michaels, and S. J. Green. "Measurements of Dynamic Properties of Materials - Vol. I - Summary of Results." Materials and Structures Laboratory, Manufacturing Development, General Motors Corporation, Warren, Michigan, MSL-70-23, Vol. 1,1970. 12. Green, S. J., and F. L. Schierloh. "Uniaxial Stress Behavior of S-200 Beryllium, Isotropic Pyrolytic Boron Nitride and ATJ-S Graphite at Strainrates to 103/Second and 700 °F." Materials and Structures Laboratory, Manufacturing Development, General Motors Corporation, Warren, Michigan. 13. Munson, D. E. "Dynamic Behavior of Beryllium." Sandia Laboratories, Albuquerque, New Mexico, SC-RR-67-368,1967. 14. McQueen, R. G., S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Cuter. 'The Equation of State of Solids from Shock Wave Studies." High-Velocity Impact Phenomena, R. Kinslow, ed. New York: Academic Press, 1970. 15. Isbell, W. M., F. H. Shipman, and A. H. Jones. "Hugoniot Equation of State Measurements for Eleven Materials to Five Megabars." Materials and Structures Laboratory, Manufacturing Development, General Motors Corporation, Warren, Michigan, MSL-68-13,1968.
6. Attenuation of Shock Waves from High Pressure 16. Royce, E. B., and W. H. Gust. "Shock Compression of the Stainless Uranium Alloy Mulberry." Lawrence Radiation Laboratory, University of California, Livermore, California, UCRL 50888,1970. 17. Al'tshuler, L. V., S. B. Kormer, A. A. Bakanova, and R. F. Trunin. Soviet Physics JETP, vol. 11, no. 4,573 (1966). 18. McQueen, R. G., W. J. Carter, J. N. Fritz, and S. P. Marsh. 'The Solid-Liquid Phase Line In Cu," presented at the Symposium on the Accurate Characterization of High Pressure Environment, U.S. Dept. of Commerce, National Bureau of Standards, Gaithersburg, Maryland, October 14-18,1968. 19. Urlin, V. D. Soviet Physics, JETP, vol. 22, no. 2, 341-346,1966.
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7. Response of Porous Beryllium to Static and Dynamic Loading 7.1
Introduction
The effectiveness of porous materials in attenuating stress pulses and in reducing the thermomechanical stresses arising from rapid energy deposition has been the subject of numerous studies. Because of the large number of manufacturing parameters available to the developers of porous materials (composition, porosity, pore size, heat treatment, etc.), extensive tailoring of properties to meet widely varying requirements is practical, and the materials manufactured and the studies to date now number in the dozens. The study described in this section involves two porous berylliums of different initial heat treatments and slightly different porosities. A theoretical model was developed and a series of measurements were made to describe the complex equation of state surfaces peculiar to porous materials. The portion of the work to be reported on involves the behavior of the materials under static and dynamic loading. A more complete documentation of the experimental and theoretical results is presented in the literature [1-5].
7.2
Equation of State Surface
Thermodynamic equilibrium properties of nonporous materials may be described by unique relationships between the thermodynamic variables: pressure, temperature, entropy, energy, volume, etc. This unique relationship between the thermodynamic variables is often represented by a three-dimensional surface showing the allowed thermodynamic
233
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Shock Waves: Measuring the Dynamic Response of Materials
equilibrium states of the material in terms of any three of the thermodynamic quantities. Figure 7.1 shows a schematic representation of the pressure, temperature, and volume relationship for a typical material, including phase changes. Each point on the non-porous equation of state surface represents a unique state-point, i.e., the pressure at a particular volume and temperature is independent of the path (or past history) used to arrive at that volume and temperature.
Figure 7.1. Schematic representation of the P-V-T relationships for typical solid ana porous materials No such unique relationship exists for porous materials. The complex non-equilibrium thermodynamic states that can be reached by a porous material are not described by a single equation of state, but rather by a
7. Response of Porous Beryllium to Static and Dynamic Loading
235
mathematical model that depends on the thermodynamic path by which the material arrived at its current state. The porous "addition" to the surface shown in Figure 7.1 represents only an initial crushing surface. Unloading and subsequent reloading of the porous material occurs on another path which penetrates below the initial crushing surface. This is shown as an intermediate crush surface in the figure. Such intermediate crush surfaces are examples of how the pressure in a porous material depends not only on the volume and temperature, but also on the past history of the material.
7.3
Dynamic Behavior
The initial compaction surface (and the intermediate unloading-reloading surfaces) shown in Figure 7.1 is valid for static or quasi-static loadings and unloadings. If the loading is rapid (as in a shock wave front, for instance) then inertial and viscous resistance to the rapid collapse of the pores will come into play to make the dynamic compaction path lie above the quasi-static path. In both impact tests and energy deposition tests, the initial stresses produce pore-collapse rates that are high enough to lead to temporary overstresses. Several models have been proposed [6-10] to describe ratedependent pore collapse. Most of these assume that a dynamic overpressure (dependent on the rate of pore collapse) exists for some characteristic time, r (dependent on material properties), with an exponential relaxation to the static crushing surface. Each of the models used can be calibrated, with varying degrees of success, to agree with the risetimes and wave velocities from plate impact experiments.
7.4
The Porous Constitutive Model
The model developed to describe the porous beryllium [3] based on the P-a-x model of Holt et al. [7] which is, in turn, a rate-dependent modification of Herrmann's model [6]. The new features in the present model are: the inclusion of deviatoric stresses, the use of a porosity-
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Shock Waves: Measuring the Dynamic Response of Materials
dependent relaxation time for pore closure, an allowance for partial reopening of the pores on unloading, and the use of improved static compaction functions. Two relations are used in the formulation of the constitutive model. The first relation is an equation which relates the pressure P, volume V, and energy E of a porous material to the equation of state (EOS) of the corresponding fully compacted solid. The porosity parameter a is defined by the relation P(V,E) = - P s ( V / a , E s )
(7.1a)
a which can be rewritten in an alternate but equivalent form a = V/Vs=Ps/P
(7.1b)
where the subscript s refers to the solid material. The second relation used in formulating the model is an equation which describes the elastic and plastic portion of paths for a during a time-dependent deformation process. Along the plastic path, the ratedependent pore-closure relation of Holt et al. is used. o! = g ( P ) - r d a / d t
(7.2)
The parameter T is a time constant describing the rate of plastic flow of material into pores, which reduces a to a final average equilibrium value given by the static compaction function g(P). To determine g(P), V in (1) is eliminated by using hydrostatic P-V data for both solid and porous beryllium. The resulting data, P and g(P), can be accurately represented by the following expression: g (P) = ax + (a0 - a^) exp (aP + bP 2 + cP 3 )
(7.3)
where a0 is the initial porosity and am is introduced to account for some residual porosity that appears to persist under extremely high pressures
7. Response of Porous Beryllium to Static and Dynamic Loading
237
[5, 11]. The constants a, b, and c are obtained by fitting the data by a least-squares method. A modification of the elastic viscoplastic, rate dependent behavior described by Holt's spherical pore model is used to describe the rate dependence of the compaction. A porosity dependent r of the form T = TQa0(a0 -ax)/[a(a
- ax)]
(7.4)
can be calibrated to fit the calculated wave profiles for as-sprayed beryllium. To obtain an equation for a along an elastic path, the porous and solid equations of state are expressed in incremental form. That is, using A to denote the difference in P (and in Ps) at two successive time steps AP = K ( A p / p)
(7.5a)
APs^Ks(Aft/ft)
(7.5b)
where K and Ks are the bulk moduli of the porous and solid materials, respectively. After lengthy algebraic manipulation, an expression is obtained for the rate of change of a with P along the elastic path of the form da/dP = a(Ks/K-a)/(aP-Ks)
(7.6)
where Mackenzie's [12] expressions for K and G are rewritten l / K = (aoo/Ks)t + ( l / K 0 ) ( l - t )
(7.7a)
G = (l/Gs)t + (l/G0)(l-t)
(7.7b)
where t is defined as (oto - a)/(cto - cu) and the subscript "o" refers to the initial porous state.
