N 

Shock Response of Snow 

Analysis of Experimental Methods and 

Constitutive Model Development 

Jerome B. Johnson, Jay Α. Brown, Edward 5. Gaftney, 

George L. Blaisdell and Daniel J. 50118 

July 1441 

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For conversion of SI metric units to U.S./British customary units 

of measurement consult ASTM Standard Ε380, Metric Practice 

Guide, published by the American Society for Testing and 

Materials, 1916 Race St., Philadelphia, Pa. 19103. 

COVER: Schematic of the experimental apparatus and 

the P-V curve. 
CRRΕ L Report 92-12 

U.S. Army Corps 

of Engineers 

Cold Regions Research & 

Engineering Laboratory 

Shock Resροnse of Snow 

Analysis of Experimental Methods and 

Constitutive Model Development 

Jerome B. Johnson, Jay Α. Brown, Edward S. Gaffney, 

George L. Blaisdell and Daniel J. Solie 

July 1992 

Prepared for 

OFFICE OF THE CHIEF OF ENGINEERS 

Approved for public release; distribution is unlimited. 
PREFACE 

This report was prepared by Dr. Jerome B. Johnson, Geophysicist, Applied Research 

Branch, Experimental Engineering Division, U.S. Army Cold Regions Research and 

Engineering Laboratory (CRREL); Dr. Jay A. Brown, Physicist, Los Alamos National 

Laboratory; Dr. Edward S. Gaffney, Geologist, Ktech Corporation; George L. Blaisdell, 

Research Civil Engineer, Applied Research Branch, Experimental Engineering Division, 

CRREL; and Dr. Daniel J. Solie, Physical Scientist, Applied Research Branch, Experimental 

Engineering Division, CRREL. The research described in this report was funded by DA 

Project 4Α762784ΑΤ42, Cold Regions Engineering Technology; Work Unit CS/012, Attenuation of Shock Waves by Snow. 

Dr. T. Day, Los Alamos National Laboratory, technically reviewed this report. 

The contents of this report are not to be used for advertising or promotional purposes. 

Citation of brand names does not constitute an official endorsement or approval of the use 

of such commercial products. 
CONTENTS 

Preface 

Introduction 1 

Experimental methods 1 

Experimental results and difficulties 3 

Model simulation of the experimental design 4 

Constitutive relationship and discussion 7 

Conclusion 10 

Literature cited 11 

Appendix A: Propagation of stress waves from the lateral edges of the aluminum 

buffer and snow 13 

Appendix B: Modified soil and crushable foams model 16 

Abstract 19 

ILLUSTRATIONS 

Figure 

1. Schematic of the snow target assembly 2 

2. Stress records from gauge 1 in the aluminum and gauges 2-6 embedded in snow 3 

3. Schematic of the experimental configuration used to generate the one-dimensional 

finite-element mesh for PRONTO 2D 5 

4. Comparison of the calculated stresses for gauge plane 0 with the measured 

stress record for gauge 1, located in the aluminum buffer 6 

5. Loading pressure—volumetric strain curve and unloading—reloading curves used in 

PRONTO 2D to model snow compression and Abele and Gow's quasi-static 

pressure—volumetric strain results 8 

6. Comparison of the calculated stresses for gauge plane 1 with measured 

stress records for gauges 2 and 3 9 

7. Comparison of the calculated stresses for gauge plane 2 with measured 

stress records for gauges 4 and 5 10 

8. Comparison of the calculated stresses for gauge plane 3 with measured 

stress records for gauge 6 1 0 

Al. Schematic of the two-dimensional axisymmetric finite-element mesh for 

PRONTO 2D 13 

A2. Pressure—time plots for three cross sections in the 2D axisymmetric model 14 

B 1. Pressure vs volumetric strain in terms of user-defined curves for pressure at 

the maximum volumetric strain, pressure upon unloading and pressure 

upon reloading or the modified soils and crushable foams material model 17 

TABLES 

Table 

1. Stress gauge positions relative to the snow/aluminum interface 2 

2. Constitutive model parameters used to develop the PRONTO 2D model calculation 5 

3. P-V curve parameters for Abele and Gow's measurements 8 

4. Snow P-V curve parameters, moduli data and unloading curve parameters used 

to define snow behavior in PRONTO 2D model calculations 9 
Shock Response of Snow 

Analysis of Experimental Methods and Constitutive Model Development 

JEROME B. JOHNSON, JAY A. BROWN, EDWARD S. GAFFNEY, 

GEORGE L. BLAISDELL AND DANIEL J. SOLIE 

INTRODUCTION EXPERIMENTAL METHODS 

Α constitutive model of the shock compression of 

snow is of interest fonts direct application to such fields 

as planetary sciences, cold regions and military engineeńng, and shock isolation. The only existing data 

obtained with reliable experimental methods are the 

high-pressure data (3.8-35.4 GPa) of Bakanova et al. 

(1975). Experiments by Napadensky (1964), Wakahama 

and Sato (1977), Sato and Brown (1983) and Sato 

(1987) did not meet criteria for steady plane-wave 

propagation even though the data were reduced using 

Α shock wave experiment was conducted using a 

200-mm-diameter gas gun to obtain uniaxial strain 

loading in snow. Α flat-faced 35.9-mm-thick, polymethyl methacrylate (PMMA) flyer plate was accelerated 

with compressed nitrogen gas to impact a snow target. 

that assumption, 

The target assembly consisted of a copper cylinder 

with sealed aluminum capping end plates to provide a 

vacuum-tight cannister (Fig. 1). Α vacuum-tight target 

assembly was required because the target chamber of 

the gas gun was evacuated prior to firing. The snow, 

which had a vapor pressure of about 300 Pa, would have 

sublimated if it had not been isolated in a sealed cannister. Α spiral of copper tubing was soldered around the 

copper cylinder, allowing refrigeration of the sample 

once the target had been mounted on the gas gun. The 

target was assembled in an adjacent coldroom with the 

target axis vertical. Α buffer plate, consisting of a 

carbon stress gauge epoxied between two aluminum 

plates (12.7-mm-thick plate epoxied to a 6.4-mm-thick 

We have conducted a test program using embedded 

stress gauges to obtain stress—strain relations for snow 

(Brown et al. 1988). The unsteady nature of shock wave 

propagation in the snow, the specialized sample preparation methods, and the large impedance mismatches 

between the snow and stress gauges resulted in complex 

stress histories. These features precluded direct application of the Rankine—Hugoniot jump conditions or the 

Lagrangian conservation equations for mass, momentum and energy (Fowles and Williams 1970, Seaman 

1976) to analyze the experimental results. Instead, a 

detailed analysis of one of our experiments, using the 

PRONTO 2D dynamic finite-element program (Taylor 

and Flanagan 1987), was conducted to determine the 

origins of distinctive features from measured stress 

histories and to determine loading, unloading and reloading paths for the snow. In this paper we discuss our 

experimental procedures and difficulties, the data records 

for a typical shot, our assumptions for determining the 

initial loading curve, and unloading and reloading curves 

for the snow, as well as our results. The overall shock 

response of snow derived from the entire shot test series 

will be presented separately. 

plate), formed the front of the target. 

