JOURNAL OF APPLIED PHYSICS 107, 043520 共2010兲
Dynamic behavior of tungsten carbide and alumina filled epoxy composites T. J. Vogler,1,a兲 C. S. Alexander,2 J. L. Wise,2 and S. T. Montgomery2 1
Sandia National Laboratories, Livermore, California 94550, USA Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
2
共Received 20 July 2009; accepted 24 December 2009; published online 24 February 2010兲 The dynamic behavior of a tungsten carbide filled epoxy composite is studied under planar loading conditions. Planar impact experiments were conducted to determine the shock and wave propagation characteristics of the material. Its stress-strain response is very close to a similar alumina filled epoxy studied previously, suggesting that the response of the composite is dominated by the compliant matrix material. Wave propagation characteristics are also similar for the two materials. Magnetically driven ramp loading experiments were conducted to obtain a continuous loading response which is similar to that obtained under shock loading. Spatially resolved interferometry was fielded on one experiment to provide a quantitative measure of the variability inherent in the response of this heterogeneous material. Complementing the experiments, a two-dimensional mesoscale model in which the individual constituents of the composite are resolved was used to simulate its behavior. Agreement of the predicted shock and release wave velocities with experiments is excellent, and the model is qualitatively correct on most other aspects of behavior. © 2010 American Institute of Physics. 关doi:10.1063/1.3295904兴 I. INTRODUCTION
The shock wave propagation characteristics of the encapsulant material alumina filled epoxy 共ALOX兲 have been studied by several researchers over the past 3 decades.1–9 Besides the overall shock response, these investigations have focused on experimental aspects such as particle morphology and volume fraction,9 pressure-shear loading,3 wave form and release velocity,7 and deviatoric response.8 Some effort has also been made to model the behavior of these materials using continuum techniques.2,4 Some of the key findings of these studies are the propagation of structured dispersive waves in these materials and surprisingly high release wave speeds compared to their shock velocities. In this study, we investigate an alternate formulation in which granular tungsten carbide 共WC兲 is used in place of alumina. These two materials are both much stiffer than epoxy, but the density is much higher for WC, 15.7 versus 3.97 g / cm3 for Al2O3. The use of the different particles in the composite allows us to investigate the effect of constituent properties on the behavior of these materials: their shock loading behavior, their unloading behavior, and the nature of large amplitude waves traveling in them. Previously, the static and dynamic behavior of WC powder has been studied experimentally10 and through mesoscale simulations.11–13 The tungsten carbide epoxy composite studied in this investigation differs by less than 10% in density from the granular WC studied previously. However, the composite is a solid material held together by a polymer matrix, while the granular WC behaves as a cohesionless material under ambient conditions. Comparison of those two cases allows us to better understand the behaviors of these two classes of materials. Through plate impact experiments using gas guns, we a兲
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measure the shock response and release velocity of tungsten carbide filled epoxy 共WCE兲 and characterize the wave structure using time-resolved diagnostics. We also utilize magnetically driven ramp loading14 to characterize the material response and spatially resolved velocity interferometry to measure local variability in the material response. Behavior during these experiments is simulated using a twodimensional 共2D兲 mesoscale model similar to that of Borg and Vogler.11 The experimental results for WCE are compared to the previous results for ALOX to gain an understanding of the role of the filler ceramic, and the mesoscale modeling approach is applied to ALOX as well to test its applicability. II. EXPERIMENTAL TECHNIQUES
In this section, we describe the WCE material studied and the experimental configurations used for the plate impact and ramp loading experiments. A. Materials
The WCE studied was fabricated using the same WC powder from Kennametal, Inc., studied by Vogler et al.10 Individual particles consist of single WC crystals produced through a melt process.15 The particles were sieved to give nominal particle sizes in the range of 20– 32 m. Epon 828 resin and Epi-Cure Z curing agent mixed 5:1 by weight were used. A nominal volume fraction of WC particles of 45.3% was used. Based on a density of 1.20 g / cm3 for the epoxy and 15.7 g / cm3 for WC, one expects a density of approximately 7.77 g / cm3. The mixture was vacuum outgassed before and after pouring into molds. The molds were heated to 93 ° C over 10 h, then held at that temperature for 10 h before cooling to room temperature over an hour. After curing, approximately 1 mm was machined from the top and bottom of each WCE disk to remove resin-rich and resin-
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B. Plate impact experiments
FIG. 1. Scanning electron microscopy image of cross-section of a WCE sample showing WC particles 共light兲 surrounded by the epoxy matrix 共dark兲.
