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EXPERIMENTAL INVESTIGATION OF HIGH VELOCITY
IMPACTS ON BRITTLE MATERIALS
by
DAVID ISAAC NATHENSON
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Thesis Advisor: Dr. Vikas Prakash
Department of Mechanical & Aerospace Engineering
CASE WESTERN RESERVE UNIVERSITY
May, 2006
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
______________________________________________________
candidate for the Ph.D. degree *.
(signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein.
Copyright © 2006 by David Isaac NathensonAll rights reserved
Dedicated to my wife,
Randi Gross Nathenson,
For her constant
Love and support.
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Table of ContentsTitle Page iSignature Sheet iiCopyright page iiiDedication Page ivTable of Contents 1List of Tables 4List of Figures 5Acknowledgements 7Abstract 8
Chapter I – Introduction 9A. Properties of Brittle Materials 10A.1. Soda Lime Glass 11A.2. Silicon Nitride 18B. Intent of This Dissertation 22References 25
Chapter II – Configuration and Procedures for Particle Impact Experiments 28A. Components 29A.1. Projectile Firing Mechanism 29A.2. Gun Barrel and Sabot Stripper 31A.3. Impact Chamber and Target Holder 32B. Measuring Systems 34B.1. High Speed Camera 34B.2. Strain Gages 35B.3. Laser Velocity System 36C. Experimental Procedures 37C.1. Series Preparation 37C.1.1. Soda Lime Glass Specimen Preparation 37C.1.2. Sabot/Projectile Preparation 38C.2. Setup Sequence 39C.2.1. Projectile Accelerator Alignment 39C.2.2. Measurement System Preparation 39C.3. Pre-Firing Sequence 41D. Summary 42References 43Figures 43
Chapter III – Particle Impact Experiments on Soda Lime Glass 48A. Background on Particle Impact of Brittle Materials 48A.1. Theoretical Calculations 49A.1.1. Hertzian Theory 50A.1.2. Computed Longitudinal, Shear and Rayleigh Wave Profiles 52A.1.2.1. Lamb’s Solution 53A.1.2.2. Mitra’s Solution 55A.1.3. Estimation of the Energy Dissipated by Elastic Waves 58A.2. Previous Research on the Cracking Patterns of Soda Lime Glass 61A.2.1. Configurations and Computations for Previous Experiments 61A.2.2. Results of Previous Experiments 64B. Design of Soda Lime Glass Particle Experiments 69C. Results and Analysis of the Soda Lime Glass Experiments 71C.1. Projectile Velocity vs. Firing Chamber Pressure Calibration 71C.2. Time of Contact Between the Sphere and the Plate 73C.3. Examination of the Impact Surface Strain Gage Records 74C.3.1. Strain Wave Components Explained Using Theory and Numerical Methods 74
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C.3.2. Description of Experimental Strain Records 77C.3.3. Correlation Between Rear Surface Vibration and Crack Oscillation 79C.4. Impact Energies and Coefficients of Restitution 81C.4.1. Determination of Experimental Impact Energy and Coefficient of Restitution 814.2. Elastic Stress Wave and Vibrations as Portions of the Impact Energy 84C.5. Cracking Pattern Behavior 85C.5.1. Details of Cracking Following Experiments 86C.5.2. Dynamic Evolution of Cracking 91D. Significance of Results 95References 99Tables 101Figures 106
Chapter IV – Planar Impact Experiments: Configuration and Procedures 124A. The Single Stage Gas Gun and Observation Systems 124A.1. Description of the Single Stage Gas Gun 125A.2. Laser Velocity, VISAR and Tilt Pin Measurement Systems 127A.2.1. Laser Velocity System 127A.2.2. Tilt Measurement and Triggering System 128A.2.3. VISAR Interferometer 129B. Experimental Procedures 132B.1. Specimen and Flyer Preparation 132B.1.1. Materials for Specimens and Flyers 132B.1.2. Assembly of the Projectiles and Specimens 133B.2. Gas Gun Setup Sequence 135B.3. Firing Sequence 137C. Summary 140References 141Figures 142
Chapter V – The Shock Response of AS800 Grade Silicon Nitride 147A. Background on Shock Compression Experimentation 148A.1. The Theory of Planar Shock Compression 148A.1.1. Elastic Impact Theory 148A.1.2. Elastic-Plastic Impact Theory and Experiments 151A.2. The Failure in Dynamic Tension due to the Spallation of Silicon Nitride 155A.2.1. The Theory of Material Spall 155A.2.2. Experimental Determination of Spall Strength 159B. Experimental Design 162B.1. Description of Materials Under Investigation 162B.2. Experimental Matrix for the Shock Compression Tests 163B.3. Experimental Matrix for the Pressure-shear Impact Tests 164C. Experimental Analysis and Results 168C.1. Elastic and Hydrodynamic Relationships in the Hugoniot State 168C.1.1. Shock Velocity vs. Particle Velocity in the Hugoniot State 168C.1.2.Difference in Hugoniot State Variables due to Pressure-Shear Loading 170C.1.3. Elastic Hugoniot Stress and Strain Relationship 170C.1.4. Calculation of the HEL 171C.1.5. Determination of the Hydrodynamic Hugoniot Stress and Strain States 173C.2. Measurement of the Spall Strength 174C.2.1. Derivation of the Spall Strength Formulae 176C.2.2. Spall Strength and Material Failure Mode during Spall in Pure Shock Compression
179C.2.3. Spall Strength and Failure Modes Under Combined Pressure and Shear Impact Loading
182C.3. Material Failure Modes during Dynamic Spall and Fractography of the Spall Surface 183
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D. Summary of Planar Results 186References 190Tables 191Figures 193
Chapter VI – The Effect of Shock Re-Shock, Shock Unloading, and Stress Reverberationson the Behavior of Silicon Nitride 206A. Theory and Computations for Multiple Shock Experiments 208A.1. Shock Re-shock and Shock Unload Experiments 208A.2. Reverberation Experiments 213B. Design of Shock Re-shock, Shock Unload, and Reverberation Experiments 221C. Results and Analyses of Multiple Shock Experiments 223C.1. Shock-Re-shock and Shock-Unload Experiments 224C.1.1. Computations for Shock Re-Shock and Shock Unload Experiments 223C.1.2. Observations and Analysis of Shock Re-shock and Shock Unload Experiments 228C.2. Reverberation Experiments 230C.2.1. Computations for Shock Reverberation Experiments 230C.2.2. Observations and Analysis of the Shock Reverberation Experiments 234D. Conclusions and Discussion 236References 238Tables 239Figures 240
Chapter VII – Investigation of Shock Induced Failure Wave Propagation in Soda Lime Glass248
A. Description of the Nature of the Failure Wave 249A.1. Experimental Determination of the Properties and Effects of the Failure Front 249A.2. Proposed Mechanisms for the Propagation of the Failure Front 254B. Design of Experiments to Investigate the Failure Wave in Soda Lime Glass 261B.1. Description of the Experimental Configurations 261B.2. Modifications to the VISAR System to Enabled the Simultaneous Measurement of theNormal and Transverse Components of the Particle Velocity Histories 263C. Experimental Results and Analysis of Failure Waves in Soda Lime Glass 265C.1. Spall Strength of Glass under Shock Compression and Pressure-Shear Loading 265C.1.1. Normal Particle Velocity Calculations 265C.1.2. Transverse Particle Velocity Calculations 268C.1.3. Spall Strength and Particle Velocity Observations for Aluminum Impacting Glass 270C.2. The Effects of the Failure Wave on the Properties of Soda Lime Glass 272C.2.1. Computations for the Elastic Predictions 273C.2.2. Experimental Observations of Spall Strength and Transverse Particle Velocity 274C.3. Observation of the Impedance in the Comminuted Material 278C.3.1. Elastic Computations for the Internal Stresses and the Acoustic Impedance in ExperimentsG/WC1 and G/WC2 279C.3.2. Measurements of the Normal and Transverse Particle Velocities and the ImpedanceChange Due to the Failure Wave 280D. Conclusions and Discussion 283References 287Tables 288Figures 289
Chapter VIII – Summary of Experimental Analyses 303A. Particle Impact Experiments on Soda Lime Glass 304B. Planar Shock Compression and Pressure-Shear Experiments on AS800 Grade Silicon Nitride
308C. The Effect of Dual Shock Loading and Shock Reverberation on the Strength of AS800 GradeSilicon Nitride 311
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D. Shock Induced Failure Waves During Dynamic Compression of Soda Lime Glass 312E. Summary 316References 318
Appendices 321Appendix 1: Equations of Wave Propagation Resulting from a Point Load on a Elastic Half Space
3211.1. Using the Equations of Motion to Generate the Radial and Vertical Surface Displacements
3211.2. Calculating the Radial and Vertical Stresses on the Surface 3251.3. Specific Solution for a Concentrated Vertical Pressure 3291.4. Generalizing the Equations of Displacement for an Arbitrary Loading 338
Appendix 2: Calculations for the Surface Wave Produced by the Loading of a Circular Area of aHalf Space by an Impulse Load. 3402.1. Solutions to the Equations of Motion Transformed using Laplace Transforms 3402.2. Cagniard’s method for obtaining the inverse Laplace transforms of, I1, I2 3462.3. Surface Displacements 351
Appendix 3: Calculation of Energy Absorbed by Elastic Waves During Impact 358References 368Figures 369
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List of TablesTable 3.1: Experiment List Including Specimen Impact Velocity and Thickness. 101Table 3.2: Experimental, Theoretical, and Numeric Peak Contact Times. 102Table 3.3: Experimental Strain Measurements. 103Table 3.4: Impact Kinetics and Coefficients of Restitution. 104Table 3.5: Post Impact Static Cracking Pattern Details 105
Table 5.1: Experimental Results for Hugoniot variables 192Table 5.2: Mohr's Circle Stress Calculations 192Table 5.3: Experimental Results for Spall Stress and Recovery Time 193
Table 6.1: Dynamic Properties for Experiments SR-1 and SU-1 239Table 6.2: Dynamic Properties for Experiments RB-1 and RB-2 239
Table 7.1: Parameters for spall strength and tungsten carbide on glass experiments 288Table 7.2: Parameters for glass on tungsten carbide experiments 288
List of FiguresFigure 2.1: Complete view of projectile accelerator 43Figure 2.2: View of firing chamber with piston and sabot loading procedure 43Figure 2.3: Sabot and projectile assembly diagram 44Figure 2.4: Sabot stripper diagram 44Figure 2.5: Impact chamber diagram and Measurement Setup 45Figure 2.6: Target holder diagram 46Figure 2.7: CEA-06-032WT-120 strain gage image 47
Figure 3.1: Hertzian approximation and Numerical Simulations for Impact Force Curve 106Figure 3.2: Surface strain profile using the equations from Lamb 107Figure 3.3: Surface strain profile using the equations from Mitra 108Figure 3.4: Calibration curve for projectile impact including all experiments 109Figure 3.5: Plot of experimental, theoretical and numerical contact time vs. velocity 110Figure 3.6: LS-DYNA surface strains profiles on a half space 111Figure 3.7: LS-DYNA surface strain profiles on plates with infinite lateral boundaries 112Figure 3.8: LS-DYNA surface strain profiles on plates with finite rectangular geometries 113Figure 3.9: Comparison plot of strain profiles in 350 m/s range for all four thicknesses. 114Figure 3.10: Variation of strain with specimen thickness and velocity 115Figure 3.11: Rear surface strains from a bending series impact of a 5 mm specimen 116Figure 3.12: Rear surface strains from a bending series impact of a 15 mm specimen 117Figure 3.13 Example of oscilloscope output from laser velocity triggering system 118Figure 3.14: Variation in rebound kinetic energy to impact kinetic energy with velocity 119Figure 3.15: Variation in coefficient of restitution with velocity 120Figure 3.16: Impact of 5 mm thick specimen at 323 m/s (BQ-13) showing conical cracks 121Figure 3.17: Impact of 3 mm thick specimen at 345 m/s (BQ-26) showing radial cracks 122Figure 3.18: Impact of 15 mm thick specimen at 371 m/s (BQ-25) showing lateral cracks 123
Figure 4.1 Overview of the single stage gas gun 142Figure 4.2: Firing chamber and air supply diagram 143Figure 4.3: Valyn VISAR interferometer design 144Figure 4.4: Fringe patterns and velocity curve for representative experiment 145Figure 4.5: Representative velocity measurement system beam intensity plot 145Figure 4.6: Specimen and flyer ring configurations for pressure-shear experiments. 146
Figure 5.1: Calculation plots for elastic compression impact experiments. 193Figure 5.2: Calculation plots for elastic-plastic compression impact experiments 193Figure 5.3: Normal experimental free surface velocity history plots 194
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Figure 5.4: Shear experiments free surface velocity history plots 195Figure 5.5: Calculation plots for pressure-shear impact experiments. 195Figure 5.6 Shock velocity plotted versus Hugoniot state particle velocity 196Figure 5.7: Hugoniot elastic and hydrodynamic stresses and strains 197Figure 5.8: Rear surface velocity history plot showing Hugoniot Elastic Limit. 198Figure 5.9: Spall strength plotted versus free surface velocity for normal impact 199Figure 5.10: Spall strength plotted versus free surface velocity for pressure-shear 200Figure 5.11: Post impact fragment images 201Figure 5.12: SEM images of SC-8, 31.8 m/s Hugoniot particle velocity 202Figure 5.13: SEM images of SC-2, 100 m/s Hugoniot particle velocity 203Figure 5.14: SEM images of SC-4, 178 m/s Hugoniot particle velocity 203Figure 5.15: SEM images of SC-12, 152 m/s 204Figure 5.16: SEM images of Polished surface of specimen, pre-impact 205
Figure 6.1: Theoretical stress vs. strain diagram 240Figure 6.2: Reverberation experiment design 241Figure 6.3: Time vs. distance and stress vs. velocity diagrams experiment RB-2 241Figure 6.4: Design for shock re-shock and shock release experiments 242Figure 6.5: Time vs. distance and stress vs. velocity diagrams experiment RB-1 242Figure 6.6: Time vs. distance and stress vs. velocity diagrams experiment SR-1 243Figure 6.7: Time vs. distance and stress vs. velocity diagrams experiment SU-1 243Figure 6.8: Velocity vs. time profiles for experiments SR-1 and SU-1. 244Figure 6.9: The stress vs. strain plot for reloading and unloading experiments 246Figure 6.10: Velocity vs. time profiles experiments RB-1 and RB-2. 246Figure 6.11: Reverberation experiments stress and strain levels 247
Figure 7.1: Specimen configuration for normal spall strength experiment. 289Figure 7.2: Specimen configuration for pressure-shear spall strength experiment. 289Figure 7.3: Time vs. distance and stress vs. velocity diagrams for spall experiments 290Figure 7.4: Specimen configuration for WC/Glass experiments 290Figure 7.5: Time vs. distance and stress vs. velocity diagrams exp. WC/G1, WC/G3 291Figure 7.6: Time vs. distance and stress vs. velocity diagrams exp. WC/G2, WC/G4 291Figure 7.7: Specimen Configuration for Glass/WC Experiments 292Figure 7.8: Time vs. distance and stress vs. velocity diagrams exp. G/WC1, G/WC 292Figure 7.9: Experimental measurement technique 293Figure 7.10: Velocity Profile for Spall experiment Al/G1 294Figure 7.11: Velocity Profile for Spall experiment Al/G2 at 18 degrees 295Figure 7.12: Velocity Profile for Experiment WC/G1 296Figure 7.13: Velocity Profile for Experiment WC/G2 297Figure 7.14: Velocity Profile for Experiment WC/G3 298Figure 7.15: Velocity Profile for Experiment WC/G4 399Figure 7.16: Velocity Profile for Experiment G/WC1 300Figure 7.17: Velocity Profile for Experiment G/WC2 301Figure 7.18: Transverse velocity measurements 302
Figure A.1: Surface displacement as described by Lamb’s equations 369Figure A.2: Schematic solutions of entire wave based on Lamb’s equations 370Figure A.3: Mitra’s surface displacement in the radial direction 371
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Acknowledgments
I would like to acknowledge the assistance of Processor Vikas Prakash and J Michael Pereirafrom NASA without whom this work would not have been possible.
I would also like to thank both the NASA Glenn Research Center and Case Western ReserveUniversity for financial support.
Additionally, I wish to thank the members of the D. K . Wright lab for their assistance and support:Mostafa Shazly, Liren Tsai, Guodong Chen, George Sunny, Fu Ping Yuan, Naoto Utsumi, andTang Xin.
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Experimental Investigation of High VelocityImpacts on Brittle Materials
Abstract
by
DAVID ISAAC NATHENSON
Experiments were conducted on soda lime glass and AS800 grade silicon nitride. Soda lime glassis often used in windows of military vehicles and aircraft where integrity in the event of shrapnelimpacts is of vital concern. AS800 grade silicon nitride is considered one of the leading materialcandidates for the next generation of aircraft engine turbine blades because of its superior hightemperature properties when compared with nickel based super-alloys. The suitability of thesematerials for their applications depends upon their response to point and planar dynamic impactloading. An experimental apparatus was constructed to fire one-sixteenth inch diameterhardened chrome steel ball bearings at 50 mm square soda lime glass blocks of thicknessesbetween 3 mm and 25.4 mm. Inelasticity due to the crushed zone effects the coefficients ofrestitution and the surface strains. The change in severity of cracking with velocity and specimenthickness is observed. Shock compression and pressure-shear experiments were conducted bymeans of a single stage gas gun capable of attaining impact velocities of 600 m/s. High velocityplanar shock compression experiments on soda lime glass reveal a lack of spall strength, and adecrease of shear impedance and shear strength in the presence of a failure wave. Thelongitudinal impedance remains nearly constant. The spall strength of glass is 3.49 GPa and issensitive to the presence of shear. Shock compression studies on silicon nitride using normalshock compression show that the material has a Hugoniot Elastic Limit of 12 GPa and that thespall strength decreases with increasing impact velocity due to damage below the HEL. Thepresence of inelastic deformation stops this trend, while the presence of shear increases the rateof spall strength drop by five times because of more severe microscopic damage. Experimentsinvolving multiple shocks on silicon nitride show that material loading and unloading follows theshock Hugoniot closely. The HEL of the shocked material is decreased by 6%, while the residualstrength remains high. This indicates that the longitudinal properties of silicon nitride including itsshock impedance are not dramatically affected during the initial shock compression.
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Chapter I: Introduction
Brittle materials have many uses in aerospace and military applications. Glass is
one brittle material that is utilized in vision elements of both aircraft and combat
vehicles. In these situations glass is subjected to impact conditions from objects
such as projectiles or runway debris. Due to the lack of plastic energy
dissipation, glass fails primarily by brittle cracking. Ceramics such as AS800
grade silicon nitride are candidates for the next generation of aircraft engine
turbine blades and casings because they have higher strength properties at
elevated temperatures than current nickel based super-alloys. However, ceramics
are more brittle than these alloys.
In order to determine the suitability of these brittle materials for applications
where resistance to fracture is important, their response to dynamic impact
loading must be studied. Both small particle impact and planar shock
compression loading are among current testing methods. By impacting soda lime
glass with spherical particles, the three dimensional wave and cracking behavior
can be examined. Shock compression involves a state of plane strain in which the
behavior of glass and silicon nitride can be quantitatively examined. Through
these methods the dynamic response of the material can be discerned. This
assists in the determination of their suitability for aerospace applications.
In this chapter, the specific properties of the two materials are discussed. The
specifics of each material’s brittle behavior are explored. The cracking behavior
in soda lime glass under particle impact and the failure wave propagation under
shock compression are summarized. The material properties and point impact
classification of this specific silicon nitride are reviewed. From this, the reasons
for this study emerge. The specifics of the study and its goals are then explained.
Finally, the point impacts on soda lime glass, and the various shock compression
experiments that are performed in this study are categorized.
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A. Properties of Brittle Materials
Brittle materials have many qualities that recommend them for aerospace and
military applications. Ceramics for example when compared with ductile
materials retain more strength at elevated temperatures. Typically, their melting
temperatures are higher [ASM, 1999]. They also have less creep than ductile
materials. Glasses on the other hand are commonly used in windows because of
their clarity. They also have high strength and high shock compressive yield
limits [Rosenberg et al., 1985; Bourne et al., 1996]. These properties recommend
ceramics and glasses for their respective applications.
However, brittle materials have a distinct downside in their lack of ductility. In
broad terms, brittle cracking and fracture occurs when irreversible damage occurs
before the yield stress is reached, while in a ductile failure, the material will yield
before failure. The brittle fracture leaves a glittering, crystalline appearance, or
one covered with river lines caused by tearing between slip planes. High strength
materials, such as ceramics, like silicon carbide and high strength steels such as
18 Ni maraging steel are examples of brittle materials [Dowling, 1999]. One
method of quantifying brittle versus ductile materials is by examining the plastic
strain to failure. Ceramics and glasses have almost no plasticity before failure.
As a result, brittle fracture occurs at all normal temperatures and strain rates in
these materials. For some of these materials such as ferritic steels raising the
temperature above a critical ductile to brittle transition temperature (DBTT)
causes a substantial increase in ductility [Anderson, 1995]. Ductile failure can
show both global and local necking. Micro-voids typically expand and coalesce
causing the failure. This results in a dimpled appearance either locally or
globally. Examples of ductile materials include lower strength steels such as AISI
1020 and aluminum 7075-T6 [Dowling, 1999]. Brittle fractures are generally less
desirable in applications subjected to large static or dynamic stresses occur due to
the lack of warning before failure and the catastrophic nature of the failure.
However, ductile materials usually have lower strength and higher creep rates,
especially at elevated temperatures. Thus, brittle materials are preferred in
applications where strength is the overriding concern. For this reason, any
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attempt to characterize the acceptability of brittle materials for aerospace or
military applications must include an assessment of the dynamic fracture/spall
strength. In the following sections, the relevant behavior and properties are
reviewed for both soda lime glass and silicon nitride.
A.1. Soda Lime Glass
Soda lime glass has many uses and similar properties to many glass ceramics. It
is employed as the vision elements of military and aviation vehicles. Glass is also
employed as a modeling material for glass ceramics that have similar brittle
behavior. Because soda lime glass is brittle, it has practically no plastic
deformation. Rather, stress waves and failure waves are responsible for most of
the energy dissipation upon impact. These effects can be studied using particle
impact and shock compression experimentation.
Experimentally and theoretically, cracking and stress behavior in glasses have
been explored. Under lower velocity dynamic loading a combination of stress and
cracking systems dominates the material response. Two types of stress fields are
applicable to small particle impacts. The first is Hertzian which occurs for a
blunt tipped projectile and produces a cone crack that follows the path of the
smallest principal stress [Evans and Wilshaw, 1976]. The other field is
Boussinesq, which involves point impacts and generates stresses which cause half
penny radial cracks on loading and upon unloading saucer shaped lateral cracks
[Lawn and Fuller, 1975]. At higher impact velocities the stress waves have a
more pronounced effect on the cracking patterns [Field, 1988; Field et al., 1989].
Using high speed photography and strain gages, these stress and cracking
patterns are observed in particle indentation experiments.
As described by Hertz and others, the crack formation in the case of a blunt
spherical indenter follows a distinct path. The crack path is defined by principal
stress trajectories. This path is, to a first approximation, orthogonal to the
highest principal stress in the stress field prior to the cracking. The crack thus
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propagates along the planes defined smaller principal stresses. This was
determined by comparing experimental crack paths with contours of the principle
stresses [Frank and Lawn, 1967]. Ideally, the crack tries to propagate in a
straight line. However, oblique stresses are generated on the crack when the
direction of the principal stresses changes. These stresses force the crack to
correspond to the principal stress directions. [Frank and Lawn, 1967].
The cracking is divided into four parts: nucleation, formation, propagation, and
unloading [Lawn et al., 1974]. The initial crack forms just outside the contact
circle radius where the stresses are tensile. There may be more than one crack
initiated. These cracks form at existing flaws at the surface, which extend as the
stress level increases. After the cracks initiate, the dominant crack extends in a
circular manner around the region of contact. Thus, a shallow surface crack is
created. For any stable extension of the surface crack a critical level of energy
must be overcome by increasing the contact force. Overcoming this energy
results in stable downward motion and a ring-like crack [Lawn et al., 1974]. The
crack continues downward to a critical depth of about 0.1 times the contact
radius. Then, the crack begins to propagate in an unstable manner. This
propagation occurs in the form of a Hertzian cone with a depth about equal to
the contact radius.
Sharp indenters, such as a Vikers pyramid or a cone have distinctly different
stress fields and hence cracking patterns when compared with blunt indenters.
Theoretical elastic stress intensity varies inversely with the square of the distance
from the tip. This is commonly known as the Boussinesq field [Lawn and Fuller,
1975]. These sharp indenters are more often employed because the indentation
size has no effect on the contact pressure [Lawn and Wilshaw, 1975].
The largest tensile stress is now immediately beneath the tip. Micro-cracking
initiates there [Lawn and Fuller, 1975]. These micro-cracks are then extended to
form macroscopic cracks in near-penny shapes. The cracks will grow during
loading in the direction of the trajectories of the smallest principal stress as in
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the blunt sphere case. The crack will propagate both downward, in a median
pattern both downwards and outwards orthogonal to the hoop stress [Lawn and
Swain, 1975]. These cracks, called radial cracks, because of their extension on
radial (constant θ) planes will grow in a stable fashion and full-penny geometry
until they reach a depth of approximately one contact radius [Lawn and Fuller,
1975]. Following this, the cracks will break through compressive stress lobes in
the hoop direction that were restricting the propagation. The restrictive
compressive stress lobes occur at an angle greater than or equal to 51.8º in the
second principle stress field [Lawn and Swain, 1975]. The median cracks will
then reach the free surface and continue to propagate in a stable fashion. They
then have half penny geometry [Lawn and Fuller, 1975].
Additionally, during unloading, the residual stress field causes the creation of
cracks in the direction parallel to the free surface. These lateral cracks nucleate
at the edge of the deformation zone close to the surface and propagate in a saucer
shape. This can cause chipping of the surface [Lawn and Fuller, 1975]. This
chipping is the removal of material from the surface because of the lateral crack
extension [Evans and Wilshaw, 1976]. Thus two types of crack systems,
median/radial and lateral, exist for brittle materials under the Boussinesq stress
field.
Under short duration and high amplitude loading inertia effects start to dominate
the stress field patterns. For 3 mm and 5 mm diameter steel projectiles on soda
lime glass this shift in stress patterns is observed to occur at around 250 m/s
[Field, 1988; Field et al., 1989]. The internal fractures become very chaotic and
the surface cracks are caused by Rayleigh surface waves. Back surface damage is
caused by the reflection of the compressive loading wave as a tensile wave from
the back surface [Field et al., 1989]. It is apparent that this is a different
category of cracking than that obtained at lower impact speeds, where cracking
patterns similar to those that exist in quasi-static indentation testing are
observed [Field, 1988].
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In glasses under planar shock compression, a failure wave has been observed by
several investigators. This failure wave has been deduced from the gradual
propagation of failed material resulting in a comminuted state [Kanel et al.,
1992]. It has also been observed directly using high speed photography [Brar et
al. 1992; Bourne et al., 1995]. Many researchers like Clifton et al. [1997] have
investigated glass using shock compression experiments. Espinosa et al. [1997]
and Feng [1999] have suggested models for failure wave.
Failure waves have been identified in several brittle materials. Brar et al. [1992]
noted a failure wave propagating in pyrex bars and soda lime glass plates under
impact loading. Photographic records were recorded during the pyrex bar
experiments showing the propagation of this wave. It is noted in these
experiments that impacting with steel plates instead of pyrex bars resulted in a
higher failure wave velocity; this is understood to be due to the larger impact
stress that is generated with the steel bar at the same impact velocity [Brar et
al., 1992].
Brar et al. [1992] and Kanel et al. [1992] suggest that the generation of a
compression wave at the intersection of the release wave in shock compression
experiments on soda lime glass means that the failed material has a slightly lower
impedance than the intact material. Brar et al. [1992] suggest that this is due to
the comminution of the glass, a fact which is confirmed by the lack of spall
strength. The spall strength drops from 30 kbar before the failure wave to
almost zero behind the failure wave. Also, transverse strain gages indicate a
jump in the transverse stress level behind the failure wave at experiments with
impact stresses greater than 38 kbar. This increased stress reflects a drop in the
shear strength of the material behind the failure wave.
The failure wave has been observed to travel at a speed of about 1.5 to 2.5 km/s
[Ginzberg and Rosenberg, 1998] which is lower than the elastic wave speed of 5.7
km/s. The resulting material alteration causes the maximum shear stress the
material can withstand and the spall strength to drop. However, the longitudinal
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wave speed and the elastic impedance remain relatively unchanged [Ginzberg and
Rosenberg, 1998].
This failure wave has been observed to initiate during shock compression at
impact stresses between half of the Hugoniot Elastic Limit (HEL) of the material
and the HEL [Ginzberg and Rosenberg, 1998]. The HEL of the material
represents the uniaxial stress component in the impact direction during shock
compression at which the inelastic deformation begins. This failure wave is
reported in experiments throughout the literature, but the cause of its formation
remains uncertain. The systematic study of this failure wave is necessary in
order to determine its nature and causes.
Additional images of the propagating failure wave were taken by Bourne et al.
[1995]. These plate impact experiments employ copper flyer plates and soda lime
and pyrex glass specimens. The wave front for soda lime glass consist of many
bifurcating cracks. In pyrex, a greater number of small cracks are observed. In
both glasses, crack nucleation is observed at a small number of sites just before
the front. Crack propagation first links up the initiation sites and following
coalescence the entire front propagates [Bourne et al., 1995].
Also in Bourne et al. [1995], the failure wave propagation velocities were observed
to increase with impact velocity. For soda lime glass, the failure wave was
observed to travel at 1.8 km/s for a 250 m/s (2.2 GPa) impact and to increase to
3.6 km/s for a 760 m/s (6.8 GPa) impact. The smaller failure wave velocity is
from about half of the shear wave speed while the higher wave velocity is about
the shear wave speed. This variation is not seen over the same range of impact
velocities in the pyrex glass where for the same set of impact velocities the failure
wave only increases from 3.4 km/s to 3.8 km/s, which is close to the shear wave
velocity in the material. From the study it was concluded by Borne et al. [1995]
that the failure wave is associated with the shock front shear stresses.
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Several models exist describing the failure wave including that of Espinosa et al.
[1997]. There the suggestion that this failure front in plates is due to
inhomogeneous shear flow is discussed. The compressive shock wave causes this
flow on fault planes. The observed micro-cracks then nucleate along shear plane
intersections. This model suggests that the damage is a function of the stress
level [Espinosa et al., 1997].
A comprehensive model for the failure wave is discussed by Feng [2000]. This
model classifies the failure propagation as a diffusive phenomenon. The shock
compression of the material causes heterogeneous microfissuring, shear dilatancy
and void collapsing under a confining stress all at once. Thus the micro-scale
damage is controlled by the heterogeneous deformation. This process causes the
mean stress to increase and the devatoric stress to decrease. The failure wave
results in the conversion of the devatoric strain energy into volumetric potential
energy [Feng, 2000].
In this model the propagation of the failure wave occurs due to the “progressive
percolation of microfissures” [Feng, 2000]. The failure process initiates at the
impact surface. Flaw nucleation occurs rapidly followed by material
comminution without substantial crack growth. The presence of damage at the
surface causes stress concentrations within the material, which provide local
initiation sites and the microfissures spread. The failure propagates laterally
much faster than longitudinally. This spreading of cracks accounts for the
diffusive nature of the model [Feng, 2000].
A comparison of Feng’s [2000] model to experimental results indicate that the
model predicts the longitudinal stress and particle velocity measurements.
Additionally using the time between loading and the initiation of damage as an
alternative to inferring the variations in the failure wave velocity by measuring
the rise time of the waves across the strain gages is suggested. This suggestion
provides better agreement with experimental results. Some questions are left
17
unanswered by the model, however such as inconsistencies in experimental
longitudinal strains and stresses [Feng, 2000].
One set of experiments that have been carried out on soda lime glass use
Hampden steel flyer plates to elevate the stress level to 5 GPa [Clifton et al.,
1997]. Other shock compression tests were carried out by Dandekar [1998] and
Sundram and Clifton [1998]. In Clifton’s Hampden steel experiments, the glass
was impacted at 302 m/s and exhibited a clear spall signal. No failure wave was
observed in this experiment. At higher velocities, near 400 m/s, a failure wave
was exhibited by means of a recompression of the material. No spall strength
behind the failure wave front was observed in these experiments. The transverse
compressive stress was also observed to increase behind the failure front,
representing a decrease in the shear strength of the material [Clifton et al., 1997].
However, the steel exceeds its HEL at this stress level. By using tungsten
carbide flyers which will remain elastic, the soda lime glass response can be
observed with greater accuracy. By using shock compression-shear
experimentation at an angle of 18° and varying the thickness of the specimen and
flyer the spall strength and the shear strength of the glass can be observed.
In examining soda lime glass both point and planar impacts will be studied. The
point impacts enable the material cracking patterns and stresses of three
dimensional impact to be characterized at varying velocities and thicknesses.
The energy dissipation is also determinable from these experiments. This allows
for the characterization of the impact process on a brittle material that has
practical aerospace and military applications. In examining the failure wave at
high velocities, the characterization of this phenomena is extended. Again,
knowing the effects of this failure wave on material properties such as the spall
strength, and the shear stress provides designers with information on the
limitations of the material for its applications.
18
A.2. Silicon Nitride
AS800 grade silicon nitride is a high strength ceramic that is under consideration
to replace current nickel based superalloys in aircraft engine turbine blades due
to its superior properties at high temperatures. For example, the melting
temperature of silicon nitride is around 1800-1900°C [Brandes and Brook, 1999]
whereas the melting temperature for INCONEL MA 956, a nickel superalloy
designed for use in aircraft engines by Special Metals is only between 1311°C and
1400°C [Matweb, 2005]. In creep testing, at 1350°C, the bulk material samples of
AS800 silicon nitride have a lifetime of 100 hours at a tensile stress of about 220
MPa and a lifetime of 1000 hours at about 170 MPa [Lin et al., 2001]. Whereas,
INCONEL MA 754, which is also used in aircraft engine components, has a
lifetime of 100 hours at about 110 MPa rupture strength and a lifetime of 1000
hours at about 100 MPa at 1093°C [Brandes and Brook, 1999]. Silicon nitrides
thus have superior properties of melting temperature and creep resistance to
nickel based superalloys.
However, the ductility of silicon nitrides is less than that of the superalloys.
Ceramics in general are also not as thermally conductive and more likely to
exhibit thermal shock than metallic based high temperature materials. However,
silicon nitride in particular has a thermal shock resistance and toughness that
recommends it for aerospace components [ASM, 1999]. The brittle nature of
failure in the ceramic is still of concern. For this reason, the specific dynamic
strengths of the proposed material must be explored through high velocity shock
experiments.
The AS800 grade silicon nitride was developed by Honeywell Engines. It was
designed to have elongated grain structures and has been toughened [Choi et al.,
2002]. It was also annealed prior to testing at 1200 °C for 2 hours to reduce any
residual machining stresses. [Choi et al., 2002]. Most of the grains were fine,
19
having a size of about 0.5 μm. About 15 to 20 % of the grains were larger and
elongated, with sizes of 1.5-2.0 μm. The aspect ratio of these grains was between
5 and 12. Also, a second phase in the material was observed to be about 10 % of
the total material. This second phase consists of Y10Si3N4O23. No differences
were observed in the structures of the elongated grains and the second phase
grains [Lin et al., 2001]. This grade of silicon nitride has been demonstrated to
have favorable properties under high temperatures such as creep and high
strength retention. Lin et al. [2001] studied these properties for AS800. The
result is a material that has favorable properties for aerospace applications [Choi
et al., 2002].
Silicon nitride has been examined under particle impact and its superiority to
various other brittle materials has been established [Choi et al., 2002]. In this
and other studies, silicon nitride was subjected to impact by 1.59 mm diameter
steel spheres. The AS800 grade silicon nitride was compared with two other
grades: SN282 and NC132. Flexure specimens were impacted at velocities
ranging up to about 440 m/s. The residual strength of the material was then
observed by means of a bending test [Choi et al., 2002].
A critical velocity for impact failure was observed for these three silicon nitride
grades. The experiments indicate that AS800 has a critical velocity of 400 m/s, as
opposed to 300 m/s for SN282, and 230 m/s for NC132. Also, for the specimens
that did not fail under impact, the post-impact strength of AS800 is larger than
that of SN282 between 220 m/s and 440 m/s. The key parameter in the
increased foreign impact damage resistance is the fracture toughness. In fact, the
fracture toughness is linearly related to the critical impact velocity. The AS800
has a fracture toughness of 8.1 ± 0.3 MPa-m1/2. Additionally, an increase of
fracture resistance is recorded when a copper layer is employed to reduce
Hertzian contact stresses [Choi et al., 2002]. This and other studies have
characterized the particle impact behavior of silicon nitrides.
20
Shock compression studies on silicon nitride have been performed. However they
are not extensive and leave unanswered questions. Studies, such as Nahme et al.
[1994] have been performed on silicon nitrides of varying densities. Similar
experimental techniques as those utilized in the present work, a gas gun and
VISARTM interferometer, was employed. One of Nahme’s materials had similar
density, 3.15 g/cm3, and longitudinal wave speed, 10.7 km/s, to the current
study’s AS800. These grades had small particles with 1 μm size. Voids with
diameters between 10 μm and 100 μm are responsible for the density difference.
The resultant spall strength was measured and determined to be between 0.5
GPa and 0.8 GPa [Nahme et al., 1994]. However, the relationship of this spall
strength to velocity was not discussed. Also, these experiments were normal
shock compression with no shear effects.
The Hugoniot Elastic Limit of the material has also been estimated using shock
compression experimentation. In Nahme’s paper, an HEL of 12.1 GPa is
reported for silicon nitride with a density of 3.15 g/cm3 [Nahme et al., 1994].
Other papers have noted a variation of the HEL with grain size. For example,
Mashimo [1998] quoted three HEL ranges for silicon nitrides that have grain sizes
varying from 0.15 μm to 1.2 μm. The HEL is observed to decrease from 17-20
GPa in the smaller grain sizes (0.15 μm to 0.3 μm) to 10-12.5 GPa in the larger
grain sizes (0.5 μm and 1.2 μm) [Mashimo, 1998]. This variation due to grain
size means that establishment of the HEL for each grade of silicon nitride is
necessary.
Silicon carbide (SiC) is another ceramic material with properties that recommend
it for aerospace uses. Silicon carbide exhibits a high HEL. This varies with
microstructure, but is 11.5 GPa forα -SiC (6H) [Feng et al., 1998] and between
13.2 and 15.7 GPa for three other types whose densities vary from 3.16 g/cm3 to
3.23 g/cm3 [Bourne and Millet, 1997]. In shock compression experiments
performed by Bourne and Millet [1997], no spall strength was observed in
specimens impacted at stress levels exceeding the HEL. The spall strength
decreased as the stress approached the elastic limit [Bourne and Millet, 1997].
21
The shear strength of the SiC was shown to increase from 4.5 GPa at the HEL,
when the stress level was 11.5 GPa, by a factor of two at 23 GPa [Yuan et al.,
2001]. However, in Bourne and Millet’s experiments, a failure wave, such as that
seen in the glass plate impact experiments was also seen [Bourne and Millet,
1997]. This failure wave caused reductions of the shear stress in two of three
specimen microstructures of SiC that they studied. One microstructure shows a
near constant shear level and another shows a decrease of 20% from the HEL to
1.4 times the HEL. During the same interval the shear stress in the material
increases by 30 to 40% for all three microstructures before the failure wave and
in the third specimen type after the failure wave [Bourne and Millet, 1997]. This
failure wave degradation of the shear strength and the drop in the spall strength
with increasing impact speed are of concern. Therefore, experiments examining
the degradation in spall strength and observing the shear behavior of silicon
nitride are performed in the current study.
These and other papers leave open several issues of the dynamic strength which
are examined in this study. First, the spall strength variation with both
increasing velocity, is important to investigate. Also the inclusion of a skew
angle and its effects on the spall strength are studied. The material HEL is
examined and from it evaluate the dynamic yield strength is found. Also the
strength of the shocked state of the material is determined. These results provide
a significant increase to the available knowledge of the dynamic behavior of
aerospace grade AS800 silicon nitride.
22
B. Intent of This Dissertation
The study of brittle materials under point impact loading and planar shock
compression is vital to the understanding of dynamic material behavior. The
velocities necessary to perform experiments which test these conditions can be
generated by means of compressed air gas guns, which fire projectiles at hundreds
of meters per second. The result is the capability to examine the material
properties under dynamic loading conditions that approximate real world
impacts. In this dissertation, the effect of dynamic point loading on soda lime
glass and that of planar shock compression on soda lime glass and AS800 grade
silicon nitride are examined.
Small particle impacts at speeds approaching sonic velocity conducted in this
study are similar to real world impacts and can be used to predict damage
patterns. Examples of these real world impacts include foreign impact damage in
aircraft engines and projectile or runway debris impact on windows. The
resulting damage and stress patterns in the materials are critical to determining
the degradation in strength of the material.
In order to examine the properties of soda lime glass under particle impact
conditions, a projectile accelerator system was designed and constructed. This
system, employing a compressed nitrogen firing system and a specially designed
impact chamber allowed for measurements to be taken using several methods.
The records included high speed photographs and surface strain readings. This
system is discussed in Chapter II. Experimental results are covered along with
theoretical and numeric studies in Chapter III. Previous research related to the
impact of the materials is summarized and its relation to the current work is
expressed. The various data collected provides a comprehensive picture of the
variations with velocity and thickness in the cracking patterns, the energy
dissipation, and the surface strain wave patterns.
23
Planar shock compression impacts are used to study the material properties in a
state of uniaxial strain. This condition is achieved by the impact of two precisely
parallel plates under normal impact loads. The condition enables the study of
several basic properties such as the materials’ HEL and spall strength. The
shock experiments were carried out on the existing single-stage gas gun. The
experimental design and procedures are described in Chapter IV. Using the gas
gun facility, three different sets of experiments were carried out.
The shock response of silicon nitride is examined in Chapter V by employing
shock compression experiments. The material begins to exhibit inelastic
deformation in compression at the Hugoniot Elastic Limit, which is proportional
to the dynamic yield stress. The Hugoniot Elastic Limit, or HEL, is an extrinsic
property that represents the maximum stress to which the material can be loaded
prior to plastic deformation. This represents the component of stress generated
in the target plates by the impact in the direction of compression. The dynamic
yield, on the other hand accounts for the HEL stress and the stress components
orthogonal to the direction of impact. In other words, the dynamic yield stress
represents the combination of the longitudinal and transverse stresses necessary
to cause yielding. Also, the limit in resistance to tensile rarefaction wave induced
spallation otherwise known as the spall strength is examined. The current study
evaluates the change in spall strength with impact velocity under shock
compression and under combined shock compression-shear loading. Additionally,
from the data generated by the shock compression impacts, equations are written
relating the material’s elastic and hydrodynamic stress and strain response in the
elastic and the beginning of the plastic regimes. Also the shock velocity versus
particle velocity relationship in the elastic regime is examined. These two curve
types are known as the equation of state and the Hugoniot curve respectively.
In Chapter VI, further study of the dynamic yield stress of silicon nitride was
conducted by employing shock re-shock and shock release experiments. These
experiments make use of multiple loading states to examine the response of the
deformed material to shock loading. In the shock re-shock and shock unload
24
experiments a two material flyer is used with the impedance of the second
material being greater and less than the silicon nitride. By this means, two
velocity profiles are generated which correspond to the reloading and unloading
states. From this response the yield strength of the deformed material can be
calculated. Another method involves a thin specimen of silicon nitride impacted
by a relatively thick flyer plate. This causes reverberations within the flyer,
which successively unload the specimen. This method also enables the
examination of the deformed material. Reinhart and Chhabildas [2002] and
Dandekar et al. [2003] used these methods, respectively to examine aluminum
oxide and glass-fiber-reinforced polyester.
Experiments using soda lime glass specimens and tungsten carbide flyers under
shock compression-shear at 18° are conducted to observe the effect that the
failure wave has on the spall strength and the shear stress. The failure wave data
is discussed in Chapter VII. By employing angled projectiles, a shock
compression-shear loading is produced. Varying the thickness of the specimens
and flyer enable the examination of the spall strength and shear stress both
before and after the passage of the failure wave. Additionally, the material
impedance behind the failure wave is investigated by impacting tungsten carbide
specimens with soda lime glass flyers. These experiments detail the behavior
behind the failure front.
The data collected provides a critical step in several areas. First, the soda lime
glass particle impact study provides a critical look at the damage patterns at
varying velocities and thickness, which can be used both to calibrate numerical
models and to compare with other brittle materials. Second, the shock
experiments provide a profile of the two materials. From these experiments, the
effects of shear loading and velocity on various strength measures are observed.
This information will help predict the dynamic response of AS800 grade silicon
nitride and soda lime glass and help determine their suitability for aerospace and
military applications.
25
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Compression of Condensed Matter - 1997. The American Institute of Physics. 1-56396-738-3/98.
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28
Chapter II – Configuration and Procedures for Spherical Particle ImpactExperiments
Aerospace and military vehicles are subjected to impacts under normal operating
conditions. For example, windows on aircraft and ground vehicles can be
impacted by shrapnel or runway debris. The response of brittle materials like
soda lime glass to dynamic impacts is therefore critical to their use as windows.
In the current study, spherical ball bearings are employed to impact soda lime
glass. Spheres cause stress waves and cracks that are easier to predict using
closed form equations than those from a more complex geometry of impacting
particle. The experimental behavior of the simple geometry may be used to
perfect a numerical model for more complex projectiles. Correlation between the
equations, numerical modeling, and the experimental results can be used to
explore the material behavior.
Studying brittle materials under dynamic impact conditions required the creation
of a mechanism to accelerate projectiles consistently to given velocities and to a
precise location on the target. The specimens that were chosen for this phase of
the study were soda lime glass plates 50 mm square with thickness of 3 mm, 5
mm, 15 mm and 25.4 mm. Soda lime glass is a brittle, transparent material.
Velocities of between 300 and 400 meters per second were desired to simulate real
world collisions with debris. An appropriate particle delivery system needed to
be determined to accelerate steel ball bearings of 1/16th inch diameter. It is the
focus of this chapter to describe this system.
By observing the impact directly, the time of contact of the projectile, its impact
velocity, and rebound velocity can be determined. Additionally, the cracking
patterns can be tracked as they propagate. To accomplish this, a high speed
camera is employed. This camera takes pictures with microseconds between
frames in order to capture the motion with sufficient accuracy. Surface strain
profiles indicate the propagation of stress waves. To observe these strains, gages
are placed on the front or rear surfaces. On the front surface, the gages measured
strains both along the expected direction of wave propagation and perpendicular
29
to the direction of propagation. The rear surface gages are located opposite the
impact site.
These two data collection methods provided the means to examine the energies,
cracking patterns and surface strains involved in the impact. From the impact
and rebound velocities, the coefficient of restitution and total impact energy were
determined. Combined with theoretical equations and numerical modeling, these
values are used to examine the partitioning of energy during the impact process.
The velocity of impact and the thickness of the specimen categorized the damage
done by cracking patterns. Experimental strain records were compared with
those from the closed form solutions and the numerical model. The effects of
velocity and finite thickness on the strains are determined. These results are
discussed in Chapter III.
A. Components
To this end, a system was designed in several phases (Figure 2.1). First, a firing
chamber and gun barrel were designed to accelerate the desired projectiles to the
specified velocities. Secondly, the target was surrounded with a chamber to
contain the impact while allowing for visual observation of the impact and access
for other measuring elements. An impact system was designed that consists of
both the afore mentioned impact chamber, as well as mechanisms for orienting
the specimen and safely stopping the projectile’s components.
A.1. Projectile Firing Mechanism
In order to carry out these requirements, it was decided to utilize an existing
design for a Hopkinson compression bar firing chamber. The design consists of a
flanged cylinder capped on both ends by 12 inch square steel plates which are one
inch thick. These end pieces are connected with 12 5/8 inch diameter bolts to
the flanged cylinder. Polytetrafluroethylene gaskets seal the pieces. These
plates contain various connections to allow the inflow and outflow of high
pressure air. One end piece contains two holes machined with a three-quarter
30
inch and a one inch NPT thread for air intake and exhaust respectively. The
other plate has a one inch through hole and four 5/8 inch connection bolt holes
for the barrel. It also has a ¼ inch NPT threaded hole with a 1/8 inch through
hole inside of it for a pressure gage and trim valve. Contained within the
chamber is a six inch diameter piston, which is constructed of aluminum to
reduce mass and has two rubber o-rings to provide a seal along its outer rim.
This piston is connected to a shaft with a plug on the other end, which when the
chamber is filled seals the connection to the gun barrel.
This design arrangement allows for a reservoir of high pressure air to build up in
a chamber on both sides of the piston (Figure 2.2). Filling the chamber is
accomplished by means of standard plumbing connections. A compressed
nitrogen tank is utilized to provide an air source. Typical loading pressures are
between 100 and 500 psi. Two 1/16 inch diameter holes bored through the
piston allow the equalization of pressure between the two sides of the piston
during the pressure buildup stage. A valve on the back plate is then opened to
atmosphere, causing the piston to pull back and open the breach. The remaining
air in front of the piston then pushes the projectile down the barrel.
This firing chamber is mounted upon a wheel and track setup which allows the
whole firing chamber and barrel arrangement to be adjusted in distance from the
impact chamber. The track is made of L section steel beams and is 24 inches
long. The four wheels are four inches in diameter and are anchored to both sides
of the end plates. This track system allows for easier loading of the projectile
and unloading of the components for maintenance.
The several components of the firing chamber are the cylinder, which holds the
piston, the end plates, and the wheel and track system. The firing chamber
design provides a simple and repeatable method of providing the force for
accelerating the projectile. The entire firing chamber and its track are supported
on an aluminum plate above a set of I-beams. The gun barrel and sabot stripper
are also attached to this surface. These I-beams are attached to the table upon
31
which the impact chamber is located. This arrangement accounts for the
differences in height between the axis of the gun barrel and the target location.
A.2. Gun Barrel and Sabot Stripper
The gun barrel was designed to use a three-quarter inch diameter cylindrical
sabot and carry it for four and a half feet (1.37 meters). The length of 4.5 feet
was chosen by calculation of the best length for achieving a final velocity in the
300 to 500 m/s range. The barrel was honed at Commercial Honing and
machined with a 1 inch NPT thread at one end to screw into a flange connected
to the front of the firing chamber. This connection, the “breach,” is unscrewed
to load the projectile (Figure 2.2). Velocity is confirmed by means of a laser
velocity system that measured the sabot emerging from the barrel end and by
camera measurement of the projectile in flight. This method enables a
calibration curve that refines the calculated firing pressure versus velocity
measurement. The calibration curve is discussed further in Chapter III.
The system of sabot and projectile was chosen because it is more versatile than
using a straight projectile arrangement. With a cylindrical sabot, the projectile
can be any shape and size (smaller than the sabot). A hole that conforms to the
projectile shape is machined into the center of the sabot in order to contain the
projectile. A three-quarter inch diameter sabot was chosen (Figure 2.3). When
combined with the actual projectiles; in this study, spheres of diameter one
sixteenth of an inch, the combination masses about 15 grams. This arrangement
requires that only the projectile itself continues to the specimen and the sabot be
removed along the way.
In order to separate the sabot from the projectile, a “sabot stripper” was devised
that uses a steel piece called an “upright” with a 7/16 inch diameter hole in it to
block the sabot while allowing the projectile to continue unimpeded. The
stripper consists of this “upright” and a supporting piece (Figure 2.4). The
combination of both provides both the mass to stop the sabot and the area to
prevent the fragments of the sabot from hitting the specimen. Both of these
32
components are made from 1040 steel. The sabot stripper prevents the sabot
from entering the impact chamber, while passing the actual projectile through.
A.3. Impact Chamber & Target Holder
The impact chamber is constructed of 1040 steel and clear polycarbonate. The
side walls of the chamber are constructed of half inch steel to prevent shrapnel
from escaping. The top, front wall, rear wall, and the portholes in the sides are
made of clear shatterproof Lexan, a polycarbonate material. The chamber is 30
inches long, 17.5 inches wide and 19.75 inches high. This design contains room
for the sabot stripper, target holder, and its support apparatus (Figure 2.5). The
front of the chamber has a 2.25 inch diameter hole for the sabot and projectile to
pass through. Portholes are located in each side wall to allow for the viewing of
the impact. These portholes are covered by Lexan shields to prevent shrapnel
from damaging cameras and other important instruments. The holes are covered
in such a way that they can be opened to allow the entry of strain gage wires and
other equipment. Holes are also machined in the front lexan plate to allow for
the entry of the laser beams of the velocity system. A steel shelf is included for
the placement of optical instruments such as mirrors and beam splitters. A
Lexan shield protects optics on the shelf from stray shrapnel.
The impact chamber contains the aforementioned sabot stripper as well as a
target holding apparatus that consists of multiple stages allowing for both
angular and linear alignment (Figure 2.6a). The target holder can articulate
angularly in two directions using pivots. The connections between the first three
stages employ such pivot points that enable coarse rotation of the specimen
around the vertical axis and around the horizontal axis perpendicular to the
projectile motion. Fine angular alignment occurs by means of three adjustment
screws that connect the third stage to the linear adjustment stages. Linear
alignment is carried out on the fourth, fifth and sixth stages by means of screw
locked slides that enable the target to be placed precisely in the path of the
projectile. The entire target holder can also be adjusted in distance along the
33
projectile path by means of a slide at its base.
The target itself is held in a two screw clamping arrangement that does not
impede measurements from the side or from the rear (Figure 2.6b). This device
is shaped as a square C-section of metal. The two prongs of the “C” are 1.5
inches long by ¼ inch thick. From these ends, smaller C-sections protrude. Two
screws threaded through the back surface of the “C” hold the specimen in place
against these smaller C-sections. The base of the “C” has a two inch diameter
circle machined in it. This circle is used to observe the rear of the specimen.
Side observations can also be made because the sides of the “C” are open. This
arrangement allows the specimen to be securely clamped while allowing it to be
viewed from three directions (both sides and the back). In this way, the
specimen is secured and properly aligned with the path of the projectile.
34
B. Measuring Systems
Observations of specimens are made by two real time methods: crack pattern
investigation by high speed camera and point-wise deformation examination by
strain gauges. DRS Hadland provided high speed cameras that are used to
capture the internal cracking patterns of the soda lime glass during the impact.
This allows for comparison to static cracking patterns and their effect on stress
fields. Strain gages give surface measurements of strains on both the front and
rear surfaces. With these methods, the state of the deforming material is
examined.
B.1. High Speed Camera
The DRS Hadland ULTRA 17 and IMACON 200 high speed cameras allow for a
small set of pictures to be taken within a very short span of time. The ULTRA
17 camera takes pictures at a rate of up to 150,000 frames per second. That is,
one frame every 6.67 microseconds. The IMACON 200 has a maximum framing
rate of 200,000,000 frames per second and takes 16 pictures. The frame rate is
adjustable to allow for the best capturing of the material behavior. These
cameras provide a series of images of the changes in the material from which the
impact and rebound velocities as well as the contact times can be observed.
In order to best capture the various images, the high speed camera can be
configured for different types of inputs. For simple visual imagery, the camera
uses a Photogenic Power Light 2500 DR flash which delivers 1 kJ within the span
of about 200 microseconds. This flash illuminates the target either from the side
(Figure 2.5). The camera is then placed at an optimum distance for the
required magnification and configured with appropriate Nikkon lenses for the best
magnification.
The camera and flash are connected to a triggering system that is setup to allow
for the maximum amount of light to illuminate the target while the pictures are
being taken at the earliest possible time while allowing for accurate timing. The
laser velocity system, discussed in section B.5., is used to send a trigger signal for
35
the camera and through a delay generator to the strain gages. The optimal delay
time varies with velocity. For example, it is about 310μs for a 150 m/s shot. In
order to account for some degree of inaccuracy arising from fluctuations in
velocity in each individual experiment, the camera is set to a small frame speed,
such as 200,000 frames per second or 150,000 frames per second to ensure capture
of the entire impact.
Once the imagery is obtained, it is processed through a proprietary program of
DRS Hadland. This program outputs either single images for each of the
seventeen frames, or a movie file. The images can be specifically analyzed in the
program to obtain the impact and rebound velocities of the ball bearing. The
images and their results are discussed further in chapter III.
B.2. Strain Gages
Stacked rosette strain gages are utilized to gain a picture of the radial and
circumferential (hoop) strains on the surface of the glass during the impact
process. These gages, CEA-06-032WT-120, from Vishay Measurements Group,
use two perpendicular gages placed one on top of the other in order to get the
measurement of the strains in two perpendicular directions at the same point.
(Figure 2.7). These gages are made of constantan alloy with self temperature
compensation close to the soda lime glass’ compensation, 120 ohm resistance, and
copper tabs for soldering.
Placed upon all of the specimens at a radial position 5-20 mm away from the
projected point of impact, these gages provide readings of strain very close to the
impact location. This provides a time history of the strains moving parallel to
the surface of the specimen upon impact. By positioning the gage so that one
component of the rosette is parallel to the radius, the radial and circumferential
(hoop) surface stresses are measured.
36
B.3. Laser Velocity System
It is extremely important to ensure the velocity of each projectile due to
individual variations in sabot firing conditions. Therefore, in addition to the
approximate velocity determined by the calibration curve between velocity and
firing pressure, actual velocity measurements are taken during each shot.
A laser system was chosen to accurately measure the velocity.
The laser trigger for the camera provided by DRS Hadland was converted into a
velocity measurement system by means of a pair of 5 mm long cube beam
splitters that create two parallel beams which are projected across the path of the
firing sabot. The beams are 5.9 mm apart. This is done despite the slight drop
in velocity caused when the impacting particle separates from the sabot at the
sabot stripper because the one-sixteenth inch diameter particle is much more
difficult to measure with a laser beam than the three quarter inch diameter
sabot. The sabot passes through the two beams in sequence and the drop in
intensity each time is recorded by means of a high speed photodiode (Figure 2.5).
These drops in light intensity both trigger the camera system and also provide
accurate velocity measurement.
37
C. Experimental Procedures
An experimental procedure was developed in order to perform the experiments in
the most effective fashion. This procedure contains instructions for the
preparation of the specimens with the application of the strain gages. It also
provides a method for standardized sabot production. Both of these steps are
done in groups so as to minimize the time of preparation. Also, the method of
aligning the specimen and setting up the projectile accelerator and the various
measuring systems is discussed. Finally, the procedure for firing the projectile is
elucidated. This total procedure allowed for efficient experimental performance.
C.1. Series Preparation
Before each experiment, both the specimen and the projectile were configured.
The specimen was prepared with a strain gage to measure the surface strains.
The small projectile was glued to the larger sabot. These processes are described
in detail below.
C.1.1. Soda Lime Glass Specimen Preparation
The specimens were provided by American Precision Glass. These specimens
were procured as a group of 60 and are made from soda lime glass. The
specimens are rectangular prisms with two inch by two inch (50.8 mm) square
sides as the large faces. The thickness of the specimens comes in four varieties:
3mm, 5mm, 15mm and 25.4mm. The specimens were required to have
dimensional tolerances of ±0.1mm, flatness of 5 to 25 wavelengths of light and a
parallelism of 3 to 5 arc minutes.
In order to prepare the glass specimens, several steps are taken. First, both the
rear and front surfaces of the specimens are cleaned. Next, strain gages are
attached to the front face of each specimen. The gages are glued to the front
surface in precise orientation to the center of the specimen. After a 24 hour
curing time, wires are soldered to the gages. Care is taken during this process
38
not to contaminate the other surfaces.
C.1.2. Sabot/Projectile Preparation
The projectiles that were used to impact the specimens are made of hardened
chrome steel obtained from the Bearing Service Company in 1/16 inch diameter
ball bearing form. These projectiles are encased in sabots that are stripped off
prior to impact. The sabots allow a wide range of projectile sizes (anything
below 7/16 inch diameter) and shapes to be fired from the same gun.
The sabots are constructed of nylon 6/6 so as to be easily breakable. They are
made of a cylindrical piece that is one and a half inches long with a three quarter
inch outer diameter. These parts are machined from nylon bar stock obtained
from McMaster Carr. The individual pieces are then cut from the bar and
machined so that the front end is perpendicular to the sides and to remove rough
edges left by the cutting process. The design is such that the sabot hits the
sabot stripper and splits apart.
After the sabot pieces are machined, a hole of diameter 40/1000 inches is drilled
in the exact center of the forward face. This hole has a depth of 1/16 inches so
as to allow the projectile to be centered on the sabot’s front face. Following this,
the hole is filled with glue and the projectile and sabot are pressed together.
Once the adhesive is dry, the sabot is ready for use.
The sabot and projectile are then covered with vacuum grease to reduce friction
in the barrel. The gun breach (the connecting piece between the firing chamber
and the barrel) is then unscrewed and the sabot is inserted into the barrel,
projectile end facing towards the impact chamber. The breach is then sealed.
39
C.2. Setup Sequence
Following the preparation of the specimen and the sabot/projectile combination,
and the loading of each of these into their proper places, each experiment
proceeds along a similar process. This process includes the alignment of the
various components of the projectile accelerator. Also included is the setup of
the measurement systems.
C.2.1. Projectile Accelerator Alignment
First, the alignment between barrel and sabot stripper is made placing a sabot
with its glued on projectile in the front end of the barrel and pushing the barrel
and firing chamber forward until there is contact. The best alignment is when
the projectile is centered inside of the sabot stripper’s 7/16” diameter hole. In
order to adjust the alignment, the steady rests (which hold the barrel in a specific
alignment by means of three connecting rods) are unlocked. The barrel is then
adjusted so that the projectile is centered on the sabot stripper’s hole, and the
steady rests are re-tightened.
Next, the alignment between the target and the sabot stripper occurs. The
specimen is placed in the C-section and the screws are tightened. The C-section
is then attached to the rest of the target holder. The target holder is manually
adjusted angularly first until it is parallel with the sabot stripper. Then, the
linear slides are adjusted until the specimen center is along the axis of the barrel
in both the vertical and horizontal directions. Finally, the screws for the linear
slides are locked in place.
C.2.2. Measurement System Preparation
For velocity measurement and triggering of the flash prior to impact, the laser
trigger is used. As mentioned above, the velocity measurement system is a pair
of beam splitters that are employed to split a single laser beam into two useful
parallel beams (and one perpendicular beam that is not used). This system is set
as close to the impact chamber as possible in order to minimize flight time for
40
the projectile. For this reason, the beams pass in front of the sabot stripper.
The system is setup according to the diagram in Figure 2.5. The elements must
be properly positioned so that both laser beams meet at the receiving diode.
Following the laser setup, the exact distance between the laser beams and the
specimen is measured. From this and the anticipated speed of the projectile the
time from triggering to impact can be determined.
When the camera is going to be used, the velocity measuring oscilloscope must be
hooked up to the delay generator which triggers the camera and the oscilloscope
attached to the strain gage at the appropriate time. The delay generator has
multiple channels. One channel, for the strain gage oscilloscope, is set to a delay
time ten to 15 microseconds shorter than the estimated impact time. This allows
for the proper triggering of the strain gages. Another channel is set to 10
nanoseconds of delay time, and is used to trigger the camera. This system
ensures that the camera triggering and the strain gage triggering are always
separated by a known time factor.
