EFFECT OF LOADING RATE AND PEAK STRESS ON THE ELASTIC LIMIT OF DYNAMICALLY COMPRESSED BRITTLE SINGLE CRYSTALS By BRANDON MICHAEL LALONE A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Physics AUGUST 2011 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of BRANDON MICHAEL LALONE find it satisfactory and recommend that it be accepted. Yogendra M. Gupta, Ph.D., Chair Matthew D. McCluskey, Ph.D. Choong‐Shik Yoo, Ph.D. ii ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Yogendra M. Gupta, for his guidance, support, and patience throughout the course of this work. I have benefitted greatly from our interactions and will always be thankful for the opportunities that he provided to me. I would also like to thank the other committee members, Dr. Matthew D. McCluskey and Choong‐Shik Yoo, for their encouragement and their willingness to serve on my committee. I would like to thank the technical and administrative staff of the Institute for Shock Physics, especially Cory Bakeman and Kent Perkins whose assistance made my experimental work possible. I am very grateful for the help of the administrative staff in the Department of Physics, especially Sabreen Dodson for her commitment to my well being. I wish to thank my friends and family, particularly my parents, Joanne and Michael LaLone, for their endless encouragement, and my wife Hui for her patience and support. This work was supported by the DOE/NNSA Grant. iii EFFECT OF LOADING RATE AND PEAK STRESS ON THE ELASTIC LIMIT OF DYNAMICALLY COMPRESSED BRITTLE SINGLE CRYSTALS Abstract by Brandon Michael LaLone, Ph.D. Washington State University August 2011 Chair: Y. M. Gupta To examine the effect of loading rate on the compressive elastic limit of brittle solids, shockless and shock compression experiments were conducted on x‐cut and z‐cut quartz crystals. A compact pulsed power generator was used to achieve shockless compression at stresses up to 16 GPa in x‐cut quartz and 18 GPa in z‐cut quartz. Plate impacts generated shock compression to elastic impact stresses up to 16 GPa in x‐cut quartz and 37 GPa in z‐cut quartz. Transmitted wave profiles were recorded at the interface between the quartz crystals and LiF windows using a velocity interferometer. The shockless (loading rate ~ 105‐106 s‐1) elastic limit of x‐cut quartz was ~ 11 GPa, a significant increase over the shock wave (loading rate > 107 s‐1) elastic limit of ~ 6 GPa. The shockless elastic limit of z‐cut quartz, ~ 14 GPa, was also higher than the shock wave elastic limit of ~ 12 GPa. The increase in elastic limit with a decrease in loading rate is contrary to the expected loading rate dependence of material strength, and cannot be explained by usual inelastic deformation models. A strain energy localization model is proposed to explain the observed loading rate dependence for the quartz elastic limit. Strain energy localization is incorporated using dynamic stress concentrations in a shear cracking model. Constant loading rate calculations, performed using the model, provided results that were in qualitative agreement with the measured elastic limit‐loading rate dependence. iv To examine the effect of impact stress on the elastic limit of brittle solids, shock wave experiments were conducted on 3.2 mm thick, [100] and [111] gadolinium gallium garnet (GGG) crystals. Elastic impact stresses ranged from 15 to 50 GPa. Velocity interferometry measurements at the interface between GGG samples and LiF windows, together with impact pins, were used to measure transmitted shock and particle velocities. The measured elastic limits were strongly dependent on the impact stresses for both GGG orientations, and ranged from 13 GPa at an elastic impact stress of 17 GPa to 34 GPa at an elastic impact stress of 50 GPa. v TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .............................................................................................................. iii
ABSTRACT
........................................................................................................................iv
LIST OF TABLES ........................................................................................................................xi
LIST OF FIGURES ...................................................................................................................... xii
CHAPTER
1. Introduction 1
1.1
Objectives and Approach........................................................................................... 2
1.2
Organization of Subsequent Chapters ....................................................................... 4
References for Chapter 1 ..................................................................................................... 6
2. Background ........................................................................................................................ 8
2.1
Elastic Limit of Brittle Solids – Loading Rate Dependence ...................................... 8
2.2
Dynamic Compression of Quartz ............................................................................ 14
2.2.1 Discussion of Quartz Properties .................................................................. 14
2.2.2 Dynamic Compression of X-Cut Quartz ...................................................... 16
2.2.3 Dynamic Compression of Z-Cut Quartz ...................................................... 19
2.3
Summary of Findings from Existing Work ............................................................. 23
References for Chapter 2 ................................................................................................... 25
3. Experimental Methods ...................................................................................................... 28
3.1
Sample Characterization and Preparation ................................................................ 28
3.1.1 X-Cut Quartz Preparation ............................................................................ 28
3.1.2 Z-Cut Quartz Preparation ............................................................................. 30
3.1.3 Optical Windows .......................................................................................... 32
3.2
Particle Velocity Measurements ............................................................................. 32
3.3
Shockless Compression Measurements ................................................................... 32
3.3.1 Compact Pulsed Power Generator................................................................ 33
vi 3.3.2 Target Construction ...................................................................................... 35
3.3.3 Instrumentation Setup .................................................................................. 39
3.4
Plate Impact Experiments ........................................................................................ 41
3.4.1 Target Construction ...................................................................................... 41
3.4.2 Projectile Preparation ................................................................................... 46
3.4.3 Instrumentation Setup .................................................................................. 48
References for Chapter 3 ................................................................................................... 51
4. X-Cut Quartz Experimental Results and Analysis ........................................................... 53
4.1
Shockless Compression Experiments ...................................................................... 53
4.1.1 10 GPa Peak Stress Experiments ................................................................. 55
4.1.2 16 GPa Peak Stress Experiments ................................................................. 59
4.2
Shockless Compression Analysis ............................................................................ 61
4.2.1 Shockless Elastic Limits .............................................................................. 70
4.2.2 Loading Rates............................................................................................... 73
4.2.3 Error Analysis for Shockless Compression .................................................. 73
4.3
Shock Wave Compression Experiments .................................................................. 77
4.3.1 10 GPa Peak Stress Experiments ................................................................. 77
4.3.2 16 GPa Peak Stress Experiments ................................................................. 80
4.4
Shock Wave Compression Analysis ........................................................................ 83
4.4.1 Shock Compression Elastic Limits .............................................................. 84
4.4.2 Peak State ..................................................................................................... 87
4.4.3 Loading Rates............................................................................................... 92
4.5
X-Cut Quartz Experimental Findings ...................................................................... 92
References for Chapter 4 ................................................................................................... 97
5. Z-Cut Quartz Experimental Results and Analysis............................................................. 99
5.1
Shockless Compression Experiments ...................................................................... 99
5.1.1 13 GPa Peak Stress Experiments ............................................................... 101
5.1.2 18 GPa Peak Stress Experiment ................................................................. 104
5.2
Shockless Compression Analysis .......................................................................... 106
5.2.1 Shockless Elastic Limits ............................................................................ 109
vii 5.2.2 Loading Rates............................................................................................. 110
5.3
Shock Wave Compression Experiments ................................................................ 110
5.4
Shock Wave Compression Analysis ...................................................................... 115
5.4.1 Shock Compression Elastic Limits ............................................................ 116
5.4.2 Peak State ................................................................................................... 118
5.4.3 Loading Rates............................................................................................. 121
5.5
Z-Cut Quartz Experimental Findings .................................................................... 121
References for Chapter 5 ................................................................................................. 126
6. Phenomenological Model and Calculations for Quartz ................................................... 127
6.1
Need for the Phenomenological Model ................................................................. 127
6.1.1 Continuum Approach to Inelastic Deformation ......................................... 128
6.1.2 Shear Stress Dependent Inelastic Strain Rate Models................................ 130
6.1.3 Dislocation Model for Iron......................................................................... 131
6.1.4 Conclusions Regarding Previous Models .................................................. 137
6.2
Phenomenological Model ...................................................................................... 137
6.2.1 Energy Localization ................................................................................... 138
6.2.2 Rate Dependent Energy Localization ......................................................... 143
6.2.3 Inelastic Strains from Shear Cracks ........................................................... 144
6.2.4 Elastic – Inelastic Model ............................................................................ 147
6.2.5 Model Parameters....................................................................................... 148
6.3
Calculated Results ................................................................................................ 151
6.3.1 Stress-Strain Response ............................................................................... 151
6.3.2 Elastic Limit-Loading Rate ........................................................................ 154
6.3.3 Inelastic Strains .......................................................................................... 157
6.3.4 Shear Cracks............................................................................................... 159
6.3.5 Parameter Variation ................................................................................... 161
6.4
Summary and Main Findings ................................................................................ 170
References for Chapter 6 ................................................................................................. 172
7. Shock Wave Compression of Gadolinium Gallium Garnet Crystals .............................. 174
7.1
Introduction .......................................................................................................... 174
viii 7.2
Experimental Method ............................................................................................ 175
7.2.1 Material Characterization ........................................................................... 175
7.2.2 Experimental Details .................................................................................. 178
7.3
GGG Experimental Results and Analysis.............................................................. 178
7.3.1 Results for [100] GGG ............................................................................... 181
7.3.2
Results for [111] GGG .............................................................................. 183
7.3.3 Elastic Shock Velocity ............................................................................... 185
7.3.4 Elastic Limit Determination ....................................................................... 188
7.3.5 Elastic Hugoniots ....................................................................................... 192
7.3.6
Inelastic Wave Velocity ............................................................................ 194
7.3.7 Peak Stress ................................................................................................. 196
7.3.8 Intermediate Disturbance Wave ................................................................. 200
7.4
Discussion.............................................................................................................. 202
7.4.1 HEL: Impact Stress Dependence................................................................ 202
7.4.2 Strength in the Peak State .......................................................................... 206
7.5
Summary and Main Findings ................................................................................ 208
References for Chapter 7 ................................................................................................. 212
8. Summary and Conclusions .............................................................................................. 214
8.1
Shockless and Shock Wave Compression of X-Cut and Z-Cut Quartz ................. 214
8.2
Shock Wave Compression of Gadolinium Gallium Garnet (GGG) Crystals ........ 217
References for Chapter 8 ................................................................................................. 219
APPENDIX
A. MATLAB® Code ‘forwardcalc’ for Forward Wave Propagation Using the Method
of Characteristics ............................................................................................................. 220
A.1 Overview of forwardcalc ...................................................................................... 220
A.2 Operation of forwardcalc ....................................................................................... 222
A.3 MATLAB® Code for forwardcalc ........................................................................ 223
References for Appendix A ............................................................................................. 229
ix B. Jump Conditions for the Peak State Calculations in Chapters 4, 5, and 7 with a
Lagrangian Inelastic Wave Velocity ............................................................................... 230
B.1
Coordinate Change from Lab Frame (Eulerian) to Lagrangian Inelastic Wave
Velocity ................................................................................................................. 231
B.2
Jump Conditions with a Lagrangian Inelastic Wave Velocity .............................. 233
References for Appendix B ............................................................................................. 235
x LIST OF TABLES
Page
2.1 Elastic response of x‐cut quartz to shock compression ....................................................... 18
2.2 Elastic response of z‐cut quartz to shock compression ....................................................... 22
4.1 Ambient measurements on x-cut quartz crystals ................................................................ 54
4.2 Experimental details ........................................................................................................... 56
4.3 Results of x-cut quartz experiments .................................................................................... 74
4.4 Peak state results for shock compressed x-cut quartz ......................................................... 88
5.1 Ambient measurements on z-cut quartz crystals ............................................................... 100
5.2 Experimental details.......................................................................................................... 103
5.3 Results of z-cut quartz experiments .................................................................................. 111
5.4 Peak state results for shock compressed z-cut quartz ....................................................... 119
7.1 Ambient measurements on GGG crystals ......................................................................... 177
7.2 Calculated and reported elastic constants ......................................................................... 177
7.3 Experimental details.......................................................................................................... 180
7.4 Elastic wave properties ..................................................................................................... 189
7.5 Inelastic wave and peak state results................................................................................. 197
7.6 Elastic impact stress and the elastic limit .......................................................................... 203
xi LIST OF FIGURES
Page
2.1
A typical yield stress‐loading rate curve for a brittle solid ................................................ 10
2.2 The confinement stress at the yield point as a function of loading rate .............................. 12
2.3 Diagrams of α-quartz .......................................................................................................... 15
3.1 Photograph of equipment for cutting quartz disks .............................................................. 31
3.2 Photograph of the CPPG facility.........................................................................................34
3.3 Diagram of CPPG experiments and photograph of CPPG panels ...................................... 36
3.4 Photograph of completed panel........................................................................................... 38
3.5 Photograph of panel loaded into the CPPG facility ............................................................ 40
3.6 Schematic view and photograph of standard plate impact experiment ............................... 42
3.7 Schematic view and photograph of target for wave speed measurements .......................... 45
3.8 Photographs of 100 mm diameter projectiles ..................................................................... 47
3.9 Photographs of powder gun projectiles ............................................................................... 49
4.1 X-cut quartz velocity profiles for 10 GPa shockless compression experiments ................. 57
4.2 X-cut quartz velocity profile and velocity interferometer contrast ..................................... 58
4.3 X-cut quartz velocity profiles for 16 GPa shockless compression experiments ................. 60
4.4
X-cut quartz velocity profile and interferometer contrast for 16 GPa experiment ............ 62
4.5 Illustration of Lagrangian analysis for in-material shockless profiles ................................ 64
4.6 The Cu(u) curves after consecutive iterations in the computer program Charice 1.1 ......... 66
4.7
Measured and calculated interface profiles ........................................................................ 67
4.8 Calculated stress‐particle velocity and wavelet speed‐particle velocity curves .................. 69
4.9 Experimental wavelet velocity‐particle velocity curve and fit............................................ 71
4.10 Time difference between opposing point panel free surface measurements ....................... 76
4.11 Typical experimental wavelet velocity uncertainties .......................................................... 78
4.12 Velocity histories for 10 GPa shock wave experiments...................................................... 79
4.13 Velocity history and interferometer contrast for 10 GPa shock wave experiment ............. 81
4.14 Velocity histories for 16 GPa shock wave experiments..................................................... 82
4.15 Stress‐particle velocity diagram ......................................................................................... 85
4.16 Stress-particle velocity diagram for peak state calculation ................................................. 90
xii 4.17 Peak state (inelastic) stress‐volume compressions .............................................................. 91
4.18 Typical shockless and shock wave profiles for x‐cut quartz ............................................... 93
4.19 The measured elastic limits of x‐cut quartz as a function of applied loading rate .............. 94
5.1 Z-cut quartz velocity profiles for 13 GPa shockless compression experiments ............... 102
5.2 Velocity profiles for 18 GPa shockless compression experiments ................................... 105
5.3 Velocity profile and interferometer contrast for 18 GPa shockless compression ............. 107
5.4
Calculated stress‐particle velocity and wavelet speed‐particle velocity curves ............... 108
5.5 Velocity profiles from the shock wave experiments on z‐cut quartz ................................ 113
5.6 Stress‐particle velocity diagram ........................................................................................ 114
5.7 Elastic limit stresses as a function of the theoretical maximum elastic stress .................. 117
5.8
Peak (inelastic) stress‐volume compressions ................................................................... 120
5.9 Typical shockless and shock wave profiles for z‐cut quartz ............................................. 122
5.10 Elastic limits of z‐cut quartz as a function of the applied loading rate ............................ 123
6.1 Stress-strain relations for the dislocation based model ..................................................... 134
6.2 Elastic limit-loading rate curve for dislocation based model ............................................ 136
6.3 Shear crack extension due to dynamic compressive load ................................................. 140
6.4
Dynamic stress concentration factor at the crack tip ....................................................... 142
6.5 Stress-strain curves for the phenomenological model for quartz ...................................... 152
6.6 Elastic limit – loading rate for the phenomenological model for quartz ........................... 155
6.7
Inelastic shear strain as a function of longitudinal strain ................................................. 158
6.8 The crack density at 20% strain is plotted as a function of loading rate ........................... 160
6.9
The K and c0 parameter variation results ........................................................................ 163
6.10 The C parameter variation results ..................................................................................... 164
6.11 The B, β, and τ1 parameter variation results ..................................................................... 166
6.12 The H parameter variation results .....................................................................................168
6.13 The Emax parameter variation results ............................................................................. 169
7.1 Experimental configuration for the GGG experiments ..................................................... 179
7.2
Velocity histories for [100] GGG experiments ................................................................ 182
7.3 Velocity histories for [111] GGG experiments ................................................................. 184
xiii 7.4
Illustration of impact tilt corrections to wave speed measurements ................................ 186
7.5
Time-distance and stress-particle velocity diagrams ....................................................... 190
7.6 Elastic shock‐particle velocity values ............................................................................... 193
7.7
Time vs Lagrangian distance diagram ............................................................................. 195
7.8 Stress-particle velocity diagram for peak state calculations ............................................. 199
7.9 Time vs Lagrangian distance diagram .............................................................................. 201
7.10 Elastic limit as a function of impact stress ........................................................................ 205
7.11 Stress-density compression results for GGG ................................................................... 209
B.1 Time-lab frame distance and time-Lagrangian distance diagrams ..................................... 232
xiv Chapter 1 Introduction Understanding and modeling the response of brittle solids to dynamic loading is important for a broad range of scientific interests and applications.1 An important aspect of developing realistic material models for brittle solids is to determine and understand the compressive elastic limit (or yield strength) under different loading conditions, and to determine the factors that govern these values.1 Likely, the simplest continuum models for strength are the well known Tresca and von Mises yield stress conditions where the onset of inelastic deformation takes place once a critical shear or deviatoric stress is exceeded.2 While these yield strength models are reasonable for many ductile materials, they are often insufficient, especially for brittle solids.1,2 Under certain loading conditions, large deviations from von Mises or Tresca yield stresses have been observed.1 Three factors that are known to influence the yield strength of brittle solids are: the loading rate, the load path, and the peak stress.1,3 For most brittle solids, large deviations from the quasi‐static yield strengths are observed at loading rates above 103‐104/s.1 This is attributed to a competition between the rate of stress increase and the rate of stress relaxation (inelastic deformation rate), and many strength models incorporate time dependent effects at high loading rates.1,3‐5 It has been observed experimentally that the load path (particularly, confining stress) can modify the yield strength of brittle solids. Quasistatic measurements under compressive tri‐
axial stress loading have shown significantly higher yield strengths than under uniaxial stress loading.6 Similarly, yield strength under uniaxial strain is generally higher than under uniaxial stress.1 In making this last comparison, care needs to be exercised because uniaxial strain 1 experiments are performed at significantly higher loading rates than uniaxial stress measurements.1 Since the load path, to a large degree, determines the shear and confinement stresses that a material is subjected to, some studies have incorporated load path effects of by including the dependence confining stress into strength models.3,7,8 Others have restricted the strength model to a particular load path.1,4 For a given load path and comparable loading rates, different peak stresses may also alter the yield strength value.7,8 This has been observed in shock compression measurements, and has been incorporated into strength models by defining a stress dependent strength function.7,8 Due to experimental limitations, it is difficult to separate the relative effects of the above cited factors on the yield strength.1 For example, as indicated above, the load paths may be different in experimental work that compares strength at different loading rates.1 These difficulties will be discussed in detail in the next chapter. 1.1 Objectives and Approach The main objectives of this work were to study the effects of loading rate and peak stress on the elastic limits of brittle crystals. The research effort was divided into studies on two separate materials. The first study, the majority of the work presented here, focused on the loading rate dependence of the elastic limit for x‐cut and z‐cut quartz crystals for the same load path (uniaxial strain) and comparable peak stresses. The second study, a considerably smaller effort, focused on the peak stress dependence of the elastic limit of [100] and [111] oriented gadolinium gallium garnet (GGG) crystals for the same load path (uniaxial strain) and comparable loading rates (shock wave loading). 2 The loading rate dependence of the x‐cut and z‐cut quartz elastic limits were examined by comparing measurements under shockless and shock wave compression. The shockless compression experiments were conducted using the Washington State University compact pulsed power generator (CPPG), at the Institute for Shock Physics (Chapter 3).9 The shock compression experiments were conducted using plate impact facilities at the Institute. In both sets of experiments, transmitted particle velocities were recorded using a velocity interferometer (VISAR).10 These data were analyzed to obtain the elastic limits and loading rates. The specific objectives of the quartz study were as follows. 1. Determine the elastic limits of x‐cut and z‐cut quartz subjected to shockless and shock wave uniaxial strain compressions. 2. Compare the shockless and shock wave response and analyze the results in terms of existing models. 3. Develop a phenomenological model to provide a qualitative explanation of the observed loading rate behavior. Crystalline quartz was chosen for this investigation for several reasons. 1. Quartz is a strong brittle crystal with shock wave elastic limits between 6 – 15 GPa;11‐13 stresses that are within the range of the CPPG capabilities. 2. Quartz has been well characterized in the literature, including under shock wave compression, and a large body of past work can be drawn upon for comparison and analysis. 3. The material is also a major constituent of the Earth’s crust and is used in a variety of applications;14 its shockless response may have a broader scientific interest. 3 The peak stress dependence of the [100] and [111] oriented GGG elastic limits was examined by comparing shock wave (plate impact) compression measurements at different impact stresses. The elastic impact stresses ranged between 15 GPa and 50 GPa, a shock stress region that had not been examined previously for this brittle material.15,16 This study is presented in its entirety in Chapter 7; the background, specific objectives, and motivation for the GGG work are provided there. 1.2 Organization of Subsequent Chapters The Chapters are organized in the following manner: 
Chapter 2 reviews past work relevant to the loading rate dependence of the strength of brittle materials and summarizes previous dynamic compression studies on x‐cut and z‐cut quartz crystals. 
Chapter 3 provides details of the experimental methods used for the shockless and shock wave compression experiments on quartz. A summary of the CPPG operation and capabilities is also presented. 
Chapter 4 presents the results and subsequent analysis of shockless and shock wave compression experiments on x‐cut quartz. 
Chapter 5 presents the results and subsequent analysis of shockless and shock wave compression experiments on z‐cut quartz. 
Chapter 6 provides a discussion of phenomenological models in the literature, and presents a new phenomenological model for quartz that is qualitatively consistent with the experimental results. 4 
Chapter 7 presents the GGG investigation in its entirety. 
Chapter 8 provides a summary of the main conclusions in this work. 5 References for Chapter 1 1. D. E. Grady and R. E. Hollenbach, Tech. Rep. SAND76‐0659, Sandia National Laboratories, 1977. 2. M. A. Meyers, Dynamic Behavior of Materials, (John Wiley & Sons Inc, NY, 1994). 3. G. R. Johnson and T. J. Holmquist, High‐Pressure Science and Technology‐1993, edited by S. C. Schmidt, J. W. Shaner, G. A. Samara, and M. Ross, (AIP, New York, 1994), pp. 981‐
984. 4. G. E. Duvall, Stress Waves in Anelastic Solids, H. Kolsky and W. Prager, Eds. (Springer‐
Verlag, Berlin, 1963). 5. D. R. Curran, L. Seaman, and D. A. Shockey, Phys. Reps. 147, 253 (1987). 6. J. M. Christie, H. C. Heard, and P. N. LaMori, Am. J. Sci. 262, 26 (1964). 7. R. Feng, G. F. Raiser, and Y. M. Gupta, J. Appl. Phys. 79, 1378 (1996). 8. C. Hari Manoj Simha and Y. M. Gupta, J. Appl. Phys. 96, 1880 (2004). 9. T. Jaglinski, B.M. LaLone, C.J. Bakeman, and Y.M. Gupta, J. Appl. Phys. 105, 083528 (2009). 10. L. M. Barker and R.E. Hollenbach, J. Appl. Phys. 43, 4669 (1972). 11. J. Wackerle, J. Appl. Phys. 33, 922 (1962). 12. R. Fowles, J. Geophys. Res. 72 5729 (1967). 13. R. A. Graham, J. Phys. Chem. Solids, 35, 355 (1974). 14. J. Kimberley, K. T. Ramesh, and O. J. Barnouin, J. Geophys. Res. 115, B08207 (2010). 15. Y. Zhang, T. Mashimo, K. Fukuoka, M. Kikuchi, T. Sekine, T. Kobayashi, R. Chau, and W. J. Nellis, in Shock Compression of Condensed Matter‐2003, edited by M. D. Furnish, Y. M. Gupta, and J. W. Forbes (AIP, New York, 2004), pp. 127‐128. 6 16. T. Mashimo, R. Chau, Y. Zhang, T. Kobayoshi, T. Sekine, K. Fukuoka, Y. Syono, M. Kodama, and W.J. Nellis, Phys. Rev. Lett. 96, 105504 (2006). 7 Chapter 2 Background This chapter presents an overview of previous studies on the yield strength of brittle solids subjected to dynamic compressions; particularly, the dependence of the yield strength on the loading rate. Since the present work is on the dynamic compression of quartz, an overview of dynamic compression studies on single crystal quartz is also presented. Because both of these topics have a vast body of literature, only studies that are relevant to the current work are discussed. 2.1 Elastic limit of Brittle Solids – Loading Rate Dependence Until the recent developments in pulsed power capabilities (discussed in section 3.3.1) to produce shockless uniaxial strain compression,1‐3 the loading rate dependence of the compressive elastic limit for brittle solids had been measured using only a few different methods. For loading rates below ~104 s‐1, measurements were generally obtained under uniaxial stress loading using a combination of quasi‐static (loading rates < 100 s‐1) and split‐
Hopkinson bar (loading rates ~ 102 s‐1 – 104 s‐1) methods.4 For loading rates above ~ 104 s‐1, the only methods used were shock wave uniaxial strain loading (> 106 s‐1),5,6 or ramp loading using fused silica as a ramp compression generator.5,6 In general, a combination of these uniaxial stress and uniaxial strain loading techniques have been used to determine the elastic limit of brittle solids over a broad range of loading rates.5,6 Since quasi‐static and Hopkinson bar measurements have been obtained for a wide range of brittle solids, the present discussion will be restricted to solids that were also examined under shock wave compression. A good review of many of these material studies is presented in Ref. 6; the review covers geologic solids such as dolomite7,8 and granodiorite,7 and ceramics such 8 as silicon carbide6,9 and aluminum oxide,6,9 that were previously measured under both Hopkinson bar and shock wave loading.6 The results generally give a yield strength – loading rate curve with the features shown in Fig. 2.1 (similar to Figure 5b in Ref. 5). To account for differences in the load path between uniaxial stress and uniaxial strain in such a plot, a yield strength would be specified for which these differences were corrected reasonably.6 For low loading rates, corresponding to uniaxial stress, the yield strength increases slowly with increasing loading rate. At a certain transition loading rate of around 103 – 104 s‐1, the yield strength first increases rapidly with loading rate,6 then begins to level off with further loading rate increase as it asymptotes toward an upper limit yield stress.6 This upper limit yield stress corresponds to the HEL under shock wave compression.6 In this generalized picture, the elastic limit never decreases with an increase in loading rate, regardless of the loading type or loading rate.6 The behavior shown in Fig. 2.1 was explained by Grady6 using arguments that are summarized as follows: the lower limit of the yield strength (low loading rates) is controlled by the quasistatic fracture limit. In this region, there is a weak dependence of the elastic limit on the loading rate, possibly due to thermally activated subcritical crack growth. At a certain threshold, the elastic limit increases rapidly with loading rate; this threshold region is governed by the transition to inertia‐dominated fracture damage kinetics. In this region, dynamic loads can exceed the quasi‐static fracture value because of time‐dependent fracture processes. At the highest loading rates, approaching the upper yield value, the yield strength is again nearly constant with loading rate changes, and an alternative failure mechanism is activated. 9 Yield Stress
Shock Wave HEL
Transistion Region
Quasistatic Fracture
Loading Rate
Figure. 2.1. A typical yield stress‐loading rate curve for a brittle solid is shown as the solid black line, which is similar to Fig. 5b in Ref. 6. 10 Arguments similar to Grady’s have been provided by other authors to explain the behavior in Fig. 2.1.10,11 An alternative explanation for the transition region in Fig. 2.1 is a structural or specimen size effect.10 Here, the concept is that the specimen size will determine the time scale for crack propagation across the sample.10 For compressive loading times longer than this characteristic time, the yield strength would be approximately the same as the quasi‐static value.10 For compressive loading times shorter than this characteristic time, the yield strength will increase rapidly with loading rate.10 The structural transition loading rate is therefore given by,10 (2.1)
In the above equation, is the transition loading rate, is the quasistatic elastic limit, is the Young’s modulus, is the crack propagation speed, and is the characteristic length of the sample. The term is the strain to failure, and is the time for a crack to propagate across the sample. While the arguments presented by Grady and others are reasonable,6,10,11 it is important to note that the confining pressure (mean compressive stress) under uniaxial strain is significantly higher than under uniaxial stress.5 This is indicated in Fig. 2.2 which shows the confinement stress at the elastic limit as a function of loading rate. The confinement stress increases as the increase in loading rate changes the load path from uniaxial stress to uniaxial strain. It has been demonstrated previously that the elastic limit can depend strongly on the 11 Mean (Confinement) Stress
Uniaxial Strain
Uniaxial Stress
Loading Rate
Figure 2.2 The confinement stress at the yield point as a function of loading rate. 12 mean stress for some brittle solids.12‐14 Since the loading rate comparison (Fig. 2.1) combines the uniaxial stress and uniaxial strain data, it is not possible to distinguish between loading rate dependence and mean stress dependence from such a comparison. This difficulty, though recognized by Grady,5,6 was not considered, in general, in the preceding arguments for the yield strength – loading rate behavior. To separate the loading rate and mean stress dependence, the different loading rate comparisons need to be made for the same load path, i.e. uniaxial stress or uniaxial strain.5,6 For loading rates above ~ 104 s‐1, the measured compressive elastic limits in the literature are restricted to uniaxial strain compression. However, measurements of non‐
shock wave, uniaxial strain loading of brittle solids at loading rates above 104 s‐1, are extremely limited. Shockless uniaxial strain experiments on brittle solids have been performed using fused silica as a ramp compression generator.15,16 A initial single step shock input to fused silica will spread into a ramp wave as it propagates due to the material’s unique stress‐strain relationship, and the rise time of the ramp can be controlled by changing the thickness of the fused silica sample.15,16 Using this technique to achieve loading rates of 104 – 106 s‐1, experiments were conducted previously on dolomite15 and aluminum oxide.16 However, the results of these experiments did not reveal a significant change in the elastic limit when compared with shock compression results.15,16 However, because the ramp compression using fused silica is restricted to stresses below 3.5 to 4 GPa,17 the use of this technique for strong brittle solids (elastic limit higher than ~ 3.5 GPa) is not valid. 13 2.2 Dynamic Compression of Quartz Since the current work is on the study of quartz compressed along the x‐cut and z‐cut directions, the following overview will focus on the existing compressive studies that were performed on these orientations. A summary of general crystal quartz properties is provided in section 2.2.1, and a summary of previous dynamic compression studies on x‐cut and z‐cut quartz are provided in sections 2.2.2 and 2.2.3 respectively. 2.2.1 Discussion of Quartz Properties Crystalline quartz is composed of a network of interlocking SiO4 tetrahedra with each oxygen atom being shared by two tetrahedra,18 so that the overall composition is SiO2.18 At ambient pressure and temperature, quartz exists in the α‐quartz phase and possesses either trigonal P3221 or P3121 crystal symmetry depending on if it is right or left handed quartz, respectively;18 the lattice constants are a1 = a2 = 0.491 nm, and z0 = 0.540 nm.18 A schematic drawing of α‐quartz, extracted from Ref. 19, is shown in Fig. 2.3a. This drawing shows the m, r, and z faces which are the most common crystal faces encountered in natural quartz,19 and shows the Cartesian x, y, and z axis. A projection of the atomic positions for right handed quartz onto the basal plane (normal to the z‐axis) is shown in Fig. 2.3b. Three of the four conventional20 lattice vectors are shown in Fig. 2.3b as a1, a2, and a3, with the fourth being the out of plane vector along the z‐axis. In the current work, the relevant crystallographic directions are 1120 and 0001 ; the x and z directions, respectively.18 At atmospheric pressure and an elevated temperature of 573 0C, α‐quartz transforms reversibly to the β‐quartz phase.18 Several high pressure phases of quartz are also known to exist.18 The high pressure phases are not expected to be relevant for dynamic compressions at 14 z‐axis m
m x‐axis y‐axis b
a Figure 2.3. (a) Diagram of an α‐quartz crystal displaying the m,r, and z faces; extracted from Ref. 19. (b) Looking down the z‐axis at the right handed, α‐quartz lattice projected onto the basal plane. The silicon atoms are shown as filled circles and the oxygen atoms are shown as open circles. The Cartesian x and y directions are shown as are three of the four lattice vectors. 15 stresses below ~ 15 GPa.21 Since the main focus of this work is the room temperature elastic limit of quartz under dynamic compressions, the high temperature and high pressure phases of quartz are not relevant to the present work. In the remainder of this work, the term ‘quartz’ refers to the α‐quartz phase only unless otherwise specified. 2.2.2 Dynamic Compression of X‐Cut Quartz The compressive response of room temperature X‐cut quartz has been measured previously in three loading rate regimes: quasi‐static loading22 (loading rate of 10‐3 s‐1), split Hopkinson bar loading,22 (loading rate of 103 s‐1), and shock wave loading (loading rate > 106 s‐1).23‐26 Prior to the current study,27 and the recent work by Root and Asay,28 the dynamic uniaxial strain response under shockless loading conditions had not been examined previously for x‐cut quartz. Quasi‐static and split Hopkinson bar (uniaxial stress) measurements on x‐cut quartz were performed most recently by Kimberley et. al.22 Under quasi‐static compression, the crystals exhibited no measureable ductility and failed catastrophically when the elastic limit was exceeded, fragmenting into many small pieces.22 This behavior is typical of brittle solids under uniaxial stress.22 The longitudinal stress at the elastic limit was recorded in each experiment and had an average value of 2.5 GPa.22 X‐cut crystals subjected to unconfined split Hopkinson bar (Kolsky bar) loading also failed catastrophically when the elastic limit was exceeded,22 similar to the quasi‐static measurements. These dynamic uniaxial stress measurements revealed an average longitudinal stress elastic limit of 2.8 GPa,22 which is a small increase in the elastic limit over the quasi‐static measurements.22 The compressive response of x‐cut quartz to shock wave loading was first examined in the 1960’s by Wackerle23 and Fowles,24 and later examined by Graham.25,26 The HEL values of x‐
16 cut quartz, determined from the stress amplitudes of the elastic precursor waves, were generally measured to be between 5 and 7 GPa.23‐24,26 These HEL values were reported to be nearly independent of the peak stress and the sample thickness; both parameters spanned a wide range. 23‐24,26 Peak shock stresses where elastic waves had been observed ranged from ~ 6 GPa in the Graham26 investigation to as high as 27 GPa in the Wackerle study.23 Sample thicknesses ranged in thickness from 4.6 mm to 25.4 mm (both in Ref. 23). It is noted, however, that the HEL had not previously been measured on samples thinner than 4.6 mm. Wackerle23 mentioned that the yield strength under shock compression, corresponding to the shock wave HEL, was somewhat higher than the quasi‐static yield strengths measured for x‐cut quartz, though no shock strength values were provided. Graham26 reported that the maximum shear strength (difference between longitudinal and mean stress) at the HEL was 5.5% of C44, or 3.2 GPa. This is about a factor of two increase in shear strength over the quasi‐
static and Hopkinson bar measurements on x‐cut quartz.22 In the Wackerle23 and Fowles24 investigations, all of the peak stresses were above the elastic limit. Nevertheless, some elastic properties were extracted from the data using the measured shock and particle velocities of the elastic precursor waves.23,24 Shock wave compressions created by planar impacts above and below the elastic limit were performed by Graham25,26 from which the elastic properties were determined. Below the elastic limit, the response x‐cut quartz is non‐linear elastic.23‐26 The expressions for elastic Hugoniots from Refs. 23‐26 are listed in Table 2.1. Discrepancies between the measured elastic shock stress‐strain curve and the predicted curve from 2nd and 3rd order elastic constants29‐30 were used to evaluate fourth order elastic constants, C1111, in the Fowles,24 and Graham25 investigations. These values 17 Table 2.1 Elastic response of x‐cut quartz to shock compression. Ref. Elastic Hugoniota Upper Limit to Elastic Fourth Order Constant C1111 Hugoniots (GPa) (GPa) Wackerle23 σ = 14.86u + 2.10u2 ~ 7 ‐ 24
2
Fowles σ =15.23u – 0.4u ~ 7 15930 ~ 6 7500 Graham25‐26 σ = 15.16u + 0.83u2 a
The stress, σ, is in units of GPa, and the particle velocity, u, has units of mm/µs. ‐ Quantity not reported. 18 are also listed in Table 2.1. The elastic properties were fairly consistent among the investigations, though there are some noticeable differences. Above the elastic limit, shock compressed x‐cut quartz undergoes inelastic deformation in a heterogeneous manner.31‐32 This inelastic deformation is thought to occur through cohesion loss at highly localized planar deformation features.31 Planar features were reported in Ref. 32 in which the authors recorded images of the x‐cut quartz emission pattern under shock compression above the elastic limit. The density of the planar features increased with the impact stress.32 This type of heterogeneous yielding is thought to be typical of shocked brittle solids.31 2.2.3 Dynamic Compression of Z‐Cut Quartz The compressive response of room temperature z‐cut quartz has been measured previously under quasi‐static33 and tri‐axial stress12 loading at loading rates near (10‐4 s‐1), under shockless uniaxial strain loading3 at loading rates near 105 s‐1, and under shock wave loading23,24,34 at loading rates > 106 s‐1. High loading rate uniaxial stress measurements on room temperature z‐cut quartz, such as split‐Hopkinson bar measurements, were not available in the literature. The quasi‐static, uniaxial stress, response of compressed z‐cut quartz crystals was examined by Griggs et. al.33 Similar to x‐cut quartz,22 the crystals showed no measureable ductility and fractured when the elastic limit was exceeded.33 The elastic limits in these measurements were reported to be between 2.2 and 2.5 GPa,33 comparable to the x‐cut direction.22 Quasi‐static, tri‐axial stress measurements on z‐cut quartz were reported in Ref. 12. The lateral stresses in these measurements were between 2.7 and 3.0 GPa at the moment of failure due to confinement in pressurized bismuth.12 As with the uniaxial stress measurements,33 19 the failure mechanism under the higher confining stress was brittle fracture.12 The longitudinal stress elastic limit was reproducible and had an average value of 7.4 GPa,12 corresponding to a factor of ~ 2 increase in yield strength over the uniaxial stress measurements.33 The increase in yield strength was attributed to the increase in confining stresses.12 The shockless, uniaxial strain compressive response of z‐cut quartz (below the HEL) was examined previously by Jaglinski et. al.3 using a compact pulsed power source. The authors examined the shockless response in the nonlinear elastic regime (to about 7 GPa) using a velocity interferometer, and the elastic properties were determined from the measured wave profiles.3 The elastic limit of z‐cut quartz under shockless uniaxial strain compression had not been determined previously. Like x‐cut quartz, the first shock wave uniaxial strain studies on z‐cut quartz were performed in the 1960’s by Wackerle23 and Fowles24 using plane wave explosive lenses to create the pressure pulses. Both authors measured rear free surface and shock wave velocities from which they determined the mechanical properties.23,24 Later, the nonlinear elastic response was extensively characterized under plate impact loading using a velocity interferometer system to record the transmitted wave profiles.34 The experiments by Wackerle23 and Fowles24 provide most of the available Hugoniot elastic limit information on z‐cut quartz. Both authors report a peak stress dependent elastic wave amplitude;23,24 the elastic wave stress amplitude increased from as low as 10.5 GPa at a peak stress of 13 GPa to as high as 14.8 GPa at a peak stress of 22 GPa for 3.4 mm thick samples.24 The elastic precursor amplitude increased by approx. 40% with a 70% increase in the peak stress. The elastic wave was also reported to decrease in amplitude with increasing propagation distance.23,24 For example, one experiment on a 3.4 mm thick sample of z‐cut quartz had an elastic wave amplitude of 14.8 GPa when shock compressed to 20 21.5 GPa.24 Experiments on ~ 6.5 mm thick samples had an average elastic wave stress amplitude of 13.0 GPa when compressed to a peak stress of near 20 GPa;24 a 12% decrease in the elastic wave amplitude for a factor of two increase in the propagation distance. Since the elastic wave amplitude was dependent on both peak stress and sample thickness, a single, or minimum, elastic limit stress could not be readily identified in the Wackerle and Fowles investigations.23,24 Wackerle23 estimated that the minimum shock wave elastic limit would be between 6.5 and 8.0 GPa for z‐cut quartz. A later experiment by Jones and Gupta34 demonstrated that z‐cut quartz shock compressed to 5.9 GPa responds elastically which supports the lower end of Wackerle’s estimate. Gallivan and Gupta35 reported indirect evidence that the elastic limit was exceeded at a shock stress of 6.9 GPa based on excess luminescence from the crystal at this stress; this result also supports the low end of Wackerle’s estimate. Wackerle mentioned that the yield strength of z‐cut quartz under shock compression, corresponding to the shock wave HEL, was somewhat higher than the quasi‐static yield strengths,23 though no shock strength values were reported. Below the elastic limit, the response of z‐cut quartz is non‐linear elastic;3,23,24,34 the elastic stress‐particle velocity Hugoniots from Ref. 23, 24, and 34 are listed in Table 2.2. The fourth order elastic constant values, C3333, reported by both Fowles24 and Jones and Gupta34 are also listed. The elastic properties are in excellent agreement amongst all of the investigations. The elastic shockless compression response was also reported to be in excellent agreement with the shock wave response.3 Overall, the elastic response of z‐cut quartz subjected to uniaxial strain has been well characterized. 21 Table 2.2 Elastic Response of z‐cut quartz to shock compression Ref. Elastic Hugoniota Upper Limit to Fourth Order Elastic Hugoniots Constant C3333 (GPa) (GPa) 23
2
Wackerle σ = 16.84u + 3.60u ~ 15 ‐ Fowles24 σ =16.74u + 3.12u2 ~15 18490 Jones and Gupta34 σ = 16.74u + 3.31u2 + 1.48u3 ~6 17481 a
The stress, σ, is in units of GPa, and the particle velocity, u, has units of mm/µs. ‐ Quantity not reported. 22 2.3 Summary of Findings from Existing Work The main findings of this chapter are summarized as follows: 
The elastic limit, or yield strength, of brittle solids is thought to increase with loading rate, and the highest value is expected be the HEL observed in shock compression experiments.6 Experimental support for this conclusion is provided by comparing the yield strength under quasistatic and Hopkinson bar measurements (uniaxial stress) to the elastic limit under shock compression loading (uniaxial strain).6 
The confining stresses are significantly higher under uniaxial strain (inertial confinement) than under uniaxial stress and yield strengths of brittle solids are known to increase with confining stresses.5 Therefore, it is difficult to separate the effect of a loading rate increase on yield strength from the effect of a confinement stress increase when comparing uniaxial stress and uniaxial strain data.5 To address this difficulty, yield strength ‐ loading rate studies should be along the same load path, such as uniaxial stress or uniaxial strain. 