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Shock Waves: Measuring the Dynamic Response of Materials
To complete the formulation, the deviatoric stress s must be specified. Then, the axial stress G\ (along direction 1) in a 1-D strain deformation is given by ax = - P +
Sl
(7.8)
where Sl=-Gln(V/V0),if|Sl|<-Y
2 = ± — Y otherwise ( + for loading 3 V and — for unloading)
(7.9a)
(7.9b)
and Si is positive for loading and negative for unloading. The shear modulus G in (7.9a) is calculated from (7.7b). Y may be calculated from 2/3 Y = o\ + P if experimental data are available for P and
Y = Max [Yo Y1 + Y2 e + Y3 e2 ]
(7.10)
where s = ln(V/V0) and small elastic strain is neglected.
7.5
Description of the Porous Berylliums
Two porous beryllium materials were studied, both plasma-sprayed by Union Carbide Corporation from powders supplied by Kawecki Berylco Industries, Inc. The materials were prepared in accordance with Kaman Science Corporation specifications for Models 67 and 68 beryllium. The plasma spraying process involves the ejection of metallic powder using a jet of inert gas (argon) from a nozzle. The stream of powder passes through an electric arc and is melted to form a stream of molten droplets. These droplets land on a rotating aluminum mandrel (or
7. Response of Porous Beryllium to Static and Dynamic Loading
239
turntable). The nozzle moves radially in and out across the mandrel face during the spraying process so that the droplets form a spiral pattern in and out across the mandrel, building up a plate, layer upon layer. Two grades of beryllium powder were used to manufacture the plates, P-l and P-10. Specimens made from P-10 powder were tested in the "as-sprayed" condition while specimens from P-l powder were sintered for 2 hours at 1175°C, producing a less porous and stronger material. Table 7.1 summarizes the chemistry of the two powders used and the densities of the resulting plasma-sprayed materials. Table 7.1. Chemical Analysis of Beryllium Specimens P-10 (-325 mesh, not sintered) wt% Chemistry BeO 0.66 Fe 0.075 Al 0.024 C 0.029 Mg 0.026 Si 0.010 Density, g/cm3 1.587 Porosity 14.2%
7.6
P4 (-325mesh, sintered) wt% 0.72 0.035 0.006 0.026 0.002 0.008 1.647 11.0%
Experimental Data
To perform meaningful calculations, a computational model must first be fitted to accurate and reasonably complete experimental data. In the present study, impact and static pressure-volume data provided paths for mechanical loading and unloading and rate-dependency of yielding as well as providing a check on the calculational ability of the model to predict shock wave attenuation. Ultrasonic measurements were used to provide elastic moduli at standard temperature and pressure. Microscopic
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Shock Waves: Measuring the Dynamic Response of Materials
examination of compressed specimens yielded insight into deformation and mechanisms of the pore collapse process. Effective Gruneisen and expanded volume states were measured using an electron-beam machine to provide nearly constant-volume thermal heating in short deposition times. A large amount of experimental data were generated on the two berylliums during this program. A brief synopsis of the data follows.
7.6.1 Static Compression Data Throughout the study, quasi-static deformation data [2, 13] served to guide the modeling effort. Considerable detail is contained in the data concerning: compression under hydrostatic and one-dimensional conditions, release behavior of the compressed materials, residual porosity at high pressures (>4GPa), and elastic yield in compression and release. The relatively low cost of this data, when compared to shock wave data, makes the static technique an attractive alternative for characterizing porous materials. The static data, when combined with a modest number of shock wave tests to determine deviatoric and timedependent behavior, have proven adequate for modeling several materials of interest. Two types of experiments were performed: loading and unloading under conditions of uniaxial strain and under conditions of hydrostatic pressure. In uniaxial strain loading, an axial stress was applied to a cylindrical sample with the condition that the radial strain remain constant. This was achieved by control of the lateral confining pressure and resulted in a loading path similar, except for time-dependent flow, to that of plane shock-loading. In the hydrostatic testing, an axial stress was applied to a medium surrounding a cylindrical sample. The plastic flow of the surrounding medium (fluid at low pressure and tin at high pressure) insured that the loading was nearly hydrostatic in nature. Details of the uniaxial strain loading data are shown in Figure 7.2. The initial slope of the unsintered curve is seen to be much lower than in the sintered material, showing that the unsintered material is originally more compressible. This may be the result of some or all of the following
7. Response of Porous Beryllium to Static and Dynamic Loading
241
effects: (1) the sintering process, which produces more spherical and hence stronger pores; (2) the existence of numerous microcracks in the unsintered material; (3) the decreased porosity and thus greater strength in the sintered beryllium; and (4) the effect of residual stresses in the unsintered material, lowering the applied shear stress necessary to cause yielding. Also shown in Figure 7.2 are Hugoniot points taken over the same stress range as the laboratory uniaxial data. Within experimental error, for the unsintered beryllium the points are coincident with the path that defines the stress-volume curve, hi contrast, the shock wave data for the sintered beryllium lie consistently above the static data, most probably indicating increased strain-rate behavior for that material.
Figure 7.2. Comparison of quasi-static uniaxial stress-strain data on two porous berylliums with corresponding Hugoniot data
7.6.2 Shock Wave Tests Shock wave generation and propagation under conditions of uniaxial strain were measured [1, 5] using gas gun-launched impactors and a variety of experimental configurations. Data obtained included: Hugoniots to 3.2 GPa (32 kb), release adiabats from shocked states, shock wave profiles for attenuated and unattenuated waves, and compressional and release velocities over a range of stresses. A separate series of explosive tests [11] extended the range of Hugoniot measurements to over 33 GPa (330 kb).
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Shock Waves: Measuring the Dynamic Response of Materials
Most data were used directly in developing the material models, although independent check data (primarily attenuated wave profiles) were used to determine the accuracy of the model's predictive capability. This section summarizes the data obtained and experimental techniques used. Hugoniot data and unattenuated wave profiles were obtained primarily with the three measurement techniques shown in Figure 7.3. A 90 mm diameter gas gun was used to launch four porous beryllium specimens into four materials of different impedances mounted on quartz stress gages (Figure 7.3a), or reversing the procedure, to launch four materials of different impedances into beryllium specimens mounted on quartz gages (Figure 7.3b). The "direct impact" technique (a) provided Hugoniot data in the form of stress-particle velocity points calculated by knowledge of the Hugoniots of the four impactor materials plus the known characteristics of the quartz stress gages and buffers. This multiple gage technique is particularly efficient in reducing ambiguities among data points in that all specimens are impacted at the same impact velocity and at the same impact tilt. Cross correlation between data points is thus facilitated and results in relatively accurate data. Figure 7.4 shows Hugoniot data obtained to 4.5 GPa (45 kb). The relatively low compaction wave and high initial release wave velocities account for the very rapid attenuation rates customarily found in porous materials. Although quartz stress gage measurement techniques have advantages in simplicity and reasonably well-known gage characteristics, the (usually) large change in shock impedance at the gage-specimen interface reflects a portion of the shock wave and may considerably complicate analysis. An ideal measurement technique would require in situ measurement of the passage of the shock wave without disturbance of the wave itself. At present, "in-material" piezoresistive stress gages meet this requirement most closely.
7. Response of Porous Beryllium to Static and Dynamic Loading
243
Figure 7.3. Schematic of the measurement techniques used to obtain Hugoniot data and the stress-time profiles of transmitted waves, (a) Hugoniot points from four specimens launched into four quartz gages with buffer plates of various shock impedances; (b) Four cornpressive wave profiles at different material thicknesses; (c) Attenuating wave profiles measured with in-material piezoresistive gages.
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Figure 7.4. (a) Hugoniot points and their analytical representations; (b) Compaction wave speeds vs. velocity Us obtained from Hugoniots and initial release wave velocities calculated using Equations 7.3 and 7.7.