The DRS Hadland cameras are used for impact and rebound velocity
measurement, as well as qualitative cracking imagery. The camera is setup by
first selecting the appropriate lens and extension tube combination. The lens and
however many doublers and extension tubes needed to focus at the proper
distance and magnification are attached to the camera. The camera is then
connected to its trigger, the flash, and the computer. The camera computer
recording program is then setup. The delay time to firing is input here and the
frame rate and exposure time are also programmed. Finally, the camera is tested
to insure the aim and focus are correct.
Another system that is always in use are the strain gages. The strain gages are
already attached to the specimen. There are three wires for each gage; one red,
black and white wire. The black and white wires have previously been soldered
to the negative terminal on the gage and the red wire is connected to the positive
terminal. All that is required is for the six wires to be threaded out of the
41
chamber in such a manner that they do not pass in front of any other recording
device.
The gage wires then connected to the correct terminals on the circuit box. This
box contains Whetstone quarter-bridge circuits that accurately determine the
strain. This circuit box is connected to the amplifier which magnifies the output
voltage one thousand times. Once this is done, the amplifier is turned on. An
input voltage of 3 volts is used for the experiment. Using the knobs on the
amplifier with calibration switches off, Whetstone bridge circuits are balanced.
The shunt voltage is checked (the calibration coefficient is about 1000 με - per
1.5 volts).
C.3. Pre-Firing Sequence
Following the attachment of the strain gages and the setup of the laser velocity
system and the camera, the two recording oscilloscopes are set for the
experiment. The oscilloscope settings are checked, confirming the voltage and
time ranges are appropriate for the experiment. The trigger levels are checked to
ensure triggering at the appropriate time. Once this is complete, the
oscilloscopes are set for single acquisition and the ready lights are is ensured to
be on. At this time, the high speed camera is also armed.
Next, the fire, input and trim valves on the firing chamber are all closed. The
compressed air tank, which has two valves, the main valve and the pressure
control value, is opened. The high pressure valves, both the main valve and the
pressure control valve, and the firing chamber intake valve are opened and the
load chamber is filled to the desired pressure. The load pressure must be greater
than the desired firing pressure by about 50 to 100 psi to ensure rapid loading.
The tank pressure and the load pressure are recorded. Once the desired firing
chamber pressure has been reached, the main tank valve, the tank pressure valve
and the intake valve are closed in sequence. All recording devices are checked to
confirm that they are armed and ready. The pressure is verified to be at the
42
desired firing level. The firing pressure is recorded. The firing valve is opened,
releasing the sabot and performing the experiment. Following the experiment all
data is recorded from the oscilloscopes and the camera computer for later
analysis.
D. Summary
This design of equipment, experiments and setup procedure were established in
order to examine the impacts of spherical steel projectiles on brittle materials.
The gas gun system utilizes compressed nitrogen to propel a nylon sabot down a
4.5 foot barrel at speeds up to 400 m/s. Following impact with a sabot stripper,
which removes the nylon sabot, the steel projectile impacts the specimen. This
specimen, held and aligned by a target holder, is located inside of a protective
impact chamber. Laser velocity measurements are taken of every experiment.
Instruments such as a high speed camera and strain gages record the strains of
the specimens and velocities of the projectiles as well as the cracking patterns of
the specimens.
The laser velocity measurement system and the high speed camera provide a
comprehensive picture of the relationship between the firing pressure of the
chamber and the resulting velocity of the sabot and projectile. With this
calibration curve, target velocities can be reached consistently. Also, in
observing the velocities of sabot versus projectile, it can be seen that little to no
energy is lost during the sabot stripping process. This verifies that almost all the
kinetic energy is retained by the projectile following the impact with the sabot
stripper.
Armed with this data, the primary experiments of the soda-lime glass phase were
undertaken. Experiments utilizing the camera and the strain gages in the
discussed configurations were performed. This data provides a fundamental
understanding of the process of dynamic impact useful as a comparison to
numeric simulations of dynamic impacts of this and other brittle materials.
43
References
Vishay Measurements Group online handbook, 2002.
www.vishay.com/brands/measurements_group/guide/500/gages/032wt.htm
Figures
Figure 2.1: Complete view of projectile accelerator
Figure 2.2: View of firing chamber with piston and sabot loading procedure
44
Figure 2.3: Sabot and projectile assembly diagram
Figure 2.4: Sabot stripper diagram
45
Figure 2.5: Impact chamber diagram and Measurement Setup
46
a)
b)
Figure 2.6: Target holder diagram: (a) Target holder overview. (b) Close up of specimen holder
47
Figure 2.7: CEA-06-032WT-120 strain gage image (Vishay Measurements Group, 2002).
48
Chapter III – Spherical Particle Impact Experiments on SodaLime Glass
In this chapter, the results of a series of experiments are discussed that were
carried out in order to better understand the impact of spherical ball bearings on
soda lime glass plates. For the study four different velocities and plates of
different thickness of soda lime glass were chosen. These results are compared
with theoretical analysis, numerical simulations [Nathenson et al., 2005], and
previous experimental work. Three specific types of results are analyzed: surface
strain profiles, impact kinetics, and cracking patterns. These results of the
experimental study are consistent with the previous experimental work in the
literature, but provide some unique insights when coupled with the numerical
study and theoretical equations.
A. Background on Particle Impact of Brittle Materials
Stress waves are created in any impact. At low impact velocities, these waves
have small amplitudes and their effects are negligible. However, when the impact
velocity is sufficiently large the inertia of the material plays a role in controlling
its response [Spath, 1961]. There are two ways to approach the study of dynamic
impacts. In the long time solution, stress waves have been given sufficient time
to reverberate and bring the entire specimen to a state of equilibrium. This
method uses quasi-static methods to study the impact while accounting for the
impact velocity. The partitioning of impact energy and the coefficient of
restitution can be studied with this method. The second method involves the
study of stress waves. This analysis considers the transient effects of the stress
waves generated by the impact. The impact of the sphere on a half space
generates waves that travel outwards from a central point either as circular
waves on the surface, or as spherical waves within the body. There are three
main types of waves, the Rayleigh surface wave, the longitudinal (pressure), and
the transverse (shear) waves [Lamb, 1904]. The combined effects of these waves
are used to develop theoretical models to predict the material behavior under
impact. These models will be compared with numerical simulations and
experimental measurements of surface strains generated by the waves.
49
For the long time solution, the results from high speed photography of the
experiments are compared to predictions of the Hertzian theory, along with a LS-
DYNA 3D computer model. In this way, the long time solution is probed and
the coefficient of restitution and impact energy partitioning are examined. The
high speed photography and post impact examinations are used to study the
cracking patterns. These findings are compared with those described in the
literature. By correlating the results from the surface strain gages with the
results from three theoretical models for the surface strains and the LS-DYNA
numerical simulation, the stress wave response is categorized. Before the results
are discussed, the previous literature on theoretical formulations of particle
impact and their predictions are reviewed.
A.1. Theoretical Calculations
The first step in understanding high velocity particle impacts is to discuss the
varied existing theoretical models and their predictions for understanding
transient and long time solutions for various quantities of interest. In particular,
these models will provide the framework for interpreting the data collected in the
experiments. Hertzian theory for quasi-static impacts provides a method of
examining the impact and rebound energy, coefficient of restitution, peak force,
and contact time. The experimental data is compared to the results of these
equations and to numerical simulations. In Appendices 1 and 2 the equations for
the surface displacements generated by the dynamic stress waves are derived. By
using the derivatives of these equations two estimates of the surface strain can be
obtained. These formulae from Lamb [1904] and Mitra [1964] provide estimates
for different geometrical conditions. Plots of these equations for strain versus
time are compared to the experimental strain records and to numerical
simulations.
50
A.1.1. Hertzian Theory
The Hertzian theory for a spherical particle impacting on a half space is subject
to the restrictions of the Love Criterion [Love, 1944]. This criterion defines the
limiting maximum impact velocity for which the quasi-static assumptions of
Hertzian theory are accurate.
( )15 1.v C {III.A.1}
In equation {III.A.1}, v, represents the impact velocity and, C, is the longitudinal
wave speed. Quantities that can be estimated by means of this theory include
the peak force and total contact time. Also, the impact energy and coefficient of
restitution can be determined [Hunter, 1957]. These are determined by the
equations shown in their respective sections. The predictions of these equations
can then be compared to experimental measurements determined by means of the
strain gage and the high speed camera, which are discussed in Section C below.
The Hertzian force versus time profile for a spherical projectile on a half space
specimen is approximated as a sine curve [Goldsmith, 2001]. This is for elastic
conditions where neither the sphere nor the half space deform plastically. The
equations still approximate the true behavior even if a small amount of plasticity
in the projectile occurs, as long as the specimen remains elastic. The following
equations describing this force versus time profile are taken from Goldsmith
[2001]. The subscript 1 refers to the sphere and subscript 2 refers to the half
space in the equations.
( )
2
1
1.140 1.068sin , 01.068
0, 1.068
m
m m
m
v vt tk vF t
tv
παα α
πα
⎧⎪⎪ ≤ ≤⎪⎪⎪= ⎨⎪⎪ ≥⎪⎪⎪⎩
{III.A.2}
Here, Goldsmith approximates the force between two impacting bodies as a
symmetric sine curve. In equation {III.A.2}, the impact velocity is, v, and the
51
time is, t. Here and always in this thesis, π, is the numeric quantity Pi. Also, k1
and, mα , are defined by equations {III.A.3} and {III.A.4}, respectively, and are
given below:
1 31 1 1
1 34
km Rπρ
= = , {III.A.3}
and,
( )252
1 1 2
1
1516m
v mR
π δ δα
⎡ ⎤+⎢ ⎥= ⎢ ⎥⎣ ⎦
. {III.A.4}
In the above equations, m1, is the mass of the sphere, ρ1, is the density, and, R1,
is the radius of the sphere [Goldsmith, 2001]. The quantities, δ1, and, δ2, for the
sphere and half space respectively are defined as follows:
21
11
1 1Eμδ
π−= , {III.A.5}
22
22
1 1Eμδ
π−= , {III.A.6}
In equations {III.A.5} and {III.A.6}, E, is modulus of elasticity, μ, is Poisson’s
ratio, and, ρ, is the mass density [Goldsmith, 2001]. For the experiments, the
mass density, Young’s moduli and Poisson’s ratio values are 2,500 kg/m3, 79.2
GPa and 0.19 respectively for the soda lime glass plates, and 7,800 kg/m3, 201
GPa and 0.3 for hardened chrome steel spheres [Matweb, 2005].
The Hertzian force versus time equation {III.A.2} results in a function that is
plotted in Figure 3.1. As can be seen, the impact force rises to a maximum force
at half of the total time of contact and then falls to zero symmetrically. This
results in a maximum force when the time is half of the total contact time, tMAX:
52
2 2.136MAX mt
vπ α= . {III.A.7}
The corresponding maximum force is:
( )2
12 1.140MAX MAXm
m vF F tα
= = . {III.A.8}
The total contact time of the sphere and half space can also be found using the
densities, Young’s moduli, and Poisson’s ratios of the projectile and specimen:
( )25
1 2 1
1
4.53CONTACT MAXI
mT T
R vδ δ⎡ ⎤+⎢ ⎥= = ⎢ ⎥
⎣ ⎦. {III.A.9}
The results from these theoretical equations for the forcing versus time curve,
contact time, and peak force are compared with the force vs. time function
determined in a numerical study using LS-DYNA performed by Guodong Chen
[Nathenson et al., 2005]. These curves are also plotted in Figure 3.1. Close
agreement between the theoretical and numerical results is an indication of the
accuracy of the numerical model. The forcing function could not be determined
experimentally, but the contact time could be estimated with the high speed
camera. The numerical, theoretical, and experimental contact times are
discussed further in the Section C.
A.1.2. Computed Longitudinal, Shear and Rayleigh Wave Profiles
The use of strain gages to measure strain waves on the impact surface of the
material provides a method for examining the stress wave propagation within the
specimen. These experimental results can be directly related to both theoretical
and numerical calculations. In measuring the strain profiles, certain waves are
expected. The longitudinal and shear waves in soda lime glass, which are
spherical in shape and disperse with increasing distance travel approximately at
speeds of 5300 m/s and 3500 m/s respectively [Field, 1988]. The Rayleigh wave
53
does not disperse with increasing propagation distance and is cylindrical in
nature. It travels at the slower velocity of 3150 m/s [Goodier, 1959]. On the
surface, an effect of the three stress waves is strain waves propagating in the
radial direction outwards from the impact point. The strain waves, which are
measured by the surface gages in the experiments, are discussed in this chapter.
In Appendices 1 and 2 two different theoretical estimates of the surface
displacements associated with these stress waves are calculated. Each estimate
has its own impact geometry. By taking the time derivative of the equations for
the surface displacements, the surface strains are estimated. These strains,
estimated at the strain gage location on the surface of the specimen, are
compared with the strains measured in the experiments and simulated
numerically.
A.1.2.1. Lamb’s Solution
Lamb’s examination of a point loading on the surface of a half-space is the first
geometry [1904] discussed in this section. This geometry is the simpler of the
two discussed. A result of Lamb’s [1904] analysis using approximations
(Appendix 1) is an integral equation for the radial displacement at a the strain
gage location along the surface due to the Rayleigh wave:
( )
( ) ( ){ }
32
32
4 22
34 22
0
2 sin cos4
2 sin cos
oRcq H
c
U db
χπ
χπ
τ χπμτ ϖ
θθ χ θπ μ ϖ
∞
≈ − − −
−∫, {III.A.10}
where,
( )( ) ( ) ( )
( ) ( )( )
3 2 2 2 2 2 2
42 2 4 2 2 2 2
2
2 16
b b a bU
b a b
θ θ θ θθ
θ θ θ θ
− − − −=
− + − −, {III.A.11}
2 2 2x yϖ = + , {III.A.12}
54
tθϖ
= , {III.A.13}
and,
1tan t θϖχτ
− ⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠ . {III.A.14}
For a specific loading function at a point:
( ) 2 2
RR ttτ
π τ=
+, {III.A.15}
The terms in these equations are defined as follows: qo, represents the Rayleigh
wave solution for radial displacement, x, and, y, are coordinates on the plane of
the half space, t, is time; and,R , and, τ, are variables of the loading function
{III.A.15}. Also, H, is a function of the two Lame constants, μ , and, λ ; c, is the
inverse of the Rayleigh wave speed; a, is the inverse of the longitudinal wave
speed, and, b, is the inverse of the shear wave speed. Solving these equations
numerically produces an estimate for the radial surface displacement. In order to
obtain the strain, the equation must be differentiated by time. The one-
dimensional nature of the equation means that differentiating by radial distance,
ϖ , is the same as differentiating by time, t.
q t q ϖ∂ ∂ = ∂ ∂ . {III.A.16}
The equation is one-dimensional because there is no estimated wave propagation
in the hoop or vertical directions. Because of the complexity of this equation,
numerical differentiation was employed. The numerical differentiation of
equation {III.A.10} yields the dynamic surface radial strain [Goodier, 1959].
55
Lamb’s [1904] solution for strain at various radii from the impact point results in
an estimated wave pattern. Plotting this wave shows one peak (Figure 3.2) that
decreases in amplitude and spreads out with increasing radius. Also, the initial
depression attenuates, as the distance from the impact point radius increasing.
Lamb [1904] also states that the calculation in equation {III.A.10} represents
only the Rayleigh wave. Because no information is contained in the solution
about the longitudinal or shear waves, no statements can be made excepting
what is in Lamb’s paper as to their intensity relative to the main pulse. Lamb
[1904] states that the amplitudes of the shear and longitudinal waves are much
smaller than that of the Rayleigh wave. This case represents the general solution
for the Rayleigh wave generated by a loading at a point on a half space.
A.1.2.2. Mitra’s Solution
The closest case to the experimental geometry used in the current study that has
been computed in closed form was by Mitra [1964]. This is the case of an
impulse loading on a circle on the surface of a half space. The solutions are
presented in terms of elliptical integrals. The coordinate system for this solution
is cylindrical. The solution, ( ), 0,u r z t= , is on the surface, so the z coordinate is
zero in the following equations. It is axi-symmetric so there is no variation in the
θ direction. The entire solution, ( ), 0,u r t , on the surface is given in terms of
both the Rayleigh wave component, ( ), 0,ou r t , and the shear and longitudinal
components, which involve, U(τ):
( ) ( )( )
0 20
cos,0, , 0, P Uu r t u r t dR
παβ τ φ φπ μ
= + ∫ . {III.A.17}
The terms are defined:
56
( ) ( ) ( )[ ]
( ) ( )( )
( )
0
2 2 2 22 2 2 1 1
2 2 2
0
2,0, 2
k k
Pa Vu r t E k K kar
V r a V tV t r a E Kar V t r a
πμ
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪Γ ⎪ ⎪= −⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤⎪ ⎪⎪ ⎪+ −⎢ ⎥⎪ ⎪⎡ ⎤− − −⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎢ ⎥⎡ ⎤− −⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭( )
( ) ( )
( )
0 -
when, -
t r a V
r a V t r a V
r a V t
⎧ ⎫⎪ ⎪< <⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪< < +⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪+ <⎪ ⎪⎪ ⎪⎩ ⎭
, {III.A.18}
and,
( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )2 1
3
2 2
1 38 2 2
1 1 1
14 1
0
6 18 8 , 6 4 3 20 12 3 ,
6 4 3 20 12 3 ,
3 9 8, 2 3 3 20 12 3 ,
2 3 3 20 12 3 ,
l l l
l
K l l l l lU
l l
Kτ
τ
−
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤⎪ ⎪⎡ ⎤− Π + − Π −⎪ ⎪⎢ ⎥⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎢ ⎥= ⎨ ⎬⎪ ⎪⎢ ⎥⎡ ⎤+ + Π +⎪ ⎪⎢ ⎥⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎪ ⎪⎪⎪ ⎡ ⎤⎡ ⎤⎪ − Π − − Π −⎢ ⎥⎪ ⎢ ⎥⎣ ⎦⎪ ⎢ ⎥⎪⎪ ⎢ ⎥⎡ ⎤⎪ + + Π +⎢ ⎥⎢ ⎥⎪ ⎣ ⎦⎣ ⎦⎪⎩ ⎭
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪0< <1 3
for, 1 3 1
1
τ
ττ
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪< <⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪<⎪ ⎪⎪ ⎪⎩ ⎭
. {III.A.19}
The Rayleigh wave speed, V, is:
2
3 3V β=
+{III.A.20}
Also:
57
( )2
2 2
0
1 sinE xdxπ
θ θ= −∫ , {III.A.21}
( )2
2 20 1 sin
dxKx
π
θθ
=−∫ , {III.A.22}
( )( )
2
2 2 20
,1 sin 1 sin
dn kn k
πθ
θ θΠ =
+ −∫ , {III.A.23}
tRβτ = , {III.A.24}
2 2 2 cosR r a ar φ= + − , {III.A.25}
23 12
l τ⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎟⎜⎜⎝ ⎠, {III.A.26}
( )2 2 2
4V t r ak
ar⎛ ⎞− − ⎟⎜= ⎟⎜ ⎟⎟⎜⎜⎝ ⎠
, {III.A.27}
and,
( ) ( ) ( )( ) ( ) ( )
2 2 2
2 2 2
2 2 4 2 2 2
2 2 2 2 2 2 23 3 2
2 1 1
4 4 2 2 1 1
V V V
V V V V V V
β α β
α β α β β α β
π − − −Γ =
− − − − − − −. {III.A.28}
The coordinate system is: r, is the radial direction, z, is the vertical direction,
and, t, is time. The other variables in equations {III.A.17} through {III.A.28}
are: P, the impact force, α, and, β, the compression and shear wave velocities for
the half-space, μ, a lame constant, φ, a variable of integration, and the area of
loading, a. As in Lamb’s solution, Mitra’s solution is obtained by solving
58
Equation {III.A.17}, then differentiating numerically with respect to time.
Equation {III.A.16} still holds equating the radial dimension derivative and the
time derivative. This yields the strain in the radial direction [Goodier, 1959].
In the Mitra [1964] solution, as the wave travels further from the impact point,
the relative amplitudes of the three stress waves shift. There is a distinct
increase in the amplitudes of the Rayleigh and shear pulses relative to the
longitudinal pulses with increasing radial distance as well as an alteration in the
way the shear and Rayleigh waves combine (Figure 3.3). This is in addition to
the decreasing absolute amplitude and increasing duration, of all three of the
waves with time. This solution contains geometry and loading conditions that
match closely with the experiments and should therefore provide an accurate
theoretical approximation of the impacts.
A.1.3. Estimation of the Energy Dissipated by Elastic Waves
Another quantity that is observed directly from the experiments is the amount of
energy absorbed in the impact. This is found by subtracting the remaining
kinetic energy during the rebound of the projectile from the initial kinetic energy
of the projectile. Many processes of energy dissipation occur within the brittle
material during the impact. Elastically, stress waves and vibrations dissipate
most of the energy and most of the remaining inelastic energy goes into cracking
[Nathenson et al., 2005]. Hunter [1957] provides a method for determining the
portion of the impact energy that is dissipated by the stress waves. This
provides an estimate of the elastic coefficient of restitution of the impact
assuming only stress waves dissipate energy. Hunter concludes that for the
Hertzian case of the particle impacting a half-space only “a negligible portion of
the original kinetic energy of the small body is transferred to the specimen by the
collision.” That this conclusion is accurate is one reason that the Hertz theory is
valid for the quasi-static cases [Hunter, 1957].
The estimate for energy dissipated by the stress waves, W, is found starting from
a differential equation for the rate of work done on the specimen:
59
( )2dW dua P tdt dt
π= . {III.A.29}
Here, a, is the contact radius, P(t), is the pressure on the surface and, u , is the
Fourier synthesis of the mean surface displacement. By the derivation in
Appendix 3 the energy is found to be:
( )1
2 2
30
1 1 .1 2 o oW M
Cβ ν ν ωρ ν
⎧ ⎫+ −⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪−⎪ ⎪⎩ ⎭{III.A.30}
Hunter specializes this equation using a transient pulse force vs. time function of:
21o o oM m Z ω= − , {III.A.31}
The energy due to vibrations normalized by the pre-impact kinetic energy is
estimated as follows.
( )1 1 6 35 5 5 5
13
2 21
2 4 /312
g vWCm v
τ π ρρ
− −
= , {III.A.32}
( ) ( )1 62 5
5 22
2
1 161.068 11 2 15
μτ μ βμ
⎛ ⎞− ⎛ ⎞⎟⎜ ⎟⎜⎟= + ⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟ ⎝ ⎠⎜ −⎝ ⎠, {III.A.33}
2 22 1
2 1
1 1gE Eμ μ⎧ ⎫⎪ ⎪− −⎪ ⎪= +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭
. {III.A.34}
Here, m, is mass, ρ, is density, C, is the longitudinal wave speed, v, is the initial
velocity of the projectile, E, is the modulus of elasticity, μ, is Poisson’s ratio and,
β, and, τ, are constants related by the listed formula. A subscript 1 indicates the
projectile, while subscript 2 indicates the specimen. For a glass where the
60
Young’s modulus is 70 GPa Hunter estimated, 0.5374β ≈ , and, 1.38τ = , for,14ν = . The coefficient of restitution is found by:
( )12
2112
1o
Wm V
e = − . {III.A.35}
Using equation {III.A.35} the coefficient of restitution, e, was evaluated to be, e
= 0.996, at an impact speed of, v = 0.09 m/s [Hunter, 1957]. This value of, e, is
slightly higher when compared to an experimental value of 0.985 estimated at the
same impact speed [Tillet, 1954]. Increasing the impact velocity from 0.01 m/s
to 0.3 m/s decreased the coefficient of restitution by about 0.5 % [Hunter, 1957].
Tillet [1954] observed the same variation over the same range of impact
velocities. This equation will be employed to estimate the coefficient of
restitution considering only stress waves as an energy dissipation mode.
61
A.2. Previous Research on the Cracking Patterns of Soda Lime Glass
Projects conducted at the Cavendish Laboratory of the University of Cambridge
produced several papers concerning experimental impact studies of soda lime
glass. In these papers, two of which are from the late 1980s and two of which
are from 1977, tests were carried out on a large variety of specimen types. In
fact, the paper from 1988 is the final report on an extensive U.S. Army research
project [Field, 1988]. The additional studies were carried out by Field’s group,
by Knight, Swain and Chaudhri [1977], and by Chaudhri and Walley [1978].
These studies all employed spherical projectiles and included soda lime glass as a
specimen material.
A.2.1. Configurations and Computations for Previous Experiments
Two main types of specimens were discussed in these studies: glass ceramics and
glasses are one type and alumina and boron carbide are the other type. These
two categories are distinguished mainly by their young’s moduli, E, and their
longitudinal wave speeds, C. The glasses and glass ceramics have, E < 100 GPa
and, C < 6500 m/s. The alumina and boron carbide specimens have, E > 250
GPa and, C > 8500 m/s. Soda lime glass, falls into the first class with, E of 79.2
± 3.0 GPa and C of 5600 ± 100 m/s [Field, 1988].
These studies were conducted on a gas gun capable of using either helium or
nitrogen gas to fire a projectile down its 2 meter barrel length with a 13.3 mm
bore diameter. The projectile is attached to sabot in configuration similar to
that discussed in chapter II. Their sabot is of 25 mm length. A muzzle block
stops the sabot allowing the projectile to go through. The gun can either be
triggered at low velocities by a solenoid valve, or at higher velocities with a
double diaphragm technique. The second technique involves pressurizing the
reservoir to the firing pressure and causing rupture by venting the area between
the two diaphragms which was pre-pressurized to half the firing pressure. This
produces velocities up to 1 km/s [Field, 1988].
62
The specimens were rectangular blocks, and their thickness dimensions were
between 3 mm and 35 mm. Impacts were made on the large square face for most
of the specimens, but on the largest thickness, some impacts were made on the
smaller sides. The papers discuss many salient properties of the material
including: density, 2500 kg/m3, hardness, 4.5 ± 0.4 GPa, and fracture toughness
0.75 MN/m3/2 [Field, 1988]. In another study on soda lime glass and Pyrex the
impacted soda lime glass specimens were 40 mm x 40 mm x 10 mm and 50 mm x
50 mm x 25 mm [Knight, Swain and Chaudhri, 1977], respectively.
In addition to varying the specimen dimensions, multiple projectile geometries
were employed. Chaudhri and Walley [1978] used both glass beads and hardened
steel spheres of 1 mm diameter. Field [1988] and Field, Sun, and Townsend
[1989] used larger projectiles of hardened steel. These were spheres with diameter
of 3 mm and 5 mm. This apparatus provided inspiration for parts of the
projectile accelerator used in the current study.
The three papers focus mainly upon the changes in crack propagation with
respect to velocity and projectile size. The range of experimental velocities goes
from 100 m/s to over 400 m/s. Key highlights are presented in the following
paragraphs. There is little discussion of change in crack pattern with respect to
thickness of the specimen in the papers, except when examining the effects of
placing another layer of material either in front or behind the glass plates.
By the use of an IMACON high speed camera, with an image recorded every 0.95
microseconds, the cracking patterns were well resolved. Also, with the
measurement feature of the IMACON, they were able to measure the crack
velocity of the fastest internal cracks directly. This crack velocity was, 1550 m/s
[Chaudhri and Walley, 1978] and alternatively, 1500 m/s [Field, 1989] and, 1480
m/s ± 50m/s [Field, 1988].
In one set of experiments [Chaudhri and Walley, 1978] the contact time, that is
the time that the spheres were in contact with the glass specimens, was measured
63
from the high speed camera images. A computation similar to that described in
equation {III.A.9} was used to calculate estimates, which were compared to the
observed contact times. This calculation is given in equations {III.A.41} below.
Chaudhri and Walley’s [1978] paper also contains theoretical calculations for the
maximum loads. These are found by the following equations when the impact is
elastic:
2/53/52 6/5
max, 12
5 43 3elastic
kF R vE
πρ−⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟= ⎜⎟ ⎟⎜ ⎟ ⎜⎝ ⎠ ⎟⎜⎝ ⎠
, {III.A.36}
2/52 22 1
1 1/52 1
5 1 12.94 .4elastic
RtE E vμ μπρ
⎡ ⎤⎛ ⎞− − ⎟⎜⎢ ⎥⎟= +⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎣ ⎦{III.A.37}
Where,
( ) ( )2 2 22 1
1
9 1 116
EkE
μ μ⎛ ⎞⎟⎜ ⎟= − + −⎜ ⎟⎜ ⎟⎜⎝ ⎠
. {III.A.38}
In these three equations, ρ1 is the density; μ1 and E1, are the Poisson’s ratio and
elastic modulus for the projectile respectively; R, is the projectile radius; μ2 and
E2, are the specimen Poisson’s ratio and elastic modulus respectively; and v is the
impact velocity.
In the case when the projectile deforms, plasticity causes alterations in the
contact time and peak force. The total time of contact for the elastic-plastic case
can be approximated as the sum of the plastic loading time {III.A.41} and the
elastic unloading time. Examination of the specimens impacted by 1 mm
diameter steel balls showed no evidence of plastic flow in the specimen. The steel
projectile did deform, however, and that is why the governing equations
{III.A.36} to {III.A.38} must be modified to incorporate the effects of plasticity
of the projectile [Chaudhri and Walley, 1978]:
64
2max,plastic plastic dynamicF a Yπ= , {III.A.39}
3dynamic staticHY Y≈ = , {III.A.40}
1/2
.12 2plastic loading
dynamic
mtRY
ππ
⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠. {III.A.41}
In equations {III.A.39} through {III.A.41}, ap, is the radius of the post impact
dent on the projectile, Y, is the flow stress of the projectile and, H, is the
hardness of the projectile.
The theoretical and experimental contact times, quoted from Chaudhri and
Walley [1978] are for a steel projectile, were 1.9 μs and 1.5 ± 0.3 μs, respectively.
As these values reflect, the calculated contact times are 0.1 μs higher than the
range on the experimental contact times. This disagreement indicates that while
the theory gives reasonably accurate estimates of the contact time, it is not
exact.
A.2.2. Results of Previous Experiments
In all of the experiments on soda lime glass, the same types of cracking patterns
were observed. The impact velocity has a large effect on which of these types of
cracking patterns are dominant. This holds true for the 1 mm, 3 mm and 5 mm
diameter projectiles, with critical speeds for changes between the different
cracking patterns depending upon the diameter of the projectile. Cracks begin to
be seen at a threshold velocity of about 50-100 m/s. Between the threshold
velocity and about 200 m/s conical cracks are dominant with smaller radial and
lateral cracks. This threshold velocity varies with projectile diameter. It is
about 100 m/s for balls of diameter 1 mm and 50 m/s for balls of diameter 5 mm
[Field, 1988; Chaudhri and Walley, 1978]. The effect of kinetic energy is therefore
an important parameter in the altering cracking patterns. There is also a
65
hemispherical crushed contact zone under the impact site. As the velocity
increases beyond 100-140 m/s the radial and lateral cracks begin to increase in
size. Eventually, at velocities between 180 and 200 m/s they dominate the
cracking [Field, 1989; Chaudhri and Walley, 1978]. The crushed zone increases in
size and material ejects from the surface increasing the severity of the radial
cracks. Eventually the cone crack becomes unclear and difficult to see. In one
paper it was remarked that with the limitation on frame rate they could no
longer see the conical cracks [Chaudhri and Walley, 1978]. Finally, at velocities
at and above 250 m/s stress wave behavior becomes dominant [Field, 1988; Field,
1989]. Once the stress waves dominate internal cracking patterns are disordered,
no quasi-static cracking feature are visible, and the Rayleigh surface waves
produce cracking patterns on the surface [Field, 1988; Field, 1989].
The conical cracks form during the loading of the specimen by the projectile.
They initiate at ring cracks just outside of the contact zone [Field, 1988]. The
radius of the ring cracks appears constant with changing velocity because the
stresses, which generate them, are mainly a function of contact area. The cracks
then propagate along a conical path shaped by principal stresses [Field, 1988].
With increasing velocity, the cone cracks angle to the impact axis decreases.
This is due to the final contact area of the projectile being larger. If the glass is
thin enough, then the crack curves to meet the rear surface at right angles due to
interaction with the reflecting waves [Field, 1988]. If there is sufficient thickness,
oscillations in the crack size can be seen as it attempts to reach an equilibrium
state. This can cause the crack to angle back towards the front face in a ‘skirt’
[Chaudhri and Walley, 1978]. This type of crack dominates at speeds of about
100 to 180 m/s for the 1 mm steel projectiles [Field, 1989; Chaudhri and Walley,
1978]. For the larger steel projectiles, other studies show that the cone’s angle to
the impact axis decreases with velocity for increasing velocity for similar
materials such as Pyrex and fused silica glasses [Field, 1988]. Also, with the
larger 5 mm diameter projectiles, the cone produced by the crack ejects from the
specimen at speeds as low as 30 m/s [Field, 1989].
66
The other major types of cracks are radial cracks, which propagate on the surface
and lateral cracks which propagate initially parallel to the surface, and then
curve up to meet it. The radial cracks form during the loading process and,
because there is no plastic deformation, are caused by circumferential stresses
generated from frictional forces in the surface ring cracks due to the surface
deformation [Chaudhri and Walley, 1978]. The lateral cracks form during
unloading from tensile stresses at the contact zone interface, which cause small
tensile cracks to initiate [Field, 1988]. They can also form from the turn up of
the ‘skirt’ of the conical crack [Chaudhri and Walley, 1978]. Both types of cracks
are present from about 100 m/s; they appear at significant sizes in the impact
velocity range between 100 and 140 m/s. The smaller the projectile, the larger
the impact velocity that is needed to produce the same size crack. This cracking
occurs at a speed of 207 m/s for a 5 mm steel projectile that impacted an 8 mm
thick specimen. With such a large projectile diameter to plate thickness ratio,
the radial cracking was caused by plate bending [Field, 1989].
Another type of cracking is observed for smaller projectiles. Knight et al. [1977]
observed the impact of 0.8 mm and 1 mm diameter steel spheres on soda lime
glass and Pyrex at speeds between 30 and 300 m/s. The soda lime glass impacts
showed cracking patterns similar to that of a quasi-static indentation of a point
load. Point indentations, create a Boussinesq stress field. Instead of just the
cone crack, a group of “splinter cracks” beneath the indentation point is observed
to initiate. These cracks grow normal to the principal tensile stress direction
[Knight, Swain and Chaudhri, 1977]. The cone crack was observed in some cases,
starting to grow before the splinter cracks.
Crushing of the contact area or plastic deformation at the projectile are suggested
as possible causes of the splinter cracks. These cracks propagate at the
maximum crack velocity for soda lime glass ~ 1500 m/s [Knight, Swain and
Chaudhri, 1977]. During the unloading process, the cracks closest to the surface
can keep growing and bend towards the impact face in the same manner as
lateral cracks. Other cracks closer to the impact axis continue to grow towards
67
the rear surface for a distance on the order of millimeters. However, with
increasing velocity more of the splinter cracks alter direction. The unloading
cracks travel at the much slower speed of about 300m/s [Knight, Swain and
Chaudhri, 1977].
Additionally in this study, measurements of the strain produced by the stress
waves on the surface of the specimens occurred. These strain records indicated
that dynamic cracking is reflected in changes to the strain records. In this case,
compressive strain pulses were associated with well developed conical cracks
[Knight, Swain and Chaudhri, 1977]. In the present study, the relationship
between cracking patterns and the stress waves in the material, which in turn
alter the surface strains, is further explored.
At speeds of about 250 m/s, for the 3 mm and 5 mm diameter steel projectiles,
stress waves begin to dominate [Field, 1988; Field, 1989]. The internal fractures
become very chaotic and the surface cracks are caused by Rayleigh surface waves.
Back surface damage is caused by the reflection of the compressive loading wave
as a tensile wave from the back surface [Field, 1989]. A specimen impacted by a
5 mm diameter projectile at 402 m/s shows short circumferential bands of surface
cracks. These cracks are the result of reinforcement of the Rayleigh surface wave
and reflected tensile waves from the back surface. This is a different category of
cracking than at lower speeds where the same patterns that exist in quasi-static
indentation testing are observed [Field, 1988]. The current study examines these
impact velocities, using 1 mm diameter steel projectiles.
To summarize, the cracking patterns of the specimens indicate a distinct
dependence on both impact velocity and specimen thickness. In research
conducted by Chaudhri and Walley [1978], the specimens impacted by 1 mm
diameter projectiles exhibit dominant Hertzian cone cracks for velocities up to
about 180 m/s. Studies by Field [1988] showed that conical cracks are seen to
initiate at front surface ring cracks and then propagate along a path shaped by
the principle stresses. At velocities above 100 to 120 m/s radial and lateral
68
cracking formed and grew more dominant with increasing velocity in impacts of 1
mm balls [Chaudhri, 1978]. Front surface radial cracks formed during the
loading and unloading process due to tensile surface circumferential stresses
[Chaudhri, 1978]. Rear surface radial cracking was seen to be caused by the
reflecting longitudinal stress wave occurred in specimens impacted with 3 mm
balls [Field, 1989]. The lateral cracks formed during unloading from tensile
stresses on the contact zone interface from small tensile cracks [Field, 1988].
Front surface chipping represents either lateral cracks that have ejected from the
front surface or, at higher velocities, damage due to the stress waves [Field,
1989]. In cases of small balls impacting, 0.8-1 mm diameter, instead of just the
cone crack, a set of splinter cracks beneath a crushed indentation zone were seen
to initiate in a study by Knight et al. [1977]. These cracks initiated at tensile
stresses produced by the crushing of the contact zone and grew normal to the
principal tensile stress direction. A cone crack was observed in some cases,
starting to grow before the splinter cracks [Knight, 1977]. At high velocities, the
larger 3 mm and 5 mm projectiles produced chaotic internal damage and different
forms of surface damage [Field, 1988; Field, 1989]. The current study expands
upon these observations and studies the relationship between impact velocity and
thickness on the cracking patterns.
69
B. Design of Soda Lime Glass Particle Experiments
Within the soda lime glass phase of the project, experiments were designed to
vary the thickness of the specimen and the velocity of the impact. Four
velocities and four plates of varying thickness were studied in this phase. The
velocities in the experiments are designated in four categories: low or around 150
m/s, medium low or around 230 m/s, medium high or around 300 m/s, and high
or around 350 m/s. The glass targets are 3 mm, 5 mm, 15 mm and 25.4 mm
thick. Also factored into the experimental matrix are the methods of
measurement. The experimental apparatus is that discussed in chapter II. A
preliminary set of tests provided the loading pressures necessary to achieve the
desired impact velocities. Two sets of measurement setups were employed. The
basic quantities configuration (BQ) and supplemental experiments (SP)
contained all but two of the experiments. The bending configuration (BN)
measures the rear surface vibrations along orthogonal directions.
The basic quantities (BQ) set of experiments is designed to measure several
quantities. The contact time, the impact velocity, the rebound velocity, and the
general cracking pattern are determined from the IMACON high speed cameras.
To ensure that the entire impact and rebound is recorded, a low framing rate of
150,000 frames per second is used. The strain gages on the surface measure the
dynamic wave patterns of the radial and hoop strains. These strain records can
be compared with numerical and theoretical strain wave estimates. Supplemental
experiments (SP) were performed with the IMACON 200 camera using a framing
rate of 400,000 frames per second. The greater framing rate was used for greater
accuracy of velocity measurement. The strain gages were also used in the
supplemental experiments.
The second series of experiments (BN) observed vibrations on the rear surface.
Strains in two orthogonal directions were observed. This configuration in called
bending because the plate flexes due to these strains. Since the strain gages in
this set are located on the rear surface along the axis created by the impact of
the projectile, the vibrations of the rear surface can be measured. The same
70
camera system is used as in the BQ experiments to obtain impact and rebound
velocities as well as general cracking patterns. The vibrations in the strains on
the back surface represent the dynamic flexing of the entire specimen. By using
high-speed data collection the oscillations in the strain can be measured. Two
oscilloscopes collect data; one with a 50 microsecond recording window and one
with a 500 microsecond window. This is to gain a clear picture of the dynamic
vibration history of the specimen. These two experiments were carried out after
the BQ experiments and were targeted at specific velocity and thickness
combinations that would produce the best data on dynamic specimen flexing.
Each experiment is designated by the order in which it was performed and by a
configuration type. For example BQ-12 indicates the twelfth experiment
performed in the set. Most experiments were conducted using the basic quantities
setup. The bending setup was used only to measure in two experiments. These
are designated BN-01 and BN-02 respectively. Table 3.1 contains the set of
experiments sorted primarily by type, impact velocity, and finally by specimen
thickness.
The experiments are divided up in the table by two different parameters because
each different experimental setup has experiments with varying impact velocities
and thickness. For the basic quantities, four different velocity ranges are explored
for each of the four thickness plates. These different velocities represent various
regimes of behavior in the previously studied soda lime glass experiments. The
four thickness plates represent a variation from thin panels to near half space
thickness. Some experiments were repeated due to either clarification issues or
unclear measurements. This design of experiments provided the ability to observe
the transient and equilibrium behavior of soda lime glass under various impact
conditions.
71
C. Results and Analysis of the Soda Lime Glass Experiments
In the present study, several series of experiments were performed. The first,
series involved a set of experiments to determine the pressures required to obtain
the desired projectile velocities. The basic quantities (BQ) set of experiments
was the second series conducted. Following this was the third experimental
group, the bending (BN) experiments. Finally, the supplemental experiments
(SP) were performed. From the collected data, various impact properties of the
material can be deduced. These properties can be divided into four categories.
First, the contact times, which can be inferred from high speed camera images,
are compared with the theoretical Hertzian contact times and the numerical
contact times from LS-DYNA [Nathenson et al., 2005]. Second, the realm of the
stress waves and material behavior before equilibrium is reached is examined by
means of the strain gage signals. These are compared with the theoretical
estimates from Lamb and Mitra, as well as strain waves generated by the LS-
DYNA numerical studies [Nathenson et al., 2005]. Also from the bending
experiments, the vibration of the specimens is analyzed. Third, the kinetics of the
impact, the elastic energy absorbed by the specimen and the coefficient of
restitution of the glass are discussed. Fourth, the records of the cracking
patterns are examined with respect to changes in the thickness and the impact
velocity. By examining these areas a comprehensive picture of soda lime glass’
behavior under high velocity particle impact is assembled.
C.1. Projectile Velocity vs. Firing Chamber Pressure Calibration
Before the projectile accelerator could be used for impact studies, the loading
pressure to firing velocity relationship had to be determined. The projectile was
accelerated using various loading pressures and removed from its sabot. The
velocity of the projectiles were determined using the high-speed camera images.
The position of each projectile in succeeding frames was used along with the time
between the frames to estimate the velocity. In order to measure the sabot
velocity following its exit from the gun barrel, the laser system was used. The
sabot velocity is determined by placing two laser beams a fixed distance apart
72
across the path of the sabot and using a diode to measure the time of the
intensity drop caused by the passage of the sabot though each beam.
The high speed camera is triggered by the laser velocity system when the first
beam is passed. This laser then triggers the flash and the ULTRA 17 camera
that is set to take exposures of ten nanoseconds duration at the frame rate of
150,000 frames per second. This camera takes seventeen pictures, which is more
than sufficient to establish the velocity. The first exposure begins after a delay
determined by the distance between the laser velocity system and the specimen
surface, allowing for the recording of images to commence when the projectile
passes in range of the camera.
In order to insure that the camera accurately measures the distance moved by
the projectile between two frames, an object of known length in one of the frames
is measured. This object was either the width of the front prong of the specimen
or the projectile diameter. The distance traveled by the projectile, and the inter-
frame time allows the camera’s dedicated computer to determine the velocity by
selecting one point in space at two different times that is, two separate frames.
This method yielded the curve for sabot and projectile velocity versus firing
pressure. Measurements were taken at pressures of 100, 150, 200, 250, 300, 350
and 400 pounds per square inch. As shown, velocities as high as 311 meters per
second were achieved (Figure 3.4). By curve fitting it was determined that the
equation with the greatest R2 value for firing pressure in terms of desired velocity
is a polynomial function. This relationship is specified by the following equation:
2.1209Pressure = 0.0019 Velocity× . {III.C.1}
This equation has a R2 value of 0.871. With this equation, the necessary pressure
for a given projectile velocity can be determined. The full set of experiments is
also shown in Figure 3.4.
73
C.2. Time of Contact Between the Sphere and the Plate
The contact times were obtained from the experiments using the camera, from
the theoretical Hertzian approximation [Goldsmith, 2001], and the numerical
study using LS-DYNA [Nathenson et al., 2005]. The accuracy of the camera
contact times was limited accurate due to the 6.6 microsecond or 2.5 microsecond
inter-frame time and debris obscuring the point at which the ball left the surface.
Accepting this limitation, the contact time estimates from the camera are
contrasted with those estimates using the theoretical Hertzian approximation and
the LS-DYNA numerical model. All of the contact time values are tabulated in
Table 3.2.
The Hertzian approximation [Goldsmith, 2001] shows a decrease in contact time
from 2.61 μs at 150 m/s to 2.20 μs at 350 m/s (Figure 3.5). This trend of
decreasing contact time is also reflected in the LS-DYNA numerical modeling,
where the contact times are estimated to decrease from 2.61 μs at 150 m/s to
2.16 μs at 350 m/s. According to the LS-DYNA results at higher impact
velocities, the 3 mm thick plates have slightly smaller contact times, on the order
of 0.05 microseconds when compared with the other three thickness plates
[Nathenson et al., 2005]. The three thicker specimens have nearly identical
contact times according to the numerical model. Specifically, the simulation at
150 m/s in the 5 mm plate was off by 0.01 μs from the simulation of the 15 mm
and 25.4 mm plates. The other velocity simulations matched exactly for all three
thicker plates. Including the 3 mm plates, the numerical model contact times are
all within 3% of the theoretical Hertzian contact times. Both the model and
Hertzian theory are elastic, with no cracking so this congruence is expected. This
close agreement confirms that the numerical model is accurately predicting the
theoretical behavior.
In examining the camera records, it is evident that the contact times can be
estimated to within an accuracy of 6 μs, as this is the inter-frame time for the
ULTRA 17, and to within 2.5 μs for the IMACON 200. The time at which the
specimen leaves the surface is uncertain in several of the experimental images due
74
to debris ejecting from the surface. This makes identifying contact time to be
uncertain to within 1-2 frames. However, in most of the images, the contact
times can be seen to be within one frame. The resulting contact times for the
experiments where the IMACON 200 is employed are between 0 and 2.5 μs
(Figure 3.5). This estimate fits well with the Hertzian estimate of contact times
between 2.61 μs and 2.20 μs. The literature also shows close but not perfect
agreement, within 0.4 μs, between theoretically and experimentally estimated
contact times [Chaudhri and Walley, 1978].
In summary, close agreement is found between the contact times estimated from
the experimental camera images and the two models. The theoretical Hertzian
equation {III.A.9} and the LS-DYNA numerical model contact time data (Table
3.2) indicate agreement within 0.1 μs, or 3 %, at each velocity for all of the
thickness plates. The experimental estimate of 1 frame, or 2.5 μs falls within the
range of both the theoretical and numerical estimates. This agreement of the
contact time estimates confirms that the Hertzian theory is applicable to impacts
of hardened chrome steel on soda lime glass and that the LS-DYNA numerical
model is accurate.
C.3. Examination of the Impact Surface Strain Gage Records
Using stacked tee rosette strain gages from Vishay Measurements Group, the
dynamic surface strain wave profiles were captured. Specifically, the specimen’s
the radial and hoop strains are measured. These experimental results show
distinct variations with specimen thickness, with the same trend for each
velocity. The resulting records also show similar characteristics to the LS-DYNA
numerical and the theoretical results from Lamb and Mitra.
C.3.1. Strain Wave Components Explained Using Theory and NumericalMethods
In the numerical study performed by Guodong Chen [Nathenson et al., 2005] in
conjunction with the current set of experiments, the material stresses and
deformations were examined using LS-YDNA 3D. In this study, the effects of a
75
steel ball hitting three different cases of soda lime glass plates were studied. The
response of a half space, plates of finite thickness but infinite lateral size, and
rectangular geometry that corresponded to the experimental specimens were
examined. The strains in the LS-DYNA model were measured at a point 10 mm
from the impact location in order to simulate the results observed in the
experimental study. Lamb’s [1904] and Mitra’s [1964] theoretical models were
also solved for the 10 mm radial distance and are compared to the half space LS-
DYNA simulation.
In the LS-DYNA numerical study on the half space the structure of the radial
wave is that of a pulse first in tension, then compression. Smaller reverberations
follow this large pulse (Figure 3.6). The LS-DYNA simulation includes all three
of the stress waves: the longitudinal wave, the shear wave, and the Rayleigh
wave.
The strains generated from the equations by Lamb [1904] (Figure 3.2) and Mitra
[1964] (Figure 3.3) are both compatible in form with the LS-DYNA solution at
350 m/s impact velocity. Lamb’s solution specifies that the Rayleigh wave is
dominant. The solution in Figure 3.2 has the same general form as the LS-
DYNA solution for the half space in Figure 3.6. The initial compressive strain
pulse is followed by a tensile strain pulse of greater amplitude and then a
compressive strain pulse.
Mitra’s [1964] solution includes the longitudinal and shear waves. Comparing
Mitra’s solution (Figure 3.3) to the approximation to the LS-DYNA study at 350
m/s (Figure 3.6), shows a similar initial pulse, this time attributed to the
longitudinal wave. The pulse’s maximum strain is normalized to the largest
maximum experimental strain in the plot. The plot of Mitra’s solution at this
radial distance contains a large longitudinal pulse and much smaller shear and
Rayleigh wave pulses, which is consistent with the experimental strain records.
76
Both theories yield solutions that are of similar form to the LS-DYNA
simulation. However, Mitra’s solution (Figure 3.3) is based on more realistic
assumptions and loading conditions. Mitra’s solution also appears more similar
to the numeric results for the half space (Figure 3.6). The initial compressive
pulse is smaller than that of Lamb’s (Figure 3.2). The LS-DYNA strain arrives
at the time for the longitudinal wave, not the Rayleigh wave. However, some
differences are visible from the records. For example, the fact that the Mitra
solution includes an impulse forcing function with zero duration and infinite
magnitude, and the numeric solutions do not. The greater duration and lower
absolute magnitude of the experimental and numeric pulses are evidence of this
difference.
For the case of finite thickness, variations exist within the strain records. The
changes in the strain occur around the time of the arrival of the reflected waves
from the rear surface or following their arrival (Figure 3.7). In the simulation of
the 3 mm and 5 mm specimens the tensile portion of the strain pulse is increased,
while the compressive part disappears. This is not present in the 15 mm and
25.4 mm thick specimens, where the reflected wave does not return until after the
pulse has passed. In all cases, the reverberations following the pulse were altered
somewhat due to the passing of the reflected wave.
In the case of a finite rectangular geometry, some further variations in the
reverberations following the pulse were measured (Figure 3.8). Because the wave
reflecting from the lateral boundary did not return until the main strain pulse
passed the simulated recording location, no major effect was observed on the
surface strain. Minor deviations are visible in the reverberations following the
passage of this wave. The same thickness effect was observed as in the
simulations with infinite boundaries.
77
C.3.2. Description of Experimental Strain Records
All of the experimental radial strain records showed one distinct spike and a
smaller spike in the hoop strain record. The highest velocity experiments show
the largest amplitude strains. Looking at one experiment from each thickness at
the 350 m/s range (Figure 3.9) it can be seen that as the thickness increases from
3 mm to 15 mm, the peak amplitude decreases. This is due to the fact that the
thicker materials more closely approximate the half space assumption of the
calculations. The 15 mm and 25.4 mm thick plates have very similar strain
profiles. The radial profiles show a tensile spike in the case of the thicker
specimens, 15 mm and 25.4 mm, that quickly returns to a near zero level with
some reverberations. In the thinner two specimens, this tensile pulse is of much
longer duration. Before these large tensile pulses, there is always a small
compressive pulse. These effects of thickness are also seen in the numerical and
theoretical work.
In the 3 mm experiments, at all impact velocities, the strain pulses are observed
in the radial direction to have an average duration of 34 microseconds (Table
3.3). The smallest velocity (BQ-18) has a duration of 44 microseconds, while the
300 m/s (SP-1) has a duration of 27 microseconds, and the maximum velocity
has a duration of 33 microseconds. The pulse magnitude increases from 67
micro-strains at 148 m/s (BQ-18) to 151 micro-strains at 300 m/s (SP-1).
However, the magnitude decreases to 93 micro-strains at 345 m/s (BQ-26).
In the 5 mm thick specimens, the strain pulses have an average duration of 13
microseconds (Table 3.3). At 174 m/s (SP-8) the duration is 6.2 microseconds.
This increases to 26 microseconds at 323 m/s (BQ-13). The average magnitude
is 49 micro-strains. At 174 m/s (SP-8) the amplitude is 46 micro-strains and this
increases to 57 micro-strains at 299 m/s (SP-9) before returning to 48 micro-
strains at 323 m/s (BQ-13).
78
Pulses of shorter duration occur in the 15 mm thick specimens (Table 3.3). Here
the average duration is 3.28 microseconds and the average amplitude is 26 micro-
strains. The shortest pulse occurs at 164 m/s (SP-6) and is 1.6 microseconds
long. The longest pulse is 6.2 microseconds long and occurs at 370 m/s (BQ-25).
The maximum strain of 37 micro-strains occurs at 164 m/s (SP-6). The
minimum strain is 20 micro-strains and is at 231 m/s (SP-13).
Similar behavior is observed in the 25.4 mm thick specimens where the average
duration is 4.5 microseconds (Table 3.3). The corresponding average peak strain
is 25 micro-strains. The smallest pulse is 2.6 microseconds long in the 287 m/s
experiment (SP-11). The largest pulse happens in the 336 m/s experiment (BQ-
11) and is 9 microseconds long. The highest peak strain is 29 micro-strains and
occurs in the 287 m/s experiment (SP-11). The smallest peak strain occurs at
336 m/s and is 21 micro-strains.
There is considerable variation in the amplitude and pulse duration recorded by
the strain gages. However, certain conclusions can be drawn from the data. The
average amplitude of the stress pulses decrease with thickness from 104 micro-
strains to 25 micro-strains when the thickness is increased from 3 mm to 25.4 mm
(Figure 3.10a). The average duration of the stress pulses also decreases over the
same thickness change from 34 microseconds to 4.5 microseconds (Figure 3.10b).
The strain amplitude shows an increasing trend over the velocity range, but with
wide variation (Figure 3.10c). The duration of the pulse shows no trend with
increasing velocity (Figure 3.10d). Therefore, varying the specimen thickness
correlates with a quantifiable effect on both the strain magnitude and pulse
duration while varying the impact velocity does not.
In light of the LS-DYNA numerical study, certain conclusions can be drawn from
the experimental strain gage data. In all of the experiments, the strain profile is
that of a small compression dip followed by a large tensile pulse. This is in
agreement with the LS-DYNA simulations. Unlike the simulations, there is no
rebound into compression in the experiments, suggesting that the material’s
79
fracture strength is low enough that cracking occurs, preventing compressive
strain from occurring on the surface. The low fracture strength of the specimen is
also responsible for the fact that the numeric peak strains are an order of
magnitude greater than that of the experiments. Additionally, in the LS-DYNA
simulations at 350 m/s, the tensile strain pulse with the 3 mm plate is seen to
have twice the duration as in the 25.4 mm plate (Figure 3.8). The experimental
tensile strain pulse in the 3 mm specimen has 4.5 times the duration of the strain
pulse in the 25.4 mm experiment for the same velocity range (Figure 3.9). This
additional distortion of the duration of the strain pulse in the experiment is also
due to cracking.
C.3.3. Correlation Between Rear Surface Vibration and Crack Oscillation
Measuring the strains of the rear surface of the specimen with the strain gages
allows for observation of dynamic oscillations in the strain gage profiles. The
oscillation in the strain profiles reflect the dynamic flexing or bending of the
plate. The measurements were conducted on two specimens; one of which was 5
mm thick and the other was 15 mm thick. These impacts occurred at speeds of
284 m/s and 290 m/s, respectively. These are designated by experiment
numbers BN-1 and BN-2. The experiments were conducted at these speeds in
order to examine the regime of velocity and thickness where the cracking
patterns are most evident. That is, experiments in which the largest size cone
cracks were observed to form and oscillate.
There is a correlation between the oscillations in the strain histories and the
cracking pattern oscillations observed using the high speed camera. The strain
gage histories indicate oscillations, always in the tensile range during the first 50
μs of the experiment. Both the 5 mm (BN-1) and 15 mm (BN-2) thick strains
have several wave oscillations during this time. The 5 mm experiment (BN-1)
has longer and more irregular oscillations of about 12-18 μs (Figure 3.11). The
overall crack in the 5 mm experiment (BN-1) is a crushed zone with many small
splinter cracks. These splinter cracks appear to oscillate with a frequency of 6-12
μs. The 15 mm experiment (BN-2) has shorter oscillations of about 6-12 μs
80
(Figure 3.12). The 15 mm experiment (BN-2) shows a crushed zone and a
partially formed cone crack. This crack’s edges distinctly show oscillations of 12-
24 μs. In the camera images, the cracking patterns in these two experiments are
somewhat less severe than those in the BQ series of experiments of approximately
the same velocities. This may be due to the strain gage glued to the rear
surfaces. The added strength imparted by the glue may reduce stresses and
retard crack growth.
The cracking patterns and the strain records in both experiments show
oscillations. In both BN-01 and BN-02, the cracking patterns oscillate at the
same period that the strain gage records vibrate, within the experimental error.
It is apparent from the geometry of the specimens that cracks, which are within
the specimen, should not be subject to exactly the same stresses as the rear
surface. However, given the thin nature of the specimens, any oscillatory
behavior of the stresses and hence the strains should be similar. In fact, this is
seen in the experiments. Therefore, it is likely that the dynamic vibration caused
by the reflection of stress waves from the rear surfaces is connected to the
oscillations in the cracks.
81
C.4. Impact Energies and Coefficients of Restitution
The impact energy imparted to the specimen is dissipated by elastic and inelastic
processes. A certain portion of this impact energy is dispersed through elastic
processes such as stress waves and vibrations, while most of the remaining energy
either goes into crack generation or the rebound of the projectile. This section
discusses quantifying the relationships between these different forms of energy.
First, the experimental method for determining the impact energy and the
corresponding coefficient of restitution is explained. Next, these quantities are
compared with the elastic energy lost and resulting coefficient of restitution
estimated from the numeric simulations [Nathenson et al., 2005]. The energy
dissipated due to stress waves is also estimated using theory [Hunter, 1957].
C.4.1. Determination of Experimental Impact Energy and Coefficient ofRestitution
The impact velocity was determined in two ways. First, the laser velocity system
measured the sabot velocity by determining the time of travel between two laser
beams set a fixed 5.9 mm apart. These beams produce a set of velocities before
the sabot stripper. Error in this measurement comes from both the exact spacing
of the laser beams, and the definition of the drops in the pulse measurement
(Figure 3.13). The measurements of the pulses are taken from the beginning, at
the end, and at the center of the first drop to the same point on the second drop.
The second velocity measurement comes from the high speed camera. By
measuring the distance traveled by the projectile between two of the camera
frames, a fixed time interval, the velocity is measured. This measurement is
based on a calibrated distance, specifically, the projectile’s diameter. This is
known to be 1.588 mm for the hardened chrome steel ball bearings. In order to
improve accuracy, four sets of two frames are used. These measurements are
averaged in order to compute the impact velocity.
82
The rebound velocity is computed in the same manner as the camera
computation of the impact velocity. In this case, debris from the impacts of the
specimens can result in obscuring of the rebounding projectile. Thus, usually
fewer sets of frames are used in this computation. Because the camera image is
two dimensional, any sideways motion of the projectile is not measurable.
However, the measurement of rebound velocity is still reasonably accurate
because any motion perpendicular to the axis of impact is minimal given proper
alignment of the specimen.
The laser impact velocity, camera impact velocity, and rebound velocity are all
measured from each experiment. The actual impact velocities indicate a spread
around the intended velocities (Figure 3.4), but this spread is acceptable as the
velocities still resolve into four distinct categories. By measuring the average
impact and rebound velocities and then comparing them, the kinetic energy that
is absorbed by the specimen, . .K EΔ , can be calculated. This is done by means
of the following equation:
( )2 21
1. .2 RK E m v vΔ = − . {III.C.2}
In equation {III.C.2}, m1, is the mass of the projectile, v, is the impact velocity
and, vR, is the rebound velocity. This equation is used because the specimen can
be treated as not moving for the duration of the contact. Also, the rebound
energy can be normalized to the initial impact energy:
2
2
. .. .
Rrebound
impact
vK EK E v
= . {III.C.3}
The projectiles’ rebound kinetic energy decreases with increasing impact velocity
(Figure 3.14). This indicates that as the impact velocity increases, a larger
percentage of the impact energy is absorbed by the specimen. The ratio of
rebound kinetic energy to impact kinetic energy corresponds to a coefficient of
restitution which, is expressed in the following formula:
83
RE
ve v−= . {III.C.4}
The coefficient of restitution is symbolized by, eE. These values are tabulated
(Table 3.4). With increasing impact velocity, the coefficients of restitution
decreases (Figure 3.15). The thickness of the impacted specimen has no effect on
the coefficient of restitution.
84
C.4.2. Elastic Stress Wave and Vibrations as Portions of the Impact Energy
As discussed above, the portion of the impact energy absorbed by the specimen
manifests in elastic deformation and cracking processes. The dissipation of elastic
energy occurs in two main modes: stress waves and vibrations. The amount of
energy that goes into the stress waves can be estimated by means of the analysis
presenter by Hunter [1957]. In equation {III.A.32} this relationship is given.
The resultant values are plotted in Figure 3.14. The corresponding coefficient of
restitution is given in equation {III.A.35} and these values are plotted in Figure
3.15. Furthermore, results of the numerical simulations conducted using LS-
DYNA enable the estimation of the change in kinetic energy and the
corresponding coefficient of restitution. However, no cracking is considered in the
numerical model, and thus these estimates only include the effects of elastic
processes. These two estimates and the experimental values for the coefficients
and change in kinetic energies are also tabulated in Table 3.4 and plotted in
Figures 3.14, and 3.15.
The coefficient of restitution, based on the measurements recorded via the high-
speed camera, show a drop over the four different thickness plates from 0.50 at
150 m/s to 0.19 at 350 m/s (Table 3.4). Hunter’s [1957] estimates for the stress
wave coefficient of restitution show a drop from an average over the thickness of
0.94 at 150 m/s to 0.89 at 350 m/s (Table 3.4). The LS-DYNA estimates for the
elastic coefficient of restitution also exhibit some thickness dependence (Figure
3.15). The three thicker specimens have near identical coefficients as they
decrease from 0.89 at 150 m/s to 0.85 at 350 m/s (Table 3.4). The maximum
difference between the estimates for the three thicker specimens at the same
velocity is 0.01. The 3 mm specimens decrease from 0.81 to 0.77 as the velocity
increases over the same range (Table 3.4). No such correlation between thickness
and the experimental coefficient of restitution is visible due to scatter in the
experimental data.