Yield strength measurements on brittle solids subjected to loading rates between that achieved in Hopkinson bar measurements (loading rate < 104 s‐1) and that achieved in shock compression measurements (loading rate > 106 s‐1) are extremely limited. Conclusions regarding the yield strength in this loading rate regime are generally not supported by experimental evidence. 
X‐cut quartz has been subjected previously to compression in quasi‐static, Hopkinson bar, and shock wave experiments.22‐26 The crystal is strong and brittle and the elastic limits (longitudinal stress at yield) ranged from 2.5 GPa in quasistatic 23 tests to ~ 6 GPa under shock wave loading.22‐26 The yield strength under shock compression (uniaxial strain) was higher than under Hopkinson bar or quasi‐static (uniaxial stress) compressions; consistent with other brittle solids in the literature.5 The shockless uniaxial strain response (loading rates of 104 s‐1 ‐ 106 s‐1) had not been examined previously. 
The elastic limit of z‐cut quartz crystals was measured previously when subjected to quasi‐static and shock wave compressions.12,23‐24,33‐34 Both uniaxial and tri‐axial stress experiments were performed under quasi‐static loading.12,33 Similar to x‐cut quartz, the z‐cut crystal orientation is strong and brittle.12,33 The measured quasi‐
static uniaxial stress elastic limits ranged from 2.2 GPa to 2.5 GPa,33 and the tri‐axial stress elastic limit was 7.4 GPa with about a 3 GPa lateral stress confinement.12 This corresponds to a factor of ~ 2 increase in yield strength under the higher confining stress.12 The shock wave HEL of z‐cut quartz was dependent on peak stress and sample thickness and has been observed to be between 10.5 and 15 GPa.23‐24 This is an increase in yield strength over the uniaxial stress experiments, consistent with other brittle solids in the literature. The elastic limit of z‐cut quartz under shockless uniaxial strain compression had not been measured previously. 
The mechanical response of both x‐cut and z‐cut quartz crystals, subjected to non‐
linear elastic compression has been well characterized in the literature.3,23‐26,34 24 References for Chapter 2 1. G. Avrillaud, J. R. Asay, M. Bavay, M. Delchambre, J. Guerre, F. Bayol, F. Cubaynes, B. M. Kovalchuk , J. A. Mervini, R. B. Spielman, C. A. Hall, R. J. Hickman, T. Ao, M. D. Willis, Y. M. Gupta, C. J. Bakmeman, in Shock Compression of Condensed Matter – 2007 edited by M. Elert, M. D. Furnish, R. Chau, N. C. Holmes, J. Nguyen (American Institute of Physics, New York, 2007) p. 1161. 2. T. Ao, J. R. Asay, S. Chantrenne, M.R. Baer, and C.A. Hall, Rev. Sci. Instr. 79, 013903 (2008). 3. T. Jaglinski, B.M. LaLone, C.J. Bakeman, and Y.M. Gupta, J. Appl. Phys. 105, 083528 (2009). 4. M. A. Meyers, Dynamic Behavior of Materials, (John Wiley & Sons Inc, NY, 1994). 5. D. E. Grady and R. E. Hollenbach, Tech. Rep. SAND76‐0659, Sandia National Laboratories, 1977. 6. D. E. Grady, Shock Waves in Condensed Matter, edited by S. C. Schmidt and W. C. Tao (AIP, New York, 1996), p. 9. 7. R. N Shock and H. C. Heard, J. Geophys. Res. 79, 1662 (1974). 8. D. E. Grady, R. E. Hollenbach, K. W. Schuler, and J. F. Callender, J. Geophys. Res. 82, 1325 (1977). 9. J. Lankford, J. Amer. Ceramic. Soc. 64, C33 (1981). 10. G. Ravichandran and G. Subhash, Int. J. Solids Struct. 32, 2627 (1995). 11. B. Paliwal and K. T. Ramesh, J. Mech. Phys. Solids. 56, 896 (2008). 12. J. M. Christie, H. C. Heard, and P. N. LaMori, Am. J. Sci. 262, 26 (1964). 25 13. R. Feng, G. F. Raiser, and Y. M. Gupta, J. Appl. Phys. 79, 1378 (1996). 14. C. Hari Manoj Simha and Y. M. Gupta, J. Appl. Phys. 96, 1880 (2004). 15. D. E. Grady, Geophys. Res. Letts., 1, 263 (1977). 16. J. Cagnoux and F. Longy, Shock Waves in Condensed Matter, edited by S. C. Schmidt and N. C. Holmes (Elsevier, New York, 1988), p. 293. 17. L. M. Barker and R. E. Hollenbach, J. Appl. Phys. 41, 4208 (1970). 18. H. Megaw, Crystal Structures: A Working Approach (Saunders, Philadelphia, 1973) p. 263. 19. C. Frondel, Dana’s The System of Mineralogy, 7th ed. (Wiley, NY, 1962). 20. A. R. Lang, Acta Cryst. 19, 290 (1965). 21. L. C. Chhabildas, Shock Compression of Condensed Matter‐1985 edited by Y. M. Gupta (Plenum, NY, 1986). 22. J. Kimberley, K. T. Ramesh, and O. J. Barnouin, J. Geophys. Res. 115, B08207 (2010). 23. J. Wackerle, Appl. Phys. 33, 922 (1961). 24. R. Fowles, J. Geo. Phys. Res. 72, 5729 (1967). 25. R. A. Graham, Phys. Rev. B, 6, 4779 (1972). 26. R. A. Graham, J. Phys. Chem. Solids, 35, 355 (1974). 27. B. M. LaLone and Y. M. Gupta, J. Appl. Phys. 106, 053526 (2009). 28. S. Root and J. R. Asay, J. Appl. Phys. 106, 056104 (2009). 29. H. J. McSimin, P. Andreatch Jr., and R. N. Thurston, J. Appl. Phys. 36, 1624 (1965). 30. R.N. Thurston, H.J. McSkimin, and P. Andreatch, J. Appl. Phys. 37, 267 (1966). 31. D. E. Grady, J. Geophys. Res. 85, 913 (1980). 26 32. P. J. Brannon, C. Konrad, R. W. Morris, E. D. Jones, and J. R. Asay, J. Appl. Phys. 54, 6374 (1983). 33. D. T. Griggs, F. J. Turner, and H. C. Heard, Geol. Soc. Am. Mem. 79, 39 (1960). 34. S. C. Jones and Y. M. Gupta, J. Appl. Phys. 88, 5671 (2000). 35. S. M. Gallivan and Y. M. Gupta, J. Appl. Phys. 78, 1557 (1995). 27 Chapter 3 Experimental Methods This chapter details the techniques used to examine the shockless and shock wave compressive response of x‐cut and z‐cut quartz crystals. Sample characterization and preparation is discussed in section 3.1. Time‐resolved particle velocity measurements, used to determine the continuum response of the compressed quartz crystals, are discussed in section 3.2. The shockless compression methods are discussed in 3.3, and the shock wave methods in 3.4. 3.1 Sample Characterization and Preparation The x‐cut and z‐cut quartz samples were characterized using density, longitudinal and shear sound speeds, and x‐ray Laue measurements. Densities were determined using the Archimedean method with typical uncertainties of about ± 0.004 g/cm3. Sound speeds were measured using ultrasonic transducers and the pulse‐echo technique; timing uncertainties were typically 2‐3 ns resulting in typical sound speed uncertainties of about 0.04 mm/µs. Crystalline quality and axial alignment were judged solely using Laue x‐ray diffraction images. Samples showing highly elliptical or split Laue spots and samples whose crystalline axis was misaligned by more than 3 degrees from the normal were rejected. 3.1.1 X‐Cut Quartz Preparation Optically polished and well oriented x‐cut quartz disks were received from Boston Piezo Optics. The disks ranged in thickness from 1 to 3.5 mm and in diameter from 8 to 25 mm. The 28
samples had vendor quoted flatness and parallelism values of better than 5 fringes/inch and 3 minutes of arc, respectively. The surface normals were aligned to the x‐axis to within 20 minutes of arc, as measured by the vendor. The average measured density and longitudinal sound speed of the samples were 2.648 g/cm3, 5.80 mm/µs, respectively. The average shear sound speeds for the two different shear modes were 5.15 mm/µs, and 3.31 mm/µs. All of the ambient values were consistent with prior results.1,2 The density and sound speed values for each individual sample are listed in Table 4.1 of the next chapter. The sample surfaces were generally used as‐received from the vendor. However, one sample was lapped and polished to the final thickness; x‐cut quartz sample 6 was prepared this way. Lapping was performed using 20 µm alumina grit, and a step wise polishing procedure was used ; first 15 µm, then 6 µm, 3 µm, and finally 1 µm diamond grit in an oil suspension. Typically smaller disks, 8 mm or 11 mm diameter, were required for the experiments due to restrictions in sample geometry; a smaller size than some of the as‐received disks. The procedure used to prepare smaller disks from larger diameter samples is discussed next. Disk Cutting The quartz sample, from which the disks were to be cut, and a lapped aluminum plate were heated to about 150 degrees C on a hot plate. Once the samples reached temperature, hot melt glue was applied liberally to the aluminum plate until a sufficient puddle of glue was formed. The quartz sample was then lightly pressed into the puddle until all visible air bubbles in the glue were gone; this formed a temporary bond between the quartz sample and the aluminum plate. Hot melt glue was then further applied to the upper surface of the quartz sample; this layer of glue protected the surface from scratching during subsequent hole 29
grinding. The aluminum plate/quartz assembly was then removed from the hot plate and left to cool. A hole grinding tool of appropriate inner diameter (8 mm or 11 mm), consisting of a slotted brass pipe, and a water/SiC (120 grit) grinding compound slurry were used to cut the disks from the as‐received sample mounted to the aluminum plate. A Teflon retaining ring was used to contain the slurry during grinding on a machining mill. A photograph of the grinding tool, the SiC grit and the retaining ring are shown in Figure 3.1. Once the hole grinding tool had completely penetrated the quartz sample for each disk, the assembly was reheated on a hot plate to melt the glue bond, and the newly cut quartz disks were removed from the aluminum plate. Soaking the newly cut disks in acetone removed any remaining hot melt glue. When completed, the faces of the disks were free of scratches and the edges were smooth, resembling the surface of ground glass. 3.1.2 Z‐Cut Quartz Preparation Polished z‐cut quartz disks, 25.4 mm in diameter and 2 to 3.2 mm thick were received from Meller Optics. The samples had vendor quoted flatness and parallelism values of 0.5 µm and 20 seconds of arc, respectively, and were typically aligned to the crystalline axis to within 15 minutes of arc. The average measured density and the longitudinal and shear sound speeds of the samples were 2.647 g/cm3, 6.37 mm/µs, and 4.71 mm/µs respectively, consistent with prior results.2 The density and sound speed values for each individual sample are listed in Table 5.1 in chapter 5. The sample surfaces were generally used as received. However, sample 1 was prepared by lapping and polishing the sample surfaces to obtain the desired thickness, the procedure used was identical to the x‐cut quartz sample preparation. Most disks were used as 30
Figure 3.1: Equipment for cutting quartz disks. The quartz piece was glue bonded to the aluminum mounting plate. The Teflon retainer surrounds the disk and contains grinding compound/water slurry. The brass cutting tool is mounted to an end mill for grinding through the quartz to make smaller sized disks. 31
received, with respect to their diameters. However, some smaller diameter disks were required and these were prepared identically to the x‐cut quartz disks. 3.1.3 Optical Windows All shockless compression experiments utilized an optical window to nearly match the sample mechanical impedance and obtain a near in‐material measurement using a laser‐
interferometer. This window was made from a high purity [100] LiF disk. The windows were characterized as discussed in 3.1 and further prepared by vapor depositing a thin Al mirror to one side to serve as a reflector for the interferometer laser. The Al vapor deposition was conducted using a LADD vacuum evaporator. 3.2 Particle Velocity Measurements Particle velocity histories were measured in all experiments to determine the continuum response. Although several experimental techniques exist for wave profile measurements,3 the current measurements were made exclusively using a velocity interferometer system for any reflector (VISAR).4‐5 Transmitted wave profiles were measured with the VISAR system at the interface between the sample material and an optical window. A multi‐point VISAR system, capable of measuring up to four points simultaneously, was used. The details of the VISAR system have been described elsewhere.6 The multi‐point VISAR system used in this work was capable of velocity determination to better than 0.6 m/s. For some of the z‐cut quartz experiments, a single point VISAR system was used, and the details of this system can be found in Ref. 7. 3.3 Shockless Compression Measurements Shockless, uniaxial strain compression measurements allow for a continuous loading curve (stress – density) to be determined at loading rates lower than those present in shock 32
wave experiments.8‐12 The method for determining the loading curve from shockless compression data will be discussed in Chapter 4. The remaining details pertaining to shockless compression measurements are discussed in this section. 3.3.1 Compact Pulsed Power Generator Shockless compression measurements were performed utilizing the WSU compact pulsed power generator (CPPG). The CPPG system operation has been discussed in detail elsewhere,6,13‐14 and will only be described briefly here. A photograph of the CPPG is shown in Fig. 3.2 and the CPPG has a footprint of about 10 m2. The usage of CPPG for shockless compression studies is based on a similar approach developed at the Sandia Z facility.15 The CPPG can produce 2‐3 MA of electrical current with a rise time (0‐100%) of ~ 550 ns by discharging 8 capacitors (Fig. 3.2a). The current flows across the surfaces of opposing conductive aluminum panels that are 11 or 15 mm in width, Fig. 3.3. The electrical current flow across the panel surfaces creates a magnetic pressure which drives a large amplitude compression wave into the panels. Samples are mounted to the panels on the faces that are opposite to the faces with the drive current. Thus, the compression wave travels into the samples after first propagating through the thickness of the panels. With this experimental configuration, shockless compression waves can be propagated in sample materials to a peak stress of up to 18 GPa. Detailed discussions of the magnetic field uniformity and drive panel designs can be seen in Ref. 14 for a nearly identical system built at Sandia National Laboratories. In all shockless compression experiments performed in this work, the dynamic switch (Fig. 3.2b) was filled with 100% Argon at 1 atm of pressure and the capacitor bank was charged to 70 kV. Different panel widths were used to obtain different peak stresses. 15 mm wide panels 33
A
C
B
Figure 3.2: Photograph of the CPPG facility. (A) The 8 main capacitors topped with the main switches. (B) The dynamic switch. (C) The peaking capacitors. 34
were used to obtain a peak stress of up to 13 GPa, and 11 mm wide panels were used to obtain a peak stress of up to 18 GPa. 3.3.2 Target Construction The 15 mm wide panels for these experiments were machined exclusively from 1100 H14 Al by the instrumentation shop. This relatively pure aluminum was used to reduce the amplitude of the “foot” of the wave due to the aluminum elastic limit. The importance of this small elastic amplitude will be discussed in Chapter 5. The 15 mm panels (Fig. 3.3b) were from a Sandia design.14 These panels are 34.6 mm in length and have a 13 mm diameter circular sample area with a floor thickness of ~ 2 mm. The bottom surfaces of the machined panels were hand sanded flat using 320 grit SiC paper on a steel flat. The circular sample areas were then sanded flat and parallel to the rear surface, using a vertical mill with 400 and 600 grit sand papers. Final polishing was done using either 3 μm or 1 μm diamond particles in an oil suspension. Due to the procedure, the outer ~ 1mm radius of the sample area could not be made flat and was not usable. The usable sample area had a floor thickness of ~ 2 mm with a uniformity of 2 μm or less. To ensure that the wave input from both panels in an experiment was the same, the floor thickness of each panel pair was consistent to within ~ 5 μm. The 11 mm panels (Figure 3.3b) were constructed using a slightly different design. To utilize a greater sample area, the panel floors were prepared separately out of 1100 H14 Al and then bonded to 6061 Al panel bodies. This was done out of necessity as edge wave considerations become more important for the smaller diameter samples, so the usable sample area had to be maximized. The panels were 32 mm in length and had a usable circular sample area of 10 mm diameter. 35
Lens Tube
Aluminum Panels
To/From VISAR
Current Out
End Short
Current In
Sample
Window
Figure 3.3a: Schematic diagram of CPPG experiments. Current from the CPPG flows through conductive opposing panels. The large current creates a magnetic pressure that sends a uniaxial strain compression wave into the samples backed by optical windows. The VISAR system monitors the sample/window interface velocities. Figure 3.3b: Photograph of the two panels used in the shockless compression experiments on the CPPG. The 11 mm wide panels are used for peak stresses to 18 GPa. The 15 mm wide panels are used for peak stresses up to 13 GPa. 36
The 11 mm wide panels were prepared as follows. The panel floors were first prepared by bonding a pair of them to a flat soda lime glass window using hot melt glue. Once cooled, the soda lime glass window was fastened to a steel polishing jig using double sided tape. The panel floors were then polished using an auto‐polisher (Allied High Tech Products ‐ Multiprep) to keep the polished surface flat and parallel to the soda lime glass surface. A step wise polishing procedure was used; first 15 μm, then 6 μm, and finally 3 μm diamond grit in an oil suspension. The panel floors were then removed from the glass, flipped over, and the process was repeated on the opposite side. This procedure brought the panel floors to a thickness of ~ 2 mm with a parallelism of ~ 2 μm and a pair wise floor thickness consistency of 2 μm. Once the sample area was prepared, as described above for the 15 mm and 11 mm panels, the samples were bonded to the sample area using a 5:1 ratio of 815 epoxy resin and HD 3475 hardener. A sample press was used to add a consistent force of 40 lbs to the sample while the epoxy bond cured for ~ 24 hrs. After the sample was bonded to the 11 mm panel floor, the panel floor was glued to the panel body using either a quick curing adhesive or the epoxy mixture and the procedure described above. The optical window (with the mirror for the interferometer laser) was bonded to the sample last, using the 815 epoxy mixture above, and allowed to cure for 24 hrs while the window was being pressed with 20‐40 lbs. of force. The next step in the shockless target construction, after the sample and window bonds had cured, was to attach optical lens tubes to direct laser light from the VISAR fibers to/from the sample/window interface mirror (see Fig. 3.4). The lens tubes were of a standard design made from ¼ inch ID brass tubing with 3/16 ‐ 40 male threads, and end fittings for the optical fibers. 10 mm focal length collimating and focusing lenses were used exclusively in the lens tubes. The lens tubes were fastened to the target using a threaded lens unit holder that was glue bonded 37
Figure 3.4: Photograph of a completed panel showing the panel, lens tube holder, lens tube (covered with pipe thread tape), and the optical fiber connector at the top of the lens tube.