Snow was sieved into the copper cylinder in stages, 

with stress gauges clamped in place at specific distances 

from the aluminum/snow interface (Table 1). The stress 

gauges were placed off-axis so that the shock wave had 

a direct, unobstructed path from the buffer through the 

snow to each gauge. Our snow had approximately 

millimeter-sized grains. Consequently our uncertainty 

of gauge position was about ±1 mm, which for this 

experiment corresponds to an uncertainty in arrival time 

of about ±5 μs. 

Thermocouples were placed in the snow and were 

used to monitor the sample temperature. The copper 

cylinder was completely filled with snow, and the 

instrumentation wires were run along the inside wall of 
110 

111111111111111 

• • • 

Figure 1. Schematic of the snow target assembly, consisting of an aluminum plate (A 

and B), a carbon gauge (1), a cooling coil (C) 

soldered to a copper cylinder, an aluminum 

back support plate (D), gauge and thermocouple leads exiting the rear surface (E), and 

a gauge supportpedestal mounted to the inside 

wall of a copper cylinder (F). 

4 ^ 

Im pact Direction 

Snow 

6 

267 mm 

5 3 

A 

I 

E Β 

F 

1 

the copper cylinder and through vacuum-sealed holes in 

the back plate (12.7-mm aluminum). Six shorting pins, 

each with a different length, were mounted concentrically on the outer edge of the aluminum. These shorting 

pins were used to determine the flyer plate impact 

perature was controlled by circulating cold nitrogen gas 

through the copper tubing attached to the outer wall of 

the target. 

velocity and to trigger data acquisition, 

Stress—time records were measured using 50-ohm 

carbon-film piezoresistive gauges (Krehl 1978). Their 

high sensitivity allows for a relatively low excitation 

power and less Joule heating of the gauge compared to 

other piezoresistive gauges. The gauge has a calibration 

uncertainty on loading and unloading of between 5 and 

After the target was assembled, the sample was "cold 

soaked" in a refrigerated room overnight before being 

mounted on the end of the gas-gun barrel. Snow ternTable 1. Stress gauge positions relative to the snow/aluminum interface. 

Relative Gauge Average relative 

Gauge Gauge position thickness position 

plane number (mm) (mm) (mm) 

—6.45±0.01 0.1778* —6.45 

13.2±1 0.8128 14.0 

14.8 ± 1 0.6604 

2 27.2 ± 1 0.6858 26.0 

24.8 ± 1 0.5080 

3 37.2 ± 1 0.5824 37.2 

*Includes epoxy and mylar tape (the unclad thickness was 0.0762 mm). 

2 
10% of the stress at stresses below about 2.0 GPa 

(Gourdin and Weinland 1986, King and Jande 1987). 

Hysteresis effects range from 0 to 9% of the stress on 

partial or complete unloading (King and Jande 1987). 

Our experience in using the gauge in soils and aluminum have not shown evidence of measurable hysteresis. 

The active element for the carbon gauge (0.75 x 1.25 cm) 

forms a single, continuous, wide strip rather than a grid, 

as is the case for the more commonly used manganin 

and ytterbium gauges; it is thus less susceptible to 

puncturing by individual snow grains. The gauges were 

encapsulated between 0.025-mm-thick layers ofkapton. 

Recording life was extended by using a 0.19- to 0.34- 

output signals were recorded digitally using CAMACbased waveform digitizers with a band width of 1 MHz 

and a sampling rate of 2 MHz. 

EXPERIMENTAL RESULTS 

AND DIFFICULTIES 

mm layer of mica as armor, 

Figure 2 shows data from the six stress gauges in the 

target assembly (Fig. 1): one in the aluminum buffer and 

the remainder in the snow (compressive stresses are 

treated as positive). For this shot the snow had an initial 

density of 400 kg m-3and a temperature of -8 °C. The 

flyer plate impact velocity was 150.7 ms-1, resulting in 

about a 0.4-GPa peak impact stress with four reverberations in the target buffer (Fig. 2, gauge 1 at gauge plane 

Pulsed Wheatstone bridge power supplies were used 

to provide 70 V of gauge excitation for 200 !s. Bridge 

400 

Gauge 1 t 

200 

0 

30 2 

20 

10 

0 d 

30 

20 

3 

10 

0 

30 

20 

r 

4 

10 h - 

0 

30 

20 5 

10 _ 

20 

10 

0 

Time (μs) 

Figure 2. Stress records from gauge 1 in the aluminum and gauges 

2-6 embedded in snow. 

3 
0). Shock propagation velocities in the snow were 242 

± 15 m s-1at gauge plane 1 (gauges 2 and 3), 230 ± 15 

m s-1at gauge plane 2 (gauges 4 and 5) and 205 ± 15 m 

s-1at gauge plane 3 (gauge 6). Gauge records from the 

first gauge plane show an initial spike at shock arrival, 

a gradual decrease to a minimum value of about 7-8 

MPa followed by a shallow rise to about 8-10 MPa. Α 

slight dip in stress level then occurs, with a spike at 135 

μs before a decrease in stress below 7 MPa. The stress 

records at the second and third gauge planes show a 

significantly longer initial rise time and smaller spikes. 

Rankine-Hugoniot theory predicts a stress level of 

about 12 MPa for a steady wave traveling at 240 m s -1 

in the snow if the snow particle velocity was equal to the 

maximum possible buffer particle velocity after impact 

constructed a simple model incorporating only the 

flyer, buffer and snow in a one-dimensional (1 D) geometry. Ultimately it was necessary to include the stress 

gauges and the failure strength of the epoxy bond 

between the two plates that made up the buffer in the 1 D 

model. Figure 3 shows the final model geometry used in 

our 1 D analysis. In the model a PMMA flyer impacted 

an aluminum buffer that had a no-friction contact with 

the snow. The aluminum buffer, in the model, consisted 

of a mylar layer sandwiched between two plates of 

aluminum tooling plate, which simulated the epoxy, 

mylar tape and stress gauge between the two buffer 

plates used in the test experiment. Mica-clad gauges 

were simulated by layers of mica at the average position 

of gauges 2 and 3 for the first gauge plane and gauges 3 

and 4 for the second gauge plane. The third gauge plane 

was placed at the same position in the finite-element 

mesh as the gauge 6 position in the actual test. Gauge 

dimensions used in the model were taken from gauges 

with the PMMA flyer plate (about 120 m s -1 ). 