85
Similar patterns are observed for the particle’s kinetic energy change due to the
impact (Figure 3.14). For the estimated coefficient of restitution using Hunter’s
[1957] equations, the rebound kinetic energy drops from 87 % of the input energy
at 150 m/s to 78 % at 350 m/s. The LS-DYNA simulations for 5 mm, 15 mm,
and 25.4 mm exhibit similar decreases in rebound kinetic energy over the same
impact velocity range from an average of 80 % to an average of 72 %. The
deviation among these values is 1 %. The numeric estimate of the rebound
energy is 66 % at 150 m/s and 59 % at 350 m/s in the 3 mm thick specimens.
There is a larger decrease in the rebound kinetic energy in the experiments. At
impact velocities around 150 m/s the average rebound kinetic energy is 25 %.
When the impact velocity is about 350 m/s, the average rebound energy is 3.8 %.
The absolute rebound energy is smaller in absolute terms, and the rate of
decrease in the rebound energy with increasing impact velocity is much greater
than that for the LS-DYNA numeric simulation or Hunter’s [1957] stress wave
computation (Table 3.4).
The coefficient of restitution and percent rebound energy both decrease with
increasing velocity in the experiments. Additionally, most of the impact energy
is unaccounted for by elastic processes. The ratio of the inelastic energy to the
elastic energy increases with increasing velocity. Glass is a brittle material, and
almost no plasticity occurs during the impact. Therefore it can be assumed that
the inelastic energy is applied to the crushing of the contact zone and the various
cracking patterns. These cracking patterns are discussed in the following section.
C.5. Cracking Pattern Behavior
Cracking patterns were examined both in situ and post test. According to high
speed camera observations the crack propagation speed is about 1500 m/s [Field,
Sun, and Townsend, 1989]. This is much slower than the stress waves traveling
through the material. The longitudinal wave speed is 5600 ± 100 m/s [Field,
1988]. As a result the stress waves themselves may alter the stress distributions
within the material, and thus cause alterations in the cracking patterns. The
86
sections below describe the observations of the cracking patterns and the
variations due to thickness and impact velocity.
C.5.1. Details of Cracking Following Experiments
Following each experiment an investigation of the cracking patterns was carried
out. The cracking patterns were found to change significantly with both impact
velocity and thickness. The variations in these patterns with thickness and
velocity are detailed for representative experiments in Table 3.5. For specimens
impacted below about 130 m/s, either no damage occurred at the impact site, or
only a small circular indentation of 1 mm to 3 mm diameter appeared. This
indentation size is of the same order as the 1.588 mm diameter projectiles. At
speeds of about 130-150 m/s a crushed zone appeared in all specimens. From
this, it can be argued that the threshold velocity of crack initiation for this
geometry of projectile and these impact velocity ranges is therefore 140 ± 10 m/s.
More severe forms of damage accumulated with velocity. The resultant cracking
patterns are described below.
The experiment with a 3 mm thick plate, at 149 m/s impact velocity (BQ-18)
resulted in a 2 mm diameter, 1 mm deep crushed zone. 3 mm diameter chipping
occurred on the front surface. According to previous studies, a series of ring
cracks should appear outside of the crushed zone [Field, 1988]. In the current
experiments, surface chipping usually, but not always occurred on the front
surface obscuring the area where the ring cracks should be located. This made
determination of the ring crack diameters impossible. In BQ-18, three short
radial cracks appeared. For the purposes of this discussion, the definition of a
short radial crack is that it has not propagated completely through the specimen,
while the long radial cracks have propagated completely through the specimen.
Also, a partial cone crack with maximum diameter of 9 mm appeared.
At higher impact velocities, the variation in cracking patterns due to thickness
became more apparent. For the 3 mm specimens, the impacts produced radial
cracks of increasing number and severity as the impact velocity increased. A
87
Hertzian cone was also ejected from the rear surface. At 238 m/s (BQ-3), in
addition to ring cracks and surface chipping of maximum diameter 9 mm, there
were four long and three short radial cracks. The ejected cone crack had a
diameter of 10 mm. At 292 m/s (BQ-9) 8 long radial cracks and 4 short radial
cracks were present. Again, there were ring cracks and surface chipping this time
8mm in diameter. The cone that is ejected had a 10 mm diameter. At 300 m/s
(SP-1) the cone crack had a 6 mm base diameter. Six large and 2 small radial
cracks were observed. The 345 m/s experiment (BQ-26) resulted in 4 small and
4 large radial cracks. There was no observed surface ring cracks or chipping.
The ejected cone had a diameter of 11 mm.
The 5 mm specimen produced a crushed zone 2 mm in diameter and 1 mm deep
when impacted at 155 m/s (BQ-14) and shallow ring cracks and chipping of 6
mm diameter. A lobe of partial lateral cracking appeared, spreading from the
crushed zone. A maximum radius of 2.5 mm from the axis of impact was
observed. Also, 1 mm long splinter cracks were observed extending from the base
of the crushed zone. These splinter cracks extended perpendicular from the
surface of the crushed zone. They formed at or near the bottom of the crushed
zone. At an impact velocity of 174 m/s (SP-8) a crushed zone of 1 mm diameter,
an 8 mm diameter lateral crack and 4 mm surface chipping formed. Also splinter
cracking was observed with a maximum length of 1.5 mm. Some of the smaller
splinter cracks curved towards the surface of the specimen, while the largest, 1.5
mm, splinter cracks remained perpendicular to the crushed zone.
At 234 m/s impact velocity (BQ-8), the crushed zone deepened to 1.5 mm while
the diameter remained 2 mm. Ring cracking was not observed, but some
chipping of 9 mm diameter did occur from the front surface. One short radial
crack and partial lateral cracking formed. The lateral crack is considered partial
because it did not completely surround the crushed zone. Its largest radius was
4.5 mm from the impact point. Splinter cracks of 1 mm length were also seen.
88
In BQ-7, at 300 m/s impact velocity, the crushed zone increased in diameter to
2.5 mm and depth to 2.5 mm. Also, 5 mm diameter surface chipping was seen.
Four short radial cracks and a cone crack with maximum diameter of 8 mm were
visible in the 5 mm thick specimen, but the cone crack did not eject. No lateral
cracking or splinter cracks were observed. At 295 m/s (SP-9) a crushed zone of
2.5 mm diameter was also observed with 12 mm diameter surface chipping.
Lateral cracks of 15 mm diameter were seen and 3 small radial cracks initiated at
the rear surface.
At an impact velocity of 311 m/s, the crushed zone was observed to be 2.5 mm in
diameter and 1 mm deep. A partial cone crack 6 mm in diameter formed but did
not eject. Two large radial cracks also initiated from the rear surface. At 323
m/s (BQ-13), the 5 mm specimen exhibited a crushed zone of diameter 2 mm and
depth 2 mm, large partial shallow ring cracks, and chipping from the front
surface of 16.5 mm diameter. Also visible were one large radial crack, two cone
cracks, one full cone crack but not ejected of diameter 8 mm and one partial cone
crack. Partial lateral cracking of 5 mm radius could be seen. No splinter cracks
were visible.
Different damage patterns occurred in the 15 mm specimens in the same velocity
range than in the 3 mm and 5 mm thick specimens. At 137 m/s impact velocity
(BQ-19) there was a crushed zone of diameter 3 mm and depth 1 mm. The
surface ring cracks and front surface chipping extended to 5 mm diameter. No
radial, conical, or lateral cracks were seen. There were splinter cracks of length 1
mm. At 164 m/s (SP-6), a crushed zone of 1.5 mm diameter and 1 mm deep was
observed. Surface chipping of 9 mm diameter occurred. Also 1 mm deep splinter
cracks were visible.
At 230 m/s impact velocity (BQ-6) the crushed zone was 2.5 mm in diameter
and 1.5 mm deep. Material was ejected from the front surface and there was
shallow surface ring cracking and chipping of 8 mm diameter. There were again
splinter cracks of length 1 mm but no radial, cone or lateral cracks. Another
89
experiment at the velocity of 231 m/s (SP-13) resulted in a 1 mm diameter
crushed zone. A 9 mm completely circular lateral crack was observed along with
7 mm radius partial surface chipping. Splinter cracks of 1 mm length also
occurred. At 239 m/s (SP-5) a crushed zone that was 2.5 mm diameter and 1.5
mm deep was seen. Some surface chipping occurred with a 3 mm radius. Five
partial lateral cracks initiated but only propagated 1 mm. Also, 1 mm deep
splinter cracking was observed.
In the 15 mm specimens at higher impact velocities, more severe cracking is
observed. At 279 m/s (SP-7) a 1.5 mm diameter crushed zone is generated by
the impact. A lateral crack results in surface chipping of 12 mm diameter. At
300 m/s (SP-4) a crushed zone of 2 mm diameter is observed. Surface chipping
of 14 mm diameter occurs. Additionally partial lateral cracks of 5 mm and 10
mm radius are seen. At 307 m/s (BQ-5) the crushed zone was 3 mm in diameter
by 1 mm deep. The surface cracking was partial but 10.5 mm in diameter. No
radial or cone cracks were observed, but a complete lateral crack of diameter 13
mm that surrounded the crushed zone and traveled in a saucer like fashion
towards the surface was seen. Splinter cracks of length 0.5 mm were also seen.
At 371 m/s (BQ-25) the impact resulted in a crushed zone of 3 mm diameter and
2 mm depth. Large shallow ring cracks and chipping of 20.5 mm diameter were
observed on the front surface. A complete lateral crack of 11 mm diameter was
seen. No radial cracks or vertical splinter cracks were observed.
The largest, 25.4 mm thick, specimens showed the smallest amount of cracking.
At 139 m/s (BQ-21) a crushed zone 3 mm in diameter and 1 mm deep was
present. 5 mm diameter chipping existed. A partial lateral crack of 2.5 mm
radius appeared. Small splinter cracks 0.5 mm long appeared. No radial or
conical cracking occurred. An impact at the velocity of 172 m/s (SP-10) resulted
in a crushed zone of 2 mm diameter and 1 mm depth. 3 mm diameter surface
chipping was observed. 1 mm deep splinter cracks also occurred. At 229 m/s
(SP-12) a 2.5 mm diameter crushed zone occurred. There was 10 mm long
chipping to one side of the impact point in a semi-circular pattern. When
90
measured through the point of the impact the maximum chipping diameter was
also about 10 mm. A 9 mm diameter lateral crack was seen and 1 mm deep
splinter cracks occurred.
At 242 m/s (BQ-4), there was a crushed zone of 2 mm diameter and 1 mm deep
and chipping of 10 mm diameter from the front surface. No radial, cone, or
lateral cracking was present. However, splinter cracking 1 mm in length
occurred. At 287 m/s (SP-11) a 2.5 mm diameter crushed zone was observed. A
maximum diameter of 11 mm was observed in the surface chipping. 11 mm was
also the size of the lateral crack diameter. Splinter cracks of 1 mm depth were
also seen.
Increasing the impact velocity had little effect on the cracking patterns in
thickest specimens. At an impact velocity of 307 m/s (BQ-10) a crushed zone of
diameter 2 mm and depth of 1.5 mm was seen with shallow surface ring cracks
and material chipping of 8.5 mm diameter from the front surface. As at lower
velocities 1 mm long splinter cracks were seen. However, increasing the impact
velocity to 336 m/s (BQ-11) did affect the cracking pattern. In experiment BQ-
11 the crushed zone was 3 mm in diameter and 2 mm deep, while the surface ring
cracks increased to 18 mm diameter. Chipping occurred from the surface. Both
0.5 mm long splinter cracks and a sub surface lateral crack 6.5 mm radius
formed.
To summarize, several types of cracking are found in the specimens following the
impacts. Front surface damage includes ring cracks and chipping. Internal
damage consists of a hemispherical crushed zone as well as other forms of
cracking including radial, conical, lateral, and splinter cracks. The types of
cracking patterns that dominate and the size and shape of these patterns are
effected by both specimen thickness and impact velocity. It is apparent that the
conical and radial cracks were most prevalent in thinner specimens and caused
more damage at higher velocities. Rear surface radial cracking appeared where
the dynamic bending stresses were greatest, that is in the thinner specimens.
91
The lateral cracks and surface chipping, on the other hand do not show an effect
of thickness, but increase in diameter with velocity. The splinter cracks were
visible in thinner and thicker specimens at low velocity, but as velocity increased,
were visible only in the thicker specimens. Changing the thickness and velocity
has different effects on the internal stress state during the impact, which causes
the variations in the cracking systems.
C.5.2. Dynamic Evolution of Cracking
As discussed above, the impacts produced several different types of cracking
including radial, conical, and lateral cracks. During the experiments, high-speed
observations were made of the cracking patterns. These images allow for the
formation of cracking to be examined with respect to variations in the impact
velocity and specimen thickness. The dynamic crack observations are
summarized below.
In all cases above a threshold impact velocity, a roughly hemispherical crushed
zone formed first directly below the impact site. This crushed zone varied from 1
mm to 3 mm in size. Multiple ring cracks also formed on the surface, although
this was difficult to see from the camera images. Debris ejected from the surface
was clearly visible in all cases. Most of this debris was generated by chipping
that occurred from the surfaces of the specimens. The amount of debris
increased with velocity.
Cone cracks were formed for specific cases where the thickness of the specimen is
small and the impact velocity is large. These cracks were seen to originate either
at the surface of the specimen at the ring cracks (BQ-3, BQ-26) or towards the
bottom of the crushed zone (BQ-18, BQ-13, and BQ-7) and to proceed
downwards towards the rear surface following the smallest principle stress
trajectory as in Field’s paper [1988]. They formed within 6 μs of the impact.
The result of the cone crack is a section of the specimen ejected from the rear
surface when sufficient impact velocity is reached. For the three higher impact
velocities of the 3 mm plates (BQ-3, BQ-9, and BQ-26) the cone crack did eject.
92
The minimum impact velocity in these experiments was 238 m/s (BQ-3). For
two of the 5 mm specimens (BQ-13, BQ-7) and for the lowest impact velocity 3
mm (BQ-18) the cone formed but did not completely eject. This is referred to
from now on as partial cone formation. In fact two partial cones formed for the 5
mm at 323 m/s (BQ-13) case. Oscillations were seen to occur during formation
of the partial cone crack at 5 mm and 300 m/s (BQ-7) and in the outer of the
two cone cracks in the 5 mm, 323 m/s (BQ-13) specimen.
The radial cracks also appear in the 3 mm specimens at all velocities and in some
5 mm thick specimens (BQ-8, BQ-7, and BQ-13). The slowest 3 mm (BQ-18) and
the 5 mm, 234 m/s (BQ-8) & 300 m/s (BQ-7) impacts resulted in small radial
cracks, cracks that did not propagate through the entire specimen. The 5 mm,
323 m/s (BQ-13) impact produced one large radial crack that propagated to the
side boundary of the specimen. The remaining impacts (BQ-3, BQ-9, and BQ-
26), generated multiple long radial cracks and fractured. In some cases the radial
cracks originated at the front surface. These impacts include the 5 mm, 234 m/s
(BQ-8) where the cracks appeared within 6 μs & the 5 mm, 300 m/s (BQ-7)
where the cracks were seen to appear after 54 μs. These cracks did not
propagate to the sides. On the other hand, most of the radial cracks appear to
originate on the rear surface. These include the rest of the above mentioned
specimens. The radial cracks appeared at the following times following the 3 mm
impacts: 149 m/s (BQ-18), 36 μs; 238 m/s (BQ-3), 12 μs; 300 m/s (BQ-9), 12 μs;
345 m/s (BQ-26), 12 μs. The radial crack in the 5 mm, 323 m/s (BQ-13)
specimen was out of the focal plane and thus unseen by the camera. All of the
radial cracks opened in a half penny fashion leaving visible marks of this shape
on the crack surfaces. In most cases the radial cracking propagated for far
greater distances than the conical cracks.
The lateral cracks, like those discussed in the literature [Field, 1988] formed upon
unloading and nucleated under the surface at the crushed zone boundary. The
cracks formed within 12 μs of the impact in all cases where they were observed.
These cases are: 5 mm at 155 m/s (BQ-14), 234 m/s (BQ-8), & 323 m/s (BQ-13);
93
15 mm at 137 m/s (BQ-19), 307 m/s (BQ-5), & 370 m/s (BQ-25) and 25 mm at
139 m/s (BQ-21) & 336 m/s (BQ-11). Lateral cracks were seen to propagate in
saucer like patterns angling towards the front surface. In some cases, they
reached the surface, resulting in chipping from the front surface. Also, in several
experiments, the lateral cracks were seen to oscillate. Specifically, oscillations
occurred in the following experiments: 5 mm at 323 m/s (BQ-13); 15 mm at 307
m/s (BQ-5) & 370 m/s (BQ-25); and 25mm at 336 m/s (BQ-11). The lateral
and cone cracks oscillated because of stress wave reflections from the surfaces of
the specimens. This is discussed further in the bending section.
Splinter cracks were seen in several experiments. They propagated from near the
bottom of the crushed zone, as mentioned in the static section. The sizes of the
splinter cracks are typically 0-1 mm. The cracks started out traveling
perpendicular to the point in the crushed zone that they nucleate from. Some of
them curved back towards the front face, but the ones that propagate from the
very bottom of the crushed zone remained straight. These cracks are usually
observed in the thicker specimens at all velocities. Approximately 1 mm long
splinter cracks were seen in the following specimens: 5 mm at 155 m/s (BQ-14) &
234 m/s (BQ-8), 15 mm at 137 m/s (BQ-19) & 230 m/s (BQ-6), and 25 mm at
242 (BQ-4) & 307 m/s (BQ-10). Smaller, about 0.5 mm long splinter cracks
were observed in the 15 mm thick specimens at 307 m/s (BQ-5) and the 25 mm
thick specimens at 139 m/s (BQ-21) & 336 m/s (BQ-11). Some of these cracks
(BQ-10) were observed to oscillate. These cracks while less severe than other
forms of damage are still distinct.
Inaccuracies in this measurement develop because of the narrow field of focus and
the 6.6 or 2.5 microsecond minimum time between camera images. The
observations of the dynamic cracking behavior are limited by the inter-frame
time. The dynamic evolution of two cone cracks can be seen in Figure 3.16.
Radial cracking from both the front and rear faces were observed in all the
thinner (3 & 5 mm) specimens and became more severe as the velocity increased.
In the 3 mm specimens, the cracks all initiated from the rear surface, and in the
94
three experiments at higher velocities (BQ-3, BQ-9, and BQ-26) specimen
fractured into several pieces. The nature of the rear surface cracking can be seen
in Figure 3.17. In the 5 mm specimens, no radial cracks were observed in the
slowest impact (BQ-14), radial cracking from the front was observed in the
middle velocities (BQ-7, BQ-8), and 1 radial crack was observed from the rear
surface in the fastest velocity impact (BQ-13). Lateral cracks did not appear in
the 3 mm impacts, but occurred in most of the 5 mm, 15 mm, and 25.4 mm
impacts. One full lateral crack can be seen in Figure 3.18. The cracks appear to
increase in size and fullness with velocity, but not thickness. Surface chipping,
which appeared in almost every experiment, appeared to increase in diameter
with velocity but not thickness. Splinter cracks are visible in the 5 mm, 15 mm
and 25.4 mm thick specimens at the slowest (BQ-14, BQ-19) and medium slow
velocities (BQ-6, BQ-8). As the velocity increases, the splinter cracks are only
seen in the 15 mm and 25.4 mm thick specimens for the medium high velocity
(BQ-5, BQ-10) and only in the 25.4 mm for the highest velocity (BQ-11)
impacts.
95
D. Significance of Results
The current study on soda lime glass has resulted in a number of correlations
between the experimental data and estimates from theoretical equations and the
LS-DYNA numerical model. First, the contact time provides a means of ensuring
that the experiments are accurately predicted by estimates from Hertzian theory
and the numerical model. Second, the numerical model and theoretical
predictions for this wave are compared to the experimental records. The models
and the experiments provide information about the effects of impact velocity and
specimen thickness on the strain. Third, observations concerning the kinetics of
the impact provide information about the coefficient of restitution of the
specimen as well as the partitioning of impact energy. This allows for an
examination of the relative severity of the variety of damage modes. Finally, the
cracking patterns provide an insight into the differing state of damage in the
material due to changes in thickness and impact velocity. Knowledge of the
types of cracking and their specific patterns can provide insights into the
weakening of the material.
The experimental contact times are within 3 % of both the LS-DYNA numerical
simulations and Hertzian theory. This precise of a correlation indicates a strong
match between the three methods. This fact indicates that the assumptions upon
which Hertzian theory and the LS-DYNA model are based are applicable to the
impact of steel ball bearings on soda lime glass blocks. As a result, both the
theory and numerical methods can be used to predict the elastic impact behavior
as observed in the experiments.
The front and rear surface strains were measured and compared to the LS-DYNA
simulations and plots of theoretical equation. In the experiments, the strain on
the front surface consists of one large tensile pulse. The initial pulse increases in
size and duration with decreasing specimen thickness. The complete pulse can be
approximated by the solution provided by Mitra [1964] for the impact over a
circular area on the surface of a half space. Additionally, LS-DYNA numerical
simulations using the same geometry as the specimens indicate that cracking and
96
the material strength place a limiting factor on the elastic deformation. The LS-
DYNA numeric simulations with different boundary conditions show that the
strain profile at a specific location on the front surface is altered by the return of
waves from these boundaries. Additionally, the simulations suggest that the
reflections of the stress waves have an effect on the crack formation. Specifically,
the dynamic vibrations of the rear surface show oscillation periods in tension that
are within error to the oscillation periods of the cracks observed by the high
speed camera. This indicates that the stress waves influence the crack
propagation.
In the LS-DYNA numerical model for an impact on an infinite half space the
strain is observed at a location on the front face of the specimen. This record
indicates the arrival of a single pulse and several smaller reverberations.
Modifying the model geometry to include a finite and decreasing thickness has
the following effects on the radial strain: an increasing magnitude of the tensile
part of the pulse, the disappearance of the compressive part of the pulse, and
alteration of the reverberations. These alterations occur around and after the
time at which the stress pulse should reflect from the rear surface and arrive at
the measurement location. Adding a finite lateral boundary to the LS-DYNA
model geometry has lesser effect on the surface strains and only changes the
reverberations following the major strain pulse.
The experimental surface strain profiles agree with the LS-DYNA numerical
simulations and Mitra’s [1964] and Lamb’s [1904] theories on both the structure
and the nature of the waves. Experimental strain measurements in the thicker
plates have a similar tensile portion of the stress pulse, but no compressive
portion. The thinner specimens show an increased tensile magnitude as
compared to the thicker specimens, and also no compressive strain. In the thin
specimens, the experimental pulses have a greater duration than the models
because of the effects of cracking. The magnitude of the experimental strains is
an order of magnitude lower than the numerical elastic strains suggesting that
97
cracking prevents the material from straining as much as it would under pure
elastic conditions.
The kinetics analysis reveals the long time impact behavior of the material to be
within the regime of quasi-static understanding. The experimental coefficients of
restitution are always much lower than the theoretical or LS-YNDA elastic
coefficients and they decrease at a greater rate than the models. The same
pattern is seen where the fraction of rebound energy over impact kinetic energy is
plotted. Here, increasing velocity correlates with decreasing rebound kinetic
energy, more of which is taken up by cracking and elastic energy dissipation at
higher velocities. Inelastic processes, such as crack formation account for most of
the loss of energy and the amount of inelasticity increases with increasing
velocity. Within the elastic portion of the impact, theoretical and numerical
estimations show that part of the energy goes to the formation of stress waves.
Most of the remainder of the energy goes into vibrations.
The experiments show that the radial and lateral cracks propagate the farthest
distance parallel to the impact surface and that the conical cracks typically
penetrate to the rear surface of the thin specimens causing cone ejection. Of
these three cracking patterns, the radial cracks are observed to cause the most
severe damage. The conical and radial cracks are most prevalent in the thinner
specimens, while the lateral cracks did not appear in the 3 mm thick specimens
but occurred in the other specimens. All three types of cracking increased in
severity with velocity. Surface chipping and vertical splinter cracking were also
observed, although they did not cause the as much damage as the other three
cracking types.
It is apparent that altering the impact velocity and the geometry of the
specimens has effects on the surface strains and the cracking patterns. Cracking
patterns present in previous experimental studies are clearly observed exist in the
current study. This study provides a description of the changes in cracking
patterns due to changes in impact velocity and thickness. The dynamic stress
98
waves, do not dominate the internal cracking patterns at these impact velocities,
although they have an effect as seen by oscillations in some cracking patterns.
The stress waves do cause surface strains, which are also estimated by numerical
simulations in LS-DYNA and by theory. These results indicate that quasi-static
and stress wave theories both apply to impacts of this geometry.
99
References
Chaudhri, M. M. and S. M. Walley, 1978. “Damage to Glass Surfaces by the Impact of Small
Glass and Steel Spheres.” Philosophical Magazine. A 37(2): 153-165.
Field, J. E., 1988. “Investigation of the Impact Performance of Various Glass and Ceramic
Systems.” U.S. Army: Final Technical Report – Contract Number DAJA45-85-C-0021.
Field, J. E., Q. Sun, and D. Townsend, 1989. “Ballistic Impact of Materials.” Inst. Phys Conf.
Ser. No 102: Session 7. Oxford.
Goldsmith, W. 2001. Impact: The Theory and Physical Behavior of Colliding Solids, Dover
Publications.
Goodier J. N.; W. E. Jahsman; and E. A. Ripperger. 1959. “An Experimental Surface-Wave
Method for Recording Force-Time Curves in Elastic Impacts.” Journal of Applied Mechanics. 3-7.
Hunter, S. G. 1957. “Energy Absorbed by Elastic Waves During Impact.” Journal of the
Mechanics and Physics of Solids. 5 162-171.
Knight, C. G.; M. V. Swain; and M. M. Chaudhri. 1977. “Impact of Small Steel Spheres on Glass
Surfaces.” Journal of Materials Science. 12: 1573-1586.
Lamb, H., 1904. “On the Propagation of Tremors over the Surface of an Elastic Solid.
Philosophical Transactions of the Royal Society. A 203 1-42.
Love, A. E. H. 1944. The Mathematical Theory of Elasticity 4th Ed. Cambridge University Press:
London.
Matweb. 2005. www.matweb.com
Mitra, M. 1964. “Disturbance Produced in an Elastic Half-Space by Impulsive Normal Pressure.”
Proceedings of the Cambridge Philosophical Society. 69: 683-696.
Nathenson, D., G. Chen, and V. Prakash. 2005. Dynamic Response of Soda Lime Glass to Small
Particle Impacts. Society for Experimental Mechanics Conference Proceedings.
Spath, W. 1961. Impact Testing of Materials. Gordon and Breach: New York.
100
Tillet J. P. A. 1954. A Study of the Impact of Spheres on Plates. Proceedings of the Physical
Society. B 67: 677-88.
101
Tables
Table 3.1: Experiment List Including Specimen Impact Velocity and Thickness.
Experiment # Thickness (mm) Impact Velocity (m/s)
BQ-3 3 238
BQ-4 25.4 242
BQ-5 15 307
BQ-6 15 230
BQ-7 5 300
BQ-8 5 234
BQ-9 3 292
BQ-10 25.4 307
BQ-11 25.4 336
BQ-13 5 323
BQ-14 5 155
BQ-18 3 149
BQ-19 15 137
BQ-21 25.4 139
BQ-25 15 371
BQ-26 3 345
BN-1 5 284
BN-2 15 290
SP-1 3 300
SP-4 15 300
SP-5 15 239
SP-6 15 164
SP-7 15 279
SP-8 5 174
SP-9 5 295
SP-10 25.4 172
SP-11 25.4 287
SP-12 25.4 229
SP-13 15 231
102
Table 3.2: Experimental, Theoretical, and Numeric Peak Contact Times.
Experiment # Contact Time (μs) Calculation Label Contact Time (μs)
BQ-3 ≤ 6.67 Hertzian 150 m/s 2.61
BQ-4 ≤ 6.67 Hertzian 230 m/s 2.40
BQ-5 ≤ 6.67 Hertzian 300 m/s 2.27
BQ-6 ≤ 6.67 Hertzian 350 m/s 2.20
BQ-7 ≤ 6.67 Num. 3 mm 150 m/s 2.61
BQ-8 ≤ 6.67 Num. 3 mm 230 m/s 2.32
BQ-9 ≤ 6.67 Num. 3 mm 300 m/s 2.22
BQ-10 ≤ 6.67 Num. 3 mm 350 m/s 2.13
BQ-11 ≤ 6.67 Num. 5 mm 150 m/s 2.60
BQ-13 ≤ 6.67 Num. 5 mm 230 m/s 2.37
BQ-14 ≤ 6.67 Num. 5 mm 300 m/s 2.28
BQ-18 ≤ 6.67 Num. 5 mm 350 m/s 2.18
BQ-19 ≤ 6.67 Num. 15 mm 150 m/s 2.61
BQ-21 ≤ 6.67 Num. 15 mm 230 m/s 2.37
BQ-25 ≤ 6.67 Num. 15 mm 300 m/s 2.28
BQ-26 ≤ 6.67 Num. 15 mm 350 m/s 2.18
SP-1 ≤ 2.50 Num. 25.4 mm 150 m/s 2.61
SP-6 ≤ 2.50 Num. 25.4 mm 230 m/s 2.37
SP-7 ≤ 2.50 Num. 25.4 mm 300 m/s 2.28
SP-8 ≤ 2.50 Num. 25.4 mm 350 m/s 2.18
SP-9 ≤ 2.50
SP-10 ≤ 2.50
SP-11 ≤ 2.50
SP-12 ≤ 2.50
SP-13 ≤ 2.50
103
Table 3.3: Experimental Strain Measurements.
Experiment # Amplitude (με) Duration (μs)
BQ-3 107 31.2
BQ-4 16.0 4.63
BQ-5 128 3.90
BQ-6 26.7 2.10
BQ-7 58.7 6.10
BQ-8 45.3 10.7
BQ-9 115 16.1
BQ-10 32.0 4.40
BQ-11 21.3 9.00
BQ-13 48.0 26.0
BQ-14 128 9.61
BQ-18 66.7 44.0
BQ-19 13.3 4.20
BQ-21 16.0 5.00
BQ-25 24.0 6.20
BQ-26 93.3 32.9
SP-1 151 26.8
SP-6 37.1 3.56
SP-7 22.7 2.60
SP-8 45.8 2.90
SP-9 57.1 2.40
SP-10 22.7 1.63
SP-11 28.6 2.90
SP-12 25.7 6.24
SP-13 20.0 8.00
104
Table 3.4: Impact Velocities, Rebound Velocity, Kinetic Energies, and Coefficients of Restitution.
Experiment
#
Impact Velocity
(m/s)
Rebound Velocity
(m/s)
Rebound Energy/
Impact Energy
Coefficient of
Restitution
BQ-3 238 71.7 0.091 0.30
BQ-4 242 87.3 0.13 0.36
BQ-5 307 65.7 0.046 0.21
BQ-6 230 83.9 0.13 0.36
BQ-7 300 43.0 0.021 0.14
BQ-8 234 58.0 0.061 0.25
BQ-9 292 70.7 0.059 0.24
BQ-10 307 94.5 0.095 0.31
BQ-11 336 84.8 0.064 0.25
BQ-13 323 61.9 0.037 0.19
BQ-14 155 72.4 0.22 0.47
BQ-18 149 46.2 0.097 0.31
BQ-19 137 94.5 0.48 0.69
BQ-21 139 63.2 0.22 0.46
BQ-25 371 78.9 0.045 0.21
BQ-26 345 38.1 0.012 0.11
SP-6 164 98.0 0.36 0.60
SP-8 174 88.6 0.26 0.51
SP-9 295 92.0 0.097 0.31
SP-10 172 81.9 0.23 0.48
Calculation Label Energy Ratio Coefficient Calculation Label E. Ratio Coefficient
Hertzian 150 0.87 0.94 Num. 5 mm 300 0.72 0.85
Hertzian 230 0.83 0.91 Num. 5 mm 350 0.72 0.85
Hertzian 300 0.80 0.90 Num. 15 mm 150 0.80 0.89
Hertzian 350 0.78 0.89 Num. 15 mm 230 0.74 0.86
Num. 3 mm 150 0.66 0.81 Num. 15 mm 300 0.71 0.84
Num. 3 mm 230 0.62 0.79 Num. 15 mm 350 0.71 0.84
Num. 3 mm 300 0.53 0.73 Num. 25.4 mm 150 0.80 0.89
Num. 3 mm 350 0.59 0.77 Num. 25.4 mm 230 0.74 0.86
Num. 5 mm 150 0.79 0.89 Num. 25.4 mm 300 0.72 0.85
Num. 5 mm 230 0.74 0.86 Num. 25.4 mm 350 0.72 0.85
105
Table 3.5: Post Impact Static Cracking Pattern Details
106
Figures
Time / Arrival of Longitudinal Wave at 10 mm Strain Gage
Pea
kFo
rce
/Max
Forc
efo
r350
m/s
sim
.
0 0.5 1 1.50
0.25
0.5
0.75
1Target 5 mm 350 m/sTarget 5 mm 300 m/sTarget 5 mm 230 m/sTarget 5 mm 150 m/sHertz 350 m/sHertz 300 m/sHertz 230 m/sHertz 150 m/s
Sphere Diameter: 1/16"Plate Thickness: 5 mm
Figure 3.1: Hertzian approximation [Goldsmith, 2001] and numerical simulations for impact force
Curve [Nathenson, 2005]. The solid “Target” curves are the asymmetrical simulations, while the
dotted Hertzian curves are symmetric.
107
Time / Arrival of Rayleigh Wave
Stra
in/M
axS
train
0.98 0.99 1 1.01 1.02 1.03 1.04
-0.5
0
0.5
1
Lamb
Arrival ofRayleighWave
Figure 3.2: Surface strain profile for Rayleigh wave of the general point impact of a half space,
using the equations from Lamb [1904]. The numerical differentiation of the theoretical equation
{III.A.10} is shown here. The strain is normalized to the peak tensile strain and the time is
normalized to the arrival of the longitudinal wave.
108
Time / Arrival of Longitudinal Wave
Stra
in/M
axS
train
1 1.5 2-0.5
0
0.5
1
Mitra
Longitudinal Wave
Shear WaveArrival (t2)
Rayleigh WaveArrival (t3)Lo
ngitu
dina
lWav
eA
rriv
al(t1
)
Figure 3.3: Strain profile for impulsive impact of a half space in a circular area, after Mitra [1964].
The strain is normalized to the peak tensile strain in the longitudinal wave. The longitudinal
wave arrives at t1, the shear wave at t2 and the Rayleigh wave at t3. These correspond to limits
on the calculation in equations {III.A.18} and {III.A.19} as follows:
13
Rtβ
= , {III.F.1}
2 Rtβ
= , {III.F.2}
3 r atV−= . {III.F.3}
The symbols are the same as those from equations {III.A.18} and {III.A.19}.
109
Velocity (m/s)
Pre
ssur
e(p
si)
0 50 100 150 200 250 300 350 4000
50
100
150
200
250
300
350
400
450
500
550
600
BendingBasic QuantitiesCalibrationSupplamental Exps.Best Fit Equation
Pressure = 0.0019 * Velocity 2.1209
Figure 3.4: Calibration curve for projectile impact including all experiments. Also specified is the
best fit curve, which happens to be polynomial. The symbols indicate the range of velocities for
each target velocity.
110
Impact Velocity (m/s)
Con
tact
Tim
e(μ
s)
0 50 100 150 200 250 300 350 4001.5
1.75
2
2.25
2.5
2.75
3
Hertz TheoryNumeric 3 mm ThickNumeric 5 mm ThickNumeric 15 mm ThickNumeric 25.4 mm ThickAverage of Experiments
Figure 3.5: Plot of experimental, theoretical [Goldmsith, 2001], and numerical [Nathenson, 2005]
contact time variation with velocity. The numeric values for 5 mm, 15 mm, and 25.4 mm are
within 0.01 μs for all impact velocities. The accuracy of the experimental contact times are
limited by the fastest camera framing rate. The plot shows the maximum contact time for the
experiments to be 2.5 μs.
111
Time / Arrival Time of Longitudinal Wave
Stra
in/M
axS
train
0 1 2 3 4 5 6-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Radial Strain 350 m/sHoop Strain 350 m/s
She
arW
ave
Arr
ival
Ray
leig
hW
ave
Arr
ival
Long
itudi
nalW
ave
Arr
ival
Figure 3.6: LS-DYNA simulations of the surface strains profiles at 10 mm from the impact
location on a half space [Nathenson, 2005]. The impact speed is 350 m/s. This strain is similar
in form to the theoretical Mitra wave (Figure 3.3), except that the strain wave has a greater
duration and encompasses all three stress waves. This is a result of a finite loading time rather
than the impulsive loading used in Mitra’s computations.
112
Time / (Arrival of Longitudinal Wave)
Stra
in/(
Max
Nor
mal
Stra
info
r3m
mP
late
s)
0 1 2 3 4 5 6-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radial 3 mm 350 m/sRadial 5 mm 350 m/sRadial 15 mm 350 m/sRadial 25.4 350 m/s
She
arW
ave
Rea
rSur
face
Wav
e3
mm
Rea
rSur
face
Wav
e5
mm
Rea
rSur
face
Wav
e15
mm
Rea
rSur
face
Wav
e25
.4m
m
Ray
leig
hW
ave
Finite Thickness Simulations
Figure 3.7: LS-DYNA simulations of the surface strain profiles at 10 mm on plates with infinite
lateral boundaries but finite thickness [Nathenson, 2005]. Impact speed is 350 m/s. The 25.4 mm
and 15 mm signals overlap during the initial pulse, while the 5 mm and 3 mm initial pulses have
greater peak strains and longer duration.
113
Time / (Arrival of Longitudinal Wave)
Stra
in/(
Max
Stra
info
r3m
mP
late
Sim
ulat
ion)
0 1 2 3 4 5 6-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radial 3 mmRadial 5 mmRadial 15 mmRadial 25.4 mm
She
arW
ave
Ray
leig
hW
ave
Rea
rSur
face
Wav
e3
mm
Rea
rSur
face
Wav
e5
mm
Rea
rSur
face
Wav
e15
mm
Rea
rSur
face
Wav
e25
.4m
m
Late
ralB
ound
ary
Wav
e
Finite Thickness andLateral BoundariesImpact Velocity: 350 m/s
Figure 3.8: LS-DYNA simulations of the surface strain profiles at 10 mm on plates with finite
rectangular geometries identical to the experiments [Nathenson, 2005]. The impact velocity is 350
m/s. Until the arrival of the lateral boundary wave, no differences are seen from Figure 3.5.
114
Time / (Arrival of Longitudinal Wave)
Stra
in/(
Max
Stra
inof
3m
mP
late
Sim
ulat
ion)
0 1 2 3 4 5 6
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Exp 3 mm 345 m/sExp 5 mm 323 m/sExp 15 mm 371 m/sExp 25.4 mm 336 m/s
She
arW
ave
Ray
leig
hW
ave
Rea
rSur
face
Wav
e3
mm
Rea
rSur
face
Wav
e5
mm
Rea
rSur
face
Wav
e15
mm
Rea
rSur
face
Wav
e25
.4m
m
Late
ralB
ound
ary
Wav
e
ExperimentalRadial Strain
Figure 3.9: Comparison plot of experimental strain profiles in 350 m/s range for all four thickness
specimens. No compressive strains are visible. As in the LS-DYNA simulations, the 3 mm and 5
mm experiments show a greater rise in the peak strain than the 15 mm and 25.4 mm specimens.
However, the duration of the 3 mm and 5 mm pulses is much greater than in the numeric
simulations. Additionally, the absolute magnitude of the strains is a factor of 10 lower than
predicted by the numerical simulations.
115
Specimen Thickness (mm)
Max
imum
Am
plitu
de(μ
-stra
in)
0 5 10 15 20 25 300
25
50
75
100
125
150
175
200(a)
Specimen Thickness (mm)
Prim
ary
Tens
ileP
ulse
Dur
atio
n(μ
s)
0 5 10 15 20 25 300
10
20
30
40
50(b)
Impact Velocity (m/s)
Max
imum
Am
plitu
de(μ
-stra
in)
0 50 100 150 200 250 300 350 4000
25
50
75
100
125
150
175
200(c)
Impact Velocity (m/s)
Prim
ary
Tens
ileP
ulse
Dur
atio
n(μ
s)
0 50 100 150 200 250 300 350 4000
10
20
30
40
50(d)
Figure 3.10: (a) The strain amplitude decreases with increasing thickness. (b) Also, the pulse
duration decreases with specimen thickness. (c) Variation of strain amplitude with impact
velocity. An increasing trend is visible in the strain amplitudes. (d) Plot of pulse duration with
impact velocity. No conclusions can be drawn from this plot.
116
Camera Frames
Vol
tage
(V)
0 2 4 6 8 10 12 14 16-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Figure 3.11: Rear surface strains from a bending series impact of a 5 mm specimen at 284 m/s.
Two oscilloscopes recorded the strains that register tensile strains for at least the first 50
microseconds (7.5 camera frames). Note the period of the oscillating strains are about 12 to 18
microseconds.
117
Time (s)
Vol
tage
(V)
0 2 4 6 8 10 12 14 16-0.1
-0.075
-0.05
-0.025
0
0.025
0.05
0.075
0.1
Figure 3.12: Rear surface strains from a bending series impact of a 15 mm specimen at 290 m/s.
Two oscilloscopes recorded the tensile strains that, considering an offset of about 25 μe, register
tensile strains for the entire recording period. Note the period of the oscillating strains are about
6 to 12 microseconds.
118
Time (seconds)
Bea
mIn
tens
ity(V
olts
)
0 1E-05 2E-05 3E-05 4E-05-0.16
-0.15
-0.14
-0.13
-0.12
-0.11
-0.1
-0.09
Start of Second Drop
Midpoint ofSecond Drop
End ofSecond Drop
Start of First Drop
Midpoint of First Drop
End of First Drop
Figure 3.13: Example of oscilloscope output from laser velocity triggering system. The three
locations in each of the two drops where the time is measured are indicated. These are divided
by a fixed distance of 5.9 mm and the resulting speeds are averaged.
119
Impact Velocity (m/s)
Reb
ound
Ene
rgy
/Im
pact
Ene
rgy
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hunter's TheoryNumeric 3 mmNumeric 5 mmNumeric 15 mmNumeric 25.4 mmExp. 3 mmExp. 5 mmExp. 15 mmExp. 25.4 mm
Figure 3.14: Variation in ratio of rebound kinetic energy to impact kinetic energy with velocity.
The experimental rebound energy is lower than the LS-DYNA or the stress wave estimations,
which are both elastic.
120
Impact Velocity (m/s)
Coe
ffici
ento
fRes
titut
ion
0 50 100 150 200 250 300 350 4000
0.25
0.5
0.75
1Hunter's TheoryNumeric 3 mmNumeric 5 mmNumeric 15 mmNumeric 25.4 mmExp. 3 mmExp. 5 mmExp. 15 mmExp. 25.4 mm
Figure 3.15: Variation in coefficient of restitution with velocity. The numeric estimates for the
thicker three specimens are within 0.01. As in Figure 3.14, the experimental coefficients are much
less than those estimated elastically.
121
1 2 3
4 5 6
7 8 9
Figure 3.16: Impact of 5 mm thick specimen at 323 m/s (BQ-13) showing two conical cracks.
Images are 6.67 microseconds apart. Note the formation of a lateral crack near the front surface
above the crushed zone.
122
1 2 3
4 5 6
7 8 9
Figure 3.17 Impact of 3 mm thick specimen at 345 m/s (BQ-26) showing radial cracks
propagating from the rear surface. Images are 6.67 microseconds apart. Note the cone crack that
forms in frame 2 and ejects from the rear surface. Also, developing radial cracks that start in
frame 3 have half-penny shape squished by the small thickness of the specimen.
123
1 2 3
4 5 6
7 8 9
Figure 3.18: Impact of 15 mm thick specimen at 371 m/s (BQ-25) showing lateral cracks. Images
are 6.67 microseconds apart.
124
Chapter IV –Planar Impact Experiments: Configuration and Procedures
High velocity planar impacts were carried out using the single stage gas gun
facility in the Department of Mechanical and Aerospace Engineering at Case
Western Reserve University. These experiments examined the effects of dynamic
stress loading on AS800 silicon nitride and soda lime glass under plain strain
conditions. The single stage gas gun is designed to accelerate projectiles using
compressed air or helium, down an 82.5 mm gun barrel. The specimen is located
in an impact chamber; an expansion chamber facilitates soft recovery of post
impact debris. Both pure compression and pressure shear experiments at
multiple impact velocities were conducted. A Velocity Interferometer System for
Any Reflector (VISAR), contact pin tilt system, and laser velocity system are
used to obtain the projectile velocity at impact, the particle velocity of the free
surface of the target plate and the impact parallelism during the impact process.
From these measurements, the dynamic spall strength and Hugoniot Elastic
Limit of the material was determined. The effect of impact velocity and skew
angle on spall strength was also examined.
In this chapter, the single stage gas gun is described, the method of preparing the
specimen, projectile and the gas gun for the experiments are explained, and the
procedures for conducting the experiments are discussed.
A. The Single Stage Gas Gun and Observation Systems
The single-stage gas-gun in the D. K. Wright Laboratory was employed in the
planar impact experiments (Figure 4.1). This gas gun utilizes compressed
nitrogen or helium gas to accelerate the flyer plate up to 550 m/s by means of a
fiberglass projectile down the gun barrel. The barrel and impact chamber are
evacuated to a pressure of 50 μm of Hg to avoid the effects of air shock waves
inside the target chamber. The specimen is located in a target alignment fixture
within an airtight impact chamber. The impact velocity of the flyer plate is
measured by using a laser velocity system, the target’s free-surface particle
velocity is measured by a laser reflected from this surface and analyzed using the
125
VISAR, and the parallelism between the flyer and target at impact by the tilt
pin system. Following impact, soft recovery of debris is accomplished by means of
a momentum balance and de-acceleration in soft cloths to absorb the impact. A
steel cylinder, which is the momentum balance, catches the debris in the
expansion chamber. It transforms the linear kinetic energy of the projectile and
the impacted specimen into rotational kinetic energy and gravitational potential
energy. This section describes the physical components of the single stage gas gun
and the laser and tilt pin based diagnostics.
A.1. Description of the Single Stage Gas Gun
The mechanism for rapidly accelerating projectiles using high pressures consists
of a piston in a seal and firing chamber, connected by a rod to a cap in the
loading chamber (Figure 4.2). The seal chamber is pressurized to seal the cap
against the breach between the loading chamber and the gun barrel. The load
chamber is then pressurized with the gas that will accelerate the projectile.
During the firing sequence, the firing chamber gas is released from a reservoir to
create a pressure gradient that causes the piston to open the breach. This
process is discussed further in the procedure section below. The setup is
controlled externally by means of a series of valves in a control panel (Figure
4.2). This system allows the firing to be operated from a distance.
The single stage gas gun barrel has an 82.5 mm inner diameter and is 4.57 meters
long. It is honed and has a square cross section key-way, which traverses the
length of the bore. The fiberglass projectile is accelerated down this barrel, and a
Teflon key in the key-way prevents the projectile from rotating while it is
accelerating down the gun barrel. A rubber o-ring, at the rear of the projectile,
creates an airtight seal between the projectile and the gun barrel. This barrel is
secured to the ground by means of a concrete block that prevents gun barrel
recoil during the experiment. The gun barrel is connected to both the impact
chamber and high pressure chamber by means of bolted flanges. These bolts and
126
o-ring seals maintain a vacuum within the gun barrel and the impact chamber
during the experiment.
The impact chamber contains the specimen and its holder. It is rectangular in
shape and constructed of ½ inch thick steel plates welded together (Figure 4.1).
Internal to this chamber is a shelf to support the specimen holder, a tray to catch
falling debris and a rod to hold the alignment system. The specimen holder
consists of a base plate that adjusts for the skew angle and three rectangular
rings, which enable the tilt alignment of the specimen. Glass windows allow for
laser measurements to be taken within the chamber. The measurement systems
include the VISAR interferometer, laser-based velocity system, and a series of
pins that measure tilt between the flyer and the target and indicate the onset of
impact. Vacuum feed-throughs are employed for the VISAR interferometer’s
fiber optic cables and the triggering pins’ relay cables. These feed-throughs are
necessary because prior to impact, the gun barrel and impact chamber are
evacuated to a pressure of 50 μm of Hg. A square guide tube is also bolted to
the shelf behind the specimen holder so as to contain and guide the projectile and
the impact debris towards an expansion chamber at the end of the impact
chamber. A circular opening in the rear of the impact chamber, covered with a
Mylar sheet during the experiment, enables the post impact debris to enter the
expansion chamber while maintaining a vacuum in the impact chamber during
the experiment. The Mylar sheet is destroyed during each experiment. Moreover,
the expansion chamber is not evacuated during the experiment.
The expansion chamber is considerably larger than the impact chamber. It is
also rectangular and made of ½ inch thick steel plates welded together (Figure
4.1). Inside of the chamber, a cylinder, which weighs on the order of 2000 lbs., is
suspended from the top plate of the chamber. This balance is filled with cloth
rags that, along with a transfer of kinetic energy, absorb the impact and allow
soft recovery of the sample. Unlike the rest of the gun, the expansion chamber is
not evacuated prior to the experiment. The experiment takes place in the
impact chamber, so it is unnecessary a vacuum to be maintained in the expansion
127
chamber. A transitional flange projects from the front of this chamber. This
flange prevents debris from ricocheting outside of the expansion chamber during
the transfer of the speeding debris. The expansion chamber is mounted on
wheeled tracks, which allow the chamber to move away from the impact chamber
during setup to allow access to the rear of impact chamber. Prior to the
experiments, the transitional flange of the expansion chamber is placed against
the rear wall of the impact chamber, covering the Mylar sheet. Post impact
specimen recovery is through the transitional flange on the front end of the
expansion chamber and also through a circular access hatch on the rear side.
A.2. Laser Velocity, VISAR and Tilt Pin Measurement Systems
Measurements for the gas gun consist of three types. The first data type is the
impact velocity, recorded by means of a set of parallel laser beams across the
projectile’s path. The second type is the determination of the parallelism of the
specimen and flyer interface, as indicated by the staggered triggering times of the
tilt pins. The third type of measurement is of the free surface velocity, carried
out by interferometer readings. The details of the observation mechanisms for
each of these measurements are discussed in this section.
A.2.1. Laser Velocity System
The velocity of the projectile is measured by the time the projectile takes to
traverse the distance between three laser beams set a distance of 9 mm apart. A
UNIPHASE Helium-Neon 5mW laser (Model 1125p) is used to generate these
three beams. These beams enter and exit the impact chamber by means of a pair
of glass windows. The beams are focused by a lens and collected using a high
frequency photo-diode. The path of the beams is across the path of the projectile.
The beams are interrupted in sequence by the passing projectile. The signal is
amplified and the history of the drop in intensity as the projectile cuts each beam
is recorded on an oscilloscope (Figure 4.5).
128
The time for the projectile to pass between beams is measured from the plot of
the beam intensity versus time (Figure 4.5). This measurement is repeated six
times; once for the beginning and end of each of the three intensity drops. The
times between the initial and final drop as well as the times between the middle
drop and the two other drops are recorded. Dividing the appropriate distances
by the recorded times gives six different velocity measurements. These six
velocities are averaged in order to specify the impact velocity.
A.2.2. Tilt Measurement and Triggering System
In order to determine the exact moment of impact between the flyer and
specimen, contact pins are employed. The electrical circuit that is completed
when contact occurs also triggers the recording systems for the VISAR. The
triggering system consists of five conductive pins placed into machined holes in
the specimen holding ring. Four of the pins are electrically isolated from the
aluminum ring, while the fifth acts as a ground. The pins are lapped until they
are flush with the specimen surface. Once any pin is in contact with the flyer
ring, which is also in contact with the specimen ring, the electrical circuit is
complete.
The other three isolated pins however are not impacted at the exact same time as
the pin that triggers the system. This delay, measured in tens of nanoseconds
indicates the parallelism of the flyer and specimen at the time of impact.
Therefore, the triggering system doubles as a tilt measuring system for the
purpose of determining any deviation from a parallel impact of the flyer and
target plates. The optical alignment procedure imparts an alignment of 0.2 milli-
radians [Kim et al., 1977]. However, a deviation of less than 0.5 milli-radians is
acceptable [Prakash, 1998]. The tilt is measured and checked after each
experiment to ensure that the deviation from the parallel remains below this
level.
129
A.2.3. VISAR Interferometer
The Valyn VISAR interferometer system is used to measure the rear, or the free
surface particle velocity of the specimen. The acronym VISAR abbreviates the
full name: Velocity Interferometer System for Any Reflector. This system uses a
COHERENT VERDI 5 Watt solid-state diode-pumped frequency doubled
Nd:YVO4 CW laser. This laser is monochromatic and has a wavelength of 532
nm. The original VISAR interferometer dates from 1972 and was created by
Barker and Hollenbach [1972]. Their design enabled the collection of velocity
measurements from surfaces that were both specular and diffuse. The current
generation of interferometer is multi-beam, and up to seven independent points
can be measured simultaneously during impact [Barker, 2000]. The VISAR
system uses four glass etalons, which provide a range of velocity per fringe
constants from 99.2 m/s/fringe to 1874 m/s/fringe. The minimum resolution is
approximately ±2 m/s using the 99.2 m/s/fringe etalon setup.
One of the features of this system is that it uses fiber optic cable to channel the
laser beam between the laser and the probe and the probe and the interferometer.
A beamsplitter directs the laser light into a 60 micrometer fiber, which is
connected, through a fiber splice into an optical fiber attached to a probe. The
probe focuses the light on to a pre-determined spot on the target surface 30 mm
away from the probe tip. The motion of the specimen causes a Doppler shift in
the light. When positioned at this height over a diffuse or specular surface, the
Doppler shifted light returning to the probe is collected with a lens into a 300
micrometer optical fiber. This fiber is connected to a second fiber through
another splice. The use of splices allows a probe to be destroyed in each
experiment, without damaging the fibers permanently attached to the laser’s
beamsplitter and the interferometer. [Barker, 2000].
The fiber carrying the Doppler shifted light enters the interferometer, which is
mounted on an aluminum frame to keep the components aligned (Figure 4.3).
The incoming light is first filtered through an iris. The intensity of the laser
130
beam is monitored using a small portion of the light that is removed with a
beamsplitter. This observation ensures that any errors due to a loss of beam
intensity during the experiment are recognized. Possible causes of loss of beam
intensity include poor alignment of the probe to the specimen, and damage to the
free surface of the specimen. Following this, the primary laser beam is split using
a 50/50 beamsplitter into two beams. One beam is reflected off a mirror
mounted on a piezoelectric aligner/translator (PZAT) that vibrates, creating a
motion that allows the interferometer to be aligned, using the bull’s eye
technique. Three potentiometers within a Burleigh RG-93 Programmable Ramp
Generator control the bias of three separate crystals that comprise the PZAT
system [Barker, 2000].
The second beam travels through glass etalons, which delay the beam by an
integer number of wavelengths. Combinations of four different length etalons
enable the resolution of the VISAR to be altered from 99.2 m/s per fringe to over
1 km/s per fringe. A 1/8th wave plate, which only retards the p-orientation of
the light but not the s-orientation, is placed in front of a mirror off which the
beam reflects. These orientations are perpendicular. The retardation caused in
the p-component of the beam is one quarter of a wavelength. This beam then
traverses the etalons a second time. The two primary beams are interfered and
the resulting beam is again split into two components using the 50/50
beamsplitter [Barker, 2000].
In the alignment process, the PZAT provides pulses in only the one beam that
does not pass through the etalons. During the experiment, both of the beams
contain temporal pulses, but the pulses are not synchronized because of the delay
influence of the etalons. The resulting light and dark fringes are directly related
to the rear surface particle velocity. The beams are optimized by the bulls’ eye
method to provide the best fringe contrast. Then, they are each fed into a
beamsplitter that directs the p and s components into different fibers. Because
the p-wave was retarded twice by the 1/8th wave plate, the p-wave beam is 90o
out of phase with respect to the s-wave. Each of these four components enters a
131
fiber and is directed to the HP oscilloscope. The scope allows the beam
intensities to be optimized by plotting them as Lissajous curves [Barker, 2000].
The optimized beams are subtracted to increase the signal to noise ratio
[Hemsing, 1979]. The s and p oriented components are recorded separately in
order to remove the inaccuracy caused by the ambiguity of in the sign of the
acceleration and to increase the accuracy of the data reduction. This is known as
quadrature coding. The signals are strengthened using 1.2 GHz bandwidth
amplifiers and are recorded on a Tektronix oscilloscope [Barker, 2000] with a
sampling rate of 2 GHz.
Post impact, these signals are analyzed using proprietary software enabling the
determination of the particle velocity from the fringe patterns (Figure 4.4a)
[Barker, 2000]. The signals indicate an abrupt jump in free surface particle
velocity of the sample corresponding to the arrival of the first compressive wave
(Figure 4.4b). A deviation from this sharp rise indicates the elastic limit has
been exceeded and the material is deforming plastically. In the spall
experiments, after the signal has reached a plateau, a dip in the free surface
particle velocity is recorded indicating the spall of the material. The depth of
this dip indicates the spall strength. The shape of the dip is also indicative of the
amount of damage present in the material. The effects of impact velocity and
skew angle on the spall strength of silicon nitride and soda lime glass will be
discussed further in the Chapters V, VI, and VII.
132
B. Experimental Procedures
The experimental apparatus described above was employed in experiments on
AS800 silicon nitride and soda lime glass. The VISAR rear surface particle
velocity measurements enabled the determination the Hugoniot Elastic Limit,
dynamic spall strength, and other strength observations as described in Chapters
V through VII. In order for the experiments to be carried out, a standardized
procedure was developed for the preparation of the specimen, flyer, and the gas
gun itself. The specimen and the flyer were placed within aluminum 7075-T6
rings. The projectile must be assembled. The gas gun must be properly cleaned
and prepared for the experiment. Following the setup and the alignment, the
experiment is performed using high pressure air or helium to fire the gas gun.
B.1. Specimen and Flyer Preparation
In Chapters V through VII, the planar impact experiments on AS800 grade
silicon nitride and soda lime glass are described. In these experiments, GC103
tungsten carbide and aluminum 6061-T6 were employed as additional flyer and
target materials. In Section 1.1, the configuration of these flyers and specimens is
briefly described. More extensive discussions of the materials including the
material wave speeds and densities are reserved for the individual chapters.
Section 1.2 explains the process for assembling any of the configurations of
specimens and flyers.
B.1.1. Materials for Specimens and Flyers
The experiments described in Chapter V were conducted on AS800 grade silicon
nitride. Specifically, for the shock compression and pressure-shear experiments at
a 12 degree skew angle, single layer flyers of 3 mm or 4.5 mm thick silicon nitride
or 2.83 mm thick CG103 tungsten carbide impacted 8 mm and 5.5 mm thick
silicon nitride plates. In the pressure shear experiments, a 12 mm thick
133
polymethylmethacrylate (PMMA) window was attached to the free surface of the
specimen.
For the multiple shock experiments, as described in Chapter VI, the target
(specimen) plate was impacted by dual flyer plates. The flyers consisted of 2.83
mm thick GC103 tungsten carbide or 3 mm thick aluminum 6061-T6 glued to a 3
mm thick silicon nitride flyer. The glue that was used was a TRA-CON binary
epoxy. The specimens were 4.5 mm thick silicon nitride with 10 mm thick and
25.4 mm diameter silica glass windows. The reverberation experiments employed
4.5 and 5.5 mm thick silicon nitride flyers and 0.5 mm thick tungsten carbide
specimens.
In Chapter VII, the soda-lime glass planar-impact experiments are described. In
this series of experiments both shock compression and combined pressure and
shear at a skew angle of 18 degrees, were conducted. These experiments used
soda-lime glass and tungsten-carbide as both flyer and target plates. The
experiments employed 5.9 mm thick aluminum 6061-T6, 6 mm thick tungsten-
carbide, and 12.5 mm thick soda-lime glass as flyer disks. The specimens were
6.5 mm and 12.5 soda-lime glass, or 4 mm thick tungsten-carbide disks. All of
these disks were 62.6 mm in diameter. Also, in these experiments, the specimens
and flyers were single layers.
B.1.2. Assembly of the Projectiles and Specimens
The specimens’ free surfaces are coated with a 150 nm layer of aluminum using
the AUTO 306 vacuum coating machine in order to provide for a reflective
surface. If the specimen and flyer are layers of multiple plates they are glued
together using TRA-CON BIPAX BA-2115 binary epoxy and set to cure in a vise
for 24 hours. Both the specimen and the flyer need to be encircled by aluminum
7075-T6 rings in order to facilitate their attachment to the target holder and
projectile respectively (Figure 4.6). Holes are drilled around the edges of the
specimen ring for the four plastic holding pins, for the alignment bolts, and a
134
tapped hole for a ¼-20 bolt is drilled through the ring. This bolt enables the
attachment of the VISAR probe. The specimen ring is also machined with five
holes in which metallic pins are inserted for the triggering and tilt measurement
system. One is a ground pin and the other four are electrically isolated from the
ring. These pins are glued into place in the holding ring using Hardman Red
04001 binary epoxy, which cures in 45 minutes.
The impact faces of both the flyer and target rings are then lapped on a
Lapmaster machine in order to produce a flatness of 1 to 3 light rings. The
surface finish is held to the order of 15 microns by the diamond slurry. A high
strength binary epoxy is then used to glue the flyer and specimen to their
respective aluminum rings. This is done in another vise arrangement so that the
specimen and flyer are aligned precisely in the center of their rings. Before the
rings and specimens are secured, Loctite™-1711 mold release is used to ensure
that the vice is not glued to the specimens. The binary epoxy consists of 35.0 ml
of Loctite™-9412 Hysol resin and 10.0 ml of Loctite™-9412 Hysol hardener.
After curing period of 24 hours, the impact surfaces are lapped again on the
Lapmaster in order to ensure that the impact surfaces, the epoxy surfaces, and
the trigger pins still have the prescribed flatness. Wires of 16 inch length are
soldered to the trigger pins on the side away from the impact face. Four plastic
holding pins are then glued into the holes in the outer surface of the specimen
ring. These pins are of lengths ranging from 39 mm to 23 mm and are designed
to align the target to the center of the target holder (Figure 4.6). These pins
break during the experiment preventing damage from occurring to the permanent
components of the target holder.
The projectile is a hollow tube of fiber glass with diameter 3.25 inches and length
1 foot. The outer surface of this projectile is machined to a tolerance of +0 and –
0.005 inches in order to allow for smooth fitting in the barrel of the gun. Angled
projectiles are marked indicating the orientation of the key-way in the barrel
with respect to the major axis of the elliptical front of the projectile. An
135
aluminum cap is glued using the Hardman binary epoxy into the circular groove
machined into one end of the projectile’s inner diameter. This cap is aligned to
the location of the key-way. A machined o-ring grove encircles the cap.
Additionally, a hook allows the projectile to be pulled through the barrel.