38
to the panel. Focusing was accomplished by maximizing the laser light returning from the interface mirror into the optical fiber. After the focusing was completed, the lens tubes were glued into place using a quick drying epoxy. The completed panels were loaded into the CPPG by the person operating the system. A photograph of one of the panels after being loaded into CPPG is shown in Fig. 3.5. 3.3.3 Instrumentation Setup The instrumentation for the shockless compression measurements consisted of the multipoint VISAR system, triggering, timing laser, and diagnostic measurements. The VISAR system and diagnostics were located in the instrumentation room, and all information from the CPPG facility was transmitted via fiber optic cables to minimize any electrical noise that may arise in coaxial cables. The return light from each of the two VISAR optical fibers that monitor the two sample/window interfaces were split into 2 light paths using a 2 to 4 fiber beam splitter. This light was fed into the VISAR interferometer which was operated in a dual velocity per fringe (VPF) configuration. The two VPF’s were 0.1724 mm/µs in leg 1 and 0.0935 mm/µs in leg 2. The splitting was done so that high velocity precision was obtained for smooth profiles with the small VPF, and contrast was maintained with the large VPF when there were rapid velocity fluctuations due to the samples deforming heterogeneously. A digital timing delay generator, Stanford Research Systems (SRS) – DG 535, located in the CPPG facility provided the various instrument time delays when the CPPG was fired. Two channels from this SRS were fed to the instrumentation room via electric to optical fiber relay cables. Once in the instrumentation lab, the optical signals were converted back into electric 39
Figure 3.5: Photograph of a completed panel after being loaded into the CPPG for an experiment. The panel is bolted to the anode and bolted to the opposing panel (not shown) via the end short. The opposing panel is bolted to the cathode to complete the circuit. Kapton insulation separates the two panels except at the end short. The lens unit which transfers the VISAR laser light to/from an optical fiber (not connected in the photo) to the sample window interface is also shown. 40
signals. The first channel sent a trigger signal ~ 170 μs before the main trigger; this signal triggered the timing laser which has about a 170 μs delay between when it is triggered and when it emits a laser pulse. (The timing laser pulse appeared in each of the VISAR oscilloscope records and was used for time correlation of the VISAR records during analysis). The second channel from the SRS sent a trigger signal to the VISAR digitizing oscilloscopes, the acoustic optic modulator to turn on the VISAR laser, and a diagnostic scope. Simultaneously to this trigger signal, in the CPPG facility, a trigger was sent to fire the main switches. The diagnostic scope monitored the timing laser trigger, the main trigger, the VISAR digitizing oscilloscopes triggers, and the VISAR laser return light intensity. 3.4 Plate Impact Experiments Planar impacts were used to create the uniaxial strain condition in the shock wave experiments. The details of the experimental arrangements used in the impact experiments are discussed in this section. 3.4.1 Target Construction Although multiple target designs were used, they can generally be described by a single experimental schematic view, shown in Fig. 3.6. A flyer plate, launched by a 100 mm light gas or 30 mm powder gun, impacted the sample backed by an optical window. A VISAR laser, coupled using an optical fiber, monitored the velocity at the interface between the sample and the optical window. The prepared samples were bonded to the optical windows using a 5:1 ratio of warmed EPON 815 epoxy resin and HD 3415 hardener and the sample/window assembly was pressed using 30‐40 lbs. of force while the bond cured for at least 24 hrs. Typical bond thicknesses were 41
Figure 3.6: Schematic view and photograph of a standard plate impact experiment. An impactor (launched from a gas or powder gun) impacts a sample backed by an optical window. The sample/window interface velocity is monitored using a VISAR laser; the laser light was fed to the interface using fiber optical cable connected to a lens tube containing a pair of lenses. Common features in both the schematic view and the photograph are the sample backed by a window, the target plate and the target ring. 42
~ 2 μm. Prior to bonding, a thin (~ 200 nm) aluminum mirror was vapor deposited onto the LiF optical window to serve as a reflector for the VISAR laser light. The window backed samples were glued to lapped aluminum target plates using a two‐part, fast curing, epoxy. The target plate served to keep the samples in position during an experiment. For some of the experiments, two targets were mounted to the target plate to obtain two measurements in a single experiment. The target plate was then mounted to an aluminum target ring using three concentric screws and compressible Bellville washers as spacers. The samples were aligned parallel to the target ring by adjusting the amount of compression of the spacers using screw tension. Optical components, which transmit the light from the optical fibers to/from the sample/window interface mirror, were mounted to the target. Lens units, constructed from threaded brass tubes and fitted with collimating and focusing lenses, were used to image the laser light from the fiber onto the interface. The optical fiber tip could be reversibly threaded into the top of the lens unit. Symmetric lens pairs were used for one‐to‐one imaging of the optical fiber tip to interface. The focal lengths of the lenses were either 10 mm or 18 mm depending on the distance from the lens unit to the interface. Threaded lens unit holders, constructed of aluminum or plastic, were used to center the lens units over the samples. The lens unit holder was attached to the back of the target plate using either a fast curing epoxy and/or machine screws. The laser light was focused onto the sample/window interface mirror by adjusting the height of the lens unit until the return light was maximized. Once the lens unit alignment was complete, all of the optical components were secured in place with quick drying epoxy. 43
In a few experiments, shock wave speeds were measured. This was accomplished by monitoring the impact surface with either a VISAR probe and lead zirconate titanate (PZT) electrical impact pins, or multiple VISAR probes. The experimental arrangement for such measurements is shown in Fig. 3.7. The impact surface probes (VISAR and PZT pins) provided an impact time which, together with the sample/window interface probes, provided wave transit times that could be used to calculate shock wave speeds. Because of the small impact tilt, the multiple impact probes were arranged so that the tilt could be accounted for and the impact time in‐line with the sample/window interface probe could be determined. This approach minimizes the wave speed uncertainty associated with the timing error due to the impact tilt. The impact surface VISAR probes could be easily incorporated into the experiments because the quartz samples were transparent, Fig. 3.7. During target preparation, an aluminum mirror was vapor deposited onto the impact side of the sample to serve as a reflector for the laser light. The probes themselves were constructed identically to the interface probes. PZT electrical pins, obtained from Dynasen, were 1.6 mm in diameter. The pins were first pressed and epoxied into an aluminum ring in a concentric arrangement of four pins. The ring was then lapped so that all pins were in the same plane. Copper was vapor deposited over the lapped surface to provide electrical contact between the outer electrodes and the impact surface of the PZT crystals. Once the ring was completed, the target sample was potted into the ring such that the surface of the target was in the same plane as the PZT pins. Any deviations from planarity (typically < 10 μm) were measured and recorded so that timing corrections could be incorporated later, as needed. Coaxial cables were connected to the PZT pins using connecting clips for the center electrode, and grounding wires attached directly to the target plate for the outer electrode. 44
Figure 3.7: Schematic view and photograph of target for measuring shock wave speed and particle velocity. In addition to the standard target which monitors the sample/window interface velocity, VISAR probes and (optional) PZT impact pins monitor the impact surface to obtain an impact time. The photograph shows the back of such a target mounted to the 100 mm light gas gun for an experiment. Three VISAR fibers and lens tubes are seen in the photograph, two monitor the impact surface (outer probes), and the center probe monitors the sample/window interface. In this particular experiment, PZT pins were not used. 45
3.4.2 Projectile Preparation Several different projectiles were used in these experiments and were accelerated in either a 100 mm diameter light gas gun or a 30 mm diameter powder gun. Photographs of two different 100 mm diameter light gas gun projectiles, used in the experiments, are shown in Fig. 3.8. The 200 mm long projectiles, machined from 6061‐T6 aluminum, were used for projectile velocities < 0.8 mm/µs. The 175 mm long projectiles, machined from 7075 aluminum, were used for projectile velocities between 0.8 mm/µs and 0.94 mm/µs. The faces of these projectiles were lapped to remove machining marks. The projectile alignment was typically measured to be within 0.2 milliradians of square after being lapped. The 175 mm long projectiles were fitted with a threaded back plate which was epoxied in place prior to the experiments for added structural integrity. Impactors in the experiments were generally oxygen free high conductivity (OFHC) copper disks, nominally 25 mm in diameter and 2 mm to 4 mm in thickness. One experiment (z‐
cut quartz sample 8) utilized a z‐cut quartz impactor that was 25 mm in diameter and 6.4 mm thick. These impactors were held in an impactor mount that was machined from 6061‐T6 aluminum. The impactor was bonded to the mount using a 2 part (815/3475) epoxy resin and cured while being pressed with ~ 40 lbs. of force for 24 hrs. After the 24 hour curing period, the rear surface of the impactor mount was made parallel to the impactor surface using a lathe followed by hand lapping on a steel flat with sand paper. The finished impactor mount was then screwed to the face of the 100 mm diameter projectile. The screw heads were covered with a quick drying epoxy to prevent gas flow through the screw holes in the projectile during launch. The impactor tilt was typically better than 0.2 milliradians with respect to the projectile axis. 46
Figure 3.8: Photographs of the 100 mm diameter gas gun projectiles. The upper photo shows both the 200 mm long, 6061‐T6 aluminum, projectile and the 175 mm long, 7075 aluminum, projectile. The lower photograph shows a copper impactor mounted to a 100 mm diameter projectile. 47
Photographs of two different 30 mm diameter powder gun projectiles, used in the experiments, are shown in Fig. 3.9. The 86 mm long projectiles, machined from lexan, were used for projectile velocities between 1.1 mm/µs and 2.0 mm/µs. The 57 mm long projectiles, also machined from lexan, were used for projectile velocities between 2.0 mm/µs and 2.4 mm/µs. A recess was cut into the face of the projectile to fit the dimension of the impactor. The impactor, an OFHC copper disk, was glued into the recess using a two part epoxy resin and pressed with ~ 40 lbs. of force during a 24 hrs. curing period. The impactor tilt was usually measured to be better than 1 milliradian with respect to the projectile axis. The impact surface of the copper disks was polished to a mirror finish using a step wise hand polishing procedure; first 9 µm, then 3 µm, and then 1 µm diamond grit in an oil suspension. The finished impactor surface was flat to within ± 5 µm. For the experiment which utilized the single crystal z‐cut quartz impactor, no surface preparation was done as the vendor prepared surface finish was acceptable. 3.4.3 Instrumentation Setup The instrumentation for the shock compression measurements was similar to the shockless compression measurements, but differed in the triggering of the diagnostics and in the VISAR timing. Impact shorting pins on the target were generally used to trigger the VISAR system and the diagnostic oscilloscope. The pins were mounted on the target plate above the sample such that the impactor would short out the pin about 5 µs before impacting the target. A coaxial cable, attached to the trigger pin, fed the shorting signal to a shorting pulse generator, which in turn, sent a trigger signal to the digital delay generator to trigger all of the diagnostics. 48
Figure 3.9: Photographs of powder gun projectiles with copper impactors. The projectile on the left is for projectile velocities between 1.1 mm/µs and 2.0 mm/µs, the projectile on the right is for projectile velocities between 2.0 mm/µs and 2.4 mm/µs. 49
Some of the experiments were triggered off of the drop in the VISAR return laser light that occurs late after impact (~ 3‐8 µs) due to the destruction of the interface mirror. This was done when there was not sufficient space for trigger pins on the face of the targets. Time correlation among the VISAR signals was accomplished by measuring the various delay times before each experiment; this differs from the time correlation in the shockless compression experiments that utilized a timing laser pulse that provided a fiducial time in the VISAR records. 50
References for Chapter 3 1. R. A. Graham, Phys. Rev. B. 6, 4779 (1972). 2. H. J. McSkimin, P. Andreatch, R. N. Thurston, J. Appl. Phys. 36 1624 (1965). 3. D. E. Grady, in Shock Compression of Condensed Matter – 1995, edited by S. C. Schmidt and W. C. Tao, American Institute of Physics, Woodbury, NY, pg. 9 (1996). 4. L. M. Barker and R. E. Hollenbach, J. Appl. Phys. 43, 4669 (1972). 5. L. M. Barker, K. W. Schuler, J. Appl. Phys. 45, 4208 (1974). 6. T. Jaglinski, B.M. LaLone, C.J. Bakeman, and Y.M. Gupta, J. Appl. Phys. 105, 083528 (2009). 7. W. F. Hemsing, Sci. Instrum. 50, 73 (1979). 8. R. Fowles and R. F. Williams, J. Appl. Phys. 41, 360 (1970). 9. M. Cowperthwaite and R. F. Williams, J. Appl. Phys. 42, 456 (1971). 10. L. Seaman, J. Appl. Phys. 45, 4303 (1974). 11. J. B. Aidun and Y. M. Gupta, J. Appl. Phys. 69, 6998 (1991). 12. S. D. Rothman and J. Maw, J. Phys. IV France, 134, 745 (2006). 13. G. Avrillaud, J. R. Asay, M. Bavay, M. Delchambre, J. Guerre, F. Bayol, F. Cubaynes, B. M. Kovalchuk , J. A. Mervini, R. B. Spielman, C. A. Hall, R. J. Hickman, T. Ao, M. D. Willis, Y. M. Gupta, C. J. Bakeman, in Shock Compression of Condensed Matter – 2007 edited by M. Elert, M. D. Furnish, R. Chau, N. C. Holmes, J. Nguyen (American Institute of Physics, New York, 2007) p. 1161. 14. T. Ao, J. R. Asay, S. Chantrenne, M.R. Baer, and C.A. Hall, Rev. Sci. Instr. 79, 013903 (2008). 51
15. J. R. Asay, in Shock Compression of Condensed Matter – 1999 edited by M.D. Furnish, L. C. Chhabildas and R. S. Hixson (American Institute of Physics, New York, 2000) p. 261. 52
Chapter 4 X‐Cut Quartz Experimental Results and Analysis This chapter describes the results and analysis for the shockless and shock wave experiments performed on x‐cut quartz. A total of 4 shockless compression experiments were performed on 8 x‐cut quartz samples, and 6 shock wave experiments were performed using another 6 x‐cut quartz samples. Two different peak stress states were chosen for both the shockless and shock wave compression experiments: ~ 10 GPa and ~ 16 GPa. Sample characterization results at ambient conditions are displayed in Table 4.1, and are in good agreement with previously published measurements.1 The dynamic compression results are organized according to the type of experiment. The shockless compression results and analysis are presented in sections 4.1 and 4.2, respectively. The shock wave compression experimental results and analysis are in sections 4.3 and 4.4, respectively. A summary and a comparison between the two sets of measurements are presented section 4.5. 4.1 Shockless Compression Experiments The shockless compression experimental results are organized according to the peak stresses. 15 mm wide CPPG panels were used for the ~ 10 GPa experiments which give an expected elastic stress of 11.1 GPa. 11 mm wide panels were utilized to obtain an expected elastic stress of 16.1 GPa. These panels and their role in achieving the applied stress were discussed in Chapter 4. The velocity history measurements were made at the interface between the x‐cut quartz samples and [100] oriented LiF optical windows using a velocity interferometer (VISAR). 53 Table 4.1 Ambient measurements on x‐cut quartz crystals
Sample. No. (Exp. No.) Loading Condition ‐ Peak Stress Thickness
(mm) Density ρ0 (g/cm3) Sound Speed
Longitudinal (mm/µs) 1 (R8016) 2 (R8016) 3 (R8015) 4 (R8015) 5 (R9025) 6 (R9025) 7 (R9034) 8 (R9034) 9 (08‐047) 10 (08‐052) 11 (08‐045) 12 (09‐001) 13 (10‐603) 14 (10‐605) Average Std. Dev. (Ref. 1) Shockless‐10 Shockless‐10 Shockless‐10 Shockless‐10 Shockless‐16 Shockless‐16 Shockless‐16 Shockless‐16 Shock‐10 Shock‐10 Shock‐10 Shock‐10 Shock‐16 Shock‐16 All Samples All Samples 1.378
3.535
1.562
2.018
1.204
2.261
1.205
2.269
1.540
1.878
2.018
2.026
0.999
1.999
2.643
2.647
2.647
2.647
2.649
2.650
2.650
2.650
2.646
2.650
2.647
2.647
2.646
2.649
2.648
(0.002)
2.6485
5.81
5.80
5.78
5.80
5.77
5.85
5.77
5.76
5.79
5.82
5.80
5.80
5.79
5.79
5.80
(0.02)
5.7492
54 Sound Speed Shear 1 (mm/µs) 5.16 5.18 5.17 5.13 5.15 5.16 5.17 5.16 5.14 5.16 5.17 5.13 5.12 5.16 5.15 (0.02) 5.1140 Sound Speed Shear 2 (mm/µs) 3.31
3.33
3.30
3.30
3.30
3.30
3.32
3.30
3.30
3.31
3.32
3.31
3.32
3.32
3.31
(0.01)
3.2975
4.1.1 10 GPa Peak Stress Experiments X‐cut quartz/LiF Interface velocity histories for the ~ 10 GPa shockless compression experiments are shown in Fig. 4.1. The waves entered the x‐cut quartz samples at 0 µs on the time scale shown. The expected elastic interface profiles, plotted in Fig. 4.1, were calculated using a forward propagating characteristic calculation (Appendix A). The calculations utilized a previously measured 15 mm panel free surface velocity to estimate the particle velocity history in the aluminum panel. This in‐panel profile was then computationally propagated with the computer code across the panel/x‐cut quartz sample interface and then the x‐cut quartz/LiF interface using an aluminum model,2 the quartz elastic constants to fourth order,1,3‐4 and a [100] LiF model.5‐6 The peak in‐material stresses in these calculations were recorded and are listed later in Table 4.2 under ‘maximum elastic stress’. The measured interface velocity profiles in Fig. 4.1 are smooth up until the point where the records are truncated, with no indication of elastic to inelastic transitions throughout the smooth portion of the profiles; an elastic to inelastic transition would cause a disruption in the continuity of the velocity profile and a significant deviation from the expected elastic response. The slight deviation seen in samples 1 and 2 is a result of the difference between the Al panel input profile used for the expected elastic response calculations and the actual input in the experiment; this deviation is not evidence of an elastic‐inelastic transition. The features observed at velocities of around 0.01 mm/µs were due to the elastic precursor in the aluminum panel and are well understood.7‐8 The measured interface profiles were truncated because of the contrast loss in the velocity interferometer signals, observed for all four samples in the 10 GPa shockless measurements; a typical example of the contrast loss is shown in Figure 4.2. The interface 55 Table 4.2 Experimental Details Sample. No. (Exp. No.) Loading Type Panel Width (mm) 1 (R8016) 2 (R8016) 3 (R8015) 4 (R8015) 5 (R9025) 6 (R9025) 7 (R9034) 8 (R9034) 9 (08‐047) 10 (08‐052) 11 (08‐045) 12 (09‐001) 13 (10‐603) 14 (10‐605) Approximate Shockless Shockless Shockless Shockless Shockless Shockless Shockless Shockless Shock Shock Shock Shock Shock Shock Error 15 15 15 15 11 11 11 11 56 Impactor Projectile Thickness Velocity (mm/µs) (mm) 3.123 0.902 3.130 0.901 3.306 0.902 3.050 0.901 3.150 1.431 3.213 1.394 0.003 0.002 Theor. Max Elast. Stress (GPa) 11.1 11.1 11.1 11.1 16.1 16.1 16.1 16.1 10.0 10.0 10.0 10.0 16.4 16.0 0.1 Interface Velocity (mm/s)
1.0
0.9
Expected Elastic Response
0.8
0.7
Shockless Elastic Limits
0.6
Sample 1
(R8016)
0.5
0.4
0.3
0.2
Expected Shock Elastic Limits
Sample 3
(R8015)
Sample 2
(R8016)
Sample 4
(R8015)
0.1
0.0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Time (s)
Figure 4.1. X‐cut quartz/LiF interface profiles for shockless compression of x‐cut quartz to 11 GPa peak stress. The expected elastic response profiles are shown as dashed grey lines. The range of expected elastic limits from shock compression experiments4,10‐11 are indicated with dashed lines. The shockless elastic limits from the present experiments are also indicated. 57 1.6
1.0
1.4
Sample 3
(R8015)
0.8
1.2
0.7
1.0
0.6
0.8
0.5
0.4
0.6
0.3
0.4
0.2
Interferometer Contrast
Interface Velocity (mm/s)
0.9
0.2
0.1
0.0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0
1.0
Time (s)
Figure 4.2. X‐cut quartz/LiF interface velocity profile (Black line) and velocity interferometer contrast (Grey) from the shockless experiment on sample 3. The contrast drops abruptly at about 0.65 µs, indicating a deviation from an elastic response. 58 velocities at the time of contrast loss ranged from 0.63 to 0.69 mm/µs, near the peaks of the velocity profiles, and are attributed to a deviation from an elastic response. The contrast loss indicates that multiple velocity components are present in the probe region9 which would not be expected for elastic behavior. Furthermore, Lagrangian analyses of the profiles, discussed in section 4.2, demonstrate that the samples were loaded elastically up to these identifiable times. Therefore, the contrast losses in the shockless compression measurements indicate elastic‐
inelastic transitions. The range of expected elastic limits from previous shock compression experiments4,10‐11 are indicated with dashed lines in Figure 4.1.The measured interface velocity records reveal a significantly higher elastic limit under shockless compression than reported for shock compression of x‐cut quartz. The present results demonstrate that the elastic limit of x‐cut quartz increased significantly with decreasing loading rate (from shock wave to shockless compression); this is the most significant finding of this work. The signal above the elastic limit was not obtained due to contrast and return laser light intensity loss. Therefore, nothing was inferred about the behavior above the elastic limit from these experiments. 4.1.2 16 GPa Peak Stress Experiments The x‐cut quartz/LiF interface velocity profiles for the ~ 16 GPa shockless compression experiments are shown in Fig. 4.3. Expected elastic interface profiles, calculated using a previously measured 11 mm panel free surface velocity, are also shown. For 3 out of the 4 measured velocity histories, the initial portion of the loading profiles is smooth and continuous up to a velocity of about 0.68 mm/µs, with no indication of an elastic to inelastic transition. One velocity history (sample 6) shows deviation from elastic behavior at a lower velocity of about 0.5 59 1.0
Expected Elastic Profiles
Interface Velocity (mm/s)
0.9
0.8
0.7
11 GPa shockless elastic limits
Sample 5
Sample 6
0.6
Sample 7
0.5
0.4
Sample 8
Expected shock wave elastic limits
0.3
0.2
0.1
0.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Figure 4.3. X‐cut quartz/LiF interface velocity histories for the 16 GPa x‐cut quartz experiments. The expected elastic responses are indicated with dashed lines. The shockless elastic limits from the 11 GPa experiments are indicated with horizontal dashed lines. The range of elastic limits from shock compression experiments4,10‐11 are also indicated with horizontal dashed lines.
60 mm/µs; the results for sample 6 will be discussed later in this subsection. After reaching these velocities, large deviations from the expected elastic behavior are observed, indicating an elastic‐inelastic transition. The measured interface velocity at the elastic limit for 3 of the 4 samples is consistent with the 10 GPa peak stress experiments, indicating similar elastic limits at higher peak elastic stresses. However, unlike the 10 GPa experiments, the VISAR interferometer contrast is maintained for some time after the elastic limit, as shown in Fig. 4.4. During this extra recording time, a large drop (~ 0.25 mm/µs) in the interface velocity is observed indicating a significant stress relaxation. Interferometer contrast is eventually lost before a second wave structure is observed, as shown in Fig. 4.4. The shockless compression experiment on sample 6 showed an elastic limit that was significantly lower than all of the other shockless compression samples. Although it is not fully clear why this sample failed at a lower stress, two possible explanations are: 1. This was the only sample that was lapped and polished in‐house, all other samples had vendor polished surfaces; the surface finish may influence the elastic limit. 2. The compression waves for the 2.3 mm thick samples (samples 6 and 8) had very large slopes, so large that sample 6 may have been ‘shocking up’ or transitioning from shockless to shock wave compression. Since the expected shock wave elastic limits of x‐cut quartz are lower than the measured shockless elastic limits, a possible transition to shock compression could have lowered the elastic limit for sample 6. 4.2 Shockless Compression Analysis Under shockless compression, a continuum loading curve (stress‐density) can be determined from a single experiment for certain wave shapes.12 A simple case to illustrate this 61 1.6
1.0
1.4
0.8
Sample 8
1.2
0.7
1.0
0.6
0.8
0.5
0.4
0.6
0.3
0.4
0.2
Interferometer Contrast
Interface Velocity (mm/s)
0.9
0.2
0.1
0.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
0.9
Time (s)
Figure 4.4. X‐cut quartz/LiF interface velocity profile (Black line) and velocity interferometer contrast (Grey) from the shockless experiment on sample 6. Unlike the 11 GPa shockless measurements, the contrast is maintained for some time after the elastic limit. The contrast is still lost before the arrival of a second wave. 62 feature is to consider in‐material particle velocity histories obtained at two or more Lagrangian positions in an elastic solid subjected to shockless compression. In this situation, corresponding to simple wave propagation shown in Figure 4.5, the wavelet speed, Cu, can be determined at every particle velocity value through the relationship, Cu = h/Δtu, where h is the Lagrangian separation in space between two measurement planes and Δtu is the time separation between the two velocity histories at a given particle velocity. In this example, the wavelet speed is equivalent to the Lagrangian sound speed, CL,13‐15 and the CL – u curve can be integrated to give stress and density compression through the relationships /
, and ,13‐15 where u is particle velocity, is the longitudinal stress, is the ambient density, and is the specific volume. In the present experiments, the VISAR measurements were not in‐material measurements and were performed at the interface between the sample and an optical window. In this situation, the wave interactions due to impedance mismatch between the two materials must be accounted for and a different approach for determining the σ‐ρ loading path is required. To handle the impedance mismatch between the x‐cut quartz samples and the [100] LiF optical windows in analyzing the data, a computer code (Charice 1.1 written by J.P. Davis)16‐17 was used that utilizes an iterative approach developed previously.13‐15, 18‐19
An important requirement for this type of an analysis is that the sound speed and density are functions of stress only, independent of wave interactions or loading history. The technique is briefly outlined here. The computer code first uses the method of characteristics to propagate backward or convert the interface velocity profiles to in‐material profiles. The backward propagation uses a material model for the LiF window5‐6 and an assumed sample material model, , where denotes the zero order iteration of the stress‐particle 63 Particle Velocity, u
Lagrangian
Position 1
Lagrangian
Position 2
tu
Cu = h/tu
h = Pos. 2 - Pos. 1
Time, t
Figure 4.5. Illustration of Lagrangian analysis for in‐material particle velocity histories measured at two sample positions. The wavelet velocity, Cu, is calculated at every particle velocity, u, by dividing the separation distance by the transit time at that particle velocity value, Δtu. 64 velocity curve as shown in Fig. 4.6. Since the measurements were taken at two Lagrangian positions, the calculated in‐material profiles are then used to generate a new material model through the Lagrangian CL ‐ u relation described in the preceding paragraph. The new material model is then used to regenerate the in‐material profiles from the interface profiles using the method of characteristics. This procedure is repeated until satisfactory convergence is reached between the material model used to propagate backward the interface profile and the model obtained from the calculated in‐material profiles. This procedure typically takes 2‐3 iterations (see Fig. 4.6). With this code, a continuous wavelet velocity‐particle velocity7 (Cu – u) loading curve for the elastic portion of the shockless wave profiles is generated. Only the elastic portion of the profiles satisfies the assumptions required for this procedure.16‐17 The computer code also integrates the Cu curve to generate stress – particle velocity, and stress‐density compression curves along the experimental loading path. To check for self consistency, the in‐material profiles generated by Charice 1.1 were forward propagated to x‐cut quartz/LiF interface profiles using a forward characteristics code described in Appendix A. The forward propagated velocity history, shown in Fig. 4.7 was always a perfect match to the original interface profiles demonstrating that the Charice 1.1 program was working in analyzing the present data. For the above mentioned iterative Lagrangian analysis to give the correct sample loading curve, the window used must have a single valued, time independent, σ‐u relationship.14,16 Since the [100] LiF windows used in these experiments have some, likely small, time dependence, they do not rigorously satisfy this condition. Therefore, the x‐cut quartz loading curves reported in this thesis, while consistent with the chosen [100] LiF model,5,6 may not be rigorously correct. It is, however, expected that errors from this assumption are small 65 7.4
Wavelet Velocity (mm/s)
7.2
Iteration 3
7.0
Iteration 1
6.8
Iteration 2
6.6
6.4
6.2
6.0
Iteration 0, Cu(u) = C0
5.8
5.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Particle Velocity (mm/s)
Figure 4.6. The Cu(u) curves after consecutive iterations in the computer program Charice 1.1. Starting from an initial guess of Cu(u) = C0, satisfactory convergence of the calculated Cu(u) curve is generally reached within 3 iterations for the shockless compression measurements in x‐cut quartz. 66 1.0
Interface Velocity (mm/s)
0.9
Measured
Calculated
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
12.5
12.6
12.7
12.8
12.9
13.0
13.1
13.2
Time (s)
Figure 4.7. Measured interface profiles (thin black lines) and those determined by forward propagating the calculated in‐material profiles (thick black lines). The excellent match between the profiles demonstrates that the Charice 1.1 code16‐17 is working properly for the analysis presented in this work. 67 and likely influence the calculated loading curve only in the low stress regime. It will be demonstrated in Chapter 5, that the z‐cut quartz loading curves calculated from experimental data using [100] LiF windows are consistent with prior data using z‐cut quartz windows (elastic) justifying the validity of this procedure for [100] LiF windows. The stress‐particle velocity (σ‐u) curves determined from two of the CPPG experiments (sample pairs 1‐2 and 7‐8 in Table 5.1) using Charice calculations, are plotted in Fig. 4.8. (The loading curve for sample pair 3‐4 is not shown because the Lagrangian separation was small and as a result there is significant noise, section 4.2.2. The loading curve for sample pair 5‐6 only extends to ~ 8 GPa, and is not shown for clarity.) Also shown in Fig. 4.8 are the published HEL values from shock wave experiments,4,10‐11 the calculated elastic response using the elastic constants for quartz (to fourth order),1,3‐4 and the x‐cut quartz elastic Hugoniot from Ref. 11; the elastic Hugoniot from Graham11 has been extrapolated to higher stresses than reported in Ref. 11 so that it can be made visible in the figure. Also shown in Fig. 4.8 are the experimental wavelet velocity – particle velocity (Cu‐ u) curves from the CPPG experiments, the Cu‐u curve calculated from elastic constants to fourth order,1,3‐4 and the curve calculated from the elastic Hugoniot in Ref. 11 (extrapolated). Compared to the σ‐u curves, the Cu – u curves show more noise and a larger deviation from the behavior predicted from elastic constants. Integration of the Cu‐u curves to generate the σ‐u curves greatly smoothes the data, eliminating many of the details that were present in the Cu – u curve; this effect has also been recognized by other authors.20 The Cu – u curves are a more stringent measure of the material response, and will be used frequently throughout this work. The experimentally determined curves (σ‐u and Cu‐ u) in Fig. 4.8 are very similar to the calculated elastic curves from previous work, demonstrating that the samples remained elastic 68 14
R9034
R8016
Elastic Constants
Elastic Hugoniot (Ref. 11)
(Extrapolated)
Shock Wave Elastic Limit
(Ref. 4)
12
Stress (GPa)
10
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Particle Velocity (mm/s)
7.4
R9034
R8016
Elastic Constants
Elastic Hugoniot (Ref. 11)
(Extrapolated)
Wavelet Velocity (mm/s)
7.2
7.0
6.8
6.6
6.4
6.2
6.0
5.8
5.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Particle Velocity (mm/s)
Figure 4.8. The calculated stress‐particle velocity and wavelet speed‐particle velocity curves from two of the CPPG experiments (sample pairs 1‐2 (R8016) and 7‐8(R9034)). Also shown are the shock wave elastic limits from Ref. 4, the regressions from the elastic constants (to fourth order) and the elastic Hugoniot relation from Graham.11 The elastic Hugoniot from Graham has been extrapolated to higher stresses than reported in Ref. 11 so that it can be made visible in the stress‐particle velocity figure, the particle velocity derivative of this relation is shown in the wavelet speed‐particle velocity figure. 69 throughout the measureable loading region of the velocity profiles, and that the load path under shockless compression is consistent with the load path under shock wave compression. The shockless loading curves extend to approximately 11 GPa, well beyond the published HEL values under shock wave compression. Also, the elastic loading curves for the 10 and 16 GPa experiments are consistent. The experimental Cu – u curves from the two x‐cut quartz experiments, sample pairs 1‐2 and 7‐8, were averaged to obtain the experimental average curve shown in Fig. 4.9. This curve was used to obtain a new value for the fourth order elastic constant C1111 for quartz using methods outlined in Refs. 4 and 21. The new value of C1111 was found to be 14980 ± 460 GPa (fitting uncertainty) which is slightly lower than 15930 ± 3200 GPa that was reported by Fowles4 and substantially higher than the 7500 GPa value that was reported by Graham22.. 4.2.1 Shockless Elastic Limits The x‐cut quartz elastic limit stresses could not be determined directly from the Lagrangian analysis results given by the Charice 1.1 computer code calculations. For the present experiments, the computer code would only perform the Lagrangian analysis correctly if the x‐
cut quartz/LiF interface profiles, from both samples in an experimental pair, corresponded to completely elastic behavior from the quartz samples. Therefore, the interface profiles were truncated slightly below the samples’ elastic limits prior to the iterative characteristics calculation with the code. The in‐material particle velocity and stress at the elastic limit had to be determined separately from the calculation in the code. 70 7.4
Average Experimental Data
Wavelet Velocity (mm/s)
7.2
C1111 = 15930 GPa (Fowles)
7.0
C1111 = 14980 GPa (Best Fit)
6.8
6.6
6.4
6.2
6.0
5.8
5.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Particle Velocity (mm/s)
Figure 4.9. The average experimental wavelet velocity‐particle velocity curve (solid line) shown with the curve calculated using C1111 = 15930 GPa.4 A best fit to the experimental data yielded a value of C1111 = 14980 ± 460 GPa (dotted line). 71 To determine the in‐material particle velocity at the elastic limit, the experimental stress‐particle velocity, , loading curve, calculated by the Charice 1.1 code, was extrapolated to slightly higher stresses, using a fourth order polynomial, and the interface velocity at the elastic limit was propagating backward using the method of characteristics,13‐14,18‐
19
the extrapolated loading curve, and the [100] LiF model.5‐6 The method of characteristics gives the following solution for the in‐material particle velocity at the elastic limit in terms of the interface velocity.13‐14,18‐19 /2
(4.1)
In the above equation, is the particle velocity at the elastic limit, corresponding interface velocity, and the interface velocity, and on stress. The is the is the interfacial stress calculated from the LiF model is the x‐cut quartz material state function16 which is dependent function is the inverse of the loading curve (extrapolated) for x‐cut quartz.16 The LiF model used was taken from a shock compression study in Ref. 5, the relation between the interfacial stress and velocity is5 13.59
3.573
(4.2)
The interface stress in Eq. 4.2 is in units of GPa and the interface velocity is in units of mm/µs. The in‐material elastic limit stress is then determined from the extrapolated loading curve. (4.3)
72 The measured elastic limit interface velocities, the calculated shockless elastic limit particle velocities, and the elastic limit stresses for each x‐cut quartz sample are listed in Table 4.3. The values ranged from 8.8 to 11.1 GPa and are 60 – 100 percent higher than the published shock compression elastic limits for x‐cut quartz.4, 10‐11 4.2.2 Loading Rates The maximum loading rates in the shockless compression experiments were calculated by converting the elastic portion of the interface velocity profiles to in‐material strain‐time profiles using the x‐cut quartz elastic constants1 ,3‐4 and the [100] LiF Hugoniot.5,6 The maximum slopes in the strain‐time profiles were recorded and are listed in Table 4.3 under the ‘Maximum Loading Rate’ column. 4.2.3 Error Analysis for Shockless Compression The shockless compression error analysis was first presented in Ref. 7, and is reproduced here with little modification. There were several sources of uncertainty in the experimental Cu – u results: particle velocity measurements, input pulse symmetry of the opposing drive panels, VISAR timing, and Lagrangian positions. All of the uncertainties, except positions, manifest themselves as a time uncertainty in the Cu calculations. The VISAR timing uncertainty contributes to both the particle velocity and directly to the time error, though these are assumed independent for the analysis. The time uncertainty, δt, is directly linked to the particle velocity uncertainty, δu, through the u‐t slope in the wave profile, δt = δu/slope. Assuming that each source of error is random and uncorrelated, the total uncertainty in the time difference between the profiles is 73 Table 4.3 Results of x‐cut quartz experiments Max Theor. Max Elast. Limit Elast. Limit Elast. Limit Int. Vel. Loading Part. Vel. Elast. Stress Stress Ratea (
(GPa) (GPa) ) (
) (s‐1) (mm/µs) (mm/µs) 1 (R8016) Shockless 11.1 0.654 0.642 10.7 7.3 x 105 2 (R8016) Shockless 11.1 0.663 0.651 10.8 1.7 x 106 3 (R8015) Shockless 11.1 0.683 0.668 11.1 9.4 x 105 4 (R8015) Shockless 11.1 0.627 0.615 10.1 8.3 x 105 5 (R9025) Shockless 16.1 0.678 0.666 11.1 1.5 x 106 6 (R9025) Shockless 16.1 0.547 0.536 8.7 4.5 x 106 7 (R9034) Shockless 16.1 0.670 0.658 11.0 1.5 x 106
8 (R9034) Shockless 16.1 0.686 0.674 11.2 4.8 x 106
9 (08‐047) Shock 10.0 0.352 0.362 5.7 4.4 x 107 10 (08‐052) Shock 10.0 0.318 0.328 5.1 4.1 x 107
11 (08‐045) Shock 10.0 0.394 0.405 6.4 4.9 x 107 12 (09‐001) Shock 10.0 0.333 0.342 5.4 4.2 x 107 13 (10‐603) Shock 16.4 0.462 0.452 7.2 5.0 x 107
14 (10‐605) Shock 16.0 0.473 0.463 7.4 5.0 x 107 Approximate Error 0.1 0.004 0.005 0.1 4% b
a
The max loading rates for the shock compression measurements represent lower limits since the measured rise time is limited by the VISAR interferometer delay time. b
The shock compression loading rate error is the percent error of the measured value which is a lower limit loading rate. Sample. No. (Exp. No.) Loading Type 74 (4.4)
∆
where the subscripts refer to the first and second Lagrangian positions; δtv is the VISAR timing uncertainty and δts is the input pulse time asymmetry. Since Cu = h/Δt,13,14 the wavelet velocity uncertainty is (4.5)
∆
∆
∆
where δ(h) is the uncertainty in the Lagrangian separation distance. In the present experiments, only one Lagrangian position is measured per drive panel side. Thus, the input pulse symmetry is important. Temporal differences between the opposing points in panel free surface measurements served as a metric for input pulse symmetry, since the time difference would be zero ideally. Fig. 4.10 shows the measured time difference plots between panel free surface velocity profiles for the 15 mm, and 11 mm wide panels. The input pulse time asymmetries were difficult to quantify since panel surface velocity measurements contain an aggregate of uncertainties. The largest deviations from zero, labeled A and B in Fig. 4.10, are caused by velocity errors that are magnified by the relatively small slope in the u‐t profiles near the elastic limit of 1100 Al, and near the maximum of the profiles where the u‐t slope again becomes low. VISAR induced oscillations,7,23 also a particle velocity error, are present in the time difference plots and are labeled C in Fig. 4.10. From the measurements, it 75 Time Difference, t(ufree surface), (mm/s)
0.004
0.003
0.002
A
15 mm Panel
0.001
C
B
0.000
-0.001
-0.002
-0.003
-0.004
0.0
11 mm Panel
0.5
1.0
1.5
2.0
Panel Free Surface Velocity (mm/s)
Figure 4.10. Time difference between opposing point panel free surface measurements as a function of panel free surface velocities. The time difference for the 15 mm panel is shown in black, and for the 11 mm panel in grey. The large time differences at low velocities (A) and at high velocities (B) are a result of the relatively low slope in the u‐t profiles. VISAR induced oscillations7,23 are labeled C. From these plots, it was estimated that less than 1 ns of timing uncertainty is caused by panel asymmetries. 76 was estimated that < 1 ns of uncertainty was caused by actual drive pulse asymmetries for both the 11 mm and 15 mm panels. The rest was attributed to velocity measurement error. The source of the input pulse time differences are unknown, but could be caused by current asymmetry, or slight differences in panel composition. Typical Cu measurement uncertainties are plotted along with the corresponding Cu ‐ u curves in Fig. 4.11 for experiments on 15 mm and 11 mm wide panels. The estimated particle velocity uncertainties were 0.6 m/s for the present measurements. These values dominated the Cu uncertainties in regions of low u‐t slope near the panel elastic limit and at the foot and peak of the wave profiles. Throughout the rest of the wave profiles, uncertainties were dominated by the Lagrangian position measurements, δ(h) ~ 4µm, and the input pulse asymmetries (1 ns). 4.3 Shock Wave Compression Experiments Similar to the shockless compression experiments, the shock wave compression experiments were performed at two different peak stresses, 10 GPa and 16 GPa (maximum lastic stresses or elastic impact stresses). Also, the sample thicknesses were comparable to the shockless experiments. OFHC copper impactors, launched at a velocity of 0.90 km/s were used to produce the 10 GPa drive stress, and the same impactors launched at a velocity of 1.4 km/s were used to produce the 16 GPa drive stress. 4.3.1 10 GPa Peak Stress Experiments X‐cut quartz/LiF interface velocity histories for the ~ 10 GPa shock wave compression experiments are shown in Fig. 4.12. A two wave structure, consistent with an elastic to inelastic 77 Wavelet Velocity (mm/s)
7.2
0.4
7.0
6.8
0.3
6.6
6.4
0.2
6.2
6.0
0.1
5.8
5.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.8
Wavelet Velocity Uncertainty (mm/s)
0.5
7.4
Particle Velocity (mm/s)
0.5
7.4
Wavelet Velocity (mm/s)
7.2
0.4
7.0
6.8
0.3
6.6
6.4
0.2
6.2
6.0
0.1
5.8
5.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.8
Wavelet Velocity Uncertainty (mm/s)
Particle Velocity (mm/s)
Figure 4.11. Typical experimental wavelet velocity uncertainties as a function of particle velocities for experiments using 15 mm panels (upper graph) and 11 mm panels (lower graph).