The generally good agreement, within experimental 

uncertainty, between the stress records in the same 

plane for gauges 2 and 3, and gauges 4 and 5, and the fact 

that pseudo-steady stresses forgauges 2 and 3 are on the 

same order as estimated using Rankine-Hugoniot theory, 

2, 4 and 6 (Table 1). 

suggest that the measured records are valid. 

The PMMA and aluminum were modeled as an 

elastic-plastic-hydrodynamic material. This model 

combines a Von Mises yield condition including strain 

hardening to describe the deviatoric response. Volumetric response is represented by a Mie-Gruneisen 

equation of state (EOS) of the form 

The stress records from all of the gauges in the snow 

are both complex and unsteady. Two factors—target 

design and impedance mismatching between the snow 

and stress gauges—contribute to the complexity. Our 

target design was constrained by the need for a vacuumtight container that would still produce a one-dimensional 

strain shock when impacted. We achieved a flat impact 

surface by using a thick buffer of sufficient strength to 

prevent any significant deflections due to the pressure 

difference between the interior of the target and the 

target chamber. Upon impact from the flyer plate, the 

buffer imparted stresses to the snow, which gradually 

decreased as the momentum was transferred from the 

buffer to the target by multiple reverberations in the 

buffer. The rate of decrease in applied stress was controlled by the impedance mismatch between the buffer 

and the snow. A large impedance mismatch between the 

stress gauges and the snow produced large-amplitude 

reflected pulses, further complicating the signal (Brown 

et al. 1988, Gaffney 1989). Shock wave stress records 

were sufficiently long that waves generated at the edge 

of the target propagated to the stress gauges prior to the 

end of the experiment. 

p = k p η (1 + k 1 η + k2 η2 ) (1 — 

Γ0 ) + Γ0 Po Ε^ 

2 Ρ 

(1) 

where 

Ev energy per unit volume 

ρο initial density 

ρ density at pressure P 

n (1- Ρp /Ρ) 

and the remaining parameters are material-dependent 

properties and are given in Table 2. Mylar was modeled 

as a pure hydrodynamic material using the EOS parameters tabulated. Mica was treated as an elastic material. 

Although both deviatoric behavior and volumetric behavior of the materials were included in the model, 

changes in deviatoric parameters had a negligible effect 

on the calculated results. 

Modulation of the uniaxial strain conditions in our 

impact experiment by release waves, generated at the 

lateral edge of the target, was examined using an 

axisymmetric two-dimensional (2D) model (Appendix 

A). Material parameters and the constitutive model for 

MODEL SIMULATION OF THE 

EXPERIMENTAL DESIGN 

snow were the same for both the 2D and 1 D models. 

We used the PRONTO 2D transient solid dynamics 

finite-element program as a tool to understand the 

complex wave records and to construct constitutive 

models of shock propagation in snow. We initially 

In constructing our model we used the measured 

stresses as a guide to establish the validity of calculated 

results. The measured stress-time record for gauge 1 

(Fig. 2) was used to determine if the calculated stresses 

4 
Mylar 

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Figure 3. Schematic of the experimental configuration used to generate the one-dimensional finiteelement mesh for PRONTO 2D. Gauge planes 0, 1, 2 and 3 are designated by GPO, GPl , GP2 and 

GP3 in the figure. The mesh was one element wide, with the number of lateral elements for each 

component shown in the figure. 

from the model produced by the impact of the PMMA 

flyer and aluminum buffer were realistic. Figure 4 

shows a comparison of the model-calculated stresses in 

the buffer, at the location of gauge 1, and the measured 

stresses determined from gauge 1. Calculated stresses 

in the buffer were not significantly influenced by the 

snow behavior for any reasonable choice of snow Pressure vs volumetric strain (P—V) curve. The calculated 

and measured stresses agree very well for the first peak 

and fall. Thereafter there is fair agreement in the timing 

of the respective maxima and minima. The four reverberations calculated by the model are caused by reflected waves from the aluminum/snow interface and 

the PMMA/aluminum interface. The reverberations 

end when the unloading wave from the back of the 

PMMA flyer arrives at the mylar layer, causing 

Table 2. Constitutive model parameters used to develop the PRONTO 2D model calculation. 

Young's Yield Hardening Pressure 

Density modulus Poisson's strength modulus 

(kg m 3) (GPa) ratio (MPa) (MPa) 

cutoff ko 

Material Model type Beta (MPa) (GPa) k1 k2 G0 

PMMA* Elastic/plastic- 1184 6.3 0.4 375 500 0.5 —100 5.83 4.42 13.04 0.7 

hydrodynamic 

Aluminumt Elastic/plastic- 2828 72.5 0.33 270 1150 0.5 —1000 81.86 2.81 7.25 2 

hydrodynamic 

Mylar** Hydrodynamic 1390 6.73 1.196 2.29 1 

Mica i i Elastic 2844 69.6 0.26 

*PMMA EOS data (Rice 1980) were reformulated to conform to the PRONTO 2D representation of the Mie-Gruneisen EOS, and the hardening 

modulus was estimated. Beta is a parameter that determines the amount of isotropic and kinematic hardening and was estimated. 

i Aluminum EOS from Rice (1980). The hardening modulus was determined from data given by Herrmann and Lawrence (1978). Beta was 

estimated. 

**Mylar EOS from Louie et al. (1970). 

ttCalculated from isotropic elastic moduli (Vaughan and Guggenheim 1986). 