Once this glue has cured, four small holes are machined in the projectile. Two of
these holes are along the line of the key-way near to the end cap and are
designed equalize pressure within the projectile. The other two holes are opposite
each other and are located about 5 mm from the open end of the projectile.
These are used in the alignment procedure. One hole is machined 45 degrees off
the key-way line and the other is 180 degrees opposite the first hole. Following
this, the outside of the cap and projectile are sanded to remove any glue and to
reduce the diameter of the projectile near the capped end, which typically swells
slightly once the cap has been glued into it. The projectile is then pulled through
the barrel to ensure to ensure a tolerance on the fit that allows the projectile to
slide freely, but not slip. Following this, the flyer ring is glued with the
Hardman binary epoxy to the front of the projectile ensuring that no glue
contaminates the outside of the projectile. After this glue has cured, a rubber o-
ring is placed in the o-ring grove on the cap, and the machined Teflon key is
placed in the appropriate slot on the cap. The key and o-ring are secured with
vacuum grease.
B.2. Gas Gun Setup Sequence
The single stage gas gun is now prepared. First, acetone soaked cloths are pulled
through the barrel until any dirt or debris from previous experiments is removed.
The VISAR interferometer probe is then threaded through the vacuum feed-
through plate and the probe is then inserted into the probe holder, which in turn
is screwed into the appropriate hole on the specimen ring. This probe holder
allows the VISAR probe to be aligned to the center of the free surface, with the
proper distance of 30 mm between the probe tip and the free surface. The
VISAR probe is adjusted until the maximum return signal intensity is achieved.
136
As mentioned above, the free surface is coated with aluminum so as to be more
reflective and provide a stronger return signal.
The projectile and flyer assembly is then placed into the impact chamber end of
the barrel with a rope connected so that it can be pulled back through the barrel.
The key on the cap is inserted into the barrel’s key-way ensuring that the flyer
will remain in the same orientation throughout the alignment process. The
specimen is attached to the specimen holder. The specimen holder is then placed
inside of the chamber and bolted down. The trigger wires are attached to the
appropriate connections on the holder, which connects them to the triggering
system.
Once this is complete, the alignment of the specimen and the flyer commences.
The flyer and specimen are aligned to each other by using circular paper guides
and adjusting the specimen’s vertical and horizontal location in the specimen
holder. Once the specimen and flyer are centered on each other, the alignment
pins are locked down holding the specimen assembly in place. The paper guides
are then removed, the specimen and flyer impact surfaces are cleaned with
acetone, and the projectile is pushed into the barrel until it is protruding only
enough to let the two holes drilled opposite each other to be accessible. Rubber
bands are stretched from screws along these holes and the corresponding ones in
the specimen ring. Dust free disk mirrors are placed in these rubber bands over
the flyer and projectile. A mirror prism on an adjustable stand is then placed in
the chamber along with a free standing mirror. Together, this system allows the
angular alignment of the specimen with the flyer in three stages. First, a light
bulb is employed for rough alignment. Then, two auto-collimators are employed
successively to bring the angular alignment to 2×10-5 radians. This optical
alignment process employs a technique that was pioneered by Kim et al. [1977].
The trigger and velocity systems are tested. The intensity of the VISAR signal is
also checked.
137
Following this, a square guide tube is placed into the chamber to ensure that
once fired, the projectile will be directed into the expansion chamber. This tube
is bolted into place. The projectile is then pulled through the barrel to the
breach end. After this, the breach between the high pressure chamber and the
barrel, which is sealed with an o-ring, is bolted closed. The rear wall of the
impact chamber is sealed, by bolting down a circular plate, secured with an o-
ring, and a Mylar sheet. This Mylar sheet covers the opening that allows the
projectile and flyer to proceed into the expansion chamber following the impact
without causing damage to a permanent component of the gas gun. This seal
also permits a vacuum in the impact chamber during the experiment. This
completes the setup procedures for the system.
B.3. Firing Sequence
The pre-firing sequence consists of vacuuming the chamber and setting up the
measurement systems. First, the PRAXAIR compressed air or helium tank is
connected to the regulator of the control system (Figure 4.2). The regulator is
not opened yet. All three of the vent valves; the seal vent valve, fire/reservoir
vent valve, and load vent valve; are opened and any remaining air from previous
tests is allowed to escape. The firing valve, and the three regulator valves; the
seal regulator valve, fire/reservoir regulator valve, and load regulator valve; are
also opened. The system is thus clear of any remaining pressurized air from the
previous experiments.
The seal vent valve is closed first, followed by the load vent valve. The
fire/reservoir vent valve is left open for the moment. The seal regulator,
fire/reservoir regulator and load regulator valves are also closed. The fire valve
is ensured to be open. The compressed air tank is then opened and the tank
pressure is recorded. The air tank regulator valve is then opened to above the
desired loading pressure. This pressure is also recorded. The seal regulator valve
is then opened until the desired seal pressure, usually 60 psi, is reached. This
causes the piston to seal the breach between the loading chamber and the gun
138
barrel (Figure 4.2). The seal regulator valve is then closed. The impact chamber
pressure measurement panel power is engaged at this point.
The vacuuming of air from the impact chamber is the next phase of the setup.
First, the exhaust valve on the top of the impact chamber is closed. Then, the
check valve between the impact chamber and the impact chamber vacuum gage
is opened. The vacuum pump is then filled with oil to the appropriate level and
the pump is started. After a few moments, the check valve between the vacuum
pump and the gun barrel is opened. This is done first in order to keep the
projectile from being pulled down the barrel towards the impact chamber by the
pressure differential. A minute later, the check valve between the impact
chamber and the vacuum pump is opened. The pressure drop rate is monitored
by the impact chamber vacuum gage and if it slows below a certain rate, the
chamber is checked for leaks. Tightening bolts, or adding additional vacuum
grease typically stops any leaks. Once the chamber pressure is down to around
100 milli-torr, the VISAR system is recalibrated. Following this, the velocity
system is set up using its oscilloscope to check for proper beam intensity and
contrast.
The next step is to pressurize the load chamber. First the air tank pressure is
confirmed to be sufficient for the desired loading pressure. Then the load
regulator valve is opened to the desired pressure, which is recorded. The pressure
necessary for a certain velocity is a factor of flyer weight and projectile tightness
in the barrel. As a result, each experiment requires an estimation based upon
past experimental velocities and pressures. For example, at a speed of 65 m/s,
with an average amount of friction between the barrel and projectile, a load
pressure of 30 psi is typically used.
The final firing procedure commences after the loading chamber is pressurized.
First the firing valve is closed and then the fire/reservoir vent valve is closed.
After the vacuum pressure is at firing level, which is 80 milli-torr or less, the
fire/reservoir regulator valve is opened and the pressure in the auxiliary reservoir
139
chamber is set equal or higher than the load pressure. This pressure is noted.
Then the fire/reservoir valve is closed. The compressed air tank main valve is
then closed. The seal pressure is dumped by opening the seal vent valve, which
removes most of the pressure holding the breach closed. The remaining pressure
comes from the loading chamber pressure.
The check valve between the vacuum gage and the impact chamber is closed.
Then the check valve between the vacuum pump and the gun barrel is closed.
All of the optical and electronic recording systems are confirmed to be aligned
and armed at this time. The check valve between the vacuum pump and the gun
barrel is closed. The vacuum pump is then shut off. A second check of all
systems is performed ensuring that all recording devices are armed, that the tilt
system is functioning, and that the VISAR intensity as measured by the
Lissajous figure on the oscilloscope is a circle of greater than 100 mV diameter.
This ensures a strong VISAR signal. After a five count the firing valve is pulled.
This releases the air in the firing reservoir into the firing chamber, which moves
the piston, allowing the air in the load chamber to escape down the barrel.
Following the experiment, all of the records are transferred to disk and then
analyzed by means of dedicated programs. The VISAR software produces
velocity versus time curves for the rear surface velocity (Figure 4.4). The laser
velocity system measures beam intensity, which is translated into velocity by
observing the drops in intensity (Figure 4.5). The tilt system is recorded and the
drops in voltage indicate the amount of parallelism between the flyer and the
target. As mentioned above, the acceptable parallelism is 0.5 milli-radians or
fewer [Prakash, 1998]. The data analysis will be discussed further in the next
three chapters.
140
C. Summary
The single stage gas gun is employed to create plane strain shock compression
and pressure shear impacts on silicon nitride and soda lime glass. This gas gun
consists of a high pressure chamber, a 4.572 m long barrel with an 82.55 mm
diameter bore, an impact chamber, and an expansion chamber. The impact
chamber contains a specimen holder, which is used to align the specimen and
flyer. The experiment is conducted in vacuum of less than 80 milli-torr to avoid
the effects of shock waves in air. Measuring systems include the VISAR
interferometer, a laser velocity system, and a tilt system built into the triggering
mechanism. The VISAR uses a probe system, which allows precise alignment to
the free surface for the strongest signal.
The experimental procedure involves the preparation of a projectile using the
flyer plate, preparation and alignment of the specimen, and operation of the gas
gun. The flyer and specimen are placed in aluminum 7075-T6 rings and secured
using epoxies. These rings enable the flyer to be attached with the fiber glass
projectile and the specimen to be connected to the specimen holder and the
VISAR probe. The gas gun is cleaned, and the specimen and flyer are aligned to
within 0.5 milli-radians before the vacuuming process. The firing sequence
involves a rapid release of the loading gas, which expands into the gun barrel and
accelerates the projectile.
Planar impact experiments were chosen because of the goals of this study, among
which the determination of the effects on the spall strength and the dynamic
strength of various impact velocities and skew angles are the most important. By
means of the VISAR, these quantities were determined. The results, which are
discussed in the next three chapters, provide insights into the behavior of the
AS800 silicon nitride and soda lime glass under plane strain shock compression
and pressure-shear.
141
References
Barker L.M, Barker V.J., Barker Z.B., 2000. Valyn VISAR User’s Handbook. Albuquerque, New
Mexico, USA.
Barker L.M., Hollenbach R. E., 1972. Laser interferometer for measuring high velocities of any
reflecting surface. Journal of Applied Physics, 43(11), 4669-4675
Hemsing W.F., 1979. Velocity sensing interferometer (VISAR) modification. Review of scientific
Instrumentation, 50 (1), 73-78.
Kim K. S., Clifton R.J., Kumar P., 1977. A combined normal and transverse displacement
interferometer with an application to impact of Y-cut Quartz. Journal of Applied Physics, 48,
4132-4139.
Prakash V., 1998. Time-resolved friction with applications to high speed machining: experimental
observations. Tribology Transactions, 41 (2), 189-198.
142
Figures
Figure 4.1: Overview of the single stage gas gun. The high pressure chamber, gun barrel, impact
chamber and expansion chamber are shown. Inside the impact chamber, the specimen holder and
guide box are indicated. The momentum balance inside the expansion chamber is shown. The
control and observation equipment is also indicated.
143
Figure 4.2: Firing chamber and air supply diagram. The loading paths from the compressed gas
tank to the seal, fire/reservoir, and load chambers are indicated along with the pressure gages.
The regulator and vent valves are also shown.
144
Figure 4.3: Valyn VISAR interferometer design. The fiber optic cables are terminated in the laser
collimator. Part of the beam is split off and directed into the beam intensity monitor. The main
beam travels through the large beamsplitter, and is split 50/50. One path reflects from the
mirror fixed to the piezo-electric crystals. The other path is delayed in the etalons and the 1/8th
wave retardation plate further delays the p-orientation of this beam. Next, recombining and
splitting the beams into the left and right exit beams occurs. The polarizing beamsplitters
separate the s and p orientations of the beams. The resulting four beams are collected by output
fibers.
145
Figure 4.4: The output of the VISAR interferometers is shown. (a) The fringe patterns for a
representative experiment as recorded by the oscilloscopes. (b) The processed velocity curve for
this representative experiment, using the VISAR analysis program.
Figure 4.5: The velocity measurement system beam intensity plot for a representative experiment.
The blue circles represent the points at which the time is measured at the beginning and end of
each intensity drop. The time between succeeding drops and between the first and last drops are
used to calculate the impact velocity.
146
Figure 4.6: The specimen and flyer ring configurations for a representative experiment. The
images show a top view and a cross section view through the center of the assemblies. A
representative single stage flyer is shown on the left side. The aluminum 7075-T6 ring is secured
to the flyer by Hysol resin. On the right side is a representative specimen. The specimen has a
window attached by TRA-CON binary epoxy. The specimen is secured to the ring with Hysol
resin. The large tapped hole in the lower left corner of the aluminum ring is for the probe holder.
The remaining five holes are for the triggering system pins. Six holes are drilled in the side of the
ring. The four plastic pins, which connect the specimen assembly to the specimen holder are
shown in their holes. The remaining two holes are for the bolts which anchor the rubber bands
during the fine alignment of the specimen and target.
147
Chapter V –The Shock Response of AS800 Grade Silicon Nitride
Plate impact shock wave experiments were conducted on AS800 aerospace grade
silicon nitride in order to examine the effects of the state of stress on its dynamic
material behavior. The use of nominally plane wave loading conditions on the
impacted specimens enable quantitative determination of the material response.
In order to understand the motivation for the experiments described herein,
previous theoretical and experimental work on AS800 silicon nitride is first
discussed. The equipment and procedures involved in conducting these
experiments were discussed in detail in Chapter IV. However, the specific
theoretical background that is useful in the analysis of these experiments is
discussed in detail in Section A of this chapter. Following this, in Section B, the
specimen and flyer properties are provided and the experiments are categorized.
In Section C, details of the experimental results and analysis are provided. The
experimental data include the elastic and hydrodynamic Hugoniot state
parameters, the spall strength, and the post impact microstructure. This set of
dynamic material properties provides the basis for characterizing the dynamic
material response of AS800 silicon nitride under intense shock wave loading
conditions.
148
A. Background on Shock Compression Experimentation
In order to discuss the dynamic response of the material, first the basic theory of
high speed planar impacts is reviewed. This includes the theory of stress wave
propagation from impacting surfaces under both elastic and elastic-plastic cases.
In this discussion, the definitions of the Hugoniot Elastic Limit and dynamic
yield stress are provided. Also, the process of spallation, or the delamination of
the material due to internal tensile stresses resulting from rarefaction waves is
discussed. Previous shock compression experiments on silicon nitrides from the
literature are examined. Additionally, the motivation for the current study is
developed.
A.1. The Theory of Planar Shock Compression
The impact of two plates under planar shock compression results in the
propagation of stress waves in the two materials. The amplitude of the resulting
shock waves and the particle velocities are proportional to the impact velocity.
Eventually, if failure does not occur, the reverberations of the stress waves within
the two plates result in an equilibrium state of stress. According to one
dimensional wave theory, the stress waves cause the material stress and particle
velocity states to alter in a predictable fashion, which enables the computation of
the state of stress in the material from the measurement of the particle velocity
at the free surface of the target plate. This predictability allows the
determination of the dynamic material properties under both elastic and elastic-
plastic conditions. The equations that describe both of these regimes and the
Hugoniot Elastic Limit that marks their separation are discussed in Sections
A.1.1 and A.1.2.[VP2]
A.1.1. Elastic Impact Theory
The study of the transient stress waves in solids is its material response before
stress equilibrium is reached. Because this is the case, the material response is
determined by the propagating compression/tension and/or rarefaction waves
within the solid. The velocity at which these waves propagate is unique to each
149
material. For example, when two material plates with parallel surfaces impact
each other in the elastic regime, both longitudinal compression and shear waves
are generated. The longitudinal compression waves propagate through the
material at the longitudinal wave speed, CL, which is calculated using the elastic
modulus, E, density, ρ, and Poisson’s ratio, v, through the well known relation
(1 - )(1 - 2 )(1 )L
EC νρ ν ν
=+
. {V.A.1}
The shear wave velocity, CT, is given by
(2(1 - )/(1 - 2 ))L
TCCv v
= . {V.A.2}
Because these waves have well defined propagation velocities within the material,
a time-distance diagram (T-X), as shown in Figure 5.1a, can be produced which
determines the wave propagation in both time and depth in the material. The T-
X diagram contains information on the temporal and spatial wave propagation in
the flyer and specimen [Boslough, 1993].[VP3]
Figure 5.1a shows the time distance diagram in a material subjected to combined
compression and shear loading. The longitudinal waves, which travel faster than
the shear waves, have lesser slope because the slopes of the lines in this diagram
are equal to the inverse of the propagation wave velocity. The waves propagate
both forward (into the specimen) and backwards (into the flyer) from the point
of impact, causing the two material plates to compress. The free surfaces at the
far ends of the two plates cause the compression waves to reflect as tensile
rarefaction waves, which unloads the two impacting plates [Boslough, 1993].
Since the flyer plate is thinner than the target plate (specimen), the two
rarefaction waves meet inside of the specimen and can result in spall of the target
plate if the stress level is high enough [Grady, 1988]. This spall process and the
critical conditions for it to occur are discussed in more detail in the next section.
150
In a pressure-shear experiment, the shear waves also propagate from the impact
point and reflect from the free surfaces. However, because of the elastic nature of
impact, the shear waves and the longitudinal waves propagate independently of
each other. That is, when the waves intersect, neither is affected by the other.
The longitudinal and shear waves each produce a stress consistent with the wave
type. The longitudinal wave causes compression, and the shear wave causes
shear. These stresses combine to produce the overall stress state in the material.
The stresses produced in the materials can be determined by characteristic
equations [VP5] which are governed by one-dimensional elastic hyperbolic wave
theory [Achenbach, 1975]. These characteristic equations are also referred to in
this chapter as line equations because the hyperbolic wave theory can be
approximated as linear relations between the stress and the particle velocity.
Continuum theory states that the sum of the stress, σ, and the quantity
material’s acoustic impedance times the particle velocity, V, remains constant
during the propagation of the stress wave. The acoustic impedance is defined by
the wave speed, c, multiplied by the density, ρ. The line equation can be written
as
constantcVσ ρ± = . {V.A.3}
If this equation is applied to each of the stress states in the T-X diagram (Figure
5.1a), a set of stresses and particle velocities can be built up. The stress states
before impact can be obtained from the flyer (State 1: v= impact velocity, σ = 0)
and the specimen (State 2: v = 0, σ = 0). Using the equation twice results in a
pair of equations for the impacted state in both the flyer and the specimen,
known as state 3:
1 1 3 3 - cV = - cVσ ρ σ ρ , {V.A.4}
2 2 3 3 + cV = + cVσ ρ σ ρ . {V.A.5}
151
Solving this pair of equations for the two unknowns yields the stress state in the
impacted material. This stress state is also known as the Hugoniot state. In
theory, this process can be repeated for each different stress state resulting in a
stress velocity pairing for each state which can be plotted on a stress versus
particle velocity ( S-V) diagram (Figure 5.1b). Here, the slopes of the lines
correspond to the materials’ impedances. If the flyer and target have two
different impedances’, the slope will be different, as shown in Figure 5.1b.
To reiterate, the stress and particle velocity states of the material that are
depicted in Figure 5.1b are caused by the propagation of stress waves shown in
Figure 5.1a. The initial states of the material before impact and after impact, but
before the longitudinal wave fonts arrive are State 1 for the flyer and State 2 for
the specimen. These are represented in the triangular regions in Figure 5.1a
under the first pair of lines representing stress waves. State 3 is the compressive
state that occurs following the impact in the interior of both plates. State 4
occurs on the free surface of the specimen where measurements can take place.
State 5 is the corresponding state at the free surface of the flyer. Both of these
states are completely unloaded and have zero stress in the absence of shear.
State 6 is created by the combination of the two rarefaction waves. This is the
state where spallation may occur. State 7 is the state that is created on the
specimen’s free surface by the interaction of the two stress waves at State 6..
Any spall in State 6 will affect the particle velocity and the stress in State 7.
These states define the material behavior. The stresses and particle velocities of
these states can be determined if the free surface particle velocity time history
and flyer impact velocity are known from experiments.
A.1.2. Elastic-Plastic Impact Theory and Experiments
The characteristic line equations can also be used for describing the material
deformation in the regime of elastic-plastic deformation. However, the elastic-
plastic wave speed is slower than the corresponding elastic wave speed. As a
result, the plastic impedance is smaller than the elastic impedance. The plastic
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waves on the time-distance diagram propagate from the impact point in a similar
manner to the elastic waves, with a smaller slope due to the slower plastic wave
speed. The Hugoniot Elastic Limit is the point where the material response
reaches the elastic limit under dynamic plane strain loading conditions. When
this stress level is exceeded on the stress-velocity diagrams, the impedance
difference produces a slope change. This point, hereafter referred to as the HEL,
is directly related to the dynamic yield strength of the material by the expression
1 21O HELY νσ
ν−=−
. {V.A.6}
In this equation, the material’s HEL, HELσ , is related to the dynamic yield stress,
Y0, by a function of the Poisson’s ratio, v [Reinhart and Chhabildas, 2002]. The
HEL represents the dynamic tensile strength in plane strain. In the plate impact
experiments, there is no strain orthogonal to the impact direction. The dynamic
yield strength Y0, can be measured directly under plane stress conditions, such as
those obtained in a Split Hopkinson bar type experiment. Additionally, the HEL
can vary with grain size [Mashimo, 1994] [Mashimo, 1998] and material density
[Nahme, 1994].
In the present study, three different monolithic silicon nitrides were examined.
The specimens varied in grain size and porosity, but the porosity variation was
small. All the materials had Al2O3, Y2O3, and other binders in their makeup.
The first group of specimens had a grain diameter between 0.5 μm and 1.2 μm,
an un-shocked density of about 3.16 g/cc, a porosity of 4%, and an HEL between
10 GPa and 12.5 GPa. The second group had a grain size between 0.25 μm and
0.4 μm, an un-shocked density of about 3.22 g/cc, a porosity of 1%, and a HEL
between 14 GPa and 16.5 GPa. The third set had grain diameters of 0.15 μm to
0.3 μm, the same density as the second group, a porosity of 1% and a HEL
between 17 GPa and 20 GPa. Mashimo’s [1998] study suggests that the larger
the grain size, the smaller is the HEL.[VP7]
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It is important to note that the HEL manifests itself as a kink in the rise time of
the free surface particle velocity versus time profile during the loading from State
2 to State 4. The HEL in a silicon nitride of similar density to AS800 (3.15
g/cm3) was observed to be 12.1 GPa [Nahme, 1994]. A less dense silicon nitride
(2.28 g/cm3), in contrast, had a lower HEL of 1.9 GPa. At applied compressive
stresses greater than 5.2 GPa, a second higher slope particle velocity region was
also observed. The authors state that this is similar to the “double-HEL”
behavior of TiB2. The collapse of voids is suggested as an explanation for this
phenomenon [Nahme, 1994].
The elastic-plastic impedance can be estimated by employing the bulk modulus,
K, of a material. The bulk modulus is related to the elastic Young’s modulus E,
and Poisson’s ratio υ, as
( )[ ]3 1 2K E ν= − . {V.A.7}
In terms of the bulk modulus, the plastic or the bulk wave speed, cP, can be
defined as
( )12
Pc K ρ= . {V.A.8}
In Equation {V.A.8}, ρ is the density of the material. The elastic-plastic
impedance is calculated by multiplying by the density with the plastic wave
speed as
P PZ cρ= . {V.A.9}
This elastic-plastic impedance can be used to estimate the stress and the particle
velocity states when the state of stress is close to the HEL. This alters the single
line estimation of the loading of the silicon nitride from State 2 to State 3 on the
stress and velocity diagram (Figure 5.1b) to a double line (Figure 5.2b). The line
154
connecting State 2 to the HEL has a slope equal to the elastic impedance. The
line connecting the HEL to State 3 has the slope equal to the elastic-plastic
impedance. Note that the tungsten carbide flyers depicted in both diagrams are
represented by a double lines while going from State 1 to State 3. This is
because the HEL of tungsten carbide is smaller than that of silicon nitride. The
tungsten carbide flyer plates are employed in the experiments due to the higher
impedance of tungsten carbide when compared with silicon nitride. This
produced a higher stress level for the same impact velocity. The Hugoniot Elastic
Limit of tungsten carbide has been reported to be 7.2 ± 0.8 GPa [Dandekar,
2004].
To use the double lines to predict State 3, the HEL must first be determined.
Knowing the HEL, the particle velocity corresponding to the HEL can be
calculated using
2 /HEL HEL TV Zσ= . {V.A.10}
In Equation {V.A.10} the target impedance is represented by ZT. For the silicon
nitride described by Nahme [1994] with a similar density to AS800, the HEL
velocity is 682 m/s. For the tungsten carbide in Dandekar’s [2004] work the HEL
velocity is 136 m/s. Using the [VP9]stress and particle velocity state of (VHEL ,
HELσ ) instead of the zero stress and zero particle velocity for State 2, the particle
velocity V3, and the stress σ3, in State 3 can be determined as
3 F 1 T-P HEL T HEL T-P FV =(Z V +Z /Z - )/(Z +Z )σ σ , {V.A.11}
3 T-P 3 HEL T-P HEL T=Z V +( -Z /Z )σ σ σ . {V.A.12}
In Equations {V.A.11} and {V.A.12}, the impact velocity from State 1 is V1, the
flyer impedance is ZF, and the elastic-plastic impedance of the target is ZT-P.
From these estimates of the stress and particle velocity in State 3 particle
velocity and stress estimates in the remaining states can be determined. It should
155
be noted that the unloading to State 4 is also elastic-plastic, but the remaining
states are elastic (Figure 5.2b).
A.2. The Failure in Dynamic Tension due to the Spallation of SiliconNitride
As stated above, when the rarefaction induced tensile stress reaches a level
beyond a material’s ability to resist decohesion, it fails in a process known as
spallation. This occurrence represents a dominant failure mode in materials
subjected to dynamic loading in dynamic tension. In section A.2.1, the process of
spall failure [Dremin and Molodets, 1990] and estimation of the spall strength
[Grady, 1988] are discussed. Following this, in Section A.2.2, previous
experimental works conducted to determine the spall strengths of tungsten
carbide [Dandekar, 2004] and silicon nitride [Nahme et al., 1994] are discussed.
A.2.1. The Theory of Material Spall
The spall of a material is the decohesion of a material during its interaction with
tensile stress waves. The spall of a material results in the formation of new
material surfaces in the interior of the specimen (for example, State 6 in the T-X
and S-V diagrams (Figure 5.1)). Here, two studies are presented that describe
the spall process. The first [Dremin and Molodets, 1990] indicates that the spall
in metals occurs as nucleation and propagation of flaws. The second [Grady,
1988] takes a more mathematical approach and describes the criterion that must
exist for spall to occur.
A. N. Dremin and A. M. Molodets discuss spall as a two step process [1990].
They describe the process as one where nucleation and unstable propagation are
divided into distinct stages where the stress is increasing and decreasing
respectively. Then, they discuss the micro-mechanism of damage. The
nucleation of microscopic flaws occurs irregularly via a thermo-activation process
with local stresses causing a decrease in the inter-atomic bond activation energy.
The propagation of spall in metals was observed to occur by voids in aluminum
156
and copper and by micro-cracking in iron and low carbon steels. Either, the
voids or the micro-cracks coalesce to form the spall plane [Dremin and Molodets,
1990].
In Grady’s paper, spall is acknowledged to involve both energy methods and
micro-structural details [Grady, 1988]. The requirements for the initiation of
spall according to both energy and microstructure based models are discussed.
Flaw induced spall, where sufficient energy is available for the spall process, but
the lack of material flaws is the limiting factor is one model. In this case, the
stress waves impart sufficient energy in the material to drive the generation of
the new surfaces, which are created by the spall process. However, the spall
process can only initiate when a flaw such as a void or a micro-crack is created.
Energy induced spall, where the microstructure is assumed to be flawed such that
when the critical energy criterion is reached the material immediately spalls is
the main focus of Grady’s [1988] paper.[VP12] This criterion assumes that flaws
already exist in the material to such an extent that when the stress waves have
imparted enough energy for the formation of the new surfaces, spall initiates at
one of these pre-existing flaws.
The spall of a material is defined by Grady [1988] to be an internal fracture
caused by a dynamic stress state higher than the material tensile stress. A
theoretical spall strength, Pth, can be determined by equating the cohesive energy
with the elastic energy that is imparted to the lattice due to the tensile loading
2 coh oth
o
U BPv
= . {V.A.15}
This theoretical spall strength is defined in terms of the bulk modulus, Bo, the
specific cohesive energy, Ucoh, and the specific volume, vo [Grady, 1988].
The spall strength of brittle materials is limited by either a minimum energy
condition or by a maximum pre-existing flaw size. The energy criterion states
157
that the fracture surface energy must be equaled or exceeded by the sum of the
kinetic energy and the elastic energy. A minimum time criterion, ignoring the
kinetic energy, which is of an order of magnitude smaller than the elastic energy,
U, results in equations for the brittle spall strength, Ps, time to spall, ts, and
fragment size, s. The values used for fracture toughness, Kc, were quasi-static.
The equations are as follows
( )1 323s o cP c Kρ ε= , {V.A.16}
( )2 31 3s c oo
t K cc
ρ ε= , {V.A.17}
( )2 32 3 c os K cρ ε= . {V.A.18}
In Equations {V.A.16} through {V.A.18}[VP13], ρ, is the density, c0, is the
longitudinal wave speed, and, ε , is the strain loading rate [Grady, 1988]. The
criterion for energy limited spall can be written directly as
3 / oU t cγ⋅ ≥ , {V.A.19}
Where,
2 2c oK cγ ρ= . {V.A.20}
In Equations {V.A.19} and {V.A.20} the material spall value is independent of
the loading strain rate. The energy limited spall criterion is met when the elastic
energy times the spall time, U t× , meets this value. This indicates a minimum
value for the spall, which necessitates a “favorable condition” of microstructural
flaws. That is, that a flaw of sufficient size exists such that spall occurs as soon
as the spall criterion is met. Experimental comparisons show overlap between
experimental and calculated data for this criterion [Grady, 1988]. This indicates
158
that this energy criterion is a good predictor of the spall strength for brittle
materials.
Ductile spall on the other hand occurs through micro void coalescence and the
fracture energy, W, can be found approximately by the multiple of the flow stress
Y, times a critical void volume fraction, cε . This ductile fracture energy has no
dependence on fragment size due to the ductile process. Equations can be
written for the spall strength Ps, fracture time ts, and fragment size s, as for the
brittle case in terms of the flow stress and critical void volume fraction
( )1 22s o cP cYρ ε= , {V.A.21}
( )1 22 21 2s c o
o
t Y cc
ε ρ ε= , {V.A.22}
( )1 228 cs Y ε ρε= . {V.A.23}
Also noted in Grady’s paper [1988] are effects due to strain rate, temperature,
scale, and pre-compression. The strain rate during the loading and the energy of
fracture effects the spall properties, but this effect is small. The flow stress
increases with strain rate slowly while the fracture toughness decreases slowly.
Analogous effects on Kc, and Y, are caused by increasing temperature. Also, if
the spall process occurs on the same length scale as the process zone ahead of the
crack, the energy release rate can be altered. Elastic-plastic compressive shock
loading can cause material damage prior to spall, which affects the spall strength.
These factors can all effect the material spall properties [Grady, 1988]. However,
these effects are small when compared with the effects of impact velocity and the
skew angle of impact during combined pressure and shear loading.
159
A.2.2. Experimental Determination of Spall Strength
In previous studies, the spall strength of materials was found to vary with impact
velocity, or equivalently the maximum compressive stress. A study by Dandekar
et al. [2004] in tungsten carbide found that the spall stress decreases with impact
velocity. This study found that variation of the skew angle or the duration of
the stress pulse produced no effect on the spall stress.[VP15] The dynamic spall
stress of tungsten carbide was also much greater than the quasi-static spall stress
[Dandekar, 2004]. However, the presence of high hydrostatic pressures in the
shock experiments is likely to cause the increasing of tensile strengths [Dandekar,
2004].[VP16] The quasi-static spall strength of silicon nitride is reported by
Nahme et al. [1994] to be 0.59 ± 0.06 GPa. This value is within the range of
reported dynamic spall strengths of 0.5 GPa and 0.8 GPa [Nahme, 1994]. Nahme
et al. [1994] examined the dynamic spall strength of silicon nitride, but did not
systematically examine the effect of impact velocity.
The high velocity shock properties of tungsten carbide were examined in depth
by Dandekar [2004]. Hot pressed tungsten carbide made by Cercom that is
97.2% WC and 2.8% W2C by weight was employed. A hexagonal structure exists
for both of these materials. The density was 15.530 Mg/m3, the elastic wave
speed was 7.05 km/s and the elastic shear wave speed was 4.32 km/s. The
average grain size was 0.9 μm and the void volume fraction was 0.01. This
material was subjected to planar impact experiments using a 100 mm diameter
bore light gas gun. Flyer plates of the same material and of c-cut sapphire and
x-cut quartz were employed.
A total of seventeen experiments were carried out with maximum compressive
stresses between 6.88 GPa and 24.27 GPa. The material spall strength was
calculated by measuring the drop in the free surface particle velocity
corresponding to the time when the two reflected rarefaction waves meet and
cause State 6 (Figure 5.1). The velocity difference between the initial State 3
value and the bottom of the dip was examined. An elastic release impedance was
assumed. The spall strength was observed to decline rapidly from 2.06 ± 0.08
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GPa at a compressive stress of 3.4 GPa to a spall strength of 1.38 GPa at a
compressive stress of 7 GPa, after which the spall strength is observed to
decrease more slowly. The experiments indicated an elastic spall strength of 1.22
± 0.45 GPa at around a compressive strength of 24 GPa. This behavior is
understood to be the result of crack blunting and shock induced plastic
deformation, which decreases the weakening of the spall strength caused by
compressive stress levels greater than the HEL. The experiments also indicate
that there is not a large dependence on the duration of the tensile stress loading
between 0.5 μs and 1.1 μs. Shock pressure-shear experimentation did not show a
major difference in the spall strength at a skew angle of twelve degrees
[Dandekar, 2004].
Nahme, Hohler, & Stilip subjected silicon nitride specimens of two different
densities to both planar impact and depth of penetration experiments with the
goal of increasing the available knowledge of the relationship between high speed
impact behavior and material microstructure [Nahme et al., 1994]. The two
silicon nitride materials had densities of 3.15 g/cm3 and 2.28 g/cm3 and had
longitudinal wave speeds of 10.6 km/s and 8.6 km/s. These materials were
subjected to plate impacts using steel disks in a 70 mm diameter bore gas gun. A
VISARTM interferometer was employed to measure free surface particle velocities.
Tungsten sinter alloy rods were used in the depth of penetration experiments,
which hammered the ceramic plates into a steel backing.
Among the results from the plate impact experiments was the determination of
the spall strength of the materials. The spall strength was observed to be
between 0.5 GPa and 0.8 GPa. Several experiments were carried out at varying
velocities, but this range is the only quoted value [Nahme et al., 1994].
SEM analysis of the impacted fragments indicated that the two silicon nitrides
had quite similar features at the scale of 1 micron and above[VP18]. The particles
were mostly less than 1 μm in diameter. Only a very few particles larger than
10 μm in diameter were seen. Voids, with diameters between 10 μm and 100 μm
161
were understood to be the main cause for the variation in density. Areas of
“glassy resemblance” were observed in both materials, with the denser silicon
nitride having larger regions. Also, at velocities less than 400 m/s inter-granular
fracture was the preferred failure mechanism, with only a very few trans-
crystalline cracks [Nahme et al., 1994].
The depth of penetration experiments indicated that both grades of silicon nitride
offered the same level of resistance to penetration based on areal density as
silicon carbide, and were only slightly worse than TiB2. Both silicon nitride
grades were observed to have a greater resistance than Al2O3. This paper fails
to find a single parameter that can explain the differences in dynamic and
terminal ballistic behavior, but does contribute by suggesting that the similarities
in microstructure above the level of a few microns may be related to the
similarity in terminal impact behavior [Nahme et al., 1994].
162
B. Experimental Design
In this series of experiments both normal and combined compression and shear
plate-impact experiments were conducted to investigate the HEL of Si3N4 and
also to examine the effects of the state of stress on its spall strength. The normal
plate impact experiments were conducted in the impact velocity range of 50 m/s
to 550 m/s. The 550 m/s impact velocity with a tungsten carbide flyer plate
corresponds to a predicted elastic stress of 14.8 GPa, which is above the HEL of
Si3N4. In the combined compression and shear set of the plate impact
experiments, a twelve degree skew angle was utilized. The impact velocity for
this set of experiments was kept between 115 m/s and 300 m/s; the state of
stress corresponding to this range of impact velocities and the 12 degree skew
angle was such so as to not cause premature disintegration of the specimen in
tension due to the pure-shear state of stress in Si3N4. These sets of experimental
parameters had not been previously investigated for this grade of silicon nitride.
B.1. Description of Materials Under Investigation
The material under study is AS800 grade silicon nitride, manufactured by
Honeywell and provided by the NASA Glenn Research Center. This material
was obtained in disks of 50 mm diameter that had thickness of 5.5 mm or 8 mm.
These disks were impacted by either Si3N4 or tungsten carbide flyer plates. The
silicon nitride has a longitudinal wave speed, CT, of 10.9 km/s and a density, ρ,
of 3.27 g/cm3 [Choi, 2002]. Multiplying the density times the wave speed gives
the acoustic/longitudinal impedance, ZT, which for silicon nitride is 35.5x106
Pa/(m/s). The corresponding shear impedance is found by multiplying the
density times the shear wave speed, CTS, which is 6.10 km/s. This results in a
shear impedance, ZTS, of 19.9 x106 Pa/(m/s). The elastic-plastic wave speed of
silicon nitride is 8.27 km/s. The corresponding elastic-plastic impedance, ZTP, is
27.1x106 Pa/(m/s).
Two materials were used for these flyers: the AS800 grade silicon nitride and a
CG103 tungsten carbide. This grade of tungsten carbide has 3% by weight of
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cobalt as a binder. Tungsten carbide has a longitudinal wave speed of 6.99 km/s
and a density of 15.2 g/cm3 which resulted in a longitudinal impedance, ZF, of
106x106 Pa/(m/s) [Dandekar, 2004]. The shear wave speed in tungsten carbide is
4.19 km/s, which results in a shear impedance of 63.6 x106 Pa/(m/s). This is
under elastic conditions. The elastic-plastic wave speed of tungsten carbide is
5.05 km/s. The elastic-plastic impedance, ZFP, is thus 76.6x106 Pa/(m/s).
For the shear experiments, a polymethylmethacrylate (PMMA) window material
is employed in order to reduce the tensile stress level during refraction. This
PMMA has a density of 1.19 g/cm3 and a longitudinal wave speed of 2.00 km/s.
This gives it an elastic material impedance, ZW, of 2.38x106 Pa/(m/s). The shear
impedance, ZWS, is 1.14 x106 Pa/(m/s), which is found from the shear wave speed
of 0.96 km/s. There are two reasons why this material is used for a window.
The low impedance allows a portion of the compressive wave to propagate, while
the remainder is reflected from the boundary as a tensile wave. The low wave
speed means that the wave propagation through the material is slow enough that
the waves returning from the free surface of the window do not return during the
measurement time.
B.2. Experimental Matrix for the Shock Compression Tests
Nine experiments were conducted in shock compression in order to determine the
particle velocity of the free surface and thus the state of stress in the interior of
the specimen (Figure 5.3). These experiments are summarized in Table 5.1.
From these experiments, the Hugoniot Elastic Limit (HEL) and the spall
strength as a function of shock compression could be determined. The
characteristics of the dip in the particle velocity profile caused by the release
waves emitted during the spall event also indicate the mode of material failure
that occurs. The significance of these quantities will be discussed later.
The free surface velocity records for all the experiments are summarized in Figure
5.3. In these experiments, 5.5 mm thick silicon nitride flyer plates were used to
164
impact 8 mm thick silicon nitride target plates at impact velocities of 65 m/s, 66
m/s, 107 m/s, 208 m/s, 355 m/s, and 528 m/s . These experiments were labeled
SC-1 through SC-5 and SC-8. Two experiments were run at low impact speeds,
65 (SC-8) and 66 m/s (SC-5), in order to examine the velocity time profiles at
stress levels close to the minimum spall strength.
Besides this set of experiments, three experiments were conducted at higher
impact speeds using the tungsten carbide flyer plates in order to examine the
inelastic behavior of Si3N4. Tungsten carbide, because of its higher acoustic
impedance when compared to silicon nitride, produces a higher compressive stress
for the same impact velocity. Two of these experiments were conducted at impact
velocities of 494 m/s (SC-7) and 546 m/s (SC-6) that resulted in free surface
particle velocities of 708 m/s, and 792 m/s respectively. The free surface particle
velocity profile for the experiment conducted at 792m/s (SC-6) indicates a kink
during the rise time of the shock wave. This kink in the particle velocity indicates
a transition from elastic to inelastic response during its shock compression.
One experiment was conducted using a tungsten carbide flyer plate at 417 m/s
(SC-9), inducing a rear surface particle velocity of 599 m/s. This experiment was
performed to examine the differences in stress pulse duration between the two
flyer types. The time which the initial stress wave takes to reflect off of the flyer
free surface and return to the impact face is known as the pulse duration.[VP19]
The silicon nitride flyers of 5.5 mm thickness and tungsten carbide flyers of 2.83
mm thickness generated wave pulses with durations of 1.03 μs and 0.81 μs
respectively.
B.3. Experimental Matrix for the Pressure-shear Impact Tests
A set of four plate impact experiments were conducted under combined pressure-
shear conditions. In these experiments tungsten carbide flyer plates of thickness
2.83 mm were used to impact silicon nitride disks of 5.5 mm thickness. The
primary objective of these experiments was to examine the effects of combined
165
pressure and shear loading on the spall strength of Si3N4. The experiments were
conducted at impact velocities in the range from 115 m/s to 300 m/s (Figure
5.4), and were labeled SC-10 through SC-13.
One of the concerns in conducting the plate impact spall experiments under
combined pressure and shear loading is the possibility of material disintegration
prior to the spall event due to development of a state of pure shear within the
specimen. Because the normal stress wave travels faster than the shear stress
wave, there is a region after the reflection of the incident longitudinal stress pulse
but before the reflection of the shear wave from the free surface of the specimen,
where a part of the specimen is in a state of pure shear. This state is shown as
State 4* in Figure 5.5. The state of pure shear can lead to the generation of
principal tensile stresses at 45 degrees to the impact direction that are large
enough to cause material failure.
A Mohr’s circle analysis was used to determine the maximum velocity at which
the tensile stress in State 6 would exceed the expected spall strength in Si3N4 but
the maximum tensile stress in state 4* would not. The stress-velocity analysis
technique of the line equations works equally well for both normal and shear
stresses. Therefore, from the impact velocity, the normal and shear velocities can
be obtained as
1 cosN IV V θ= , {V.B.1}
1 sinS IV V θ= . {V.B.2}
In the above equations, the skew angle, θ, is 12 degrees. Using the line equations,
the normal stress in state four is found in terms of the normal impedance values
for the specimen, ZT, the flyer, ZF, the window, ZW, and the normal impact
velocity, V1N is given as
166
( )( )2 2
4 12 2T F T W T F
N W NT F T W
Z Z Z Z Z Z Z VZ Z Z Z
σ⎡ ⎤+ − −⎢ ⎥= ⎢ ⎥+ −⎢ ⎥⎣ ⎦
. {V.B.3}
This is a compressive stress, which is taken to be positive in this analysis. The
shear stress can be determined in a similar fashion to be
4-No Window 1TS FS
STS FS
Z Z VZ Z
τ =+
. {V.B.4}
In Equation {V.B.4} the shear impedances, as specified in Section B.1 are
employed. Using these stress values, the maximum principal stress for the
material with and without the window can be calculated as
22
1 2,2 2
X Y X YXY
σ σ σ σσ σ τ+ −⎛ ⎞⎟⎜= ± +⎟⎜ ⎟⎝ ⎠ . {V.B.5}
In Equation {V.B.5} σX is the normal stress, and τXY represents the shear stress.
The solutions to equations {V.B.3} and {V.B.4} are used for to obtain the
magnitudes for σX and τXY, respectively. Moreover, σY is assumed to be zero for
this calculation. Thus, the principal stress formula takes the form
22
2 2 2X X
XYσ σσ τ⎛ ⎞⎟⎜= − +⎟⎜ ⎟⎝ ⎠ . {V.B.6}
Substituting in the normal stress value, Equation {V.B.3} and the shear stress
value, Equation {V.B.4} enables the calculation of the maximum tensile stress for
a given impact velocity. These values are listed in Table 5.2. They are
compared with estimations of the spall strength of the material at a given normal
velocity calculated using Equation {V.C.35}. When the spall strength was
exceeded by the principal tensile stress, the material disintegrated in State 4*.
167
To avoid disintegration of the specimen due to tension in pure shear, a low
impedance transparent window made from PMMA is glued to the rear of the
specimen. The presence of the relatively low impedance PMMA window causes a
part of the initial compressive wave to transmit into the PMMA window, such
that the Si3N4 specimen adjacent to the PMMA window does not reach a state of
pure shear. The normal stress in State 4* can still be calculated using Equation
{V.B.3}. However, the shear stress in State 4* changes to
( )( )2
4-Window 12 2TS FS TS FS WS WS TS
WS STS FS TS WS
Z Z Z Z Z Z Z Z VZ Z Z Z
τ⎡ ⎤+ + −⎢ ⎥= ⎢ ⎥+ −⎢ ⎥⎣ ⎦
. {V.B.7}
Using the principal stress Equation {V.B.6}, the new tensile principal stress can
be determined.
Based on the above analysis and the estimate for the spall strength of Si3N4
under nominal shock compression conditions, an estimate for the limiting impact
velocity under combined compression and shear impact conditions that does not
lead to disintegration of the specimen can be made for both with and without the
presence of the window. When no window is used, the principal tensile stress is
larger than the spall strength at impact velocities of 250 m/s and higher. For the
case of the tungsten carbide flyer, this corresponds to a maximum 350 m/s free
surface particle velocity. However, in presence of the PMMA window, the
principal tensile stresses are insufficient to cause spall in Si3N4 up to an impact
velocity of 300 m/s and higher. This impact velocity corresponds to a 420 m/s
free surface particle velocity. In view of these impact velocity constraints, the
chosen impact velocities for these experiments were 115 m/s, 186 m/s, 233 m/s,
and 300 m/s. The impact angle was maintained at 12 degrees in all experiments.
In this way, the matrix for the pressure-shear experiments was determined.
168
C. Experimental Analysis and Results
Using the single stage gas gun, the two series of experiments described above
were performed on AS800 silicon nitride. The experiments yield several results
that elucidate the material behavior under dynamic tensile conditions. The
equation of state and Hugoniot curve are investigated. The HEL of the material
is determined by analyzing the free surface particle velocity of the target plate by
impacting it to a level of shock compressive stress that is higher than the
expected level for inelastic deformation in Si3N4 . From the HEL the dynamic
yield stress of Si3N4 under uniaxial loading conditions can be determined. Also,
by observing the spall signal in the free surface particle velocity versus time the
spall strength in Si3N4 could be determined. In these experiments both the effects
of increasing impact velocity and state of stress (combined pressure and shear
loading) on the spall strength were investigated.
C.1. Elastic and Hydrodynamic Relationships in the Hugoniot State
Observing the state of stress of the initial shocked state of the material enables
the determination of the equation of state and the Hugoniot curve. Specifically,
the material shock velocity was computed and related to the particle velocity to
determine the EOS of Si3N4 that is applicable to the range of the impact stresses
employed in the present experiments. Additionally, the maximum compressive
stress and strain are compared to obtain the Hugoniot curve. The results of
these experiments offer insight into the suitability of AS800 grade silicon nitride
for use in impact related aerospace applications.
C.1.1. Shock Velocity vs. Particle Velocity in the Hugoniot State
Figure 5.3 summarizes the free surface velocity profiles for all of the shock
compression plate impact experiments. In examining the shock response of a
material, an important parameter is the equation of state, which gives the
relationship between the shock velocity and the particle velocity under
hydrodynamic loading conditions. Looking at the S-V diagram for the shock
169
compression case (Figures 5.1b, 5.2b, 5.5b), the particle velocity in State 3,
hereafter referred to as uP, is one-half of the rear surface particle velocity in State
4, V4 , i.e.
4 2Pu V= . {V.C.1}
The shock velocity, US, represents the rate at which the shock propagates
through the material. In principle this is the same as the elastic wave speed for
impacts below the HEL. However, there is some variation (Table 5.1) due to the
uncertainty in the measurement of the shock arrival times. The shock velocity is
given by
2S arrivalU T t= . {V.C.2}
The thickness of the specimen T, is measured by a micrometer prior to the
assembly of the specimen and flyer. The time when the flyer impacts the
specimen is the trigger time for the experiment and corresponds to the zero time
on the particle velocity record. The arrival of the stress wave also represents the
beginning of State 3. The arrival time, tarrival, is the difference between these two
moments.
Figure 5.6 shows the plot between the shock speed and the measured particle
velocity obtained from the present series of experiments. From the plot it can be
seen that the shock velocity is nearly a constant and is equal to the elastic wave
speed of longitudinal waves in Si3N4. This is to be expected in the present study
since Si3N4 remains nearly elastic in all the plate impact experiments except for
the two highest impact velocity experiments conducted with the tungsten carbide
flyer plates. In fact, the average of the shock velocities is 10.7 km/s, which is
within 1.8% of the elastic wave speed, 10.9 km/s. This close value indicates
good agreement with the wave speed of AS800 silicon nitride calculated by
equation {V.A.1} from the density, elastic modulus, and Poisson’s ratio [Choi,
2002].
170
C.1.2. Difference in Hugoniot State Variables due to Pressure-ShearLoading
Figure 5.4 summarizes the measured surface velocity profiles of all the pressure-
shear plate impact experiments conducted in the present study. Because a
PMMA window was employed in the pressure-shear experiments, the stresses
within the material need to be evaluated keeping the new experimental
configuration in mind (Figure 5.1b). The Hugoniot state, State 3 remains the
same as without the PMMA window. However, the State 4 does not return to
zero compressive stress. This is because the measured surface of the silicon
nitride is not a free surface, but the interface between the Si3N4 and the PMMA.
By calculating the intersection of the line with slope ZW, and connecting the line
from State 3 to what was State 4 without the window, the new relationship
between the stress and particle velocity in States 3 and 4 can be determined.
The evaluation of the particle velocity in State 3 from the free surface particle
velocity can therefore be carried out by using the relationship
( )4 2
W TP
T
Z Zu V
Z+
= . {V.C.3}
The rest of the calculations of the shock velocity, the Hugoniot stress and strain
remain the same as for the normal shock compression case. The resulting
experiments are summarized in Table 5.1.
C.1.3. Elastic Hugoniot Stress and Strain Relationship
The Hugoniot curve is a stress versus strain relationship that describes the
material deformation above the HEL. It is technically, the set of all possible end
states to the shock process in a given material. The material actually deforms
along the Rayleigh line from the HEL to the final shocked state. In the current
study, the curve cannot be determined in its entirety because only the elastic
171
regime is covered in depth by the experiments. However, an elastic stress versus
strain relationship can be estimated from the data.
The elastic stress and strain represent the material behavior under elastic
deformation conditions. By means of the line equations, this stress can be found
simply by multiplying the particle velocity, uP, by the material impedance, ZT:
E O T P T PC u Z uσ ρ= = . {V.C.4}
In Equation {V.C.4}, ρ, is the density. The corresponding strain value is simply
the ratio of the particle velocity to the elastic wave speed, CT:
E P Tu Cε = . {V.C.5}
The elastic compressive stresses and strains for each experiment can be
determined by these equations. Combining Equations {V.C.4} and {V.C.5}, the
elastic stress and strain are directly related by
2 386E O T E ECσ ρ ε ε= = (GPa). {V.C.6}
This Hugoniot stress and strain relationship represents the loci of all the shocked
states in Si3N4 under elastic compression (Figure 5.7). The resulting values are
tabulated for the experiments in Table 5.1. The onset of plasticity alters the
stress and strain away from the elastic case. This is discussed in more detail in
Sections C.1.4 and C.1.5.
C.1.4. Calculation of the HEL
In the event that the impact velocity is high enough to cause either the flyer or
the specimen to exceed its HEL, the loading shock profile shows a sharp change
of slope (kink) in the free surface particle velocity diagram at a point defined by
the HEL. The stress at the HEL can be determined from the corresponding free
172
surface particle velocity by multiplying by the impedance of the target plate and
dividing by two, that is by reversing Equation {V.A.10}. As can be seen from
Figure 5.8, in Experiment SC-6, where the free surface particle velocity reaches
792 m/s, the curve reduces slope at about 692 m/s. The elastic wave speed is
used to estimate the acoustic impedance and calculate the HEL by the well
established formula [Nahme et al., 1994]:
4 2HEL TZ Vσ = . {V.C.7}
The HEL estimated by Equation {V.C.7} is approximately 12 GPa. This is
consistent with the material behavior documented in other studies, which show
similar grades of silicon nitride to have HEL’s of 12.1 GPa and 10-12.5 GPa
respectively [Nahme et al., 1994; Mashimo, 1998]. Most of the other experiments
conducted in the present study were in the elastic deformation regime and thus
show no change in slope in their particle velocity versus time profile. The state
of stress in Experiment SC-7 was nearly elastic, and the measured particle
velocity versus time profile showed a maximum particle velocity of 708 m/s.
This value of the particle velocity was too close to the particle velocity
corresponding to the HEL (~692 m/s) found in Experiment SC-6 for Experiment
SC-7 to be used in determining the HEL.
From the HEL, the dynamic yield strength, Yo, of Si3N4 can be estimated by the
well known relationship [Reinhart, 2002]
1 21O HELY νσ
ν−=−
. {V.A.6}
This dynamic yield strength represents the strength of Si3N4 in compression
under plane stress conditions beyond which plastic yielding begins, as opposed to
the HEL which represents the onset of inelastic deformation under fully plane-
strain conditions. In Equation {V.A.6} the Poisson’s ratio, v, of the AS800 silicon
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nitride is taken to be 0.27 [Choi, 2002], and therefore, the dynamic yield strength
of Si3N4 can be estimated to be 7.6 GPa.
C.1.5. Determination of the Hydrodynamic Hugoniot Stress and StrainStates
Because the inelastic material behavior alters the relationship between the stress
and the particle velocity, Equation {V.C.7} needs to be modified for stresses
above the HEL. In the present experiments, all three of the normal impact
tungsten carbide flyer experiments exceed the HEL of the tungsten carbide. Also,
the two highest velocity impacts exceed the HEL of the silicon nitride as
established in the literature [Nahme et al., 1994; Mashimo, 1998]. However, even
though both the tungsten carbide and the silicon nitride become inealstic at the
highest impact velocity experiment, it can be seen (Figure 5.2) that the
compression and release of the silicon nitride is always symmetric, maintaining
Equation {V.C.7} as the means for determining the HEL.
One method of approximating the Hugoniot stress and strain relationship of the
material under elastic-plastic loading is to treat it as if it were deforming in a
hydrodynamic fashion. The hydrodynamic Hugoniot stress can be experimentally
determined for each experiment by using the shock velocity, US. In the present
case Equation {V.C.4} takes the form
( )H HEL O S P P HELU u uσ σ ρ −= + − . {V.C.12}
The hydrodynamic strain at this Hugoniot state can also be found from the shock
and particle velocities in State 3. The strain is in fact the ratio of the particle
velocity to the shock velocity
/H P Su Uε = . {V.C.13}
In Equation , ρO, is the initial density of Si3N4, which as mentioned above is 3.27
g/cm3.
174
This hydrodynamic stresses and strains differ from the elastic line once plasticity
has occurred. This fact is reflected in the experimental results (Table 5.2). The
hydrodynamic strain values for the experiments where the HEL is exceeded are
plotted versus the stress to yield an equation for the relationship between them
(Figure 5.7). A relationship can be found for these two points with a correlation
coefficient:
H H = 149 + 6.46σ ε (GPa). {V.C.14}
Another relationship can be determined between the HEL and the highest
velocity experiment (SC-6).
H H = 87.2 + 9.29σ ε (GPa). {V.C.15}
These stress versus strain relationships apply only within the range of the elastic-
plastic experiments. In order to examine the hydrodynamic curve more fully,
more experiments must be performed at greater impact velocities. However, this
is beyond the capabilities of the single stage gas gun.
C.2. Measurement of the Spall Strength
The wave recorded on the free surface of the target plate is a compilation of the
time history of the dynamic material response (Figures 5.3, 5.4) under shock
wave loading conditions. Upon reaching State 4, the free surface particle velocity
is at its maximum point, Vmax. The rarefaction waves caused by the compression
waves reflecting from the free surfaces of the flyer and specimen intersect within
the specimen to produce a state of tension, State 6 (Figures 5.1a, 5.2a, and 5.5a).
Without spall, the stress and particle velocity levels will reach State 6, as
specified by the line equations. This is depicted in Figures 5.1b, 5.2b, and 5.5b
as the dotted lines and the circle representing the “no spall” state. This stress
level is not reached if the tensile stress at the spall plane is greater than the
material’s spall strength, in which case the material delaminates. In the case of
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the material spall, the tensile stress can only reach the spall strength, which is
indicated in Figures 5.1b, 5.2b and 5.5b by the circle showing the intersection of
solid blue line in the tensile region and horizontal spall strength line. This state
is known as State 6’. In the absence of spall, once the intersection of the release
waves propagates to the free surface of the material it completely unloads to,
State 7. This state is indicated in Figures 5.1b, 5.2b, and 5.5b by the circle
marked,Vno spall. If there is spall, the material unloads partially, causing a
decrease in free surface particle velocity and then it reloads, which produces a re-
acceleration.
This de-acceleration and re-acceleration produces a dip in the free surface particle
velocity, from which the spall strength is calculated. The spall signal causes a
decrease in free surface particle velocity from Vmax to Vmin,. This corresponds to
the bottom of the dip. As the spall progresses, the free surface velocity
accelerates again to another value Vo.
In the shock compression experiments Vmax and Vo are the same. Thus, Vo, is not
marked in Figures 5.1b or 5.2b. Using two of these three velocities, the material’s
spall strength, σspall, can be calculated by
( )min12Spall T oZ V Vσ = − . {V.C.16}
In Equation {V.C.16} ZT represents the specimen’s acoustic impedance. For the
pressure-shear experiments the spall strength calculation must be altered to
include the effects of the PMMA window, and is given by
( )( )min12Spall T W oZ Z V Vσ = + − . {V.C.17}
In the pressure-shear experiments, the two velocities Vo and Vmax are different.
The rebound velocity, Vo, provides the more accurate measure of the spall
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strength. This is because the material does not fail completely during the tensile
pulse, but rather exhibits partial or incipient spall.
C.2.1. Derivation of the Spall Strength Formulae
In order to derive formulae {V.C.16} and {V.C.17}, relations must be found
linking the spall strength to the particle velocity on the rear surface. These
analyses start from the line equations that connect the stress state and particle
velocity through the material impedance. The following process is then used to
determine the connection between the observations of the free surface particle
velocities in States 4 and 7’ and the spall strength. This discussion includes the
effects of a window covering the free surface of the specimen, which makes the
equations valid for the pressure-shear experiments as well as the shock
compression experiments.
First, it can be noted from Figures 5.1b and 5.2b that the line connecting State 6
(no spall) and State 4 (Vmax) passes through the unknown spall point (Vspall, σspall).
Thus, the line equation for the spall strength can be written in terms of State 4
where Vmax is obtained from the experiment as
( )max maxTspall SpallZ V Vσ σ− = − . {V.C.18}
Also, using the line between State 7’-s [VP25] and State 6,[VP26]
( )min minTspall SpallZ V Vσ σ− =− − . {V.C.19}
State 7’-s is marked on Figures 5.1b, 5.2b, and 5.5b by, Vmin, which is known
from the experiment. Next, because State 4 is on a line crossing the origin with a
slope equal to the impedance of the window, ZW, the State 4 stress, σmax, can be
written in terms of the state 4 velocity Vmax, as
max maxWZ Vσ = . {V.C.20}
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The same applies to State 7’-s, because it is also lies on this line, i.e.
min minWZ Vσ = . {V.C.21}
Substituting Equations {V.C.20} and {V.C.21} into Equations {V.C.18} and
{V.C.19}, yields
( )max maxW Tspall SpallZ V Z V Vσ− = − , {V.C.22}
( )min minW Tspall SpallZ V Z V Vσ− =− − . {V.C.23}
Rearranging equation {V.C.22} to solve for Vspall, gives
( ) maxWSpall
T spall
T
Z VV
ZZ
σ− += . {V.C.24}
Now, substituting Equation {V.C.24} into Equation {V.C.23} yields
( )min
maxminW
T W spallTspall
TZ V
Z Z VZ V
Zσ
σ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠
− +− =− − . {V.C.25}
Rearranging Equation {V.C.25} leads to
( ) ( )min max12 W W TT spallZ V Z Z VZ σ⎡ ⎤+ + −⎣ ⎦ = . {V.C.26}
Because Vmax is greater than Vmin, this will be a negative number, which
corresponds to tension. This equation is in terms of Vmax and Vmin, both of which
are known quantities from the experiments.
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Another equation for spall, relating Vo and Vmin, can be found in the following
fashion. Using the same arguments as above, the stress σo in State 7’-e can be
found from the velocity at state 7’-e,Vo, as
o oWZ Vσ = . {V.C.27}
Next, the velocity intercept along the line connecting State 7’-e to the line
between State 4, V2, will be calculated. The intercept is referred to as Point A.
Because this point is on the velocity axis, the stress, σA, is zero. The line
equation uses the elastic impedance of the specimen, ZT:[VP27]
A T A o T oZ V Z Vσ σ+ = + . {V.C.28}
Since the stress at Point A is zero and substituting Equation {V.C.27} for the
stress at State 7’-e σo, yields
( )T A W T oZ V Z Z V= + . {V.C.29}
Also, using the line equations to connect Point A to the spall plane, yields
T A spall T spallZ V Z Vσ− = − . {V.C.30}
Substituting Equation {V.C.29} into Equation {V.C.30}, yields
( )W T o spall T spallZ Z V Z Vσ− + = − . {V.C.31}
Using equation {V.C.23}, which provides the relationship between, Vmin, and the
spall strength and velocity, Equation {V.C.23} can be rearranged to give
( ) minspall T spall W TZ V Z Z Vσ + = + . {V.C.32}
Adding equations {V.C.32} and {V.C.31} yields,
179
( ) ( )min2 spall W T W T oZ Z V Z Z Vσ = + − + . {V.C.33}
Finally, simplifying Equation {V.C.33}, gives
( )( )min12spall W T oZ Z V Vσ ⎡ ⎤= − + −⎣ ⎦ . {V.C.34}
This is a second equation that can be used to calculate the spall strength. Again,
the answer is negative because tension is negative on the S-V diagram. This
equation is equivalent to equation {V.C.17}. Note that in the case of the normal
impact experiments, the window impedance is zero, and Equation {V.C.34}
reduces to Equation {V.C.16}.
C.2.2. Spall Strength and Material Failure Mode during Spall in Pure ShockCompression
SiN exhibited spall in all experiments conducted in this series of experiments and
a variation of the spall strength with impact velocity was recorded (Table 5.3).