78 0.9
0.8
0.7
Shockless Elastic Limits
0.6
0.4
Interface Velocity (mm/s)
Expected Shock Elastic Limits
0.3
0.2
0.1
0.0
-0.05
Samp. 9
0.5
1.0
0.9
0.8
0.7
Shockless Elastic Limits
0.6
0.5
0.4
0.2
0.00
0.05
0.10
0.15
0.0
-0.05
0.20
Time (s)
1.0
0.9
0.9
0.8
0.7
Shockless Elastic Limits
0.6
0.4
0.3
Samp. 10
Samp. 11
Expected Shock Elastic Limits
0.00
0.05
0.10
0.15
0.20
Time (s)
1.0
0.5
Expected Shock Elastic Limits
0.3
0.1
Interface Velocity (mm/s)
Interface Velocity (mm/s)
1.0
Interface Velocity (mm/s)
0.8
0.7
Shockless Elastic Limits
0.6
0.5
0.4
Samp. 12
Expected Shock Elastic Limits
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.05
0.00
0.05
0.10
0.15
0.20
-0.05
0.00
0.05
0.10
0.15
0.20
Time (s)
Time (s)
Figure 4.12. X‐cut quartz/LiF interface velocity histories from the 10 GPa shock wave experiments. The first wave amplitudes are consistent with the expected range of shock elastic limits from Refs. 4, 10‐11. The measured range of shockless elastic limits is also shown. 79 transition, can be seen in only one of the experiments. The remaining experiments showed a distinct elastic wave, but the return laser beam intensity and/or interferometer contrast were lost before arrival of the inelastic wave. The typical interferometer contrast loss for one of these experiments is shown in Fig. 4.13. The first wave or the elastic precursor had an amplitude which varied between 0.33 and 0.41 mm/µs for all four measurements, consistent with the expected elastics limit from prior investigations.4,10‐11 Such variations are typical of brittle solids.4,24 Velocity fluctuations are apparent for times after the elastic limit, and together with the poor interferometer contrast (Fig. 4.13) indicate that a distribution of velocity values were present in the probe area.9 Similar features have been observed by other authors in wave profiles on brittle solids and are generally attributed to heterogeneous deformation processes.25 4.3.2 16 GPa Peak Stress Experiments X‐cut quartz/LiF interface velocity histories for the ~ 16 GPa shock wave compression experiments are shown in Fig. 4.14. A two wave structure, consistent with an elastic to inelastic transition, can be seen in both experiments. Unlike the 10 GPa peak stress shock experiments, the VISAR interferometer contrast was lost only partially during the arrival of the second wave but recovered to enable the acquisition of the peak velocities. The elastic wave amplitudes were consistent between the two experiments and occurred at an interface velocity near 0.47 mm/µs, higher than in the 10 GPa experiments. Velocity fluctuations are apparent for times after the elastic limit indicating velocity distributions caused by heterogeneous deformation. Because the interferometer contrast was not completely lost, it appears that the samples shock compressed to 16 GPa are more “homogeneous” above the elastic limit than samples 80 1.6
0.5
0.4
Expected Shock Elastic Limits
1.2
1.0
0.3
0.8
0.2
0.6
0.4
0.1
Interferometer Contrast
Interface Velocity (mm/s)
1.4
0.2
0.0
-0.05
0.00
0.05
0.10
0.15
0.0
0.20
Time (s)
Figure 4.13. Laser interferometer contrast (grey curve) is shown along with the measured interface velocity for one of the 10 GPa shock experiments on x‐cut quartz (sample 12). The interferometer contrast is generally lost in these experiments before arrival of the second wave, truncating the velocity record. This interface velocity curve is shown on an expanded scale compared to Fig. 4.12 for clarity. 81 1.0
Interface Velocity (mm/s)
0.9
0.8
Sample 14
Sample 13
0.7
Shockless Elastic Limits
0.6
0.5
0.4
Expected Shock Elastic Limits
0.3
0.2
0.1
0.0
-0.05
0.00
0.05
0.10
0.15
0.20
Time (s)
Figure 4.14. X‐cut quartz/LiF interface velocity profiles for the 16 GPa shock experiments. Both elastic wave amplitudes from previous shock wave studies,4,10‐11 and the measured shockless elastic limits are indicated with dashed lines. 82 compressed to 10 GPa. This result is consistent with the ideas discussed by Grady26 and the experimental findings by Brannon et. al.27 where deformation features (inelastic compression) in shocked quartz become more closely spaced with increasing drive stress which, in turn, increases the homogeneity. 4.4 Shock Wave Compression Analysis The theoretical maximum elastic stresses for the shock compression experiments were calculated to compare the peak elastic stresses in the shock experiments with the shockless experiments. The theoretical maximum elastic stresses are the peak stresses that would occur if the x‐cut quartz samples responded purely elastically. The values were calculated using a x‐cut quartz elastic stress‐particle velocity (
) relation determined from the quartz elastic constants (to fourth order)1,3‐4, the copper impactor Hugoniot,28 and the projectile velocity (Table 4.2). The x‐cut quartz elastic relation was generated using a procedure outlined by Jones and Gupta21 for elastic uniaxial strain compression. Using the procedure for the present work, the elastic constants, C11 = 86.8 GPa1, C111 = ‐210 GPa,3 and C1111 = 15930 GPa,4 and the ambient density,1 ρ0 = 2.6485 g/cm3, were used to generate the following relation, 15.16
0.387
7.077
5.737
1.675
(4.6)
The copper impactor Hugoniot is the left going copper Hugoniot centered at the projectile velocity. The copper Hugoniot is28 35.2
13.51
83 (4.7)
Therefore, the impactor Hugoniot is 35.2
13.51
(4.8)
The longitudinal stresses, , in the above relations have units of GPa and the particle velocities, u, have units of mm/µs. The projectile velocities are listed in Table 4.2. The theoretical maximum elastic stresses were determined by finding the intersection of the curves given by Eq. 4.6 and 4.8. The theoretical maximum elastic stresses for the shock compression experiments are listed in Table 4.2 and are consistent with the maximum stresses in the shockless compression experiments. 4.4.1 Shock Compression Elastic Limits The elastic limit values were obtained for each sample from the elastic precursor amplitudes, measured at the x‐cut quartz/LiF window interfaces (
), and impedance matching.29 The impedance matching utilized the [100] LiF Hugoniot5,6 (Eq. 4.2) and the elastic relation for x‐cut quartz derived from the elastic constants (Eq. 4.6). The impedance matching procedure is illustrated using the stress‐particle velocity diagram shown in Fig. 4.15. First, the interface particle velocity for the elastic precursor, interface stress, the , was used to determine the , by using the [100] LiF window Hugoniot. Second, the particle velocity on relation for x‐cut quartz that corresponds to the interface stress was found, . The average of the these quantities was the in‐material particle velocity at the elastic limit, /2. Third, the stress at the elastic limit was found using the in‐
material particle velocity, , and the relation for x‐cut quartz; 84 . X-Cut Quartz Curve
from Elastic Constants
limit=(ulimit)
[100] LiF Hugoniot
Stress
limit
in
limit
limit
ulimit=[uint + u(int )]/2
limit
u(int )
limit
uint
Particle Velocity
Figure 4.15 Stress‐particle velocity diagram that illustrates the impedance matching procedure used to determine the x‐cut quartz shock compression elastic limits from the x‐cut quartz/LiF interface velocity profiles. The x‐cut quartz elastic stress – particle velocity relation, determined from the elastic constants, and the LiF [100] Hugoniot curve are shown as solid lines. The curves are shown on an expanded scale so that the separation between the x‐cut quartz and LiF loading curves could be observed easier. The various terms in the diagram are described in the text. 85 compression measurements, C1111 = 14980 GPa, instead of the constant found by Fowles,4 decreased the calculated values by less than 0.4%. The calculated shock wave elastic limits are listed in Table 4.3 and ranged from 5.1 to 6.4 GPa (5.7 ± 0.6 GPa) for the 10 GPa peak stress experiments and were about 7.2 GPa for the 16 GPa peak stress experiments. The elastic limits from the 10 GPa peak stress experiments were consistent with the range of values reported in Refs. 1,3‐4 but the 16 GPa measurements gave unexpectedly higher elastic limits. The increase in elastic limit with drive stress for x‐cut quartz was unexpected as it was not observed by previous authors.1,3‐4 However, the previous authors only examined significantly thicker samples so the current results could be a consequence of precursor decay. As indicated earlier, the scatter observed in the HEL values is typical of brittle solids. The estimated errors for the shock wave elastic limits (Table 4.3) are propagated from the uncertainty in determination of the elastic wave amplitudes in the interface velocity histories. These errors do not account for uncertainties in the material models used for compressed x‐cut quartz and [100] LiF. The elastic shock velocity was measured in one 10 GPa experiment (sample 11) and in one 16 GPa experiment (sample 13), using impact surface VISAR probes to obtain impact fiducial times. The elastic shock wave speeds were measured to be D = 6.032 ± 0.02 mm/µs for the 10 GPa experiment, and Delastic = 6.064 ± 0.04 mm/µs for the 16 GPa experiment. These values are consistent with those predicted from elastic constants:1,3‐4 Delastic = 5.96 mm/µs for the 10 GPa experiment and Delastic = 6.030 mm/µs for the 16 GPa measurement. 86 4.4.2 Peak State Peak state information was determined from the wave profiles when the second wave was observed, specifically samples 9, 13, and 14. In these experiments, the inelastic shock wave speed was determined from the time separation between the midpoints of the first and second waves and an assumed elastic wave speed calculated using the elastic constants; = 1/(1/Delastic + Δt/h), where the superscript L denotes a Lagrangian speed, Δt is the time separation between the two waves, and h is the sample thickness. Since the second wave arrival was obscured due to poor velocity interferometer contrast, the arrival time of the second wave has a large uncertainty based on the extremes of possible times (max and min recorded arrival times). Thus, the inelastic wave speeds, shown in Table 4.4, have relatively large uncertainties. The calculated inelastic wave speeds are used in the jump conditions to relate the elastic limit and peak compression states. The momentum and mass Rankine‐Hugoniot jump conditions that relate these two states are:29 (4.9)
(4.10)
For the above expressions, the lab frame wave velocity in Ref. 29 was replaced with the Lagrangian inelastic shock wave velocity (see Appendix B). 87 Table 4.4 Peak State Results for Shock Compressed X‐Cut Quartz Sample. No. (Shot No.) 9 (08‐047) 13 (10‐603) 14 (10‐605) Max. Elastic Stress (GPa) 10.0 16.4 16.0 Inelastic Wave Speeda (mm/µs) 4.27 ± 0.35 5.20 ± 0.16 5.16 ± 0.23 Peak Particle Velocity (mm/µs) 0.67 ± 0.01 1.05 ± 0.01 1.02 ± 0.01 a
Listed speeds are Lagrangian inelastic wave speeds. 88 Peak Stress (GPa) 9.0 ± 0.2 15.5 ± 0.2 15.0 ± 0.2 Peak Volume Compression (V/V0) 0.868 ± 0.007 0.811 ± 0.004 0.815 ± 0.006 A stress‐particle velocity diagram, Fig. 4.16, shows the elastic limit and peak state for sample 13 (chosen arbitrarily) to illustrate the analysis. Since the x‐cut quartz samples were impacted by a copper impactor, the peak state must lie on the copper impactor Hugoniot, Eq. 4.8, as shown in the figure. The slope of the line in σ‐u space that connects the elastic limit and peak state is given by Eq. 4.9. Therefore, the peak state must fall along the line given by, (4.11) The peak state in each experiment is found graphically by intersecting the above line with the impactor Hugoniot. The resulting peak stress and particle velocities are listed in Table 4.3. Also listed is the material density in the peak state obtained from in Eq. 4.10. The calculated peak state stress‐volume compression data from this work are plotted in Fig. 4.17 together with other single crystal quartz data from Wackerle10 and Fowles.4 The present data are consistent with the previous quartz results despite using different measuring techniques (both Wackerle and Fowles utilized explosive drives and made free surface measurements). The key assumptions used to obtain peak state information from the shock wave data are listed below: 1. The interaction of the inelastic wave with the reflection of the elastic wave from the [100] LiF window does not alter the inelastic wave speed. 2. The inelastic wave is a steady wave. 89 Peak State
L
in
e
D
0
e
op
10
Sl
Stress (GPa)
=

la
st
ic
15
Elastic Limit
5
Copper Impactor Hugoniot
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Particle Velocity (mm/s)
Figure 4.16 Stress‐particle velocity diagram showing the elastic limit and peak stress states for sample 13. The figure illustrates the peak state calculation described in the text. 90 30
Present X-Quartz Inelastic
Wackerle Quartz Inelastic
Fowles Quartz Inelastic
Elastic Constants (to Fourth Order)
25
Stress (GPa)
20
15
10
5
0
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
Volume Compression (V/V0)
Figure 4.17 Peak state (inelastic) stress‐volume compressions from present shock wave experiments (solid squares). The peak state data from Fowles4 (open circles) and Wackerle10 (open squares) is also shown from experiments on multiple orientations of crystal quartz. The curve calculated from elastic constants is also shown. 91 It is difficult to quantify the possible errors due to the above assumptions, which are quite common in shock wave data analysis. It is hoped that the relatively large error bars used in the analysis will encompass possible values of the true peak state. 4.4.3 Loading Rates Determination of the loading rate to the elastic limit in shock wave experiments was limited by the VISAR interferometer delay time τ, as this defines the minimum observed rise time of a step wave.9 Thus, only the lower limit to the loading rates can be quantified. The lower limit is taken as the strain at the elastic limit divided by the rise time of the wave to the elastic limit (always measured to be τ). The lower limit loading rates are listed in Table 4.3 and ranged between around 107 – 108 s‐1. 4.5 X‐Cut Quartz Experimental Findings Fig. 4.18 illustrates a key result from the x‐cut quartz experiments. In this figure, velocity profiles from one shockless compression experiment (sample 3) and from one shock wave compression experiment (sample 9) are plotted to show the significant difference in the elastic limits for these two different loading histories. Both experiments were performed under uniaxial strain loading, used comparable sample thicknesses, and were performed to the same peak stress. Yet, the shockless elastic limit is much higher than the shock wave elastic limit. The elastic limit as a function of loading rate, for every sample, is plotted in Fig. 4.19 to illustrate the role of loading rate on the elastic limit. The x‐cut quartz results clearly show a significant increase in the elastic limit, from ~ 6 GPa to 10.5 GPa, despite a loading rate decrease 92 1.0
Interface Velocity (mm/s)
0.9
0.8
0.7
Shockless Elastic Limits
0.6
0.5
Shock Elastic Limits
0.4
0.3
0.2
0.1
0.0
-0.1
0.0
0.1
0.2
0.3
Time (s)
Figure 4.18. Typical shockless (sample 3) and shock wave (sample 9) profiles for x‐cut quartz are shown together. This figure illustrates the large difference in observed elastic limits for the two different loading histories, shockless compression and shock wave uniaxial strain compressions. The range in measured shockless elastic limits is indicated with dashed lines as is the range of measured shock wave compression elastic limits. 93 12
Elastic Limit (GPa)
10
8
Shockless
Compression
6
Shock Wave
Compression
4
2
0
6
7
10
8
10
10
-1
Loading Rate (s )
Figure 4.19. The measured elastic limits of x‐cut quartz as a function of applied loading rate. The arrow for the shock wave compression measurements indicates that the measured loading rates represent lower limits for these experiments and the actual loading rate is likely higher. 94 of more than an order of magnitude. This finding, a significant increase in the elastic limit due to a large decrease in the loading rate, directly contradicts previous conclusions regarding the loading rate dependence of the elastic limit discussed in Chapter 2. Because both sets of measurements reported here were carried out for uniaxial strain loading (same load path), effects of loading rate variations were directly evaluated. In contrast, previous studies on brittle solids, discussed in Chapter 2, compared dynamic uniaxial stress measurements to uniaxial strain measurements; the effect of loading rate could not be separated from the effect of the loading path. The present findings suggest unique loading rate dependence for the elastic limit of x‐cut quartz. A physical picture and a phenomenological model to explain these findings will be discussed in Chapter 6. An elastic stress‐particle velocity loading curve to ~ 10 GPa for x‐cut quartz was determined form the present shockless compression experiments. This curve was in good agreement with the predicted curve from the elastic constants to fourth order,1,3‐4 even though the fourth order constant was determined previously from shock compression experiments at elastic stresses below 6 GPa.4 A new value for the fourth order elastic constant was determined from the present shockless compression data to be C1111= 14980 ± 460 GPa, which is within the experimental error of the previous value of 15930 ± 3200 GPa reported by Fowles.4 The shockless compression elastic limits did not depend significantly on the peak elastic stress or the sample thickness. The shock wave elastic limits, from the present work, for samples shock compressed to ~ 10 GPa were between 5.1 GPa and 6.4 GPa which is consistent with the values reported previously from a wide range of impact stresses and samples thicknesses greater than 4.6 mm.4,10‐11 The shock wave elastic limits on samples shock compressed to ~ 16 GPa had slightly higher values of around 7.3 GPa on sample thicknesses of 1‐2 mm, suggesting 95 some peak stress dependence of the elastic limit. Since the previous work performed on much thicker samples and at similar peak stresses gave lower elastic limits than the present 16 GPa measurements,4 the present results also suggest that the shock wave elastic limit of x‐cut quartz is dependent on the sample thickness. The main findings of the current work are summarized below: 1. The elastic limit of x‐cut quartz crystals, subjected to uniaxial strain, is higher under shockless compression than under shock wave compression. 2. The elastic stress‐particle velocity loading curve for x‐cut quartz, determined to 10 GPa from the shockless compression measurements in the present work, was in good agreement with the predicted curve from the elastic constants determined previously. 3. The shockless compression elastic limits did not depend significantly on peak stress or sample thickness. The shock compression elastic limits of x‐cut quartz are somewhat dependent on the peak stress and sample thickness; increasing with an increase in peak stress and a decrease in sample thickness. 96 References for Chapter 4 1. H. J. McSkimmin, J. Appl. Phys. 34, 1271 (1962). 2. W. Mamun, J. M. Winey, and Y. M. Gupta, ISP Internal Report, Washington State University, (2006). 3. R.N. Thurston, H.J. McSkimin, and P. Andreatch, J. Appl. Phys. 37, 267 (1966). 4. R. Fowles, J. Geo. Phys. Res. 72, 5729 (1967). 5. W. J. Carter, High Temp. – High Press. 5, 313 (1973). 6. J. L. Wise and L. C. Chhabildas, Shock Waves in Condensed Matter‐1985, edited by Y.M. Gupta (Plenum, New York, 1986), p. 441. 7. T. Jaglinski, B.M. LaLone, C.J. Bakeman, and Y.M. Gupta, J. Appl. Phys. 105, 083528 (2009). 8. T. Ao, J. R. Asay, S. Chantrenne, M.R. Baer, and C.A. Hall, Rev. Sci. Instr. 79, 013903 (2008). 9. D.H. Dolan, Tech. Rep. Sand2006‐1950, Sandia National Laboratories, 2006. 10. J. Wackerle, J. Appl. Phys. 33, 922 (1961). 11. R. A. Graham, J. Phys. Chem. Solids, 35, 355 (1975). 12. J. R. Asay, in Shock Compression of Condensed Matter – 1999 edited by M.D. Furnish, L. C. Chhabildas and R. S. Hixson (American Institute of Physics, New York, 2000) p. 261. 13. R. Fowles and R. F. Williams, J. Appl. Phys. 41, 360 (1970). 14. M. Cowperthwaite and R. F. Williams, J. Appl. Phys. 42, 456 (1971). 15. J. B. Aidun and Y. M. Gupta, J. Appl. Phys. 69, 6998 (1991). 97 16. J.‐P. Davis, CHARICE 1.0: An IDL Application for Characteristics‐Based Inverse Analysis of Isentropic Compression Experiments, Sandia National Laboratories Report SAND2007‐4984, September 2007. 17. J.‐P. Davis, CHARICE Version 1.1 Sandia National Laboratories Report SAND2008‐6035, October 2008. 18. L. Seaman, J. Appl. Phys. 45, 4303 (1974). 19. S. D. Rothman and J. Maw, J. Phys. IV France, 134, 745 (2006). 20. D. B. Hayes, C. A. Hall, J. R. Asay, and M. D. Knudson, J. Appl. Phys. 94, 2331 (2003). 21. S. C. Jones and Y. M. Gupta, J. Appl. Phys. 88, 5671 (2000). 22. R. A. Graham, Phys. Rev. B. 6, 4779 (1972). 23. W. H. Gourdin, Rev. Sci. Instrum. 60, 754 (1989). 24. O. V. Fat’yanov, R. L. Webb, and Y. M. Gupta, J. Appl. Phys. 97, 123529 (2005). 25. T. J. Vogler and J. P. Clayton, J. Mech. Phys. Solids. 56, 297 (2008). 26. D. E. Grady, J. Geophys. Res. 85, 913 (1980). 27. P. J. Brannon, C. Konrad, R. W. Morris, E. D. Jones, and J. R. Asay, J. Appl. Phys. 54, 6374 (1983). 28. M. Van Thiel, A. S. Kusubov, A. C. Mitchell, and V. W. Davis, Compendium of Shock Wave Data, Lawrence Radiation Laboratory, Livermore, CA. (1966). 29. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in High‐Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970). 98 Chapter 5 Z‐Cut Quartz Experimental Results and Analysis This chapter describes the results and analysis for the shockless and shock wave experiments performed on z‐cut quartz. A total of 3 shockless compression experiments were performed on 6 z‐cut quartz samples, and 7 shock wave experiments were performed using another 8 z‐cut quartz samples. Two different peak stress states were chosen for the shockless compression experiments: ~ 13 GPa and ~ 18 GPa, and peak stresses in the shock compression experiments ranged from 13 GPa to 37 GPa. Sample characterization results at ambient conditions are displayed in Table 5.1, and are in good agreement with previously published measurements.1 The dynamic compression results are organized according to the type of experiment. The shockless compression results and analysis are presented in sections 5.1 and 5.2, respectively. The shock wave compression experimental results and analysis are in sections 5.3 and 5.4, respectively. A summary and a comparison between the two sets of measurements are presented section 5.5. 5.1 Shockless Compression Experiments The shockless compression experimental results are categorized according to the peak stresses. 15 mm wide CPPG panels were used for the ~ 13 GPa experiments which gave an expected elastic stress of 13.1 GPa. 11 mm wide panels were utilized to obtain an expected elastic stress of 18.0 GPa. The role of panel width on the peak driving stress was discussed in Chapter 4. The velocity history measurements were made at the interface between the z‐cut quartz samples and [100] oriented LiF optical windows using a velocity interferometer (VISAR). 99 Table 5.1 Ambient measurements on z‐cut quartz crystals Sample. No. (Exp. No.) 1 (R9012) 2 (R9012) 3 (R9016) 4 (R9016) 5 (R10007) 6 (R10007) 7 (05‐045) 8 (06‐601) 9 (06‐605) 10 (10‐602) 11 (10‐602) 12 (05‐636) 13 (05‐644) 14 (05‐645) Average Std. Dev. (Ref. 1) Loading Condition‐
Peak Stress Shockless‐13 Shockless‐13 Shockless‐13 Shockless‐13 Shockless‐18 Shockless‐18 Shock‐12 Shock‐16 Shock‐18 Shock‐19 Shock‐19 Shock‐20 Shock‐27 Shock‐37 All Samples All Samples Thickness (mm) Density ρ0 (g/cm3) Sound Speed Longitudinal (mm/µs) 1.870 3.197 2.028 3.199 2.035 2.037 3.192 3.197 3.197 2.027 2.029 3.197 3.197 3.198 2.650 2.648 2.650 2.649 2.649 2.650 2.647 2.640 2.640 2.651 2.650 2.639 2.645 2.648 2.647 (0.004) 2.6485 6.32 6.34 6.37 6.34 6.34 6.34 6.34 6.39 6.39 6.34 6.34 6.43 6.41 6.43 6.37 (0.04) 6.3185 100 Sound Speed Shear (mm/µs) 4.72 4.75 4.69 4.68 4.70 4.75 NA NA NA 4.72 4.70 NA NA NA 4.71 (0.03) 4.6867 5.1.1 13 GPa Peak Stress Experiments Z‐cut quartz/LiF Interface velocity histories for the ~ 13 GPa shockless compression experiments are shown in Fig. 5.1. The waves entered the z‐cut quartz samples at 0 µs on the time scale shown. The expected elastic interface profiles are also plotted in Fig. 5.1. The calculations were performed in a similar manner to the x‐cut quartz elastic curves (Chapter 4), with the computer code in Appendix A, and utilized a previously measured 15 mm panel free surface velocity, an aluminum model,2 the quartz elastic constants to fourth order,1,3‐4 and a [100] LiF model.5‐6 The peak in‐material stresses in these calculations were recorded and are listed in Table 5.2 under ‘maximum elastic stress’. For 3 out of the 4 measured velocity histories (samples 2‐4), the loading profiles are in good agreement with the expected elastic profiles; they are smooth and continuous with no indication of an elastic to inelastic transition. Similar to the x‐cut quartz/LiF shockless interface profiles (Chapter 4), the feature in the profiles at velocities around 0.01 mm/µs is due to the elastic in the aluminum panel. The velocity histories for these samples were truncated at the expected arrival time of edge waves, not because of a loss of contrast. These samples appear to have been loaded completely elastically. One velocity history, sample 1, shows deviation from elastic behavior at a velocity of about 0.62 mm/µs; indicating an elastic‐inelastic transition for this sample. The velocity record for this sample was truncated due to a loss of contrast. This was the only sample that was lapped and polished in‐house to achieve the final thickness. Therefore, as with the x‐cut quartz experiments, the surface finish appears to have affected the value of the elastic limit. Also shown in Fig. 5.1 is the range of expected elastic limit interface velocities from measurements in previous shock compression studies.7‐8 As discussed in Chapter 2, the z‐cut 101 1.0
Interface Velocity (mm/s)
0.9
Expected Elastic Response
0.8
0.7
0.6
0.5
Expected Shock Elastic Limits
0.4
0.3
0.2
Sample 4
(R9016)
0.1
0.0
0.2
Sample 3
(R9016)
Sample1
(R9012)
0.3
0.4
0.5
0.6
Sample2
(R9012)
0.7
0.8
0.9
Time (s)
Figure 5.1 Z‐Quartz/LiF [100] interface velocity profiles for the shockless z‐cut quartz measurements to 13 GPa peak stress (solid black lines). The expected elastic response curves for each sample are also shown (dashed grey lines). The range of expected shock wave elastic limits from previous studies7‐8 is indicated with horizontal dashed lines. 102 Table 5.2 Experimental Details Sample. No. (Exp. No.) Loading Type Panel Width (mm) 1 (R9012) 2 (R9012) 3 (R9016) 4 (R9016) 5 (R10007) 6 (R10007) 7 (05‐045) 8 (06‐601) 9 (06‐605) 10 (10‐602) 11 (10‐602) 12 (05‐636) 13 (05‐644) 14 (05‐645) Approximate Shockless Shockless Shockless Shockless Shockless Shockless Shock Shock Shock Shock Shock Shock Shock Shock‐ Error 15 15 15 15 11 11 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ a Impactor Projectile Thickness Velocity (mm/µs) (mm) ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ 4.496 0.946 1.558 6.355a 4.494 1.289 3.139 1.342 3.139 1.342 2.010 1.395 4.415 1.797 1.935 2.30 0.003 0.002 Theor. Max Elast. Stress (GPa) 13.1 13.1 13.1 13.1 18.0 18.0 12.3 15.7 17.7 18.6 18.6 19.5 26.8 37.3 0.1 This experiment utilized a z‐cut quartz impactor, all others utilized a copper impactor. ‐ Quantity not relevant to the type of experiment. 103 quartz shock wave elastic limits are dependent on the peak stress. Therefore, the range shown in Fig. 5.1 encompasses previous elastic limit values measured only at peak shock stresses below ~ 13 GPa in Ref. 7‐8 and the “infinite‐run” (lower limit) estimate made by Wackerle.7 Similar to x‐cut quartz, the shockless profiles for z‐cut quartz (excluding sample 1) extend elastically to interface velocities beyond those expected from shock compression studies. Thus, the shockless elastic limit is larger than the shock wave elastic limit for z‐cut quartz also. However, since an elastic limit was not observed in the 13 GPa shockless experiments (excluding sample 1), the elastic limit was not determined. An experiment was performed to 18 GPa peak stress to determine the shockless elastic limit. 5.1.2 18 GPa Peak Stress Experiment The z‐cut quartz/[100] LiF interface velocity profiles for the ~ 18 GPa shockless compression experiment on samples 5 and 6 are shown in Fig. 5.2. The expected elastic interface profile for the samples, calculated using a previously measured 11 mm panel free surface velocity, is also shown. The measured profiles show deviation from elastic behavior at a similar value of around 0.78 mm/µs, indicating an elastic‐inelastic transition. The range of expected elastic limit interface velocities from measurements in previous shock compression studies is shown in Fig. 5.2 along with the 18 GPa shockless velocity profiles. The range shown encompasses previous elastic limit values measured only at peak shock stresses below ~ 18 GPa in Ref. 7‐8 and the “infinite‐run” elastic limit estimate made by Wackerle.7 From comparison of the measured profile with the expected range of shock wave elastic limits, it is clear that the shockless elastic limit is larger than the shock wave elastic limit. This finding for z‐cut quartz is qualitatively consistent with x‐cut quartz results (Chapter 4). 104 1.0
Interface Velocity (mm/s)
0.9
0.8
Sample 5
(R10007)
Expected Elastic
0.7
0.6
0.5
Sample 6
(R10007)
Expected Shock Elastic Limits
0.4
0.3
0.2
0.1
0.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Figure 5.2 Interface velocity profiles for shockless z‐cut quartz measurements to 18 GPa peak stress. The expected elastic response is indicated with a dashed line. The range of expected shock wave elastic limits is indicated with horizontal dashed lines. 105 The z‐cut quartz/LiF interface velocity in the 18 GPa experiment, corresponding to the elastic‐inelastic transition, was just above the peak velocity obtained for samples 2‐4 (0.72 mm/µs) which responded completely elastically in the 13 GPa experiments. This observation shows that, under shockless compression, z‐cut quartz can support elastic stresses just below its expected elastic limit for an extended time; the interface velocities for samples 2‐4 were within 90% of the elastic limit for over 80 ns without any indication of an inelastic response. In the 18 GPa experiment, velocity interferometer contrast (Fig. 5.3) was maintained for times after the elastic limit, indicating a somewhat homogeneous inelastic behavior.9 For x‐cut quartz, the contrast was maintained above the elastic limit only for the 16 GPa experiments and was still eventually lost. Beyond the peak, a drop of about 0.12 mm/µs was observed in the z‐cut quartz/LiF interface velocities, indicating significant stress relaxation after the elastic‐inelastic transition. After the stress relaxation, the profiles remain fairly flat with superimposed velocity fluctuations until the approximate arrival time of edge waves where the records were truncated. The origin of the relatively flat region is not clear. 5.2 Shockless Compression Analysis The elastic portion of the shockless measurements on z‐cut quartz were analyzed using methods that were identical to the x‐cut quartz experiments (Section 4.3), only the results of the analysis are presented here. The elastic σ – u and Cu – u loading curves for the 13 GPa experiments are shown in Fig. 5.4. Also shown are the published HEL values from shock wave experiments below ~ 13 GPa,8 the calculated elastic response using the elastic constants for quartz (to fourth order),1,3‐4 and the z‐cut quartz elastic Hugoniot from Ref. 4 (extrapolated). The elastic constants that were used for the elastic response curve were: C33 =105.75 GPa1, C333 = 106 1.6
1.0
Sample 5
(R10007)
0.8
1.4
1.2
0.7
1.0
0.6
0.8
0.5
0.4
0.6
0.3
0.4
0.2
0.2
0.1
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0.2
Interferometer Contrast
Interface Velocity (mm/s)
0.9
0.3
0.4
0.5
0.6
0.7
0.8
0.0
0.9
Time (s)
Figure 5.3 Z‐cut quartz/LiF interface velocity profile (black line) and velocity interferometer contrast (grey line) from the shockless compression experiment on sample 5. 107 16
Elastic Constants (to Fourth Order)
Hugoniot Relation from Ref. 4
14
Sample Pairs 3-4
Stress (GPa)
12
10
Sample Pairs 1-2
8
Shock HEL (Fowles)
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Particle Velocity (mm/s)
9.5
Elastic Constants (to Fourth Order)
Wavelet Velocity (mm/s)
9.0
8.5
Sample Pairs 1-2
8.0
Sample Pairs 3-4
7.5
7.0
6.5
6.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Particle Velocity (mm/s)
Figure 5.4 The calculated stress‐particle velocity and wavelet speed‐particle velocity curves from the shockless 13 GPa z‐cut quartz experiments. Also shown are elastic limits from Ref. Fowles8 (solid squares), the regression line calculated from the fourth order elastic constants,1,3‐4 and extrapolated Hugoniot relation from Ref. 4. 108 ‐815 GPa,3 and C3333 = 17481 GPa4. The experimental loading curves are in excellent agreement with the elastic response determined in previous studies. The experimental curves shown in Fig. 5.4 are also in good agreement with previous shockless compression measurements to 7 GPa;10 the results from Ref. 10 are not shown because the extremely good overlap with the current data makes them difficult to discern. The excellent agreement demonstrates that the elastic load path under shockless compression is consistent with the load path under shock wave compression. Furthermore, the results demonstrate that the time‐dependence of the LiF windows used in the experiments does not significantly affect the results of the Lagrangian analysis. Since both samples in the 18 GPa shockless experiment were the same thickness, experimental σ – u and Cu – u loading curves could not be determined from these measurements through the use of a Lagrangian analysis. 5.2.1 Shockless Elastic Limits The shockless elastic limits for samples 1,5 and 6 were determined by first identifying the peak interface velocity for each sample, and then using this value in the analysis presented in section 4.2.1. It is reasonable to assume that the peaks correspond to peak elastic velocities since they all occur near the value where the profiles first deviate from the expected elastic response (Figs. 5.1 and 5.2). For determination of the elastic limit for sample 1, the experimental stress‐particle velocity curve was used in the analysis (Eq. 4.1 and 4.3). This stress‐particle velocity curve was determined from the Lagrangian analysis of sample pair 1‐2. For determination of the elastic limit for samples 5 and 6, the following stress‐particle velocity curve from the elastic constants (to fourth order) was used.4 109 16.74
3.31
1.48
(5.1)
In the above z‐cut quartz σ – u relation, stress is in units of GPa and particle velocity is in units of mm/µs. The same [100] LiF stress‐particle velocity relation5‐6 that was used for the x‐cut quartz analysis (Eq. 4.2) was again used for the present analysis. Samples 2‐4 did not show evidence of failure so the shockless elastic limit for these samples is unknown, but they would have been greater than 13.1 GPa (maximum elastic stress). The shockless elastic limits for z‐cut quartz are listed in Table 5.3, and (excluding sample 1) are at least 15% larger than the previously published shock wave elastic limits (6.5 GPa to 12.4 GPa)7‐8 at comparable peak stresses. This finding is qualitatively consistent with the results on x‐cut quartz. 5.2.2 Loading Rates The maximum elastic loading rates for the shockless z‐cut quartz measurements were calculated in the same manner as the shockless x‐cut quartz loading rates (section 4.2.2). The maximum loading rates are listed in Table 5.3. 5.3 Shock Wave Compression Experiments Unlike the shock wave experiments for x‐cut quartz, performed at only two peak stresses, shock wave measurements for z‐cut quartz were obtained for the stress range 12 GPa to 37 GPa peak (elastic impact or maximum elastic) stress. This was done to examine the influence of impact stress on the elastic limit, a dependence observed previously in z‐cut 110 Table 5.3 Results of z‐cut quartz experiments Sample. No. (Exp. No.) Loading Type 1 (R9012) 2 (R9012) 3 (R9016) 4 (R9016) 5 (R10007) 6 (R10007) 7 (05‐045) 8 (06‐601)c 9 (06‐605) 10 (10‐602) 11 (10‐602) 12 (05‐636) 13 (05‐644) 14 (05‐645) Approximate Shockless Shockless Shockless Shockless Shockless Shockless Shock Shock Shock Shock Shock Shock Shock Shock‐ Error Theor. Max Elast. Limit Elast. Limit Elast. Limit Stress Elast. Stress Int. Vel. Part. Vel. (GPa) (GPa) (