For most of the wave profiles used for model development and for checks on predictive capability, carbon-Kapton in-material stress gages were used in the configuration shown in Figure 7.3c. Since the gage is quite thin (0.1 mm), it usually comes into pressure equilibrium with the specimen material within a few tens of nanoseconds by a series of shock reverberations across the gage, even though the shock impedances of the gage and test specimens may be very different. Thus, if a suitable calibration has been performed on the gage, one obtains a direct measure of the stress in the text specimen. It should be noted, however, that the impedance difference does indeed affect the profile shape somewhat, and accurate work requires the ability to calculate profiles using equations of state of the gage and of the material under test. Using in-material gages placed between successive layers of the specimen, the evolution of waves as they progressed through the material was studied. Elastic and plastic compressive and release waves were measured, and, by using careful timing techniques, wave speeds were obtained.
7. Response of Porous Beryllium to Static and Dynamic Loading
7.7
245
Comparison of Predictions with Data
Experimental wave profiles for comparison were obtained at two stress levels, approximately 0.6 and 1.7 GPa. Profiles were obtained by impacting a PMMA plate onto either sintered or as-sprayed porous beryllium having carbon-Kapton piezoresistive gages embedded at as many as six different depths within the material. Thus a single experiment took data on wave profiles at several different levels. The gages were located as deep as 1.9 cm from the impact surface and some measurement times extended up to 4 us. The resulting 21 wave profiles — both unattenuated and attenuated — represent large variations in the physical parameters and enabled us to extensively test the model. Measured and computed wave profiles are compared in Figure 7.5 for the sintered material and in Figure 7.6 for the as-sprayed material. Agreement between the experimental wave profiles and those predicted from the present model is good both quantitatively and qualitatively. For sintered porous beryllium (Figure 7.5) an elastic precursor of amplitude -0.4 GPa, which corresponds to the "shoulder" in the hydrostat in Figure 7.2, precedes the main plastic wave. At the foot of the precursor the velocity is close to the longitudinal sound speed, but it becomes slower at higher stresses. The 0.62 cm/us value at P = 0.2 GPa chosen for the calculation of the first arrival times agrees reasonably well with the velocity of the precursor at this level. A faster-rising shock front, the lack of an elastic precursor, and a higher predicted attenuation rate are the chief distinguishing features of compressive profiles in as-sprayed porous beryllium. Other tests at 1.7 GPa (17 kb) showed similar behavior. Figures 7.5 and 7.6 demonstrate that the arrival times of the shocks (and their precursors in the case of sintered specimens) agree satisfactorily. The relaxation time T = 0.04 us matches the observed risetimes of the shocks in the case of the sintered specimens, while the porosity-dependent r from (4) adequately describes the risetimes of the shocks in the as-sprayed specimens. In the latter case, it is noteworthy that T changes from 0.060 us at the foot of the shocks to 0.015 us) had been used for r, the foot of the shocks would be traveling too fast, resulting in much longer risetimes than those in the observed profiles.
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Figure 7.5. Full and attenuated wave profiles for a low-stress test in sintered porous beryllium. Theoretical predictions are shown by solid lines. Velocity of the 0.124 cm thick PMMA impactor was 0.0262 cm/us.
7. Response of Porous Beryllium to Static and Dynamic Loading
247
The largest deviation occurs in the calculation of the peak stresses, with the experimental data lying 10% or less below computer predictions. Either experimental uncertainties or approximations in the model or a combination of the two could account for these relatively small differences.
Figure 7.6. Wave profiles for as-sprayed porous beryllium at a similar impact velocity. PMMA impactor thickness was TX1245 cm and impact velocity was 0.0258 cm/us.
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7.8
Electron-Beam Tests
This study included the development of experimental measurement techniques particularly suited for obtaining porous material property data in hostile electron-beam environments. New techniques were developed to measure optically the free-surface motion of exposed porous samples with a DISAR (Displacement Interferometer System for Any Reflector). Existing techniques for diagnosing electron-beam fluence and energy deposition in materials were modified to measure the energy density in samples on every data shot. Experiments using high-energy electrons (-2.2 MeV) gave very flat deposition profiles in porous beryllium and successfully demonstrated both the DISAR and the energy-deposition monitoring techniques. As expected, pronounced differences were observed in the responses of the two types of porous beryllium to the rapid (-70 ns) deposition of energy.
7.8.1
Experiment Design
Although it is complex and its results can be uncertain, pulsed electronbeam exposure is one of the few techniques available for laboratory study of the properties of rapidly heated, porous materials. Reliable EOS data for porous materials have been difficult to obtain with conventional electron-beam experiments for several reasons: • •
•
The incident fluence is usually obtained from shot-to-shot repeatability information (± 15-20%). Electron absorption as a function of depth typically generates an energy-deposition profile with a gradient of energies in the sample, instead of a more desirable uniform heating of the material to one P-V-E point. Porous materials typically produce low initial stresses and strongly attenuate the waves, so that extensive knowledge of the properties of the material is required to interpret the propagatedwave profiles.
7. Response of Porous Beryllium to Static and Dynamic Loading
249
A schematic of the experiment configuration used for these tests is shown in Figure 7.7. This design includes: • •
•
•
In-line calorimeters for every data shot, consisting of thin depthdose monitoring foils in front and back of the sample. A totally absorbing, segmented carbon calorimeter behind the sample to measure the spatial distribution of the "shine-through" fluence. Flat deposition profiles obtained by using high-energy electrons (-2.2 MeV) with a sample that is thin with respect to the electron range, but thick with respect to the distance a relief wave would travel during deposition time. Free-surface velocity measurement via DISAR, a displacement interferometer system designed to work with diffuse reflecting surfaces.
Figure 7.7. Schematic of electron-beam-experiment configuration showing the location of the energymonitoring system and the diffuse surface displacement interferometer (DISAR).
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Shock Waves: Measuring the Dynamic Response of Materials
The thin depth-dose foils and the segmented calorimeter were adaptations of existing electron-beam diagnostic equipments designed to make the energy-density measurement independent of the shot-to-shot variability of pulsed electron beams. The displacement interferometer system was an innovation in the optical measurement of free-surface motion of porous or irregular surfaces. Three configurations of a DISAR were developed and demonstrated. These displacement-measuring interferometers, which relied on the slight degree of optical coherence remaining in a laser beam reflected from a presumably "diffuse" surface, differed from the velocity-measuring interferometer (VISAR) developed at Sandia National Laboratory. Such coherence is unnecessary in the VISAR system, which measures substantially higher velocities than those attained in the electron-beam experiments. The DISAR measures velocities in the 0-0.05 mm/us range, using the short coherence-path length (typically a few cm) of a laser beam reflected from a porous surface. The instrument combines the reflected beam with a "reference" beam in an adaptation of a conventional Michelson interferometer. In laboratory tests of these systems, better than 50% modulation of the light signal was obtained when sample motion shifted the fringe pattern. An overall schematic of the system showing the location of the photomultiplier tube, light-detector system with respect to the electronbeam machine is shown in Figure 7.8.
7.8.2 Experimental Results Displacement vs. time data, as measured with the DISAR on a plasmasprayed, sintered porous beryllium sample, are shown in Figure 7.9. The peak deposition level was ~20 cal/g for this data shot. This energy level is clearly in the category of very low-energy density, compared to the melt energy of 878 cal/g for beryllium, and the displacement record agrees qualitatively with the expectation that the material would act in a manner similar to an elastic solid at this energy density.
7. Response of Porous Beryllium to Static and Dynamic Loading
Figure 7.8. Schematic of the optical system used to monitor free surface motion
Figure 7.9. Displacement vs. time measured by DISAR on rear surface of sintered beryllium sample after exposure to -20 cal from ~1.7 MeV electrons deposited in ~25 ns
251
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Shock Waves: Measuring the Dynamic Response of Materials
The initial deposition in the porous sample takes place at essentially constant volume. As seen in Figure 7.9, the free surface moves out after the energy is deposited as the stress-relief wave travels through the material. After one complete transit time across the sample, the free surface is "pulled back" by the tensile strength of the material. The sample continues to "ring" much the same way an elastic solid would respond to such loading. Numerical derivatives of successive displacement points were used to obtain the velocity vs. time data shown in Figure 7.10.