As per the discussion in Section C.3.1, the spall strength can be directly related
to the spall dip (Figure 5.3) in the free surface particle velocity profile as
( )min12Spall T oZ V Vσ = − . {V.C.16}
In the case of the normal loading in the elastic material range, Si3N4 indicates a
drop in the spall strength with increasing impact velocity (Figure 5.9), and the
spall strength is observed to decrease from an average of 895 MPa at 65 m/s
(SC-8) to 564 MPa at 528 m/s (SC-3). Statistically, ceramic materials possess a
large scatter in the material failure data. This variation can also be seen in the
present experiments, where spall strengths of 908 MPa and 881 MPa are recorded
in two experiments at impact velocities of 65 m/s (SC-8) and 66 m/s (SC-5),
respectively. A linear fit to the spall strength data can be represented as
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9-0.0383 10 (Pa)spall Hσ σ= + , {V.C.35}
This linear fit is plotted along with the experimental data in Figure 5.9. The
linear fit line had a correlation coefficient of R2 = 0.976. It is interesting to note
that the experimental spall strengths were very close to the predictions of the
straight line, especially given the fact that ceramic materials tend to have large
statistical scatter.
As the material behavior becomes inelastic under compression prior to spall, the
spall strength reaches a plateau. This effect is seen clearly in the two
experiments with the highest impact velocity. In these experiments, the spall
strengths were 640 MPa at 494 m/s (SC-7) and 622 MPa at 546 m/s (SC-6)
(Figure 5.9). These two values are only 18 MPa apart from each other, which is
smaller than the scatter of 27 MPa between experiments SC-5 and SC-8, both
conducted at 65.5 ± 0.5 m/s. The near constant level of spall strength in
experiments SC-7 and SC-6 suggests that the applied compressive stresses results
in some degree of inelasticity in Si3N4 that retards the spall initiation.
The pull-back time, which is the time that the velocity takes in recovering from
Vmin to VO, changes distinctly with impact velocity (Figure 5.3). In the lowest
impact velocity experiment, at 65 m/s (SC-8), this recovery time is about 90 ns.
At an impact velocity of 107 m/s (SC-1) the pull-back time drops to 30 ns, which
suggests a more brittle type of material failure. The increase in the recovery time
with an increase in impact velocity indicates a higher degree of the accumulation
of damage during the spallation process. In experiments conducted at the higher
impact velocities (528 m/s for Experiment SC-3) the pull-back time was again
observed to be 90 ns.
Within the four experiments using the tungsten carbide flyers, the pull-back time
is also seen to increase. The impact velocity in Experiment SC-9 is 419 m/s and
the pull-back time is 80 ns. By an impact velocity of 546 m/s (SC-6) in tungsten
181
carbide, the pull-back time has expanded to 130 ns. This, along with the
increase in pull-back time with impact velocity in the experiments with silicon
nitride flyers, confirms the trend of increasing pullback time with velocity. This
increase in pull-back time suggests a damage accumulation mode that increases in
duration with increasing impact velocity rather than an instantaneous brittle
fracture.
An interesting feature is observed in the effect of stress pulse loading time on the
pull-back time. The impact velocity in Experiment SC-9, 419 m/s is lower than
in Experiment SC-3, but a tungsten carbide flyer was used, so the compressive
stress in Experiment SC-9 is higher than in Experiment SC-3, With a higher
compressive stress, the pull-back time in Experiment SC-9 should be higher than
in Experiment SC-3, However, in Experiment SC-9, the pull-back time is 80 ns.
This is slightly lower than in Experiment SC-3, where the pull-back time is 90 ns.
The reason this is occurs is that in the experiments with the tungsten carbide
flyer, a shorter compressive pulse duration, i.e. 0.81 microseconds instead of 1.03
microseconds, was attained in the experiments. This difference in pulse duration
was due to the differences in the impedance and thickness of the WC flyer when
compared to the Si3N4 flyer used in the other spall experiments. Thus, the
noticed decrease in pull-back time between experiments SC-3 and SC-9 is due to
the shortening of the compressive stress pulse loading time.
C.2.3. Spall Strength and Failure Modes Under Combined Pressure andShear Impact Loading
The shock compression-shear experiments resulted in substantial degradation of
the spall strength at the relatively small angle of obliquity (12 degrees). This
degradation of the spall strength is seen by the change in the level of the free
surface particle velocity, following the spall dip (Figure 5.4).
The presence of a PMMA window results in the velocity pull-back following the
spall not recovering to the level of Vmax. This creates a distinction between the
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pre-spall peak velocity Vmax, and the post-spall peak velocity Vo, which was not
evident in the pure shock compression experiments (Figure 5.1b). This being the
case, the relation for the spall strength in terms of the pull-back particle velocity
Vo, and the minimum velocity Vmin, is
( )( )min12Spall T W oZ Z V Vσ = + − . {V.C.17}
Table 5.3 provides the details of the experiments and the experimental results
obtained from the pressure-shear experiments conducted in the present study.
When compared to the plate impact experiments, the spall strength in the
presence of a modest amount of shear is observed to drop more rapidly with an
increase in impact velocity. The spall strength decreases from 803 MPa at an
impact velocity of 115 m/s (SC-11) to 249 MPa at an impact velocity of 233 m/s
(SC-12) (Figure 5.10). As the velocity is increased still higher, a complete loss of
spall strength in Si3N4 is observed. This is evidenced by the lack of spall
recovery in the 299 m/s impact velocity experiment (SC-13).
As with the shock compression data, a linear fit was determined for the pressure-
shear spall strength data
9H-0.1751 + 10 (Pa)spallσ σ= . {V.C.36}
This fit has a correlation coefficient of R2 = 0.989. As mentioned in Section
C.2.2 the R2 value represents a good fit considering the inherent uncertainty in
material failure data for ceramics. The slope of the linear fit is almost five times
as steep as that of the slope for the shock compression experiments. Also, note
that the highest velocity experiment has no spall strength. This can be
determined from Figure 5.4, because following the drop in free surface particle
velocity due to the spall, there is no pull back in the particle velocity. The exact
compressive stress for which the spall strength becomes zero is 5.71 GPa. This is
between Experiment SC-13, 299 m/s, and Experiment SC-12, 233 m/s.
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The steeper drop in slope in Equation {V.C.36} when compared with Equation
{V.C.35} indicates that the shear loading creates a severe degradation of the
spall strength even at a modest skew angle of twelve degrees (Figure 5.10). It is
evident that this shear loading causes the material to lose its strength because
the material is damaged in shear prior to the spall. The shear stress, while not
enough to cause complete failure in shear, does result in damage in the material,
making it easier to spall. Pull-back times during the spall are about 100 ns for
the lower impact velocity experiments (SC-10, SC-11), and could not be
determined for the other experiments (SC-12, SC-13).
C.3. Material Failure Modes during Dynamic Spall and Fractography of theSpall Surface
The recovered silicon nitride target and the flyer materials were heavily
fragmented in all the experiments. The expansion chamber makes possible the
soft recovery of these fragments. Moreover, the nature of this fragmentation is
altered with the impact velocity and the stress state. At low impact velocities
around 65 m/s (SC-8, SC-5), the specimen breaks up along the spall plane and in
a star shaped pattern generated by stresses resulting from the boundary waves
(Figure 5.11a). The spall plane is directly observable in the fragments from these
experiments. At higher impact velocities, such as 107 m/s (SC-1), the specimen
breaks into smaller fragments (Figure 5.11b). At still higher impact velocities
beyond 356 m/s, the material completely disintegrates, becoming essentially a
powder (Figure 5.11c).
In order to examine the micro-structural effects of impact velocity and the state
of stress on the material, fragments with a face that is a surface of the spall plane
were examined under the scanning electron microscope (SEM). Four
representative experiments were evaluated. To observe the microstructure in low
and high impact velocities, experiments SC-8, SC-2, and SC-4 were chosen.
These have impact velocities of 64.9 m/s, 208 m/s, and 355 m/s, respectively.
184
To study the effect of angle, Experiment SC-12 was examined. Experiment SC-
12 was conducted at an impact velocity of 152 m/s.
For the low impact velocity experiment conducted at 65 m/s (SC-8), which
corresponds to a particle velocity of 31.8 m/s in the material, the SEM reveals a
relatively smooth fracture surface with several large intact hexagonal grains
(Figure 5.12). The Hugoniot state particle velocity is relevant here because that
is the velocity of the material at the spall plane prior to the arrival of the spall
waves. These grains are at their largest greater than 100 μm long. At higher
magnification of 2000X and 3500X, the fracture mode is seen to be
predominantly inter-granular fracture. However, some intragranular fracture
through the larger grains is also detected.
At 208 m/s impact speed, which corresponds to a particle velocity of 178m/s
(SC-2), the overall fracture surface appears to have become rougher (Figure
5.13). No grains of 100 μm size were detected, but several holes for grains of
about 50 μm can be seen. These grains are imaged at 2000 times magnification
as well. A larger fraction of the 50 μm grains are fractured. This image shows
the fracture surface to be rougher on all scales but still brittle; both inter-
granular and intra-granular fracture modes were observed.
Shot SC-4, which has an impact speed of 356 m/s and a particle velocity of 178
m/s, shows two distinct regions of fracture, one rough and one smooth (Figure
5.14). More highly magnified images show a larger grain sticking out of a clump
of smaller grains in the rough fracture area. The smooth area is smooth even at
3500 times magnification. The fracture mode in the rough zone is still brittle,
though it is difficult to ascertain if it is intra-granular or inter-granular. The
smooth zone is likely due to inelastic failure processes.
The effect of skew angle on the modes of fracture during spallation can be
examined from the recovered specimens in Experiment SC-12 (Figure 5.15). This
experiment has an impact velocity of 233 m/s with a skew angle of 12 degrees;
185
the corresponding normal component of the particle velocity in Si3N4 was 152
m/s. The overview at 250 times magnification shows a brittle fracture surface,
similar to that in SC-2 (Figure 5.13). A few grains on the order of 50 μm are
seen. Closer views reveal grains on the order of 10 μm long sticking out of the
fracture surface, mostly intact.
These images can be compared with the SEM pictures of the microstructure of
the intact Si3N4 ceramic taken from the surface of a sample that was not
impacted (Figure 5.16). The examined face was polished. This surface shows the
granular structure of the material at 2000 times magnification. Holes in the
surface show where large grains, on the order of 10 μm have been pulled out of
the material during the polishing process.
The SEM images indicate that spallation occurs by a primarily planar inter-
granular brittle fracture mode at the low impact velocities of ~ 60 m/s. 100 μm
grains are visible, many clearly intact. However, some of these grains are
fractured. At the higher impact velocities, on the order of 200 m/s, the surface
becomes rougher and several 50 μm grains are no longer intact, suggesting both
inter-granular and intra-granular fracture modes to be active. At around 350 m/s
impact velocity, the material shows both brittle fracture and smooth areas.
These smooth areas could represent inelastic failure processes beginning to occur.
At higher velocities, the material is reduced to powder, and so no SEM analysis
was possible. The presence of shear stresses in the 200 m/s impact is not reflected
by a change in failure mode during spallation. No additional surface features in
the shear experiment are noticed and the fracture mode is primarily brittle. This
suggests that the addition of shear does not cause a different type of
microstructural damage.
186
D. Summary of Planar Results
The shock compression and shock compression shear experiments discussed in
this chapter examine the dynamic behavior of silicon nitride using one
dimensional wave theory. The elastic and hydrodynamic Hugoniot and the
elastic equation of state of the material are observed in the range of impact
velocities up to and slightly above the Hugoniot Elastic Limit. The dynamic
yield strength of the material is evaluated from the HEL. The spall strength
under shock compression and shock compression-shear loadings is also described.
The elastic equation of state of silicon nitride was determined, and the portion of
the elastic-plastic regime near the HEL was also examined. The average level of
the elastic shock velocity, 10.7 km/s, is reasonably close, 1.8%, to the elastic
wave speed of the material, 10.9 km/s (Figure 5.6). Likewise, the elastic
Hugoniot of the material, or the stress-strain relationship is found for the elastic
regime. The hydrodynamic Hugoniot equation, which is used above the HEL,
indicates divergence from the elastic equation as the material begins to deform
plastically (Figure 5.7). These relationships enable the prediction of the stresses,
strains, and shock velocities within the impact velocity range of the experiments.
The Hugoniot Elastic Limit can be determined from the VISAR records of the
experiments that exceed the predicted HEL from the literature. The HEL is
found by examining a sudden kink, a decrease in the loading rate of the initial
rise in free surface particle velocity. This indicates the onset of plastic
deformation. By observation, the HEL is found to be 12 GPa, which concurs
with previously recorded HEL’s for similar materials of 12.1 GPa and 10-12.5
GPa [Nahme et al., 1994; Mashimo, 1998]. From the HEL the dynamic yield
strength in compression can be found to be 7.6 GPa. The compressive elastic-
plastic loading results in a residual compressive stress within the material.
The spall strength is determined by the magnitude of the drop and pull-back in
velocity observed in all of the experiments at the time when waves reach the free
surface from the intersection of the two tensile rarefaction waves. The
187
experimental Formula {V.C.17} uses the minimum velocity, Vmin, which
corresponds to State 7’-s and the re-acceleration velocity, Vo, which corresponds
to State 7’-e. Each experiment has a unique spall strength, which varies both
with impact velocity and with the addition of a skew angle of 12 degrees. The
pull-back time from the spall increases with increasing velocity, except for the
experiments at impact velocity of about 65.5 ± 0.5 m/s (SC-8, SC-5) in which the
pull-back times are larger than the pull-back time for experiment SC-1 at an
impact velocity of 107 m/s. This indicates that insufficient stress exists in
experiments SC-8 and SC-5 to cause sudden brittle fracture. Above 66 m/s,
sudden brittle fracture occurs. However, as the velocity increases in the
experiments with silicon nitride flyers, the accumulation of damage process
resulting in spall becomes slower. Within the group of experiments with tungsten
carbide flyers, the pull-back time also increases with increasing impact velocity.
However, the alteration of the flyer material from silicon nitride to tungsten
carbide and the resulting decrease in compressive stress pulse loading time causes
a decrease in the pull-back time.
The spall strength under elastic shock compression drops 39% between
experiments conducted at impact velocities of 65.5 ± 0.5 m/s (averaging SC-8,
SC-5) and the experiment conducted at an impact speed of 528 m/s (SC-3)
(Figure 5.9). This drop indicates that with increasing severity of impact stress,
the material has lesser ability to resist spall. This indicates that inelastic micro-
cracking is occurring, damaging the silicon nitride even below the HEL. The
correlation between spall strength and impact velocity, Equation {V.C.35}
indicates a difference in spall strength with velocity that is substantially larger
than the scatter expected of a ceramic material. This scatter can be quantified
by the 27 MPa difference in the two experiments conducted at an impact velocity
of 65.5 ± 0.5 m/s (SC-8, SC-5).
Under plastic deformation, the material exhibits a near constant spall strength.
In fact, in the two elastic-plastic experiments (SC-6, SC-7) the spall strength only
changes by 18 MPa, which is 2.8% of the average spall strength of 631 MPa, and
188
less than the scatter in experiments SC-5 and SC-8. From this, it appears that
the compressive residual stress generated by the elastic-plastic loading retards the
initiation of spall. The effect of velocity is noticeable at the microscopic level in
the SEM images. The planar brittle fracture at the spall plane at an impact
velocity ~60 m/s gives way to rough brittle fracture at and impact speed of ~ 200
m/s and to the beginnings of non-elastic fracture at an impact speed of ~350 m/s
and higher velocities.
The addition of a twelve degree skew angle to the shock compression impacts
results in a pressure-shear loading which causes the material to exhibit more
severe degradation of spall strength (Figure 5.10). The spall strengths in these
experiments are nearly comparable to the shock compression experiments at
lower velocities. At an impact velocity of 115 m/s (SC-11), the spall strength is
803 MPa, which is 10% off of 893 MPa, the calculated value from Equation
{V.C.35} for the normal spall strength at this impact velocity. However, at an
impact velocity of 233 m/s (SC-12), the spall strength is only 249 MPa, which is
68% smaller than the calculated normal spall strength of 780 MPa for this impact
velocity. At an impact velocity of 299 m/s (SC-13), the spall strength disappears
altogether.
Clearly, the addition of even a mild skew angle causes severe spall strength
degradation. Although the material does not fail in shear, it does experience
brittle mode damage and therefore strength reduction prior to the arrival of the
spall wave. This has the opposite effect as the residual compressive stress
generated by plasticity. However, the microscopic damage patterns as recorded
by the SEM images at the spall plane remain the same as in the shock
compression impact case at about 200 m/s impact velocity. This indicates that
the same fracture modes are present in shock compression and pressure-shear.
189
To summarize, AS800 grade silicon nitride exhibits a high HEL of 12 GPa. The
spall strength decreases under shock compressive loading in the elastic regime.
This indicates that some damage is accumulating below the HEL. However,
under elastic-plastic loading, the spall strength remains roughly constant,
suggesting a moderating trend in the elastic spall strength’s decrease with
increasing impact velocity due to inelastic shock compression. The high HEL and
moderating influence of inelastic shock compression favor AS800 silicon nitride as
a potential aircraft engine turbine blade material. Pressure-shear loading, on the
other hand causes a five times greater decrease in the spall strength over the
same elastic region compared with the shock compression experiments. This
strong effect of pressure-shear loading on spall strength must be considered a
drawback to using AS800 silicon nitride.
190
References
Achenbach, J. D. 1975. Wave Propagation in Elastic Solids. Elsevier Science Publishers.
Boslough M. B. and J. R. Asay. 1993. Basic Principles of Shock Compression. High-Pressure
Shock Compression of Solids. Springer-Verlag. 7-42.
Choi et al. 2002. Foreign Object Damage of Two Gas-Turbine Grade Silicon Nitrides by Steel
Ball Projectiles at Ambient Temperature. NASA/TM-2002-211821.
Dandekar, D. 2004. Spall Strength of Tungsten Carbide. U.S. Army Research Laboratory.
ARL-TR-3335.
Dremin A. N.; A. M. Molodets. 1990. Metal Spall and Fracture Mechanisms. Shock Compression
of Condensed Matter. Elsevier Science Publishers. 415-420.
Grady, D. E. 1988. The Spall Strength of Condensed Matter. J. Mech. Phys. Solids Vol. 36 No. 3
pp 353-384.
Mashimo, T. 1994. Shock Compression of Ceramic Materials: Yielding Property. American
Institute of Physics.
Mashimo, T. 1998. In: High-pressure shock compression of solids III. L. Davison, M. Shahinpoor,
editors New York : Springer 101-146.
Nahme, V. Hohler, A. Stilp. 1994. Determination of the Dynamic Material Properties of Shock
Loaded Silicon-Nitride. American Institute of Physics. 765-768.
Reinhart, W. D.; L. C. Chhabildas. 2002. Strength Properties of Coors AD995 Alumina in the
Shocked State. International Journal of Impact Engineering. 29 [1-10] 601-619.
191
Tables
Table 5.1: Experimental results for Hugoniot state shock velocities, US, particle velocities, uP,
elastic stress, σE, and strain, εE, and hydrodynamic stress, σH, and strain, εH. The table shows the
values determined using Equations {V.C.1} through {V.C.6}.Exp.
#
Impact
Velocity
Flyer
Type
θ US uP σE εE σH εH
SC-1 107 m/s SiN 0° 1.21 km/s 53.1 m/s 1.88 GPa 4.89E-3 N/A N/A
SC-2 208 m/s SiN 0° 1.02 km/s 100 m/s 3.56 GPa 9.22E-3 N/A N/A
SC-3 528 m/s SiN 0° 1.10 km/s 251 m/s 8.91 GPa 2.31E-2 N/A N/A
SC-4 355 m/s SiN 0° 9.93 km/s 178 m/s 6.32 GPa 1.64E-2 N/A N/A
SC-5 66.0 m/s SiN 0° 8.52 km/s 31.5 m/s 1.12 GPa 2.90E-3 N/A N/A
SC-6 546 m/s WC 0° 8.57 km/s 394 m/s N/A N/A 13.3 GPa 4.60E-2
SC-7 494 m/s WC 0° 9.14 km/s 353 m/s N/A N/A 12.2 GPa 3.86E-2
SC-8 64.9 m/s SiN 0° 1.07 km/s 31.8 m/s 1.13 GPa 2.93E-3 N/A N/A
SC-9 417 m/s WC 0° 9.60 km/s 298 m/s 10.5 GPa 2.74E-2 N/A N/A
SC-10 186 m/s WC 12° 1.15 km/s 118 m/s 4.20 GPa 1.09E-2 N/A N/A
SC-11 115 m/s WC 12° 1.13 km/s 73.2 m/s 2.60 GPa 6.74E-3 N/A N/A
SC-12 233 m/s WC 12° 1.74 km/s 152 m/s 5.38 GPa 1.39E-2 N/A N/A
SC-13 299 m/s WC 12° 9.27 km/s 195 m/s 6.93 GPa 1.80E-2 N/A N/A
Table 5.2: Mohr's Circle Stress Calculations. The spall strength as determined using Equation
{V.C.35} is contrasted with the tensile principal stress in State 4* from Equation {V.B.6}, which
is pure shear without a low impedance window. For the experiment not to fail in shear, the spall
strength must be greater than the tensile principal stress with the PMMA window, which uses the
same equation, but a different shear stress. For it to fail at the spall plane, the spall strength
must be smaller than the maximum tension in State 6.
Impact
Velocity
(m/s)
Free Surface
Velocity
(m/s)
Spall Strength
(Pa)
Principal
Tension in
State 4*, no
window
(Pa)
Principal
Tension in
State 4*,
window
(Pa)
Maximum
tension in
State 6
(Pa)
100 140.41 8.8470E+8 3.16E+08 2.10E+08 1.91E+09
150 210.61 8.4805E+8 4.73E+08 3.14E+08 2.86E+09
200 280.82 8.1140E+8 6.31E+08 4.19E+08 3.81E+09
250 351.02 7.7475E+8 7.89E+08 5.24E+08 4.76E+09
300 421.23 7.3810E+8 9.47E+08 6.29E+08 4.83E+09
350 491.43 7.0145E+8 1.10E+09 7.37E+08 6.59E+09
192
Table 5.3: Experimental results for spall stress and pull-back time. The impact velocity, skew
angle, θ, and free surface particle velocities, V4, are used to index the data.Exp.
#
Impact
Velocity
Flyer
Type
θ V4 Spall
Stress
Pull-Back
Time
SC-8 65 m/s SiN 0° 64.6 m/s 908 MPa 0.082 μs
SC-5 66 m/s SiN 0° 64.9 m/s 881 MPa 0.100 μs
SC-1 107 m/s SiN 0° 107 m/s 890 MPa 0.032 μs
SC-2 208 m/s SiN 0° 208 m/s 835 MPa 0.052 μs
SC-4 355 m/s SiN 0° 356 m/s 709 MPa 0.062 μs
SC-3 528 m/s SiN 0° 502 m/s 572 MPa 0.086 μs
SC-9 417 m/s WC 0° 599 m/s 564 MPa 0.081 μs
SC-7 494 m/s WC 0° 708 m/s 640 MPa 0.14 μs
SC-6 546 m/s WC 0° 792 m/s 622 MPa 0.13 μs
SC-11 115 m/s WC 12° 147 m/s 803 MPa 0.11 μs
SC-10 186 m/s WC 12° 239 m/s 557 MPa 0.09 μs
SC-12 233 m/s WC 12° 306 m/s 249 MPa N/A
SC-13 299 m/s WC 12° 396 m/s 0 MPa N/A
193
Figures
1 23
45
6, 6’7,7’
Figure 5.1: Calculation plots for elastic compression impact experiments. (a) Time versus
distance diagram. The longitudinal stress waves propagate outwards from the point of impact,
reflect from the free surfaces as tensile rarefaction waves, and intersect defining the spall plane.
On this diagram, the locations of the various stress states are marked. (b) Stress versus velocity
diagram. The red line indicates the tungsten carbide which (WC) exceeds its HEL. The blue line
indicates the silicon nitride. The lines show the paths between the stress states indicated in (a).
The spall strength is measured from velocities on the particle velocity axis and Equation
{V.C.16} is based on the relationships in this plot.
Figure 5.2: Calculation plots for elastic-plastic compression impact experiments. The
configuration is the same as in Figure 5.1. (a) Time versus distance diagram (b) Stress versus
velocity diagram. The HEL of silicon nitride is exceeded and the two lines used to estimate the
Hugoniot state (State 3) are depicted. Even with plastic deformation, the loading and unloading
of the silicon nitride is symmetric.
194
Time (μs)
Free
Sur
face
Par
ticle
Vel
ocity
(km
/s)
0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
SC-5, SC-8SC-1
SC-2
SC-4
SC-3
SC-9
SC-7
SC-6
Figure 5.3: Shock compression free surface particle velocity history plots. The initial rise is due to
the arrival of the compressive wave at the free surface. From the plot of SC-6, the HEL of silicon
nitride can be seen. The spall signal arrives at the free surface from the intersection of the tensile
rarefaction waves at a time determined by the thickness of the flyer and its impedance. The pull-
back time can be seen to change with free surface particle velocity, which is directly related to
impact velocity.
195
Time (μs)
Free
Sur
face
Vel
ocity
(m/s
)
0 1 2 30
50
100
150
200
250
300
350
400
SC-13
SC-12
SC-10
SC-11
Figure 5.4: Pressure shear experiment silicon nitride-PMMA interface surface particle velocity
history plots. The velocities before and after the spall signal are distinct. As the impact velocity,
and thus measured surface particle velocity, increases, the pull-back velocity decreases.
Figure 5.5: Calculation plots for compression-shear impact experiments. (a) Time versus distance
diagram. State 4* is shown. It is a state of pure shear following the reflection of the compressive
stress wave. (b) Stress versus velocity diagram. The green line indicates the PMMA window.
4*
196
Particle Velocity (km/s)
Sho
ckV
eloc
ity(k
m/s
)
0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Elastic Wave Speed 10.9km/s
Figure 5.6 The shock velocity plotted versus the Hugoniot state particle velocity. In the elastic
regime, the average shock velocity of 10.7 km/s matches with the elastic wave speed. However,
as the particle velocity increases, the shock velocity decreases.
197
Strain
Stre
ss(G
Pa)
0 0.01 0.02 0.03 0.04 0.050
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
ElasticHydrodynamic
Stress (GPa) = 386*Strain
Stress (GPa) = 149*Strain + 6.46HELStress (GPa) = 87.2*Strain + 9.29
Figure 5.7: Plotting the Hugoniot elastic and hydrodynamic stresses and strains. Above the HEL,
the hydrodynamic predictions diverge from the elastic prediction. Two correlations are offered,
one from the HEL to the highest velocity experiment (SC-6), and one connecting both
experiments above the HEL.
198
Time (μs)
Free
Sur
face
Par
ticle
Vel
ocity
(km
/s)
0.8 0.9 1 1.1 1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VFS at HEL
Figure 5.8: Rear surface velocity history plot for 546 m/s impact velocity (SC-6) experiment
showing the decrease in slope of the velocity rise which indicates the Hugoniot Elastic Limit. The
velocity at which this occurs is indicated on the plot.
199
Free Surface Particle Velocity (km/s)
Spa
llS
treng
th(M
Pa)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
100
200
300
400
500
600
700
800
900
1000Normal SiN-SiNNormal WC-SiN
(a) Spall = -0.0383*Stress+ 1E+09 (Pa)
Figure 5.9: Spall strength plotted versus free surface velocity for normal impact experiments. In
the elastic regime, there is a linear relationship between stress and free surface particle velocity.
However, with the onset of elastic-plastic deformation, the spall strength plateaus.
200
Free Surface Particle Velocity (km/s)
Spa
llS
treng
th(M
Pa)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
100
200
300
400
500
600
700
800
900
1000Shear WC-SiN
Spall =-0.1751*Stress+
1E+09(Pa)
Figure 5.10: Spall strength plotted versus free surface velocity for pressure shear impact
experiments. The lower three velocity experiments show a linear trend. However, the highest
velocity experiment (SC-13) had no pull-back following the drop in free surface particle velocity
that indicates spall (Figure 5.4). Therefore, the spall strength of experiment SC-13 was
negligible.
201
Figure 5.11: Post impact fragment images. The scale is in inches. (a) 65 m/s impact speed (SC-
8). The specimen is still contained within the specimen ring. The spall plane and star shaped
fracture are visible. (b) 107 m/s impact speed (SC-1). The fragments are smaller than those in
(a). (c) 355 m/s impact speed (SC-4). The debris is mostly powder, but a few fragments of
sufficient size to use the SEM remain.
202
Figure 5.12: SC-8, 31.8 m/s Hugoniot particle velocity. Top left: overview at magnification 250
showing brittle fracture and several 100 μm hexagonal grains and many smaller ones. Top right:
magnification 2000 showing intra-granular brittle fractures of two large grains in different
orientations. Bottom left: magnification 3500 with differing size grains and brittle fracture.
Bottom right: magnification 3500 with a hole where a large grain has been removed.
203
Figure 5.13: SC-2, 100 m/s Hugoniot particle velocity. Left, overview at 250 times magnification
showing no 100 μm grains, but some 50 μm grains. Right: 2000 times magnification showing
fracture of 50 μm smaller grains.
Figure 5.14: SC-4, 178 m/s Hugoniot particle velocity. Top left: overview at 250 X showing two
distinct regions, rough and smooth, and no visible large grains. Top right: 2000 X view showing
rough region, but no grains as large as 50 μm. Bottom left: Close up at 3500 X of large grain in
rough region. Bottom right: Close up at 3500 X showing smooth region with no large grains.
204
Figure 5.15: SC-12, 152 m/s Hugoniot particle
velocity. Top left: Magnification 250 Overview
showing rough brittle fracture, same as SC-2.
Top right, 2000 X view of ~50 μm grains and
brittle fracture surface. Bottom left: 3500 times
magnification showing fractured large grains.
205
Figure 5.16: Polished surface of specimen, pre-impact. Top left: 250 X overview, showing no
distinguishing features. Top right: magnification 2000 showing polished grain faces and a hole
where a ~10 μm grain was removed by the polishing process. Bottom left: 3500 times
magnification showing polished ~ 20 μm grain. Bottom right: 3500 X showing grain hole on the
order of 10 μm.
206
Chapter VI – The Effect of Shock Re-Shock, Shock Unloading,and Stress Reverberations on the Behavior of Silicon Nitride
The propagation of a shock wave through a material can cause inelastic
deformation that directly affects material properties such as the dynamic yield
strength. The material subjected to a subsequent shock or series of shocks may
behave differently from unshocked material. In the present study plate impact
shock compression experiments are employed using a dual layer flyer plate to
determine the shock-release and shock-re-shock response of silicon nitride. By
studying the off-Hugoniot stress states generated in experiments of this type, the
material strength behavior of the shocked material can be examined [Asay and
Chhabildas, 1981; Reinhart and Chhabildas, 2003]. The determination of the
dynamic yield strength depends on inelasticity in both the re-shock and release
experiments.
A second set of experiments is conducted using a thick flyer of silicon nitride and
a thin target of tungsten carbide. Multiple unloading states are generated in the
flyer as stress waves reverberate in the target. Using equations developed by
Hall et al. [Preliminary], the progressive unloading of stress in the silicon nitride
flyer is examined. These experiments also provide observations of the material’s
behavior in the shocked state..
The theoretical background for these experiments and their applications to
specific materials are reviewed in Section A. The assumptions of both elastic and
inelastic deformation underlie the specific equations that describe the material
behavior in these experiments. Also, equations that use the inelasticity of the
material to determine the dynamic yield stress are provided. In Section B, the
design of experiments is presented. The two types of experiments discussed in
Section A are modified for the silicon nitride experiments in this study.
Following this, in Section C, the VISAR results are analyzed and the material
response is discussed. The experimental results are restricted by the maximum
impact velocity achievable in a single stage gas gun. Within this range of impact
207
velocities, the analysis indicates a strong agreement between the particle velocity
and stress state predictions of the equations in Section A and the experimental
free surface particle velocities presented in Section C.
208
A. Theory and Computations for Multiple Shock Experiments
Observing the effects of an additional shock to an already shocked material can
provide insights into inelastic damage within the material generated by the first
shock. A dual layer flyer can be utilized to provide a shock re-shock or a shock-
unload condition. This method has been detailed in Section A.1. The basis for
this method is explained in two papers -- Asay and Chhabildas [1981] and
Reinhart and Chhabildas [2003]. Additionally, with a shock reverberation plate
impact experimental configuration, a series of unloading waves can be designed to
propagate through the flyer. From this type of experiment, the release behavior
of the material can be studied [Dandekar et al., 2003]. This method is explained
more mathematically in the paper by Hall et al. [Preliminary].
A.1. Shock Re-shock and Shock Unload Experiments
A method for studying the dynamic behavior of a material involves shocking the
material twice-- once to load it to a predetermined shock state and then again
shock it either to a higher or lower stress state. From these experiments the
dynamic material strength of a material in the shocked state can be evaluated
without recourse to the knowledge of the material’s dynamic hydrostat. This is
accomplished by employing flyers with two layers that have differing impedances.
The result of using a higher impedance backing material is that the second wave
has a higher compressive stress level, which causes re-shock in the material. A
lower impedance backing plate reduces the stress, causing unloading after the
initial shock. By employing matching impact velocities in both of these
experiments the dynamic yield strength as well as the shear stress in the shocked
stste can be estimated [Reinhart and Chhabildas, 2003][Asay and Chhabildas,
1981].
A study on alumina that was performed by Reinhart and Chhabildas [2003]
demonstrates the dual flyer technique. Through the use of a pair of experiments,
one shock-reshock and one shock-unload, the dynamic yield strength of the
alumina was determined. Aluminum oxide (Al2O3) with a density of 3.89 g/cc
209
and with longitudinal and shear wave speeds of 10.56 km/s and 6.24 km/s
respectively was employed. A two stage light gas gun with a 28 mm bore was
employed to accelerate the specimens to about 8 km/s.
These experiments employed a flyer backed with either a higher or lower
impedance material, and a window backing the specimen. For the window,
lithium fluoride was used. Polymethylpentene or TPX was used as the low
impedance flyer backing material for the shock release experiments. Tantalum
and copper were used to back the flyer in the shock re-shock experiments. The
resulting shock waves created off-Hugoniot conditions which are used to calculate
the dynamic hydrostat, the shear stress in the shocked state, and the maximum
and the minimum stress levels, and the dynamic material strength [Reinhart and
Chhabildas, 2003].
From the knowledge of te impact velocity and the free surface velocity versus
time histories, the Hugoniot stress versus strain states can be examined. Using
conservation laws, the stress and strain Hugoniot states are given by
( )H HEL O SP H HELU u uσ σ ρ= + − , { .A.1}
( )H HEL L H HEL SPu C u u Uε = + − . { .A.2}
In Equations { .A.1} and { .A.2}, the Hugoniot stress, strain, and particle
velocity are represented by, σH, εH, and, uH, respectively. The stress and velocity
at the HEL are, σHEL, and uHEL. CL represents the Lagrangian wave speed and,
USP, the plastic shock velocity. These values are determined from the initial
velocity jump corresponding to the first shock. One important note about these
equations is that the stress in the initial Hugoniot state exceeds the Hugoniot
Elastic Limit of the material [Reinhart and Chhabildas, 2003].
Once the second shock has propagated, the material either unloads or loads
further. At the point where the elastic wave gives way to the bulk wave, the
210
second state can be defined for each of these experiments. By using incremental
forms of the conservation equations, the final stress, σ, and strain, ε, can be
found as
O LC uσ ρ= Δ∑ , { .A.3}
Lu Cε = Δ∑ . { .A.4}
At these points, the Lagrangian wave speed, CL, changes with the changing
particle velocity, uΔ [Reinhart and Chhabildas, 2003]. The density is, ρO. Before
the second shock, the Lagrangian wave speed is found by
( ) ( )T
L fsfs
C uT t u
δ=⎡ ⎤+ Δ⎣ ⎦
. { .A.5}
In Equation { .A.5}, T + Δt, is the time from the velocity profile, corresponding
to each, ufs, which are the free surface velocities. The target thickness is, δT.
After the second shock, the Lagrangian wave speed is determined by
( ) ( ) 2 /T
L fsfs F eff
C uT t u U
δδ
=⎡ ⎤+ Δ −⎣ ⎦
. { .A.6}
In Equation { .A.6}, δF, is the flyer thickness and, Ueff, is the effective velocity of
the second shock wave.
The shear stresses generated in the material due to the shocks can be calculated
from the longitudinal stresses in the material based on elementary theory [Asay
and Chhabildas, 1981]. The planar impact experiments are in a state of uniaxial
strain, and the stress can be written as
2X Yσ σ τ− = . { .A.7}
211
Here, σX, represents the longitudinal stress, σY, the lateral stress, and, τ, the
resolved shear stress. Also, the mean stress, σ , for the shocked state can be
written as
( )1 23 X Yσ σ σ= + . { .A.8}
From Equations { .A.7} and { .A.8}, the longitudinal stress, Xσ , can be written
as a function of the mean and shear stresses
43Xσ σ τ= + . { .A.9}
Differentiating Equation { .A.9} with respect to, ε, the engineering strain
43
Xd d dd d dσ σ τε ε ε
= + . { .A.10}
The engineering strain is also related to the ratio of the initial density, ρO, and
the shocked density, ρ [Asay and Chhabildas, 1981]
1 Oε ρ ρ= − . { .A.11}
At the yield point during the reshock or unload, the shear stress, τ, is maximized
and is equal to, τC, the critical shear strength. After the initial stress pulse in the
experiment, the material has a stress level, σO, which is beyond the HEL. This
stress has a factor of, (4/3) τO that separates it from the mean stress, σ
43O Oσ σ τ= + . { .A.12}
In a material that follows an ideal elastic-plastic model, the shear stress, τO, at
the end of the initial stress pulse, which is past the yielding point is equal to, τC,
212
but in actuality these two shear stresses may be different [Reinhart and
Chhabildas, 2003].
During the second shock, the material unloads elastically at first and yields again
at Point 1 or reloads in the same process to Point 2 (Figure 6.1), where inelastic
behavior begins. In either case, once the stresses and strains from the initial
impact up to Points 1 and 2 have been determined numerically from the velocity
plot, Equation { .A.9} can be integrated also numerically over the rising stress
pulse from the initial Hugoniot state to the final state where yielding occurs.
The resulting equations relating the maximum shear strength, τC, to the shear
stress, τO, are as follows [Asay and Chhabildas, 1981]. For the unloading
experiment
( )1
2 234
O
C O O L BC C dε
ε
τ τ ρ ε−+ = −∫ . { .A.13}
For the re-loading experiment
( )2
2 234
O
C O O L BC C dε
ε
τ τ ρ ε− = −∫ . { .A.14}
In Equations { .A.13} and { .A.14}, the Lagrangian wave velocity, CL, varies
instantaneously and is related to the stress and the strain as:
2 1 XL
O
dCdσ
ρ ε= . { .A.15}
This wave velocity is measured experimentally. The bulk wave velocity, CB, is
assumed to be constant and represents the wave velocity after the transition to
plastic behavior [Asay and Chhabildas, 1981]. It can be written as:
2 1B
O
dCdσ
ρ ε= . { .A.16}
213
The factor, τC + τO, is detrmined numerically using shock velocity information
from the release experiment. From the reloading experiment, τC – τO, is
determined. The points can also be determined graphically on the stress versus
strain plot (Figure 6.1) [Reinhart and Chhabildas, 2003]. These two quantities
are then related to the maximum and minimum longitudinal stress, maxσ , and,
minσ , as follows
( )min43O C Oσ σ τ τ= − + , { .A.17}
and,
( )max43O C Oσ σ τ τ= + − . { .A.18}
These values can be related to, YC, the critical dynamic yield strength of the
material:
( )max min2C CY τ σ σ= = − . { .A.19}
The series of experiments were all performed above the HEL of the alumina,
which was found to be between 6.7 and 7.9 GPa. Through multiple experiments,
the dynamic yield strength was determined to increase with increasing pressure
[Reinhart and Chhabildas, 2003].
A.2. Reverberation ExperimentsThe second method for determining the state of the shocked material employs the
use of successive unloading. This method uses a thick plate of test material as
the flyer and a very thin plate of a material with known properties including
higher impedance as the target. The reverberations of the stress in the target
causes a series of unloading waves to propagate in the specimen. The calculation
method by Hall et al., [Preliminary] examines the successive unloading states.
214
Dandekar examined the shock and release of glass-fiber-reinforced polyester
(GRP) using this plate reverberation method [Dandekar et. al., 2003]. The
material response of GRP was characterized by means of VISAR free surface
velocity measurements and by x-cut quartz strain gages. These measured the
free surface particle velocity and the internal flyer stress respectively. In this
experiment series, copper and tantalum, which have higher impedances than the
GRP, were employed as the target material. The experiments covered the range
of Hugoniot state stresses between 1.3 and 20.3 GPa.
The initial compressive shock wave causes the flyer material and the target to
deform, reaching the Hugoniot stress state. However, because the target is much
thinner than the flyer, by the time the wave has propagated to the rear surface of
the specimen, the waves have reflected off both surfaces of the target several
times (Figure 6.2). Each reverberation from the impacted surface results in
partial wave propagation causing a partial unloading in the specimen. The
weaker shock in the target continues to reflect. Each reflection from the free
surface causes a drop in the stress in the specimen and the target and a
corresponding rise in the free surface particle velocity. This velocity rise is
measured by the VISAR. As a result of using the shock relations, the stress
states in the specimen can be determined from the free surface particle velocity
[Dandekar et al., 2003].
The Hugoniot state for the specimen can be found from the Rankine-Hugoniot
jump conditions [Asay and Chhabildas, 1993]. The stress, σH, and strain, εH, at
this state are given by
H OUuσ ρ= , { .A.20}
H u Uε = . { .A.21}
215
In Equations { .A.20} and { .A.21} the initial density is, ρO, the shock velocity is,
U, and the particle velocity is, u. Often, a linear relationship between the shock
velocity and the particle velocity (Equation of State) is used, i.e.
OU C Su= + . { .A.22}
In Equation { .A.22}, CO, and, S, are parameters. By combining equations
{ .A.20} and { .A.21} for stress and strain, with the linear relationship between,
U, and, u, Equation { .A.22}: the stress as a function of the strain can be written
as follows [Dandekar et al., 2003]
( )
2
21O O H
HH
CS
ρ εσε
=−
. { .A.23}
This equation was employed to plot versus the stress versus strain data for
Dandekar et al.’s [2003] experiments.
The following analysis is applicable if there is equivalent loading and release
behavior in the flyer and target [Hall et al., Preliminary]. In determining the
stress and strain in the flyer at the Hugoniot state in the case where the
impedances of the flyer and target differ, the stress in the target, σT, is found
from Equation { .A.20},
( )2 / 2T OT TU uσ ρ= . { .A.24}
The unshocked density of the target is, ρOT. The particle velocity of the target,
uT, is one half of the recorded velocity on the free surface, u2, after the first stress
wave has passed,
2 2Tu u= . { .A.25}
216
This stress state on the free surface is State 2. The shock velocity in the target
is, UT. The shock velocity of the target plate is determined from Equation
{ .A.22}. The stress in the flyer and the target are identical, because the stress is
continuous over the plane of impact. Thus, the stress in the flyer, σS, is also
given by
( )2 / 2S T OT TU uσ σ ρ= = . { .A.26}
The subscript, S, refers to the specimen being the flyer, not the target, which is
indicated with the subscript, T. The particle velocity of the flyer is the difference
between the flyer impact velocity, VI, and the target’s velocity at the Hugoniot
state, uT, which was found in Equation { .A.25}:
S I Tu V u= − . { .A.27}
The relation in Equation { .A.27} enables the determination of the shock velocity
of the flyer, US, from Equation { .A.20}:
( )S S OS SU uσ ρ= . { .A.28}
In Equation { .A.28}, the initial density of the flyer is, ρOS, and the particle
velocity is, uS. The strain in the specimen, εS, can be found by using the shock
velocity from Equation { .A.28} and Equation { .A.21}:
S S Su Uε = . { .A.29}
Equations { .A.24} through { .A.29} determine the Hugoniot state stress strain,
shock velocity, and particle velocity in the flyer and the target [Hall et al.,
Preliminary]. For each successive release state, incremental forms of these
equations are used to determine the stress, shock velocity, and strain in the flyer.
The change in the particle velocity on the free surface of the target, 2X Xu u+ − , is
employed to determine the change in the particle velocity in the specimen, SuΔ .
217
Here, X, represents the states on the free surface. For example, State 2 is, X=2,
and State 4 is, X=4.
First, from the stress velocity diagram in Hall’s paper [Preliminary], the velocity
difference between States 1 and 3 is half the velocity difference of States 2 and 4,
assuming equivalent shock and release behavior as above. This leads to the
following expression:
( ) ( )3 1 4 23 112Tu u u u u−Δ = − = − . { .A.30}
Generalizing this expression for any two points:
( )212T X Xu u u+Δ = − . { .A.31}
The particle velocity jump in the flyer is the same as that for the target [Hall et.
al., Preliminary], so the relation can be rewritten:
( )212S T X Xu u u u+Δ = Δ = − . { .A.32}
The stress increment is found by using the incremental form of Equation { .A.26}
:
S T OT T TC uσ σ ρΔ = Δ = Δ . { .A.33}
This can be rewritten in terms of the free surface particle velocity jump from
Equation { .A.31}:
( )212S OT T X XC u uσ ρ +Δ = − . { .A.34}
218
This is the equation used in [Hall et. al., Preliminary]. However, during the
course of the experiments in the current study, a different relationship between
the stress increment in the flyer and the free surface velocities was derived:
( ) ( ) ( ) ( )( 2) ( 3) ( ) ( 1)2σ ρ ρ− − −−Δ = − − −S OS T X T X OS T X T XT X T XC u u C u u . { .A.35}
This modification is computed directly from the stress-velocity diagram for one of
the reverberation experiments on silicon nitride (Figure 6.3). The quantity, CT,
represents the average Lagrangian wave speed in the target during unloading,
2TC h t= Δ . { .A.36}
The target thickness is, h, and the time it takes the wave to transit this distance
is, tΔ . The thickness is measured prior to the experiment, and the time is
determined from the distance between the two velocity jump midpoints on the
free surface time history [Hall et. al., Preliminary].
With the changes in stress and particle velocity, the longitudinal unloading
average wave velocity, CS, in the specimen can be determined by employing the
incremental form of Equation { .A.28} [Hall et. al., Preliminary], even though the
release velocity is not technically a shock [Dandekar et al., 2003]:
( )S S OS SC uσ ρ= Δ Δ . { .A.37}
Then, the strain increment in the flyer, SεΔ , can be calculated from Equation
{ .A.37} and the incremental form of equation { .A.29} [Hall et. al., Preliminary]:
S S Su CεΔ = Δ . { .A.38}
Finally, the increments in stress and strain are combined with the stress and
strain levels at the previous state ( ,X Xσ ε ) to determine the absolute stress and
strain for each new state ( 2 2,X Xσ ε+ + ):
219
2X Xσ σ σ+ = − Δ , { .A.39}
2X Xε ε ε+ = −Δ . { .A.40}
The stress versus strain and the stress versus particle velocity relationships for
these experiments based on the method just described were presented in Hall et
al.’s [Preliminary]. Also in this paper, a plane acceleration method was employed
to determine the same relations. That method used Isaac Newton’s 2nd Law of
Motion. Both methods agreed well in experiments with Hugoniot pressures up to
10 kbars prompting the suggestion that permanent compacting of the flyer was
occurring in those experiments [Hall et al., Preliminary].
The dynamic yield strength in the material can also be estimated from Hall et
al.’s method when the HEL of the specimen material is exceeded. If the material
is considered elastic-perfectly plastic, then the shear stress at the Hugoniot state,
τH, is equivalent to the critical shear stress, τC. This results in the dynamic yield
stress upon unloading being twice,τC, at the point where the deformation along
the unloading slope changes from elastic to elastic-plastic [Reinhart and
Chhabildas, 2003]. The graphical method described in this paper can be applied
using the curve connecting the unloading states on the stress-strain plot. Or,
numerically, the method from Asay and Chhabildas [1981] can be used. At the
point where the deformation changes from elastic to elastic-plastic, the shear
stress in the specimen, τCS, is measured by modifying Equation { .A.13}. Since,
the dynamic yield stress, YCS, is twice the critical shear stress:
( )1
2 2324
S
OS
CS CS OS S BS SY C C dε
ε
τ ρ ε−= = −∫ . { .A.41}
In Equation { .A.41} the elastic-plastic (bulk) wave speed in the flyer is given as,
CBS. Integrating Equation { .A.41} numerically results in an expression for the
220
dynamic yield stress of the material. This can be compared with the other
method of calculating dynamic yield stress outlined in Section A.1.
221
B. Design of Shock Re-shock, Shock Unload, and ReverberationExperiments
The experiments in this study were performed in the highest impact velocity
range, i.e. 500 m/s, of the 3.25 inch diameter single stage gas gun at CWRU.
The specifications of this gas gun and of the experimental procedures were
discussed in Chapter IV. The high HEL of silicon nitride, determined in Chapter
V to be 12 GPa, requires a relatively high impact velocity in order to study
inelasticity in Si3N4. The specific design of each of these experiments are
discussed below.
As mentioned in Section A.1 dual shocks are created by using a dual component
flyer. For the shock-reshock experiment, SR-1, the flyer was composed of silicon
nitride and backed by a higher shock impedance tungsten carbide plate (Figure
6.4). The HEL of the silicon nitride was exceeded in this experiment. This is
due to the higher impedance of the tungsten carbide. Silica glass, which has an
impedance of 12.8 MPa/(m/s) is used as a window behind the Si3N4 specimen. in
order to prevent spallation of Si3N4 during the experiment. The unloading
experiment, SU-1 was backed with a Al 6061 plate, which has a relatively lower
shock impedance of 16.6 MPa/(m/s) when compared to Si3N4, to create
unloading. Again, a silica glass plate was used as a window material behind the
Si3N4 target. In this way, the experimental configuration requires two plates for
the flyer and two for the target (Figure 6.4).
The reverberation experiments, patterned after those discussed in Section A.2,
involve a thick flyer of silicon nitride impacting a thin plate of tungsten carbide.
In experiment RB-1 the flyer was 4.5 mm thick while the target is only 1 mm
thick (Figure 6.5). This configuration allows two reverberations in the WC
target before the free surface boundary wave from the flyer arrives at the target’s
free surface. The second experiment, RB-2, (Figure 6.2) involved a 5.5 mm thick
flyer, which allowed for three reverberations to occur before the arrival of the
boundary wave.
222
For these four experiments, predictions were determined by using elastic and
elastic-plastic line equations, such as those used in Chapter V to approximate
one-dimensional hyperbolic wave theory. These equations and their results are
discussed in Section C. As in Chapter V, the VISAR interferometer system was
employed to measure the in-material free surface particle velocity histories at the
interface of the target and the window to provide input for the equations. The
results are compared to the elastic and elastic-plastic predictions from the
equations in Section A.
223
C. Results and Analyses of Multiple Shock Experiments
The four experiments were carried out according to the design in Section B.
From the VISAR measurements of particle velocities, the internal stress
behaviors are determined and related to the elastic-plastic predictions. The
particle velocities indicate behavior which is close to that predicted by the
elastic-plastic estimates in Section A and by the line equations derived in this
section. The VISAR measurements and the internal stress levels are shown in
Tables 6.1 and 6.2. Any changes from the behavior of unshocked material are
documented in Sections C.1 and C.2.
C.1. Shock Re-shock and Shock Unload Experiments
In the dual shock experiments, the materials were subjected to either a shock
followed by a re-shock from a higher impedance material or an unload from a
lower impedance material. This resulted in an initial Hugoniot state followed by
a velocity rise or drop to a second level. Two experiments were conducted with
the impact velocity of 479 ± 4 m/s, one shock re-shock and one shock unload. At
this velocity, the HEL of silicon nitride is not exceeded in the first shock. In the
experiment with the re-shock, however the HEL is exceeded during the second
shock. In the unload experiment, the HEL is not exceeded. In Section C.1.1, the
elastic and elastic-plastic computations for these experiments are given, and then
in Section C.1.2, the free surface velocity profiles for the re-shock and unload
experiments are discussed.
C.1.1. Computations for Shock Re-Shock and Shock Unload ExperimentsThe flyer is accelerated to an initial velocity and has zero stress, State 1 (Figure
6.6). The initial shock comes from the impact of the two silicon nitride plates,
which in this study are 4.5 mm thick for the target and 3 mm thick for the flyer.
Before this, the target is in a state of zero stress and zero particle velocity, known
as State 2. Because both the flyer and the specimen are silicon nitride, the
impact is initially symmetric and produces a particle velocity, V3, in State 3 that
is one-half the free surface particle velocity, VFS.
224
= 123 FSV V . { .C.1}
The stress for this state is calculated by characteristic line equations that
approximate 1-D hyperbolic wave theory, the use of which is described in
Chapter V. Because the HEL is not exceeded, the stress in State 3 is found,
σ =3 3TZ V . { .C.2}
Equation { .C.2} contains the shock impedance of the silicon nitride, ZT.
Because a second flyer material is employed with a different impedance, a second
stress pulse is generated from the interaction of the wave with the boundary
between the two pieces. This stress lasts until the reflection of the pulse from
the free surface of the backing plate. The stress state caused by this second
shock is represented by State 4.
In the loading experiment, SR-1, State 4 has greater stress and particle velocity
than State 3 (Figure 6.6). Since the HEL is exceeded by this stress level, elastic-
plastic waves are generated and the stresses are obtained by employing the
equations of state of the tungsten carbide and the silicon nitride. The tungsten
carbide HEL, HEL WCσ − , is only 7.2 GPa, whereas the HEL of silicon nitride is 12
GPa, HEL SiNσ − . The velocity at the HEL, HEL WCV − , of the tungsten carbide backing
plate is found:
1 /σ− −= −HEL WC HEL WC BV V Z . { .C.3}
The shock impedance here, ZB, is that of the backing plate, which for tungsten
carbide is 106 MPa/(m/s). Using the bulk modulus of tungsten carbide, the
elastic-plastic shock impedance, ZBP, can be approximated as 76.6 MPa/(m/s).
The solution to Equation { .C.3} is 407 m/s. From this, a line can be drawn
from the HEL stress and velocity of the backing plate to State 4 (Figure 6.6).
This line has equation:
225
( )σ σ− −= − +4 4BP HEL WC HEL WCZ V V . { .C.4}
Note that, the HEL stress of silicon nitride is 12 GPa and the HEL velocity is the
HEL stress divided by the elastic impedance, ZT, which is 35.5 MPa/(m/s).
σ− −=HEL SiN HEL SiN TV Z . { .C.5}
The HEL velocity is thus 338 m/s. Using the elastic-plastic wave speed of silicon
nitride, 8.27 km/s, the plastic impedance, ZTP, can be approximated as 27.1
MPa/(m/s). Because the equation of state is unknown, this estimate is used as
the slope:
( )4 4TP HEL SiN HEL SiNZ V Vσ σ− −= − + . { .C.6}
Knowing this, Equations { .C.4} and { .C.6} can be rearranged to provide
estimates of the particle velocity of State 4,
σ σ− − − −+ + −=
+4BP HEL WC HEL WC TP HEL SiN HEL SiN
TP BP
Z V Z VVZ Z
. { .C.7}
The stress at State 4 is then found using either Equation { .C.4} or Equation
{ .C.6}. The stress and particle velocity equations for State 4 are used to
determine further relationships between the measured velocity and the internal
stress behavior.
For the shock-unloading experiment, SU-1, the calculation of State 4 is simpler
(Figure 6.7). The material remains elastic, so the elastic wave speeds and
impedances suffice for the calculations. Thus, State 4 can be found as the
intersection of the lines on the stress-velocity diagram connecting States 2 and 3
the line from State 1 that has the negative of the impedance of the aluminum
226
backing plate, ZB, as its slope. The stress can be found from the following
equations:
( )4 4 3 3TZ V Vσ σ= − + , { .C.8}
( )4 4 1BZ V Vσ = − − . { .C.9}
Rearranging Equations { .C.8} and { .C.9}, the stress and particle velocity at
State 4 are found:
( ) ( )4 1 3 3B T B TV Z V Z V Z Zσ= + − + , { .C.10}
( ) ( )4 1 3 3 3 3T B T B T TZ Z V Z V Z Z Z Vσ σ σ= + − + − + . { .C.11}
The silica glass window material causes a state of compressive stress at the back
of the specimen following the passage of the initial wave (State 5). This is
similar to the spall pressure shear experiments, where the window was used to
prevent failure due to shear stress. In this case, the addition of compressive
stress prevents spallation in State 6. To find the stress at State 6, first the stress
and particle velocity for State 5 need to be found. Using the lines in Figure 6.7,
from States 2 and 3, State 5 is generated:
5 5WZ Vσ = , { .C.12}
( )5 5 3 3TZ V Vσ σ= − − + . { .C.13}
In Equation { .C.12}, ZW, represents the window impedance. Rearranging
Equations { .C.12} and { .C.13} leads to:
( ) ( )5 32 W TV Z Zσ= + . { .C.14}
( ) ( )5 32 W W TZ Z Zσ σ= + . { .C.15}
227
The in-material particle velocity at the interface between the window and the
specimen in State 5 is the first measured state. Using this velocity enables the
determination of the state of stress in State 3 by reconfiguring Equation { .C.14}:
( )3 5 2W TV Z Zσ = + . { .C.16}
The state of stress when the two rarefaction waves meet is given by State 6 in
both experiments (Figures 6.6, 6.7). The equations for the determination of the
stress and particle velocity of this state are the same for both cases. Obviously,
in the shock-unload experiment, the HEL is not exceeded, so the unloading is
elastic. In the shock-re-shock experiment, the unloading is also elastic. The
equations to proceed from State 4 to State 6 and State 5 to State 6 are as
follows:
( )6 6 4 4TZ V Vσ σ= − − + . { .C.17}
( )6 6 5 5TZ V Vσ σ= − + . { .C.18}
By rearranging Equations { .C.17} and { .C.18}, the stress and particle velocity
can be found for State 6:
( ) ( )6 5 5 4 4 2T T TV Z V Z V Zσ σ= − + + . { .C.19}
( ) ( )6 5 5 4 43 2T TZ V Z Vσ σ σ= − + + + . { .C.20}
State 7 is a measured state on the interface between the specimen and the
window. Its stress and particle velocity can be determined from States 5 and 6:
7 7WZ Vσ = . { .C.21}
( )7 7 6 6TZ V Vσ σ= − − + . { .C.22}
228
Because the particle velocity in State 7 is directly measured, the stress of State 6
can be found by combining equations { .C.18}, { .C.21}, and { .C.22}:
( ) ( )6 7 5W T T WZ Z V Z Z Vσ = + + − . { .C.23}
Thus, the elastic and elastic-plastic predictions for dual shock experiments are
established. The results of these experiments are detailed in Section C.1.2.
C.1.2. Observations and Analysis of Shock Re-shock and Shock UnloadExperiments
The shock re-shock and shock unload experiments conformed well to the elastic
and elastic-plastic predictions. This is expected because the HEL of the silicon
nitride is barely exceeded during the re-shock. The measured velocity profiles
show this agreement. Both Experiments SR-1 and SU-1 are discussed in this
section. Some inelasticity was observed in Experiment SR-1, while Experiment
SU-1 conformed entirely to the elastic prediction.
The shock re-shock experiment, SR-1, shows good agreement with the elastic
prediction in the Hugoniot state (Figure 6.8). The impact speed is 475 m/s. The
Hugoniot state velocity is around 358 m/s, while the elastic value is 348 m/s,
which is only 3% away. The material begins its increase to the off-Hugoniot
state at 0.55 microseconds. Once the off-Hugoniot state is reached, it remains at
about 510 m/s, which is within 1% of the elastic-plastic prediction of 506 m/s
until boundary waves arrive at about 1 microsecond. Thus, the second shocked
state also conforms to the predictions.
However, during the re-loading to the second shocked state, a slope decrease is
seen at 467 m/s. This is indicative of the HEL limit being exceeded.
An estimate for the HEL velocity as measured on the surface between the flyer
and the window can be obtained from Figure 6.8. This is found by the following
equation:
229
σ −− =
+2 HEl SiN
HEL measuredW SiN
VZ Z
. { .C.24}
This value is 501 m/s. Thus the initial shock, although elastic and below the
HEL, has reduced the velocity where the onset of inelastic behavior occurs. The
corresponding HEL stress determined from the second shock is found by reversing
Formula { .C.24}:
( )2
HEL measured W SiNHEL SiN
V Z Zσ −
−
+= . { .C.25}
The resulting HEL stress is 11.3 GPa, which is reduced by 7% from the un-
shocked value of 12 GPa. This suggests that the material is permanently
deformed and weakened by the initial compressive pulse, but only slightly.
The shock unload experiment, SU-1, has an impact velocity of 483 m/s. The free
surface particle velocity ascends initially to the Hugoniot state sharply to 372
m/s, which is within 5% of the elastic-plastic prediction of 355 m/s (Figure 6.8).
The velocity remains constant at this level until the unloading wave arrives at
0.53 microseconds, at which time it drops to 230 m/s, which is about 1% away
from 233 m/s, which is the predicted value. This time, the velocity drop is quite
sharp, indicating elastic behavior. Again, the velocity remains constant until the
arrival of the boundary wave at 1 microsecond. Thus, the times and magnitudes
of the rises and drops in Experiment SU-1 correspond well to the predictions.
Using the methods of Asay, Chhabildas, and Reinhart [Asay and Chhabildas,
1981; Reinhart and Chhabildas, 2003], the elastic-plastic deformation of the
material can be examined. Equations { .A.3} through { .A.6} were used to
numerically evaluate the stress and strain from the time and free surface particle
velocity history. The resulting stress vs. strain plot shows that the material
behavior is elastic under unloading (Figure 6.9). As can be seen, the off-Hugoniot
stress and strain caused by the reloading in Experiment SR-1 follows the
230
hydrodynamic Hugoniot curve. This confirms that little permanent deformation
has occurred. Also, the residual shear stress is computed using Equations
{ .A.13} through { .A.19} and numerically integrating. The resulting residual
shear stress is 1.21 GPa. This confirms that little permanent deformation has
occurred. The unloading in Experiment SU-1 proceeds in a linear fashion, which
indicates elastic unloading.
In summary, these impacts do not see large changes in the material behavior or
the HEL due to in a material deformed by dual shocks. No visible behavior
changes are noted in Figure 6.9. However, because the HEL is barely exceeded,
no large variation is expected from the elastic behavior. The only indication of
damage is that the HEL as calculated using the second shock wave in Experiment
SR-1 is 6% lower than that calculated in Chapter V.
C.2. Shock Reverberation Experiments
The reverberation experiments have multiple shocks that unload the material in
increments. The impact velocities of experiments RB-1 and RB-2 were also
around 500 m/s. In this case, because of the tungsten carbide target plates the
HEL of the silicon nitride is exceeded at this impact velocity. Elastic and
linearly elastic-plastic predictions are used to estimate the behavior of the
material. These are discussed in the Section C.2.1 below. Then, these values are
compared with the experimental VISAR free surface velocity data in Section
C.2.2. The close correlation between the estimates and the actual velocities
indicate that the material behavior is well predicted by the equations in Sections
A.2 and C.2.1.