) (
) (mm/µs) (mm/µs) 13.1 0.620 0.570 10.9a b
b
b
13.1 b
b
b 13.1 b
b
b 13.1 18.0 0.781 0.721 14.3 18.0 0.782 0.722 14.3 12.3 0.678 0.625 12.1 15.7 0.627 0.578 11.1 17.7 0.693 0.639 12.4 18.6 0.693 0.639 12.4 18.6 0.712 0.657 12.8 19.5 0.753 0.696 13.8 26.8 0.794 0.734 14.7 37.3 0.817 0.754 15.1 0.1 0.004 0.005 0.1 a
Sample failed at a much lower stress, possibly due to the surface preparation. b
Max Loading Rated (s‐1) 4 x 105 5 x 105 4 x 105 5 x 105 6 x 105 6 x 105 5 x 107
5 x 107
6 x 107 5 x 107
6 x 107 6 x 107 1 x 108 6 x 107 4% These samples remained completely elastic so the elastic limit was at least 13.1 GPa. This experiment utilized a z‐cut quartz impactor, all others utilized a copper impactor. d
The max loading rates for the shock compression measurements represents lower limits since the measured rise time is limited by the VISAR interferometer delay time. c
111 quartz.7‐8 The z‐cut quartz/LiF interface velocity profiles are shown in Fig. 5.5. An elastic precursor was observed in all of the measurements and the corresponding interface velocities ranged from 0.63 to 0.82 mm/µs. For experiments above 20 GPa peak elastic stress, an inelastic wave was observed. Below 20 GPa peak stress the inelastic wave was not observed. However, the samples were clearly not behaving elastically as evidenced by the velocity fluctuations after the elastic precursor wave, and by comparison with the expected elastic response, Table 5.3. (the elastic limit was always lower). The inelastic waves for the low stress experiments can be obscured by velocity fluctuations, not observed due to early velocity interferometer contrast loss (before arrival of the inelastic wave), or the interface velocity at the peak state could possibly be equal to or lower than the elastic precursor amplitude. In the third case, the inelastic wave would not have been observed at the quartz/LiF interface even if the velocity interferometer contrast was maintained and the velocity fluctuations were at a minimum. The possibility for the third situation is shown graphically in Fig. 5.6 using a σ‐u diagram. If the inelastic deformation processes are sufficiently slow, then the elastic precursor decays little with propagation and can have an in‐material stress amplitude very close to the elastic impact stress. This is state 1 in Fig. 5.6, the intersection of the elastic z‐cut quartz Hugoniot4 and the copper impactor Hugoniot.11 The measured interface velocity corresponding to state 1 is state 1’; the intersection of an elastic release from state 1 with the LiF [100] window Hugoniot.5 After the initial elastic response, the material relaxes (due to inelastic deformation) from state 1 (elastic) to some final state 2 (inelastic) which is also on the copper impactor Hugoniot due to momentum conservation (the copper impactor is in intimate contact with z‐cut quartz). The in‐material particle velocity in state 2 is higher than in state 1, but the 112 Interface Velocity (mm/s)
1.6
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Figure 5.5 Z‐cut quartz/LiF [100] interface velocity profiles from all of the shock wave experiments on z‐cut quartz. Peak stresses increase from sample 7 to sample 14. A clear two‐
wave structure is observed only for samples 13 and 14, and is somewhat observed for sample 12. 113 State 1
State 2
Co
pa
Im
er
pp ot
ni
go
cto
r
El
a
(E stic
xt
ra Z-Q
po
la uar
te
t
Li
d) z H
F
ug
H
ug [10
on
on 0 ]
io
t
io Wi
t
nd
ow
State 2'
Hu
Stress
State 1'
Particle Velocity
Figure 5.6 σ‐u plot for a shock wave experiment just above the elastic limit. The elastic z‐cut quartz target,4 LiF [100] window,5 and copper impactor11 Hugoniots are scaled correctly in this figure. The copper Hugoniot is centered and reflected about an arbitrary projectile velocity. The in‐material (quartz) are indicated as unprimed states. The corresponding LiF interface velocities are primed states. As described in the text it is possible for the final inelastic state to have a quartz/LiF interface velocity that is lower than the elastic wave amplitude velocity. 114 corresponding measured interface velocity (state 2’) could actually be lower than state 1’ as indicated in Fig. 5.6. The line connecting state 2 to 2’ is related to the modulus of the inelastically compressed quartz, and has a slope that is lower than the slope of the elastic z‐cut quartz Hugoniot. As can be seen in the Fig. 5.6, the situation where the inelastic interface velocity is equal to or lower than the elastic precursor interface velocity is possible when the elastic precursor value is near the elastic impact stress, and the impactor Hugoniot is steeper than the quartz target Hugoniot. Both of these conditions are true for the low stress z‐cut quartz measurements. Elastic shock wave speed was measured in one of the experiments, sample 7, to verify consistency with measured fourth order elastic constants.1,3‐4 The measurement was performed using multiple VISAR probes at the impact surface as well as the z‐quartz/LiF interface (Chapter 3). The measured value was found to be 7.30 mm/µs, in excellent agreement with the predicted value of 7.32 mm/µs from elastic constants.1,3‐4 5.4 Shock Wave Compression Analysis The theoretical maximum elastic stresses for the shock compression experiments were calculated to compare the peak elastic stresses in the shock experiments with the shockless experiments. The theoretical maximum elastic stresses are also used to examine the peak stress dependence of the shock wave elastic limits. The maximum elastic stress values were calculated using the same method as for the x‐cut quartz shock wave experiments (Section 4.4); by intersecting the impactor Hugoniot (Eq. 4.8) with the z‐cut quartz stress‐particle velocity relation (Eq. 5.1). In one experiment (sample 8), a z‐cut quartz impactor was used and the maximum elastic stress was calculated using the following relation, 115 16.74
3.31
2
2
1.48
2
(5.2)
In the above equation, , or one half the projectile velocity, is equal to the particle velocity for a symmetric impact.12 Therefore, was substituted for the particle velocity in Eq. 5.1. The calculated theoretical maximum elastic stresses for the shock compression experiments are listed in Table 5.2 and 5.3. 5.4.1 Shock Compression Elastic Limits The z‐cut quartz shock wave elastic limits were calculated from the interface velocity profiles using the method described for the x‐cut quartz measurements (Section 4.4.1), using an elastic stress‐particle velocity relation for z‐cut quartz4 (Eq. 5.1) and the [100] LiF Hugoniot5 (Eq. 4.2). The experimental elastic limit stresses are listed in Table 5.3. The elastic limits range from 12.1 GPa to 15.1 GPa, consistent with the values from previous investigations.7‐8 The elastic limit stresses are plotted as a function of maximum elastic stress (elastic impact stress) in Fig. 5.7. There is a clear trend of an increasing elastic limit with increasing peak stress, which is qualitatively consistent with results reported previously.7‐8 The maximum elastic stress dependence of the elastic limit is strongest for maximum stresses between 12 GPa and 20 GPa, then only increases slightly with a further increase in the maximum elastic stress. These results show that the elastic limit of z‐cut quartz, along the same load path (uniaxial strain) and comparable loading rates (shock wave compression), is dependent on the peak stress. Most samples in the current experiments had the same nominal thickness of 3.2 mm, so thickness dependence of the shock compression elastic limit was not determined. Samples 10 116 Shock Elastic Limit (GPa)
15
14
13
12
11
10
0
5
10
15
20
25
30
35
40
Theoretical Maximum Elastic Stress (GPa)
Figure 5.7 The measured shock compression elastic limit stresses as a function of the theoretical maximum elastic stress (elastic impact stress) for z‐cut quart. A general trend is an increase in the elastic limit with increasing impact stress. 117 and 11 were slightly thinner at 2 mm, but had similar elastic limit values (~ 13 GPa) as experiments on the 3.2 mm samples at comparable peak stress (~ 19 GPa). Likely, the small thickness difference between the 2 and 3.2 mm samples was not sufficient to show a thickness dependent elastic limit as observed by previous investigators.7‐8 5.4.2 Peak State As with the x‐cut quartz experiments, peak state information was determined from the z‐cut wave profiles when the second (inelastic) wave was observed; specifically samples 12, 13, and 14. The method used to obtain the peak state information was identical to the x‐cut quartz experiments (section 4.4.2): The inelastic wave speed was determined from the profiles, and jump conditions12 were used to determine the peak state from the inelastic wave speed, the elastic limit state, the copper impactor Hugoniot (Eq. 4.8), and the projectile velocity (Table 5.2). The calculated inelastic wave speeds, peak particle velocities, peak stresses, and peak volume compressions for samples 12‐14 are listed in Table 5.4. The calculated peak (inelastic) stress‐volume compression values are plotted in Fig. 5.8 along with peak state data from the Wackerle7 and Fowles8 articles, the peak state values from the x‐cut quartz experiments (Chapter 4), and the elastic loading curves for both x‐cut and z‐cut quartz calculated from the elastic constants. The Wackerle and Fowles values shown were for compressions along the x‐cut, y‐cut and z‐cut orientations of quartz. The peak state values for z‐
cut quartz, from the present measurements, are in good agreement with the previous peak stress measurements. This figure demonstrates that the peak states lie along a similar curve, irrespective of the initial quartz orientation, even though the elastic curves are highly dependent 118 Table 5.4 Peak state results for shock compressed z‐cut quartz Sample. No. (Shot No.) Peak Elastic Stress (GPa) 12 (05‐636) 13 (05‐644) 14 (05‐645) 19.5 26.8 37.3 Inelastic Wave Speeda (mm/µs) 3.46 ± 0.46 4.14 ± 0.25 5.18 ± 0.05 a
Peak Particle Velocity (mm/µs) 0.996 ± 0.005 1.297 ± 0.007 1.667 ± 0.004 The inelastic wave speed is a Lagrangian wave speed. 119 Peak Volume Compression (V/V0) 0.819 ± 0.012 0.770 ± 0.010 0.724 ± 0.002 Peak Stress (GPa) 16.2 ± 0.1 20.9 ± 0.3 27.7 ± 0.1 30
Present Z-Quartz Inelastic
Present X-Quartz Inelastic
Wackerle Quartz Inelastic
Longitudinal Stress (GPa)
25
Fowles Quartz Inelastic
Elastic Constants (to Fourth Order)
20
Z
15
X
10
5
0
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
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Volume Compression (V/V0)
Figure 5.8 Calculated peak (inelastic) stress‐volume compressions from present shock wave experiments on z‐cut quartz (filled circles). Also shown are the peak states from the x‐cut quartz measurements (filled squares), peak states from previous experiments on multiple orientations of quartz (open squares7 and open circles8), and the elastic loading curves for z‐cut and x‐cut quartz calculated from the elastic constants1,3‐4 (solid lines). 120 on the orientation. This finding is consistent with previous observations for shock compressed quartz crystals.7‐8 5.4.3 Loading Rates The loading rates for the shock compression measurements on z‐cut quartz were determined using the same method as for x‐cut quartz in section 4.4.3: The strain at the elastic limit divided by the rise time of the wave to the elastic limit (always measured to be the VISAR delay time). Since the rise time was limited by the VISAR delay time, the calculated loading rates were lower limits. The calculated loading rates are listed in Table 5.3 and are between 107 s‐1 and 108 s‐1. 5.5 Z‐Cut Quartz Experimental Findings Fig. 5.9 illustrates a key result from the z‐cut quartz experiments. In this figure, velocity profiles from one shockless compression experiment (sample 5) and from one shock wave compression experiment (sample 10) are plotted to show the significant difference in the elastic limits for these two different loading histories. Both experiments were performed under uniaxial strain loading (same load path), used comparable sample thicknesses, and were performed to the same peak stress. Yet, as with the x‐cut quartz results, the shockless elastic limit is higher than the shock wave elastic limit. The elastic limit as a function of loading rate for the samples is plotted in Fig. 5.10 to illustrate the role of loading rate on the elastic limit. Since the shock compression elastic limits were dependent on the maximum elastic stress, only results from experiments performed at 121 1.0
Shockless Elastic Limit
Interface Velocity (mm/s)
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0.4
Sample 10
Sample 5
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Figure 5.9 Shockless (Sample 5) and shock wave (Sample 10) profiles for z‐cut quartz are shown together. The z‐cut quartz/LiF interface velocities at the elastic limits of the samples are indicated with dashed lines. 122 15
14
Shockless
Compression
Elastic Limit (GPa)
13
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11
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Compression
10
Sample 1
9
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Figure 5.10 The measured elastic limits of z‐cut quartz as a function of the applied loading rate. The shock compression elastic limits shown were for experiments at peak stresses below ~ 18 GPa. The arrow for the shock wave compression measurements indicates that the measured loading rates represent lower limits for these experiments and the actual loading rate is likely higher. The anomalously low shockless elastic limit for sample 1 is indicated. 123 and below ~ 18 GPa maximum elastic stress (samples 7‐11) are shown in the Figure. The shockless elastic limit for sample 1 was anomalously low and, while indicated in the figure, is not considered in the present discussion. The z‐cut quartz results clearly show an increase in the elastic limit, from ~ 12 GPa to ~ 14 GPa, despite a loading rate decrease of about two orders of magnitude. This finding, an increase in the elastic limit due to a large decrease in the loading rate, is consistent with the x‐cut quartz results and directly contradicts previous conclusions regarding the loading rate dependence of the elastic limit discussed in Chapter 2. The present findings suggest unique loading rate dependence for the elastic limit of quartz crystals which will be discussed further in Chapter 6. An elastic stress‐particle velocity loading curve to ~ 13 GPa for z‐cut quartz was determined form the present shockless compression experiments. This curve was in excellent agreement with the predicted curve from the elastic constants to fourth order,1,3‐4 even though the fourth order constant was determined previously from shock compression experiments at elastic stresses below 6 GPa.4 The loading curve was also consistent with elastic states measured in shock compression experiments,7‐8 and previously measured shockless compressions.10 Therefore, aside from the elastic limit values, the elastic uniaxial strain states of z‐cut quartz are, not surprisingly, independent of the loading history. It was not determined in the present measurements, due to a limited amount of data, if the shockless compression elastic limits were dependent on the maximum elastic stress or the sample thickness. The shock wave elastic limits showed a significant dependence on the maximum elastic stress; elastic limits increased from 12 GPa to 15 GPa as the maximum elastic stresses increased from 12 GPa to 37 GPa. Since most shock compression samples had similar thicknesses, sample thickness dependence of the elastic limit was not observed. 124 The main findings of the current work are summarized below: 1. The elastic limit of z‐cut quartz crystals, subjected to uniaxial strain, is higher under shockless compression than under shock wave compression. This finding is qualitatively consistent with the x‐cut quartz results. 2. The elastic stress‐particle velocity loading curve for z‐cut quartz, determined to 13 GPa from the shockless compression measurements in the present work, was in good agreement with predicted curves from elastic state measurements. 3. It was not determined if the shockless compression elastic limits were dependent on peak stress or sample thickness. The shock compression elastic limits of z‐cut quartz were dependent on the peak stress; sample thickness dependence of the elastic limit was not determined. 125 References for Chapter 5 1. H. J. McSkimmin, J. Appl. Phys. 34, 1271 (1962). 2. W. Mamun, J. M. Winey, and Y. M. Gupta, ISP Internal Report, Washington State University, (2006). 3. R. N. Thurston, H.J. McSkimin, and P. Andreatch, J. Appl. Phys. 37, 267 (1966). 4. S. C. Jones and Y. M. Gupta, J. Appl. Phys. 88, 5671 (2000). 5. W. J. Carter, High Temp. – High Press. 5, 313 (1973). 6. J. L. Wise and L. C. Chhabildas, Shock Waves in Condensed Matter‐1985, edited by Y.M. Gupta (Plenum, New York, 1986), p. 441. 7. J. Wackerle, J. Appl. Phys. 33, 922 (1961). 8. R. Fowles, J. Geo. Phys. Res. 72, 5729 (1967). 9. D.H. Dolan, Tech. Rep. Sand2006‐1950, Sandia National Laboratories, 2006. 10. T. Jaglinski, B.M. LaLone, C.J. Bakeman, and Y.M. Gupta, J. Appl. Phys. 105, 083528 (2009). 11. M. Van Thiel, A. S. Kusubov, A. C. Mitchell, and V. W. Davis, Compendium of Shock Wave Data, Lawrence Radiation Laboratory, Livermore, CA. (1966). 12. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in High‐Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970). 126 Chapter 6 Phenomenological Model and Calculations for Quartz In this chapter, a phenomenological model is developed and used to explain the loading rate dependence of elastic limits observed in the quartz experiments. Section 6.1 presents the motivation for the phenomenological model. The model development is described in section 6.2. Constant loading rate calculations performed using the phenomenological model are discussed in section 6.3. The main findings of this chapter are summarized and discussed in section 6.4. 6.1. Need for the Phenomenological Model It was experimentally demonstrated in the previous two chapters that the elastic limits for both x‐cut and z‐cut quartz decrease with an increase in the loading rate. This finding contradicts previous experimental observations regarding the loading rate dependence of the elastic limit (chapter 2). To better understand the quartz behavior in the present work, a phenomenological model based on a reasonable physical picture is needed. Many phenomenological models for elastic‐inelastic deformation have been developed previously for solids subjected to dynamic uniaxial strain compressions, for examples see Ref. 1‐
7. These models have been used to describe the response of both ductile and brittle solids subjected to shock wave loading.1‐7 Similar models have successfully described the shockless response of ductile solids.8,9 For the current work, the interest is in understanding and modeling the dynamic response of quartz (brittle solid), particularly the elastic limit, when subjected to both shock wave and shockless uniaxial strain compression. It will be demonstrated below that phenomenological models developed previously for both ductile and brittle solids subjected to 127
dynamic compression do not predict an elastic limit decrease with an increase in the loading rate. 6.1.1 Continuum Approach to Inelastic Deformation In most of the phenomenological models in the literature, inelastic strain is modeled as a continuum parameter.1‐4,6‐9 For ductile materials, inelastic strain is modeled as a continuum variable through the use of dislocation dynamics.2‐4,8‐9 For brittle materials, inelastic strain is again modeled as a continuum variable and is obtained using shear cracking or shear banding.6‐7 shear cracks and shear bands are modeled as a continuum by introducing shear crack or shear band density parameters.6‐7 In either case, the inelastic strains in these continuum approaches are usually incorporated into a phenomenological model using a method first presented by Duvall.1 The underlying assumptions in this method are:1 1. Stresses are supported by solely elastic strains. 2. Macroscopically, uniaxial strain is maintained and the total uniaxial strain is the sum of elastic and inelastic strains. 3. Inelastic strains are incompressible. Duvall applied the above assumptions to an arbitrary isotropic material and obtained the following relation,1 2/3
128
/
(6.1) In the above equation, longitudinal strain increment, is the inelastic longitudinal strain increment, is the total is the increment in the maximum resolved shear stress, and is the shear modulus. The term ‘inelastic’ in Eq. 6.1 has been used in place of the term ‘plastic’ that was used in Ref. 1. For anisotropic materials, tensor quantities would be required and Eq. 6.1 would take on a different form but would still be separated into inelastic and total strain components in a similar manner.10 It is assumed that anisotropic material models will have display the same elastic limit‐loading rate dependence as the isotropic material models, and anisotropic models will not be considered further. A particular material model is introduced into Eq. 6.1 by defining a function for the partial time derivative of the inelastic strain:1 ,
/2 (6.2)
The function could, in general, be dependent on many variables including strain rate.1 However, both the ductile and brittle material models in the literature generally consider to be a function of inelastic strain and shear stress only as written in Eq. 6.2.1‐4, 6‐9 Additionally, the function increases with an increase in for all of the models examined in the literature.1‐4,6‐9 In Ref. 1, shear stress and inelastic strain in the function are written in terms of longitudinal stresses and strains. The present discussion will continue using the variables introduced in Eq. 6.1: , , and , because is simpler to express the elastic limit in terms of the shear stress. By combining Eq. 6.1 and 6.2, a complete mechanical model for an arbitrary isotropic solid is obtained which can be written in the following form: 129
3
4
,
(6.3)
The above expression is equivalent to Eq. 8 in Ref. 1:1 4
3
,
where is the longitudinal stress, is the density, and is the mean stress. The loading rate dependence of the elastic limit for material models of the form in Eq. 6.3 will be obtained in section 6.1.2 assuming the function, , does not depend on the inelastic strain, . Elastic limit‐loading rate calculations are performed in section 6.1.3 for a specific material model2 which includes inelastic strain dependence in the function ,
. Although the model for iron2 is a dislocation based model for a ductile metal, it is similar to brittle material models in that the inelastic strain rate is ultimately only a function of shear stress and inelastic strain.1,6‐7 6.1.2 Shear Stress dependent Inelastic Strain Rate Models Models which do not include phenomena such as strain hardening, strain softening, dislocation multiplication, etc. will have the function be independent of the inelastic strain; an example is the overstress model in Ref. 1 for Sioux Quartzite. This group of material models will have the form: 130
3
4
(6.4)
where the function is dependent only on the shear stress, . For models having the form in Eq. 6.4, that is ‐‐ no strain hardening, the elastic limit can be defined under compression ( > 0) as when the maximum shear stress, , which happens when 0. As can be seen from Eq. 6.4, this occurs when: 3
4
At (the elastic limit), the function (6.5)
must be larger at higher loading rates, , for the equality in Eq. 6.5 to hold. Furthermore, since in Eq. 6.5 can only increase with an increase in , than must also increase with loading rate. Therefore, ductile and brittle material models of the general form in Eq. 6.4, have an elastic limit that can only increase with increasing loading rate. Physically, this is a statement that inelastic processes take time to relax the shear stress so the elastic limit or yield strength will increase with the applied loading rate. 6.1.3 Dislocation Model for Iron The model developed by Taylor2 for iron is an example of a material model that has explicit inelastic strain dependence in the function . Taylor incorporated dislocation motion and multiplication, and strain hardening into material model having a linear elastic stress‐strain relationship using the same assumptions as Duvall;1 same form as Eq. 6.3. The dislocation dynamics in the Taylor model are dependent on shear stress and inelastic strain only. Instead of 131
using inelastic longitudinal strain, Taylor2 used inelastic shear strain, . The two quantities are .2 Although not written explicitly in Ref. 2, the function in Taylor’s related by model would have the following form: ,
8
3
exp
(6.6)
In the above equation, is the initial density of mobile dislocations, is the dislocation multiplication term, is the maximum dislocation velocity, is a model parameter with units of stress, and is a strain hardening term. The only terms in Eq. 6.6 that are not constants are , and . Using the model developed by Taylor, constant loading rate calculations were performed. The constant loading rate calculations consider only one material element (no wave . While propagation) and fix the time derivative of total strain to be a constant, these calculations do not replicate an actual dynamic compression experiment, they are sufficient and convenient to show the dependence of the elastic limit on the loading rate. Taylor’s model for iron2 is obtained by combining Eq. 6.3 and 6.6. Expressed in terms of the longitudinal stress and the total longitudinal strain, the model is given by,2 2
8
3
3
8
132
2
,
(6.7)
,
3
4
3
8
2
2
3
(6.8)
To perform the constant loading rate calculations, numerical values were needed for the various constants in Eq. 6.7 and 6.8. Values for iron were used; most of these values were taken from Ref. 2 except  , the linear dislocation multiplication term, which was taken from Ref. 11.  = 1.198 x 1012 dyn/cm2 = 0.814 x 1012 dyn/cm2 b = 2.5 x 10‐8 cm v = 3.22 x 105 cm/s N0 = 2.0 x 108 cm‐2  = 3 x 105 cm‐2  0 = 1.98 x 1010 dyn/cm2  = 0 (The strain hardening term was assumed to be zero for the present calculations). Using the above constants, stress and strain in Eq. 6.7 and 6.8 were numerically integrated in time. A different loading rate was used for each calculation, and the loading rates were varied from 10^0/s to 10^8/s. Longitudinal stress‐strain relations from some of the calculations are plotted in Fig. 6.1. Also shown in Fig. 6.1 is a line with a slope equal to the longitudinal modulus, λ + 2 , and a line with a slope equal to the bulk modulus, λ +2/3 . The stress‐strain curves have an initial slope that is equal to the longitudinal modulus, corresponding to elastic compression. After elastic compression, the curves display stress relaxation where the 133
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Longitudinal Stress (GPa)
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1E3/s
0.4 1E0/s
0.2
0.0
0.000
0.002
0.004
0.006
0.008
0.010
Total Longitudinal Strain
Figure 6.1. Longitudinal stress‐strain (uniaxial strain) relations calculated using the model developed in Ref. 1 and assuming a constant loading rate. Also shown is a dashed line with a slope equal to the elastic longitudinal modulus, and a dashed line with a slope equal to the bulk modulus. 134
stress‐strain slope is negative. Following stress relaxation, the stress again begins to increase with increasing uniaxial strain but with a slope corresponding to the bulk modulus (λ +2/3G). During stress relaxation, the inelastic strain rate causes the stress to decrease more rapidly than the increase due to the constant loading rate. At high strains, an equilibrium value the shear stress is reached, and the stress‐strain curve has a slope equal to the bulk modulus. For the calculated stress‐strain curves in Fig. 6.1, the highest stress between elastic loading and the stress relaxation region is taken as the elastic limit, this is also where the shear stress reaches its maximum value, . The elastic limit stresses are plotted as a function of loading rate in Fig. 6.2. The elastic limit increases continuously with the loading rate over the entire loading rate region examined; there is no region where it decreases with increasing loading rate. The dependence on loading rate is more pronounced at higher loading rates, > 105 s‐1, than at lower loading rates. In total, the elastic limit increases by a factor of ~ 5 across an 8 order of magnitude increase in the loading rate. The findings in this subsection show that even with the incorporation of inelastic strain dependence in the function , models having the form of Eq. 6.3 will have elastic limits that increase with an increase in the loading rate. Although this example was for a ductile metal, brittle material models in the literature also have the form given by Eq. 6.3,1,6‐7 where the inelastic strain rate increases with shear stress and inelastic strain. Therefore, the elastic limit is still expected to increase with increasing loading rate for dynamic compression models developed previously for brittle materials. 135
2.0
1.8
Elastic Limit (GPa)
1.6
1.4
1.2
1.0
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0.6
0.4
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0.0
0
10
1
10
2
10
3
10
4
10
5
10
6
-1
10
7
10
8
10
Loading Rate (s )
Figure 6.2. Elastic limits at different loading rates calculated using the phenomenological model developed in Ref. 1. 136
6.1.4 Conclusions Regarding Previous Models As demonstrated in this section, phenomenological models of the form in Eq. 6.3 will display an elastic limit that increases with loading rate under uniaxial strain. This behavior is the opposite of that observed experimentally in Chapters 4 and 5 for quartz. Because the behavior observed in this work cannot be reconciled with previous phenomenological models, a different modeling approach is required. It is expected, based on the results in this section, that for the elastic limit to decrease with increasing loading rate, the inelastic strain rate needs to be dependent on the loading rate in addition to being a function of the shear stress and/or the inelastic strain. The incorporation of this dependence in a physically meaningful manner is the motivation for the phenomenological model discussed next. 6.2 Phenomenological Model The first step in the development of the desired phenomenological model is to consider a physical picture12 that may explain the observed elastic limit – loading rate behavior. Brittle solids, such as quartz, are thought to deform inelastically under compression in a heterogeneous manner:5 through the formation of shear cracks or shear bands. Phenomenologically, shear cracks and bands form due to the localization or trapping of mechanical energy in a small region.5 The concentration of strain energy at a crack tip is what causes it to continue to propagate. The concentration of energy at a shear band elevates its temperature and greatly reduces its viscosity, allowing shear flow.5 In both cases, the rapid energy concentration causes a preferential site for inelastic deformation which further concentrates the energy at that site. The rate of energy concentration or localization at a deformation site will depend on the applied loading rate. However, as the mechanical energy causing inelastic deformation is being 137
localized, a competing mechanism will attempt to redistribute the energy away from the region. Two examples of energy redistribution mechanisms are: heat conduction during shear deformation, and radiative acoustic decay of dynamic stress concentrations during shear cracking. The latter example has to do with the stress field surrounding a crack tip in a material subjected to a dynamic load and will be discussed in section 6.2.1. In either case, the degree of mechanical energy localization and the resulting inelastic behavior will depend on two competing rates: the rate at which energy is concentrated and the rate at which energy is redistributed. In this manner, the inelastic strain rate in a phenomenological model becomes a function of the loading rate in addition to the usual parameters of shear stress and inelastic shear strain. At high loading rates, the energy concentration at a deformation feature is likely more rapid than energy redistribution resulting in the energy concentration being higher than at lower loading rates. The higher energy concentration will result in a higher rate of inelastic deformation. At lower loading rates, the redistribution rate likely exceeds the localization rate; the energy concentration does not get as high, and the result is a lower rate of inelastic deformation. The qualitative picture regarding these two competing phenomena, energy localization and redistribution, may explain the loading rate dependent elastic limit for quartz. The mathematical aspects of the physical picture just outlined are presented next. To simplify the phenomenological development, quartz will be treated as an isotropic solid. This simplification is not expected to change the qualitative behavior of the model. 6.2.1 Energy Localization Under quasistatic compression, quartz crystals are known to undergo inelastic deformation through the formation and propagation of shear cracks.13 Although it is possible 138
that shear bands may be formed at high loading rates with high confining stresses, they will not be considered in the present model. The present phenomenological model is developed using the idea that only shear cracks (mode II) are causing inelastic deformation and that dynamic stress concentration during shear cracking is what leads to the loading rate dependent energy localization. Tensile cracks (mode I) are not considered in uniaxial strain experiments due to the large compressive confining stresses. Consider an isolated shear crack of initial length c0 as shown in Fig. 6.3a. It is well accepted that a critical stress is required at the crack tip for the growth of the crack.14 Once this critical stress is reached, the crack grows with time as shown in Fig. 6.3b. It is also well accepted that stresses near a crack tip can be larger than the far‐field applied stresses as a result of the geometry of the crack tip; the ratio of crack tip stress to the far‐field applied stress is the stress concentration factor.14 The stress concentration factor for a dynamic load can differ significantly from a static load; time‐dependent stress concentrations, of relevance to the present work, are referred to as dynamic stress concentrations.14‐16 The actual value of dynamic stress concentration is dependent on many parameters: crack geometry and orientation, elastic constants, as well as the applied load history.14‐16 In certain situations, the dynamic stress concentration can be much higher than the corresponding static stress concentration. For example, Schroll and Cheng14 have shown that for an elliptical cavity in an elastic medium (Poisson ratio = 0.25) a plane shear wave gives a stress concentration (hoop stress) greater than 16 for specific orientations and for an ellipse eccentricity (B/A) of 0.46. Compared to the static value of ~ 3, the dynamic value is substantially larger.14,15 Therefore, at a given applied stress, the crack tip stress could be higher than the threshold to propagate the crack depending on the rise time of the wave. 139
A
Crack Tip Stress
> Critical Stress
Dynamic
Load
Shear Crack
(Length = co)
B
Crack Extension
Shear Crack
(Length = c(t))
Figure 6.3. A shear stress due to a dynamic compressive load is applied to a crack of initial length c0. Once a critical stress is reached at the crack tip (due to applied stresses and dynamic stress concentration factors) the crack grows to a length c(t) resulting in inelastic strains. 140