Figure 7.10. Velocity vs. time measured by DISAR on rear surface of sintered beryllium sample and calculated using the model described in this report. Energy density is ~20cal/g from -1.7 MeV electrons deposited in -25 ns.
The calculational model developed to fit the wave-propagation properties of this beryllium was also used to calculate the free-surface motion under the same loading conditions. The curve labeled "KO prediction" in Figure 7.10 shows the results of applying the calculational model. As shown by this figure, the peak velocity is accurately predicted by the model. The model, however, does not handle the wave timing as well.
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253
Under these loading conditions, at least two possible inaccuracies in the model could explain the differences between the two curves. First, the stress levels involved are quite low (< 1 kb). This low stress has the highest uncertainty in both the static and gas-gun tests used to calibrate the model — therefore, larger errors may be expected to occur at low stress levels. Second, the material is unloading to volumes that are larger than the initial volume of the sample. Because no data existed for expanded states (beyond the initial distension), the calculational model used mathematical extrapolations of the compaction behavior in this expansion region. Figure 7.11 shows the displacement vs. time data for one of the assprayed samples. Of particular interest in this curve is a partial relief wave producing the definite change in the free-surface velocity after -0.35 us. Such changes in free-surface velocity (occurring 0.4-0.7 us after pulsing) were observed on most of the as-sprayed samples. The sample-to-sample variability in the arrival time (and the existence) of this two-wave structure is not fully understood at this time. The pressures involved are quite low (< 1 kb), and the model calibrated to the gas-gun and static P-V tests does not include any structure below 1 kb: therefore, this two-wave structure was not predicted by the hydrocode model.
Figure 7.11. Displacement vs. time measured by the DISAR on the rear surface of as-sprayed beryllium after exposure to ~80 cal/g from ~2 MeV electrons
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Shock Waves: Measuring the Dynamic Response of Materials
Data from the two series of electron-beam exposure tests are summarized in Table 7.2. These data clearly indicate the differences observed between the as-sprayed and the sintered material. The higher free-surface velocities, coupled with faster relief velocities for the sintered material, imply that significantly greater pressures were generated in that material. Table 7.2. Electron-beam data
Shot No.
Energy Density Material cal/g
Free Surface Velocity Transit Vfs Time, T m/s us
2nd Wave 1st FreeSurface Wave Velocity Transit m/s
Predictions for Each Group of Berylliums
6524 As-sprayed
-70
8.4
-
0.63
6.3
£ ~ 100 cal/g
6526 As-sprayed
-
7.3
1.6
0.8 a
6.4
P~0.1GPa
6527 As-sprayed
-80
9.9
1.5
0.45 a
8.4
V f s - 1 8 m/s
6537 As-sprayed
-80
10.0
1.4
0.55
6.6
T-l.Ous
5201 As-sprayed
-40
4.4
1.45
0.5a
4.0
6528 6529 6531 5202
-70 -70 -70 -18
-39 -38 -39 -9.8
0.50 0.50 0.51 0.36
a
Sintered Be Sintered Be Sintered Be Sintered Be
8 - 100 cal/g P~ 0.33 GPa Vfs - 33 m/s T - 0.62 us
Existence of two waves is questionable when the velocity difference is 10% or less.
NOTE: T above refers to the time, from shot time, until the free-surface velocity changes direction — indicating the arrival of a complete relief wave from the other free surface (sample thickness = 3.7 mm).
In addition to the calculation shown in Figure 7.10, two other hydrocode calculations were performed, using the models developed from the mechanical tests, one for as-sprayed beryllium and one for sintered beryllium. Both calculations assumed a flat energy-deposition profile with an energy density of -100 cal/g in the beryllium.
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255
In the table, the column labeled "Predictions" summarizes the results of these two calculations. Considerable difference is evident between the calculated and measured values for the transit time and the free-surface velocities. Possible reasons for these differences have already been given. Of significance, however, is the difference between the response of the two berylliums; the model as well as the data show considerable differences in the responses of the two materials at these low pressures.
7.8.3 Applicability of Experimental Techniques to Other Materials The DISAR was successfully demonstrated in these experiments. The energy monitoring system operated satisfactorily and demonstrated that the desired flat-deposition profile in the samples had been achieved. Although the configuration used for these tests was designed specifically for porous beryllium below melt, many of the measurement techniques can be readily adapted to other materials and energies. The following limitations, however, were noted: •
•
•
The thin-depth, dose foils are applicable only at energy densities below melt; above that, only the "shine-through" measurement of the segmented calorimeter would provide deposition information on data shots. (Few, if any, electron-beam facilities in the country can produce high enough fluences of high energy electrons to melt beryllium.) Flat depositions, in samples that are thick with respect to relief distances during deposition time, are almost unobtainable in high-density materials; this desirable condition, therefore, is limited to low-density materials, such as carbon or Al, and some medium density materials, such as Ti or Zr. The DISAR is readily adaptable to any situation where motion of a diffuse surface is to be monitored. In this test series, the velocities were relatively low -.05 mm/us. For higher velocities, an interferometer capable of being used in the "velocity" mode would be required.
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Shock Waves: Measuring the Dynamic Response of Materials
Summary
1.9
The study has shown that significant differences exist between the porous as-sprayed and the sintered beryllium materials examined in this study. It demonstrated that the wave propagation properties of both of the materials tested can be described by the porous material model developed. Both the compressive characteristics (first-wave arrival time, risetime of shock, and peak stress) and the release characteristics (arrival time of release wave and general attenuated shape) of shocks have been reproduced satisfactorily for different stresses and pulse durations as well as for different thicknesses of the porous specimens. Slight deviations between the experimental and computed wave profiles lie mostly within the combined errors of the model and the experimental data.
7.10 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
References
Isbell, W. M., O. R. Walton, and F. H. Ree. Lawrence Livermore Laboratory Rept. UCRL-51682, Part 1 (1977). Shock, R. N., A. E. Abey, and A. G. Duba. Lawrence Livermore Laboratory Rept. UCRL-51682, Part 2 (1974). Ree, F. H., W. M. Isbell, and R. R. Horning. Lawrence Livermore Laboratory Rept. UCRL-51682, Part 4 (1974). Hanafee, J. E., and E. O. Snell. Lawrence Livermore Laboratory Rept. UCRL-51682, Part 6 (1974). Horning, R. R., and W. M. Isbell. Rev. Sci. Instr. 46 (10), 1374 (1975). Herrmann, W. / Appl. Physics 40, 2490 (1969). Holt, A. C , A. S. Kusubov, D. A. Young, and W. H. Gust. "Thermomechanical Response of Porous Carbon." Lawrence Livermore Laboratory Rept. UCRL 51330 (1973). Carroll, M. M., and A. C. Holt. J. Appl. Phys. 43, 1626 (1972). Butcher, B. M. "Numerical Techniques for One Dimensional Rate Dependent Porous Material Compaction Calculations." SC-RR-710112, Sandia Laboratories Report (April 1971). Seaman, L., R. E. Tokheim, and D. R. Curran. "Computational Representation of Constitutive Relations for Porous Materials." Agency, DNA 3412F (May 1974). Gust, W. H., private communication. Mackenzie, J. K. Proc. Phys. Soc, B63, 2 (1950). Schock, R. N., A. E. Abey, and A. G. Duba. J. Appl. Phys., 47, 53 (1976).
8. Interferometric Methods 8.1
Foreword
As evidenced by the several hundred papers and reports written about laser interferometers and their applications over the past 30 years, interferometric techniques for measuring the velocity-time history of specimens set into sudden motion have become a standard in laboratories around the world. Development of the instruments and measurement techniques has proceeded in three phases: Displacement Interferometers, Velocity Interferometers, and the VISAR Interferometer, which represents the current state of the art. In contrast to previous interferometric techniques, the VISAR provides: • Relative insensitivity to surface tilt, allowing measurements to be made of two-dimensional shock waves • Increased velocity range, allowing measurements to higher pressures; and • Greatly increased depth of field, allowing measurements of structural motion such as projectiles accelerating down the barrel of a gun.