C.2.1. Computations for Shock Reverberation Experiments
In the reverberation experiments, the initial loading is expected to exceed the
HEL of both the silicon nitride and the tungsten carbide. The unloading from
this state is also elastic-plastic. However, the reverberations are all elastic. The
231
stress velocity and time distance diagrams for the target are used to determine
the internal stress states from the free surface velocities (Figure 6.3).
First, the initial states of the flyer and target are specified. For the flyer, State
A has velocity, VI, which is the initial impact velocity, and zero stress. The
target has State B which is a state of zero velocity and stress (Figure 6.3). The
impact of the flyer and target produces State 1, which occurs in the plastic
regimes of both materials. As above, the tungsten carbide HEL, HEL WCσ − , is 7.2
GPa and the HEL of silicon nitride, HEL SiNσ − , is 12 GPa. The velocity of the flyer
at the HEL of silicon nitride, HEL SiNV − , is found:
/σ− −= −HEL SiN I HEL SiN SiNV V Z . { .C.26}
In Equation { .C.26}, ZSiN, represents the elastic impedance of silicon nitride. For
the tungsten carbide target, the velocity is found,
/σ− −=HEL WC HEL WC WCV Z . { .C.27}
In Equation { .C.27}, ZWC, is the elastic impedance of tungsten carbide. From
the HEL points of both materials, the elastic-plastic assumption of linear
behavior is used. Thus, the two equations that determine State 1 are:
( )σ σ− − −= − +1 1WC P HEL WC HEL WCZ V V , { .C.28}
and,
( )σ σ− − −= − +1 1SiN P HEL SiN HEL SiNZ V V . { .C.29}
In Equations { .C.28} and { .C.29} the additional subscript, –P, represents
estimates for the plastic impedance from the elastic-plastic wave velocities of the
silicon nitride and tungsten carbide. Combining these equations to get the
velocity in State 1:
232
σ σ− − − − − −
− −
+ + −=
+1SiN P HEL SiN WC P HEL WC HEL SiN HEL WC
WC P SiN P
Z V Z VVZ Z
. { .C.30}
The stress at State 1 in the target is then found by substituting equation
{ .C.30} into either equation { .C.28} or { .C.29}.
From this point unloading in the target proceeds in an elastic-plastic fashion.
Thus, State 2 has zero stress and exactly double the velocity of State 1 (Figure
6.3).
=2 12V V . { .C.31}
State 2 is measured by the VISAR as the first velocity rise on the free surface.
All of the reverberations are assumed to occur elastically, since they are below
the HEL stress of either material. State 3 is the intersection of the line defined
by the elastic equation from State A to the HEL of silicon nitride (Figure 6.3):
( )σ = −3 3SiN IZ V V , { .C.32}
and a line rising from State 2 in the tungsten carbide, with slope, ZWC:
( )σ = −3 3 2WCZ V V . { .C.33}
Equations { .C.32} and { .C.33} are then combined to determine the particle
velocity of State 3:
+=
+2
3SiN I WC
WC SiN
Z V Z VVZ Z
. { .C.34}
233
State 4, which corresponds to the first reverberation on the free surface of the
target, is another state of zero stress (Figure 6.3). Its velocity can be found
simply by connecting a line with slope, -ZWC, from State 3 to the zero stress axis:
( ) σ= − − +4 3 30 WCZ V V . { .C.35}
Rearranging Equation { .C.35}:
σ= +4 3 3 / WCV V Z . { .C.36}
The second and third reverberation free surface velocities and internal stresses in
the target can be determined by successively applying Equations { .C.32}
through { .C.36} to find States 5, 6, 7 and 8 (Figure 6.3). These formulae enable
the determination of the elastic and elastic-plastic prediction of the reverberation
experiments.
C.2.2. Observations and Analysis of the Shock Reverberation Experiments
The impact velocities were 471 m/s and 511 m/s respectively for experiments
RB-1 and RB-2. Other experimental values are located in Table 6.2, including
the free surface particle velocities and internal stresses. First, the free surface
velocity results are examined and the relationship between them and the elastic-
plastic predictions as detailed in Section C.2.1 are examined. The analysis
discussed in Section A.2 is then used to determine the material stress vs. strain
behavior.
Experiment RB-1 has an impact velocity of 471 m/s. The free surface particle
velocity shows an initial jump to 240 m/s, which is the first shocked state. After
0.21 microseconds, the velocity rises to 247 m/s (Figure 6.10). This is off of the
elastic-plastic prediction of 257 m/s for the Hugoniot by 4%. Then, the first
reverberation causes the velocity to jump to 356 m/s. Another slow rise occurs
for 0.12 microseconds to 363 m/s. This is the same as the prediction of 364 m/s
234
for the first reverberation to within experimental error of 2%. Following this, the
second reverberation occurs, which brings the velocity to 417 m/s. The elastic
prediction for this velocity is 417 m/s. The velocity remains at about this level,
rising only to 420 m/s until the arrival of the boundary wave at 0.975
microseconds.
In experiment RB-2, the impact velocity is 512 m/s. The initial free surface
particle velocity rise is to 263 m/s due to the higher impact velocity (Figure
6.10). The elastic-plastic prediction is 277 m/s, from which the actual velocity is
off by 5%. Then the velocity slowly rises for 0.17 microseconds to 267 m/s before
jumping to 384 m/s. The predicted velocity level for the first reverberation is
395 m/s. This first reverberation rises to 390 m/s over 0.16 microseconds before
jumping to the second reverberation level of 448 m/s. The velocity level rises to
455 m/s, 0.23 microseconds after the beginning of the reverberation. This
compares well to the elastic-plastic prediction of the velocity level: 453 m/s. The
final reverberation begins at 478 m/s and ends 0.1 microseconds later at 483 m/s.
Its predicted level is 483 m/s.
From these experimental plots (Figure 6.10) the materials can be seen to follow
the elastic-plastic predictions quite well. The level of error of the VISAR is 2%.
The Hugoniot and the reverberation free surface particle velocities are predicted
to within 5% by the elastic-plastic theory derived in Section C.2.1. Additionally,
the rise times of the prediction agree closely with the times of the jumps in the
particle velocities between the states. The extended nature of these rise times
indicates elastic-plastic behavior in the tungsten carbide specimen.
By use of the equations provided by Hall [Hall et. al., Preliminary] and Dandekar
[Dandekar et al., 2003], the stress and strains of each reverberation state can be
observed. The midpoints of the velocities in each state are employed for this
purpose. The stress and strains in the silicon nitride are plotted versus the
elastic Hugoniot curve established in Chapter V (Figure 6.11). As can be seen
from the figure, the stresses and strains cluster along the elastic predicted line,
235
within the experimental error, indicating that the behavior of the silicon nitride is
primarily elastic. This, combined with the elastic-plastic behavior of the
tungsten carbide, means that the material behavior during the reverberation
experiment is well predicted by the current methods.
236
D. Conclusions and Discussion
Two methods for evaluating the elastic and elastic-plastic material behavior and
dynamic yield strength subjected to successive shocks from the literature are
presented in Section A. Using the single stage gas gun and appropriate choices of
flyer, target, backing, and window materials, experiments using these methods
were conducted at impact velocities around 500 m/s. However, the high HEL of
silicon nitride prevented the experimental Hugoniot stress from greatly exceeding
the HEL in the reverberation experiments. In the shock re-shock and shock
unload pair of experiments, the HEL was only exceeded in the second shock of
Experiment SR-1. An estimate of the residual shear stress is 1.21 GPa.
Within the obtained range of stresses, however the material behavior in the four
experiments is well predicted by the equations describing elastic-plastic motion
presented in Sections A and C of this chapter. Shock re-shock Experiment SR-1
indicates that the silicon nitride deforms mainly in a hydrodynamic fashion, but
reaches the HEL near the end of the re-shock (Figure 6.9). Shock unloading
Experiment SU-1 initially follows the hydrodynamic curve as well, but unloads in
an elastic manner, (Figure 6.9) as is expected below the HEL. In the two
reverberation experiments, the tungsten carbide behaves in a predictable elastic-
plastic manner, while the silicon nitride exceeds its HEL during the initial shock,
but the reverberations remain elastic. Because of this, the stresses and strains of
the reverberation states conform to the elastic relationship established in Chapter
V (Figure 6.11).
As a result of the high HEL of silicon nitride, the experiments were unable to
probe deeply into the nature of inelastic multiple shock behavior. However, one
indication of a small amount of inelastic deformation from the first shock is that
a simple calculation of the HEL in experiment SR-1 shows the HEL to be
reduced 6% from 12 GPa to 11.3 GPa. The equations in Section A [Asay and
Chhabildas, 1981; Reinhart and Chhabildas, 2003] indicate a residual shear stress
of 1.21 GPa. The material behavior does not exhibit a significant change from
237
the hydrodynamic prediction along the lines of those described in the literature.
Similarly, the reverberation experiments indicate elastic behavior. Because the
initial Hugoniot state in the re-shock/unload experiments is below the HEL, the
effect of subsequent shocks on the dynamic strength of the material above the
HEL remains largely unexplored. A gas gun with the capability of a greater
maximum impact velocity is necessary to further investigate this area.
238
References
Asay, J. R. and L. C. Chhabildas, 1981. Determination of the Shear Strength of Shock
Compressed 6061-T6 Aluminum. In: Shock Waves and High-Strain Rate Phenomena in Metals.
Plenum Publishing Corporation. New York, New York. 417-431.
Dandekar, D. P., C. A. Hall, L. C. Chhabildas, W. D. Reinhart. 2003. Shock Response of a
Glass Fiber-Reinforced Polymer Composite. Composite Structures. 61 [1-2] 51-59.
Hall, C. A., L. C. Chhabildas, and W. D. Reinhart., Preliminary Report. Shock Hugoniot and
Release States in GRP Composite From 3 to 20 GPa. Sandia National Laboratories.
Reinhart, W. D.; L. C. Chhabildas. 2002. Strength Properties of Coors AD995 Alumina in the
Shocked State. International Journal of Impact Engineering. 29 [1-10] 601-619.
239
Tables
Table 6.1: Dynamic Properties for Experiments SR-1 and SU-1. The abbreviation Hydro. stands
for hydrodynamic.
Free Surface State 5 Hugoniot State Free Surface State 7
Exp
#
Impact
Velocity
Backing
Plate
Measured
Velocity
Elastic
Velocity
Hydro.
Stress
Hydro.
Strain
Measured
Velocity
Elastic
Velocity
SR-1 475 m/s WC 358 m/s 348 m/s 8.0 GPa 2.6E-2 510 m/s 506 m/s
SU-1 483 m/s Al 372 m/s 355 m/s 8.6 GPs 2.5E-2 230 m/s 233 m/s
Table 6.2: Dynamic Properties for Experiments RB-1 and RB-2. Note, stress and strain in States
1, 3, 5, and 7 are computed respectively from measured velocity in States 2, 4, 6, and 8.
Exp # RB-1 RB-2
Impact Velocity 471 m/s 512 m/s
Flyer Thickness 4.5 mm 5.5 mm
State 1: Actual Stress 12.4 GPa 13.4 GPa
State 1: Actual Stain 3.19E-2 3.48E-2
State 2: Measured Velocity 245 m/s 265 m/s
State 2: Predicted Velocity 257 m/s 277 m/s
State 3: Actual Stress 6.2 GPa 7.01 GPa
State 3: Actual Stain 1.48E-2 1.56E-2
State 4: Measured Velocity 361 m/s 389 m/s
State 4: Predicted Velocity 365 m/s 395 m/s
State 5: Actual Stress 3.16 GPa 3.43 GPa
State 5: Actual Stain 6.63E-3 0.763E-3
State 6: Measured Velocity 420 m/s 452 m/s
State 6: Predicted Velocity 418 m/s 454 m/s
State 7: Actual Stress n/a 1.59 GPa
State 7: Actual Stain n/a 3.82E-3
State 8: Measured Velocity n/a 482 m/s
State 8: Predicted Velocity n/a 483 m/s
240
Figures
Figure 6.1: Theoretical stress vs. strain diagram showing graphical method for computing
dynamic yield strength using the technique from Reinhart and Chhabildas [2003]. The initial
loading occurs along the hydrodynamic Hugoniot curve from zero to the Hugoniot state. In the
re-shock experiment, the wave velocity transitions from elastic to bulk (elastic-plastic) at Point 2.
Point 1 is the location where the same occurs in the unloading experiment. By using the stress
values at these points, the dynamic yield stress can be found.
241
Figure 6.2: Reverberation experiment design showing silicon nitride flyer and tungsten carbide
target. The VISAR probe reflects off of a thin aluminum layer on the surface of the target. The
thin target and thicker flyer are visible.
Figure 6.3: (a) Time vs. distance and (b) stress vs. velocity diagrams for reverberation experiment
RB-2 showing computation of SσΔ . The change in stress between States 3 and 7 can be found
using the velocities measured on the free surface, States 2 through 8. Knowing the slope of each
purple line, which is the target’s impedance, CT, and these velocities, the stress at each state can
be found. Then, by subtracting two stresses, Equation { .A.35} is generated.
242
Figure 6.4: Specimen and flyer configuration for shock re-shock and shock release experiments.
The backing plate on the flyer is shown, as is the silica glass window.
Figure 6.5: (a) Time vs. distance and (b) stress vs. velocity diagrams for reverberation experiment
RB-1. The method of determining the stress jump is the same as in Figure 6.3. One fewer stress
reverberation occurs in this experiment than in experiment RB-2, because of the reduced
thickness of the flyer.
243
SilicaGlass
Silica Glass
Figure 6.6: (a) Time vs. distance diagram, (b) stress vs. velocity diagram for shock re-shock
experiment. The HEL of both tungsten carbide and of silicon nitride are exceeded in the re-shock
state.
SilicaGlass
Silica Glass
Figure 6.7: (a) Time vs. distance diagram, (b) stress vs. velocity diagram for shock unloading
experiment. The HEL of silicon nitride is not exceeded in either the initial shock or the
unloading states.
244
Time (seconds)
Vel
ocity
(km
/s)
-5E-07 0 5E-07 1E-06 1.5E-060
0.1
0.2
0.3
0.4
0.5
0.6
Figure 6.8: Free surface particle velocity vs. time profiles for shock re-shock and shock unload
experiments SR-1 and SU-1. The initial rise corresponds well with the predictions. The shock re-
shock experiment shows an HEL at a velocity below that which occurred without a previous
shock. The shock unload experiment appears elastic upon unloading.
HEL
Hugoniot
Re-shock
Unload
245
Strain
Stre
ss(P
a)
0 0.01 0.02 0.03 0.04 0.050
1E+09
2E+09
3E+09
4E+09
5E+09
6E+09
7E+09
8E+09
9E+09
1E+10
1.1E+10
1.2E+10
1.3E+10
1.4E+10
Hugoniot CurveRe-shockUnload
ElasticLine
ElasticUnloading
HydrodynamicLoading
1
2
Figure 6.9: The stress vs. strain plot for reloading and unloading experiments that shows the
material behavior is elastic under unloading and hydrodynamic under re-shock. The initial shock
loading is hydrodynamic. As expected from the equations, the loading under shock re-shock also
follows the hydrodynamic predictions from Chapter V. The unloading is linear and elastic.
246
Time (sec)
Vel
ocity
(km
/s)
0 2.5E-07 5E-07 7.5E-07 1E-060
0.1
0.2
0.3
0.4
0.5
RB-2RB-1
Figure 6.10: Free surface particle velocity vs. time profiles for reverberation experiments RB-1
and RB-2. The experiential velocities correspond well with the elastic and elastic-plastic
predictions.
247
Strain
Stre
ss(G
Pa)
0 0.01 0.02 0.03 0.04 0.050123456789
10111213141516
Elastic HugoniotRB-1RB-2
Figure 6.11: Stress and strain levels for the two reverberation experiments plotted against the
elastic Hugoniot curve. The stresses and strains of the unloading states are elastic to within the
experimental error.
248
Chapter VII – Investigation of Shock Induced Failure WavePropagation in Soda Lime Glass
In the present study, in order to examine the shock wave response of soda lime
glass, plate impact experiments were performed using a single stage gas gun. In
addition to the compression and shear waves that are observed in ceramics and
metals, a second type of wave, known as a failure wave is present in glasses.
This failure wave, which has been described in several studies, is a result of
damage accumulation in glasses during the initial stages of planar shock
compression. It is generally agreed that the spall strength as well as the shear
impedance behind the failure wave in glasses are negligibly small [Kanel et al.,
1992].
In order to examine the suitability of soda lime glass for use in applications
where impacts are likely, a series of experiments were performed in the present
study to characterize the failure wave and its effects. The goal of the
experiments was to measure the effects of the failure wave on spall strength,
dynamic shear strength, and the longitudinal and shear impedance under
combined compression and shear shock-wave loading. Specifically, in the study
one of the experiments was designed to evaluate the spall strength in soda-lime
glass under uniaxial shock compression. A companion experiment examined the
degradation in the spall strength with the addition of a shear stress due to a skew
angle of 18 degrees. The results of these two experiments were used to establish
the spall strength of soda lime glass. The remaining six experiments, discussed in
this chapter, were designed to study of the failure wave and its effects on the
impedance, the spall strength and the shear strength of the soda lime glass at the
eighteen degree skew angle.
In order to understand the nature of the failure wave, studies by several authors
describing previous experimentation on glasses are discussed in Section A. This
discussion is followed in Section B by a description of the experiments performed
in the current study. The newly developed method for using the VISAR
249
interferometer to measure the shear velocity is also described in Section B. Then,
in Section C, the experimental results are analyzed and compared with the
previous results from the literature.
A. Description of the Nature of the Failure WaveIn order to understand the nature of the current experiments, an explanation of
the nature of the failure wave and its effects on glasses is necessary. In Section
A.1, several previous experimental studies are reviewed. Each of these studies
provides an insight into the physical nature of the failure wave, or the change in
properties of glass such as impedance, shear strength, or spall strength behind the
failure wave. In Section A.2, three models are suggested to explain the failure
wave and its observed features. These models are compared with experiments
and numerical computations.
A.1. Experimental Determination of the Properties and Effects of theFailure Front
Several types of dynamic loading experiments have been employed in order to
examine the failure wave in soda lime glass over the past fifteen years. These
experiments have included gas gun shock compression experiments on plates such
as those performed by Kanel et al. [1992]. These plate impact experiments have
included sandwiched configurations [Espinosa et al., 1997] as well as flyer and
single target experiments [Clifton et al., 1998]. Each of the experiments
described in this section describes an aspect of the failure wave.
Kanel et al. [1992] conducted an experimental and numerical work on the shock
compression of glass and fused quartz. In this work, a shock loading of 4.5 GPa
was induced in K19 glass plate specimens by means of high speed planar impacts
using copper flyers. Shock loading of fused quartz with aluminum flyers was
carried out by similar means. Free surface velocities were measured using the
VISAR system. The experiments were conduced close to but not exceeding the
HEL for the two materials.
250
A series of normal plate impact experiments was conducted on K19 glass to
confirm the existence of failure waves in glass. In these experiments a release
wave that originates at the back of the target plate interacts with the advancing
failure wave and a small recompression was noted in the longitudinal particle
velocity history [Kanel et al., 1992]. This anomalous recompression was
explained as a wave reflection from the failure wave front due to a reduction in
the mechanical impedance of the material behind the failure wave. Varying the
thickness of the specimen showed that the depth of the failed region from the
impact face increases with time. The authors interpreted this as a failure wave
moving through the K19 glass at a slower speed than the elastic wave velocity.
They also noticed that the failure wave front velocity decreases with increasing
depth [Kanel et al., 1992]. Simulations of the experiments utilizing a one-
dimensional elastic-plastic model were able to match the anomalous
recompression in the longitudinal stress history by means of a reduction in the
shear modulus at the higher stress levels. This model indicated that the shear
stress and the spall strength of the shocked glass dropped to zero behind the
failure front. The decreasing velocity of the failure front was also accounted for
in the calculations. In this way, the model predicted the form of the stress and
free surface velocities for the propagation of failure waves in the K19 glass [Kanel
et al., 1992].
Brar et al. [1991 A], were also among the first to publish on the topic of failure
waves. They defined the failure wave front as the boundary between the intact
and the comminuted material. The definition of a comminuted material is one
that is broken into pieces that are fully compacted although heterogeneously
deformed [Feng, 2000]. In their study, Brar et al. [1991 A] conducted impact
experiments on both glass bars and plates. They employed 12.7 mm by 150 mm
long diameter pyrex glass bars and impacted them with either steel or pyrex
striker bars accelerated using a gas gun. Observations of the failure wave were
conducted using high speed cameras. Soda lime glass plates were also impacted
with flyers of copper and aluminum by means of a gas gun in a plate impact
251
configuration. Transverse strain gages embedded within the soda lime glass
specimens were used to measure the strain.
High speed camera images of the bar experiments in Brar et al.’s [1991 A] study
indicated that the failure wave propagated at differing velocities depending on
the velocity of impact. The comminution of the material was observed directly.
At an impact speed of 125 m/s, the failure wave speed was 2.3 km/s and at an
impact speed of 330 m/s the failure wave front propagated at 5.2 km/s. The
velocity of the failure wave was also greater when a steel striker bar rather than
the pyrex glass was used. This suggests that the velocity of the failure wave is
also dependent on the impact stress. Additionally, behind the failure wave, the
bar was observed to expand in the radial direction in an explosive fashion [Brar
et al., 1991 A]. On the other hand, the plate impact experiments on glass were
designed to observe any changes in the shear strength of the shocked glass behind
the failure wave front. A two wave structure was noticed in the gage record
indicating the propagation of both the longitudinal and failure waves. The
second recorded wave was not the failure wave itself, but rather the longitudinal
release wave that is reflected first off the free surface of the flyer and then off the
failed material boundary. The existence of this second wave indicated a change in
material property in the shocked region behind the failure wave. From these
measurements, the failure wave speed was estimated to be 2.2 ± 0.2 km/s. Using
the impedance matching method, the critical longitudinal stress to initiate the
failure wave was found to be 3.8 GPa (38 kbar). Moreover, below a critical
longitudinal stress, before the arrival of the failure wave, the measured shear
strength of the shocked glass was observed to increase linearly with longitudinal
stress. However, above this critical stress, the shear strength is observed to level
out at about 1.1 GPa. The comminuting of the glass is assumed to be
responsible for this loss of shear strength and a corresponding increase in the
mean stress [Brar et al., 1991 A].
252
Kanel et al. [2002] in a more recent paper studied the failure wave more closely.
In these experiments soda-lime glass were utilized instead of the K19 glass
utilized used in their previous study [Kanel et al., 1992]. Two types of
experiments were performed. In the first type, plates of glass sandwiched
between copper were impacted by aluminum flyer plates to determine the HEL of
soda lime glass. The HEL of the soda lime glass was measured to be 8 GPa. In
the second set of experiments, aluminum flyer plates were used to impact single
soda-lime glass plates. This configuration was used to study the effect of the
failure wave on the measured free surface velocity histories and to examine the
spall strength of the material ahead of the failure wave front. The spall strength
of the intact glass ahead of the failure wave was determined to be greater than 3
GPa [Kanel et al., 2002]. Moreover, like in their earlier study, a recompression
wave was observed in the measured particle velocity history at the rear surface of
the glass target plate. From these measurements Kanel determined the
propagation speed of the failure wave front to be 1.55 ± 0.06 km/s [Kanel et al.,
2002].
The Hugoniot Elastic Limit and the spall strength of glass were examined in a
study performed by Rosenberg et al. [1985]. In Rosenberg et al.’s [1985] release
wave experiments, an increase in stress was recorded upon the arrival of the
recompression wave. This indicated the presence of a failure wave. The HEL of
soda lime glass was found to be 6.4 GPa. In the spall strength experiments
above the HEL no spall strength was recorded [Rosenberg, et al., 1985]. Below
the HEL, Brar et al. [1991 B] found that at shock stresses of 4.9 GPa through 5.7
GPa behind the failure wave there is no spall strength. In front of the failure
wave, there is no spall recorded, and thus the spall strength is higher than 3 GPa
[Brar et al., 1991 B].
[VP2]
In a paper by Ginzburg and Rosenberg [1998] three plate impact experiments
were described on soda lime glass using brass flyers. These experiments used a
PMMA backing plate and an internal strain gage. The impact velocities
produced compressive stresses of about 2 GPa, 4.5 GPa and 6 GPa [Ginzburg
253
and Rosenberg, 1998]. A summary of the three commonly accepted conditions of
the failure waves are given in this paper [Ginzburg and Rosenberg, 1998]. First,
the failure waves are present at impact stresses that are greater than half of the
HEL of the glass, but lower than the HEL. Second, the failure front propagates
at a speed between 1.5 km/s and 2.5 km/s. Third, the shear strength decreases
and the spall strength disappears entirely behind the failure front. In Ginzburg
and Rosenberg’s [1998] 6 GPa experiment, the failure wave was detected by a
recompression in the velocity at the time of reflection from the rarefaction
wave/failure front interface. This anomalous recompression is not seen in the
lower stress experiments (4.5 GPa), suggesting that the threshold stress is
between these two levels [Ginzburg and Rosenberg, 1998]. This is consistent with
the previous work in the literature. Moreover, in these experiments a second
recompression wave was not detected, which according to Ginzburg and
Rosenberg [1998] implies no change in the impedance of the material. The time
for this second wave to arrive was computed by assuming the failure front stops
moving after the intersection of the unloading wave with the propagating failure
front. Also, there were no differences in the wave velocities of the longitudinal
compression and the rarefaction waves. Ginzburg and Rosenberg [1998] stipulate
that the density along the shock direction must remain unchanged, while the
material dilates in the lateral direction.
A set of experiments conducted by Clifton et al. [1998] employed Hampden steel
plates to impact soda lime glass at an oblique angle of eighteen degrees. These
experiments included three experiments in which the steel was the flyer and three
experiments where steel was the target. The set of experiments where steel was
the flyer and glass was the target was used to examine the spall strength and
residual shear strength of glass following shock compression. The objective of the
other set of experiments, where steel was the target and the glass was the flyer,
was to examine the state of stress behind the failure front. VISAR and normal
and transverse displacement interferometers were employed to examine the free
surface particle velocity histories [Clifton et al., 1998].
254
The first group of Clifton et al.’s [1996] experiments was conducted using 5.6 mm
thick soda lime glass targets and 4.4 mm thick steel flyer plates. These thickness
choices resulted in a spall plane that is formed behind the failure front in the
comminuted material. Two impact velocities were examined. At 302 m/s, or 3.3
GPa compressive stress, the failure wave did not occur and a spall strength was
recorded. In the two experiments at about 392 m/s, or 4.3 GPa compressive
stress, no spall signal was observed. A recompression signal indicated the arrival
of the longitudinal wave reflected from the failure front. Free surface transverse
particle velocity signals indicate an initial rise to near the elastically predicted
transverse velocity followed by a continuous decrease in the particle velocity
[Clifton et al., 1998].
Clifton et al.’s [1996] experiments with the glass flyers and the steel targets were
conducted at speeds such that the steel remained elastic. In the low velocity
experiment, 310 m/s (3.4 GPa), the elastic prediction corresponds well to the
measured normal and transverse particle velocities. It is noted that a long rise
time exists for the last 10% of the transverse signal. This is markedly different
from the higher velocity experiment at 397 m/s (4.4 GPa). In this experiment,
the normal velocity is smaller than the elastic prediction, although the predicted
velocity jump corresponding to the arrival of the unloading wave from the flyer is
of the same magnitude as the elastically predicted rise. The consistency in the
velocity jump magnitude indicates that the unloading response is unchanged from
the elastic behavior. The signal of the transverse velocity indicates an initial rise
to a level below the elastic prediction, then a drop to a lower constant level.
This indicates an instability in the shear stress at high levels of shear strains or
stresses [Clifton et al., 1998].
A.2. Proposed Mechanisms for the Propagation of the Failure Front
Several models have been presented in the past to explain the phenomena of
failure waves in glass. Espinosa et al. [1997] suggested that upon impact a
deviatoric stress component drives the creation of a failure front consisting of
255
shear induced flow planes. Micro-cracking was assumed to occur along these
planes, leading to microscopic fracture and the observed changes in glass
properties. Feng [2000] argued that a diffusive wave was initiated at the impact
surface due to the high transient stresses, and was propagated by deviatoric
stresses ahead of the micro-cracks. Sundaram [1998] described a atomic bond
switching model to explain the formation of the failure wave. The atomic
structure of soda lime glass is reviewed the model is tested against experiments
using numeric computations [Sundaram, 1998].
Espinosa, Xu, and Brar [Espinosa et al., 1997] performed a study on glass plates
and bars in which they examined the spall strength, shear resistance, and failure
wave propagation. In these experiments the glasses were: soda lime glass,
aluminosilicate glass, and pyrex glass. The plate experiments used both
longitudinal and lateral imbedded manganin gages for normal impacts as well as
normal and transverse displacement interferometers for pressure-shear impacts
[Espinosa et al., 1997].
In Espinosa et al.’s [1997] experiments below a compressive stress of 4 GPa, no
failure wave was observed. The HEL of soda lime glass is quoted to be 6.4 GPa.
Also, slightly above the HEL, at compressive stress of 7.6 GPa, ahead of the
failure wave front no spall was observed and the particle velocity was observed to
unload to its elastic no-spall prediction. An aluminum flyer was used in this
experiment. For almost the same conditions, at a compressive stress of 7.5 GPa,
and behind the failure wave front, a spall strength of 0.4 GPa was seen. The
same pattern was seen at 5.7 GPa stress, which is below the HEL. Before the
failure front full unloading occurs with no-spall, while behind the failure front,
there is a reduced spall strength [Espinosa et al., 1997]. This paper thus suggests
that the failure wave is present at stresses near the HEL. Also, the gage
experiments show several details about the failure front. First the propagation
speed is between 1.5 km/s and 2 km/s. Second, the shear resistance of the
material is pressure dependent and is reduced behind the failure front. The
longitudinal stress also decays following the failure front.
256
Espinosa et al. [1997] proposed a mechanism for the failure front of shear induced
flow planes. The driving force of these shear flow planes is the deviatoric stress
component. Microscopic cracks, voids or other defects located at the intersection
of these planes may propagate reducing the spall strength of the material.
Without the micro-cracks the flow surfaces retain their cohesive strength. This
model was found to be consistent with the shear and spall strength observations,
and with characteristic surface features on recovered surfaces [Espinosa et al.,
1997].
In a study conducted by Feng, a model is proposed which explains the failure
front propagating as a wave. His model is based on three assumptions: (a) the
material has a nonlinear elastic response in front of the failure wave; (b) the
longitudinal stress, strain and particle velocity remain constant through the
failure front; and (c) the thermal effects are insignificant [Feng, 2000].
The failure wave model proposed by Feng [2000] models the failure wave as a
thin layer that propagates through the material wherein microscopic failure
processes occur above a threshold stress level. Due to the lateral confinement
inherent in shock compression experiments, shear dilatancy does not cause lateral
expansion. Instead, voids collapse at the same time as shear dilatancy occurs.
The result is that the material is comminuted, which implies fragmentation into
pieces that are both fully compacted and heterogeneously deformed. The void
collapsed state and the shear dilatancy, both cause a larger compressive stress in
the comminuted material than would be present had the material remained
intact. With the assumption of constant longitudinal stress, the lateral stress
must increase resulting in a decrease in shear stress [Feng, 2000]. This
phenomenon is seen in the experiments.
The initiation of the failure front occurs at the impact surface. Feng suggests
that initial damage occurs at the impact surface due to a combination of
transient loading conditions. These transient loading conditions are caused by
257
tilt differences between the flyer and target plates, and by existing surface
roughness. The resulting micro-cracking produces deviatoric strains ahead of the
failure front which, in combination with existing microscopic flaws cause
propagation of the failure wave. The wave is therefore diffusive on the
macroscopic scale. Additionally, the modeling uses a two-dimensional
simplification using an average of the percolation of micro fissures over the lateral
direction. The percolation in the lateral direction is much faster than the
percolation in the longitudinal direction due to the presence of more sites that
meet the initiation conditions along the plane of the failure front [Feng, 2000].
The numerical model results are compared with Kanel’s [Kanel et al., 1992]
experimental results. As indicated by Feng’s theory, an increase in mean stress
occurs in the comminuted material. Additionally, the two wave structure of the
computed lateral stress profiles match the stress rise times and relative velocity
rises from the experiment. Feng [2000] argues that the transient response of the
gage may not clearly reflect the velocity of the failure front. This argument, as
well as differences in the variation of the computed failure wave velocity with
gage thickness suggested to Feng [2000] that this method of finding the wave
velocity is not reliable.
With this assumption, Feng [2000] used a measure of 5% of the second wave to
estimate a wave speed of 3.32 km/s at 4.5 mm and 3.37 km/s at 6.5 mm into the
material. These propagation speeds agree with measurements taken from Kanel
et al.’s [1992] experiments. However, Feng did not find evidence to support
Kanel’ et al.’s [1992] idea that the failure wave velocity is distance dependent. In
a comparison with a different set of experimental data [Bourne and Rosenberg,
1996], Feng’s [2000] calculations did show a decreasing failure front velocity with
increasing depth although not as large a decrease as in the data [Bourne and
Rosenberg, 1996].
Another issue discussed by Feng [2000] is that in Bourne and Rosenberg’s [1996]
experiments on soda lime glass a foil gage in the longitudinal direction indicates a
258
small strain increase that corresponds to the failure wave. Feng raises the
question as to if this is really a decrease in impedance or just the reflection of an
elastic recompression wave from the failure front. Feng’s [2000] model is valid for
experiments where little or no change in the longitudinal properties are recorded.
He also notes that while the model predicts stress and particle velocity in soda
lime glass, the inconsistency above remains to be explained [Feng, 2000].
In Sundaram’s [1998] experiments, a tungsten carbide flyer plate is used to
impact a sandwich configuration. The front plate of the sandwich was tungsten
carbide. The specimen, a 5 μm layer of soda lime glass was vapor deposited on
the Hampden steel rear plate. These experiments were conducted at a skew
angle of 22 degrees and had velocities ranging from 118 m/s to 198 m/s. The
resulting compressive stresses were 2.5 GPa, 3.5 GPa, and 5.7 GPa [Sundaram,
1998]. In Sundaram’s [1998] thesis, a model is proposed that links the failure
wave to the shear strain and to atomistic bond rearrangements. The atomic
bond changes cause the loss of shear and spall strength. The basic structure of
soda lime glass is amorphous with a network of tetrahedra with one silicon atom
surrounded by four oxygen atoms. A bond switching model is suggested. It is
suggested that below the critical stress level, modifier ion movement at defects is
responsible for causing the inelastic deformation. Other low activation energy
mechanisms such as distortion of the covalent matrix also occur causing
progressive hardening of the glass and the ramping of the stress. At large strains
a bond switching occurs in the covalent silicon-oxygen bonds and causes stress
relaxation. This switching can occur due to the decrease of potential energy
barriers [Sundaram, 1998].
Sundaram [1998] states that the deviatoric and hydrostatic shock compression
stress components combine to produce a phase transformation in the shocked
glass. The transformation then proceeds into the material as a diffusive process
because of the relaxation of the shear stresses caused by the distortion of the
network of covalent bonds. Because this process is constant volume, there is no
change in the longitudinal strain across the front. However, there is a distinct
259
shear stress change, which leads to an increase in the transverse stress.
Additionally, this model predicts a decrease in velocity of the front with
increasing depth into the material. Finally, the passage of a tensile wave through
the transformed material would, Sundaram [1998] states, cause coalesce of nano-
cracks which would cause the material to have essentially no spall strength.
The results of a numerical computation used to test this model indicate a clear
decrease in shear stress and strain with increasing compressive stress. In the
lowest velocity experiment (2.5 GPa), no loss in shear strength is observed. A
shear stress level of 510 MPa is recorded and the end of the record corresponds to
a total shear strain of 0.5. In the 3.5 GPa experiment, the shear stress increases
to 580 MPa in approximately 250 ns, which is below the elastic prediction of 780
MPa. The total shear strain to this point is 2.05. After about 300 ns, the shear
stress falls off to a level between 180 MPa and 100 MPa, where it remains steady.
In the highest velocity shot (5.7 GPa), above the 4 GPa critical limit for failure
waves, the shear stress rises to about 480 MPa before dropping off after only 150
ns to a level near 60 to 80 MPa. The shear strain to the peak is 1.97. From
these experimental results, it is suggested that a critical shear strain is
responsible for triggering the loss of shear strength [Sundaram, 1998]. The results
for shear stress are similar to Clifton’s findings [Clifton et al., 1998].
Sundaram’s experimental results fit well with this model for all three compressive
stress experiments. The ramping initial rise and plateau in the higher two
experiments is matched, as is the lower near constant behavior. Only the peak
stress in the highest velocity experiment (the experiment with an impact stress of
5.7 GPa) does not quite match. Therefore, Sundaram [1998] concludes that this
model explains the glass behavior under shock wave loading conditions.
These three papers offer substantial insight into the mechanism of the failure
wave. The models suggest micro-structural and atomic explanations for the
initiation and propagation of the failure front. However, these models and
experiments have not yet covered all of the regions where the failure wave occurs,
260
nor provided a single explanation that explains the entire process. The current
research aims to increase the experimental knowledge base of failure waves.
261
B. Design of Experiments to Investigate the Failure Wave in Soda LimeGlassIn order to build upon the research described above, a set of plate impact
experiments was conducted in the present study to understand the behavior of
glass under combined compression and shear impact loading. The present study,
in particular, builds on Clifton et al.’s [1998] work. The flyer and single target
plate configuration of Clifton’s study was chosen for the current experiments. In
order to determine the spall strength of soda lime glass, aluminum flyers were
used in both the shock compression and the combined shock compression and
shear configurations. These two experiments compared the spall strengths of
glass with and without the presence of shear. Next, four experiments were
conducted using glass plates that were impacted with tungsten carbide flyers.
Using glass specimens of two different thicknesses, changes in the normal and
shear velocities and the spall strength in the undamaged and comminuted glass
were examined. Also, in two experiments using a glass flyer and a tungsten
carbide target, the impedance change in the glass due to the failure wave was
measured. The VISAR interferometer system was used to measure the normal
and transverse components of the particle velocity histories at the free surfaces of
the target plates.
B.1. Description of the Experimental Configurations
In Clifton et al.’s [1998] paper, the single flyer and target configuration was
employed. Hampden steel was used as the material for the flyer plates while the
target plate was the glass specimens. The choice of steel limited the maximum
stresses that could be employed as the steel was required to remain elastic. The
maximum compressive stress in the experiments was 4.3 GPa, produced at an
impact speed of 407 m/s [Clifton et al., 1998]. In the current study, tungsten
carbide was chosen because it remains elastic when compared to steel under the
high velocity impact loading. In fact, the HEL of tungsten carbide is about 7.2 ±
0.8 GPa [Dandekar, 2004]. This, plus the greater impedance of tungsten carbide
allows for the higher compressive stress of 5 GPa to be reached. A lower level of
262
3 GPa was also chosen in order to compare with the elastic results in Clifton’s
paper [Clifton et al., 1998].
Two experiments were conducted to obtain the spall strength, one experiment at
the impact angle of eighteen degrees and the other under pure compression.
Experiment Al/G1 was the normal shock compression (Figure 7.1) experiment,
while Experiment Al/G2 was conducted under combined shock compression and
shear loading at an angle of 18 degrees (Figure 7.2). These experiments
employed aluminum 6061-T6 flyer plates of 5.9 mm thickness that impacted 12.5
mm thick glass specimens (Figure 7.3). The aluminum’s lower impedance of 16.6
MPa/(m/s), as opposed to tungsten carbide’s impedance of 106 MPa/(m/s),
allowed for a larger tensile stress to be developed for the same impact velocity.
By using the high normal velocity of 485 m/s, a tensile stress of around 3.5 GPa
was reached, which is in excess of the recorded data which state that the spall
strength is greater than 3 GPa [Kanel et al., 2002]. The tungsten carbide plates
used for the other experiments only produced a maximum tensile stress of 1.2
GPa.
Using 6 mm thick tungsten carbide flyers and soda lime glass targets,
Experiments WC/G1 through WC/G4 examined both shock compression and
shear loading. A shearing angle of 18 degrees was chosen, as in Clifton’s paper
[Clifton et al., 1998] to enable the study of normal and shear effects generated by
the impact. The normal compressive stresses were ~ 3 GPa and 5 GPa (Figure
7.4). At 3 GPa, according to Clifton et al. [1998] the failure wave should not be
present and the material should behave like an elastic brittle material. The 5
GPa experiments were performed to observe the fully developed failure wave.
Both of these experiments were conduced twice, using two glass targets with
different thickness of 6.5 mm and 12.5 mm. These plates were chosen so that in
the experiments with the 12.5 mm thick specimen, the spall plane would occur in
the target ahead of the failure wave front (Figure 7.5). In the other experiments,
with the 6.5 mm thick specimen, the spall occurred behind the failure front, i.e.
in the comminuted glass, following the passage of the failure wave (Figure 7.6).
263
The free surface particle velocity histories in these experiments were measured by
using the VISAR.
A second group of experiments were performed using soda lime glass as the flyer
material and 4 mm thick tungsten carbide as the target (Figure 7.7).
Experiments G/WC1 and G/WC2 were configured similar to the reverberation
experiments described in Chapter VI (Figure 7.8). The 12 mm glass flyers were
thick when compared with the WC target plate; three wave reverberations were
possible in the target before the return of the release wave from the rear surface
of the flyer plate. The key objective of these reverberation experiments was to
examine the longitudinal and shear impedance of the comminuted glass near the
impact surface.
B.2. Modifications to the VISAR System to Enable the SimultaneousMeasurement of the Normal and Transverse Components of the ParticleVelocity Histories
In the normal impact experiment the VISAR probe was used in the same
configuration as the previous chapters. However, in all of the experiments with
skew angles of eighteen degrees, both the normal and shear velocities of the
specimen surface had to be examined. In order to measure these velocities, the
multi-beam VISAR interferometer system with three probes was used. This is
because at the high velocities generated by the impact, the conventional NDI and
TDI techniques are not very suitable. The VISAR system with its variable
etalons can be configured so as to resolve the expected particle velocity with
enough accuracy. The mechanism to capture the shear and normal velocities was
modified from the standard single probe holder discussed in Chapter IV. Issues
that were considered included how to properly align multiple probes to light
reflected from a single point on target surface and how to optimize the intensity
of laser light to the probes.
The complete laser diagnostic setup is shown in Figure 7.9. The Coherent laser
was beamed into the impact chamber through a 1000 mm focal length lens and a
264
series of mirrors. This lens focused the laser light upon a grating, which was
glued onto the back surface of each specimen. The cross-line gratings were from
Photomechanics Inc. and had 1200 lines per millimeter. The laser beam was
moved 8 mm horizontally from the center of the lens in order to allow the
reflected zero beam to be separate from the input beam. The incident laser light
beam was diffracted by the grating on the target plate; the zeroth order beam
was reflected back normal to the target surface, while the plus 1 and the minus 1
beams were created at an angle of 39.7 degrees from the zeroth beam. The three
output beams were collected by mirrors and directed out of the impact chamber.
The beams were then picked up by three 120 mm focal length VISAR probes.
The fiber optics then conveyed the beams to the interferometer as described in
Chapter IV. In this way the interferometer recorded three particle velocity signals
[Barker et al., 2000]. The measured components of the particle velocity were
used to determine both the normal and the transverse components of the particle
velocity during the combined pressure and shear impacts.
265
C. Experimental Results and Analysis of Failure Waves in Soda Lime Glass
Each experiment exhibited unique features, which explored the effects of the
failure wave on soda lime glass. The aluminum on glass spall experiments
investigated the difference in the spall strength of the glass in pure shock
compression (Al/G1) from the spall strength under both compression and shear
loading (Al/G2). These experiments are discussed in Section C.1. The tungsten
carbide impacting glass experiments (WC/G1 – WC/G4) investigated the effect
of the failure wave on the spall strength and also show direct evidence of the
failure wave. The results are detailed in Section C.2. From the glass on
tungsten carbide experiments (G/WC1, G/WC2), the impedance of the failed
glass can be obtained. These results are discussed in Section C.3.
C.1. Spall Strength of Glass under Shock Compression and Pressure-Shear Loading
The spall strength indicates the resistance of soda lime glass to failure in dynamic
tension. Analysis of the relative impedances of various flyers showed that
impacting tungsten carbide flyers against glass target plates does not produce
sufficient tensile stress to cause spall. However, using Al6061-T6 flyer plates
produces a dynamic tension of 3.5 GPa at a normal impact velocity of 485 m/s.
In view of this, in the present study, two experiments were carried out using
Al6061-T6 flyer plates against glass target plates, one with no skew angle and
another at a skew angle of eighteen degrees. The normal velocity of 485 m/s was
aimed for in both experiments.
C.1.1. Normal Particle Velocity Calculations
The 6061-T6 Al alloy has an longitudinal impedance of 16.6 GPa. The
impedance is determined from its density, 2700 kg/m3 and its longitudinal wave
speed, 6.15 km/s. In the present study two experiments were conducted in order
to investigate the spall strength of soda-lime glass with and without the presence
266
of shear. For both experiments line equations were used to calculate the stress
and particle velocity histories in the various states (Figure 7.3).
The particle velocity in State 3 is calculated from the intersection of the lines
drawn from State 1 in the flyer, and State 2 in the target, both prior to impact:
3 1 F T FV =V Z /(Z +Z ) . {VII.C.1}
Here, the soda lime glass target impedance, ZT, is 14.5 MPa/(m/s) and the flyer
impedance, ZF, is 16.6 MPa/(m/s). In Experiment Al/G2, V1, is the normal
component of the impact velocity, VI, and can be calculated using
1 cosIV V φ= . {VII.C.2}
Here, φ , is the skew angle of impact and is equal to 18 degrees in the present
study. In Experiment Al/G1, φ is zero, and the impact velocity is equal to V1.
The velocity, V3, which is also the particle velocity of the Hugoniot state can be
related to the particle velocity of the rear surface of the glass in State 6, by
6 32=VV . {VII.C.3}
The stress in State 3 is simply the impedance of the glass times the particle
velocity, i.e.
3 3σ = TZ V . {VII.C.4}
Calculations for the additional states are carried out in a similar fashion. A
general formula is employed for each new state. This formula shows how to go
from any two states to a third state. If State A is to the left and below State C
on the t-X diagram and State B is the state to the right and below, then State C
can be determined as follows:
267
1 2
1 2
A B A BC
V Z V ZVZ Z
σ σ+ + −=
+, {VII.C.5}
( )2C C B BZ V Vσ σ= − + . {VII.C.6}
Here, Z1, and, Z2, are the impedances of the left and right materials respectively.
Using Equations {VII.C.5} and {VII.C.6}, the stresses and particle velocities can
be computed for all the other states in the t-X diagrams. The equations for the
important states are given below:
State 4, which represents the state in the flyer after the first release:
3 34
F
F
Z VVZ
σ−= , {VII.C.7}
4 0σ = . {VII.C.8}
State 5, which represents the stress and the particle velocity that is developed in
the flyer and the target after the release wave from the free surface of the flyer
plate reaches the flyer/target interface, i.e. the impact face
45
F
T F
Z VVZ Z
=+
, {VII.C.9}
5 5TZ Vσ = . {VII.C.10}
State 6, which is the stress and the particle velocity at the free surface of the
target following the reflection of the compressive wave back into the target
6 32V V= , {VII.C.11}
6 0σ = . {VII.C.12}
268
Stress and particle velocity in State 7, which also represents the spall plane
( )5 6 5 67 2
T
T
Z V VV
Zσ σ+ + −
= , {VII.C.13}
( )7 7 6 6TZ V Vσ σ= − + . {VII.C.14}
The stress and particle velocity in the various states for Experiments Al/G1 and
Al/G2, and the tungsten carbide on glass Experiments WC/G1, WC/G2,
WC/G3 and WC/G4, are determined from these relations. The computed stress
and particle velocity levels are shown in the figures for the particle velocity
versus time histories as dotted lines.
C.1.2. Transverse Particle Velocity Calculations
In the combined pressure and shear experiments, the shear velocity can be
calculated by simply replacing the normal component of the projectile velocity
with the transverse component of the projectile velocity, that is using sine instead
of cosine in Equation {VII.C.2}, and then using the shear wave speed, CS, instead
of the longitudinal wave speed to calculate the shear impedance, i.e.
S SZ Cρ= . {VII.C.15}
Using the shear impedances and the transverse component of the projectile
velocity instead of the normal impedances and normal impact velocity, Equations
{VII.C.1} through {VII.C.14} are then used to compute the shear velocities and
shear stresses in States 1 through 7. The elastic shear predictions are then
compared with the actual experimental free surface particle velocities. These
predictions are also shown by dotted lines on the figures, which show the
experimental free surface particle velocity versus time histories.
269
The three VISAR probes record beams reflecting from the target surface at zero,
+39.7 and -39.7 degrees. The later two are referred to as plus 1 and minus 1
diffracted beams, respectively. In order to determine the transverse (or shear)
component of the particle velocity from the measured particle velocity histories
along the directions of the plus and the minus diffracted beams, the following
relations are employed:
( )θ θ+ = + +1 1 11 cos sin2 2
N SV V V , {VII.C.16}
( )θ θ− = + −1 1 11 cos sin2 2
N SV V V . {VII.C.17}
In these equations V+1, represents the plus one order diffracted beam;V-1 represents
the minus one order diffracted beam; and VN and VS represent the normal and
transverse components of the particle velocity. The diffraction angle θ, is 39.7
degrees. By combining these relations, several combinations can be used for
calculating the shear velocity. Two are used in this chapter. When all three
beams are of good quality, an equation using the plus 1 and minus 1 velocities
can be used, i.e.
( ) ( )θ+ −= −1 1 sinSV V V . {VII.C.18}
This expression generates the least amount of noise in the transverse component
of the particle velocity. When the plus 1 beam is of questionable quality an
alternate expression can be employed
( )12 1 1 cossin 2
S NV V Vθθ
−⎛ ⎞= − − +⎜ ⎟⎝ ⎠
. {VII.C.19}
270
C.1.3. Spall Strength and Particle Velocity Observations for AluminumImpacting Glass
The spall strength experiments were conducted at a projectile velocity of
approximately 500 m/s. In Experiment Al/G1, the glass specimen was impacted
by an Al6061-T6 flyer at an impact velocity of 544 m/s; the resulting maximum
tensile stress on the spall plane was 3.89 GPa. However, no spall was observed in
this experiment (Figure 7.10). Experiment Al/G2 was conducted under combined
pressure and shear loading. In this experiment the glass target was impacted at a
speed of 497 m/s; as a result the glass was shocked in State 7 to a maximum
tensile stress of 3.59 GPa, assuming elastic conditions. However, this stress level
is not reached in the experiment, because the glass undergoes spall during the
tensile loading process (Figure 7.3). As in Chapters V and VI, the spall dip and
pull-back signals can be used to evaluate the spall strength under the various
experimental conditions. It is to be noted that the dip in particle velocity at the
time of spall was observed most clearly in particle velocity derived from the
minus 1 order diffracted beam (Figure 7.11).
The spall strength of the glass is determined in a similar manner as in Chapters
V and VI. By examining the particle velocity dip in the particle velocity of the
rear surface of the glass target in State 8, and employing the following relations,
the spall strength can be calculated
( )( )min12Spall T oZ V Vσ = − . {VII.C.20}
This formula can be used rather than the formula corresponding to the pull-back
height because the spall is elastic, as evidenced by the fact that the maximum
predicted compressive stress at 544 m/s and 0° is 4.21 GPa, which is lower than
the HEL of 6.4 GPa for glass [Rosenberg et al., 1985]. Important experimental
parameters such as impact velocity, compressive stress, and spall strength are
listed in Table 7.1.
271
Experiment Al/G1 was performed to examine the spall strength of glass under
shock compression. In this experiment no spall signal was observed and the free
surface particle velocity profile is observed to completely unload to its elastic no-
spall prediction (Figure 7.10). An Al 6061-T6 flyer plate was utilized in the
experiment. The impact velocity was 544 m/s; this results in a maximum
compressive stress of 4.21 GPa in glass, which is considerably below its HEL.
From the results of these experiments it can be inferred that the lower bound for
the spall strength of glass is at least 3.89 GPa.
The second experiment, Al/G2, (Figure 7.11) was performed at an impact
velocity of 523 m/s under a combined compression and shear loading at a skew
angle of 18 degrees. The corresponding normal velocity was 497 m/s, which
resulted in a compressive stress of 3.85 GPa, which is also below the HEL. In
this experiment a spall strength of 3.49 GPa was measured. The free surface
particle velocity profile clearly shows the shear wave arrival at 3.73 microseconds
after impact. The transverse velocity is determined by combining the normal
and minus one velocity signals using Equation {VII.C.19}. The peak transverse
velocity is 78 m/s, which is lower than the elastic shear prediction of 159 m/s.
The smaller transverse velocity is expected because similar traits occurred in the
in the literature [Clifton et al., 1998]. The behavior of the transverse velocity as
the impact progresses can not be determined because the spall wave arrives only
0.21 microseconds after the shear wave.
These two experiments determined the difference in the spall strength of the
intact soda lime glass between pure compression and combined compression and
shear loading at a skew angle of 18 degrees and impact velocities around 500 m/s.
The spall strength of the glass exceeds 3.89 GPa at 0 degrees and 4.21 GPa
compressive stress while dropping to 3.49 GPa at 18 degrees and 3.85 GPa
compressive stress. The drop in the spall strength due to a skew angle of 18
degrees shows that like ceramics, the spall strength of glass has a dependency on
shear stress. The effects of the passage failure wave on the spall strength and the
transverse velocity will be discussed in the Section C.2.
272
C.2. The Effects of the Failure Wave on the Properties of Soda Lime Glass
Four combined pressure-shear plate impact experiments were performed with
tungsten carbide flyer plates and soda-lime glass targets. Two of these
experiments occurred at a maximum compressive stress of 3 GPa (WC/G1,
WC/G2). At this stress level, the glass is expected to remain elastic and no
failure wave should be present. If there is sufficient tensile stress, spallation
should occur at the intersection of the two release waves which create State 7’.
The longitudinal wave generated by the spall event propagates to the rear surface
and is measured at State 8’ (Figures 7.5, 7.6). Experiments WC/G1 and WC/G2
were conducted to confirm the lack of a failure wave at this level of compressive
stress. The glass specimen thickness in Experiment WC/G1 was 12.5 mm, while
that in Experiment WC/G2 was 6.5 mm. With this variation in thickness, the
location of the spall plane was altered from in front of the hypothetical location
of the failure wave to behind it. The presence of a failure wave would cause
differences in the spall strength and the transverse particle velocity in the two
experiments.
The other two experiments in this series (WC/G3, WC/G4) were conducted at
stress levels below the HEL, but above the accepted critical stress limit for the
formation of the failure wave. This critical stress has a lower limit of
approximately one-half of the HEL [Ginzburg and Rosenberg, 1998]. Therefore,
the critical stress could be as low as 3.2 GPa, which is one-half of the HEL of 6.4
[Rosenberg et al., 1998]. The chosen maximum compressive stress was 5 GPa.
Experiment WC/G4 was designed so that the failure wave would propagate
through the region of the glass where the spall should occur, prior to the spall
event. The thickness of 6.5 mm was chosen for the glass specimen in order to
place the spall behind the failure wave front (Figure 7.6). This thickness was the
same as experiment WC/G2. Experiment WC/G3 employed the same impact
velocity as experiment WC/G4, but altered the specimen thickness to 12.5 mm.
This changed the spall position (Figure 7.5) so that the glass spalled first and
273
then the failure wave passed through the spalled region. However, due to
insufficient tensile stress, the glass did not reach its spall strength, and thus the
specimen behaved elastically.
C.2.1. Computations for the Elastic PredictionsThe elastic predictions for this set of experiments are found by using the line
equations as indicated on the stress versus particle velocity diagrams that are
shown in Figures 7.5 and 7.6. Because the impedance of tungsten carbide, ZF, is
106 MPa/(m/s), the impacts generate a high particle velocity as well as a high
compressive stress at State 3. The corresponding time versus distance diagrams
show the wave propagation in the target and flyer (Figures 7.5, 7.6). As in the
experiments Al/G1 and Al/G2, the impacts in experiments WC/G1 through
WC/G4 are elastic, and the normal and shear waves propagate at their wave
velocities, CL, and, CS, respectively. In soda lime glass, these velocities are, CL,
5.74 mm/μs and, CS, 3.40 mm/μs. The failure wave propagates at a slower
velocity than the shear stress. The failure wave velocity is calculated from the
longitudinal wave speed, the thickness of the specimen, δ, and the time at which
the anomalous recompression is observed in the free surface velocity profile, tR.
22
R LF L
R L
t CC Ct C
δδ−
=+
{VII.C.21}
Equation {VII.C.21} is based on formulae presented in Kanel et al. [2002].
Despite the larger impedance of tungsten carbide, its wave speeds are only
slightly larger than those of soda lime glass; CL, is 6.99 mm/μs and, CS, is 4.19
mm/μs [Dandekar, 2004]. The inverse of these wave velocities are plotted as
before in the t-X diagrams in Figures 7.5 and 7.6 to show the intersections of
these waves during the impacts.
The free surface particle velocities of experiments WC/G1 through WC/G4 are
shown in Figures 7.12 through 7.15, respectively. In all of these experiments, the
normal signals and the minus one signals are displaced in time. In the
experiments where the signal was of sufficient strength for the plus one velocity
274
to be measured, it is also displayed. The shear signal is computed using either
equation {VII.C.18} or {VII.C.19}. In each of these figures, the elastic
predictions using the line equations are included for comparison.
C.2.2. Experimental Observations of Spall Strength and Transverse ParticleVelocityIn order to fully examine the effects of the failure wave on the soda lime glass
under shock compression and shear, the state of stress as evidenced by the free
surface time histories in the material must be examined. It is evident from the
literature that a failure wave in glass results in the deterioration of the shear
strength, but the normal stress remains nearly constant. The deterioration in
shear stress is reflected in the level of the transverse component of the particle
velocity. The spall strength of the intact glass was also seen to disappear in the
comminuted material behind the failure wave in the literature.
In Experiment WC/G1, the specimen and target were impacted at 220 m/s,
which causes a compressive stress of 2.67 GPa. This experiment was designed to
be in the elastic region, with the longitudinal wave produced by the spall (the
spall signal) reaching the free surface before the rarefaction wave reflected from
the failure wave (the failure wave signal). No failure wave was observed in the
signals in Experiment WC/G1. The free surface particle velocity signals from the
normal and minus 1 beams are shown in Figure 7.12 along with the transverse
velocity, which is computed in the manner discussed above from Equation
{VII.C.19}.
In Figure 7.12, the predicted arrival of the longitudinal compressive wave
corresponds to the observed signal. The normal component of the particle
velocity reaches a level of about 371 m/s after the initial shock. This velocity
corresponds well to the elastic prediction of 368 m/s. The normal compression is
also observed in the minus one velocity signal, with the minus one particle
velocity at the longitudinal wave arrival time being reduced from the normal
particle velocity signal. The difference is calculated as:
275
( )1 1 1 cos2
NV Vθ− = + . {VII.C.22}
The signal, V-1, is the component of the normal particle velocity measured by the
minus 1 beam before the arrival of the shear wave at the free surface. In
Equation {VII.C.22}, the angle θ is 39.7 degrees. The shear wave also arrives at
the indicated time and reduces the minus 1 particle velocity by about 30 m/s,
which is in agreement with the elastic prediction of a 38 m/s drop. The
transverse velocity is calculated from Equation {VII.C.19} to be on average 104
m/s, which is within 14% of the elastic value of 120 m/s. The transverse velocity
remains constant for 0.17 microseconds until the rarefaction wave arrives. After
this time, the transverse velocity can not be determined from the normal and the
minus one beams because of the arrival of the release waves from the lateral
boundary of the target.
The interaction of the rarefaction wave from the rear surface of the flyer plate
and the rarefaction wave from the free surface of the target plate is observed in
State 8. As mentioned in Chapter V, if the resulting tensile stress is high
enough, spall is produced. However, as evidenced by the lack of immediate
reacceleration of the velocity following the velocity drop, the material has not
spalled. The minus one beam clearly shows that the velocity corresponds to the
elastic prediction (Figure 7.12). The minus one velocity drops to 76 m/s. The
corresponding normal velocity drop, by reversing formula {VII.C.22}, is 86 m/s.
The elastic prediction for the drop in normal velocity is 89 m/s. The close
agreement between the elastic prediction and the actual velocity drop indicates
that no spall has occurred. This drop is therefore distinct from the lack of
reacceleration of the spall signal in silicon nitride in Experiment SC-13 from
Chapter V, which does not correspond to the elastic prediction for no spall. In
that experiment, the spall strength is zero.
In Experiment WC/G2, the impact velocity was also 220 m/s. This resulted in a
compressive stress level of 2.67 GPa. The normal, minus one and plus one beams
276
were all recorded with sufficient intensity for the velocity to be measured. The
resulting free surface velocities (Figure 7.13) show an initial rise in the normal
component of the free surface particle velocity to about 370 m/s, which is close to
the elastic prediction of 368 m/s. At the time of the shear arrival, the plus one
and minus one signals diverge in opposite directions, from the initial value, which
represents the component due to the normal wave. This results in a transverse
velocity wave with a peak of 109 m/s, which again is close to the elastic
prediction of 120 m/s. The transverse velocity then drops off and reaches a near
zero value by 0.70 microseconds.
Although the material is supposed to be elastic, there is clearly a deflection
visible in the normal and the minus one and plus one beams corresponding to a
reflection of the compressive wave from the failed region. The signal occurs at
2.33 μs. This raises the normal velocity by about 13 m/s, to around 383 m/s.
Using Equation {VII.C.21} with 2.33 μs as, tR, the velocity of the failure wave is
computed to be 1.82 km/s. This value was used to predict the failure wave
arrival time in all the other experiments.
In Experiment WC/G2 the spall wave is clearly visible and has a velocity dip of
94 m/s. This corresponds well to the elastic prediction of 88 m/s. The normal
velocity does not immediately reaccelerate, but rather stays the same for 0.53
microseconds. The reacceleration occurs at the elastically predicted time of the
next wave reflection. This pattern corresponds to the material remaining intact
and strong enough to prevent spallation. Therefore, the small change in the
particle velocity recorded at the failure time of 2.33 μs does not reflect the total
failure of the material, but rather some non-critical damage.
In Experiment WC/G3, the impact velocity is 411 m/s and the spall is designed
to occur in the intact glass before the failure wave arrives. The impact produces
a compressive stress level of 4.99 GPa. The normal and minus one beams are
strong enough for the velocity to be determined (Figure 7.14). In the normal
velocity signal, there is a slightly lower velocity in State 6 than the elastic normal
277
prediction, about 667 m/s. The elastic prediction is 687 m/s. These values are
only off by 3% and is almost within the VISAR error tolerances of 2%. The
velocity of the minus one beam indicates that the transverse velocity reaches a
level below that of the elastic prediction of 224 m/s. The transverse velocity
records a transient pulse that has a maximum value of 147 m/s. The transverse
velocity is no longer accurately measured by the normal, plus one, and minus one
beams after the rarefaction wave reaches the free surface, as was also the case in
experiment WC/G1.