poor regions. The material produced consists primarily of isolated WC particles surrounded by epoxy with few, if any, voids present, as shown in Fig. 1. This is qualitatively similar to the ALOX studied by Setchell and Anderson.7 Though efforts were made during the mixing process to ensure uniform distribution of the particles, their distribution was not characterized in detail. Samples were machined to be flat and parallel to approximately 13 m. Contact surface profilometry indicates an average roughness value Ra of approximately 1 m with isolated low points on the order of 10 m deep that are suggestive of WC particles that have pulled out of the surface. After fabrication, each sample disk was characterized for nominal density based on volume and mass, and the longitudinal and shear wave speeds of the material were measured at the center and four points between the center and edge distributed around the disk. The average density was 7.79⫾ 0.19 g / cm3, while the average longitudinal and shear wave speeds were 2.04⫾ 0.07 and 1.12⫾ 0.05 km/ s, respectively 共second number is one standard deviation兲. The WCE samples had significantly greater scatter in their densities and wave speeds than the ALOX studied by Setchell and Anderson.7 Since very similar fabrication processes were used for both materials, the scatter is probably due to the much greater density of the WC particles which caused them to settle more than the Al2O3 particles. Also, the sample density will be more sensitive to variations in particle distribution due to the greater density of WC. WCE impactor WCE target
WCE impactor
window
Plate impact experiments were conducted using smooth bore guns of different types for different velocities: gas guns using compressed helium for velocities to about 0.5 km/s, and a propellant gun for velocities to about 1.4 km/s. The main experimental configuration, referred to as the transmitted wave 共TW兲 configuration, is shown in Fig. 2共a兲. In it, a WCE target, backed by a thin 共⬃0.5 mm兲 aluminum or fused silica buffer and a window 共LiF or fused silica兲 was impacted by a second piece of WCE or aluminum mounted on the end of a projectile. The velocity of the buffer/window interface was monitored with a laser doppler interferometer known as VISAR.16 Two small windows at the edge of the sample were impacted simultaneously with the WCE target. By monitoring the impact surface of these edge windows with the same interferometry system, the impact time could be accurately determined on the same interferometry system, thus avoiding timing issues across different instrumentation systems. Projectile velocities and tilt were measured using shorting pins. Samples used for gun experiments were typically 51 mm in diameter and 6 mm thick, while the impactors used were 64 mm in diameter and 3 mm thick. These lateral dimensions are sufficiently large so that deformation in the central part of the sample remains one-dimensional 共uniaxial strain兲 for the time of interest of the experiment. The windows used were sufficiently thick that waves in the experiment do not reach their free surface for the time of interest as that would affect the velocity history measurement. The second configuration is referred to as reverse impact 共RI兲 and consists of a WCE sample impacted directly into a buffered window, as shown in Fig. 2共b兲. Note that the TW and RI configurations were used simultaneously on certain experiments. Details for the TW and RI experiments are given in Table I. A third configuration shown in Fig. 2共c兲 was used to study the evolution of waves 共EWs兲 in WCE. In this configuration, three WCE samples with different thicknesses were impacted simultaneously with a thick aluminum plate. Again, the WCE samples were backed by a thin buffer and a transparent window. Details for the EW experiments are also given in Table I. impactor
window
Al buer
to VISAR
to VISAR
to VISAR
to VISAR
low impedance backer
(a)
buer
low impedance backer
(b)
window
buer
WCE target
(c)
FIG. 2. Configurations used in plate impact experiments for 共a兲 the TW configuration, 共b兲 the RI configuration, and 共c兲 the EWs configuration.
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TABLE I. Configurations for TW, RI, and EWs gas gun experiments.