5 
500 

Calculated GPO Record 

400 Measured Gauge 1 Record 

300 

200 

100 

) 0 

— 10 

Time (μs) 

Figure 4. Comparison of the calculated stresses for gauge plane 0 with the measured 

stress record for gauge 1, located in the aluminum buffer. 

debonding. Calculations have shown that tensile stresses 

in the buffer would have exceeded 120 MPa had the 

band between the mylar and the aluminum not been 

allowed to fail. The reduction in calculated peak stress 

for each succeeding reverberation is controlled by the 

impedance mismatches at the aluminum/PMMA flyer 

interface and the aluminum/snow interface. The most 

probable explanation for the difference between measured and calculated stresses for the later reverberations 

is that gauge 1 and/or the epoxy/mylar bond was damaged after being subjected to a tensile stress of about 25 

MPa at 4.8 μs (Fig. 4). Also the carbon gauges are not 

designed to measure tensile stresses. Epoxy resins, such 

as we used in manufacturing the buffer, have tensile 

strengths that range from 14 to 90 MPa, depending on 

their composition (Baumeister and Marks 1967). After 

the first tensile pulse the gauge was still able to sense 

compressive stresses but not tensile stresses. The gauge 

failed completely upon the arrival of the unloading 

wave from the flyer (Fig. 4). This unloading wave also 

caused the two parts of the buffer plate to separate. The 

low-amplitude stresses (less than 10 MPa) in the calcuens during all stages of its compaction and that the 

form of the dynamic P—V curve is similar to that found 

from quasi-static uniaxial strain tests (Abele and Gow 

1975). That is, the slope of the stress—strain curve is 

either increasing or constant with increasing volumetric 

strain. 

We used the Soils and Crushable Foams material 

model in PRONTO 2D to describe snow compaction. 

This phenomenological model was constructed from a 

version of the model developed by Krieg (1978) and 

described by Swenson and Taylor (1983). It consists of 

a yield function describing deviatoric deformations 

uncoupled from the P—V curve used to define volumetric deformations. The yield surface σyis a surface of 

revolution about the hydrostat specified as a quadratic 

in pressure 

σ = aj + a1 P + P 2 (2) 

lated record after the arrival of the unloading wave, 

shown in Figure 4, are caused by reverberations in 

buffer disk B (in contact with the snow, Fig. 1). These 

reverberations cause the gradual decrease in stresses in 

We used ap = 10 kPa, a j = 0.1 and a2 = 0.0 in our calculations. The choice of parameters in the yield function 

does not greatly affect the results since the deformations 

are dominated by volumetric compaction. Allowing the 

yield function to depend slightly on pressure does, 

however, greatly increase the stability of the calculation. 

the snow after shock front passage. The user-defined function fρ(ε 2 ), where the v01- 

umetric strain is 6,, = — In (pp/p), is used to define the 

P—V curve (Taylor and Flanagan 1987). This function 

incorporates both a loading curve and a linear unloading—reloading curve. The slope of the unloading curve 

Measured stress—time records from gauges embedded in the snow were used to aid in our construction of 

a constitutive relationship for snow compaction. This 

was accomplished by assuming that snow strain-hard6 
is required to be greater than the loading P-V curve. The strained in this study by several factors. We assumed 

that the slope of the P-V curve either increases or remains constant with increasing volumetric strain (which 

is consistent with observations from quasi-static tests 

on snow assuming no phase changes occur). We have 

included the further constraint that the final density for 

the snow must be less than that of our recovered postexperiment samples (860 kg m-3). In addition, our 

calculations have shown that the P-V curve can be 

broken into two regions, which affect the shock wave 

differently. The magnitude of the P-V curve in the midstress range (2.5 MPa <- P <_ 16 MPa) affects the shock 

velocity and produces moderate changes in the magnitude of post-peak stresses. The magnitude of the P-V 

curve in the high-stress range (16 MPa <_ P <- 40 MPa) 

does not affect shock velocity but is important in reproducing the shock arrival stress spikes at the stress 

gauges and does affect the post-peak stress amplitudes. 

The P-V curve definition in the low-stress range could 

not be determined directly from our tests. The curves 

were estimated from the quasi-static test results of 

Abele and Gow (1975). Unloading and reloading curves 

were constrained by the shock wave arrival times and 

stress magnitudes of waves reflected from gauges in the 

snow and from the aluminum/snow interface. The unloading and reloading curves were further constrained 

by comparing the attenuation of the calculated peak 

stresses and the magnitude and frequency of post-peak 

function 

0 p= Ρ - fρ(εm ) (3) 

where is the maximum εν experienced by each 

element, defines the motion of the end cap of the 

deviatoric yield function along the hydrostat. Tensile 

failure is assumed to occur if the pressure is less than a 

user-defined fracture pressure Pfr. Tensile failure strain 

εfr is initially negative and is set to Pfr/KO, where Κρ is 

the initial bulk modulus of the snow; εfrincreases with 

εmα` and is determined by unloading from the fρ(εrx ) 

curve along a linear unloading-reloading curve to Pfr. 

The strain at Ρ is the new value of εfr. 

The results of preliminary calculations using the 

Soils and Crushable Foams model replicated many 

features of our measured data, including arrival times 

and peak stresses that are in good agreement with 

measurements (Johnson et al. 1990). However, the 

post-peak stress histories could not be reproduced because of the restrictions that the unloading-reloading 

curve be linear, have the same value as the initial bulk 

modulus irrespective of the total volumetric strain, and 

have a slope greater than the maximum slope of the PV curve. We modified the Soils and Crushable Foams 

model to accommodate more general unloading and 

reloading behavior for snow. On initial loading, the 

pressure in the snow is a function of εmx , as was the 

case in the original model. Multiple linear or nonlinear 

unloading paths may be defined, with each unloading 

path being defined over a specified range of εν. One of 

two possible reloading schemes may be used: reloading 

along the same path as unloading or reloading along a 

chord from the last P-V state to the fρ(εmx ), εm 

state (Appendix B). 

stress oscillations with those measured. 

CONSTITUTIVE RELATIONSHIP 

AND DISCUSSION 

Initial calculations used a P-V curve from a quasistatic compaction test by Abele and Gow (1975) (Table 

3). These produced arrival times that were much later 

than measured for all gauges in the snow. Succeeding 

calculations were made after increasing the model P-V 

curve slope in a systematic manner (that is, increasing 

the stress needed to cause a given volumetric strain). 