The normal velocity signal’s dip to near the elastic level at the time when the
spall wave should occur and the long time before reacceleration indicate that
insufficient tensile stress has been generated to cause the material to spall. The
velocity dip is 145 m/s this time. The elastic prediction is 165 m/s.
There are oscillations in the measured signals at the time of the arrival of the
longitudinal wave at the rear surface of the target plate after reflecting from the
failure front, i.e. at time 4.36 microseconds, as estimated using the failure wave
velocity calculated in Experiment WC/G2. The presence of an oscillatory signal
at this time indicates that failure has occurred. These oscillations are particularly
evident in the minus one beam where the velocity rises to about 30 m/s. In the
normal beam, the oscillations only change the particle velocity by approximately
5 m/s. Thus, this failure wave is evident in the transverse motion of the plate.
Strangely, the failure wave did not greatly affect the normal velocity signal at
this time.
Experiment WC/G4, with its thinner target plate exhibits a failure wave that
clearly affects the spall strength. In this experiment the impact velocity is 408
m/s, which corresponds to a compressive stress of 4.96 GPa. The normal signal
corresponds well to the elastic prediction (Figure 7.15). The measured particle
velocity is 686 m/s and the elastic prediction is 683 m/s. The minus one signal
shows a larger discrepancy between 632 m/s, the measured value and 603 m/s,
which is the elastic prediction. However, this is still only a 5% difference. The
278
transverse velocity elastic prediction is 222 m/s. The actual transverse velocity
only reaches 148 m/s and it rises and falls over a period of 0.28 microseconds.
The failure wave occurs at, 2.34 microseconds, and is apparent as an increase in
the particle velocity of both the normal and the minus one signals. The time of
this increase approximately corresponds to the time of arrival of the failure wave
using the failure wave velocity estimate of 1.82 km/s, as obtained from
experiment WC/G2. This increase is of the order of 45 m/s for the normal
component of the particle velocity. This increase is indicative of a reduction in
impedance of the material behind the failure wave. Using Equation {VII.C.21},
and the actual time at which the velocity rise begins, the velocity of this failure
wave is calculated to be 1.63 km/s for this experiment. No spall signal is
observed indicating that the failed material has no resistance to the generated
tensile stress, and thus zero spall strength. The resulting velocity signals show no
wave reflections following the spall arrival, which confirms that the glass has
separated at the spall plane.
C.3. Observation of the Impedance in the Comminuted MaterialThe glass on tungsten carbide experiments (G/WC1, G/WC2) were performed
using the reverberation method discussed in Chapter VI to observe the change in
the impedance of glass in to the presence failure wave (Figure 7.7). The glass
flyers were three times as thick as the target plates in these experiments. The
Hugoniot state and two reverberations were measured before the arrival of the
flyer unloading wave. The experiments were conduced using 3.04 GPa and 5.51
GPa impact stresses. Experiment G/WC1, which was conducted at 3.04 GPa of
impact stress, did not show any failure wave and the normal velocity profile
corresponded well to the elastic prediction (Figure 7.16). The 5.51 GPa
experiment (G/WC2) had a failure wave, and the resulting change in the normal
velocity of the first reverberation is visible in Figure 7.17. Also, the normal and
transverse velocities do not conform to the elastic predictions.
279
C.3.1. Elastic Computations for the Internal Stresses and the AcousticImpedance in Experiments G/WC1 and G/WC2
The reverberation experiments can be analyzed in the same manner as the
reverberation experiments in Chapter VI. In Figure 7.8, the stress versus particle
velocity diagrams and time vs. distance diagrams for Experiments G/WC1 and
G/WC2 are given. The only new equations are the equations for determining the
stresses and the acoustic impedance from the free surface particle velocities. In
these equations, the flyer impedance, ZF, refers to the soda lime glass and the
target impedance, ZT, refers to the tungsten carbide.
For the elastic calculations, State 1 is the flyer’s normal impact velocity, as found
in Equation {VII.C.2}. State 2 is a state of zero stress and zero particle velocity
in the target, prior to impact. This means that to reach the same stress level in
the Hugoniot state as the tungsten carbide on glass experiments (WC/G), the
material must be impacted at the same velocity. To prove this, the general line
Equations {VII.C.5} and {VII.C.6} can be used to calculate the stress and
particle velocity in State 3:
13
F
T F
Z VVZ Z
=+
, {VII.C.23}
( )3 1 T F T F=V Z Z /(Z +Z )σ . {VII.C.24}
This is the same as substituting Equation {VII.C.1} into Equation {VII.C.4}.
The other states leading up to the velocity equation for State 6, where the
impedance is measured are also determined by the application of the line
equations. State 4, which follows the reflection of the compressive wave from the
free surface of the specimen is
4 32V V= , {VII.C.25}
4 0σ = . {VII.C.26}
280
State 5 is computed from States 3 and 4,
4 15
T F
T F
Z V Z VVZ Z
+=
+, {VII.C.27}
( )5 1 5FZ V Vσ = − . {VII.C.28}
And State 6, which is the state on the free surface of the specimen following the
first reverberation is:
5 56
T
T
Z VVZ
σ += , {VII.C.29}
6 0σ = . {VII.C.30}
These equations are then used to determine the change in the impedance of the
material due to the passage of the failure wave. The values for the normal
impact velocity, V1, and the two measured free surface velocities, V4, and, V6, are
used. Rearranging and combining Equations {VII.C.27} through {VII.C.29} to
solve for, ZF, yields:
( )( )
6 4
1 4 62−
=− −
TF
Z V VZ
V V V. {VII.C.31}
The shear impedance of the flyer is computed in the same manner as Equation
{VII.C.31}, except that the shear impedance of the target and the shear
velocities are used in place of the longitudinal values.
C.3.2. Measurements of the Normal and Transverse Particle Velocities andthe Impedance Change Due to the Failure Wave
281
Experiment G/WC1 was conducted at an impact speed of 250 m/s. This
resulted in a maximum compressive stress of 3.04 GPa. The normal components
of the particle velocity profiles are closely matched with the elastic predictions.
In fact, the estimated velocity in the Hugoniot state of the tungsten carbide is 57
m/s, which is the same as the elastic prediction. The first reverberation has a
velocity of 101 m/s and the elastic prediction is 102 m/s. The particle velocity in
the second reverberation is 134 m/s, which is also the elastic prediction.
Unlike the normal particle velocity, the velocities of the different states of the
plus one and minus one beams do not agree with the elastic predictions for these
velocities. In fact, the velocities of these two signals are closer to the average of
their elastic predictions. This is because, the plus one and minus one beams
deflect less in the presence of shear than is elastically predicted in all the cases.
As a result of the plus one and minus one beams not matching the elastic
predictions, the transverse velocity only reaches 10 m/s at about 1 microsecond,
which is lower than the elastic prediction of 18.4 m/s. This velocity slowly
ramps downwards and reaches 2.5 m/s at 1.7 microseconds.
In the 454 m/s (5.51 GPa) experiment, G/WC2, the Hugoniot velocity in the
tungsten carbide occurs at the predicted time of 0.57 microseconds, but only
reaches 95 m/s, instead of the elastic prediction of 104 m/s. The normal
component of the particle velocity in the reverberations also exhibit similar
behavior. The first reverberation only exhibits a velocity of 167 m/s instead of
the elastic prediction of 185 m/s. The second reverberation is 234 m/s, which is
lower than the elastic prediction of 244 m/s. The third reverberation is 274 m/s
instead of the elastic prediction of 289 m/s. Again, the plus one and minus one
signals are less than their elastic predictions . The transverse component of the
particle velocity peaks at 5.9 m/s instead of 33.42 m/s, which is the elastic
prediction. As in experiment G/WC1 there is a slow drop off from this peak.
The transverse velocity takes 0.63 microseconds to return to the zero level.
282
In Figures 7.16 and 7.17, the first reverberation is clearly visible. In experiment
G/WC1, using Equation {VII.C.31} results in a normal impedance of 13.6
MPa/(m/s), which is within 0.9 MPa/(m/s) of the elastic predicted value. The
shear impedance is 5.94 MPa/(m/s), which is 2.66 MPa/(m/s) lower than the
elastic value of 8.6 MPa/(m/s). In experiment G/WC2, once the reflection of
stress waves from the failed material reaches the rear surface, there is a change in
the particle velocity of 45 m/s. If the velocity for State 6 before this velocity
change is used, the normal impedance found from Equation {VII.C.31} is 13.0
MPa/(m/s). This is slightly lower than that observed in experiment G/WC1.
However, the shear impedance is 4.24 MPa/(m/s). This is 49% of the elastic
shear impedance. These results suggest that the failure wave causes only a small
change in the impedance and the longitudinal wave speed in the failed glass, but
a substantial change in the shear impedance.
283
D. Conclusions and Discussion
Based on the eight experiments conducted in this chapter, several observations
can be made that extend the existing knowledge of failure waves. These
conclusions cover the effect of the failure wave on the shear strength, spall
strength, and the impedance of soda lime glass. Also, the spall strength
sensitivity to change in the impact skew angle of soda lime glass is discussed.
The spall strength of the material has been determined from Experiment Al/G1
to be larger than 3.89 GPa at 544 m/s impact velocity in the absence of shear
stress. This is in agreement with previous results, such as Kanel [Kanel et al.,
2002] and Brar [Brar et al., 1991 B], where the spall strength was found to be in
excess of 3 GPa. The spall strength at a normal velocity of 497 m/s in the
presence of shear due to a skew angle of 18 degrees was found to be 3.49 GPa in
experiment Al/G2. This result indicates that spall strength is lost with the
application of shear stresses, as in ceramic materials like AS800 silicon nitride.
Recent unpublished work by Dandekar [2005] indicates the spall strength of glass
under shock compression is around 3.5 GPa. This suggests that the spall
strength does not change with shear stress. Further research is necessary to
clarify this point.
The spall strength is clearly altered behind the failure front. In the tungsten
carbide impacting glass experiments (WC/G), ahead of the failure wave the spall
strength is not degraded. This is evident because insufficient tensile stress exists
to cause spall and the elastic reflections are clearly visible in the free surface
velocity profiles. In experiment WC/G1 (220 m/s impact velocity), the velocity
drops 86 m/s at the time where a spall signal should occur, and in experiment,
WC/G3 (411 m/s impact velocity), it drops to 145 m/s. In these experiments
tensile stresses of 644 MPa and 1.20 GPa, were generated. Clearly neither
experiment has a tensile stress as large as the 3.5 GPa spall strength. Thus, the
material behaves elastically, and no spall occurs. The lack of spall is confirmed
by the observation that the material ahead of the failure wave is intact for
284
signals to propagate through. In Experiment WC/G3, a failure wave occurs and
is measured by oscillations in the minus one velocity signal.
As predicted, the spall strength of soda lime glass disappears behind the failure
wave. In experiment WC/G2, instead of a spall signal elastic wave reflection
occurs similar to that observed in the two experiments WC/G1, WC/G3. The
velocity drops by 94 m/s, which is close to the elastic prediction of 88 m/s.
However, some damage to the material occurs, as indicated by the small rise in
the particle velocity seen at the approximated failure wave arrival time. In
experiment WC/G4, at 408 m/s impact velocity, the material failed completely
as zero spall strength is recorded and a larger rise in the particle velocity of 45
m/s is observed at the failure wave arrival time. The average failure wave
velocity in these experiments is 1.7 ± 0.1 km/s.
The transverse particle velocity profiles are summarized in Figure 7.18. In the
experiment WC/G1 the transverse velocities are close to the elastic predictions.
The transverse free surface particle velocity is on an average 104 m/s which is
lower than the elastic prediction of 120 m/s by 14%. The value is steady until
the arrival of the rarefaction wave 0.39 microseconds later. Experiment WC/G2
has smaller normal velocities than the elastic predictions. The peak transverse
velocity is 109 m/s, which is within 9.2% of 120 m/s, which is the elastic
prediction. The transverse velocity decreases to zero after 0.7 microseconds.
There are no major differences between the transverse velocities in the two
experiments. This is also consistent with the results of Clifton et al. [1998] for the
transverse component of the particle velocities.
In measurements of transverse velocity, at higher impact velocities, the glass
material behaves differently from previous results at high velocities [Clifton et al.,
1998]. In the current experiments, the transverse velocity is about 30% smaller
than the elastic predictions. Experiment WC/G3 has a transverse velocity that
exhibits a transient pulse with a peak at 147 m/s, which is 34% smaller than the
elastic prediction of 224 m/s. The transverse particle velocity peaks just before
285
the arrival of the rarefaction wave. In experiment WC/G4, the peak in the
transverse particle velocity is 148 m/s, which is 33% less than the elastic
prediction of 222 m/s. This particle velocity pulse lasts 0.28 microseconds before
dropping to a level of about 67 m/s. The transverse component of the particle
velocity is less than the elastic prediction in both cases. In Clifton’s experiments,
at 4.3 GPa compressive stress [Clifton et al, 1998], the transverse velocities
reached a level near the elastic prediction and then fell off continuously. This is
different from the behavior seen in Figure 7.18, where the peak value is 30%
smaller than the elastic estimate. Of course, the current high velocity
experiments are conducted at a higher stress level than conducted by Clifton.
The experiments where the glass impacts the tungsten carbide also exhibit
similar trends. The normal velocity of experiment G/WC1, is predicted well by
the elastic equations. The transverse velocity reaches 10 m/s, which is down by
45% from 18.4 m/s, the elastic prediction, and then it falls slowly. In experiment
G/WC2, on the other hand, the normal velocity is distinctly below that of the
elastic prediction. In the Hugoniot state, the normal velocity is 95 m/s instead of
the elastic value of 104 m/s. The peak transverse velocity is 5.9 m/s, which 82%
lower than the elastic value of 33.42 m/s. This inconsistency between the
experimental and elastic values is much greater than the mismatch in the WC/G
experiments. This large transverse velocity is also different from the research
done by Clifton et al. [1998]. For these types of experiments, the transverse
velocity was the same as the elastic prediction for the experiment at 3.4 GPa
compressive stress and decreased from a peak level to a lower plateau at 4.4 GPa
[Clifton et al., 1998]. Neither behavior is exhibited in the current experiments.
The impedance of the material decreases slightly from Experiment G/WC1 to
Experiment G/WC2. The elastic material impedance is estimated to be 14.5
MPa/(m/s). The calculated impedance in Experiment G/WC1 is 13.6
MPa/(m/s), which is 0.9 MPa/(m/s) lower than the elastic value. The normal
particle velocity profile fits the elastic prediction in experiment G/WC1. The
difference in the calculated impedance from experiment G/WC1 to experiment
286
G/WC2 is only 0.6 MPa/(m/s). This difference, though present, and apparently
dependent on impact velocity, is only 10% at 5.51 GPa. This suggests that the
assumption of constant longitudinal properties across the failure front is nearly
valid.
287
References
Barker L.M, Barker V.J., Barker Z.B., 2000. Valyn VISAR User’s Handbook. Albuquerque, New
Mexico, USA.
Bourne, N. K. and Z. Rosenberg., 1996. Shock Compression of Condensed Matter – 1995.
American Institute of Physics. 567.
Brar, N. S., S. J. Bless, and Z. Rosenberg., 1991 A. Impact-Induced Failure Waves in Glass Bars
and Plates. Applied Physics Letters. Vol. 59. No. 26. 3396-3398.
Brar, N. S., Z. Rosenberg, and S. J. Bless., 1991 B. Spall Strength and Failure Waves in Glass.
Journal de Physique IV. Vol 1. C3. 639-644.
Clifton, R. J., M. Mello, and N. S. Brar., 1998. Effect of Shear on Failure Waves in Soda Lime
Glass. Shock Compression of Condensed Matter – 1997. American Institute of Physics. 521-524.
Dandekar, D. 2004. Spall Strength of Tungsten Carbide. U.S. Army Research Laboratory.
ARL-TR-3335.
Dandekar, D. 2005. Unpublished measurements of the spall strength of glass.
Espinosa, H. D., Y. Xu, and N. S. Brar., 1997. Micromechanics of Failure Waves in Glass: I,
Experiments. Journal of the American Ceramic Society. Vol. 80. No. 8 2061-2073.
Feng, R., 2000. Formation and Propagation of Failure in Shocked Glasses. Journal of Applied
Sciences. Vol. 87. No. 4. 1693-1700.
Ginzburg, A. and Z. Rosenberg, 1998. Using Reverberation Techniques to Study the Properties
of Shock Loaded Soda-Lime Glass. Shock Compression of Condensed Matter – 1997. American
Institute of Physics. 529-531.
Kanel, G. I., S. V. Rasorenov, and V. E. Fortov., 1992. The Failure Waves and Spallations in
Homogeneous Brittle Materials. Shock Compression of Condensed Matter – 1991. Elsevier
Science Publishers. 451-454.
Kanel, G. I., A. A. Bogatch, S. V. Razorenov, and Z. Chen., 2002. Transformation of Shock
Compression Pluses in Glass Due to the Failure Wave Phenomena. Journal of Applied Physics.
Vol. 92. No. 9. 5045-5052.
288
Rosenberg, Z., S. J. Bless, and D. J. Yaziv., 1985. Journal of Applied Physics. Vol. 58. 3249.
S. Sundaram, 1998. Pressure-Shear Plate Impact Studies of Alumina Ceramics and the Influence
of an Intergranular Glassy Phase. Doctoral Thesis. Brown University.
Tables
Table 7.1: Experimental parameters for the experiments studying the spall strength of glass
(Al/G) and those studying the effect of a failure wave on the spall strength and transverse
(Trans.) velocity of glass (WC/G). This table includes both the experimental data and elastic
estimates.
Exp # Flyer
Type
Specimen
Thickness
Impact
Angle
Impact
Velocity
Hugoniot
Stress
Peak
Trans. Vel.
Elastic
Trans. Vel.
Spall
Strength
Al/G1 Aluminum 12.5 mm 0° 544 m/s 4.21 GPa n/a n/a >3.89 GPa
Al/G2 Aluminum 12.5 mm 18° 523 m/s 3.85 GPa 78 m/s 159 m/s 3.49 GPa
WC/G1 WC 12.5 mm 18° 220 m/s 2.67 GPa 104 m/s 120 m/s n/a
WC/G2 WC 6.5 mm 18° 220 m/s 2.67 GPa 109 m/s 120 m/s n/a
WC/G3 WC 12.5 mm 18° 411 m/s 4.99 GPa 147 m/s 224 m/s n/a
WC/G4 WC 6.5 mm 18° 408 m/s 4.96 GPa 148 m/s 222 m/s 0
Table 7.2: Experimental parameters for glass on tungsten carbide experiments. Note, all
velocities and stresses are in the tungsten carbide. Only the Hugoniot velocity and stress are
given..
Normal Velocity Transverse Velocity
Exp
Number
Impact
Angle
Impact
Velocity
Hugoniot
Stress
Impedance Exp.
Hugoniot
Elastic
Hugoniot
Exp.
Hugoniot
Elastic
Hugoniot
G/WC1 18° 250 m/s 3.04 GPa 13.6 MPa/(m/s) 57 m/s 57 m/s 10 m/s 18.4 m/s
G/WC2 18° 454 m/s 5.51 GPa 13.0 MPa/(m/s) 95 m/s 104 m/s 5.9 m/s 33.4 m/s
289
Figures
Figure 7.1: Specimen configuration for normal spall strength experiment. The impact surfaces are
normal to the direction of motion. The laser beam from the VISAR probe reflects off the
aluminum coating on the free surface of the soda lime glass.
Figure 7.2: Specimen configuration for pressure-shear spall strength experiment. The impact
surfaces are at 18 degrees to the direction of motion. A grating creates three output laser beams.
290
Figure 7.3: Spall strength Experiments Al/G1 and Al/G2. (a) Time vs. distance diagram
including pressure and shear wave propagation. (b) Stress vs. velocity diagram including the
effects of spall strength.
Figure 7.4: Specimen configuration for WC/Glass experiments. This configuration is similar to
Experiment Al/G2, except that tungsten carbide replaces the aluminum as the flyer material.
The result is a greater compressive stress in the Hugoniot state, but a smaller tensile stress in
State 7.
291
Figure 7.5: (a) Time vs. distance diagram and (b) Stress vs. velocity diagram for spall ahead of
the failure front in experiments WC/G1 and WC/G3. In these experiments, the geometry is such
that any spallation occurs in the intact material.
Figure 7.6: (a) Time vs. distance diagram and(b) Stress vs. velocity diagram for spall behind the
failure front in experiments WC/G2 and WC/G4. The thickness of the specimen is smaller than
in Figure 7.5, so that any spall takes place in the comminuted glass.
292
Figure 7.7: Specimen configuration for the glass flyer and WC target experiments. In these
experiments, the velocity measurements on the tungsten carbide are used to calculate the stresses
inside of the soda lime glass.
Figure 7.8: (a) Time vs. distance diagram showing longitudinal and shear waves and (b) Stress vs.
velocity diagram for glass flyer and tungsten carbide specimen experiments. Three reverberations
are indicated on the S-V diagram.
293
Figure 7.9: Experimental measurement technique using three VISAR probes for normal and
transverse velocity measurements and velocity measurement system. The green (535 nm
wavelength) laser beam passes through a 1000 mm focal length lens and reflects off of a mirror
(M1) before reaching the specimen. The grating produces three beams, one normal and two at
±39.7°. Because the input laser beam passes through the lens at 8 mm from the center of the
lens, the zero order reflected beam from the grating actually is reflected at a slight angle. Mirrors
M2, M3, and M4 reflect the output laser beams to points outside of the impact chamber. Three
VISAR probes collect these beams. The velocity system uses three red laser beams perpendicular
to the path of the projectile. These beams are cut in sequence and as they are, the measured
intensity at the photodiode decreases. A time record of this intensity is used to determine the
impact velocity.
294
Time (seconds)
Vel
ocity
(km
/s)
1E-06 2E-06 3E-06 4E-06 5E-06 6E-060
0.1
0.2
0.3
0.4
0.5
0.6
Normal
Figure 7.10: Spall experiment Al/G1 at an impact velocity of 544 m/s and an impact angle of
zero degrees. The actual velocity follows the elastic predictions for Hugoniot state and State 8.
295
Time (seconds)
Vel
ocity
(km
/s)
1E-06 2E-06 3E-06 4E-06 5E-06 6E-060
0.1
0.2
0.3
0.4
0.5
0.6NormalMinus 1Transverse
Figure 7.11: Spall experiment Al/G2 at an impact angle of 18 degrees and an impact speed of 523
m/s. The normal beam signal is lost during the spall, but the rebound of the minus one beam
indicates that spallation is occurring. The elastic levels, shown by the dotted lines agree closely
with the actual results of the Hugoniot state. The difference between the minimum spall velocity
and the elastic prediction for State 8 is another indication that spall is occurring.
296
Time (seconds)
Vel
ocity
(km
/s)
1E-06 2E-06 3E-06 4E-06 5E-06 6E-060
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75NormalMinus 1Transverse
Arrival ofNormalWave
Arrival ofSpall Wave
Arrival ofShear Wave Failure
Wave
Figure 7.12: Experiment WC/G1 with an impact velocity of 220 m/s. The predicted arrival of
the spall signal is before the time a failure wave could reach the surface. The elastic predictions
agree well with the normal and transverse velocities. No failure wave or spall effects occur.
Instead, the elastic wave interaction causes a drop in particle velocity. No velocity change is seen
at the time of arrival of a longitudinal wave reflecting from the failure front.
297
Time (s)
Vel
ocity
(km
/s)
0 2E-06 4E-060
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
NormalMinus 1Plus 1Transverse
NormalWave
ShearWave
SpallWave
FailureWave
Figure 7.13: Experiment WC/G2 has an impact speed of 220 m/s. The elastic predictions for the
normal and transverse particle velocities predict the actual behavior well. In this experiment, a
small upturn in the material upon is seen at the predicted time of arrival of the longitudinal
rarefaction wave reflecting from the failure front (the failure wave). This wave arrival is before
the predicted arrival time of the elastic interaction, which if the tensile stress exceeded the spall
strength, would be the spall wave.
298
Time (seconds)
Vel
ocity
(km
/s)
1E-06 2E-06 3E-06 4E-06 5E-06 6E-060
0.1
0.2
0.3
0.4
0.5
0.6
0.7
NormalMinus1Transverse
NormalWave
Arrival
ShearWave
Arrival
SpallWave
Arrival
FailureWave
Figure 7.14: Experiment WC/G3 has a velocity of 411 m/s, and the elastic wave arrival at the
spall wave arrival time is before the failure wave arrival. As in the lower velocity experiments
(WC/G1, WC/G2) the material has greater spall strength than the tensile stress that the
experiment produces. Therefore, the material behaves elastically. Vibrations in the minus 1
beam appear at the time, which a longitudinal reflection from the failure wave should reach the
free surface.
299
Time (seconds)
Vel
ocity
(km
/s)
0 1E-06 2E-06 3E-06 4E-06 5E-060
0.1
0.2
0.3
0.4
0.5
0.6
0.7
NormalMinus 1Transverse
NormalWave
Arrival
ShearWave
Arrival
SpallWave/ReflectedWaveArrival
Approx.FailureWaveArrival
Figure 7.15: Experiment WC/G4 shows the arrival of the failure wave, which is the rarefaction
wave reflected from the failure front, and no spall signal at an impact speed of 408 m/s. The
normal and minus 1 beams show a drop followed by a rise in particle velocity at the time of
arrival of this failure wave.
300
Time (seconds)
Vel
ocity
(m/s
)
0 1E-06 2E-06 3E-06 4E-060
25
50
75
100
125
150
175
200
NormalPlus 1Minus 1Transverse
Elastic NormalPrediction
Elastic Shear Prediction
Plus 1 Elastic
Minus 1 Elastic
Figure 7.16: Experiment G/WC1 shows the reverberations in the tungsten carbide at an impact
velocity of 250 m/s. The elastic prediction for the normal velocity is followed precisely by the
experimental data. However, the deflections in the plus 1 and minus 1 beams due to shear are
less than the elastic predictions. As a result, the transverse velocity is smaller than predicted.
301
Time (seconds)
Vel
ocity
(km
/s)
0 1E-06 2E-06 3E-06 4E-060
0.1
0.2
0.3
NormalPlus1Minus 1Shear
Normal Elastic
Shear Elastic
Plus 1 Elastic Minus 1 Elastic
Failure WaveFrom BackOf Failed Material
Figure 7.17: Experiment G/WC2 which was impacted at a speed of 454 m/s shows the
reverberation in the tungsten carbide. The normal velocity from the experiment is lower than the
elastic prediction. The transverse velocity is also much lower than elastically predicted.
302
Time (seconds)
Vel
ocity
(m/s
)
0 2.5E-07 5E-07 7.5E-070
50
100
150
WC/G3WC/G4WC/G1WC/G2G/WC1G/WC2
Figure 7.18: Transverse velocity measurements for tungsten carbide on glass and glass on
tungsten carbide experiments. The transverse velocities in Experiments WC/G1 and WC/G2 are
smaller than the elastic predictions by between 14% and 9.2% respectively. A larger discrepancy
occurs in experiments WC/G3 and WC/G4, where the experimental velocities are off by 34% and
33% respectively. The glass impacting tungsten carbide experiments have the smallest transverse
velocities compared to the elastic predictions. Experiment G/WC1’s transverse velocity is off by
45%, while Experiment G/WC2’s transverse velocity is off by 82%.
303
Chapter VIII – Summary of Experimental Analyses
In this thesis, two materials were studied: soda lime glass and AS800 grade
silicon nitride. Soda lime glass is often used in windows of military vehicles and
aircraft where integrity in the event of shrapnel impacts is of vital concern.
Soda lime glass is also commonly used as a model material, in order to study the
behavior of glasses and glass ceramics under static and dynamic loading
conditions. AS800 grade silicon nitride is considered one of the leading material
candidates for the next generation of aircraft engine turbine blades because of its
superior high temperature properties when compared with nickel based
superalloys. Silicon nitride has a higher melting temperature [Brandes and
Brook, 1999; Matweb, 2005] better creep resistance, at elevated temperatures [Lin
et al., 2001; Brandes and Brook, 1999] than nickel based superalloys. Also, this
grade of silicon nitride has superior particle impact resistance than other silicon
nitride grades [Choi et al., 2002]. The environment of a turbine includes many
unburned fuel particles and other small pieces of debris traveling at high velocity,
so the resistance of these blades to high velocity impacts is of critical importance.
In order to examine the impact of these brittle materials, both planar shock
compression and particle impact studies were conducted. Understanding the full
extent of stress wave propagation in three dimensions requires both experimental
observations and numerical modeling. A projectile accelerator system was
designed specifically for the particle impact experiments. The particle impact
experiments on soda lime glass and the numerical study of these impacts
[Nathenson et al., 2005] provide insight into the three dimensional features of the
impacts. Planar shock compression and pressure-shear experiments enable the
determination of quantifiable measures of dynamic strength such as the Hugoniot
Elastic Limit and the spall strength. Many factors influence the material
including impact velocity, skew angle, and previous shock damage. Shock
compression experiments examine the properties of silicon nitride and glass under
these conditions.
304
A. Particle Impact Experiments on Soda Lime Glass
In order to conduct the particle impact experiments on soda lime glass, a
projectile firing system was designed and constructed. The design is described in
Chapter II. This system uses compressed air to accelerate a sabot and projectile
arrangement down a 4.5 foot long, ¾ inch diameter barrel. The sabot and
projectile are separated by means of a sabot stripper and the spherical 1/16th inch
diameter projectile impacts the plates of soda lime glass at velocities varying
from 150 m/s to 350 m/s. This impact velocity was determined using a laser
velocity measurement. Four plate thicknesses were tested, 3 mm, 5 mm, 15 mm,
and 25.4 mm. These experiments were monitored both with surface strain gages
and with high speed camera imagery. The strain gages monitored both the radial
and hoop strains on the impact surface at a predetermined point, which was
approximately 10 mm from the impact location, on the specimen surface. The
experiments and their results are detailed in Chapter III.
Stress wave theory predicts three types of waves within a elastic solid during a
typical three dimensional impact event. Two of these waves travel within the
material, expanding in a hemispherical pattern; a longitudinal compressive wave
and a shear wave. In soda lime glass, these waves travel at velocities of 5740 m/s
and 3400 m/s. On the surface of the material, the Rayleigh wave travels at
about 90% of the shear wave speed. The Rayleigh wave has a cylindrical wave
front. These three waves create a particle velocity field and a stress field in the
material. The effect of these fields can be seen on the surface from strain gage
measurements and deduced throughout the interior of the material by means of
the numerical simulations.
The numerical work using LS-DYNA 3D and performed by Guodong Chen
[Nathenson et al., 2005] built on the experiments performed in this thesis. The
experimental impact velocities and contact times were used to calibrate the
impact model. Hertzian elastic theoretical models for the peak force and contact
time also confirmed the model’s accuracy. For example, the contact times of the
305
Hertzian theoretical equations, numerical modeling, and experimental
observations all agreed to within 3%.
The LS-DYNA numerical model of an infinite half space indicates the arrival of a
single strain pulse that first exhibits tension and then compression followed by
smaller reverberations. The nature of this wave concurs with the form of the
theoretical estimates of the surface strain wave [Mitra, 1964]. The numerical
simulations of the experiments using LS-DYNA 3D, that take into account the
finite thickness and infinite lateral boundaries of the specimens indicate an effect
with decreasing thickness. Specifically, the tensile part of the strain pulse is
intensified, the compressive part disappears, and the reverberations change form.
The alterations in the strain pulse occur at or after the time of the wave arrival
from the rear surface. The addition of lateral boundaries only changes the late
time reverberations after the boundary waves have returned to the measurement
location [Nathenson et al., 2005].
The elastic strains computed in LS-DYNA at the measurement location have
features both similar to and distinct from the experimental data. The
experimental strain gage signals record a single pulse in tension. This pulse is an
order of magnitude smaller than the elastic simulations, reflecting the limiting
fracture strength of the actual glass. The LS-DYNA numerical model does not
incorporate cracking. The structure of this tensile pulse varies depending on the
thickness of the specimen more than the velocity. The thicker specimens, 15 mm
and 25.4 mm, exhibit a smaller magnitude tensile pulse than the thinner
specimens, 3 mm and 5 mm. Additionally, the thin experiments have a longer
pulse duration due to the crack propagation. With regards to the experiments
undertaken to measure rear surface strains, the cracking patterns were observed
to vibrate. The strain records in both of these experiments clearly indicated
oscillations whose periods match the cracking pattern vibrations. The LS-DYNA
simulations also show oscillations with this approximate period, especially in the
306
thinner specimens. This suggests that the internal stresses and surface strains
are related to the oscillations in the cracking patterns.
A kinetic analysis of the partitioning of energy of impact into various dissipative
modes was performed using the coefficient of restitution calculated via stress
wave theory, the LS-DYNA numerical model, and the experimental camera
images. Stress waves and vibrations take up most of the elastic energy. The
experimental coefficients of restitution are much smaller than those obtained
from the LS-DYNA simulations or the theoretical modeling. Therefore, it is
surmised that cracking is responsible for a larger fraction of energy loss when
compared with the elastic processes. The amount of energy lost due to cracking
and elastic modes increases with impact velocity, with the larger percentage of
energy being dissipated in cracking.
In the experiments four types of cracking patterns were observed in the material.
Cone cracks were observed in the thin specimens at all impact velocities. They
became more prominent with increasing impact velocity, ejecting from the rear
surface in some cases. Radial cracking from both the front and rear surface was
seen in the thinner specimens. These cracks also became more prominent with
increasing impact velocity. Lateral cracks were not recorded in the thinnest
specimens, but occurred in most of the other experiments. These cracks were
seen to increase in size and fullness with impact velocity, but not thickness. In
almost every experiment, surface chipping occurred. It also increased in diameter
with impact velocity but not thickness. The initiation of splinter cracks appears
to be suppressed in the thinner specimens at higher impact velocities. Of the four
modes of cracking observed in the experiments and in previous experiments,
radial cracks are the most critical because they propagate laterally through the
specimen. The conical and lateral cracks both can cause local material ejection,
so they are more deleterious to the material than the splinter cracks, which only
propagate for a distance on the order of 1 mm.
307
It is evident from the above study that the LS-DYNA numerical elastic model
predicts the experimental behavior well, except for the effects of cracking. This
has allowed several observations to be made. The propagation of stress waves on
the surface has been observed to be limited by the finite failure strength of the
glass. The partitioning of energy during the impact indicates energy dissipation
by vibrations as well as stress waves and cracking. The observed cracking
patterns have also been shown to vary with thickness and with impact velocity
and their relative damage potential has been discussed. These observations
provide a guide to the behavior of soda lime glass, and by extension other glasses
and glass ceramics of the same nature, under three dimensional high speed
particle impact.
308
B. Planar Shock Compression and Pressure-Shear Experiments on AS800Grade Silicon Nitride
The planar nature of the stress wave propagation in shock compression
experimentation allows for the use of one dimensional stress wave theory. The
configuration of the single stage gas gun and the preparation of experimental
specimens are discussed in Chapter IV. From these experiments, the free surface
particle velocity is observed. This in turn allows the internal stress state, the
Hugoniot Elastic Limit (HEL), and the spall strength of Si3N4 to be evaluated.
In the experiments on AS800 grade silicon nitride, discussed in Chapter V,
impacts were carried out under pure compression and combined pressure-shear
loading conditions. The shock compression experiments occurred at impact
speeds varying from 65 m/s to 546 m/s. The highest velocity experiments,
combined with tungsten carbide flyer plates allowed the elastic-plastic regime of
Si3N4 to be examined and the HEL to be found. Also, experiments were
conducted at impact velocities between 115 m/s and 300 m/s at a skew angle of
12 degrees to examine the effects of combined a pressure and shear shock loading
on the residual spall strength of Si3N4.
The shock vs. particle velocity and stress vs. strain relationships for silicon
nitride were determined. Below the HEL, the shock velocity in the material
remains nearly constant. In fact, the average shock velocity of 10.7 km/s is
within 1.8% of the elastic wave speed of 10.9 km/s, which is computed from the
elastic modulus, Poisson’s ratio, and density. At the highest particle velocities,
the shock speed is reduced indicating the onset of plasticity. However,
insufficient data exists to generate a trend in this elastic-plastic region. In
observing the stress-strain behavior, the elastic Hugoniot line and the
hydrodynamic curve diverge near the onset of plasticity. The Hugoniot Elastic
Limit for the material is calculated from the elastic equations using the highest
impact velocity Experiment SC-6, and was found to be ~ 12 GPa, which agrees
well with values found in the literature, such as 12.1 GPa [Nahme et al., 1994].
The HEL indicates the onset of inelastic behavior in the material under
309
conditions of uniaxial shock compression loading. Using the HEL, the dynamic
yield strength under plane stress was estimated to be 7.6 GPa.
The spall strength, which is the failure of the material in dynamic tension created
by the intersection of two rarefaction waves, is found from each experiment. The
elastic region under shock compression shows a steady, linear decrease in the
residual spall strength with increasing compressive stress. The spall strength
decreases from an average of 895 MPa at an impact velocity of 65 m/s (SC-5,
SC-8) by 37% to 564 MPa at an impact velocity of 417 m/s (SC-9). This
indicates inelastic microcracking even under the HEL. The microscopic modes of
failure, as observed by the Scanning Electron Microscope, indicate increasing
spall surface roughness and a transition from predominately intergranular
fracture at 65 m/s impact speed (SC-8) to a more complex dual mode fracture at
356 m/s (SC-4). The two predominant regions on the spall surface in
Experiment SC-4 are a brittle fracture area and a melted zone. In the higher
impact velocity experiments, the onset of plastic deformation counteracts the
decreasing trend in the spall strength, and the spall strength is nearly at the
same level in the two elastic-plastic experiments. No fragments suitable for post-
impact SEM analysis were recovered from these experiments.
The addition of a pressure-shear loading causes a degradation of the spall
strength of Si3N4. A twelve degree skew angle, producing a relatively moderate
level of shear stress/strain in the Si3N4, causes the material spall strength to
decrease five times as rapidly as in pure shock compression. This is due to
additional damage caused by the shear. The spall strength drops by 69% of its
level of 803 MPa for Experiment SC-11 at an impact velocity of 115 m/s, to 249
MPa at an impact speed of 233 m/s (SC-12). This decreased spall strength is not
reflected in a change of damage mode. This is also observed in the SEM pictures
of the material, where the same brittle damage mode that was seen at the 208
m/s (SC-2) impact speed under shock compression is present in the shear
experiments conducted at an impact velocity of 233 m/s (Experiment SC-12).
Under the combined pressure and shear loading at impact velocities of 233 m/s
310
(SC-12), silicon nitride is found to loose almost it’s spall strength. In fact in
Experiment SC-13 at 299 m/s impact velocity the spall strength is essentially
zero.
The results of these experiments can be summarized as follows. The AS 8000
grade Si3N4 exhibits qualities that recommend it as a candidate for aircraft
engine turbine blades. The HEL of 12 GPa is quite high, indicating that under
impact conditions Si3N4 retains its elastic behavior. This is comparable to other
silicon nitrides [Nahme et al., 1994; Mashimo, 1998] and to silicon carbide,
another aerospace material [Bourne and Millet, 1997; Feng et al., 1998]. The
spall strength of the material in the low speed elastic region is 895 MPa (SC-5,
SC-8), which compares favorably with that of other silicon nitrides [Nahme et al.,
1994]. The inelastic deformation that occurs above the HEL has the effect of
reversing the decrease in spall strength. In contrast, the spall strength of silicon
carbide disappears after the HEL [Bourne and Millet, 1997]. The spall strength
for Si3N4 is ~ 249 MPa even under combined pressure-shear loading at an impact
velocity of 233 m/s and at a 12 degrees skew angle (SC-12). Moreover, during
dynamic shock compression of Si3N4, in all the experiments conducted in the
present study, no failure waves were observed. This is in contrast to a dynamic
shock compression of silicon carbide and glass under similar shock loading
conditions, where failure wave front shave been consistently reported [Bourne and
Millet, 1997]. These observations make AS800 silicon nitride more suitable for
use in the environment of aircraft turbines than the other grades of silicon nitride
and other materials such as silicon carbide.
311
C. The Effect of Dual Shock Loading and Shock Reverberation on theStrength of AS800 Grade Silicon Nitride
Shock compressed materials that are still intact, but have deformed inelastically
exhibit different dynamic strength than the same material that has not been
shocked. Two types of multiple shock experiments were conducted in the present
on silicon nitride in order to explore these effects. The first set of experiments
involves the use of a two layer flyer where the second layer is of either higher
(tungsten carbide) or lower (aluminum) impedance than the silicon nitride. This
is modeled after the experiments performed on alumina [Reinhart and
Chhabildas, 2003]. The second method involves creating reverberations in a thin
target material, in this case tungsten carbide by impacting it with a thicker flyer
of silicon nitride. This is modeled after experiments conducted on glass-fiber-
reinforced polyester [Dandekar et. al., 2003]. These experiments are detailed in
Chapter VI.
Experiments SR-1 and SU-1, using dual flyers, were unable to generate
compressive stresses exceeding HEL in the first shock compressed Hugoniot State.
This is due to the restriction of the maximum impact velocity of the single stage
gas gun to around 500 m/s. Using both one dimensional stress wave theory and
computations based on the methods detailed in Reinhart and Chhabildas [2003]
the stress and the particle velocity states were predictable. Analysis of the stress
and strain from the re-shock impact data of Experiment SR-1, indicates that the
material follows the hydrodynamic Hugoniot curve during initial loading and
during the re-shock. A basic computation of the HEL gives a value of 11.3 GPa,
down by 6% from the value of 12 GPa for un-shocked specimens. The
deformation during the re-shock followed the hydrodynamic Hugoniot curve, and
the analysis method [Reinhart and Chhabildas, 2003] determined the residual
shear strength to be 1.21 GPa. The shock unloading in Experiment SU-1 reveals
that the unloading from the initial shocked state occurs elastically. An elastic
unloading is consistent with theory, because this experiment occurs within the
elastic range. Unlike the spall strength, these results do not indicate a decrease
in longitudinal properties at high velocities.
312
Shock reverberation experiments were conducted at impact speeds around 500
m/s using a thin tungsten carbide target and a thick silicon nitride specimen.
Because of the geometry, several unloading states were generated in the two
materials. The free surface velocity records from the two reverberation
experiments directly provided the target material’s stress state and particle
velocity. The target, which is tungsten carbide behaves elastic-linear plastically.
By using the analysis method provided by Hall et al. [Preliminary], the stresses in
the silicon nitride flyer were calculated. The silicon nitride remains elastic. The
stresses and strains during each reverberation in both experiments, i.e. RB-1 and
RB-2, follow the predictions established in Chapter V for the elastic Hugoniot
curve at the stress levels in the reverberation experiments. As a result, the
shocking and reverberations do not cause a measurable change in the dynamic
response of AS800 grade silicon nitride at the impact velocities used in the
present study.
D. Shock Induced Failure Waves During Dynamic Compression of SodaLime Glass
The failure wave is a phenomena observed in glasses and ceramics with glassy
phases. This failure wave causes the material to fail in such a way that the
longitudinal properties such as impedance and wave speed remain largely
unchanged. The shear and spall strength of the material, on the other hand are
both adversely affected. The specific nature of this wave and its effects on
glasses is discussed in several papers such as [Kanel et al., 2002].
In the present study, in order to understand the initiation of propagation of
failure in soda lime glass, shock compression and pressure-shear experiments were
conducted. These experiments are detailed in Chapter VII. Three types of
experiments were conducted: spall strength experiments with and without the
presence of shear, spall and shear strength measurement experiments, and
impedance measurement experiments. One of the spall strength experiments
(Al/G1) was conducted under pure compression. The remaining experiments
were conducted under combined pressure and shear loading at a skew angle of 18
313
degrees. These experiments were designed to extend the findings of Clifton et al.
[1998] to higher impact velocities.
Aluminum flyers were used in the spall strength experiments because their low
impedance enabled the creation of high tensile stresses. The spall strength of
soda lime glass under shock compression was found to be greater than 3.89 GPa
in Experiment Al/G1. This spall strength compares favorably with other
investigators who found the spall strength to be in excess of 3 GPa [Kanel et al.,
2002; Brar et al., 1991]. In Experiment Al/G2, with a pressure-shear loading at
18 degrees, the spall strength was found to be 3.49 GPa, which shows that the
spall strength is degraded by the presence of shear. Thus, the soda lime glass
behaves like ceramics such as silicon nitride where a skew angle of 12 degrees was
observed to decrease the spall strength significantly.
The second set of experiments used tungsten carbide flyers was to increase the
compressive stress level. The failure wave effects of shear were determined by
conducting experiments at two impact velocities, 220 m/s (WC/G1, WC/G2)
and 410 m/s (WC/G3, WC/G4), and using two thicknesses of glass targets so
that the spall plane would be either in front or behind the propagating failure
front. The corresponding shock compression stress were 2.67 GPa and 4.98 GPa.
Ahead of the failure wave, in both the low and high velocity shots, an elastic
wave reflection occurs. Thus, in these experoments the glass remains intact
because the spall strength is greater than the applied tensile stresses, which were
644 MPa (WC/G1) and 1.20 GPa (WC/G3).
Experiment WC/G2 was designed such that the spall plane was formed behind
the failure wave front. The impact velocity was 220 m/s. The measured free
surface particle velocity profile showed a drop in particle velocity at the time
corresponding to the arrival of the spall signal. However, like in experiment
WC/G1, the particle velocity does not show the rapid pull-back consistent with
the spall event and is consistent with a no-spall level prediction of the particle
velocity. A small velocity rise of 13 m/s that was seen at the approximated
314
arrival time of the failure wave signal suggests that some damage occurred.
However, this damage was not enough to effect the spall strength, which
remained higher than the maximum tensile stress generated by the impact.
Experiment WC/G2 is contrasted by Experiment WC/G4 at 410 m/s where no
velocity drop is seen at the time of the spall signal. This lack of spall strength
and the particle velocity rise of 45 m/s at the anticipated arrival time of the
failure wave indicated that the glass had failed completely. From Experiments
WC/G2 and WC/G4 an estimate for the failure wave velocity was made. This
estimate is 1.7 ± 0.1 km/s, with the slightly higher velocity occurring for
experiment WC/G2. This finding is close to previous estimates for the failure
wave velocity.
The shear strength of the material has been observed to change with increasing
impact velocity. This shear strength is reflected in the transverse particle
velocity measured during impact. At the lower impact velocities, the level of the
measured transverse velocity is near the predicted elastic level. Experiment
WC/G1 shows the transverse velocity to be on average 104 m/s. This is 14%
lower than the elastic value of 120 m/s. In Experiment WC/G2, the peak
transverse velocity is 109 m/s. The elastic transverse velocity is still 120 m/s, so
this value is 9.2% off. In the higher impact velocity experiments, the transverse
velocity is much smaller a percentage of the elastic prediction. In Experiment
WC/G3, in front of the failure wave a peak transverse velocity of 147 m/s is
seen. This is 34% below the elastic value of 224 m/s. Experiment WC/G 4
exhibits a peak transverse velocity of 148 m/s. The elastic level is 222 m/s, so
the actual value is 33% below it. This suggests that the initiation of the failure
wave, which according to Feng [2000] occurs at the impact surface, may affect
the level of the shear wave prior to its propagation through the material.
Clifton et al. [1998] also found that the transverse velocity rose to a level at or
just below the elastic prediction and then decreased. Clifton’s experiments were
at stress levels of 4.3 GPa, lower than the highest levels conducted in this study.
315
His results agree with the lower velocity shear results from the experiments in
Chapter VII, but do not show the 30% drop in from the elastic velocities seen in
the current study’s the higher velocity experiments.
Two experiments were conducted to measure the impedance of the damaged
material. These experiments used tungsten carbide as targets and glass as the
flyer plates, and were conducted at impact velocities of 250 m/s (G/WC1) and
454 m/s (G/WC2). The corresponding compressive stresses were 3.04 GPa and
5.51 GPa, respectively. The configuration created reverberations similar to those
in the silicon nitride experiments RB-1 and RB-2, described in Chapter VI. In
Chapter VII, the behavior of the tungsten carbide was analyzed. The normal
material behavior is well predicted by elastic theory in the lower velocity case
(G/WC1). The measured transverse velocity peaks at 10 m/s and then decreases
to 5 m/s in about 0.6 microseconds. This is 45% lower than 18.4 m/s, the elastic
prediction. Clifton et al. [1998] saw a near constant level of shear stress at his
elastic prediction. In Experiment G/WC1, the longitudinal impedance of soda
lime glass is measured to be 13.6 MPa/(m/s). This is within 7% of the elastically
computed value of 14.5 MPa/(m/s). The shear impedance is 5.94 MPa/(m/s),
which is 69% of the elastic shear impedance, 8.6 GPa.
In the high speed experiment (G/WC2), the normal velocities are below the
elastic prediction. For example, the first stressed state is 95 m/s, which is down
8.6% from the elastic value of 104 m/s. The transverse velocity is 82% lower
than predicted, being only 5.9 m/s instead of 33.42 m/s. It decreases to zero
after about 6 microseconds. This indicates that the material has failed and that
the interaction of the shear waves is not predicted by elastic computations from
the intact material parameters. Clifton et al.’s [1998] experiments showed
oscillatory transverse particle velocity profiles at the higher impact velocities,
which do not match the near zero behavior seen in Experiment G/WC2. The
normal impedance in this experiment is measured to decrease to 13.0 MPa/(m/s).
This is only 10% lower than the elastic longitudinal impedance, suggesting that
the assumption of constant longitudinal properties that several of the theories
316
depend upon is valid. The shear impedance is 4.24 MPa/(m/s) which is 49% of
the elastic shear impedance. This is further evidence that the shear properties
are effected by the failure wave, but that the longitudinal properties are nearly
unchanged.
317
E. Discussion of Results
This dissertation has explored the effects of impact on two brittle materials,
AS800 grade silicon nitride and soda lime glass. In both cases, the material has
been examined under planar shock compression. The spall strength of both
materials was examined, as was the HEL of silicon nitride. Additional particle
impact tests were carried out on soda lime glass. These tests examined the three
dimensional nature of the impact process. Similar tests had already been carried
out on AS800 grade silicon nitride by Choi et al. [2002].
The particle impact experiments and a numerical model for the internal stress
propagation [Nathenson et al., 2005] resulted in the examination of the behavior
of soda lime glass under three dimensional impact. The surface strains show an
increase in tensile stress with decreasing thickness and a limit on the strain
imposed by the finite failure strength of the material. The amount of energy
consumed by cracking is larger than that dissipated by elastic processes. The
material cracking patterns were also evaluated and radial cracking causes the
greatest detriment to the integrity of the glass. In shock compression, the spall
strength of intact soda lime glass decreases with the addition of shear stress. At
high compressive stress levels, the region in the failure wave shows no spall
strength and a reduced shear strength. Also, the normal impedance of the
comminuted glass changes only 10% from the impedance of the intact glass, while
the shear impedance changes by 51%.
AS800 grade silicon nitride is a leading candidate for aircraft engine turbine
blades due to its desirable high temperature properties and its strong resistance
to damage from particle impact. In this work, AS800 grade silicon nitride was
studied using both shock compression and pressure-shear at a skew angle of 12
degrees. In the shock compression experiments, the HEL of the material was
found to be 12 GPa, with a corresponding dynamic yield strength of 7.6 GPa.
The material’s spall strength decreases linearly with increasing impact velocity,
but to ceases decreasing upon the onset of plastic deformation. The addition of
318
shear caused the material to lose spall strength with increasing impact velocity at
a rate five times as fast as under normal shock compression. Experiment SR-1
using dual shocks showed that the path between increasing stress states follows
the hydrodynamic curve. Unloading after an initial shock proceeds elastically in
Experiment SU-1, a fact which is confirmed by the unloading in reverberation
Experiments RB-1 and RB-2. As a result of these experiments, the material is
shown to have superior or comparable strength properties to other grades of
silicon nitride, silicon carbide, and nickel based superalloys. These observations
on the dynamic strength of AS800 grade silicon nitride make it a leading
candidate for the next generation of turbine blades.
319
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Glass. Shock Compression of Condensed Matter – 1997. American Institute of Physics. 521-524.
Dandekar, D. P., C. A. Hall, L. C. Chhabildas, W. D. Reinhart. 2003. Shock Response of a
Glass Fiber-Reinforced Polymer Composite. Composite Structures. 61 [1-2] 51-59.
Feng, R.; G. F. Frasier; and Y. M. Gupta. 1998. Material Strength and Inelastic Deformation of
Silicon Carbide Under Shock Wave Compression. Journal of Applied Physics. 83 [1] 79.
Hall, C. A., L. C. Chhabildas, and W. D. Reinhart., Preliminary Report. Shock Hugoniot and
Release States in GRP Composite From 3 to 20 GPa. Sandia National Laboratories.
Lin H. T. et al. 2001. Evaluation of Creep Property of AS800 Silicon Nitride from As-Processed
Surface Regions. Ceramic Engineering and Science Proceedings. 22 [3] 175-182.
Kanel, G. I., A. A. Bogatch, S. V. Razorenov, and Z. Chen., 2002. Transformation of Shock
Compression Pluses in Glass Due to the Failure Wave Phenomena. Journal of Applied Physics.
Vol. 92. No. 9. 5045-5052.
Mitra, M. 1964. “Disturbance Produced in an Elastic Half-Space by Impulsive Normal Pressure.”
Proceedings of the Cambridge Philosophical Society. 69: 683-696.
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editors New York : Springer 101-146.
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Particle Impacts. Society for Experimental Mechanics Conference Proceedings.
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Shocked State. International Journal of Impact Engineering. 29 [1-10] 601-619.
321
Appendices
Appendix 1: Equations of Wave Propagation Resulting from a Point Loadon a Elastic Half Space
This discussion is taken from Lamb [1904]. In his paper, Lamb described the
Rayleigh wave component of the radial surface displacement. The following
manipulations are used to derive his result for the surface displacement.
1.1. Using the Equations of Motion to Generate the Radial and VerticalSurface Displacements
The equations of motion for an isotropic elastic solid are:
( )2
22
u ut x
ρ λ μ μ∂ ∂Δ= + + ∇
∂ ∂, {A.1.1}
( )2
22
v vt y
ρ λ μ μ∂ ∂Δ= + + ∇
∂ ∂, {A.1.2}
and,
( )2
22
w wt z
ρ λ μ μ∂ ∂Δ= + + ∇
∂ ∂. {A.1.3}
In Equations {A.1.1} through {A.1.3},
u v wx y z∂ ∂ ∂
Δ = + +∂ ∂ ∂
. {A.1.4}
In Equations {A.1.1} through {A.1.4}, u, v, and, w, are the displacements in the
x, y, and, z directions respectively; ρ , is the density; and λ, and μ, are the
Lame’ constants. For simple harmonic motion of the form ipte , where t is time, i
is the square root of negative one, and p is the frequency, the solutions for the
displacements in terms of a potential φ, can be written as:
322
'u uxφ∂
= +∂
, {A.1.5}
'φ∂= +∂
v vy
, {A.1.6}
and,
'φ∂= +∂
w wz
. {A.1.7}
Equations {A.1.5} through {A.1.7} explicitly state the homogenous component of
the solution. The particular solutions are not explictly stated, but are identified
as, u’, v’, and, w’. These equations satisfy differential Equations {A.1.1} through
{A.1.3} if the following conditions are met:
( )2 2 ' 0k u∇ + = , {A.1.8}
( )2 2 ' 0k v∇ + = , {A.1.9}
( )2 2 ' 0k w∇ + = , {A.1.10}
( )2 2 0h φ∇ + = , {A.1.11}
and,
' ' ' 0u v wx y z
∂ ∂ ∂+ + =
∂ ∂ ∂. {A.1.12}
In Equations {A.1.8} through {A.1.11},
323
22
2ph pa ρ
λ μ= =
+, {A.1.13}
and,
22 pk pb ρ
μ= = . {A.1.14}
The inverse of the wave velocities are defined for the longitudinal and shear
waves, respectively, as follows:
1L
a C= , {A.1.15}
and,
1=T
b C . {A.1.16}
In the above equations, the longitudinal wave velocity is CL, and the shear wave
velocity CS.
In order to solve partial differential Equations {A.1.5} through {A.1.7},
particular solutions, u’, v’, and, w’, must be specified. These particular solutions
can then be written in terns of one variable, χ , as:
2
'ux zχ∂
=∂ ∂
, {A.1.17}
2
' χ∂=∂ ∂
vy z
, {A.1.18}
and,
324
22
2' χ χ∂= +∂
w kz
. {A.1.19}
Equations {A.1.17} through {A.1.19} satisfy Equations {A.1.1} through {A.1.7}
if the following condition is true
( )2 2 0k χ∇ + = , [VP1] {A.1.20}
Substituting into Equations {A.1.5} through {A.1.7} yields:
2φ χ∂ ∂= +∂ ∂ ∂
ux x z
, {A.1.21}
2φ χ∂ ∂= +∂ ∂ ∂
vy y z
, {A.1.22}
and,
22
2
φ χ χ∂ ∂= + +∂ ∂
w kz z
. {A.1.23}
Symmetry is then considered about the Z-axis.
Defining the radial direction, ϖ ,
2 2 2x yϖ = + , {A.1.24}
In terms of ϖ , the Laplacian can be written as
2 22
2 2
1zϖ ϖ ϖ
∂ ∂ ∂∇ = + +
∂ ∂ ∂. {A.1.25}
Generalizing the u and the v coordinates as q and w, and re-writing Equations
{A.1.21} to {A.1.23}:
325
2φ χϖ ϖ∂ ∂
= +∂ ∂ ∂
qz, {A.1.27}
22
2
φ χ χ∂ ∂= + +∂ ∂
w kz z
. {A.1.28}
The particular solutions, ,&,φ χ are now written explicitly:
( )zoAe Jαφ ξϖ−= , {A.1.29}
( )zoBe Jβχ ξϖ−= . {A.1.30}
In Equations {A.1.29} and {A.1.30}, A, and, B, are constants, ξ is a variable
used in the integration, and[VP2] J0 is the Bessel’s function of the first kind. In
the exponents of Equations {A.1.29} and {A.1.30}:
( )2 2hα ξ= − , {A.1.31}
and,
( )2 2kβ ξ= − . {A.1.32}
Plugging these back into Equations {A.1.27} and {A.1.28} for q & w, yields
( ) ( )' 'z zo oq Ae J Be J
zα βξϖ ξ ξϖ ξ− −∂
= +∂
, {A.1.33}
( ) ( ) ( )' 'z zo oq Ae J BJ eα βξϖ ξ ξϖ ξ β− −= + − , {A.1.34}
( )'z zoq Ae Be Jα ββ ξϖ ξ− −⎡ ⎤= −⎣ ⎦ , {A.1.35}
326
( )1z zq Ae Be Jα βξ ξβ ξϖ− −⎡ ⎤= − +⎣ ⎦ . {A.1.36}
( )( ) ( ) ( ) ( )2z z zo o ow Ae J B e J k Be J
zα β βξϖ α β ξϖ ξϖ− − −∂ ⎡ ⎤= − + − +⎣ ⎦∂
, {A.1.37}
( ) ( ) ( ) ( ) ( )2 2z z zo o ow A e J B e J k Be Jα β βα ξϖ β ξϖ ξϖ− − −= − + − + , {A.1.38}
( ) ( ) ( )2 2z z zo o ow Ae J Be J k Be Jα β βα ξϖ β ξϖ ξϖ− − −= − + + . {A.1.39}
In Equation {A.1.36} J1 is the Bessel’s function of order ??? Additionally,
2 2 2β ξ+ =k . {A.1.40}
Using Equation {A.1.40}, the following simplification can be made
( ) ( )2z zow Ae Be Jα βα ξ ξϖ− −= − + . {A.1.41}
1.2. Calculating the Radial and Vertical Stresses on the Surface
Starting from the differential Equation {A.1.42} for the radial stress zϖσ , and
substituting from Section 1.1:
zq wzϖσ μ
ϖ∂ ∂⎡ ⎤= +⎢ ⎥∂ ∂⎣ ⎦
, {A.1.42}
( )( )
21
21
α β
ϖ α β
αξ ξβ ξϖσ μ
α ξ ξϖ ξ
− −
− −
⎧ ⎫⎡ ⎤−⎪⎣ ⎦ ⎪= ⎨ ⎬⎡ ⎤− − +⎪ ⎪⎣ ⎦⎩ ⎭
z z
z z z
Ae Be J
Ae Be J, {A.1.43}
{ } ( )2 31
z z z zz Ae Be Ae Be Jα β α βϖσ μ αξ β ξ αξ ξ ξϖ− − − −= − + − , {A.1.44}
{ } ( )2 212 z z z
z Ae Be Be Jα β βϖσ μ αξ β ξ ξ ξϖ− − −⎡ ⎤= + − −⎣ ⎦ , {A.1.45}
327
( ){ } ( )2 212 z z
z Ae Be Jα βϖσ μ αξ β ξ ξ ξϖ− −⎡ ⎤= + − −⎣ ⎦ , {A.1.46}
( )2 2 2 2 2 2 22k kβ ξ ξ ξ ξ+ = − + = − , {A.1.47}
( ){ } ( )2 212 2z z
z Ae k Be Jα βϖσ μ αξ ξ ξ ξϖ− −⎡ ⎤= − −⎣ ⎦ . {A.1.48}
But, z, is equal to 0 on the surface, so
( ){ } ( )2 212 2z A B k Jϖσ μ αξ ξ ξ ξϖ⎡ ⎤= − −⎣ ⎦ . {A.1.49}
Next, calculating the vertical stress, zzσ , from the differential Equation {A.1.50}:
2zzwz
σ λ μ ∂⎡ ⎤= Δ +⎢ ⎥∂⎣ ⎦, {A.1.50}
2zzq w w
z zσ λ μ
ϖ⎡ ∂ ∂ ∂ ⎤⎛ ⎞= + +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦
, {A.1.51}
( )2 2z zo
w Ae Be Jz
α βα ξ β ξϖ− −∂ ⎡ ⎤= −⎣ ⎦∂, {A.1.52}
( ) ( )1z zq w Ae Be J
zα βλ λ ξ ξ ξ ξϖ
ϖ− −∂ ∂⎛ ⎞ ′+ = − +⎜ ⎟∂ ∂⎝ ⎠
. {A.1.53}
Using the properties of Bessel’s functions,
( ) ( ) ( )11 1oJ J Jξϖξϖ ξϖ ξϖ′= + . {A.1.54}
When the radius from the impact point becomes large, 1ϖ :[VP5]
( ) ( )1oJ Jξϖ ξϖ′≈ , {A.1.55}
328
( ) ( ) ( ) ( )1 0z z z zAe Be J Ae Be Jα β α βλ ξ ξ ξ ξϖ λ ξ ξ ξ ξϖ− − − −′− + = − + , {A.1.56}
( )( )
( )2 22
α β
α β
λ ξ ξ ξσ ξϖ
λ μ α ξ β
− −
− −
⎧ ⎫− +⎪ ⎪= ⎨ ⎬⎡ ⎤+ + −⎪ ⎪⎣ ⎦⎩ ⎭
z z
zz oz z
Ae BeJ
Ae Be, {A.1.57}
( )( ) ( )( )
( )
2 2
2 2
2
2
2
2
α α
α β
β
λξ λ λ μ α
σ λ μ α λξ ξϖ
λ μ ξ β
− −
− −
−
⎧ ⎫− + +⎪ ⎪⎪ ⎪= + + +⎨ ⎬⎪ ⎪
⎡ ⎤+ + −⎪ ⎪⎣ ⎦⎩ ⎭
z z
z zzz o
z
Ae Ae
Ae Be J
Be
. {A.1.58}
Simplifying,
( ) ( ){ } ( )2 2 22 2z z zzz oAe Ae Be Jα α βσ λξ λ μ α μ ξ β ξϖ− − −= − + + − . {A.1.59}
Using Equations {A.1.13}, {A.1.14}, and {A.1.30}, the following relations can be
derived:
( )2 2 22h p kλ μ ρ μ+ = = . {A.1.60}
Therefore, Equation {A.1.59} can be simplified further,
( )( ) ( ){ } ( )2 2 2 22 2z z zzz oAe h Ae Be Jα α βσ λξ λ μ ξ μ ξ β ξϖ− − −= − + + − − , {A.1.61}
( )( ) ( ){ } ( )2 2 2 22 2z z zzz oAe k Ae Be Jα α βσ λξ λ μ ξ μ μ ξ β ξϖ− − −= − + + − − , {A.1.62}
( )( )
( )
2 2
2 2
2
2
2
α α
α
β
λξ λξ
σ μξ μ ξϖ
μ ξ β
− −
−
−
⎧ ⎫− +⎪ ⎪⎪ ⎪= + −⎨ ⎬⎪ ⎪−⎪ ⎪⎩ ⎭
z z
zzz o
z
Ae Ae
k Ae J
Be
, {A.1.63}
( ) ( ){ } ( )2 2 22 2z zzz ok Ae Be Jα βσ μξ μ μ ξ β ξϖ− −= − − , {A.1.64}
329
( ){ } ( )2 2 22 2z zzz ok Ae Be Jα βσ μ ξ ξ β ξϖ− −= − − . {A.1.65}
Setting, z, equal to 0 on the surface:
( ){ } ( )2 2 22 2zz ok A B Jσ μ ξ ξ β ξϖ= − − . {A.1.66}
1.3. Specific Solution for a Concentrated Vertical Pressure
Assuming a case of normal stress acting on the surface, i.e. z=0, the normal
component of the stress can be written:
[ ] ( )0zz ozP Jσ ξϖ
== ⋅ , {A.1.67}
and the radial stress:
[ ] 00z zϖσ
== . {A.1.68}
Then, the coefficients in Equations {A.1.29} and {A.1.31} can be found by
Newton’s second law. For the vertical stress, σ zz , this means:
( ){ } ( )2 2 22 2σ μ ξ ξ β ξϖ= − −zz odyk A B Jdx
, {A.1.69}
[ ] ( ) ( ){ } ( )2 2 20
2 2zz o ozP J k A B Jσ ξϖ μ ξ ξ β ξϖ
== ⋅ = − − , {A.1.70}
( ){ }2 2 22 2P k A Bμ ξ ξ β= − − , {A.1.71}
( )2 2 22 2P k A Bξ ξ βμ= − − . {A.1.72}
For the radial stress, ϖσ z :
330
( ){ } ( )2 212 2z A B k Jϖσ μ αξ ξ ξ ξϖ⎡ ⎤= − −⎣ ⎦ , {A.1.73}
[ ] ( ){ } ( )2 210
0 2 2z zA B k Jϖσ μ αξ ξ ξ ξϖ
=⎡ ⎤= = − −⎣ ⎦ , {A.1.74}
( ){ } ( )2 210 2 2A B k Jμ αξ ξ ξ ξϖ⎡ ⎤= − −⎣ ⎦ , {A.1.75}
( )2 20 2 2A B kαξ ξ ξ⎡ ⎤= − −⎣ ⎦ . {A.1.76}
Combining Equations {A.1.72} and {A.1.76}:
( )2 22 k PA
Fξ
ξ μ−
= , {A.1.77}
and,
( )2 PB
Fαξ μ
= . {A.1.78}
In Equations {A.1.77} and {A.1.78},
( ) ( )22 2 22 4F kξ ξ ξ αβ= − − . {A.1.79}
Thus, Equations {A.1.36} and {A.1.41} for the generalized coordinates q & w can
be re-written as follows.