In real systems, the dynamic stress concentration will be dependent on the details of the crack and the load history. To proceed with the phenomenological model, some simplifying assumptions are made in the present work. It is assumed that the dynamic stress concentration decays to the value of the static stress concentration in a time on the order of the crack length divided by the acoustic velocity, c/v. The quantity c/v, is henceforth referred to as the acoustic decay time and is assigned the parameter TD. This assumption has been justified by some calculations for the time dependence of dynamic stress concentrations.16 Therefore, for wave rise times much larger than the acoustic decay time (TD) the dynamic stress concentration is equal to the static value. For wave rise times shorter than the acoustic decay time, it is assumed that the maximum dynamic stress concentration is equal to some maximum concentration parameter (M) multiplied by the static value: for TR >> TD, Kdynamic = Kstatic for TR<< TD, Kdynamic (max) = M*Kstatic (6.9) (6.10) TR is the rise time of the loading wave, c is the crack length, v is the acoustic velocity, and M is a maximum concentration parameter which is greater than 1 but likely no larger than ~ 5, based on previous calculations of dynamic stress concentrations.14,15 The time‐dependent stress concentration at a crack tip is shown pictorially in Fig. 6.4 for the two extreme cases of a static load and a rapid impulsive load; the curve shown is a simplified version of similar figures found in Ref. 16. As shown, the impulsive load dynamic stress concentration takes on the maximum value of M*Kstatic at some time and then decays to the static value with a characteristic relaxation time TD. Due to the increase in the stress 141
Stress Concentration Factor
(Crack Tip)
Maximum Dynamic Concentration
= M*Static Factor
Dynamic Stress Concentration
(Impulsive Load)
Static Stress Concentration
(Long Loading Time)
0
TD
3TD
2TD
4TD
Time
Figure 6.4. An example of how the dynamic stress concentration factor, at the crack tip, may change with time after an applied impulsive load. (similar to Ref. 16). The dynamic stress concentration first starts at zero at zero time (difficult to see in the figure). Some short time later, the dynamic stress concentration is M times the static value. The dynamic stress concentration factor then relaxes to the static value with a characteristic decay time TD. 142
concentration factor, the criteria for crack growth can be exceeded under a dynamic load where it would not have under an equivalent static load. The dynamic stress concentration values in between the two limiting cases of a static load and a step load are presently unknown and an assumed form for the dynamic stress concentration at all loading rate values is used here. 6.2.2 Rate Dependent Energy Localization Using dynamic stress concentrations discussed above, a rate dependent energy localization function is obtained next. Similar to dynamic stress concentration arguments, deviatoric strain energy concentration is considered because this is the energy available for the formation and propagation of shear cracks. If the deviatoric strain energy density ED is available for static crack nucleation (and propagation) than M2*ED will be available in the short rise time limit due to rate dependent concentration. Thus, the loading rate dependent localization function should take values only between 1 and M2. The square arises due to an approximation of the relationship between deviatoric strain energy and shear stress, (6.9)
3
2
where is the shear stress and is the specific volume. Instead of wave rise times, energy rates, , will be used. Here, is the rate of change of the deviatoric strain energy density, far from the crack, due to the applied load. Approximately, the maximum possible rate at which a crack can radiate energy via acoustic decay of dynamic stress concentrations is the maximum , divided by the characteristic radiative decay deviatoric strain energy density at a crack tip, 143
time, TD. Therefore, when the applied energy rate is much less than the crack’s maximum possible radiative decay rate, the localization function should have a value of one. Also, when the applied energy rate is equal to or greater than the maximum radiative decay rate, the localization function should take on its maximum value of M2. Thus, the concentration function, ), should have the form, ,
,
(6.12)
1 (6.13)
The simplest form for the localization function that meets the above requirements is a piece wise, linear function: 1
for 0
1
for (6.14) (6.15) In the above expressions, , and the slope H is related to the maximum concentration factor M through the relationship, 1
= M2. 6.2.3. Inelastic Strains from Shear Cracks Instead of modeling individual cracks as discussed previously, the material is viewed as a continuum and a crack density variable is introduced, similar to what is in the literature.6,7,17 This 144
approach greatly simplifies the calculations. Following the work of Addesio and Johnson,7 the inelastic deviatoric strain for an isotropic material under compressive stress is given by  
384 (1   )
[ Nc 3 ] , 15G (2   )
(6.16) where  is Poisson’s ratio, N is the number density (volume) of cracks, and c is an average crack length. In Ref. 6, an exponential distribution of crack sizes was assumed. For simplicity here, following Winey and Gupta,17 all cracks are assumed to be of equal length. The number density, crack size, and shear stress are all time‐dependent quantities. Thus, the inelastic deviatoric strain rate is given by  
Ac 2
[3 Nc  Nc  cN ] , G
(6.17) where A 
384 (1   )
. The crack nucleation rate and crack size growth rate are taken from 15(2   )
Curran et. al.,6 and are based on empirical models. N  B exp[(   1 ) /  ] , (Crack Nucleation Rate) c  C (   2 )c , (Crack Propagation Speed) 2  K

4c
, (Threshold Shear Stress for Crack Growth) 145
(6.18) (6.19) (6.20) Where B,  , C, and K, are model parameters, and the expression for c is only valid when    2 . In the previous work,6,7,17 there was no upper bound for c . In this work, c is not allowed to exceed twice the shear wave velocity, 2c max  2
G

. The factor of two accounts for the fact that the crack can extend in two opposing directions. Deviatoric strain energy is introduced into the model by replacing shear stress in the above equations with the following expression: 3GD
  , 2V
(6.21) In the above equation, D is the deformation strain energy density available for crack nucleation and growth. As previously discussed, strain energy is allowed to localize as a function of energy rate. Therefore, the value of D is allowed to be higher than the bulk deformation strain energy density, ED, by a localization function, f ( E D ) , so that, D  f ( E D ) E D , (6.22) where f ( E D ) ≥ 1. With these substitutions (Eq. 6.21 and 6.22), the expressions for the nucleation rate and the crack growth speed have the form, 146
3Gf ( E D ) E D
N  B exp[(
  1 ) /  ] , 2V
(6.23) (6.24) c  C (
3Gf ( E D ) E D
  2 )c , 2V
Substituting the localization function (Eq. 6.14 and 6.15) into Eq. 6.23 and 6.24 completes the energy rate dependent phenomenological model that is used to determine inelastic deformation under dynamic loading. 6.2.4. Elastic‐Inelastic Model To simplify the computations, a linear elastic description is used to approximate the elastic response of quartz. Despite this approximation, the model does serve to demonstrate the inelastic response of the crystal. The linear elastic isotropic approximation utilized the aggregate properties of quartz, using a Reuss average,18 to provide the second order constants or Lame’ coefficients:  = 8.3 x 109 Pa, G = 41.7 x 109 Pa.18 The approach presented by Duvall1 (section 6.1.1) is used to incorporate inelastic strains into the dynamic response of quartz. Using this approach, the relationship between longitudinal stress and longitudinal strain is written as,1  / t  (  2G ) / t  (8 / 3)G . 147
(6.25) In the above equation, the plastic strain rate in Duvall’s original expression1 has been replaced with the inelastic shear strain rate,  , at 45o to the longitudinal axis. It is assumed here that inelastic shear deformation only occurs along this direction. The stress strain relationships (Eq. 6.25) and the equations for the inelastic strain rate (Eq. 6.17), crack nucleation (Eq. 6.23), and crack growth (Eq. 6.24) complete a description of the quartz phenomenological model within the framework of assumptions presented in this thesis. This model can be used to examine the loading rate dependence of the elastic limit under dynamic compression. 6.2.5 Model Parameters As mentioned previously, the second order elastic constants were based on Reuss averaged constants for quartz:  = 8.3 x 109 Pa, G = 41.7 x 109 Pa.18 Other model parameters can also be associated with quartz material properties. The constant K is associated with the mode II fracture toughness, and is on average ~ 1 MPa*m1/2.19 The threshold shear stress for crack nucleation,  1 , is determined from the measured shockless elastic limits. The peak elastic longitudinal stress in the shockless compression experiments was ~ 10.5 GPa for x‐cut quartz and 14.3 GPa for z‐cut quartz. Given the linear elastic model parameters, this corresponds to a  1 of 4.77 GPa and 6.50 GPa for x and z‐cut quartz, respectively. The threshold for crack growth is taken to be the same as the threshold for crack nucleation; this fixes the stress for the onset of inelastic deformation at the low end of loading rates. Since K was previously fixed, the initial crack (or heterogeneity) length, c0 in  2  K

4c
, is required to be 3.45 x 10‐8 m, or 34.5 nm for x‐cut quartz and 18.6 nm for z‐cut quartz. Although the initial crack lengths are forced 148
parameters, they are physically reasonable since they are larger than the lattice constants of quartz, but much smaller than the wavelength of visible light; keeping the cracks (or heterogeneities) invisible to optical microscopy. The initial crack density was a difficult parameter to estimate. In the present calculations, it was found that the initial value did not influence the elastic limit but did influence the calculated stress‐strain curves beyond the elastic limit. This finding will be discussed in Section 6.3.5. Since the main focus of this model was the elastic limit, the initial crack density was arbitrarily set to zero, so that all modeled cracks were nucleated via Eq. 6.23. The parameters for the crack nucleation rate, B and  are difficult to quantify since they are not directly linked to physical processes. They were chosen such that the inelastic strain rates were sufficiently rapid, after the threshold stress was reached, to relax the longitudinal stresses even at a loading rate of 1x108 s‐1. However, they were not so high that the crack density, N, became unphysical. Since the average crack length can grow to about 1x10‐6 m in these calculations, an unphysical crack density would be anything approaching 1x1018 m‐3. Values were chosen to be B = 2 x 1015 s‐1m‐3 and  = 1 x 108 Pa‐1. With these values, the crack density did not exceed 1 x 1011 m‐3 in any of the calculations. The parameter C, pertaining to the crack growth rate, was chosen so that the crack growth rate would reach the shear wave speed for most of the calculations. A value of C = 1 Pa‐
1 ‐1
s sufficed for this purpose. Although this value is difficult to justify it was found that the calculations were not very sensitive to this value. The maximum energy rate beyond which further localization does not occur is given approximately by E max ~
E max
. TD is related to the crack length through the relation TD = c/v, TD
149
where c is the crack length and v is the shear wave velocity. The maximum elastic deviatoric strain energy (without localization) in these calculations is Emax = 1.0 x 105 J/kg. Given the initial crack lengths listed above and a shear wave velocity v = 3.97 x 103 m/s, the initial relaxation time for dynamic stress concentration is about 1.0 x 10‐11 s. Thus, the maximum energy rate of interest is E max = 1 x 1016 J/kg*s. To coincide with the experimental observations, the maximum localization parameter, M, was chosen to be M = 2.7 (M2 = 7.29) to give the most satisfactory results. This value for M is within the range of values, between 1 and 5, discussed in section 6.2.1. The value of E max and M fixes the value of H to be H = 6.29 x 10‐16 kg*s/J. All of the model parameters are summarized below in mks units. Parameters  = 8.3 x 109 Pa G = 41.7 x 109 Pa K = 1.0 x 106 Pa*m1/2  1 = 4.77 x 109 Pa (x‐cut); 6.50 x 109 Pa (z‐cut) c0 = 3.45 x 10‐8 m (x‐cut); 1.86 x 10‐8 m (z‐cut) B = 2 x 1015 s‐1m‐3  = 1 x 108 Pa‐1 C = 1 Pa‐1s‐1 H = 6.29 x 10‐16 kg*s/J E max = 1.0 x 1016 J/kg*s 150
6.3 Calculated Results To examine the quartz response, using the phenomenological model outlined here, the material description needs to be combined with the mass and momentum conservation equations in a wave propagation computer code.17 Using the code, the evolution of an input wave profile in both time and space can be calculated. However, as was done in section 6.1.3, considerable insight into the loading rate‐dependence of the elastic limit can be gained by specifying a total strain vs time profile for a given material element and calculating the longitudinal stress using the phenomenological model. This approach is simpler but does not provide wave propagation information. Therefore, it is not a simulation of a shock or shockless compression experiment. In the present work, constant loading rate calculations (
constant) were performed using the phenomenological model from section 6.2. The calculations were performed to examine the elastic limit predictions at different applied loading rates. The constant loading rate calculations were performed by numerically integrating Eq. 6.17 and Eq. 6.23 – 6.25, with loading rates ranging from 101 s‐1 to 1013 s‐1. In the remainder of this section, various results using the phenomenological model (and the parameters summarized above) are presented. 6.3.1 Stress‐Strain Response Longitudinal stress‐strain curves for several loading rates are shown in Fig. 6.5 from calculations using the x‐cut quartz parameters; the loading rates shown in Fig. 6.5 range from 107 – 1010 s‐1. Also shown is a dashed line with a slope equal to the elastic longitudinal 151
16
14
Longitudinal Modulus
8 -1
10 s
12
7 -1
Stress (GPa)
10 s
10
9 -1
10 s
8
6
10 -1
10 s
Bu
4
lu
odu
lk M
s
2
0
0.00
0.05
0.10
0.15
0.20
Strain
Figure 6.5. Stress‐strain curves at several loading rates calculated using the phenomenological model for x‐cut quartz with a constant loading rate assumption. A dashed line is shown that has a slope equal to the longitudinal modulus. Another dashed line is shown that has a slope equal to the bulk modulus. 152
modulus,
2 , and a dashed line with a slope equal to the bulk modulus, . The portion of the stress‐strain path for each curve that has a slope equal to the elastic longitudinal modulus corresponds to elastic compression. The longitudinal stress where each curve deviates from the elastic response is taken as the elastic limit at that loading rate. It can be seen from the figure that starting from a loading rate of 108 s‐1, the elastic limit begins to decrease with increasing loading rates similar to the experimentally observed behavior. Significant stress relaxation occurs for the 107 s‐1 loading curve, where the longitudinal stress drops by about ~ 1 GPa immediately after the elastic limit. Similar stress relaxation was also observed for loading rates below 107 s‐1. Smaller amounts of stress relaxation can be seen for the 108 s‐1 and 109 s‐1 curves, but is not evident in the 1010 s‐1 stress‐strain curve. In the stress relaxation region, the inelastic processes are decreasing the shear stress more rapidly than the constant loading rate imposed is able to increase it. As the shear stress (deviatoric strain energy) drops due to stress relaxation, the inelastic strain rate drops via Eq. 6.23 and 6.24. Eventually a balance is reached where the shear stress, and hence the difference between the longitudinal and mean stress, , becomes constant. When the shear stress is constant with increasing strain, the stress‐strain curves have a slope equal to the bulk modulus as seen in Fig. 6.5. At higher loading rates, the balance between the inelastic strain rate and the constant loading is reached almost immediately above the elastic limit which is why very little stress relaxation is observed. The calculated stress relaxation behavior and the shear stress values above the elastic limit in Fig. 6.5 are artificial because in reality the loading rate is not constant. Wave propagation simulations of actual experiments would likely give different results in this region. The stress 153
relaxation feature would not be expected to decrease significantly with an increase in the applied loading rate. Likewise, after some wave propagation, the shear stress above the elastic limit would not be expected to depend on the applied loading rate of the input. Without the simulations, these statements are difficult to validate. 6.3.2 Elastic Limit‐Loading Rate Calculations were performed to obtain the elastic limit, under uniaxial strain loading, as a function of loading rate for both x‐cut and z‐cut quartz using the phenomenological model. The loading rates ranged from 101 s‐1 to 1013 s‐1; the choice of this range was arbitrary. The calculated elastic limits for x‐cut quartz are shown in Fig. 6.6a. As seen in the figure, the elastic limit is initially constant at around 10.5 GPa, the chosen threshold, for loading rates below about 106 s‐1, it then rises to a value of about 12 GPa at a loading rate of about 108 s‐1. The elastic limit then decreases with increasing loading rates to a minimum value of about 5 GPa at 1010 s‐1, and then starts to slowly increase with a further increase of the loading rate. The initial increase of the elastic limit in the loading rate region from 106 s‐1 to 108 s‐1 occurs because the energy localization function, Eq. 6.12 and 6.13, has not yet changed appreciably from 1. Therefore, the loading rate dependence of the inelastic strain rate is negligible and the inelastic strain is essentially only increasing with shear stress. As discussed in section 6.1, if the inelastic strain rate is only a function of shear stress than the elastic limit increases with loading rate. For loading rates between 108 s‐1 and 1010 s‐1, the energy localization function causes a rapid increase in the inelastic strain rate resulting in the elastic limit decreasing with the loading rate. For loading rates above 1010 s‐1, the localization function has reached its maximum value of M2, and the inelastic strain rate becomes independent of the 154
16
Elastic Limit (GPa)
14
12
10
8
6
4
2
0
10
10000
1E7
1E10
1E13
-1
Loading Rate (s )
a. 16
14
Elastic Limit (GPa)
12
10
8
6
4
2
0
10
10000
1E7
1E10
1E13
-1
Loading Rate (s )
b. Figure 6.6. Calculated elastic limits using the phenomenological model are shown as a function of the loading rate (filled squares) for a) x‐cut quartz and b) z‐cut quartz. The experimental shockless data for x‐cut quartz are shown as open squares and the shockwave data as open circles in Fig. 6.6a. The experimental shockless data for z‐cut quartz are shown as open squares and the shockwave data as open circles in Fig. 6.6b. The arrow indicates that the measured shock wave loading rates are lower limits. 155
loading rate. This is the reason for a further increase in the elastic limit with loading rate above 1010 s‐1. Without experimental data, it is difficult to evaluate the validity of this calculated increase. The experimental elastic limit‐loading rate measurements for x‐cut quartz are also shown in Fig. 6.6a. The arrow in the figure indicates that the measured loading rates for the experimental shock compression data are lower limits of the true loading rates due to the time resolution of the measurements. The calculated elastic limit values are reasonably consistent with the shockless compression data for loading rates below 107 s‐1 but are not consistent with the shock wave data above this loading rate. The main similarity between the calculated and measured results for loading rates above 107 s‐1 is that both data sets show a decrease in the elastic limit with increasing loading rate. The calculated elastic limits for z‐cut quartz are shown in Fig. 6.6b for loading rates ranging from 101 s‐1 to 1013 s‐1. The data trend has all of the same qualitative features as the x‐
cut quartz elastic‐limit loading rate curve from Fig. 6.6a except that the elastic limit values are about 30% larger. This increase is due to the choice of different material parameters: initial crack length, c0, and the larger shear stress required for crack nucleation, τ1. The experimental z‐cut quartz elastic limit – loading rate data are also plotted in Fig. 6.6b. The calculated values differ from the experimental data, but the calculated results, in qualitatively agreement with the experimental data, do show that the elastic limit decreases with increasing loading rate. Several significant limitations regarding the comparison between the experimental data and the calculated results need to be mentioned. The experimental loading conditions were not constant loading rate and quartz is not a linear elastic, isotropic solid. The calculations do not take into account the propagation of the waves but only consider one material element. The 156
experimentally determined shock wave loading rates are a lower limit as indicated by the arrow in the Fig. 6.6a. The fact that the measured shock wave loading rates were lower limits may account for a significant part of the two orders of magnitude loading rate discrepancy where the transition from high to low elastic limit occurs in the elastic limit – loading rate data. 6.3.3 Inelastic Strains Within the framework of the model assumptions, the inelastic shear strain is related to the total longitudinal strain and the shear stress through the equation, 1
2
(6.26)
For each loading rate calculation, the inelastic strain was calculated as a function of total strain using the above equation. The inelastic strain can also be obtained by time integrating and both methods give equivalent results as expected. However, Eq. 6.26 is more useful for the present discussion. Fig. 6.7 shows the inelastic strain as a function of total longitudinal strain at three different applied loading rates. Starting at a threshold strain for each curve, the inelastic shear strain increases rapidly with total strain. Although not shown, the stress‐strain curves (Fig. 6.5) begin to deviate from an elastic response immediately after the threshold total strain is reached. On the scale shown in Fig. 6.7, the curves appear to level off after the initial rapid increase. However, they are still increasing although at a greatly reduced rate. The leveling‐off of the curves occurs when the shear stress takes on a nearly constant value due to the balance between the inelastic strain and the applied load. From Eq. 6.26, further increase in total strain 157
0.1
Inelastic Shear Strain
1E-3
1E-5
1E-7
1E-9
7
1E-11
10
10 s
Loading Rate 10 s
-1
9
1E-13
1E-15
0.00
10 s
0.05
-1
-1
0.10
0.15
0.20
0.25
Longitudinal Strain
Figure 6.7. Inelastic shear strain as a function of longitudinal strain at three applied loading rates, 107 s‐1, 109 s‐1 and 1010 s‐1. 158
after a constant shear stress is reached results in a linear increase in inelastic strain with a slope that is 50% the slope of the total strain. The localization function, Eq. 6.14 and 6.15, causes the onset threshold value for the rapid increase in inelastic strain to be a function of the applied loading rate. As seen in Fig. 6.7, this onset strain occurs at 0.099 for a loading rate of 107 s‐1, and occurs at a lower strain of 0.040 for the higher loading rate of 1010 s‐1. This trend is directly related to the decrease in the elastic limit with increasing loading rate for loading rates between 108 s‐1 and 1010 s‐1 (Fig. 6.6). 6.3.4 Shear Cracks In the phenomenological model, the shear cracks are the only source of inelastic shear strains. Both the nucleation of new cracks and the growth of existing cracks contribute to inelastic shear. Fig. 6.8 shows the crack density and crack length at 20% total strain as a function of the applied loading rate. The dashed line indicates co, the initial crack size in the calculations. The total strain of 20% was chosen such that a significant amount of inelastic shear would have taken place for any loading rate. Only results for loading rates above 106 s‐1 are shown because numerical instabilities caused the calculations to terminate immediately beyond the elastic limits for loading rates below 106 s‐1. The crack density increases from a value of 2 x 107 cracks/m3 for a loading rate of 3 x 106 s‐1, to a maximum of approx. 1011 cracks/m3 at around 1010 s‐1. Beyond loading rates of 1010 s‐1, the crack density slowly decreases with increasing loading rate. The change in the crack length with loading rate is opposite to that of the crack density. The crack length is highest for low loading rates and decreases from a value of 2 x 10‐4 m at a loading rate of 3 x 106 s‐1 to near its initialized value, c0 = 3.45 x 10‐8 m, at high loading rates. 159
1E10
1E-5
1E9
1E-6
1E8
1E-7
1E7
10
0
2
10
4
10
6
10
10
8
10
10
12
10
Crack Length at 20% Strain (m)
-3
Crack Density at 20% Strain (m )
1E-4
1E-8
14
10
-1
Loading Rate (s )
Figure 6.8. The crack density (filled squares) at 20% strain is plotted as a function of loading rate for the x‐cut quartz calculations. Also plotted is the crack length (open squares) at 20% strain as a function of loading rate for the x‐cut quartz calculations. 160
The calculated crack density increase in the range from 106 s‐1 to 1010 s‐1 is believed to be a result of a competition between the inelastic shear strain rate and the crack nucleation rate. At lower loading rates near 106 s‐1, a smaller density of cracks is required to relax the shear stress than at higher loading rates near 1010 s‐1; thus, the increase in crack density with loading rate in this range. For loading rates above 1010 s‐1, the calculated results for the crack density and length at 20% strain are a result of the time required to attain 20% strain; as the loading rate increases, the time decreases. This time is a fixed quantity because the loading rate and the total strain are fixed in each calculation. Because the maximum crack propagation speed is a forced parameter (twice the shear wave velocity), the change in crack length is essentially proportional to the time required to attain 20% strain. At very high loading rates, this time is negligible and the cracks do not grow much beyond their initial length. Similarly, at very high loading rates, the reduced time for crack nucleation also begins to reduce the crack density at 20% strain. 6.3.5 Parameter Variation Because many of the phenomenological model parameters were arbitrarily chosen, it was important to evaluate the effect that each parameter had on the elastic limit‐loading rate curve. To examine the sensitivity of the elastic limit – loading rate curve to the various parameters, the parameter values were varied by either ± 20% and/or ± 1 order of magnitude from the original x‐cut quartz values. Using the new values, constant loading rate calculations were again performed to obtain a new elastic limit – loading rate data set. The new set was then compared with the original; these results are discussed next and are grouped based on their 161
influence. Ultimately, the parameter variation calculations provide an approach to evaluate the overall reasonableness of the phenomenological model. 1. K and c0 The fracture toughness parameter (K) and the initial crack length (c0) were varied by ± 20%, and the resulting elastic limit‐loading rate data are shown in Figs. 6.9a and 6.9b respectively. These parameters significantly influence the elastic limit values at the lower loading rates, below ~ 108 s‐1. This is reasonable since they determine the threshold strain energy for crack growth. At high loading rates, above ~ 108 s‐1, they have little influence, likely because the inelastic strains are dominated by crack nucleation and not crack growth. 2. C The parameter C determines the rate of increase of the crack propagation speed with shear strain energy. The calculations were not strongly dependent on this parameter so its influence was more easily observed when varied by an order of magnitude increase and decrease. The resulting elastic limit‐loading rate curves are shown in Fig. 6.10. C has its greatest influence at intermediate loading rates, between 105 s‐1 and 108 s‐1. The low loading rate elastic limits are dominated by the threshold strain energy values, (related to c0 and K), and the high loading rate elastic limits are dominated by the parameters in crack nucleation. Therefore, it is only in this intermediate range that the rate of increase in crack tip velocity is relevant. 162
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Figure 6.9. Elastic limit – loading rate results for the x‐cut quartz model calculations with ± 20% variation in the parameters a) K, and b) c0. The open triangles are the result of increasing the parameters by 20% and the open circles are the result of decreasing the parameter by 20%. 163
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Figure 6.10. Elastic limit – loading rate results for the x‐cut quartz model calculations with ± 1 order of magnitude variation in the parameter C. The open triangles are the result of increasing the C, the open circles are the result of decreasing C. 164
3. B, β, and τ1 B, β, and τ1 are all parameters for the crack nucleation rate, so it is not surprising that they have similar influence. B was varied by an order of magnitude; the other two parameters were varied by ± 20%. The elastic limit‐loading rate curves with the varied parameters are shown in Figs 6.11a, 6.11b and 6.11c, for B, β, and τ1 respectively. These parameters influence the intermediate and high loading rate elastic limits: loading rates above 106 s‐1. The loading rates where these parameters influence the elastic limit is opposite the loading rates that K and c0 (crack growth parameters) influence the elastic limit. Therefore in the present phenomenological model, the onset of crack propagation is the dominant factor for the low loading rate elastic limits, and the crack nucleation rate is the dominant factor for the high loading rate elastic limits. This finding is a mostly likely a result of the linear dependence of shear strain energy on crack growth rate and an exponential dependence of shear strain energy on crack nucleation rate. This finding may also be physically reasonable. At low loading rates, the extension of a small density of cracks is sufficient for quartz to yield. At high loading rates, there is less time for crack extension and nucleation dominates the inelastic strains. The finding that crack propagation dominates the elastic limit in the low loading rate regime and nucleation dominates the high loading rate regime may be the reason why the initial crack density has little influence on the elastic limit‐loading rate curve. The initial crack density can only influence the inelastic strain rate due to crack propagation which would only have an effect in the low loading rate regime. However, at low loading rates, the elastic limit occurs immediately after the threshold shear stress for crack propagation is met, more or less independent of the number of cracks available. Therefore, the initial crack density does not influence the low loading rate elastic limit. In the high loading rate regime, the inelastic strain 165
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rate is dominated by the crack nucleation rate. In this regime, the initial crack density would have no effect on the inelastic strain rate. 4. H H was varied by both ± 20% and an order of magnitude and the results are plotted in Fig. 6.12. The 20% change demonstrates how H influences the high loading rate elastic limits (above 1010 s‐1). An increase in H causes the high loading rate elastic limits to decrease in value. Since H is a component of the localization function, its effect on the maximum concentration parameter M is the cause of the change in the elastic limit at high loading rates. The order of magnitude change demonstrates how H affects the loading rate where the elastic limit decreases. An increase in H causes the transition from high to low elastic limits to occur at lower loading rates. This is because H is the slope of the localization function (Eq. 6.14 and 6.15), so the concentration parameter increases more rapidly with increasing loading rate and its influence on the elastic limit becomes apparent at lower loading rates. 5. E max E max was varied by ± 20%, and the resulting elastic limit‐loading rate curves are shown in Fig. 6.13. Changes in E max only influence the elastic limit for loading rates above 1010 s‐1. This is because E max determines the maximum cutoff loading rate for energy localization, as such it changes the value of the maximum concentration factor, M. Hence it can only influence the elastic limit high loading rates where the concentration function reaches its maximum value. 167
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Figure 6.12. Elastic limit – loading rate results for the x‐cut quartz model calculations with both ± 20% and an order of magnitude variation in the parameter H. The open triangles are the result of increasing H; the open circles are the result of decreasing H. The order of magnitude changes are visually the larger of the two deviations from the initial calculated results. 168
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Figure 6.13. Elastic limit – loading rate results for the x‐cut quartz model calculations with ± 20% variation in the parameter E max . The open triangles are the result of increasing E max ; the open circles are the result of decreasing E max . 169
6.4. Summary and Main Findings Results in Chapters 4 and 5 showed that the elastic limit of both x‐cut and z‐cut quartz decreased with an increase in the loading rate under uniaxial strain. Previous phenomenological models for dynamically compressed materials generally display the opposite response: the elastic limit increases with an increase in loading rate. Because the behavior observed in the quartz experiments could not be reconciled with previous phenomenological models, a different modeling approach was required. For a material model to show that the elastic limit decreases with an increase in the loading rate, the inelastic strain rate needs to be explicitly dependent on the loading rate. Loading rate dependence of the inelastic strain rate was introduced using dynamic stress concentrations at the crack tip; the stress concentration factor increases with the loading rate of the input stress. Using this concept, a loading rate dependent strain energy concentration function was introduced into a shear cracking model adapted from previous studies.6‐7,17 The loading‐rate dependent shear cracking model was used along with a linear elastic approximation for quartz to complete the phenomenological model. A series of constant loading rate calculations were performed using the quartz phenomenological model to calculate the dependence of the elastic limit on the loading rate. Although the constant loading rate calculations did not simulate the experimental loading conditions because they do not consider wave propagation, they were successful in demonstrating the experimentally observed elastic limit‐loading rate dependence. The main findings of the phenomenological model calculations are summarized below. 170
1. The calculated elastic limit decreased with increasing loading rate. For the x‐cut quartz model, the elastic limit decreased from 11 GPa at 106 s‐1 to 5 GPa at 1010 s‐1. For z‐cut quartz model, the elastic limit decreased from 14 GPa at 106 s‐1 to 7 GPa at 1010 s‐1. This behavior is qualitatively consistent with the experimental results. 2. In the model, the shear stress threshold for the onset of crack growth is the main determining factor for the elastic limit at loading rates below 108 s‐1. The crack nucleation rate and energy localization parameters are the main determining factors for the elastic limit at loading rates above 108 s‐1. Mathematically, this finding is a result of the linear dependence of the crack growth parameters on shear stress and the exponential dependence of the crack nucleation rate on shear stress. This result may be physically reasonable: At low loading rates, the extension of a relatively small number of cracks is sufficient for the crystal to yield. At high loading rates, time for crack extension is limited so a larger density of cracks is required for inelastic deformation. 3. Because the experimental data exist at a limited number of loading rates, the experimental elastic limit‐loading rate curves for x‐cut and z‐cut quartz are not well known. It was demonstrated that varying the model parameters from their initial values modified the elastic limit‐loading rate curve. With more experimental data, the model parameters, or possibly the model equations, could be adjusted to match the shapes of the elastic limit‐loading rate curves. 171