8.2
The Displacement Interferometer
VISAR Interferometry has, as its antecedents, two distinct types of laser interferometers; the first, a displacement interferometer, uses a lowpower laser reflected from a specular (mirror-like) surface (see Figure 8.1a). Fringes are produced as the surface moves, one fringe for each A/2 (-0.32 |j.m, for a HeNe laser of wavelength A = 632.8 nm). Velocity is obtained by differentiating the displacement-time record. 257
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Shock Waves: Measuring the Dynamic Response of Materials
Although the instrument is appealing in its simplicity, two problems prevented its widespread use. (1) Fringe frequency is proportional to velocity, producing a signal of -300 MHz at 100 m/s. Since many tests require higher velocities, detector and recording bandwidth limitations become severe. (2) The mirror surface must remain specular under high-pressure shock waves (or the intensity of the reflected light is decreased to unusability) and must not tilt substantially (or the reflected beam no longer enters the interferometer).
8.3
The Velocity Interferometer
The first problem, that of bandwidth, was solved by arranging the interferometer optics such that the fringes were proportional to velocity rather than to displacement [1]. This is accomplished by splitting the input beam and directing the beams along paths of unequal length. Upon recombination, interference fringes are formed. Doppler-derived changes in frequency, produced by reflection of the beam from the moving surface, are received at the recombining beam splitter (see Figure 8.1b) at a slightly earlier time (order 1 ns) from the "Reference Leg" than are signals from the "Delay Leg." While the surface is accelerating or decelerating, fringes are produced, proportional to the surface velocity. V(t) = N(t).(T/2)
(8.1)
T = A,c/(L D - LR)
(8.2)
where V(t) = time varying velocity, N(t) = time varying fringe count, % = delay time between the two legs, X = laser wavelength, c = speed of light, and LD and LR are Delay Leg and Reference Leg lengths, respectively. More rigorous treatments of basic velocity interferometer equations and relationships can be found in the literature [2-4].
8. Interferometric Methods
Figure 8.1. The VISAR was preceded by the two interferometers shown here.
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Shock Waves: Measuring the Dynamic Response of Materials
By adjusting the length of the delay leg, fringe frequency can be controlled. The fringe constant, K, was introduced to represent the sensitivity of the instrument: K = A,/2x
(m/s/fringe)
V(t) = K»N(t) (m/s)
(8.3) (8.4)
A large K indicates that fewer fringes are produced for a given change in specimen velocity. The second problem, that of keeping the mirror reflective and of not being affected by tilt of the specimen surface, remained unsolved for several years, during which time hundreds of one-dimensional impact tests were conducted, primarily using gas guns to impact specimens held at the muzzle [5-9], Figure 8.2 demonstrates the substantial amount of information which can be obtained in a single test. • • • • • •
Elastic and Plastic Compressive Wave Velocities Elastic and Plastic Release Wave Velocities Point on the Hugoniot Equation of State Dynamic Material Strength (proportional to the "pullback" velocity) Thickness of the spall piece (derived from the ringing frequency in the spall signal) Dislocation Multiplication (derived from the damping in the spall signal)
Few, if any, other shock wave measurement techniques furnish such a wealth of information in a single test.
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261
Figure 8.2. The wealth of information available from VISAR tests is shown in this record of plate impact at moderate velocity (aluminum impacting aluminum at 389.2m/s). (Courtesy S. Doane)
8.4
The VISAR Interferometer
The problems of mirrors and tilt were solved in the early 1970s with the introduction of the VISAR (Velocity Interferometer System for Any Reflector) [10-12]. By using a diffuse (Lambertian) surface, instead of a specular surface, reflected light was relatively unaffected by tilt (tilt to perhaps 5-10 degrees, depending on the material and the surface treatment). Moreover, high-pressure shock waves no longer destroyed the diffuse target surface, and the maximum pressure at which the interferometer could be used was raised from a few hundred kilobars to several Megabars. Figure 8.3 shows a measurement of projectile motion, one of the more useful capabilities of VISAR systems. A piece of reflecting tape is
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Shock Waves: Measuring the Dynamic Response of Materials
adhered to the front of the projectile and the light is collected near the muzzle of the gun and sent, via fiber optic cables, to the VISAR. The data indicate a discontinuous acceleration of the projectile, despite calculations which predict smooth, adiabatic expansion of the propelling gases. Such measurements help in gun and projectile design by allowing more realistic values to be placed on the forces experienced by projectiles during the acceleration process [13-16]
Figure 8.3. (Top) The VISAR is placed at a distance from the projectile to mitigate the effects of change in intensity of the reflected beam as the projectile traverses the launch tube. (Bottom) Although the final velocity is in reasonable agreement with the test results, the paths by which the projectile achieves this velocity are different. The calculation predicts substantially greater acceleration and base pressure.
8. Interferometric Methods
8.4.1
263
Glass Etalon Delay Legs
Although the problem of tilt is mitigated by using a diffuse surface, such surfaces destroy the spatial coherence of the reflected beam, resulting in severe loss of fringe contrast for the standard velocity interferometer. The basis of the VISAR delay system depends upon satisfying the seemingly incompatible requirements of: 1) Having the two legs the same length to produce fringes from an incoherent source while 2) Having the two legs of different lengths to produce the time delay necessary for producing fringes from the Doppler-shifted beam. It has long been known that fringes can be produced from an incoherent source by having the two legs of the interferometer exactly the same length. However, with the delay legs equal in length, there is no time delay and no fringes are produced. The dilemma is solved by placing a glass "etalon" in the Delay Leg path (Figure 8.4). The two images are at the same optical distance (allowing fringes from a diffuse source, where spatial coherence has been destroyed, but temporal coherence is maintained), but geometrically are at different distances (producing the necessary delay time).
Figure 8.4. Basic VISAR design, utilizing a glass etalon for the delay leg
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Shock Waves: Measuring the Dynamic Response of Materials
8.4.2 Air Delay Legs A limitation on instrument sensitivity is the length of glass which can be manufactured with sufficient wave transmission tolerance to maintain high fringe contrast. An 8.5" (21.6 cm) length of glass produced a K = 200 m/s/fringe, sufficient for tests with a peak velocity of perhaps 100 m/s and with a velocity resolution of ~4 m/s. Tests at lower velocities or tests demanding greater velocity resolution required longer glass etalons, produced at sometimes prohibitive costs. One solution to this limitation was the introduction of the "Air Delay Leg" [17-19], in which the beams were delayed by two, two-lens systems with slightly different focal lengths. This was later simplified and made more stable in the single two-lens configuration shown in Figure 8.5.
Figure 8.5. The air delay leg lowered the measurable velocity range to peak velocities less than 10 m/s.
8. Interferometric Methods
265
By folding the air delay leg over a 2 m path , VISARs with fringe constants to K = 12 m/s/fringe were constructed. The measurable velocity range was extended to peak velocities below 10 m/s with a velocity resolution of -10-20 cm/s. Another adaptation of the air delay leg design is shown in Figure 8.6. This compact, "folded mirror" design has produced sensitivities to 40 m/s/fringe, with a resolution of 1 m/s.
Figure 8.6. Compact air delay leg folded mirror design
8.4,3 Extended High Velocity Differential Etalon At the other end of the velocity scale, laser-driven foil tests have produced velocities to >20,000 m/s in times of a few tens of nanoseconds, requiring time resolutions of a few tens of picoseconds to resolve reverberations in the foil. Such velocities require very thin etalons (order 4 mm). Such thin pieces of glass are difficult to produce in the required diameters and are quite difficult to mount with sufficient stability to avoid fringe distortion and lack of fringe contrast. The problem of thin etalons was solved by the "Differential Etalon" system shown in Figure 8.7 [18]. The Reference Leg holds an etalon which is slightly shorter than the etalon in the Delay Leg. The difference
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Shock Waves: Measuring the Dynamic Response of Materials
in lengths between the two etalons produces the (small) time delay. Time resolution is shown in Figure 8.8 to be 10-15 picoseconds for velocities >10,000 m/s. Foil velocities to >20,000 m/s have been measured with such systems.