For q:
( )1z zq Ae Be Jα βξ ξβ ξϖ− −⎡ ⎤= − +⎣ ⎦ , {A.1.36}
( ) ( ) ( )2 2
12 2z zk P Pq e e J
F Fα βξ αξ ξβ ξϖ
ξ μ ξ μ− −⎡ ⎤−
= − +⎢ ⎥⎣ ⎦
, {A.1.80}
331
and thus,
( )( ) ( )
2 2
0 1
2 2z
k Pq JF
ξ ξ αβξϖ
ξ μ=
− − −= . {A.1.81}
For w:
( ) ( )2z zow Ae Be Jα βα ξ ξϖ− −= − + , {A.1.41}
( ) ( ) ( )2 2
22 2z zo
k P Pw e e JF F
α βξ αα ξ ξϖξ μ ξ μ
− −⎛ ⎞−= − +⎜ ⎟⎜ ⎟⎝ ⎠
, {A.1.82}
and thus,
( ) ( )2
0z ok Pw J
Fα ξϖξ μ= = . {A.1.83}
Using a concentrated vertical pressure with an exponential loading function,iptR e :
/ 2P R dξ ξ π= − . {A.1.84}
In Equation {A.1.84}, R, represents the force. Integrating from zero to infinity
gives the displacements due to the forcing function:
( )( ) ( )
2 2 2
10
2 22o
kRq J dF
ξ ξ αβϖξ ξ
πμ ξ
∞ − −= ∫ , {A.1.85}
( ) ( )2
002o
R kw J dFξα ϖξ ξ
πμ ξ
∞−= ∫ . {A.1.86}
Next, the Bessel’s functions are written in terms of hyperbolic cosines:
332
( )cosh cosh
0
i u i uo
iJ e e duξϖ ξϖ
π
∞−= − −∫ , {A.1.87}
( )cosh cosh1
0
1 coshi u i uJ e e uduξϖ ξϖ
π
∞−= − +∫ . {A.1.88}
Using Equations {A.1.87} and {A.1.88}, Formulas {A.1.85} and {A.1.86} for the
surface displacements become:
( )( )
2 2 2cosh
20
2 2cosh
2i u
o
kRq udu e dF
ξϖξ ξ αβ
ξπ μ ξ
∞ ∞
−∞
− −−= ∫ ∫ , {A.1.89}
( )2
cosh2
02i u
oiR kw du e d
Fξϖξα ξ
π μ ξ
∞ ∞
−∞
= ∫ ∫ . {A.1.90}
Using the method of principle values, symbolized by, P :
( )( )
( )( )( ) ( )
2 2 2cosh 2 2 2
2 cosh
2 sin cosh2 2
24
ξϖ
ξϖ
πκ κϖξ ξ αβ
ξ ξ ξ αβξ ξ
ξ ξ
∞
−−∞
⎡ ⎤− − ⎢ ⎥
= −⎢ ⎥+⎢ ⎥⎢ ⎥⎣ ⎦
∫∫
i u ki u
h
H uk
e d kF k e d
F f
P , {A.1.91}
( )
( )
( )
( )( ) ( )
2cosh 2 cosh
22 22 cosh
2 sin cosh
2
22
ξϖ ξϖ
ξϖ
πκ κϖ
ξα ξαξ ξξ ξ
ξ ξ αξ
ξ ξ
∞ ∞−
−∞
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎣ ⎦
∫ ∫
∫
i u i u
k
ki u
h
K u
k e d k e dF F
kk e d
F f
P P . {A.1.92}
In Equations {A.1.91} and {A.1.92},
333
( )( )
2 21 12 2k
HF
κ α β
ξ
− − −=
′, {A.1.93}
( )2
1kKF
αξ
−=
′, {A.1.94}
( ) ( )22 2 22 4F kξ ξ ξ αβ= − − , {A.1.95}
( ) ( )22 2 22 4f kξ ξ ξ αβ= − + , {A.1.96}
kc μρκ = . {A.1.97}
In Equation {A.1.97}, the inverse of the Rayleigh wave speed is denoted by, c.
Further simplification yields:
( ) ( )( ) ( ) ( )
2 2 22
1 1
22
k
oh
kR ik Rq HK D dF f
ξ ξ αβκ κϖ κϖ ξμ πμ ξ ξ
−= + ∫P , {A.1.98}
( )
( ) ( )
( )( ) ( ) ( )
2
0
22 22
0
2
2
22
κ κϖμ
ξα κϖ ξπμ ξ
ξ ξ ακϖ ξ
πμ ξ ξ
∞
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎣ ⎦
∫
∫
o
ok
k
h
i R KJ
ik Rw D dF
kik R D dF f
P P . {A.1.99}
In Equations {A.1.98} and {A.1.99}, ( )1 K ζ is a Bessel’s function of the second
kind and,
( ) ( ) ( )0 ζ ζ ζ= −o oD K iJ . {A.1.100}
334
Superimposing a system of free waves to the solutions in Equations {A.1.98} and
{A.1.99}:
( )12κ κϖμ
−=o
Rq HD , {A.1.101}
( )02κ δκϖμ δ
−=o
i R yw KJx
. {A.1.102}
Or writing Equations {A.1.101} and {A.1.102} in a different way,
( ) ( )( ) ( ) ( )
2 2 22
1 1
22
k
oh
kR ik Rq HD D dF f
ξ ξ αβκ κϖ κϖ ξμ πμ ξ ξ
−= − + ∫ , {A.1.103}
( ) ( ) ( )( ) ( ) ( )
22 22 2
0 0
22 2
k
ok h
kik R ik Rw D d D dF F f
ξ ξ αξα κϖ ξ κϖ ξπμ ξ πμ ξ ξ
∞ −= − −∫ ∫P . {A.1.104}
Inserting the time factor, which is the exponential term, ipte , into Equations
{A.1.103} and {A.1.104}:
( ) ( )( ) ( ) ( )
2 2 22
1 1
22
kipt ipt
oh
kR ik Rq HD e D e dF f
ξ ξ αβκ κϖ κϖ ξμ πμ ξ ξ
−= − + ∫ , {A.1.105}
( ) ( )
( )( ) ( ) ( )
2
0
22 22
0
2
22
ξα κϖ ξπμ ξ
ξ ξ ακϖ ξ
πμ ξ ξ
∞⎡ ⎤−⎢ ⎥⎢ ⎥
= ⎢ ⎥−⎢ ⎥−⎢ ⎥
⎣ ⎦
∫
∫
ipt
k
ok
ipt
h
ik R D e dF
wkik R D e d
F f
P
. {A.1.106}
Using only the most singular portion of the principle value integral,
335
( ) ( ) ( ) ( )
( ) ( )( )
( ) ( ) ( )
2 2
00
2 2 22
22 2 2 2 2 2 20
2 cos cosh 2
1 1 22 4
h iipt
ki
h
k k e dD e d i K uF F
k h c dk e d
F f k h k
ξϖ
ξϖξϖ
ξα ξα ξκϖ ξ πκ κϖξ ξ
ξ ξα ξ
ξ ξ ξ ξ ξ ξ
∞ −
−∞
−∞−
= − +
+⎧ ⎫⎪ ⎪+ − −⎨ ⎬⎪ ⎪ + − + +⎩ ⎭
∫ ∫
∫ ∫
P
. {A.1.107}
This reduces equation {A.1.107} to:
( ) ( ) ( )2
0 2 cos cosh . . .iptk D e d i K u H OTFξα κϖ ξ πκ κϖξ
∞
−∞
= − +∫P {A.1.108}
In Equation {A.1.108}, H.O.T., stands for higher order terms. Therefore,
specifying only the lower order terms
( )0 . . .2κ κϖμ
= +oRw KK H OTP {A.1.109}
Simplifying Equations {A.1.105} and {A.1.106} to show only the most important
terms:
( )1 . . .2
ipto
Rq HD e H OTκ κϖμ
= − + , {A.1.110}
( )0 . . .2
ipto
Rw KD e H OTκ κϖμ
= + {A.1.111}
When the radial distance, ϖ , is very large, the asymptotic expansion of, ( )oD ζ ,
yields approximate exponential forms:
( ) ( )
( ) ( )
2 2 242
21 1 31 ...
1! 8 2! 8i
oD ei i
ζ ππζζ
ζ ζ− +
⎧ ⎫⋅⎪ ⎪= − + −⎨ ⎬⎪ ⎪⎩ ⎭
. {A.1.112}
Equations {A.1.110} and {A.1.111} are then further simplified,
336
( )42
2i pt
oi Rq H e κϖ π
πκϖκμ
− − −≈ − , {A.1.113}
( )42
2i pt
oRw K e κϖ π
πκϖκμ
− − −≈ . {A.1.114}
1.4. Generalizing the Equations of Displacement for an Arbitrary Loading
The Equations {A.1.110} and {A.1.111} can be written using the Bessel’s
function:
( ) cosh
0
2 i uoD e duζζ
π
∞−= ∫ , {A.1.115}
as,
( )cosh
0
. . .ip t c uo
HRq e du H OTϖ
πμ ϖ
∞−∂
= +∂ ∫ , {A.1.116}
and,
( )cosh
0
. . .ip t c uo
iKRcw e du H OTt
ϖ
πμ
∞−∂
= − +∂ ∫ {A.1.117}
Equations {A.1.116} and {A.1.117} can be generalized by replacing the
exponential inside the integral with an arbitrary forcing function, R(t).
( )0
cosh . . .oHq R t c u du H OTϖπμ ϖ
∞∂= − +
∂ ∫ , {A.1.118}
( )0
cosh . . .oKcw R t c u du H OT
tϖ
πμ
∞∂ ′= − +∂ ∫ {A.1.119}
In Equation {A.1.119},
337
( ) ( ) ( )0
1 sinR t dp R p t dλ λ λπ
∞ ∞
−∞
′ = −∫ ∫ . {A.1.120}
Using a specific forcing function, R(t):
( ) 2 2
RR ttτ
π τ=
+, {A.1.121}
( ) 2 2
R tR ttπ τ
′ =+
. {A.1.122}
Where t is the time; R is a constant force; and τ is related to t by:
tant cϖ τ χ− = . {A.1.123}
When the radius, ϖ , is much greater than a characteristic length defined as ,
/ cτ [VP6], and the variable, χ , is of moderate value Equations {A.1.118} and
{A.1.119} can be approximated for the general exponential loading, eiptR .
( ) ( )4 20
2cosh cos cos2RR t c u du
cχπτϖ χ
τ ϖ
∞
− = −∫ , {A.1.124}
( ) ( )4 20
2cosh sin cos2RR t c u du
cχπτϖ χ
τ ϖ
∞
′ − = − −∫ . {A.1.125}
Ignoring the residual terms, yields the following approximations for the
displacement:
( ) 32
4 22
2 sin cos4o
Rcq Hc
χπτ χπμτ ϖ
= − − , {A.1.126}
338
( ) 32
4 22
2 cos cos4o
Rcw Kc
χπτ χπμτ ϖ
= − . {A.1.127}
These formulae describe two types of disturbances as shown in Figure A.1.
Complete expressions for the disturbances of this loading are found by not
ignoring or simplifying the higher order terms. A new variable related to the
time t, and the radial distance ϖ , is defined in order to assist with this process:
tθϖ
= . {A.1.128}
This variable is then used in the solutions to Equations {A.1.118} and {A.1.119},
( )
( ) ( )
0
20
cosh
2 cosh
ϖπμ ϖ
θ θϖ θπ μ ϖ
∞
∞
⎡ ⎤∂−⎢ ⎥∂⎢ ⎥= ⎢ ⎥⎧ ⎫∂⎢ ⎥− −⎨ ⎬∂⎢ ⎥⎩ ⎭⎣ ⎦
∫
∫ ∫o b
a
H R t c u du
qU R t u du d
b
, {A.1.129}
( ) ( )20
1 coshoa
w V R t u du db t
θ θ θϖ θπ μ
∞ ∞⎧ ⎫∂= −⎨ ⎬∂⎩ ⎭
∫ ∫P . {A.1.130}
In Equation {A.1.129},
( )( ) ( ) ( )
( ) ( )( )
3 2 2 2 2 2 2
42 2 4 2 2 2 2
2
2 16
b b a bU
b a b
θ θ θ θθ
θ θ θ θ
− − − −=
− + − −. {A.1.131}
In Equation {A.1.130} when, θ< <a b :
( )( ) ( )
( ) ( )( )
3 2 2 2 2
42 2 4 2 2 2 2
2
2 16
θ θθ
θ θ θ θ
− − −=
− + − −
b b aV
b a b . {A.1.132}
When, θ > b :
339
( )( )
( ) ( ) ( )
3 2 2
22 2 2 2 2 2 22 4
θθ
θ θ θ θ
− −=
− + − −
b aV
b a b. {A.1.133}
Equations {A.1.129} and {A.1.130} are extremely difficult to integrate, thus the
approximations used for the general loading (Section 1.3), as well as the
approximation in Equations {A.1.134} and {A.1.135} below are used.
( ) ( ) 323
4 20
cosh sin cosR t u du χπθθϖ χϖ ϖ
∞∂− − ≈ −∂ ∫ , {A.1.134}
( ) ( ) 323
4 20
cosh sin cosR t u dut
χπθθϖ χϖ
∞∂− ≈ −
∂ ∫ , {A.1.135}
Using these approximations, Equations {A.1.129} and {A.1.130} become:
( )
( ) ( )
32
32
4 22
34 22
0
2 sin cos4
2 sin cos
χπ
χπ
τ χπμτ ϖ
θθ χ θπ μ ϖ
∞
⎡ ⎤− −⎢ ⎥⎢ ⎥≈ ⎢ ⎥⎧ ⎫⎢ ⎥− −⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦
∫o
RcHc
qU d
b
, {A.1.136}
( ) ( ) 323
4 22
1 sin cosoa
w V db
χπθθ χ θπ μ ϖ
∞ ⎧ ⎫≈ −⎨ ⎬⎩ ⎭∫P . {A.1.137}
The numerically computed displacement corresponding to the Rayleigh wave
obtained by using Equations {A.1.136} and {A.1.137}, and estimates for the
location and type of the longitudinal and shear waves using the methods
suggested in Lamb’s [1904] paper, are shown in Figure A.2.
340
Appendix 2: Calculations for the Surface Wave Produced by the Loading ofa Circular Area of a Half Space by an Impulse Load.
2.1. Solutions to the Equations of Motion Transformed using LaplaceTransforms
The calculations in Appendix 2 use Cagniard’s Method as described by Mitra
[1964]. The coordinate system is cylindrical and uses r and z as spatial
dimensions and t as time. The half-space of the material is defined to be elastic
and isotropic and is located in the half of the coordinate system where, 0z ≥ .
The surface traction for the impulsive loading of a circular area of radius a, is
given as:
( ) 0 r a for
a r0δ
σ⎧ ⎫ ≤ ≤⎧ ⎫⎪ ⎪= ⎨ ⎬ ⎨ ⎬≤⎪ ⎪ ⎩ ⎭⎩ ⎭
zz
P t, {A.2.1}
0σ =rz . {A.2.2}
In Equations {A.2.1} and {A.2.2}, ( )tδ is the Dirac delta function; zzσ is the
vertical stress; and rzσ is the radial stress. Because of symmetry, there is no
stress in the hoop direction. Other important conditions are that the half-space
is at rest before the impulse and that the displacement is bounded as z →∞ .
The equations of motion for a cylindrical coordinate system with no motion in
the hoop direction are given as:
( ) ( )2 2 2 2
2 2 2 2
12u u u u u wt r r r r z r z
ρ λ μ μ λ μ⎛ ⎞∂ ∂ ∂ ∂ ∂
= + + − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠, {A.2.3}
( ) ( )2 2 2 2
2 2 2
1 1 2w u u w w wt r z r z r r r z
ρ λ μ μ λ μ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
= + + + + + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠. {A.2.4}
Here, u, and, w, are the displacement components along the, r, and, z, axes
respectively; λ, and, μ, are the lame constants; ρ, is the density and; α, and, β,
341
are the compression and shear wave velocities for the half-space respectively.
These last two variables are related to the Lame constants. The definition of the
wave velocities are:
2λ μαρ+
= , {A.2.5}
and,
μβρ
= . {A.2.6}
The solution to Equations {A.2.3} and {A.2.4} involves integral transforms and
also Cagniard’s method. First, the Laplace transform of a function can be made
for any, p, which is positive and real. The transform is defined:
( ) ( ) ( )10
, , , , , ,ptf r z p L f r z t e f r z t dt∞
−= =⎡ ⎤⎣ ⎦ ∫ . {A.2.7}
The Laplace transforms of the equations of motion, with the initial conditions
inserted are:
( )2 2 2
2 2 2 2 21 1 1 1 11 2 2 2
1u u u u wp ur r r r z r z
α β α β⎛ ⎞∂ ∂ ∂ ∂
= + − + + −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠, {A.2.8}
( )2 2 2
2 2 2 2 21 1 1 1 11 2 2
1 1u u w w wp wr z r z r r r z
α β β α⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂
= − + + + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠. {A.2.9}
The Henkel Transforms are defined:
( ) ( ) ( )2 1 10
, , , ,u z p rJ r u r z p drξ ξ∞
= ∫ , {A.2.10}
and,
342
( ) ( ) ( )2 0 10
, , , ,w z p rJ r w r z p drξ ξ∞
= ∫ . {A.2.11}
In Equations {A.2.10} and {A.2.11}, J1, and, J0, are Bessel’s functions.
Multiplying the transformed Equations {A.2.8} and {A.2.9} by ( )1 ξrJ r and
( )0 ξrJ r , respectively, and integrating with respect to r, over an interval of zero
to ∞ gives:
( ) ( )
( ) ( ) ( )
22 2 1 1 1
1 1 12 20 0
2 22 2 21 1
1 120 0
1ξ α ξ
β ξ α β ξ
∞ ∞
∞ ∞
⎛ ⎞∂ ∂= + −⎜ ⎟∂ ∂⎝ ⎠
∂ ∂+ + −
∂ ∂ ∂
∫ ∫
∫ ∫
u u up u rJ r dr rJ r drr r r r
u wrJ r dr rJ r drz r z
, {A.2.12}
( ) ( ) ( )
( ) ( )
22 2 2 1 1
1 0 00 0
2 22 21 1 1
0 02 20 0
1
1
ξ α β ξ
β ξ α ξ
∞ ∞
∞ ∞
⎛ ⎞∂ ∂= − +⎜ ⎟∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂+ + +⎜ ⎟∂ ∂ ∂⎝ ⎠
∫ ∫
∫ ∫
u up w rJ r dr rJ r drr z r z
w w wrJ r dr rJ r drr r r z
. {A.2.13}
Simplifying, Equations {A.2.12} and {A.2.13} take the form:
( )2
2 2 2 2 22 2220 p
u wuz z
β α η α β ξ∂ ∂= − − −
∂ ∂, {A.2.14}
( )2
2 2 2 2 22 2220 s
w uwz z
α β η α β ξ∂ ∂= − + −
∂ ∂, {A.2.15}
The definitions of pη and sη are:
22
2ppη ξα
= + , {A.2.16}
22
2spη ξβ
= + . {A.2.17}
343
The variables pη and sη have real and positive components when the variable ξ
is real. The Equations {A.2.14} and {A.2.15} are subject to conditions for an
infinite half space as r →∞ . Specifically, these equations should have a bounded
solution as z →+∞ , i.e.
2p sz zu Ae Beη η− −= + , {A.2.18}
2p szp z
s
w Ae Beη ηη ξξ η
− −= + . {A.2.19}
In Equations {A.2.18} and {A.2.19} A and B are constants. The stresses from
Equations {A.2.1} and {A.2.2} are defined using the well known stress-strain
relationship:
2zzu w wr z z
σ λ μ⎡ ∂ ∂ ∂ ⎤⎛ ⎞= + +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦, {A.2.20}
and,
zru wz r
σ μ ∂ ∂⎡ ⎤= +⎢ ⎥∂ ∂⎣ ⎦. {A.2.21}
Using the inverse of the Henkel transforms, these stresses become:
( ) ( )2 2 2 21 0 2
0
2zzwJ r u dz
σ ρ ξ ξ α β ξ α ξ∞ ∂⎡ ⎤= − +⎢ ⎥∂⎣ ⎦∫ , {A.2.22}
and,
( ) 21 1 2
0rz
uJ r w dz
σ μ ξ ξ ξ ξ∞ ∂⎡ ⎤= −⎢ ⎥∂⎣ ⎦∫ . {A.2.23}
The boundary conditions for Equations {A.2.22} and {A.2.23} are:
344
( ) ( )11 0
0zz
J aPa J r d
ξσ ξ ξ ξ
ξ
∞
= ∫ , {A.2.24}
and,
1 0rzσ = . {A.2.25}
These boundary conditions are satisfied when,
22
22 2 0p spA Bη η ξβ
⎛ ⎞+ + =⎜ ⎟⎝ ⎠
, {A.2.26}
and,
( )2
2 212 2 2p PaA B J aξ ξ ξ
β μ⎛ ⎞
+ + = −⎜ ⎟⎝ ⎠
. {A.2.27}
In accordance to the Equations {A.2.26} and {A.2.27}, the constants A and B
can be defined as:
( )
( )2
2 2 22
1
42
22 p s
p
p
PaJ aA
β
η η
β ξ
ξ
μ ξ⎛ ⎞+⎜ ⎟⎝ ⎠
−=
⎡ ⎤+ −⎢ ⎥
⎢ ⎥⎣ ⎦
, {A.2.28}
( )22 2
2
1
24β
ξ
η η
ξ
μ⎛ ⎞+⎜ ⎟⎝ ⎠
=⎡ ⎤⎢ ⎥−⎢ ⎥⎣ ⎦
p
p s
PaJ aB . {A.2.29}
Using these values in Equations {A.2.18} and {A.2.19} for 2u and 2w , yields:
( )
( )( )
22 22 22 2 2
2
1 12
242
22 4
p s
pp s
p p s
z z
p
PaJ a PaJ au e e
β
β
η η
ξη η
β η ηξ
ξ ξ
μ ξ μ
− −
⎛ ⎞+⎜ ⎟⎝ ⎠
⎛ ⎞+⎜ ⎟⎝ ⎠
−= +
⎡ ⎤ ⎡ ⎤+ −⎢ ⎥ ⎢ ⎥−
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
, {A.2.30}
345
( )
( )( )
22 22 22 2 2
2
1 12
242
22 4
p s
pp s
p p s
zp z
sp
PaJ a PaJ aw e e
β
β
η η
ξη η
β η ηξ
η ξ ξξξ η
μ ξ μ
− −
⎛ ⎞+⎜ ⎟⎝ ⎠
⎛ ⎞+⎜ ⎟⎝ ⎠
−= +
⎡ ⎤ ⎡ ⎤+ −⎢ ⎥ ⎢ ⎥−
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
. {A.2.31}
Using the inverse of the Henkel transforms on Equations {A.2.30} and {A.2.31}
yields:
( ) ( ) ( ) ( )2
21 1 1 2
0
, ,2 2p szp z
p s
u r z p J r J ae e d
Paη η
β
μ ξ ξ ξξ η η ξ
∞− −⎡ ⎤= − + +
⎣ ⎦Δ∫ , {A.2.32}
and,
( ) ( ) ( ) ( )2
21 0 1 2 2
0
, ,2 2p szp z
p p
w r z p J r J ae e d
Paη η
β
μ ξ ξξ η η ξ ξ
∞− −⎡ ⎤= − + +
⎣ ⎦Δ∫ . {A.2.33}
In Equations {A.2.32} and {A.2.33},
( )2
2
22 22 4p
p sβξ ξ η ηΔ = + − . {A.2.34}
In the analysis, since the desired goal is the surface displacement away from the
impact region, it is only necessary to consider the case when the radius is greater
than the contact area, >r a . For this condition, Watson’s result as referenced
by Mitra [1964] is as follows:
( ) ( ) ( )1 1 00
1 cosJ r J a J R dξ ξ ξ φ φπ
∞
= ∫ , {A.2.35}
( ) ( ) ( )0 1 10
1 cosa rJ r J a J R dR
φξ ξ ξ φπ
∞ −= ∫ . {A.2.36}
Where R is a function of the variable of integration φ,
346
2 2 2 cosR r a ar φ= + − . {A.2.37}
In accordance to Equations {A.2.35} through {A.2.37}, the displacements can be
re-written as:
( )110
, ,cos
u r z pI d
Paππμ
φ φ= ∫ , {A.2.38}
and,
( )120
, , cosw r z p a rI dPa R
ππμ φ φ−= ∫ . {A.2.39}
In Equations {A.2.38} and {A.2.39},
( ) ( )2
20 2
10
2 2p szp zp s
J RI e e dη η
β
ξ ξξ η η ξ
∞− −⎡ ⎤= − + +
⎣ ⎦Δ∫ , {A.2.40}
and,
( ) ( )2
21 2 2
20
2 2p szp zp p
J RI e e dη η
β
ξ ξξ η η ξ ξ
∞− −⎡ ⎤= − + +
⎣ ⎦Δ∫ . {A.2.41}
2.2. Cagniard’s Method for Obtaining the Inverse Laplace Transforms of, I1,and, I2
Mitra [1964] demonstrates Cagniard’s method using a typical term from
Equations {A.2.40} and {A.2.41}. This term is known as, I, and is defined:
( ) ( ) ( ) ( )2 2
2 21 02 2
0 0
12 2p pz zp p
p p
J R J RI e d e d
Rη η
β β
ξ ξξ η ξ ξ η ξ
∞ ∞− −−∂
= − + = +Δ ∂ Δ∫ ∫ .{A.2.42}
The method for transforming Equation {A.2.42} uses the following substitutions:
347
( )cosα ξ θ ψ′ = + , {A.2.43}
( )sinβ ξ θ ψ′ = + , {A.2.44}
cosX R θ= , {A.2.45}
and,
sinθ=Y R . {A.2.46}
Simplifying Equation {A.2.42} using these substitutions and the integral
representation of ( )0J Rξ :
( )2
22
2
1 22
pi X Y
zpp
eI e d dR
α βη
βξ η α β
πξ
∞ ∞ ′ ′− +−
−∞ −∞
∂ − ′ ′= +∂ Δ∫ ∫ . {A.2.47}
The next step is substituting the following trigonometric functions into Equation
{A.2.47}:
cos sinα ω θ θ′ = + q , {A.2.48}
and,
cos sinβ ω θ θ′ = − q , {A.2.49}
Substituting Equations {A.2.48} and {A.2.49} into Equation {A.2.47} and using
a formulation developed by deHoop into {A.2.47} yields:
( )2
22
2
1 22
pi R
zpp
eI e d dqR
ωη
βξ η ω
πξ
∞ ∞ −−
−∞ −∞
∂ −= +∂ Δ∫ ∫ . {A.2.50}
348
Removing the variable, 2ξ is the next step.
2 2 2qξ ω= + . {A.2.51}
Equations {A.2.50} and {A.2.51} give:
( )2
22 2 2
20 0
2 2 pi R
zpp
eq e d dqω
η
β
ωω ξ η ωπ ξ
∞ ∞ −−+ = ℑ +
Δ∫ ∫ , {A.2.52}
Also, the variable q is replaced by qp, and w by wp:
0
2I Jdqπ
∞
= ∫ . {A.2.53}
( )0
Im ,J F q dω ω∞
= ∫ {A.2.54}
In Equation {A.2.54}, Im represents the imaginary part of the integral, and
( )( ) ( ) ( ){ }
( ) ( ) ( ) ( ) ( )2 2 2
2 2 2
2 2 2 2 2 21 1 1
22 2 2 2 2 2 2 2 2 21 1 1
2 2 exp,
2 2 4
α β α
β α β
ω ω ω ω ωω
ω ω ω ω ω
⎡ ⎤+ + + + − + + +⎢ ⎥⎣ ⎦=⎡ ⎤+ + + − + + + + +⎢ ⎥⎣ ⎦
q q p i R z qF q
q q q q q .
{A.2.55}
Equation {A.2.55} has branch points at,
( ) ( )2 2 2 21 , 1i q i qω α β= ± + ± + , {A.2.56}
and poles at,
( )2 2, 1iq i q Vω = ± ± + . {A.2.57}
349
The Rayleigh wave velocity is V; it is smaller than the shear wave velocity β ,
and the longitudinal wave velocity α . The integral J, is then evaluated by
contour integration, which has the result:
( ) ( ) ( ) ( )2
22 2 2 1
1 1 , ,2
zJ L t qR F w t q q H t z R qα α
πβ δα
⎧ ⎫⎡ ⎤= − − − + − − +⎡ ⎤⎨ ⎬⎣ ⎦ ⎢ ⎥⎣ ⎦⎩ ⎭. {A.2.58}
In Equation {A.2.58} the , H, [VP7] followed by the statement in brackets
represents the Henkel transformation of that statement,
[ ]( ) ( )
( ) ( ) ( ) ( ) ( )2 2
2 2 2
2 2 2 21 1
1 22 2 2 2 2 2 2 2 2 21 1 1
2 2, Im .
2 2 4
β α
β α β
ωω ω ωω
ω ω ω ω ω
⎡ ⎤∂+ + + +⎢ ⎥∂= ⎢ ⎥
⎡ ⎤⎢ ⎥+ + + − + + + + +⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
q qtF q
q q q q q
{A.2.59}
Then, the representative term I, becomes,
( ) ( )( ) ( )2 2 2 212
1 10
2 , ,t R zH t z
I L F w t q q dqR
ααβα π
⎡ ⎤+ −⎢ ⎥⎣ ⎦⎧ ⎫−⎡ ⎤⎪ ⎪⎣ ⎦= + ⎡ ⎤⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭
∫ . {A.2.60}
The contribution of the first term in Equation {A.2.60} to the vertical
displacement is proportional to
( ) 20
cosz a rH t dR
π
αφ φ−
− ∫ . {A.2.61}
The inverse Laplace transforms for the remaining terms in Equations {A.2.40}
and {A.2.41} are computed in a similar manner. From these two equations the
following expressions are generated:
350
( )
( )( ) ( )
( )( )
2 2 2 212 2
110
2 2 2
0
122 2 0
, ,
, , cos ,2
, ,
1
α
π
α
π μ φ φεβ
α βε
α αβ
⎡ ⎤+ −⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞+⎢ ⎥− ⎡ ⎤⎜ ⎟ ⎣ ⎦⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎧ ⎫⎛ ⎞− +⎢ ⎥= ⎪ ⎪−⎜ ⎟⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎪ ⎪⎢ ⎥+ ⎡ ⎤⎨ ⎬ ⎣ ⎦⎢ ⎥⎛ ⎞⎪ ⎪−
⎜ ⎟+ − − −⎢ ⎥⎪ ⎪⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎩ ⎭⎣ ⎦
∫
∫∫
t R z
jq
R zH t F w t q q dq
w r z t a r R z dH tPa R
F w t q q dqzRH t
{A.2.62}
and,
( )
( )( ) ( )
( )( )
2 2 2 212 2
130
2 2 2
0
142 2 0
, ,
, ,cos
2, ,
1
α
π
α
π μφ φε
β
α βε
α αβ
⎡ ⎤+ −⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞+⎢ ⎥− ⎡ ⎤⎜ ⎟ ⎣ ⎦⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎧ ⎫⎛ ⎞+⎢ ⎥= ⎪ ⎪−⎜ ⎟⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎪ ⎪⎢ ⎥+ ⎡ ⎤⎨ ⎬ ⎣ ⎦⎢ ⎥⎛ ⎞⎪ ⎪−
⎜ ⎟+ − − −⎢ ⎥⎪ ⎪⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎩ ⎭⎣ ⎦
∫
∫∫
t z R
jq
R zH t F w t q q dq
u r z t R z dH tPa
F w t q q dqzRH t
{A.2.63}
In Equations {A.2.62} and {A.2.63}:
2 2
2 2
z
z
R>0for
1 R<
β
α β
β
α β
ε−
−
⎧ ⎫⎧ ⎫ ⎪ ⎪= ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎪ ⎪⎩ ⎭
, {A.2.64}
( ) ( )2 2 2 21 if 1 or ε α β αβ= = > + −jq q t R z z , {A.2.65}
( ) ( )2 2 2 22 if 0 and ε α β αβ= = < + −jq q t R z z . {A.2.66}
The functions F2, F3, & F4 are:
351
[ ]( )
( ) ( ) ( ) ( )2
2 2 2
2 21
2 22 2 2 2 2 2 2 21 1 1
2, Im
2 2 4
α
β α β
ωω ωω
ω ω ω ω
⎡ ⎤∂− + +⎢ ⎥∂= ⎢ ⎥
⎡ ⎤⎢ ⎥+ + − + + + + +⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
qtF q
q q q q ,
{A.2.67}
[ ]( )
( ) ( ) ( ) ( )2
2 2 2
2 2 1
3 22 2 2 2 2 2 2 21 1 1
2 2, Re
2 2 4
β
β α β
ωωω
ω ω ω ω
⎡ ⎤∂− + +⎢ ⎥∂= ⎢ ⎥
⎡ ⎤⎢ ⎥+ + − + + + + +⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
qtF q
q q q q ,
{A.2.68}
and,
[ ]( ) ( )
( ) ( ) ( ) ( )2 2
2 2 2
2 2 2 21 1
4 22 2 2 2 2 2 2 21 1 1
2, Re
2 2 4
α β
β α β
ωω ωω
ω ω ω ω
⎡ ⎤∂+ + + +⎢ ⎥∂= ⎢ ⎥
⎡ ⎤⎢ ⎥+ + − + + + + +⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
q qtF q
q q q q .
{A.2.69}
2.3. Surface Displacements
From Equations {A.2.62} and {A.2.63}, the surface displacements can be found
by taking the limiting values of and as 0u w z → + . However, this process is
complicated because the Rayleigh pole ( )2 21i q Vω = ± + , lies on the real axis.
The integrands of these two equations have poles at,
( ) ( )2 2 21q t R V⎡ ⎤= −⎣ ⎦ , {A.2.70}
for /t R V> . As 0z → + :
2
2 2 2
1 , and j jt itq q wR Rα
⎛ ⎞ −= → − →⎜ ⎟
⎝ ⎠. {A.2.71}
352
The Rayleigh pole is located at t:
2 22 2 2 2 2 2
1 1 1 1 1 1,t R q iz R q izV V V Vα β
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − − + − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
. {A.2.72}
Then, when t is real and z is positive and small, q is near the Rayleigh pole.
Thus, the integral on the q-axis must bypass the pole when [ ]Im 0q < :
( ) ( )2 2 21⎡ ⎤= −⎣ ⎦q t R V . {A.2.73}
Using the sign convention:
( )2
2 21when, , tR
Rt qα α> < − ,
( ) ( )2
2 2 22 2 21 1t
Rw q i q
α α+ + → − − − , {A.2.74}
( )2
2 21when, , tR
Rt qβ β> < −
( ) ( )2
2 2 22 2 21 1t
Rw q i q
β β+ + → − − − , {A.2.75}
and,
( ) ( )2 2
2 2 2 21 1
when, ,0
or ,
R R
t tRR R
t q
t q
α β
β β α
< < <
< − < < −
( ) ( )2
2 2 22 2 21 1 ,t
Rw q q
β β+ + → − + {A.2.76}.
353
Equations {A.2.62} and {A.2.63} for u and v can now be written in terms of real
values:
( )[ ]
[ ] ( )( )
2 2 21
2 2 2
2 2 2
02
1
02
12 2
,
, , cos2 ,
1
α
π
α
β
α βπ μ φ φ
β
⎡ ⎤−⎢ ⎥⎣ ⎦
⎡ ⎤−⎣ ⎦
⎡ ⎤−⎣ ⎦
⎡ ⎤⎧ ⎫⎛ ⎞⎛ ⎞⎢ ⎥− − −⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭⎢ ⎥
= ⎢ ⎥Γ −⎛ ⎞⎢ ⎥+ − +⎜ ⎟⎢ ⎥⎝ ⎠ ⎛ ⎞−⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
∫
∫∫
t R
t R
t R
R RH t H t F t q dq
u r z tdH t R VRPa R H t F t q dq
tR V
. {A.2.77}
and,
( ) ( )
[ ]
[ ]
[ ]
2 2 21
2 2 2
2 2 2
2 2 2
0
1
23
10
1
0
,
cos,,0,
2
,
α
α
π
β
β
α β
φπ μ
φβ
⎡ ⎤−⎢ ⎥⎣ ⎦
⎡ ⎤−⎣ ⎦
⎡ ⎤−⎣ ⎦
⎡ ⎤−⎣ ⎦
⎡ ⎤⎧ ⎫⎛ ⎞⎛ ⎞⎢ ⎥− − −⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭⎢ ⎥⎢ ⎥− ⎛ ⎞⎢ ⎥+ −⎜ ⎟⎢ ⎥= ⎝ ⎠⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥⎣ ⎦
∫
∫∫
∫
t R
t R
t R
t R
R RH t H t G t q dq
t a r RH t G t q dqw r tdR
Pa
G t q dqP
.{A.2.78}
In Equations {A.2.77} and {A.2.78}:
( )( ) ( ) ( ){ }
( ) ( ) ( )( )
2 2 2
2 2 2 2 2 2
2 2 2 2
2 2 2 2 2 2 2
2 2 21 1 1
4 22 2 2 2 221 1 1
2 2,
2 16
t t tR R R
t t t tR R R R
q q qF t q
q q q q
α β β
β α ββ
− − − − + − +=
⎡ ⎤− + + − − − − +⎢ ⎥⎣ ⎦
, {A.2.79}
( )( ) ( )
( ) ( ) ( ) ( )( )
2 2
2 2 2 2
2 2 2 2 2
2 2 2 2 2 2 2 2
2 221 1
4 22 2 2 2 2 221 1 1
2,
2 16
β α
β α ββ
− + − −=
⎡ ⎤− − + + − − − − +⎢ ⎥⎣ ⎦
t tR R
t t t t tR R R R R
q qG t q
q q q q q,
{A.2.80}
( )( )
( ) ( ) ( ) ( ) ( )
2
2 2
2 2 2 2 2
2 2 2 2 2 2 2 2
21
2 22 2 2 2 2 221 1 1
,2 4
α
β α ββ
− −=
⎡ ⎤− − + − − − − − +⎢ ⎥⎣ ⎦
tR
t t t t tR R R R R
qG t q
q q q q q ,
354
{A.2.81}
and,
( ) ( ) ( )( ) ( ) ( )
2 2 2
2 2 2
2 2 4 2 2 2
2 2 2 2 2 2 23 3 2
2 1 1
4 4 2 2 1 1
V V V
V V V V V V
β α β
α β α β β α β
π − − −Γ =
− − − − − − −. {A.2.82}
These terms contain the longitudinal, shear, and Rayleigh wave contributions.
The F terms give the displacements due to the longitudinal wave. The second
term in the equation for w (Equation {A.2.78}), and the second and third terms
in the equation for u (Equation {A.2.77}) give the contributions to the shear and
longitudinal waves up to the arrival of the first Rayleigh wave. The Γ term in
{A.2.77} gives the Rayleigh wave contribution to u, while the G term in
{A.2.78} gives the Rayleigh wave contribution to w. The Rayleigh wave
contribution in {A.2.77} can be written as, uo:
( ) ( )2
2 21
0
,0, cos2o
tR V
H t R Vu r td
Pa R
ππμ φ φ−⎡ ⎤⎣ ⎦=
Γ −∫ . {A.2.83}
355
This Rayleigh wave contribution can be written in a more explicit form as
( ) ( ) ( )
( ) ( )( )
( )2 2 2 2
22 2 1 1
22 2
0,0,
2 .2
πμ
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪= −⎡ ⎤⎨ ⎬⎣ ⎦Γ ⎪ ⎪⎪ ⎪⎡ ⎤⎪ ⎪+ −⎢ ⎥⎡ ⎤− − −⎪ ⎪⎢ ⎥⎣ ⎦ ⎡ ⎤⎪ ⎪− −⎢ ⎥⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭
o
k k
u r t V E k K kPa ar
V r a V tV t r a E Kar V t r a
{A.2.84}
when,
( )( ) ( )( )
0 -
-
< <⎧ ⎫⎪ ⎪
< < +⎨ ⎬⎪ ⎪+ <⎩ ⎭
t r a V
r a V t r a V
r a V t
.
In Equation {A.2.84},
( )22 2
4V t r a
kar
⎛ ⎞− −= ⎜ ⎟
⎜ ⎟⎝ ⎠
, {A.2.85}
( )2
2 2
0
1 sinE xdxπ
θ θ= −∫ , {A.2.86}
and,
( )2
2 20 1 sin
dxKx
π
θθ
=−
∫ . {A.2.87}
In the special case where λ μ= , Equations {A.2.77} and {A.2.78} for u and w,
can be further simplified as:
356
( ) ( ) ( )0 2
0
cos,0, ,0,
UPu r t u r t dR
π τ φαβ φπ μ
= + ∫ , {A.2.88}
( ) ( )2 20
cos,0, P r aw r t W dR
παβ φ τ φπ μ
−= ∫ . {A.2.89}
In Equations {A.2.88} and {A.2.89}:
for,
0< <1 3
1 3 11
τ
ττ
⎧ ⎫⎪ ⎪⎪ ⎪< <⎨ ⎬⎪ ⎪<⎪ ⎪⎩ ⎭
:
( )( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )
( ) ( )2 13
2 2
318 2
2
1 1 1
14 1
0
6 18 8 , 6 4 3 20 12 3 ,,
6 4 3 20 12 3 ,
3 9 8, 2 3 3 20 12 3 ,
2 3 3 20 12 3 ,τ
τ
−
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤⎡ ⎤− Π + − Π −⎪ ⎣ ⎦ ⎪⎢ ⎥= ⎨ ⎬⎢ ⎥⎡ ⎤⎪ ⎪+ + Π +⎢ ⎥⎣ ⎦⎣ ⎦⎪ ⎪⎪ ⎪⎡ ⎤⎡ ⎤− Π − − Π −⎪ ⎪⎣ ⎦⎢ ⎥⎪ ⎪⎢ ⎥⎡ ⎤+ + Π +⎪ ⎪⎢ ⎥⎣ ⎦⎣ ⎦⎩ ⎭
l l l
l
K l l l l lU
l l
K
{A.2.90}
and for,
12
12
0< <1 3
1 3 1
1 3 3
3 3
τ
τ
τ
τ
⎧ ⎫⎪ ⎪
< <⎪ ⎪⎪ ⎪⎨ ⎬< < +⎪ ⎪
⎪ ⎪+ <⎪ ⎪⎩ ⎭
:
357
( )2 2 21 3 3 3 3
4 4 4
23 34
3 2 3 3 2 3 34 12 123 3 3 5 3 3 5
2 3 363 3 3 5
3
0ππ τ π τ π τ
τ τ τ
π π τ
τ
π
τ− −
−
− +
− +− − −
+
+ −
⎧ ⎫⎪ ⎪− + − +⎪ ⎪⎪ ⎪= ⎨ ⎬− +⎪ ⎪⎪ ⎪−⎪ ⎪⎩ ⎭
W . {A.2.91}
In Equations {A.2.90} and {A.2.91},
23 12
l τ⎛ ⎞−= ⎜ ⎟
⎝ ⎠, {A.2.92}
tRβτ = , {A.2.93}
2
3 3
Vβ=
+, {A.2.94}
and,
( )( )
2
2 2 20
,1 sin 1 sin
dn kn k
π θθ θ
Π =+ −
∫ . {A.2.95}
Equation {A.2.88} gives Mitra’s [1964] solution for the longitudinal, shear and
Rayleigh waves contributions to the radial displacement. The numerical solution
of this equation for the radial displacements is plotted in Figure A.3.
358
Appendix 3: Calculation of Energy Absorbed by Elastic Stress WavesDuring Impact
In Chapter III, the impact energy that is imparted to the specimen are compared
with the predictions obtained from the simulations using LS-DYNA and from an
analytical estimate for the elastic energy. This analytical solution is designed to
estimate the fraction of the impact energy that is carried away as stress waves.
The derivation of this energy estimate is taken from calculations by Hunter
[1957]. The rate of work W, done on the specimen is given by:
( )2dW dua P tdt dt
π= . {A.3.1}
In Equation {A.3.1}, a is the contact radius; P(t) is the pressure on the surface; t
is time; and, u is the Fourier synthesis of the mean surface displacement.
Integration over time yields:
( )2dW duW dt a P t dtdt dt
π∞ ∞
−∞ −∞
= =∫ ∫ . {A.3.2}
Miller and Pursey [1954] provide an equation of the normal surface displacement
u in a semi-infinite solid, for a uniform pressure ( )0i tPe ω ω > , over the contact
area 0 r a≤ ≤ as:
( ) ( ) ( )( )
122 2 2
1 122
440
, , 0 ωξ ξ ξ
ξ
∞ −= = ∫i t
k J a du r t z Pe k
C F. {A.3.3}
The radial dimension is represented by r, and the vertical by z, and ξ is the
variable of integration. In Equation {A.3.3},
( )12
1 11/ /ω ρ=k C , {A.3.4}
359
( )12
2 44/ /ω ρ=k C , {A.3.5}
and,
( ) ( ) ( )( )1222 2 2 2 2 2 2
2 1 22 4F k k kξ ξ ξ ξ ξ⎡ ⎤= − − − −⎣ ⎦ . {A.3.6}
In Equations {A.3.3} through {A.3.6},C11 and C44 are the components of the
stiffness matrix, and ρ is the density of the specimen. The mean displacement
within the contact radius a, is:
( ) ( )( )
122 2 2
1 122
440 0
2 i tk J a d
u urdr Pe ka C F
ωξ ξ ξ
ξ ξ
∞ ∞ −= =∫ ∫ . {A.3.7}
This can also be re-written as,
( ) ( )( )
122 22 2
1 1 12 2
44 10
12 i t
o
k J k a da PeuC k a F
ω ξ ξ ξγξ ξ
∞ −= ∫ . {A.3.8}
In Equation {A.3.8},
( ) ( ) ( )( )1222 2 2 2 2 22 4 1oF ξ ξ γ ξ ξ ξ γ⎡ ⎤= − − − −⎣ ⎦ , {A.3.9}
and,
( ) ( ) ( ){ }11222
11 441
/ 2 1 / 1 2k C Ckγ ν ν= = = − − . {A.3.10}
In Equation {A.3.10}, υ is the Poisson’s ratio of the specimen. The integral in
Equation {A.3.8} can be re-written for 0ω > , as:
360
( ) ( )( )2 2
44
2 ωγ ω ω= +i ta Peu A iB
C. {A.3.11}
In Equation {A.3.11}, the variable ω, represents frequency, and i represents the
square root of negative one. Also,
( ) ( )( )
122 2
1 11 2 2
10
1
o
J k a dA iB k
k a Fξ ξ ξ
ξ ξ
∞ −+ = ∫ . {A.3.12}
For 0ω < ,
( ) ( )( )2 2
44
2 ωγ ω ω= − +i ta Peu A iB
C. {A.3.13}
Where, A, and, B, represent the real and imaginary parts of the integral.
This gives real values of displacement u, for pressures ( ) ( )cos , and sinP t P tω ω .
Next, an arbitrary pulse shape, P(t), is considered:
( ) ( ) i tP t P e dωω ω∞
−∞
= ∫ . {A.3.14}
The superposition principle is used to determine the mean normal displacement,
u , for this load:
( ) ( ) ( )
( ) ( ) ( )
2 20
044
2ω
ω
ω ω ω ωγ
ω ω ω ω
∞
−∞
⎧ ⎫+⎡ ⎤⎪ ⎪⎣ ⎦
⎪ ⎪= ⎨ ⎬⎪ ⎪+ − +⎡ ⎤⎣ ⎦⎪ ⎪⎩ ⎭
∫
∫
i t
i t
P e A iB dau
CP e A iB d
. {A.3.15}
Simplifying,
361
( ) ( ) ( )( )( ) ( ) ( )( )
2 2
44 0
2ω
ω
ω ω ωγ ωω ω ω
∞
−
⎧ ⎫+⎪ ⎪= ⎨ ⎬+ − −⎪ ⎪⎩ ⎭
∫i t
i t
P e A iBau dC P e A iB
. {A.3.16}
Substituting Equation {A.3.16} into the differential Equation {A.3.1}:
( )( ) ( ) ( )( ) ( ) ( )
'2 4
'44 0
2 ' 'ω ω
ω ω
ω ωπγ ω ω ωω ω
+∞ ∞
−−∞
⎧ ⎫+− ⎪ ⎪= ⎨ ⎬− − −⎪ ⎪⎩ ⎭
∫ ∫i t
i t
P e A iBdW a i d d Pdt C P e A iB
. {A.3.17}
Integrating with respect to time, t:
( ) ( )' 2 'i te dtω ω πδ ω ω∞
+
−∞
= +∫ , {A.3.18}
( ) ( )' 2 'i te dtω ω πδ ω ω∞
−
−∞
= −∫ . {A.3.19}
The Dirac delta function is δ. Therefore, Equation {A.3.17} becomes:
( ) ( ) ( ) ( ){ }2 2 4
44 0
4 aW i d P P A iB A iBCπ γ ω ω ω ω
∞−= − + − −⎡ ⎤⎣ ⎦∫ . {A.3.20}
Using the fact that the function P(ω), is equal to its complex conjugate, P*(ω):
( ) ( )P Pω ω∗= , {A.3.21}
Equation {A.3.21} can be written as:
( ) ( )2 2 4
2
44 0
8 aW B P dC
π γ ω ω ω ω∞
= ∫ . {A.3.22}
362
In simplifying, Equation {A.3.22} the frequencies, ω , are restricted to satisfy, the
condition
( )12
11/ / 1a Cω ρ . [VP8] {A.3.23}
A new quantity is defined as being equal to the quantity on the left hand side of
Equation {A.3.23}
( )12
1 11/ /k a a Cω ρ≡ {A.3.24}
For these frequencies:
( )( )
( )
12
12
2
011
1
4 / o
daA iBFC
ξ ξ ξωξρ
∞ −+ = ∫ , {A.3.25}
( )( )
12
114 /B
Cβωω
ρ= , {A.3.26}
and,
( )( )
( )( ) ( )( )
12
12 2 2
12 22 2 2 2 2 20 0
1 1Im
2 4 1o
d dF
ξ ξ ξ ξ ξ ξβ
ξ ξ γ ξ ξ ξ γ
∞ ∞⎡ ⎤− −⎢ ⎥= =⎢ ⎥ ⎡ ⎤− − − −⎣ ⎦ ⎣ ⎦∫ ∫ . {A.3.27}
The notation, Im, represents the imaginary part of the integral. Beta, β , in
Equation {A.3.27} was estimated to be a function of Poisson’s ratio such that
when 1/ 4ν = , 0.5374β ≈ , and when 1/ 3, 0.415ν β= ≈ . Using the integral
equation for W, i.e. {A.3.22}, and combining it with the equation for B, i.e.
{A.3.26}, yields :
( )( )1
2
2 2 422
044 11
2/aW P d
C Cπ γβ ω ω ω
ρ
∞
= ∫ . {A.3.28}
363
In Equation {A.3.28},
( ) 2
44
4 1C
ν γρ+
= , {A.3.29}
Simplifying Equation {A.3.28} using:
( ) ( ) ( ) ( )1 22 i tf f t e dt a Pωω π π ω∞
− −
−∞
= =∫ . {A.3.30}
yields,
( ) ( )122
223
0
8 1 11 2
β ν ν ω ω ωρ ν
∞+ ⎧ ⎫−= ⎨ ⎬−⎩ ⎭
∫o
W f dC
. {A.3.31}
In Equation {A.3.31},
( )12/oC E ρ= , {A.3.32}
In Equation {A.3.32}, the modulus of elasticity of the specimen is, E. Using the
specific pulse shape for an elastic collision, which is defined:
( ) ( )( )
cos20
2
πωω
πω
⎧ ⎫≤ =⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬=≥ ⎪ ⎪⎪ ⎪ ⎩ ⎭
⎩ ⎭
o oo
o
t f t M t
f tt. {A.3.33}
In Equation {A.3.33}, Mo, is the pulse magnitude, and ωo the pulse frequency.
Using Equation {A.3.33}, Equation {A.3.30} is redefined as:
364
( ) ( ) ( )/ 2
2 2/ 2
cos / 2cos
2
o
o
oi to o oo
o
M Mf t e dtπ ω
ω
π ω
ωπ ωωω ωπ π ω ω
−
−
= =−∫ . {A.3.34}
Plugging this into the Equation {A.3.31} for W:
( ) ( )( )
12 2 212
23 2 20
8 1 cos / 211 2 1
o o
o
z zMW dzC z
β ν πωνρ ν π
∞+ ⎧ ⎫−= ⎨ ⎬−⎩ ⎭ −
∫ . {A.3.35}
Using residue calculus, the integral in Equation {A.3.35} is the real part of the
integral I:
( )( )
2
22
114 1
i z
C
z eI dz
z
π +=
−∫ . {A.3.36}
The integral in Equation {A.3.36} has simple poles at,
1Z = ± . {A.3.37}
These poles have residues:
Res ( )1± / 4iπ= − . {A.3.38}
Cauchy’s integral theorem states:
( )( )
2 2 2
220
cos / 24 4 4 81
z z i i idzz
π π π π π∞ ⎡ ⎤= − − =⎢ ⎥⎣ ⎦−∫ . {A.3.39}
Therefore, Equation {A.3.35} for the energy produced by the impact, W, is given
by:
365
( )1
2 2
30
1 11 2 o oW M
Cβ ν ν ωρ ν+ ⎧ ⎫−
= ⎨ ⎬−⎩ ⎭. {A.3.40}
An equation for the contact indentation distance, Z, is:
( )sin 1.068o o oZ Z V t Z= . {A.3.41}
The maximum indentation, Z0, can be determined as
251 41 5 5
0 01
1516
m mZ g R Vm m
−⎧ ⎫⎪ ⎪⋅⎪ ⎪= ⎨ ⎬⎪ ⎪+⎪ ⎪⎩ ⎭. {A.3.42}
In Equation {A.3.42}, m1 is the projectile mass; R is the radius of the projectile;
andV0 is the impact velocity. The constant g is defined as:
2 21
1
1 1gE Eυ υ− −= + . {A.3.43}
The modulus of elasticity of the projectile is E1, and the Poisson’s ratio of the
projectile is υ1. The impact force is approximated as one half period of a sine
function, and is related to the acceleration in the vertical direction Z ,
( ) 1f t m Z= . {A.3.44}
For specific time during which the force acts, and the forcing function in
Equation {A.3.44} can be written as:
( ) ( )( )
21
0 sin
00,
πω ωω
πω
⎧ ⎫≤ ≤ ⎧ ⎫= −⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬=≤ ≥ ⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭
o o oo
o
t f t m Z t
f tt t. {A.3.45}
In Equation {A.3.45}:
366
( )1.068o o oV Zω = . {A.3.46}
Offsetting the zero time by / 2 oπ ω , such that the peak is at time zero, the
Equation {A.3.45} can be written as:
( ) ( )( )
21 sin2
02
πω ωω
πω
⎧ ⎫≤ ⎧ ⎫= −⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬=≥ ⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭
o o oo
o
t f t m Z t
f tt. {A.3.47}
If the magnitude of the pulse is specified as,
21o o oM m Z ω= − , {A.3.48}
then the elastic energy in Equation {A.3.40} can be written,
( ) ( )125 2
3 513
1.068 1 11 2 o o
o
W m Z VC
ν β νρ ν
−+ ⎧ ⎫−= ⎨ ⎬−⎩ ⎭
. {A.3.49}
In the limit where m →∞ , Equation {A.3.42} becomes,
{ }2 15 5 4 /515
116o oZ m g R V−= . {A.3.50}
Substituting Equation {A.3.50} for, Zo, and taking the limit gives,
3 6 1345 5 5 5
13
τρ
−
= o
o
m R g VWC
. {A.3.51}
In Equations {A.3.50} and {A.3.51},
( ) ( )61
2 55 1 161.068 11 2 15
ντ ν βν
−⎛ ⎞ ⎛ ⎞= + ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠. {A.3.52}
367
Normalizing by the initial kinetic energy, iKE , which is
112
=i oKE mV {A.3.53}
yields,
3 6 315 5 5 5
13
21
212
o
oo
m R g VWCmV
τρ
−
= . {A.3.54}
If the impacting mass is a sphere, then the mass can be defined in terms of the
radius R, and the density of the sphere 1ρ , as
( ) 31 14 3m Rπ ρ= . {A.3.55}
Substituting Equation {A.3.55} into Equation {A.3.54}, yields:
( )6 311
5 5 551
32
1
2 4 / 312
o
oo
g VWCmV
τ π ρρ
−−
= . {A.3.56}
This equation for the normalized impact energy is used in the energy analysis in
Chapter III.
368
References
Hunter, S. G. 1957. “Energy Absorbed by Elastic Waves During Impact.” Journal of the
Mechanics and Physics of Solids. 5 162-171.
Lamb, H., 1904. “On the Propagation of Tremors over the Surface of an Elastic Solid.
Philosophical Transactions of the Royal Society. A 203 1-42.
Miller, G. F. and H. Pursey, 1954. “The Field and Radiation Impedance of Mechanical Radiators
on the Free Surface of a Semi-Infinite Isotropic Solid.” Proceedings of the Royal Society of
London Series A, Mathematical and Physical Sciences. 233 [1155] 521-541.
Mitra, M. 1964. “Disturbance Produced in an Elastic Half-Space by Impulsive Normal Pressure.”
Proceedings of the Cambridge Philosophical Society. 69: 683-696.
369
Figures
Non-Dimensional Time
Nor
mal
ized
Dis
plac
emen
t
-1 -0.5 0 0.5 1-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Radial DisplacementVertical Displacement
Figure A.1: Disturbance in the surface displacement as described by Lamb’s equations [1904].
Numerical plots of Equations {A.1.126} and {A.1.127} for, q0, and, w0, are depicted.
370
Non-Dimensional Time
Nor
mal
ized
Dis
plac
emen
t
0 0.5 1 1.5 2 2.5-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Radial Displacement
Longitudinal Wave
Shear Wave
Rayleigh Wave
Non-Dimensional Time
Nor
mal
ized
Dis
plac
emen
t
0 0.5 1 1.5 2-0.25
0
0.25
0.5
0.75
1
Vertical Displacement
Longitudinal Wave
Shear Wave
Rayleigh Wave
Figure A.2: Schematic solutions of the entire displacement wave based on Lamb’s Equations
{A.1.136} and {A.1.137}. Displacements caused by the longitudinal and shear wave pulses are
estimated here only in form because they could not be determined with this method. The
distance between the waves and the relative amplitude of the waves have been altered to show
the profiles of the different waves. (a) Radial displacement. (b) Vertical displacement.
(a)
(b)
371
Time (s)
Nor
mal
ized
Dis
plac
emen
t
0 5E-06 1E-05 1.5E-05-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Longitudinal Wave
Rayleigh Wave
Shear Wave
Figure A.3: Mitra’s displacement along the impact surface of the specimen in the radial direction.
This solution was arrived at by numerically solving of Equation {A.2.88}. The longitudinal (P-
Wave), Shear (S-Wave) and Rayleigh wave (R-Wave) components are all visible. The
differentiation of this solution provides the strain in the radial direction.
[VP9]