Expt. WCE-12
Expt. type Impactor Buffer Window
o Impact velocity Impactor thickness Sample thickness Sample diameter Buffer thickness 共g / cm3兲 共km/s兲 共mm兲 共mm兲 共mm兲 共mm兲
WCE-3 WCE-10 WCE-4 WCE-11 WCE-5 WCE-18
TW RI TW TW TW RI TW RI TW RI TW EW
WCE WCE Al Al WCE WCE WCE WCE WCE WCE WCE Al
Al Al FSb FS Al Al Al Al Al Al Al FS
LiF LiF FS FS LiF LiF LiF LiF LiF LiF LiF FS
3.07 3.07 7.08 10.00 3.06 3.06 3.07 6.06 3.06 6.05 3.06 12.66
WCE-19
EW
Al
FS
FS
12.63
WCE-7
EW
Al
Al
LiF
9.93
WCE-2 WCE-1 WCE-13
3.06 N/A 6.06 6.07 3.06 N/A 6.06 N/A 6.06 N/A 6.07 3.01 6.00 9.00 3.01 4.51 6.00 2.05 3.04 4.04
25.41 63.50a 50.83 50.80 25.41 63.52a 50.82 50.84a 50.84 50.82a 50.80 19.05 25.42 38.10 19.05 21.91 25.34 25.43 25.39 25.38
0.47 0.48 0.57 0.56 0.48 0.47 0.51 0.51 0.51 0.48 0.51 0.52 0.51 0.50 0.51 0.51 0.51 0.48 0.47 0.48
8.19 7.70 7.65 7.70 8.07 7.62 7.68 7.91 7.65 8.01 7.86 7.72 7.71 7.72 7.74 7.70 7.73 7.67 7.83 7.70
0.251 0.251 0.342 0.353 0.445 0.445 0.774 1.117 1.140 1.447 1.432 0.205
0.329
0.449
a
Diameter of impactor for RI experiments. FS—fused silica.
b
The three different experimental configurations were used in order to examine different aspects of material behavior while utilizing the available material in the best manner possible. All three have been utilized in previous studies of ALOX.2,7 The TW experiment is the classic shock physics experimental configuration for a gas gun, providing accurate shock and release wave velocity. The RI impact configuration provides the most accurate measurement of release velocity but somewhat less accurate shock results. Finally, the EW configuration provides information on the evolving wave structure as it travels through the sample material, whereas the TW and RI configuration only sample the wave at a single location. It is difficult to measure the release wave speed in the EW configuration, however, and no attempt to do that was made in this study.
sapphire ones were 8 mm thick. As with the plate impact experiments, VISAR was used to monitor the velocity of the reflective surface. Uncertainty in the timing between the various interferometer measurements is estimated to be on the order of 0.5 ns. The panels used in these experiments were wider and the charge voltage lower 共55 kV兲 than in previous studies on Veloce14,17,18 in order to produce lower stress levels in the samples. During an experiment, energy released from capacitor banks produced a 2.3 MA electrical pulse with a rise time of about 450 ns. Current flow on the panels induces a magnetic field between them, and the interaction between this field and the current produces a smooth mechanical stress wave through Lorentz forces that are proportional in magnitude to
C. Ramp loading experiments
Ramp loading experiments were conducted using the Veloce pulsed-power machine.14 The experimental configuration is shown in Fig. 3 and consists of two aluminum panels separated by a 0.34 mm thick layer of Kapton 共not shown in figure兲. Panels 30 mm wide and made of 1100 aluminum were machined to near their final thickness and then finished using single-point diamond turning. The 17 mm diameter WCE samples were glued onto the panels using an ultralow viscosity, quick setting epoxy under light applied pressure. The panels in the region where the WCE was glued were approximately 1.5 mm thick. Uncertainties in the initial thicknesses were approximately 3 m. The 17 mm diameter LiF and c-axis sapphire windows glued to samples had a thin layer of aluminum vapor deposited on the side toward the aluminum panel, but no buffer was used. The LiF windows in the ramp loading experiments were 6 mm thick, while the
aluminum panels
electrical contact
samples to VISAR LiF window current
magnetic field
FIG. 3. 共Color online兲 Schematic of experimental configuration for magnetic ramp loading on Veloce. Panels are shown separated to allow current and magnetic field to be illustrated.
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Vogler et al. TABLE II. Material models and constants for materials in mesoscale simulations.
Material WC Al2O3 Epoxy Aluminum LiF Fused Silica
EOS
o 共g / cm3兲
co 共km/s兲
s
Y 共GPa兲
Mie–Grüneisen Mie–Grüneisen Tabular Mie–Grüneisen Mie–Grüneisen Tabular
15.56 3.989 1.185 2.703 2.638 2.204
5.26 8.58 N/A 5.29 5.15 N/A
1.15 1.00 N/A 1.38 1.35 N/A
8.0 16.5 0.1 0.26a 0 8.0
0.20 0.20 0.35 0.28 N/A 0.118
a
Aluminum is modeled with the Johnson–Cook strength model.