The final P-V curve and the unloading curves used to 

calculate the results presented in this paper are shown in 

Figure 5 and listed in Table 4. The calculated peak 

stresses for gauges 2, 3, 4 and 5 and the post-peak stress 

history for gauge 1 agree with measured values within 

the limits of experimental error. The calculated stresses 

for gauge 6 are larger than for the measured record but 

have the same form. Differences between the calculated 

stress histories for gauges 4, 5 and 6 and the measured 

values may result from superposition of release waves 

that originated at the edge of the aluminum, degrading 

Development of a constitutive relationship proceeded 

in two stages by first finding a P-V curve that produced 

a good agreement between calculated and measured 

shock arrival times. Next, the stress history after shock 

arrival was used to constrain the unloading and reloading curves. The P-V curve and the unloading and reloading curves found by comparing calculated and 

measured results are not unique. They can vary because 

of experimental uncertainties, the complex relationship 

of one segment of the P-V curve to another, and the 

subjective determination of what constitutes a good 

the uniaxial strain conditions. 

model simulation of the measured results, 

Comparison of the 1D calculated and measured 

stresses for gauges 2 and 3 are shown in Figure 6. This 

provides a means of identifying important processes 

affecting the shock propagation. Calculations show that 

impedance mismatch between the gauges and the snow 

The range of possible P-V curve shapes was con- produced the stress spike at shock arrival. The measured 

7 
Table 3. P-V curve parameters for Abele and Gow's measurements (1975). 

fp (ε i ) Density 2μ Κ0 Π 

(MPa) (kg nī 3) ^ (MPa) (MPa) (MPa) 

0.0 390 0.0 710* 200t —0.01 

0.07 435 0.109 

0.14 461 0.167 

0.28 570 0.379 

0.7 581 0.399 

1.4 649 0.509 

2.1 680 0.556 

2.88 719 0.612 

*The shear modulus μ was estimated from acoustic data on snow. 

j The initial bulk modulus "0 was chosen to be about the same as the bulk modulus of ice. 

stress after shock arrival decreases to that of the ambient 

snow stress. Measured stresses then increase at 95 μs and 

remain elevated until about 120 μs. Calculations indicate that the stress rise from 95 to 120 μs is due to a shock 

pulse that had been reflected from gauge plane 1 propagated back to the aluminum/snow interface and reflected 

back to gauge plane 1. The doubly reflected wave shows 

the effects of dispersion and attenuation during its 

travel. At 120 μs the measured stress begins to decrease 

because of unloading from the buffer. The measured 

stress decrease is interrupted by the arrival of a shock 

pulse that calculations indicate originated from a reflected wave at the second gauge plane (the gauge 

located at 0.0514 m in Fig. 3, gauge plane 2). This produces a sharp stress rise at 135 μs, then a stress decrease to the stress magnitude of the buffer unloading 

wave. 

Calculations using the 2D model (Appendix A) 

show that gauge plane 1 stresses are not significantly 

affected by release waves originating at the edge of the 

50 

Abele and Gow (1975), quasi-static loading 

f ρ (ε ν mom ) , shock loading 

40 o f u (ε ν ) , unloading 1 

f υ (ε ν ), unloading 2 

• f u (ε i , unloading 3 

2 

3 

30 

20 

10 

0 t 

0 0.4 0.8 

Figure 5. Loading pressure—volumetric strain curve and unloading— 

reloading curves used in PRONTO 2D to model snow compression and 

Abele and Gow's quasi-static pressure—volumetric strain results. 

8 
Table 4. Snow P-V curve parameters, moduli data and unloading curve parameters used to define snow behavior in 

PRONTO 2D model calculations. 

ρ

(ε ') fμl( εν) fμ2(ε") fμ3(εν) 

Density Pk Π1m Kim 1 2m K2m am am 

2μ Kο 1 fr 

(kg nī 3) (ΜΡα) εvk (ΜΡα) (ΜΡα) εμι (ΜΡα) (ΜΡα) (ΜΡα) (ΜΡα) (ΜΡα) (ΜΡα) (ΜΡα) 

400 0 0.0 0.681 0.718 0.764 710 4.2 —0.01 

550 0.35 0.318 

635 5.0 0.462 40 2700 

740 10.0 0.615 

790 15.0 0.681 

820 20.0 0.718 

840 25.0 0.742 

848 30.0 0.751 

859 40.0 0.764 

aluminum until about 30 μs after the peak stress arrival. 

Stresses are reduced for the following 25 μs, then increase to slightly larger magnitudes than expected for 

1 D wave propagation. This is consistent with the 

measured stresses for gauges 2 and 3, which show 

pronounced stress reductions about 20μs afterpeak stress 

arrival. Stresses increase again 20-30 μs later, j ust prior 

to the arrival of the reflected pulse from gauge plane 2. 

We cannot unambiguously interpret the measured stress 

rise for gauges 2 and 3 at about 110 μs since 1 D model 

calculations indicate that reflected waves from the 

aluminum/snow interface arrive at between 95 and 120 

μs, overlapping the arrival of the 2D release waves, 

Figure 7 compares the calculated stress history for 

gauge plane 2 with measured stresses from gauges 4 and 

5. The peak calculated stress is about the same as was 

measured. The 2D analysis indicates that release waves 

begin affecting the shock wave upon arrival at gauge 

plane 2. Differences between the post-peak calculated 

and measured stresses are attributed to release wave 

superposition on the 1 D stresses. 

Calculated stresses for gauge plane 3 have the same 

form but larger amplitude than the stresses measured by 

gauge 6 (Fig. 8). Analysis using the 2D model indicates 

that release waves propagating from the aluminum 

buffer reduce stress magnitudes at gauge plane 6 but do 

30 

Calculated GP1 Record 

Measured Gauge 2 Record 

- Measured Gauge 3 Record 

20 

10 

0 

—5 

0 

Time (μs) 

Figure 6. Comparison of the calculated stressesfor gauge plane 1 with measured stress 

records for gauges 2 and 3. 

9 
20 '^ -1 I 

Calculated GP2 Record 

Measured Gauge 4 Record 

- - - - Measured Gauge 5 Record 

10 

0 

-5 

Time 

() 

Figure 7. Comparison of the calculated stresses for gauge plane 2 with measured 

stress records for gauges 4 and 5. 

not affect the wave form significantly. This suggests 

that magnitude differences between measurements and 

calculations are caused by the 2D edge effects, 

unique and can vary because of experimental uncertainties, the interdependent influences of different regions 

of the curve, and the subjective determination of what 

constitutes a good model simulation. The range of 

possible P-V curve shapes is constrained by separate 

physical properties in two of the major stress regions. 

The mid-stress range of the curve is determined by 

shock velocity and pseudo-steady stress levels. The 

high-stress region of the curve is constrained by the 

magnitudes of stress spikes at the gauges. The lowstress region is estimated from the quasi-static results 

of Abele and Gow (1975).The shape of the P-V curve 

CONCLUSION 

The complex unsteady stress histories from a uniaxial 

strain impact on snow have been simulated using the 

PRONTO 2D transient solid dynamics finite-element 

program, 

The P-V curve estimated using PRONTO 2D is not 

10 , ι , 

8 

- - - 

Calculated GP3 Record 

Measured Gauge 6 Record ^I 

6 

4 

2 71 1 m 

0 

Time (μs) 

Figure 8. Comparison of the calculated stresses for gauge plane 3 with measured stress 

records for gauge 6. 