Figure 8.7. Differential etalon configuration for very high velocity testing. Test data are from preliminary tests, using a streak camera to record fringe movement. (Test data courtesy K. Yoshida [20])
Figure 8.8. Time resolution is inversely related to velocity resolution. For velocities > 10,000 m/s, time resolution can be <15 ps.
8. Interferometric Methods
267
8.4.4 Fringe Recording Systems Velocity interferometer systems have progressed from single photomultiplier (PMT) systems (SIN), to dual PMTs (SIN & COS), which are capable of distinguishing accelerations from decelerations, to a 3-PMT system, which added a beam intensity monitor (BIM) to adjust fringe signals modified in amplitude by shock waves affecting the reflecting surface of the target [1, 14, 21] In 1979, Hemsing of Los Alamos National Laboratory modified the 3-PMT system by combining the SIN and COS signals with their previously unused conjugate beams, -SIN and -COS, to produce signals with amplitudes 2- SIN and 2- COS. This "push-pull" configuration (see Figure 8.9) is in standard use today [3].
Figure 8.9. Modified photo-detector design makes maximum use of available light by combining the four "conjugate" beams (see text).
8.4.5 Increased Stability, Reduced Size With the introduction in 1990 of the compact VISAR [20], VISAR systems were now reduced in size from a large optical bench, filled with individual components, to a single rack-mounted chassis, with all optical
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components cemented together in permanent alignment. The prototype configuration shown in Figure 8.10 has remained aligned for over a dozen years.
Figure 8.10. Schematic of a "solid VISAR" configuration
The uitility of this configuration was further enhanced by the development of the "Fixed Cavity VISAR" system (Figure 8.11) by Sandia National Laboratory [22].
Figure 8.11. Basic design of SNL "Fixed Cavity VISAR"
S. Interferometric Methods
8.5
269
Laser Illumination
Because of the low reflectivity of diffuse surfaces, (typically <10%), VISARs inherently require high power lasers. Additionally, the laser must have a relatively long coherence path length ( » than the length of the delay leg). The latter requirement dictates a single frequency laser, rather than a single line. For argon ion lasers, this requirement is met by placing an etalon in the laser cavity. In the early days of VISAR development, only the argon ion laser met both requirements. However, the size, the cost, the requirement for special AC power, and the necessity of cooling the laser resulted in examining alternative light sources. The first laser diodes with sufficient power (-100 mW) for VISAR applications lased in the infrared and had difficulty producing sufficient coherence path length for standard VISAR systems. Nevertheless, the low cost and compactness of both the laser and the IR photodiode receivers motivated researchers to adopt this new form of illumination [23]. With the introduction of visible laser diodes with sufficient power, coherence path length, and stability, VISAR researchers and manufacturers have incorporated them into their designs.
8.6
Photodetectors and Recording Systems
Requirements for nanosecond time resolution, linear output, and low noise have placed stringent requirements on photomultipliers (PMTs). Early instruments used bulky and costly PMTs, operating at -5000V. Modern VISARs use either lower cost PMTs with specialized voltage strings or use photodiodes, placed directly in the fiber optic cables connecting to the digital recorders [19]. Higher time resolution is provided by using streak cameras to record VISAR data, at the expense, to some degree, in velocity resolution and in total recording time. The records are digitized and the individual lines averaged to obtain a single record.
270
8.7
Shock Waves: Measuring the Dynamic Response of Materials
Light Collection Systems
8.7.1 Direct Beam Technique Early VISARs used the "direct beam" configuration, in which the beam was transported through the air from laser to target and from target to VISAR. While this technique was light-efficient, the high intensity beams exposed to the air were a hazard to the eyes and skin. Additionally, frequently it is difficult to position the argon ion laser sufficiently close to the test apparatus.
8.7.2 Fiber Optic Cable Technique Fiber optic cables, from laser to target and from target to VISAR, provide a convenient and safe method for transmitting the laser beams. With such a system, the laser, the test apparatus, and the recording equipment can be placed in convenient positions, tens of meters apart. Two basic types of light collection systems are used with fiber optic cables. The small lens system shown in left side of Figure 8.12 is expended in each test, while the larger light collection system (right side) is placed behind a barrier and views the test through an expendable mirror. An advantage of the expendable lens technique is its ability to focus the laser spot to <100 fim, whereas with the longer focal length lens of the non-expendable system, spot sizes of few mm are typical.
8.7.3 Multiple Fiber Technique One disadvantage of VISARs, that only a single point could be measured on the specimen surface, was solved by devising a multiple fiber probe system. As many as 5-10 fibers are arranged in a compact array and focused on the target, either through one lens or several. At the VISAR, the beams are sent on parallel, non-interfering paths through the optical system, then collected and sent to multiple recording systems.
8. Interferometric Methods
271
Figure 8.12. Transmission of the laser beams through fiber optic cables has increased both safety and convenience.
8.8
VISAR Configurations
Since its invention, the simple VISAR has been transformed into many specialized configurations [24]. The following paragraphs describe two of the more useful and unusual designs. 8.8.1 "Line" VISAR An imaging VISAR system with quadrature coded outputs consisting of a laser illuminated line on the experimental surface, optics to image the line through the interferometer to produce quadrature (push-pull) signals, and either a streak camera or other detectors for recording data. It provides the capability to continuously measure the velocity histories of many points during a single experiment [25].
8.8.2 Imaging "White Light" VISARs An imaging white light interferometer, consisting of two imaging superimposing Michelson interferometers in series with the target,
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Shock Waves: Measuring the Dynamic Response of Materials
imprints the same delay, independent of ray angle, position, and wavelength [26, 27]. This capability has several useful chasracteristics: • • •
Any illumination source may be used, including flash lamps and multi-wavelength sets of lasers. Two-dimensional maps of moving surfaces can be measured. Both radial and transverse velocities can be measured when the illuminating and viewing beams are non-parallel.
Figure 8.13 is a line drawing of a white light VISAR, as given in the reference. If interferometer delays, Xi and x2, match within the coherence path length of the source, partial fringes are produced at the output which vary with (Xi - X2).
Figure 8.13. Line Drawing of a White Light VISAR
8.9
Summary
This brief history describes the chronology of the development of VISAR interferometry from a large, unwieldy instrument, difficult to align and maintain, to the present-day VISAR—compact, permanently aligned, and used by both trained scientists and by students. As evidenced by the rapid pace of continued development, the end of this cycle is not yet in sight.
8. Interferometric Methods
8.10
273
References
1. Barker, L. M., and R. E. Hollenbach. Rev. Sc. Instr. 36, 4208 (1965). 2. Clifton, R. J. "Analysis of the Laser Velocity Interferometer." J. Appl. Phys., vol. 41,3535(1970). 3. Hemsing, W. F. "Velocity Sensing Interferometer (VISAR) Modification." Rev. Sci. Instr. 50(1), Jan (1979). 4. Gooseman, D. R. J. Appl. Phys., vol. 45, 3516 (1975). 5. Johnson, J. N., and L. M. Barker. "Dislocation Dynamics and Steady Plastic Wave Profiles in 6061-T6 Aluminum." /. Appl. Physics, vol. 40, 4321-4334 (1969). 6. Isbell, W. M., and J. R. Christman. "Shock Propagation and Fracture in 6061-T6 Aluminum from Wave Profile Measurements." General Motors Materials and Structures Laboratory, DASA-2419 (AD705536), 1970. 7. Christman, D. R., W. M. Isbell, and S. G. Babcock. "Measurements of Dynamic Properties of Materials, Vol. V: OFHC Copper." General Motors Materials and Structures Laboratory, DASA-2501 (AD728846), July 1971. 8. Asay, J. R. "Shock and Release Behavior in Porous 1100 Aluminum." J. Appl. Phys. 46 (1975). 9. Isbell, W. M., in "Measurements of the Dynamic Response of Materials to Impact Loading." Doctoral Thesis, Shock Wave Research Center, Tohoku University, Sendai Japan (1993). 10. Gillard, C. W., G. S. Ishikawa, J. F. Peterson, J. L. Rapier, J. C. Stover, and N. L. Thomas. Lockheed Report No. N-25-67-1 (unpublished), 1968, 11. Barker, L. M., and R. E. Hollenbach. "Laser Interferometer for Measuring High Velocities of Any Reflecting Surface." /. Appl. Phys., vol. 43, no. 11, November (1972). 12. Isbell, W. M. "The Versatile VISAR: An Interferometer for Shock Wave and Gas Gun Diagnostics." Proceedings, 26th Annual Meeting of the Aeroballistic Range Association, 1976. 13. Isbell, W. M., and P. W. W. Fuller. "Wide Range, High Resolution Measurements of Projectile Motion Using Laser Interferometry." 27th Annual Meeting, SPIE and High Speed Photonics and Videography Conference, TR-16-83 (1983). 14. Isbell, W. M. "Laser Interferometry for Accurate Measurements of Projectile Motion." Proceedings, 34th Meeting of the Aeroballistic Range Association, 1983. 15. Isbell, W. M. "Initial Tests of VISAR Interferometry to Measure E.M. Launcher Projectile Motion." Proceedings, 38th Meeting of the Aeroballistic Range Association, 1987. 16. Isbell, W. M. "Interferometric In-Bore Velocity Measurements of Electromagnetically-Launched Projectiles." Proceedings, 41st Meeting of the Aeroballistic Range Association, 1990. 17. Amery, B.T. "Wide Range Velocity Interferometer." 6th Symposium on Detonation, San Diego, CA, 1976.