the current squared. The loading condition is such that the two samples on opposite sides experience identical loading histories. Ramp loading as performed here on WCE provides some advantages over impact loading and allows different aspects of behavior to be studied. In particular, the ramp loading experiments provide a continuous loading response for the material, in contrast to the gas gun experiments which provide only a single shock state per experiment.19 Also, ramp loading has been shown to be better suited than shock loading for the study of relatively subtle effects such as collapse of voids at relatively low volume fractions.20 Finally, under ramp loading, waves generally tend to steepen as they propagate due to nonlinearity in the bulk response,21 whereas impulsive loading of composites, such as WCE and ALOX, tends to lead to waves that spread out as they propagate,7 at least for low loading stresses. III. MODELING TECHNIQUES
The dynamic behavior of WCE and ALOX was simulated using mesoscale models based on those of Borg and Vogler,11 which are similar to earlier simulations by Benson 共c.f. Ref. 22兲. The models are idealized 2D microstructures consisting of circles of WC randomly distributed within an epoxy matrix. A similar modeling approach has been used23,24 for the Al+ Fe2O3/epoxy system. Although the WC particles are not particularly well represented by circles, most mesoscale simulations of this type have utilized them.11,22,24 Further, studies of particle morphology effects in granular materials have shown the results to be insensitive to particle morphology.13 Finally, it has been demonstrated experimentally9,25 that particle morphology does not affect the shock response or release velocity of ALOX, though it can affect the rise time of the waves somewhat. The initial microstructures were generated in a manner somewhat different from that used previously,11 but the results are expected to be very similar. This technique was developed to remove any directional bias due to the artificial “filling” process used previously as well as to allow generation of particles for nonrectangular shapes. First, we distributed a number of particles uniformly over the domain that gives the correct volume fraction of WC. In order to achieve a more random spatial distribution of the particles,13 each particle was then assigned a velocity in a random direction and allowed to move for a time sufficient for a particle to travel five mean free paths. Thus, each particle undergoes, on
average, five collisions with other particles. Particles interacted with one another and with the left and right boundaries in an elastic frictionless manner. Periodicity conditions were used to link the upper and lower boundaries and simulate an infinitely tall sample for both the generation step and the actual simulation. Typically, the simulation domain was 1 mm tall, which was found to be sufficient to give a result not dependent on that dimension. For these simulations, the impactor was given a prescribed initial velocity. Appropriate buffers were used in simulations intended for comparison with experimental velocity histories; otherwise they were omitted for simplicity. The volume fraction for the WC particles was taken to be the nominal value of 45%, while simulations of ALOX used a volume fraction of 43%. Both materials were assumed to have no porosity. Simulations were performed using a parallel version of the finite volume shock physics code CTH.26 As has been done previously, relatively simple constitutive models were utilized for the materials in the mesoscale model. Mie– Grüneisen equations of state 共EOS兲 were utilized for all materials except epoxy, for which a tabular EOS was used. WC, Al2O3, and epoxy were all assumed to have elastic-perfectly plastic deviatoric responses, while the aluminum buffer was modeled with the Johnson–Cook strength model. The LiF window was assumed to behave hydrodynamically 共i.e., no strength兲. The value of strength used for WC was slightly higher than that used previously11 but represents tuning of this parameter to match the shock response of WC powder.12,13 The strength of Al2O3 was obtained from the HEL measured for c-axis sapphire of 22 GPa 共Ref. 27兲 assuming an isotropic material and a Poisson’s ratio of 0.2. Since the other orientations of sapphire are generally weaker than the c-axis, this will tend to overestimate the strength of randomly oriented bulk sapphire but may be reasonable for small individual particles. The strength of epoxy is set, somewhat arbitrarily, at 0.1 GPa. Previous experiments with lateral gauges have shown strength of this order,28 but the behavior of epoxy is undoubtedly much more complicated than elastic-perfectly plastic. Here, though, we neglect pressure and rate dependence while acknowledging that they may be important in some situations. In some simulations of experiments, sapphire or fused silica windows or buffers were used. The parameters described previously for Al2O3 were used for the sapphire windows, while fused silica was used within its elastic regime with a tabular EOS describing its bulk behavior. Material constants for the six materials used in the simulations are given in Table II. The domain of