10 
is further constrained by the assumptions that slope 

either increases or remains constant with increasing 

volumetric strain and that the final density for the snow 

must be less than 860 kg m-3. Calculations have shown 

that slight variations in P—V curve shape do not significantly change the calculated stress histories. Abele 

and Gow's (1975) quasi-static P—V curve lies well 

below the range consistent with the shock data, implying a strong strain-rate dependence for shock wave 

deformation in snow. Post-peak stresses are strongly 

controlled by the unloading and reloading behavior of 

the snow. The unloading curve for snow is a nonlinear 

function of volumetric strain. Reloading is along a 

chord from the last P—V state to the maximum pressure 

[ fρ(ε )1 and maximum volumetric strain ( Σma) ) 

state (Fig. B1) producing a hysteretic unloading—reloading cycle that is consistent with quasi-static unloading and reloading in other porous media (Johnson and 

Green 1976, Larson and Anderson 1979). 

of material constitutive models on stress wave propagation calculations. ASME Transactions, Journal of Engineering Materials and Technology, 100: 84-95. 

Johnson, J.B., J.A. Brown and E.S. Gaffney (1990) 

Interpretation of the stress histories from shock impact 

tests on snow using embedded stress gauges. In Shock 

Compression of Condensed Matter 1989, Proceedings 

of the American Physical Society Topical Conference 

on Shock Compression of Condensed Matter, August 

14-17, Albuquerque, New Mexico (S.C. Schmidt, J.N. 

Johnson and L.W. Davison, Ed.). Elsevier Science 

Publishers, p. 117-120. 

Johnson, J.N. and S.J. Green (1976) The mechanical 

response of porous media subject to static loads. In 

The Effects of Voids on Material Deformation, 1976 

Applied Mechanics Division Meeting, Salt Lake City, 

Utah, 14 June (S.C. Cowin and M.M. Carroll, Ed.). 

American Society of Mechanical Engineers, p. 93-123. 

King, H.H. and H.S. Janēe (1987) Heating and stress 

release characterization of piezoresistive carbon gauges. 

Final Report to the Defense Nuclear Agency, Science 

LITERATURE CITED Applications International Corp., Final Report SAIC87/1656. 

Abele, G. and A. J. Gow (1975) Compressibility 

characteristics of undisturbed snow. USA Cold Regions Research and Engineering Laboratory, Research 

Krehl, P. (1978) Measurement of low shock pressures 

with piezoresistive carbon gauges. Review of Scientific 

Instruments, 49: 1477-1484. 

Report 336. Krieg, R.D. (1978) A simple constitutive description 

Bakanova, A.A., V.N. Zubarev, Yu.N. Sutulov and for soils and crushable foams. Sandia National LaboraR.F. Trunin (1975) Thermodynamic properties of water 

at high temperatures and pressures. Soy. Phys.-JETP, 

vol. 41, p. 544-548. (Originally published as Zh. Eksp. 

tories, Report SC-DR-72-0883. 

Larson, D.B. and G.D. Anderson (1979) Plane shock 

wave studies of porous geologic media. Journal of 

Teor. Fiz., vol. 68, p. 1099-1107). Geophysical Research, 84 (B9): 4592-4600. 

Baumeister, T. and L.S. Marks (Ed.) (1967) Standard 

Handbook for Mechanical Engineers. Seventh ed. New 

Larson, D.B (1984) Shock-wave studies of ice under 

uniaxial strain conditions. Journal of Glaciology, 30 

York: McGraw-Hill. (105): 235-240. 

Brown, J.A., E.S. Gaffney, G.L. Blaisdell and J.B. 

Johnson (1988) Techniques for gas gun studies of 

shock wave attenuation in snow. In Shock Waves in 

Condensed Matter 1987, Proceedings of the American 

Physical Society Topical Conference on Shock Waves 

in Condensed Matter, 20-24 July, Monterey, California (S.C. Schmidt and N.C. Holmes, Ed.). Elsevier 

Louie, N.A., W.W. Kinney and R.D. Reid, Jr.(1970) 

Dynamic properties of materials. US Air Force Weapons Laboratory, Report AFWL-TR-68-101. 

Science Publishers, p. 657-660. 

Napadensky, H. (1964) Dynamic response of snow to 

high rates of loading. USA Cold Regions Research and 

Engineering Laboratory, Research Report 119. 

Rice, M.H. (1980) PUFF74 EOS compilation. US Air 

Force Weapons Laboratory, Report AFWL-TR-80-21. 

Sato, A. and R.L. Brown (1983) An evaluation of shock 

waves in unsaturated wet snow. Annals of Glaciology, 

Fowles, R. and R.F. Williams (1970) Plane stress wave 

propagation in solids. Journal of Applied Physics, 41: 

360-363. 4: 241-245. 

Gaffney, E.S. (1989) Numerical models of shock waves 

in snow. Ktech Corporation, TR 89-03 (Final Report to 

Sato, A. (1987) Velocity of plastic waves in snow. In 

Proceedings of the Davos Symposium on Avalanche 

CRREL under contract DACA8988M0734). Formation, Movement and Effects. International AssoGourdin, W.H. and S.L. Weinland (1986) Performance of piezoresistive carbon sensors in contact with 

porous materials. Review of Scientific Instruments, 57: 

ciation of Hydrological Sciences, Publ. No. 162, p. 

119-128. 

1422-1426. 

Seaman, L.J. (1976) Lagrangian analysis for multiple 

stress or velocity gages in attenuating waves. Journal of 

Herrmann, W. and R.J. Lawrence (1978) The effect Applied Physics, 45: 4303-4314. 

11 
Swenson, D.V. and L.M. Taylor (1983) Α finite elerent model for the analysis of tailored pulse stimulation of boreholes. Journalf orNumerical andAnaIytical 

Sandia National Laboratories, Report Sand8 ό-0594. 

Vaughan, M.T. and S. Guggenheim (1986) Elasticity 

of muscovite and its relationship to crystal structure. 

Journal of Geophysical Research, 91 (Β5): 4657-4664. 

Wakahama, G. and A. Sato (1977) Propagation of a 

plastic wave in snow.Journal ofGlaciology, 19: 175-183. 