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Shock Waves: Measuring the Dynamic Response of Materials
18. Isbell, W. M. "A Combined Displacement/Velocity Interferometer for Impact Measurements at 0.1 to 100 m/s." Proceedings, 32nd Meeting of the Aeroballistic Range Association, 1981. 19. Froeschner, K. E., et al. "Subnanosecond Velocimetry with a New Kind of VISAR." 22nd International Congress on High Speed Photonics and Photography, Santa Fe, New Mexico, 1996. 20. Yoshida, K. private communication, National Institute for Material and Chemical Research for Material and Chemical Research, Tsukuba, Japan, 1996. 21. Isbell, W. M. "A Simplified, Compact VISAR: Concept and Construction." Proceedings, 42nd Meeting of the Aeroballistic Range Association, 1991. 22. Sweatt, W. C , P. L. Stanton, and O. B. Crump, Jr. "Simplified VISAR System." SAND90-2419C, SPIE, vol. 1346, July (1990). 23. Isbell, W. M. "An Infrared VISAR for Remote Measurement of Projectile Motion." Proceedings, 39th Meeting of the Aeroballistic Range Association, 1988. 24. Isbell, W. M. "Extending the Range of the Third-Generation VISAR from 30 m/s to 30,000 m/s." Proceedings, 47th Meeting of the Aeroballistic Range Association, 1996. 25. Hemsing, W. F., A. R. Mathews, R. H. Warnes, M. J. George, and G. R Whittemore. "VISAR: Line-Imaging Interferometer." 1991 American Physical Society Topical Conference, Williamsburg, VA, June 17-21,1991. 26. Erskine, D. J., and N. C. Holmes. "Imaging White Light VISAR." 22nd International Congress on High Speed Photonics and Photography, Santa Fe, New Mexico, 1996. 27. Gidon, S., and G. Behar. "Multiple-Line Laser Doppler Velocimetry." Appl. Optics 27, 2315-2319 (1988). 28. Isbell, W. M. "Modern Instrumentation for Measurements of Shock Waves in Solids." Proceedings, Japanese Shock Wave Symposium, Tokyo, Japan, 1999.
Appendix Compendium of Wave Profiles A.I
Summary
Shock wave profiles, especially those exhibiting spall, provide substantial detail on shock wave processes. From a single profile can be obtained a wide number of parameters, including: • • • • • • •
Elastic compressive wave velocity and amplitude Plastic compressive wave velocity and amplitude Elastic release wave velocity and amplitude Plastic release wave velocity and amplitude Dynamic fracture strength Point on the Hugoniot Release path
During the research effort, over 400 shock wave profiles were obtained. A selection of these profiles is presented in this Appendix. The materials include those presented in the Summary of Chapter 4: • • • •
Aluminum (6061-T6) Titanium (Alpha phase) Tantalum (commercially pure) Fused Quartz
These profiles represent a data base useful as inputs for models describing the response of materials to shock wave loading.
275
276
Shock Waves: Measuring the Dynamic Response of Materials
Profiles in Aluminum: 6061-T6
Appendix: Profiles in Aluminum
277
278
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
279
280
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
281
282
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
283
284
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
285
286
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
287
288
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
289
290
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
291
292
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
293
294
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
295
296
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
297
298
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
299
300
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Aluminum
301
302
Shock Waves: Measuring the Dynamic Response of Materials
Profiles in Titanium: Alpha Phase
Appendix: Profiles in Titanium
303
304
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Titanium
305
306
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Tantalum
Profiles in Tantalum: Commercially Pure
307
308
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Tantalum
309
310
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Tantalum
311
312
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Tantalum
313
314
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Tantalum
315
316
Shock Waves: Measuring the Dynamic Response of Materials
Appendix: Profiles in Fused Quartz
Profiles in Fused Quartz
317
Index Note:
Bold text = major topics Bold pp numbers = pages of major discussion
A
B
Accelerated reservoir light-gas (ARLG) gun, 33 Alpha-phase tantalum Dynamic properties of alphaphase tantalum, 85-16 (see also Test results at low and moderate stresses) Attenuation from high pressures Background, 187-190 Experimental results, 206 6061-T6 aluminum, 214 Beryllium, 223 OFHC copper, 207 Titanium, 220 Uranium alloy, 225 Impact experiments Description of, 190 Laser velocity interferometer, 192-193 Manganin wire gages, 194 Optical measurement techniques, 194-205 Target and impactor design, 191 Summary and conclusions, 226-229 Attenuation waves, 11
Bauschinger effect, 23 Bulk modulus, 30 Bulk wave velocity, 30 £ Compressed-gas gun, 33 Confined fe ^ ^ 26 Constitutive equations, 23 Constitutive relations, 12 Convention for units, 3 Crack
ropagatiori; 17 F
v
^ Debye specific heat curve, 15 Displacement interferometer, 41 Dissipative mechanisms, 7 Dugdale-MacDonald relation, 13 Dynamic property measurements Hugoniot elastic limit (HEL), 81, 101> 117 > 1 8 8 Hugoniots (high pressure, 0.2 to 5.0 TPa), 139-158, 182-183 Hugoniots (moderate pressure, to °- 2 TPa )> 7 4 - 7 5 > 9 6 - 9 8 Instrumentation
319
320
Shock Waves: Measuring the Dynamics Response of Materials
In-material gages, 51, 194 Interferometric techniques DISAR, 249-255 VISAR, 257, 261-271 Optical techniques, 25, 194-205 Free surface velocity gages, 47-58 Materials, test results 2024-T4 aluminum, 144 AZ31B magnesium, 151 Beryllium, 150 Depleted uranium, 145 Fansteel-77, 140 Nickel, 147 OFHC copper, 142 Plexiglas (PMMA), 152 Quartz phenolic, 152 Titanium, 149 Type 304 stainless steel, 148 Dynamic response of materials at low and moderate stresses, 69 _, *^ Elastic behavior of materials Elastic constants, 23, 30 Elastic precursor, 17, 208, 224-245 Time-dependent effects, 17 Elastic-plastic flow in uniaxial strain, 15 Elevated temperature effects, 37 Elevated temperature impact testing, 45 Engineering strain, definition, 24 Experimental techniques, 23 Equations of state and wave profiles Hugoniot equations of state Definition, 39 Utility of, 40 Wave profile compendium, Appendix
Gas guns Alignment techniques, 40 Compressed-gas gun, 102 mm, 40 Effect of air "cushion", 42 Impact velocity measurement, 40 Projectile design, 40 Shock isolation, 40 Impact techniques Compressed-gas gun, 40 Light gas gun, 23 Spallation testing, 60, 79-80, 109-116 Target design, 47-60, 132-137, 191, 249 Instrumentation Laser velocity interferometer, 55 Magnetic wire gage, 58 Manganin gage, 51 Slanted resistance wire, 57 Streak camera techniques, 52 X-cut quartz gages, 47 Spall test techniques, 60-65 Recovery testing, 61 Metallographic examination 64-65 W a v e pr0 fiies, 61 Elevated temperature testing, 63 Stress-strain/strain rate High-rate testing 25 One-dimensional elastic wave theory, 27 Stresses beyond yield, 27 Testing at temperature, 25 Hopkinson bar, 27 Elevated temperature testing, 27 Medium rate, 24 Medium-strain-rate machine, 24
321
Index Optical extensometer, 25 Multi-axial, 25 Multi-axial stress tests, biaxial machine, 28 Confined pressure device, 28 High heating rate tests, 29 Uniaxial stress tests, 24 Ultrasonic measurements Ultrasonics techniques and data 27, 34-38, 76-78, 90-93, 99-102 Elastic wave velocities, 35 Elastic constants, 38 Bulk modulus 38 Bulk wave velocity, 38 Lame s parameter 38 Poissons ratio, 38 Rayleigh wave or surface wave velocity, 38 Shear or rigidity modulus, „ , . Sound wave velocity, 38 Young's or elastic modulus, 38 Pulse superposition, 35 Pulse transmission, 36 Temperature and pressure dependence, 36 r Finite deformation, 15 Free surface velocity, 41 rjj. Gruneisen ratio, 13 Gun-launched, flat-plate impact techniques, 32 - 11 High heating rates, 26
High pressure region (26) Mie-Gruneisen equation of state, 12 High-strain-rate tests, 24 Hopkinson bar, 24 Elevated temperature testing, 25 Hugoniot equation of state equations, 8-9 Hugoniot equations of state to 0.5 TPa, 125 J Interferometric methods, 257-271 m S A R (Di
lacement
interferometer for diffuse surfaces), 249 Displacement interferometry, 257 Fringe recording systemS; 267
Laser illumination, 269 Modifications to increase stability, reduce size, 267-268 R x e d Cayi
yisAR
"Solid" VISAR Photodetectors and recording systemS!