References for Chapter 6 1. G. E. Duvall, Stress Waves in Anelastic Solids, H. Kolsky and W. Prager, Eds. (Springer‐
Verlag, Berlin, 1963). 2. J. W. Taylor, J. Appl. Phys. 36, 3146 (1965). 3. Y. M. Gupta, G. E. Duvall, and G. R. Fowles, J. Appl. Phys. 46, 532 (1975). 4. T. E. Arvidsson, Y. M. Gupta, and G. E. Duvall, J. Appl. Phys. 46, 4474 (1975). 5. D. E. Grady, J. Geophys. Res. 85, 913 (1980). 6. D. R. Curran, L. Seaman, and D. A. Shockey, Phys. Reps. 147, 253 (1987). 7. F. L. Addesio and J. N. Johnson, J. Appl. Phys. 67 3275 (1990). 8. M. D. Bjorkman, Effect of Risetime and Surface Hardness on Precursor Decay in Shocked Lithium Fluoride. PhD thesis, Washington State University, (1980). 9. J. L. Ding, J. R. Asay, and T. Ao, J. Appl. Phys. 107, 083508 (2010). 10. J. N. Johnson, O. E. Jones, and T. E. Michaels, J. Appl. Phys. 41, 2330 (1970). 11. C. Voltz and F. Buy, J. Phys. IV France. 134, 1035 (2006). 12. B. M. LaLone and Y. M. Gupta, J. Appl. Phys. 106, 053526 (2009). 13. C. H. Scholz, J. Geophys. Res. 77, 2104 (1972). 14. K. R. Schroll and S. L. Cheng, J. Appl. Mech. 51, 133 (1972). 15. C. O. Horgan and S. A. Silling, J. Elast. 18, 83 (1987). 16. C. C. Mow and Y. H. Pao, The Diffraction of Elastic Waves and Dynamic Stress Concentrations, USAF‐RAND Internal Report, R‐482‐PR (1971). 17. J. M. Winey and Y. M. Gupta, J. Appl. Phys. 107, 103505 (2010). 18. G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, M. I. T. Press. Cambridge, Massachusetts 172
19. N. E. W. Hartley and T. R. Wilshaw, J. Mat. Sci. 8, 265 (1973). 173
Chapter 7 Shock Wave Compression of Gadolinium Gallium Garnet Crystals In this chapter, the results of an experimental investigation on the shock wave response of Gadolinium Gallium Garnet (GGG) crystals are reported. This work is separate from the study on quartz in the other chapters, and is presented entirely in this chapter. An introduction is given in section 7.1. The experimental method is summarized in section 7.2. Experimental results and analysis are presented in section 7.3. A discussion of the results and comparison with previous experiments are presented in section 7.4. The summary and main findings of the GGG work are in section 7.5. 7.1 Introduction Gadolinium Gallium Garnet (GGG), chemical composition Gd3Ga5O12, is a transparent crystalline solid possessing cubic symmetry.1 The material does not occur naturally but is a fairly common synthetic crystal used in the optics industry2 and as a substrate for the growth of other thin films.3 It has a density of 7.08 g/cm3, and is nearly isotropic under ambient conditions with an isotropy factor of A = 1.05, (A = 1 is isotropic).1,4 GGG is strong, brittle, and generally fails by propagation of cracks from the surface that travel into the bulk during quasistatic testing.2 With a properly etched surface, bulk samples have supported tensile stresses as high as 2 GPa before failure in 4 point bending tests.2 The shock wave response of GGG has been examined only recently.5,6 These studies focused on the use of GGG as a potential anvil material in high stress shock reverberation experiments. For this reason, peak compressive stresses in these experiments ranged from 40 to 260 GPa.5,6 A two‐wave structure was observed in experiments below 65 GPa, indicating an 174
elastic to inelastic transition. The longitudinal stress and particle velocity at the Hugoniot elastic limit (HEL) were:5,6 29 GPa at u = 0.57 km/s for the [100] orientation and 31‐33 GPa at u = 0.62‐
0.66 km/s for the [111] orientation. These HEL values are the highest reported values for a material other than diamond.7 Since GGG would be expected to remain transparent under purely elastic compression, the crystals could have potential applications as high stress optical windows in dynamic compression studies.8 However, since the peak stresses in these studies were well above the reported elastic limit, it is not clear whether the ~ 30 GPa value is truly representative of the maximum elastic stress that the material can support, irrespective of the peak stress. The reason that it may not be a representative elastic limit is because the HEL of some brittle solids depends on the peak impact stress.9‐11 The objective of the present work was to examine the response of shock compressed GGG to peak stresses ranging from 15 ‐ 50 GPa, since the peak stress range below 40 GPa had not been examined previously.5,6 The emphasis of the present work was on determination of the elastic limit in this peak stress regime, the elastic Hugoniots for both [100] and [111] oriented GGG, and the post‐HEL response. 7.2 Experimental Method 7.2.1 Material Characterization The GGG single crystals used in this work were obtained from Princeton Scientific Corporation (Princeton, NJ). Average densities from 6 samples were 7.111 ± 0.014 g/cm3, as measured by the Archimedean method. As‐received samples were polished cylindrical disks oriented along the [100] or [111] crystallographic axis with a diameter of 18 mm and thicknesses 175
ranging from 3.241‐3.249 mm. The crystal orientations were confirmed using x‐ray Laue measurements which showed circular spots for each diffracted peak, typical of high quality single crystals. The average ambient longitudinal and shear wave velocities, measured using the pulse echo method, were: VL = 6.415 ± 0.013 km/s and VS = 3.604 ± 0.004 km/s respectively for the [100] oriented samples; and VL = 6.473 ± 0.002 km/s and VS = 3.531 ± 0.004 km/s for the [111] oriented crystals. The ambient measurements for each sample are listed in Table 7.1. Since GGG has cubic symmetry, there are only three independent second order elastic constants; the ambient density and sound speed measurements in Table 7.1 are sufficient to determine them.1 The second order elastic constants C11, C12, and C44 were determined from the average of the ambient measurements using the following well known relations,12 [100] [111] (7.1) (7.2) Since there are 4 sound speed measurements and only 3 independent elastic constants, Eqns. 7.1 ‐7.2 are over determined. The elastic constants were iterated numerically to determine the set that best fits all of the measured sound speed data. These elastic constants, together with those from Ref. 1 and 4 are listed in Table 7.2. The elastic constants calculated from the current sound speed measurements are slightly higher than previously reported values, but all are within ~5%. The isotropy factor A, defined as 2C44/(C11 – C12) and listed in Table 7.2, is nearly equal to 1 indicating that the crystal is elastically almost isotropic at ambient conditions.1 176
Table 7.1 Ambient measurements on GGG crystals Sample. No. Crystal Thickness (Shot No.) Orientation (mm) 1 (07‐016) 2 (07‐606) 3 (07‐607) Average 4 (07‐052) 5 (07‐610) 6 (07‐611) Average [100] [100] [100] [100] [111] [111] [111] [111] 3.249 3.245 3.247 3.245 3.245 3.246 Table 7.2 Calculated and Reported Elastic Constants Reference C11 This Study 292 (5) Ref. 4 285.6 Ref. 1
286.2 The isotropy factor A is defined as 2C44/(C11 – C12) 177
Density ρ0 (g/cm3) 7.084 7.094 7.084 7.087 7.117 7.142 7.129 7.129 C12 116 (10) 114.8 111.4 Sound Speed Longitudinal (mm/µs) 6.40 6.43 6.41 6.41 6.47 6.47 6.40 6.45 C44 92 (3) 90.2 90.8 Sound Speed Shear (mm/µs) 3.60 3.60 3.61 3.60 3.53 3.54 3.60 3.56 A 1.045 1.056 1.039 7.2.2 Experimental Details For these shock wave experiments, the configuration in Fig. 7.1 was used to obtain wave speeds as well as GGG/LiF [100] interface velocity histories. The VISAR13‐14 system (Chapter 3) was used to measure the velocity history of the impact surface and the GGG/LiF [100] interface. The impact surface probe, together with PZT electrical impact pins for tilt determination (Fig. 7.1), served to determine an impact fiducial time for wave speed measurements. The shock and interface velocity measurements were used to deduce elastic wave amplitudes and inelastic wave speeds in GGG. Oxygen free high conductivity (OFHC) copper impactors, nominally 25 mm in diameter and 3.2 mm thick, were mounted onto projectiles and accelerated to velocities ranging from 0.7 km/s to 2.0 km/s using a 100 mm light gas gun or a 30 mm powder gun. The impact velocities for each experiment are listed in Table 7.3. A total of 6 experiments were performed: 3 on [100] oriented samples and 3 on [111] oriented samples. 7.3 GGG Experimental Results and Analysis Impact Surface Probes The intensity of the return VISAR laser light from the impact surface probes decayed to zero soon after impact for all experiments on both orientations, indicating that either the sample became opaque or the mirror did not survive. The velocity from this surface was not recorded for sufficient time to be determined accurately, but did provide an impact fiducial time for wave speed calculations. 178
Impactor GGG Target [100] LiF PZT Impact Pins x 4 VISAR Probes Figure 7.1. Schematic view of the experimental configuration for the GGG experiments. An impactor impacts the GGG sample backed by a [100] LiF window. The velocity interferometer (VISAR) monitors the velocity at two locations: one is at the impact surface, the other is at the GGG/LiF interface. A set of 4 PZT impact pins, in a concentric square geometry (all pins not shown), determines the planarity of impact. 179
Table 7.3 Experimental Details Exp. No. Impactora Impact Sample Sample (Shot Number) Thickness Velocity Orientation Thickness (mm) (mm/µs) (mm) 1 (07‐016) 3.535 0.772 [100] 3.249 2 (07‐606) 2.318 1.204 [100] 3.245 3 (07‐607) 2.440 1.981 [100] 3.247 4 (07‐052) 3.240 0.772 [111] 3.245 5 (07‐610) 3.150 1.246 [111] 3.245 6 (07‐611) 3.155 2.009 [111] 3.246 a
All impactors were oxygen free high conductivity copper disks, nominally 25 mm in diameter. 180
7.3.1 Results for [100] GGG The measured [100] GGG/LiF interface velocity records are shown in Fig. 7.2 All of the wave profiles exhibited a two wave structure, indicating an elastic to inelastic transition. The elastic wave amplitude is strongly dependent on the impact stress, increasing with impact velocity. This is easily seen in Fig. 7.2; the elastic wave amplitude for sample 3 was roughly twice the amplitude for sample 1. The elastic wave is followed by a drop in the particle velocity indicating stress relaxation. This relaxation is largest for the highest impact velocity experiment on sample 3, where a drop of about 20% of the elastic wave amplitude is observed. A step like disturbance arrives after the stress relaxation and before the second wave. This disturbance is believed to originate as an elastic release from the GGG/LiF interface reflecting off of the inelastic wave and returning to the interface. Calculations performed in section 7.3.8 support this hypothesis. The inelastic wave arrives last and raises the interface velocity to its final state. The inelastic wave rises gradually (rise time ~ 150 ns) for the low impact velocity experiment on sample 1. The low amplitude of the inelastic wave for sample 1 and its long rise time make it difficult to distinguish it from the intermediate disturbance wave and to determine its arrival time with good precision. The inelastic wave rise time decreases with increasing impact velocity and is better identifiable in the higher impact velocity experiments on samples 2 and 3 with rise times of about 70 ns for both. All of the interface profiles have significant velocity fluctuations, indicating that the sample is undergoing heterogeneous deformation above the elastic limit.15 This is typical for brittle solids and is consistent with the shock profiles observed for quartz (Chapters 4 and 5). 181
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Figure 7.2. Transmitted shock velocity profiles for [100] GGG recorded at the GGG/LiF interface. 182
7.3.2 Results for [111] GGG The measured [111] GGG/LiF interface velocity records are shown in Fig. 7.3. Overall, the velocity histories are similar to the measurements for the [100] orientation. Again, the elastic wave amplitudes are strongly dependent on the impact velocity. The stress relaxation is most pronounced for the high impact stress experiment on sample 6. The intermediate disturbance wave is again observed in each of the records. The inelastic wave for the low impact velocity experiment, sample 4, has a small amplitude and a long rise time making it difficult to identify its arrival time. The inelastic wave rise times decrease with increasing impact stress. The interface profiles for this orientation also have velocity fluctuations indicating a heterogeneous response. Although the overall response of the two orientations is similar, there are some differences. For the [111] orientation, the interface velocity between the elastic wave arrival and the intermediate disturbance wave is relatively flat for the low and medium impact experiments on samples 4 and 5, indicating little to no stress relaxation in this stress range. In contrast, some velocity decay is observed over this region for the [100] orientation. The stress relaxation for the high stress experiment on sample 6 is much more pronounced than it was for the [100] oriented sample 3. For sample 6, the velocity decays to about 50% the elastic wave amplitude, compared with only a 20% drop for sample 3. The inelastic wave rise times were significantly shorter for the experiments on the [111] orientation when comparing experiments with similar impact stresses. The inelastic wave rise time for sample 4 was around 70 ns, compared with ~ 150 ns for the [100] oriented sample 1. The inelastic wave rise time for sample 5 is 30 ns compared with 70 ns for sample 2, and the rise time is only 10 ns for sample 6 compared with 70 ns for sample 3. 183
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7.3.3 Elastic Shock Velocity The elastic shock velocity is calculated by dividing the sample thickness by the wave transit time, In the above equation, (7.3)
, is the sample thickness, timpact is the time that the impactor contacts the sample, and t1, is the measured arrival time of the elastic wave at the GGG/LiF interface. Since the wave speed to be measured is in the direction of particle motion, the times of interest would ideally be measured at two points that form a line which is parallel to the propagation direction. This was not the case in the present experiments. As shown in Fig. 1, the impact VISAR probe is separated laterally from the interface probe. With this geometry, the non‐planarity of impact or the impact tilt, must be accounted for to obtain a precise wave velocity measurement. Fig. 7.4 is a schematic view of the impact showing an exaggerated tilt angle θtilt. The upper figure, Fig 7.4a is a 3‐dimensional projection of the impact. The bottom figure, Fig. 7.4b, shows only a two dimensional cutout of the impact. The coordinate system in the figures is set up so that the x‐axis is the direction parallel to the projection, onto the target face, of the line containing the VISAR probes; the y‐axis is parallel to the projectile motion and the subsequent propagation direction. The VISAR probes are marked as positions P1 and P2 in the figure; P1 is on the impact face and P2 is at the GGG/[100] LiF interface. Also shown is the position P1’, which is not recorded by the VISAR system. The positions P1’ and P2 form a line that is parallel 185
Propagation Direction Vpr Impactor Face A GGG Target Face P1’
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x z Projection of the Line Containing VISAR Probes onto the Target Face, Parallel to the x‐axis Propagation Direction Impactor B y θ’tilt x P1 P1’
Line Containing VISAR Probes P2
GGG Target Optical Window VISAR Probes Figure 7.4. The upper figure, 7.4a, is a 3 dimensional projection of the impact with an exaggerated tilt angle. The lower figure, 7.4b, is a side profile of the impact with an exaggerated tilt angle. The positions P1 and P2 mark the locations of the VISAR probes. The coordinate system is such that the –y direction is parallel to the projectile motion and the x direction is parallel to the projection of the line containing the VISAR probes onto the GGG impact face. The viewed direction in the upper and lower figures is not the same. 186
to the elastic wave propagation direction, so the impact time in Eqn. 7.3 is when the impactor first makes contact with position P1’. As impact occurs, a line of contact between the impactor and the target (Fig. 7.4a) sweeps across the target surface at a velocity Vcr in a direction perpendicular to the line of contact. This closure velocity is related to the projectile velocity through the equation Vcr = Vpr/ tan(θtilt). In all experiments, the closure velocity is orders of magnitude faster than any mechanical wave velocity so all mechanical properties of both the impactor and target are irrelevant to the tilt analysis. The set of four PZT pins (Fig. 7.1) in each experiment provide both the impactor tilt, θtilt, and the orientation angle, φ, in each experiment. The orientation angle is the angle between the direction of the closure velocity and the projection of the line containing the VISAR probes onto the GGG target face (Fig. 7.4a). The tilt angle and the orientation angle are used to obtain θ’tilt, which is the angle that the projectile makes with the target face in the x‐
y plane (see Fig. 7.4b). The relationship between these three angles is, cos
1
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1
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The impact time, (7.5)
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(7.6)
In the above equation t(P1) is the time of contact between the impactor and the VISAR probe position P1, d is the lateral separation distance between P1 and P1’. The second term on the right hand side of Eqn. 7.6 is the correction to the measured impact time. In the present experiments, typical tilt angles, θtilt, were 1‐2 milliradians and orientation angles, φ, took on values between 0 and 2π. Eqn. 7.6 was used to correct the impact time from that measured by the impact surface VISAR probe. The lateral separation distance, d, was always 9 mm and typical time adjustments were between ‐6 ns and +6 ns. The elastic wave velocities were then determined using Eqn. 7.3. The elastic wave velocities from all six experiments are listed in Table 7.4. In experiment 6, the PZT pin signals failed to be recorded so the shock velocity in this experiment was calculated by assuming a zero tilt angle. 7.3.4 Elastic Limit Determination For the wave analysis, the measured elastic shock and interface velocities are used to calculate the elastic limit stresses. Since the measurements are made at the GGG/LiF interface, the impedance mismatch between these two materials must be accounted for in the calculation. The h‐t and the σ‐u diagrams for sample 1, a typical experiment, are shown in Fig. 7.5(a) and 7.5(b), respectively, to illustrate the elastic wave interaction with the LiF window. The stress and particle velocity at the elastic limit, (
, ), are labeled as state 1 in the diagrams and is the state of the material after passage of the elastic wave. The elastic wave interacts with the GGG/LiF interface and partially reflects and partially transmits (Fig. 7.5(a)); the partial reflection is a release wave that travels back into the GGG target bringing it into state 1’, (
, ) , the partial transmission is a shock wave that travels into the LiF window bringing it also into state 1’.16 188
Table 7.4 Elastic Wave Properties Sample. No. Impact Vel. Sample [Orientation] (mm/µs) Thickness
(mm) Interface Particle Elastic Density Elastic Velocity
Shock Velocity Limit (ρ/ρ0) at Velocity HEL (mm/µs) (mm/µs) (mm/µs) (GPa) 1 [100] 0.772 3.249 6.66 0.444 0.293 13.9 1.046 2 [100] 1.204 3.245 6.75 0.552 0.366 17.5 1.057 3 [100] 1.981 3.247 7.12 0.958 0.640 32.4 1.099 4 [111] 0.772 3.245 6.65 0.410 0.270 12.8 1.042 5 [111] 1.246 3.245 6.76 0.641 0.426 20.5 1.067 6 [111] 2.009 3.246 6.94a 1.020 0.688 34.0 1.110 a
The PZT pins did not trigger in this experiment so the wave speed was calculated assuming the impact had zero tilt. 189
20
State 1'
D
State 1'
[100] LiF Hugoniot
ER
15
D ela
c
sti
State 0
GGG
Ela
stic
D
0
10
tic
Stress (GPa)
Time
State 1
D Elas
- 0
State 1
=
pe
Sl o
S lo
pe
=
State 1'
5
State 0
[100] LiF
0
0.0
Lagrangian Distance
0.2
0.4
0.6
0.8
1.0
Particle Velocity (mm/s)
a b Figure 7.5. Time vs. Lagrangian distance (h‐t diagram) (a) and longitudinal stress‐particle velocity (σ‐u diagram) (b) for the experiment on sample 1. The sample number was chosen arbitrarily; all of the experimental stress‐particle velocity diagrams for the current experiments are qualitatively similar. Only the elastic (HEL) states are shown. State 0 is when both the GGG sample and the [100] LiF optical window are at ambient conditions. State 1 is the in‐material HEL state and state 1’ is the first jump, corresponding to the HEL, measured at the GGG/[100] LiF interface. 190
The particle velocity in state 1’ is measured in the experiment and is the amplitude of the first jump in the transmitted wave profile (Fig.7.2). Since state 1’ is reached by a single shock in the LiF window, (ignoring elastic‐plastic effects in LiF) it is on the [100] LiF Hugoniot17 as shown in Fig. 7.5(b). In the σ‐u diagram, Fig. 7.5(b), the line connecting the origin and state 1 has .16 It is assumed that the release path is linear elastic and that the line a slope of . From examination of the σ‐u diagram, connecting state 1 and state 1’ has a slope of , the stress and particle velocity in state 1 (
particle velocity in state 1’ (
, ) is written in terms of the stress and ) and the elastic shock velocity, , as follows: 1/2
/
(7.7) (7.8) The interface stress is related to the interface particle velocity, , through the [100] LiF Hugoniot:17 13.59
3.57
(7.9) where (
, has units of GPa and has units of mm/µs. Therefore, the elastic limit values ) can be calculated from the measured quantities, and through Eq. 7.7 – 7.9. In experiments where stress relaxation is observed, the elastic wave amplitude was taken immediately upon arrival of the elastic wave, prior to the decay in velocity. The calculated in‐
material particle velocities and stresses at the HEL for the GGG experiments are listed in Table 7.4. Also, the material density at the HEL was calculated using the Rankine‐Hugoniot relation:16 191
(7.10) The densities, at the HEL, for each experiment are also listed in Table 7.4. 7.3.5 Elastic Hugoniots The elastic shock velocity is plotted as a function of particle velocity for both [100] and [111] oriented GGG samples in Fig. 7.6. Curves were numerically fitted to data for both orientations with the shock velocity at u=0 fixed as the average ambient sound speed from Table 7.1. A quadratic fit was needed to accurately represent the [100] data; for the [111] orientation, a linear fit was sufficient. The fits, also plotted in Fig. 7.6, are as follows: [100] DElastic = 6.415 + 0.64u + 0.73u2 [111] DElastic = 6.473 + 0.678u (7.11) (7.12) In the above expressions, both DElastic and u have units of mm/µs. As seen in Fig. 7.6, the elastic shock velocities for the two orientations deviate from each other at high particle velocity by about 3%. This difference indicates growing material anisotropy at high stresses under uniaxial strain loading, even though the crystal was nearly isotropic under ambient conditions. The shock‐particle velocity relations, together with the momentum jump condition, σ = ρ0Du,16 lead to elastic Hugoniots for the [100] and [111] orientations of GGG given by the following expressions, [100] σ = 45.6u + 4.55u2 + 5.19u3 192
(7.13) Elastic Shock Velocity (mm/s)
7.2
7.1
7.0
6.9
6.8
6.7
6.6
[100] GGG
[111] GGG
6.5
6.4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Particle Velocity (mm/s)
Figure 7.6. Elastic shock‐particle velocity values for the two crystal orientations. The solid and dashed lines are best fits to the data. 193
[111] σ = 46.0u + 4.82u2 (7.14) In the above expressions, the longitudinal stress, σ, is in units of GPa and the in‐material particle velocity, u, has units of mm/µs. 7.3.6 Inelastic Wave Velocity The inelastic wave is the final step in the velocity history (Figs. 7.2 and 7.3) and brings the material to the peak state, state 2. The inelastic wave does not travel through uniformly compressed material. This is illustrated using the h‐t diagram in Fig. 7.7. The inelastic wave initially travels through material at state 1, produced by the elastic wave. However, the elastic wave reflects as a release wave from the mismatched impedance boundary at the GGG/[100] LiF interface and travels back into the material with speed DER. This leaves part of the material in state 1’. It is expected that the interaction of this release wave with the inelastic wave causes the propagation of a new forward going wave, an elastic reloading wave which travels with speed DERL, and leaves the material in state 1’’. The arrival of the elastic reloading wave at the LiF window is believed to be the intermediate disturbance wave in Figs. 7.2 and 7.3. The elastic reloading wave, DERL, would also reflect from the GGG/[100] LiF interface, but this and higher order reflections are not considered in this work. After interaction with the elastic release wave, the inelastic wave continues forward, but begins to travel with a potentially new speed, D3 since state 1’’ is not equal to state 1. Previous researchers have attempted to correct the measured value of D2 by assuming certain speeds for D3.18‐19 To simplify the analysis, it is assumed here that D3 = D2. Therefore, the inelastic wave speed (Lagrangian) is calculated using the expression, 194
Time
D3
State 1''
DERL
State 1'
State 2
State 1
D2
DER
D elas
tic
State 0
State 0
GGG
[100] LiF
Lagrangian Distance
Figure 7.7. Time vs Lagrangian distance diagram for the various waves in a typical experiment on GGG. On the scale shown, impact occurs at the lower left hand corner of the figure. The other stress states and the various wave speeds that are shown are described in the text. 195
(7.15)
In the above expression, is the time that the final wave, D3 in Fig. 7.7, arrives at the GGG/LiF interface. For experiments on samples 1 and 4, the arrival time of the inelastic wave was not determined due to its low amplitude, relatively long rise time, and the superimposed velocity fluctuations. In all other experiments, the arrival time was taken at the midpoint of the wave amplitude. The inelastic wave speeds from 4 of the 6 experiments are listed in Table 7.5. 7.3.7 Peak Stress For this analysis it is assumed that the inelastic wave is a steady wave. Therefore, the jump conditions can be used to connect the HEL state to the peak state.16 The calculations in this section were all performed in the Lagrangian coordinate system. The momentum and mass Rankine‐Hugoniot jump conditions that relate the HEL and peak (state 2) compression states for the GGG experiments are:16 (7.16)
(7.17)
For the above expressions, the Eulerian (lab frame) wave velocity in Ref. 16 was replaced with the Lagrangian wave velocity, Eq. 7.15 (see Appendix B). 196
Table 7.5 Inelastic wave and peak state results Sample. No. Sample Impact Inelastic [Orientation] Thickness Velocity Wave (mm) (mm/µs) Velocitya (mm/µs) 1 [100] 3.249 0.772 ‐ 2 [100] 3.245 1.204 4.79 3 [100] 3.247 1.981 5.52 4 [111] 3.245 0.772 ‐ 5 [111] 3.245 1.246 4.62 6 [111] 3.246 2.009 5.90 a
The inelastic wave velocity is a Lagrangian wave speed ‐ Indicates undetermined quantity 197
Inelastic Particle Velocity (mm/µs) ‐ 0.608 1.01 ‐ 0.625 1.02 Peak Stress (GPa) Density (ρ/ρ0) at Peak ‐ 25.8 46.9 ‐ 27.1 48.0 ‐ 1.117 1.186 ‐ 1.118 1.184 A stress‐particle velocity diagram, showing the various stress states in the experiment on sample 3 is shown in Fig. 7.8. Sample 3 was chosen arbitrarily to illustrate the calculation. Since the GGG sample is being impacted by a copper impactor, the peak state must lie on the impactor Hugoniot; the left boundary in Fig. 7.7 is in state 2 and is in contact with the copper impactor. The impactor Hugoniot is shown in Fig. 7.8 and is the left going copper Hugoniot centered at the projectile velocity. The copper Hugoniot is:20 35.2
13.51
(7.18)
where has units of GPa and has units of mm/µs. Therefore the impactor Hugoniot is: 35.2
13.51
(7.19)
where is the projectile velocity. The slope of the line in σ‐u space that connects the HEL and peak state is given by Eq. 7.16, therefore, the peak state must fall along the line given by, (7.20) The peak state in each experiment is found graphically by intersecting the above line with the impactor Hugoniot; intersection Eq. 7.19 and Eq. 7.20. The resulting peak stress and particle velocities are listed in Table 7.5. Also listed is the material density in the peak state obtained from the jump conditions, Eq. 7.17. 198
60
2
State 2
op
e
40
Sl
Stress (GPa)
=
0

D
50
State 1
30
20
Impactor Hugoniot
10
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Particle Velocity (mm/s)
Figure 7.8. Longitudinal stress vs particle velocity diagram showing both the HEL, state 1, and the peak state, state 2, for the experiment on sample 3. Sample 3 was chosen arbitrarily for this figure. The stress and particle velocity in state 2 is calculated by intersecting the line emanating from state 1 with the impactor Hugoniot. The stress states, slope, and impactor Hugoniot are described in the text. 199
7.3.8 Intermediate Disturbance Wave The expected origin of the intermediate disturbance wave, seen in Figs. 7.2 and 7.3, is illustrated by the h‐t diagram in Fig. 7.9 which is a zoomed in view of Figure 7.7 showing the interface region in greater detail. The elastic wave partially reflects from the GGG/LiF interface and propagates back into the material as a release wave, traveling at a speed DER. This wave leaves the material in state 1’ as, shown in Fig. 7.9, which is a lower stress state than the elastic limit since [100] LiF has lower acoustic impedance than GGG. The elastic release (DER ) then interacts with the inelastic wave, and another elastic wave is created which propagates forward toward the GGG/LiF interface. This forward going wave travels with speed DERL, where ERL stands for elastic re‐loading. The elastic reloading wave can be thought of as either a second reflection or as the creation of a new elastic precursor since the inelastic wave is now traveling through material at a stress state lower than the elastic limit. The arrival time of the elastic reloading wave at the GGG/LiF interface can be calculated if estimates for DER and DERL are made. Since these wave speeds were unknown, it was assumed, for simplicity, that both DER and DERL are equal to the elastic wave speed measured in the experiment (Table 7.4). From the h‐t geometry of the waves in Fig. 7.9, the intersection between the elastic release and the inelastic wave occurs at the Lagrangian coordinates, 1
1
1
1
(7.21)
200
(7.22)
(h,t2)
D ERL
State 1'
Time
State 2
State 1'
D
D2
ER
(h,t1)
State 1
ic
D elast
State 0
GGG
State 0
[100] LiF
Lagrangian Distance
Figure 7.9. Time vs Lagrangian distance diagram for the various waves in a typical experiment on GGG. This figure shows the same waves as in Fig. 7.7 except the scale has been adjusted to make the interfacial region clearer. On the scale shown, impact occurs off scale to the lower left. State 0 is when both the GGG sample and the [100] LiF optical window are at ambient conditions. The other stress states and the various wave speeds that are shown are described in the text. 201
The arrival time, t2, of the elastic reloading wave at the GGG/LiF interface is, (7.23)
The calculated times, t2 – t1, for the arrival of the intermediate disturbance wave from experiments on samples 2, 3, 5, and 6 were a reasonably good match to the measured times. These calculations support the idea that the step like disturbances between the elastic and inelastic waves in Fig. 7.2 and 7.3 are a result of elastic wave reflections off of the GGG/[100] LiF interface; they are not a result of any new finding regarding the response of GGG to shock compression. 7.4 DISCUSSION 7.4.1 HEL: Impact Stress Dependence Examination of the GGG/LiF interface velocity profiles (Fig. 7.2 and 7.3) revealed that the HEL values for each orientation were strongly dependent on the impact stresses (sections 7.3.1 and 7.3.2). To further examine this dependence, elastic impact stresses were calculated for each experiment and compared with the measured HEL values (Table 7.4). Elastic impact stresses are the impact surface stresses that would occur if the GGG samples responded purely elastically and provide a convenient approach to compare the HEL values for different experiments. The elastic Hugoniots presented in section 7.3.5 (Eq. 7.13 and 7.14), together with the copper impactor Hugoniot20, Eq. 7.19, and projectile velocity were used to determine the expected elastic impact stress for each experiment. These values are listed in Table 7.6 and they range from 13 – 34 GPa, the HEL values are also listed in Table 7.6. 202
Table 7.6 Elastic impact stress and the elastic limit Elastic Impact Sample. No. Sample Impact Stress Velocity [Orientation] Thickness (GPa) (mm/µs) (mm) 1 [100] 3.249 0.772 17.0 2 [100] 3.245 1.204 28.1 3 [100] 3.247 1.981 51.2 4 [111] 3.245 0.772 17.0 5 [111] 3.245 1.246 28.9 6 [111] 3.246 2.009 50.1 203
Elastic Limit (GPa) 13.9 17.5 32.4 12.8 20.5 34.0 A plot of the experimental HEL values as a function of the elastic impact stress for both orientations is shown in Fig. 7.10. Also shown are the range of approximate HEL values from Ref. 5 and 6; because tabulated data was not provided, the range shown was extracted from the text and figures in Ref. 5 and 6. In the present data, there is a clear trend of increasing HEL with increasing elastic impact stress. The HEL values from both orientations increase by more than a factor of two over the elastic impact stress range from 15 – 50 GPa. This behavior was not previously reported for shocked GGG crystals5,6 and is not apparent from the Ref. 5 and 6 results as shown in Fig. 7.10. Previous investigators reported essentially constant HEL’s of around 30 GPa when subjected to peak stresses between 40 and 65 GPa.5,6 The elastic impact stress‐HEL values for the high stress experiments (Samples 3 and 6) in the current work are consistent with the range of values from Ref. 5 and 6. Therefore, it is expected that the trend in the current experiments of increasing HEL with impact stress will not continue, and that further experiments would reveal a more or less constant HEL for elastic impact stresses above 40 GPa. Additionally, it is likely that the peak stress dependent HEL was not observed by the previous investigators simply because peak stresses below 40 GPa had not been examined. Since GGG is a transparent solid, it could have potentially been used as an optical window in dynamic compression studies at stresses below the elastic limit.8 Above the elastic limit, the crystal will display a time‐dependent response and will likely be opaque due to inelastic deformation features that scatter light. Therefore, GGG crystals cannot be used as optical windows above the elastic limit. The current results show that the elastic limit, or the potential stress range of GGG as an optical window, is below 14 GPa. This stress is within the useful stress range of single crystal sapphire that is already in use an optical window and has a similar 204
Hugoniot Elastic Limit (GPa)
70
[100] GGG
[111] GGG
Range in Ref. 6
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
Elastic Impact Stress (GPa)
Figure 7.10 Hugoniot elastic limit (HEL) as a function of elastic impact stress from the current measurements are shown as filled (100) and open (111) squares. The shaded rectangle encompasses the reported HEL values and the corresponding impact stress range from Ref. 5 and 6. 205