Methods in Geomechanics, 7: 469-484. 

Taylor, L.M. and D.P. Flanagan (1987) PRONTO 2D: 

A two-dimensional transient solid dynamics program, 

12 
APPENDIX A: Propagation of Stress Waves from the Lateral Edges of the Aluminum 

Buffer and Snow 

Our PMMA flyer and target assembly was designed to have a relatively large diameter 

(about 200 mm) so that release waves generated at the lateral edge of the target would not 

arrive at the stress gauge locations until after the experiment ended. Stress histories recorded 

by the gauges in our experiment were sufficiently long that distortion of the uniaxial strain 

conditions by release waves before the experiment ended was likely. We examined the 

propagation of release waves from the edges of the target assembly into the center axis of the 

snow sample by constructing an axisymmetric 2D model of our experiment. The 2D model 

was about 260 mm in diameter, which corresponded to the diameter of the buffer plate (a 

radius of about 130 mm to the axis of symmetry), and had the same configuration on axis (with 

the exception that no gauges were embedded into the snow), material properties and 

dimensions as the 1D model (Fig. Al). 

The main objective of conducting the 2D modeling was to develop a qualitative 

understanding of how release waves could distort the stresses in our shock impact experiment. In our discussion of the 2D modeling results we discussed pressure rather than the 

stresses parallel and perpendicular to the center axis of the sample since those stresses are 

about equal to each other and to the pressure. 

The pressure distribution from the edge of the target to its center axis as a function of time 

for gauge plane 0 located in the aluminum buffer is shown in Figure Α2a. Pressure 

distributions for the first snow element adjacent to the aluminum buffer and at the location 

of the gauge plane 1 in the snow are shown in Figures Α2b and Α2c. Figure Α2α shows the 

four compressive pulses (also seen in the 1D model) associated with the initial impact of the 

PMMA flyer with the aluminum buffer and subsequent reflections from the aluminum/snow 

interface and PMMA/aluminum interface. The influence of the free edge of the aluminum 

buffer can be clearly seen as the stress pulses are reduced near the free edge and release waves 

move toward the center axis with time. The diagonally oriented trough in Figure Α2α and the 

pressure ridge (A) are the result of the release wave propagating towards the center axis from 

the free surface with a propagation velocity of 4.0 ± 0.5 km s -1 . The release wave reaches the 

Mylar GPO 

Free Surface - 150.7 rn/s 

 \ \ \♦ 

\ti\ \ 

\\\\\\ 

.`\\* ♦ \\\ti` 

Ι 

Snow ::. Tooling 

g 

\\\\ Plexiglas 

: ''' "'' \\\\ 

Center \`s\\4\• itk`_^j 

Axis 30e 19e le 24e 35e 

N 0 

'- 0) 

co (D CD (D 

c c 

NN 

^ σ 

N M 

iI) u) 

00 

O

0)

O 

00 

66 66 

Figure Al. Schematic of the two-dimensional axisymmetric finite-elementmesh forPRONTO 

2D. Gauge plane 0 is designated by GPO in the figure. 

13 
350 

m ,

ε0 ń d 

Οrit Ο 

AXIS 

Free 

a. At gauge plane Ο in the aluminum. 

AXIS 

Svcace ΝΟΟ 

Free 

b. In the snow adjacent to the aluminum/snow interface. 

12.2 

,n W 

Gnt ρ 

ZOO 

Free 

c. At the position of gauge plane 1 in the snow (A2c). 

Figure A2. Pressure—time plots for three cross sections in the 2D 

axisymmetric model. Features A', A, B, C, D, E and F are discussed 

in the text. 

14 
center at approximately 46 μs. The pressure enhancement at the center axis (A) results from 

superposition of the corresponding release wave from the opposite side of the aluminum 

buffer. The several release waves seen in Figure Α2a (Α', A, B, C) can be traced back to their 

origin at the free edge of the aluminum buffer and are associated with the four primary 

pressure pulses caused by the flyer's impact with the aluminum buffer. 

Figure Α2b shows that the release waves generated in the aluminum buffer are transmitted 

across the aluminum/snow interface. The pressure profile of the release waves in the snow 

is similar to that of the release waves in the aluminum buffer (see common features A, B, C 

and D in Fig. Α2α and Α2b). Release wave pressures are greatly attenuated when transmitted 

into the snow. The release wave generated at the lateral surface of the snow at feature E (Fig. 

Α2b) is propagating inward at about 300 m s. At this velocity the release wave does not 

reach the center axis until about 450 μs, well after the experiment has ended. 

Figure Α2c shows that the gross features from Figure Α2b are still evident, such as the 

trough D; however, they are smoothed and attenuated considerably. The release wave 

generated at the snow's lateral edge F can also be seen propagating toward the center axis. 

Our analysis indicates that release waves originating in the aluminum buffer arrive at the 

stress gauge planes embedded in the snow before the experiment ends. At gauge plane 1 the 

release waves arrive at about 96 μs and modulate the stress signal by about 1 MPa (the shock 

wave arrives at gauge plane 1 at 65 μs). The release waves begin modulating the shock wave 

front before it arrives at either gauge plane 2 or 3, with a maximum modulation of stresses 

of about 1 MPa. Release waves originating from the lateral edge of the snow sample do not 

affect the stresses at the gauge plane positions. The propagation time to the center axis for 

release waves generated at the snow's lateral edge is far longer than the duration of the 

experiment. 

15 
APPENDIX B: Modified Soil and Crushable Foams Model 

Preliminary efforts to analyze our experimental measurements using the standard Soil and 

Crushable Foams model indicated that its unloading and reloading paths were too restrictive 

(Johnson et al. 1990). Geological materials exhibit nonlinearities when they are unloaded 

after shock wave passage (Johnson and Green 1976, Larson and Anderson 1979, Larson 

1984). Additionally, porous materials can have a hysteresis when unloaded and then reloaded 

(Johnson and Green 1976). We modified the Soil and Crushable Foams model to include one 

or more linear or nonlinear unloading curves. Reloading can occur back along the unloading 

curve (no hysteresis) or along chords that connect the unloading curve to the f ρ(εν `) , εναx 

state. On loading, the P—V curve is defined by f ρ(ε pa) ), as was the case for the original model. 