26g
4 j, 2=8 VISAR interferometry, 261 Delay leg configurations, 261-265 Air delay leg, 264-265 Etalon delay leg, 263 Extended high velocity T . u differential etalon, 265 L l § h t c o l l f t l o n s ? f e m s ' 2O7JA D i r e c t b e a m technique, 270 Fiber optic cable technique, 270 Multiple fiber technique, 270
v d o d t y interferometry>
322
Shock Waves: Measuring the Dynamics Response of Materials
VISAR configurations, 271 "Line" VISAR, 271 Imaging "White Light" VISAR, 271 Isotherm, 0°K, 14 J^ Lame's parameter, 30 Light-gas gun, 33 T .' .. , , .. 1 O Longitudinal wave velocity, 28 Low-pressure shock wave equation of state for tantalum, g? Low strain-rate testing, 23 M Magnetic wire gage, 43 Manganin gage, 39 Medium-strain-rate testing, 23 Metallographic examination, 46 Mie-Gruneisen equation of state, 12 Multi-axial stress tests, biaxial machine High heating rate tests, 25 Q Optical extensometer, 24 -j * Planar shocks, 7 Poisson's ratio, 30 Porous material response to static and dynamic loading, 233-255 Comparison of predictions with data 245 Dynamic behavior, 235 Electron-beam tests, 248 Applicability of tests to other materials, 255
DISAR (Displacement interferometer for diffuse surfaces), 249 Test design, 249 Test results, 250-255 Equation of state surface, 233 Experiment methodology, 239 P o r o u s berylliums, description ot'238 . . ,,„,,.. Porous constitutive model, 235 m
Test results on porous berylliums 240-244 Hugomot and wave velocities, 244 „, , Shock wave data, 241 Pulse Static superposition, 28 data, 240 compression P u I s e transmission, 29 r\ ^ Quartz
Sa§es'
36~37
K. Rankine-Hugoniot relations, 8 Rayleigh Wave or Surface Wave Velocity, 30 Recovery tests, 44 Release wave behavior, 108-109 Riemann integral, 11 Room temperature testing, 44 o modulus' 30 Shear wave velocity, 28 S h o c k adiabat> 8 Shock wave formation Dissipative mechanisms, 7 S h o c k w a v e P r o f i l e s : Appendix (Compendium) Shock wave theory, 6-18 Attenuation waves, 11
S h e a r o r n&dlty
323
Index Constitutive relations, 12 Elastic-plastic flow in uniaxial strain, 15 High pressure regime, 13 Mie-Gruneisen equation of state, 13 Slanted resistance wire (SRW) technique, 43 Slater relation, 13 Sound wave velocity, 30 Spallation (See "Spall test techniques, 60-65" and "Spall fracture", 79-80) Static high-pressure apparatus, 29 Strain gages, 24 Strain-rate sensitivity, 23 Streak camera techniques, 39 Stress-strain-strain rate, 23 'J 1 Temperature and pressure dependence of elastic wave velocities, 28-29 Test results at low and moderate stresses Check data, 82 Attenuation tests, 82-84 Types and utility, 82 Degraded properties, 80 Effects on Hugoniot elastic limit, 81-82 Effects on wave profiles, 82 Influence of temperature/heating rate, OQ Elastic behavior, 76 Bauschinger effect, 77-78 Ultrasonic measurements, 76 Hugoniots for Ta, Cu, Ti, and Al, ~,A Release wave behavior, 77 Impact of a fused quartz disc, 77
Spall fracture Ductile and brittle fracture, 79 Metallographic examination, 79 Spall threshold results, 80 Suppression of spall, 194 Steady state behavior conditions, 75 Dynamic properties of alpha-phase tantalum, Elastic behavior, 90 Elastic constants, 91, 93 Wave velocities, 90 Hugoniot to 20 GPa, 85 Low-pressure shock wave equation of state, 96, 98 Material properties, 86 Spall fracture, 109 Incipient spall velocity t e s t r e s u l t s ' 11 l~l
U
SPa11w a v e
Profiles' 115-116 Stress-strain measurements, Ultrasonic equation of state' " - 1 0 2 W a v e Propagation, 104 Compressive waves, 105 E l a s t i c P^cursor decay, 105-107 Release waves and wave
attenuation, 108-109 behavior 102 Summary of results, 69 T e s t r e s u l t s o f h i 8 h P^ssure Hugoniots to 0.5 TPa Summary of materials tested, p
Yield
126
Experimental results, 139 2 0 2 4 " T 4 ^minum, 144 A Z 3 1 B magnesium, 151 Beryllium, 150
324
Shock Waves: Measuring the Dynamics Response of Materials
Depleted uranium, 145 Fansteel-77, 140 Nickel, 147 OFHC copper, 142 Plexiglas (PMMA), 152 Quartz phenolic, 152 Titanium, 149 Type 304 stainless steel, 148 Experimental techniques at high pressures Edge rarefactions, 134-135 Impact velocity, 138 Instrumentation, 131-132 Shock wave velocity,
137-138
Target design and construction, 132-137 Hugoniot data, 153-183 Compilations, 182-183 Graphs, 158-179 Tables, 153-157, 181 Test methodology, 139-140 Theoretical considerations, 127 Testing at temperatures up to 500°C, 24 Thermoelastic-plastic theory, 15 Thomas-Fermi-Dirac equation of state, 15 Time-Dependent Effects, 17
JJ Ultrasonic measurements, 28 Uniaxial and multi-axial stress, 23 Uniaxial stress testing at low rates, 23
5
U n i a x M S t r e s s Tests> 2 3 U n i t S ; c o n v e n t i o n for> 4
~ Velocity interferometer, 41 VISAR interferometer, 42 W Wave
profiles, compendium,
275-317
Summary, 275 Wave profile data Aluminum, 276 Fused quartz, 317 Tantalum, 307 Titanium, 302 Work-hardening, 16 y Young s or elastic modulus, 30