acoustic impedance to GGG.21,22 Based on the present work, GGG has no clear advantage over sapphire as an optical window in dynamic compression studies. 7.4.2 Strength in the Peak State For the present discussion, strength is defined as the difference between the longitudinal stress and the mean stress, , under uniaxial strain compressions at and beyond the HEL. In Ref. 6, GGG subjected to peak shock stresses between 40 and 65 GPa was reported to have longitudinal stresses that were essentially coincident with the 300K isotherm measured in the diamond anvil cell.6 Later in the article. the authors concluded that GGG does have strength for peak shock stresses between 65 and 120 GPa.6 These comments suggest that GGG does not have measureable strength for peak stresses between the HEL and 65 GPa. In order to infer strength in the peak state in the present experiments, the mean stress corresponding to the longitudinal stress state needs to be known. The bulk modulus, K0, of GGG and its pressure derivative, K’0, have been measured previously under hydrostatic compression using x‐ray diffraction measurements.23,24 These values can be used to generate a mean stress‐
density relationship through the following expression, 1
206
(7.24)
The above equation was derived using the definition of the bulk modulus,25 and the dependence of the bulk modulus with pressure: (7.25)
(7.26)
In the above expressions, it is assumed that the mean stress density relationships along the isotherm, isentrope, and the mean stress curve corresponding to the Hugoniot, are all coincident. Rigorously, the mean stress as a function of volume compression under uniaxial strain does not equal the mean stress as a function of volume under hydrostatic compression.26 Nevertheless, it is a commonly used approximation,18, 27 and will be used in the present analysis. Stress contributions from the temperature rise during shock compression were estimated in Ref. 6 to be less than 1.5 GPa for peak stresses under 65 GPa; the thermal contribution to stress is not considered in the present discussion. In Ref. 6, the hydrostatic compression data used for strength comparison was taken from Ref. 23. However, the bulk modulus in Ref. 23 had a reported value of K0 = 253.3 GPa which is anomalously high, as the expected value from elastic constants (Table 7.2) is (C11 + 2C12)/3 which is in the range of around 170 to 180 GPa. Therefore in the current work, the experimentally determined bulk modulus and its pressure derivative from Ref. 24 were utilized: K0 = 169 GPa and K’0 = 3.6. The bulk modulus and its pressure derivative from Ref. 24 were used together with Eq. 7.25 to generate a mean stress‐density compression data which is plotted in 207
Fig. 7.11. Density compression is defined as (ρ/ρ0 ‐1). Also shown in Fig. 7.11 is the longitudinal stress‐density compression data from the GGG experiments (Tables 7.4 and 7.5), and the elastic Hugoniots from section 7.3.5 (Eq. 7.13 and 7.14). As can be seen in the figure, the difference between the longitudinal stress and the mean stress, defined as the strength, decreases during the transition from the HEL to the peak state in all of the GGG experiments. The strength decrease between the HEL and the peak state is between 30% and 70% for all samples. Additionally, since the peak state longitudinal stresses are not coincident with the mean stress curve, there is likely still some residual strength in the peak state; all of the peak longitudinal stress values are at least 3 GPa above the mean stress curve. So while strength decrease from the HEL to the peak state in the present experiments is consistent with observations in Ref. 6, strength is likely not completely lost, as was previously implied,6 for peak inelastic stresses below 48 GPa. 7.5 Summary and Main Findings It was reported previously that GGG crystals exhibited a high HEL of around 30 GPa when shock compressed to peak stresses above 40 GPa.5,6 However, since the peak stresses in these studies were well above the reported elastic limit, it was not known whether the measured 30 GPa HEL was representative of the elastic limit at arbitrary peak stresses. Therefore, the GGG response was examined at peak shock stresses well below the range that was examined previously. 208
Longitudinal Stress  (GPa)
50
40
Elastic Hugoniots
30
20
10
0
0.00
Mean Stress Curve
Ref. 24
0.04
0.08
0.12
0.16
0.20
Density Compression / - 1
Figure 7.11 Longitudinal stress as a function of density compression at the Hugoniot Elastic Limit and at the peak state for the GGG experiments. The HEL states are shown as filled squares for the [100] orientation and filled circles for the [111] orientation. The peak states are shown as open squares for the [100] orientation and open circles for the [111] orientation. HEL and peak states from the same experiments are connected with dashed lines. The solid lines are the elastic Hugoniots from section 7.3.5. Also shown is the mean stress curves from ref. 24. The difference between the longitudinal stress and the mean stress at each point is a measure of strength. 209
In the present work, shock compression measurements on GGG were performed at peak stresses between 15‐50 GPa. The measurements were performed to determine the HEL, the elastic Hugoniot relations, and the post‐HEL response for the peak stresses examined here. Planar impacts were achieved between copper impactors and GGG crystals oriented along the [100] and [111] directions to produce uniaxial strain conditions in the samples. Measurements of both shock and particle velocity were made with a combination of PZT electrical impact pins and a velocity interferometer system. The particle velocity histories of transmitted shock waves were measured at the interface between the GGG samples and [100] oriented LiF optical windows. The main findings of the current work are summarized below: 1. The elastic wave amplitude (HEL) had a strong dependence on the impact stress for both orientations for the same load path (uniaxial strain) and comparable loading rates (shock wave loading). For the [100] orientation, the HEL increased from 14 GPa at an elastic impact stress of 17 GPa, to an HEL of 32 GPa at an elastic impact stress of 51 GPa. For the [111] orientation, the HEL increased from 13 GPa at an elastic impact stress of 17 GPa to an HEL of 34 GPa at an elastic impact stress of 50 GPa. This HEL dependence had not been determined previously. The current results suggest that the previously reported HEL values of around 30 GPa 5,6 are only valid at impact stresses ranging from 40‐65 GPa. 2. Elastic Hugoniot relations for the two crystallographic orientations were determined from the current measurements to be 210
[100] σ = 45.6u + 4.55u2 + 5.19u3 [111] σ = 46.0u + 4.82u2 , (7.13) (7.14) where the longitudinal stress σ is in units of GPa and the in‐material particle velocity u has units of mm/µs. Examination of these elastic Hugoniot relations indicate that the behavior of GGG is somewhat anisotropic at large elastic strains; there is about a 3% stress difference between the two orientations at 10% density compression, even though the GGG behavior is nearly isotropic under ambient conditions. 3. Strength, as determined from the stress difference between the longitudinal stress and the mean stress, is 30%‐70% lower in the peak state than at the HEL in each experiment. Although the strength is lower, there does appear to be some strength retained in the peak state. In contrast, total strength loss was previously reported6 for the peak stress states in GGG below 65 GPa. 4. Based on the work reported here, GGG will not make a useful optical window in shock compression measurements to stresses above 14 GPa as had been suggested previously.8 211
References for Chapter 7 1. M. Krzesinska and T. Szuta‐Buchacz, Phys. Stat. Sol. A 82, 421 (1984). 2. J. Marion, Appl. Phys. Lett. 47, 694 (1985). 3. H. Garem, J. Rabier, P. Veyssiere, J. Mater. Sci. 17, 878 (1982). 4. L. J. Graham and R. Chang, J. Appl. Phys. 41, 2247 (1970). 5. Y. Zhang, T. Mashimo, K. Fukuoka, M. Kikuchi, T. Sekine, T. Kobayashi, R. Chau, and W. J. Nellis, in Shock Compression of Condensed Matter‐2003, edited by M. D. Furnish, Y. M. Gupta, and J. W. Forbes (AIP, New York, 2004), pp. 127‐128. 6. T. Mashimo, R. Chau, Y. Zhang, T. Kobayoshi, T. Sekine, K. Fukuoka, Y. Syono, M. Kodama, and W.J. Nellis, Phys. Rev. Lett. 96, 105504 (2006). 7. J. M. Lang Jr. and Y. M. Gupta, J. Appl. Phys. 107, 113538 (2010). 8. M. Hiltl, R. Chau, W. J. Nellis, H. Nahme, Presented at Shock Compression of Condensed Matter‐Atlanta, GA (2001). APS Bulletin, Vol. 46, Abstract #H1.077 (2001). 9. R. Fowles, J. Geophys. Res. 72, 5729 (1967). 10. C. Hari Manoj Simha and Y. M. Gupta, J. Appl. Phys. 96, 1880 (2004). 11. D. R. Curran, L. Seaman, and D. A. Shockey, Phys. Rep. 147, 253 (1987). 12. J. M. Boteler Ph.D. thesis, Washington State University, 1993. 13. L. M. Barker and R. E. Hollenback, J. Appl. Phys. 43, 4669 (1972). 14. W. F. Hemsing, Rev. Sci. Instrum. 50, 73 (1979). 15. T. Vogler and J. P. Clayton, J. Mech. Phys. Solids. 56, 297 (2008). 16. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, High Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970), p. 293. 17. W. J. Carter, High Temp. – High Press. 5, 313 (1973). 212
18. J. Wackerle, J. Appl. Phys. 33, 922 (1962). 19. T. J. Ahrens, W. H. Gust, and E. B. Royce, J. Appl. Phys. 39, 4610 (1968). 20. M. Van Thiel, A. S. Kusubov, A. C. Mitchell, and V. W. Davis, Compendium of Shock Wave Data, Lawrence Radiation Laboratory, Livermore, CA. (1966). 21. S. C. Jones, B. A. M. Vaughan, and Y. M. Gupta, J. Appl. Phys. 90, 4990 (2001). 22. S. C. Jones, M. C. Robinson, and Y. M. Gupta, J. Appl. Phys. 93, 1023 (2003). 23. H. Hua, S. Mirov, and Y. K. Vohra, Phys. Rev. B 54, 6200 (1996). 24. A. Durygin, V. Drozd, W. Paszkowicz, E. Werner‐Malento, R. Buczko, and A. Kaminska, Appl. Phys. Lett. 95, 141902 (2009). 25. L. E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice‐Hall, NJ, 1969). 26. M. P. Conner, M.S. thesis, Washington State University, 1988. 27. G. R. Fowles, J. Appl. Phys. 32, 1475 (1961). 213
Chapter 8 Summary and Conclusions Two studies on dynamically compressed brittle single crystals were conducted in this work. The overall objective of the first study, the major effort, was to examine the loading rate dependence of the elastic limit of x‐cut and z‐cut quartz crystals for the same load path (uniaxial strain) and comparable peak stresses. The overall objective of the second study, the smaller effort, was to examine the effect of elastic impact stress on the elastic limit of gadolinium gallium garnet (GGG) single crystals for the same load path (uniaxial strain) and comparable loading rates (shock wave loading). 8.1 Shockless and Shock Wave Compression of X‐Cut and Z‐Cut Quartz A compact pulsed power generator (CPPG) was used to produce shockless compression to a maximum elastic stress of 11 GPa and 16 GPa in x‐cut quartz crystals. Plate impacts, with elastic impact stresses of 10 GPa and 16 GPa, were used to generate shock wave compression. The transmitted waves in all experiments were monitored at the interface between the rear surface of the x‐cut quartz samples and [100] LiF optical windows with a velocity interferometer (VISAR). The elastic limits were determined from an analysis of the transmitted profiles. The shockless elastic limit of x‐cut quartz did not depend significantly on the shockless loading rate (7 x 105 s‐1 to 5 x 106 s‐1), peak stress, or sample thickness; the average elastic limit was 10.6 GPa. The shock wave elastic limits increased with impact stress and decreased with propagation distance; measured values ranged from 5.1 GPa to 7.4 GPa at loading rates greater than 107 s‐1. 214 Overall, the shockless compression elastic limits were 70% higher than the corresponding shock wave elastic limits for x‐cut quartz. Similar to the x‐cut quartz measurements, the CPPG was used to produce shockless compression in z‐cut quartz with a maximum elastic stress of 13 GPa and 18 GPa. Plate impacts were used to generate shock wave compression ranging from 12 GPa to 37 GPa elastic impact stresses. The transmitted waves in these experiments were measured at the interface between the z‐cut quartz crystal and a [100] LiF window with a velocity interferometer (VISAR). Aside from one anomalous sample, shockless measurements to 13 GPa showed elastic response, indicating that the shockless elastic limit was higher than 13 GPa. The shockless measurements on z‐cut quartz compressed to 18 GPa had elastic limits of 14.3 GPa at a loading rate of 6 x 105 s‐1. Because the sample thicknesses in the 18 GPa experiments were identical, it could not be determined if there was significant dependence of the shockless elastic limit on the peak stress or the shockless loading rate. The shock wave elastic limits of z‐cut quartz increased with increasing elastic impact stress, and ranged from 11.1 GPa to 15.1 GPa at loading rates greater than 107 s‐1. Since the sample thicknesses were similar in the shock compression measurements, it could not be determined if the elastic limit was dependent on the propagation distance. For comparable sample thicknesses and peak stresses, the shockless elastic limit of z‐
cut quartz was 13% higher than the corresponding shock wave elastic limit. The measured shockless and shock wave elastic limits of x‐cut and z‐cut quartz were compared with existing phenomenological models in the literature. It was found that the existing models predicted an increase in the elastic limit with increasing loading rate, which is the opposite of the measured loading rate dependence. The behavior in the existing models was a result of the inelastic deformation rate being only a function of variables such as shear stress 215 or inelastic strain. It was postulated that for the elastic limit to decrease with increasing loading rate, the inelastic deformation rate would need to incorporate loading rate dependence in addition to the usual variables. To include loading rate dependence into the inelastic deformation rate description, a model for strain energy localization in quartz was developed. It was assumed that dynamic stress concentrations at crack tips cause strain energy to localize near the cracks, and cause the degree of localization to be loading rate dependent. This mechanism, incorporated into a shear cracking model developed previously,1‐3 caused the inelastic deformation rate in the material model to be loading rate dependent. Constant loading rate calculations performed using the material model provided results that were in qualitative agreement with the measured loading rate dependence of the x‐cut and z‐cut quartz elastic limits. The main findings of the x‐cut and z‐cut quartz work are as follows: 1. The elastic limits of x‐cut and z‐cut quartz crystals, subjected to uniaxial strain, were higher under shockless compression than under shock wave compression. The measured behavior is the opposite of the loading rate dependence generally accepted for the strength of brittle solids. 2. The quartz results cannot be explained using existing phenomenological models in the literature since these models show the opposite elastic limit‐loading rate dependence. 3. A quartz phenomenological model, developed using loading rate dependent strain energy localization, provided calculated results that were in qualitative agreement with the measured elastic limit‐loading rate dependence. 216 The overall result of this work is that for the same load path (uniaxial strain) and comparable peak stress, the elastic limit of dynamically compressed x‐cut and z‐cut quartz crystals (brittle solids) decrease with increasing loading rate. A phenomenological model was proposed to provide a qualitative explanation of the measured response. 8.2 Shock Wave Compression of Gadolinium Gallium Garnet (GGG) Crystals Plate impacts, with elastic impact stresses of approx. 17 GPa, 30 GPa, and 50 GPa were used to generate shock wave compression in [100] and [111] oriented GGG crystals. Measurements of both shock and particle velocities, of transmitted waves, were made with a combination of impact pins and a velocity interferometer (VISAR). The transmitted waves in all experiments were monitored at the interface between the rear surface of the GGG samples and [100] LiF optical windows. The elastic limits were determined from analysis of the transmitted waves. The elastic limits of [100] and [111] oriented GGG crystals increased significantly with increasing impact stress. For the [100] orientation, the elastic limit increased from 13.9 GPa at an elastic impact stress of 17.0 GPa to 32.4 GPa at an elastic impact stress of 51.2 GPa. For the [111] orientation, the elastic limit increased from 12.8 GPa at an elastic impact stress of 17.0 GPa to 34.0 GPa at an elastic impact stress of 50.1 GPa. This elastic limit increase with impact stress was not observed in previous shock wave studies on GGG;4,5 the earlier work reported the shock wave elastic limits to be nearly constant at around 30 GPa at impact stresses ranging between 40 GPa and 65 GPa. The shock and particle velocities of the elastic waves were used to construct elastic Hugoniots for both the [100] and [111] orientations of GGG. Comparison of the elastic response 217 indicates that the behavior of GGG is somewhat anisotropic at large elastic strains; there is about a 3% stress difference between the two orientations at 10% density compression, even though the behavior is nearly isotropic under ambient conditions. The main findings of the GGG work are as follows. 1. The elastic limit of [100] and [111] oriented GGG crystals increases significantly with elastic impact stress for the same load path (uniaxial strain) and comparable loading rates (shock wave compression). 2. The elastic response of GGG is somewhat anisotropic at large elastic strains even though the response is nearly isotropic under ambient conditions. The overall result of this work is that for the same load path (uniaxial strain) and comparable loading rates (shock wave compression), the elastic limit of [100] and [111] oriented GGG crystals (brittle solids) increase with increasing elastic impact stress. This finding provides another example where the elastic limit of a brittle solid is strongly dependent on the peak stress under dynamic loading. 218 References for Chapter 8 1. D. R. Curran, L. Seaman, and D. A. Shockey, Phys. Reps. 147, 253 (1987). 2. F. L. Addesio and J. N. Johnson, J. Appl. Phys. 67 3275 (1990). 3. J. M. Winey and Y. M. Gupta, J. Appl. Phys. 107, 103505 (2010). 4. Y. Zhang, T. Mashimo, K. Fukuoka, M. Kikuchi, T. Sekine, T. Kobayashi, R. Chau, and W. J. Nellis, in Shock Compression of Condensed Matter‐2003, edited by M. D. Furnish, Y. M. Gupta, and J. W. Forbes (AIP, New York, 2004), pp. 127‐128. 5. T. Mashimo, R. Chau, Y. Zhang, T. Kobayoshi, T. Sekine, K. Fukuoka, Y. Syono, M. Kodama, and W.J. Nellis, Phys. Rev. Lett. 96, 105504 (2006). 219 Appendix A MATLAB® Code ‘forwardcalc’ for Forward Wave Propagation using the Method of Characteristics This appendix presents a characteristics computer code, forwardcalc, written in MATLAB® 7.10.0 (R2010a),1 to propagate a one dimensional shockless uniaxial strain compression wave across an interface between two materials having different impedances. The code was used in this thesis to check that the in‐material profiles and sample loading curves obtained using the code Charice 1.1,2 were consistent with the measured shockless quartz/LiF interface velocity profiles; the consistency check was a part of the analysis of the shockless experiments on x‐cut and z‐cut quartz (sections 5.2 and 6.2). The code developed was also used to generate the theoretical elastic profiles in sections 5.1 and 6.1. A.1 Overview of forwardcalc The code forwardcalc uses the method of characteristics (for simple waves) to advance an input sample particle velocity profile across a sample/window interface and outputs an interface velocity profile. In this context, the term ‘sample’ refers to the material in which the wave originates; the term ‘window’ refers to a second material though which the wave propagates. The code corrects for the impedance mismatch between the two materials and the interaction between the forward propagating wave and the wave reflections from the interface. This method differs from a backwards calculation where an interface profile is propagated backwards in time to acquire a sample particle velocity profile.2 The computer code presented here for forward propagation is based on the methods developed by Grady and Young,3 and later refined by Maw.4 220
The particular characteristics solution used in this code requires the loading curve for the sample material to follow a single valued (history independent) stress‐density relationship corresponding to the isentrope.4 The same requirement is imposed for the window material; the window material must follow a single valued stress‐particle velocity relationship. Due to these assumptions, the particular method used here is not applicable for materials that exhibit time‐
dependent behavior, or for compressions that are not isentropic.4 Furthermore, the code in the present work will not correctly handle shock waves or waves which shock‐up during propagation, so some care must be taken when assessing the calculated results. The code requires an input wave profile (sample particle velocity vs. time), a Lagrangian distance between the input profile and the sample/window interface, Lagrangian sound speed‐
particle velocity curves for the sample and window, and ambient densities of the sample and window materials. The input profile must be a numeric array of data where the times (in units of µs) are in the first column and the particle velocities (in units of mm/ µs) are in the second column. The simulated spatial location of the input profile should be far enough from the window to be in a region of simple flow, so that all wave interactions occur between the input profile’s location and the sample/window interface. If, instead, the input profile is located at the position where the sample/window interface profile is desired, there is an option in the code to backward propagate the wave (as though the window were not there) before performing the forward propagation. The input profile should be truncated as nearly as possible to the peak particle velocity of the curve because the program will automatically truncate the curve according to the last particle velocity value in the array. Coefficients of a polynomial fit to the sample and window Lagrangian sound speed vs. particle velocity (C‐u) curves are required (see below). 221
A.2 Operation of forwardcalc When MATLAB® is opened, the ‘current folder’ box must be set to the folder containing the input sample particle velocity – time profile; the sample/window interface velocity profile will be written to this folder at the end of the calculation. The program is opened by typing ‘forwardcalc’ in the Matlab command line, and the program is run by following a series of prompts. The first prompt will be for the user to enter the filename of the sample particle velocity vs. time data. Next, the program will prompt the user to enter ‘1’ to perform a backward calculation before the forward calculation as mentioned above; if this option is not desired then the user should enter the value ‘0’. The code will then prompt the user to enter the constants for the sample C‐u curve given by the equation, 0
1
2
3
(A.1)
In the above equation, 0 has units of mm/µs, 1 is dimensionless, 2 has units of (mm/µs)‐1, and 3 has units of (mm/µs)‐2. The ambient sample density is entered next in units of g/cm3. Similar prompts will be given for the window C‐u curve and ambient density. The final prompt will be the name of the file to which the calculated interface velocity‐time profile will be written. The code will then execute the method of characteristics calculation and write the sample/window interface profile to the user specified file. 222
A.3 MATLAB® Code for forwardcalc The forwardcalc computer code is provided below. Explanatory comments are provided for clarity, all comment lines begin with a % symbol. %
function forwardcalc
%
%Input u-t profile
%
filename=input('File name of u-t curve, make sure most of zero u is
gone >> ','s');
ut=dlmread(filename);
%
%Prompt user for backward calculation before the forward calculation
%
back = input('Type 1 to perform backward calculation before forward >>
');
%
%Input Lagrangian distance to window
%
h = input('Lagrangian distance to window (mm) >> ');
%
%Input number of characteristics
%
input('# of characteristics >> ');
%
%Input sample c-u parameters: C = C0 + C1*u + C2*u^2 + C3*u^3, and
density
%
C0 =input('sample sound speed, C0, (km/s)>> ');
C1 = input('sample linear C1, ()>> ');
C2 = input('sample quadratic C2, (s/km)>> ');
C3 = input('sample cubic C3, (s/km)^2 >> ');
rho = input('sample density, g/cc >> ');
%
%Input window c-u parameters: CW = CW0 + CW1*u + CW2*u^2 + CW3*u^3, and
density
%
CW0 = input('window sound speed, C0, (km/s)>> ');
CW1 = input('window linear C1, ()>> ');
CW2 = input('window quadratic C2, (s/km)>> ');
CW3 = input('window cubic C3, (s/km)^2 >> ');
rhoW = input('window density, g/cc >> ');
%
%Generate u-t profile with crs distinct velocity intervals
%
t_ut = ut(:,1);
u_ut = ut(:,2);
utelem = numel(ut(:,2));
ustep = (ut(utelem,2) - ut(1,2))/crs;
223
newut = zeros(crs,2);
%
for i=1:crs
newut(i,2) = i*ustep + ut(1,2);
end
j=1;
%
%Linearly interpolate between the original data to obtain new time
steps
%
for i=1:(utelem - 1)
while ((u_ut(i) < newut(j,2)) && (newut(j,2) <= u_ut(i+1)))
newut(j,1)
=
(newut(j,2)
u_ut(i))*((t_ut(i+1)
t_ut(i))/(u_ut(i+1)...
- u_ut(i))) + t_ut(i);
j=j+1;
if j==(crs + 1), break, end
end
if j==(crs + 1), break, end
end
%
%Perform backward calculation, if back=1
%
if back==1
for i=1:numel(newut(:,1))
newut(i,1)
=
newut(i,1)
h/(C0
+
C1*(newut(i,2))
+
C2*(newut(i,2))^2 +...
C3*(newut(i,2))^3);
end
end
%
%Generate sample stress-particle velocity-sound speed (p-u-c) file,
sampuc
%The file, sampuc, goes to a max of double the final u
%Increment at 0.001 km/s. first value = 0, numel = temp+1
%
temp = 2*ceil(newut(crs,2)/.001);
sampuc = zeros(temp+1,5);
for i=0:temp
sampuc(i+1,1)=rho*(C0*0.001*i
+
0.5*C1*(0.001*i)^2
+
(1/3)*C2*(0.001*i)^3 +...
(1/4)*C3*(0.001*i)^4);
sampuc(i+1,2) = 0.001*i;
sampuc(i+1,3) = C0 + C1*0.001*i + C2*(0.001*i)^2 + C3*(0.001*i)^3;
end
%
%Generate window stress vs. interface velocity, winpu, increment at
0.001
%km/s
%
winpu = zeros(temp+1,2);
for i=0:temp
winpu(i+1,1) = rhoW*(CW0*0.001*i + 0.5*CW1*(0.001*i)^2 +...
(1/3)*CW2*(0.001*i)^3 + (1/4)*CW3*(0.001*i)^4);
224
winpu(i+1,2) = i*0.001;
end
%
%Generate master p, uint (interface particle velocity), F, Qint (Q or u
+ f
%at interface), C table. All as functions of p. The sample p-u-c table
is
%already in proper format except u will go to F.
%Need to generate uint from
%window p-u and sampuc p.
%
j=1;
uint = zeros(temp+1,1);
for i=1:temp
while ((winpu(i,1)<=sampuc(j,1)) && (sampuc(j,1) < winpu(i+1,1)))
uint(j,1)
=
(sampuc(j,1)
winpu(i,1))*((winpu(i+1,2)
winpu(i,2))/...
(winpu(i+1,1) - winpu(i,1))) + winpu(i,2);
j=j+1;
if j==temp + 2, break, end
end
if j==temp + 2, break, end
end
%
%Put uint on end of of sampuc
%
sampuc(:,4) = uint(:,1);
%
%Now sampuc is in the form p,F,c,uint need to add Qint
%
Qint = zeros(temp+1,1);
Qint(:,1) = sampuc(:,2) + sampuc(:,4);
sampuc(:,5) = Qint(:,1);
%
%sampuc is p, F, C, uint, Qint. p, F, c relationship always valid, uint
and Qint only
%valid at interface
%
%Initialize empty characteristic intersection points. Char(i,j,k). i is
the
%label for the positive characteristic, j is for the negative, k goes
from
%1 to 8. 1 = Q; 2 = R; 3 = F; 4 = p; 5 = u; 6 = c; 7 = H; 8 = t
%
Char = zeros(crs, crs, 8);
%
%Q = 2*newu, the starting particle velocity values %make nested for
loops
%to range i and j.
%
for j = 1:crs
225
for i=j:crs
Char(i,j,1) = 2*newut(i,2);
end
end
%
%Next get the interface values Qint = Q. From sampuc; find Qint and the
%corresponding p, c, uint, and f values. At interface i=j. sampuc goes
as
%p,F,C,uint,Qint
%
j=1;
for i=1:temp
while ((sampuc(i,5)<=Char(j,j,1)) && (Char(j,j,1) < sampuc(i+1,5)))
Char(j,j,3) = (Char(j,j,1) - sampuc(i,5))*((sampuc(i+1,2) sampuc(i,2))/...
(sampuc(i+1,5) - sampuc(i,5))) + sampuc(i,2);
%
Char(j,j,4) = (Char(j,j,1) - sampuc(i,5))*((sampuc(i+1,1) sampuc(i,1))/...
(sampuc(i+1,5) - sampuc(i,5))) + sampuc(i,1);
%
Char(j,j,5) = (Char(j,j,1) - sampuc(i,5))*((sampuc(i+1,4) sampuc(i,4))/...
(sampuc(i+1,5) - sampuc(i,5))) + sampuc(i,4);
%
Char(j,j,6) = (Char(j,j,1) - sampuc(i,5))*((sampuc(i+1,3) sampuc(i,3))/...
(sampuc(i+1,5) - sampuc(i,5))) + sampuc(i,3);
%
%Solve for R as uint - fint
%
Char(j,j,2) = Char(j,j,5) - Char(j,j,3);
%
j=j+1;
%
if j==(crs + 1), break, end
end
%
if j==(crs + 1), break, end
end
%
%Now all invariants Q and R are known for characteristics, label all
intersections
%with Q, R, F and u. p and c will come later from f calculation. u =
%(Q+R)/2, F = (Q-R)/2
%
%Make nested for loops to range i and j. starts from (2,1),(3,1)...
then
%(3,2),(4,2)... and so on since the (i,i) positions are already
determined
%and should not be changed!!
%1 = Q; 2 = R; 3 = F; 4 = p; 5 = u; 6 = c; 7 = H; 8 = t
%
for j=1:crs
226
for i=j+1:crs
Char(i,j,2) = Char(j,j,2);
Char(i,j,3) = (Char(i,j,1) - Char(i,j,2))/2;
Char(i,j,5) = (Char(i,j,1) + Char(i,j,2))/2;
%
%Get p,c, values from F [Char(i,j,3)] need another loop for sampuc.
%sampuc-2 is F, sampuc-1 is P, sampuc-3 is c
%
for k=1:temp
if
((sampuc(k,2)<=Char(i,j,3))
&&
(Char(i,j,3)
<
sampuc(k+1,2)))
%
Char(i,j,4)
=
(Char(i,j,3)
sampuc(k,2))*((sampuc(k+1,1) -...
sampuc(k,1))/(sampuc(k+1,2)
sampuc(k,2)))
+
sampuc(k,1);
%
Char(i,j,6)
=
(Char(i,j,3)
sampuc(k,2))*((sampuc(k+1,3) -...
sampuc(k,3))/(sampuc(k+1,2)
sampuc(k,2)))
+
sampuc(k,3);
end
end
end
end
%
% All values are now posted for Q, R, F, p, u, and c. the H and t
values
% are next. A special case exists for both the (i,1) values and the
(j,j)
% values. The (i,1) values will be done independently now.
%
%Intialize the (1,1) value. H = h. t=(newut(1,1) + h/c(1,0))
%1 = Q; 2 = R; 3 = F; 4 = p; 5 = u; 6 = c; 7 = H; 8 = t
%
Char(1,1,7) = h;
Char(1,1,8) = newut(1,1) + h/(C0 + C1*(newut(1,2)) + C2*(newut(1,2))^2
+ C3*(newut(1,2))^3);
%
%Now for the rest of the (i,1) intersections, sound speeds in-between
%intersecting points are taken to be the average of the sound speeds.
%ci0 is the sound speed of the positive going characteristic, ci1 the
%negative going at the intersection
%
for i=2:crs
%
%Get time first
%
ci0 = C0 + C1*(newut(i,2)) + C2*(newut(i,2))^2 + C3*(newut(i,2))^3;
ci1 = (Char(i-1,1,6) + Char(i,1,6))/2;
%
Char(i,1,8) = (newut(i,1)*ci0 + Char(i-1,1,8)*ci1 + Char(i1,1,7))/(ci0 + ci1);
%
227
%Get h value next
%
Char(i,1,7) = (Char(i,1,8) - newut(i,1))*ci0;
%
end
%
%Now to fill in the rest of them. A special case exists for the (j,j)
%characteristic
intersection
since
there
is
not
a
negative
characteristic to the
%right (in the window). The first value in this loop will be (2,2)
since
%all of the (i,1) intersections have been taken care of.
%1 = Q; 2 = R; 3 = F; 4 = p; 5 = u; 6 = c; 7 = H; 8 = t
%
for j=2:crs
for i=j:crs
if i==j
Char(i,j,7) = h;
ci0 = (Char(i,j-1,6) + Char(i,j,6))/2;
Char(i,j,8) = (h - Char(i,j-1,7))/ci0 + Char(i,j-1,8);
else
ci0=(Char(i,j-1,6) + Char(i,j,6))/2;
ci1=(Char(i-1,j,6) + Char(i,j,6))/2;
%
%Get time first
%
Char(i,j,8)
=
(Char(i-1,j,7)
Char(i,j-1,7)
+
ci0*Char(i,j-1,8)...
+ ci1*Char(i-1,j,8))/(ci0 + ci1);
%
%Get h value next
%
Char(i,j,7) = ci0*(Char(i,j,8) - Char(i,j-1,8)) + Char(i,j1,7);
end
end
end
%
%Write interface velocity to a file
%
intut=zeros(crs,2);
for i=1:crs
intut(i,1) = Char(i,i,8);
intut(i,2) = Char(i,i,5);
end
%
%Prompt user for file name to save data.
%
filenam=input('Enter File name to Save data (include extension)
>>','s');
dlmwrite (filenam, intut, 'precision',7);
228
References for Appendix A 1. MATLAB® 7.10.0 (R2010a, The MathWorksTM, Natick, MA, 2010). 2. J.‐P. Davis, CHARICE Version 1.1 Sandia National Laboratories Report SAND2008‐6035, October 2008. 3. D. E. Grady and E. G. Young, Evaluation of Constitutive Properties from Velocity Interferometer Data, Sandia National Laboratories Report SAND 75‐0650, August 1976. 4. J. R. Maw, in Shock Compression of Condensed Matter – 2003 edited by M.D. Furnish, Y. M. Gupta, and J. W. Forbes (American Institute of Physics, New York, 2004) p. 1217.
229
Appendix B Jump Conditions for the Peak State Calculations in Chapters 4, 5, and 7 with a Lagrangian Inelastic Wave Velocity This appendix presents the coordinate change from a lab frame (Eulerian) inelastic wave velocity to a Lagrangian inelastic wave velocity (second wave in two‐step shock wave), and presents jump conditions written with the Lagrangian inelastic wave velocity. The jump conditions presented here in Lagrangian coordinates are only valid for the second step of a two‐
step shock wave (two steady waves). The jump conditions, written in terms of the Lagrangian inelastic wave velocity, were used for the peak state calculations for the x‐cut quartz (Chapter 4), z‐cut quartz (Chapter 5) and GGG (Chapter 7) shock compression experiments. Terms State 0 (subscript 0): Ambient density, zero stress, and zero particle velocity. State 1 (subscript 1): The density, stress, and particle velocity state corresponding to the elastic limit. State 2 (subscript 2): The density, stress, and particle velocity state corresponding to the peak state. : Longitudinal stress. : Particle velocity. : Density. : Elastic wave velocity. : The inelastic wave velocity in Lagrangian coordinates. : The inelastic wave velocity in lab frame (Eulerian) coordinates. 230 B.1 Coordinate Change from Lab Frame (Eulerian) to Lagrangian Inelastic Wave Velocity The coordinate change for a two‐jump wave can be obtained by comparing the time‐lab frame distance and the time‐Lagrangian distance diagrams, shown in Fig. 1a and 1b, respectively. Consider an in‐material gauge that does not interact with the waves and has an initial position of Xgauge. The elastic wave reaches this gauge position at time t1, and the inelastic wave reaches this gauge position at time t2, irrespective of the coordinates. In the lab frame coordinates, the gauge moves with velocity for times after between times and , and moves with velocity . By the time the second wave has arrived at the gauge in lab frame from its starting location. Therefore, the coordinates, it has moved a distance of inelastic wave velocity in lab frame coordinates is: (B.1)
In Lagrangian coordinates, the gauge is stationary for all times, so the inelastic wave velocity in Lagrangian coordinates is: (B.2)
In both frames the elastic wave velocity and the time are related through: 231 (B.3)
In-material
gauge
State 2
Left boundary
u2
Gauge Move
= u1(t2 - t1)
t2
u1
Time
E
D2
a State 0
t1
D1
State 1
XGauge
Lab Frame Distance
a. In-material
State 2
gauge
Left boundary
Time
State 1
t2
t1
L
D2
D1
State 0
XGauge
Lagrangian Distance
b. Figure B.1. Time‐ lab frame (Eulerian) distance diagram (a) and time‐Lagrangian distance diagram (b) for a two‐step wave. The behavior of an imbedded in‐material gauge is shown in both figures. The details of the figures are discussed in the text. 232 The expression for the lab frame inelastic wave velocity, Eq. 1, can be written in terms of the Lagrangian inelastic wave velocity and the elastic wave velocity by combining Eq. 1‐3: (B.4)
Multiplying the top and bottom of the RHS of Eq. 4 by /
and rearranging terms gives the following expression for the lab frame inelastic wave velocity: (B.5)
From the jump conditions:1 (B.6)
Combining Eq. 5 and 6, the following expression is obtained which relates the Lagrangian and lab frame inelastic wave velocities: (B.7)
B.2. Jump Conditions with Lagrangian Inelastic Wave Velocity The momentum and mass jump conditions, in lab frame coordinates, that relate state 1 to state 2 are:1 233 (B.8)
(B.9)
Substituting the lab frame inelastic wave velocities in Eq. 8‐9 with Eq. 7 gives the jump conditions that relate state 1 to state 2 in terms of the Lagrangian inelastic wave velocity. (B.10)
The above expressions were used in Chapters 4, 5, and 7 for the peak state calculations. 234 (B.11)
References for Appendix B 1. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, High Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970), p. 293. 235