Pressure is a function of volumetric strain and is given by 

( fρ(εmax) 

fρ(εΤ^`) ενt= ε

ax 

εν1<εmax 

fu i (εν1) ν

u i- 1 < E< ^ ν u i i = 1, 2, 3, ... (B1) 

ε ν t < ε ν1 -1 

fri (ε ν t) Εν t < Τ ax 

ν ui-1 < Σcax <  ε

'

ui 

Εν t> Εν t-1 

where ενt is the volumetric strain at the current time step and εν1 _ 1 is the volumetric strain 

at the previous time step. The volumetric strain limits over which unloading f u i (εν t) and 

reloading fri (ε ν t) functions are valid are given by ε ν u i and ε ν u i -1 (Fig. Bi). 

For loading, the P—V curve is given by 

ff(εν) = S(ax - 

n-1 

ενn - i ;+ / S k(ενk — ενk - n = 1,2,3, ... (B2) 

k =0 

0 k= 0 

Sk = 

Pk -Pk- 1 k>- 1 

ενk — ενk - 1 

where Pk and ενk are specified pressure and volumetric strain values defining the P—V 

loading curve (Fig. Bi) and the index n is determined by the condition that 

ενn _ 1< ενt < Ενn . (B3) 

17 
(Ρ3'ε ν ) 

(Ρ2 ,ε^2) 

(P1 ,ε v 1 ) 

(P'^ 112 ν12 („) 

f (ε) 

Κ 12 f r (ε 

(Π ,ε) ^ f

u (^v) 

o 

(P1 ' Ε ) 

(P0' εν1ο ) 

ν„ 

K11 

0 

fr εν u, 

P ® — — — — — - 

ε 

Figure B1. Pressure vs volumetric strain in terms of user-defined curves 

for pressure at the maximum volumetric strain [fρ(εf ax )], pressure 

unloading [fu i ( εν t)] and pressure upon reloading [f,.1 (ε v t)] for the 

modified Soils and Crushable Foams material model. 

upon 

Tensile failure strain εfr is initially set to Pfr/KO, where Pfr is the tensile failure pressure in 

the snow and Kp is the snow's initial bulk modulus. Failure strain is recalculated for each 

increase in by unloading from fρ (e ) using fui (ε v 1)• 

Unloading P-V curves are defined by 

- 1 i = 1, 2, 3, ... (Β4) .fui(Ενi)-Kii (^ νt — Ε ii-1 )± Σ Κi m (^^ im) j

= 1, 2,3, . 

m =0 

where 

j- 1 

ε i.l-1 ε fr— fr + Σ ∆ε im 

Kil m=0 (Β5) 

is the strain at the beginning of the curve segment containing ε',1and 

0 m =0 

∆εi (Β6) 

m — Pi m-1 

m >- 0 

Kim 

18 
is the incremental strain across each curve segment in the unloading curve. The specified 

pressure Pim and slope Kim define the P—V unloading curve, where Kim = Ο when m = Ο. 

The index i specifies the unloading curve of interest and is determined by the condition 

that 

Εν ui— ι 

< Εmax< 

ν _ ε ν u i (B7 ) 

The index j specifies the curve segment over which a given modulus is defined (Fig. Bi), 

where ] is determined by the condition that 

εij_1 < ενt<_ Εii (B8) 

Reloading can be in the reverse direction along the unloading curve, for which eq B3 applies 

and there is no hysteresis. Alternatively, reloading can be along chords connecting the 

unloading curve to the loading curve, producing a hysteretic unloading—reloading cycle. The 

slope of the reloading chord is required to be greater than or equal to the maximum slope of 

fρ(ε) eween ε ν

ui- 1 < ε Χ5 εν u . . Our analysis used reloading along chords where 

the reloading curve is given by 

fr i(εν t) =fr i( ενt-1) + S ∆ε ν t 

S = fr.(εú ) — Ir(£νt — 1) (B9) 

εmax ν — ενt — 1 

∆ενt= ενt — ενt —1 

Figure 5 illustrates the loading, unloading and reloading curves used in this study, and Table 

4 gives the values for Pk, εVk, Pim and Kim. 

19 

*U.S. GOVERNMENT PRINTING OFFICE: 1992-600-063/60012 
REPORT DOCUMENTATION PAGE Β 0 704-0188 

1. AGENCY USE ONLY (Leave blank) 

July 1992 

2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 

Shock Response of Snow: 

Analysis of Experimental Methods and Constitutive Model Development 

6. AUTHORS 

Jerome B. Johnson, Jay A. Brown, Edward S. Gaffney, 

George L. Blaisdell and Daniel J. Solie 

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS 

PR: 4Α762784ΑΤ42 

WU: CS/012 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

U.S. Army Cold Regions Research and Engineering Laboratory 

72 Lyme Road 

Hanover, N.H. 03755-1290 

8. PERFORMING ORGANIZATION 

REPORT NUMBER 

CRREL Report 92-12 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING 

AGENCY REPORT NUMBER 

Office of the Chief of Engineers 

Washington, D.C. 20314-1000 

11. SUPPLEMENTARY NOTES 

12a. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited. 

Available from NTIS, Springfield, Virginia 22161. 

12b. DISTRIBUTION CODE 

13. ABSTRACT (Maximum 200 words) 

A shock impact test was conducted on snow with an initial density of 400 kg m-3using a large-diameter gas gun and 

Lagrangian stress gauges between layers of snow. The shock propagation velocity ranged from 24010207 m s -1 , the peak stresses 

in the snow were between 20 and 40 MPa, and the compacted snow density was less than 860 kg m -3 . Interpretation of the stress 

records was complicated by the unsteady nature of the shock, impedance mismatching between gauges and snow, multiplyreflected pulses, and release waves generated at the edge of the target. A dynamic finite-element analysis was used to interpret 

the data, to construct a constitutive relationship for the snow, and to examine the importance of the release waves. Model 

calculations indicate two release wave sources: the free edge of the target aluminum buffer and the edge of the snow in contact 

with the copper container. The aluminum buffer release waves contain both shear and dilatational components. Transmission 

across the aluminum/snow interface significantly attenuated dilatational waves and essentially eliminated the shear waves. The 

snow/copper release wave did not arrive at the stress gauge position until after the end of the experiment. 

With the aid of model calculations, the pressure volumetric-strain (P-V) curve for initial shock loading was determined 

from arrival time information and stress measurements at the embedded gauges. Stress signals caused by reflected waves were 

used to determine the reloading and unloading P-V curve. The P-V response for shock loading was found to be much stiffer 

than that for quasi-static loading. The unloading P-V curves used in model calculations were nonlinear functions of volumetric 

strain with linear reloading. 

14. SUBJECT TERMS 

Shock impact tests 

Snow 

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