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the simulation was discretized into 2.5 m square cells such that each WC particle had approximately 13 cells across its diameter. This has been shown to be sufficiently refined for granular WC 共Ref. 13兲 and should be sufficient in the present simulations. Two main pieces of information were extracted directly from the mesoscale simulations of shock experiments: a shock velocity and a release velocity. Because it allowed for easier and more accurate determination of the release velocity, most simulations were done in a RI configuration similar to that shown in Fig. 2共b兲. The shock wave propagation velocity, Us, was determined by monitoring the material velocity at points in the WCE impactor. Sets of ten Lagrangian tracer points distributed at 0.1 mm intervals vertically and 0.5 mm intervals horizontally in the WCE impactor were used to determine the wave velocity. Velocity histories of each set of ten points were averaged, and the midpoint of the wave was taken as the arrival time for that Lagrangian X position. A least-squares fit of Lagrangian position and arrival time provides the shock velocity. The second piece of information obtained is the release velocity, CR. This can be determined based on the time at which the particle velocity at the impact plane begins to drop minus the time required for the shock wave to traverse the sample. In practice for both the experiments and simulations, it can be somewhat difficult to determine the arrival of the release wave precisely. Thus, CR has significantly greater uncertainty associated with it than Us. The ramp loading experiments performed on Veloce were also simulated using the mesoscale model. However, the boundary conditions are more involved than those used in plate impact. To capture the effect of the current flow and Lorentz forces that occur in the experiments, a thin ideal gas layer is placed at the surface of the Al panel away from the sample. By depositing energy into this layer, the stress waves generated by the current flow can be mimicked,29 and an analytical relationship between the energy deposited and the stress loading has been derived. Although this only crudely captures such effects near the region of current flow, it can accurately replicate the stress waves that arrive at the panelsample interface. Other details of the simulations of the ramp loading experiments are the same as for the plate impact experiments.
IV. EXPERIMENTAL RESULTS A. Plate impact experiments
Plate impact experiments were conducted as described in Sec. II B. Velocity histories for the three window interfaces in a powder gun experiment are shown in Fig. 4. Using the arrival times of the waves with appropriate small corrections for the thin aluminum buffers, an average shock velocity, Us, through the sample can be determined. The particle 共mass兲 velocity, u p, in the shocked state was taken as half of the impact velocity for symmetric impact configurations; impedance matching techniques were used for experiments with aluminum impactors. Assuming that the wave propagation is
1 WCE-5
arrival of release wave
up (km/s)
edge windows
0.5 sample window
0
0
1
2
3
4
t (µ µs)
FIG. 4. 共Color online兲 Interface velocity histories for experiment WCE-5 showing shock arrival in the edge windows and the central sample window as well as release in the central sample.
steady, the longitudinal stress for the shock 共Hugoniot兲 state was determined from the Rankine–Hugoniot equation for conservation of momentum
= oU su p ,
共1兲
with stress positive in compression and where o is the initial average density of the WCE. Similarly, the density was calculated using the conservation of mass relationship as
= o
Us . Us − u p
共2兲
Values for the shock state for the plate impact experiments are given in Table III. The arrival of the release wave from the back of the impactor is characterized by a decrease in the velocity measured at the window interface. This arrival is clear in Fig. 4, but in many other experiments it is less distinct. The transit time for the release wave to travel through the impactor and the sample is found by subtracting the time for the shock to reach the back of the impactor from the time of arrival of the release wave; the Lagrangian release wave speed, CRL, is then the initial thicknesses for the impactor and target divided by this time. Note that this assumes that the release wave travels at a constant velocity and is unaffected by wave reflections from, for example, the sample-window interface. The wave speed found in this manner is Lagrangian 共based on the undeformed dimensions兲; the Eulerian value CRE can be determined from the ratio of initial to compressed densities using CRE =
o L C . R
共3兲
Release wave speeds found in this manner are given in Table III. Release wave speeds were not determined in two TW experiments because the thicknesses of the impactor and sample were the same, making the assumption of a uniform release wave speed unreasonable. In one other experiment, the velocity history was too noisy to obtain an accurate release velocity. For experiments at impact velocities above 0.45 km/s, the velocity histories measured contain an abrupt shock, but for velocities below that level dispersive waves are observed,
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Expt.
Expt. type
Impact velocity 共km/s兲
up 共km/s兲
Us 共km/s兲
共GPa兲
共g / cm3兲
CER 共km/s兲
TW RI TW TW TW RI TW RI TW RI TW EW EW EW
0.251 0.251 0.342 0.353 0.445 0.445 0.774 1.117 1.140 1.447 1.432 0.205 0.329 0.449
0.126 0.128 0.168 0.173 0.223 0.218 0.387 0.482 0.570 0.607 0.716 0.106 0.166 0.219
2.01 1.83 1.99 2.01 2.16 2.06 2.45 2.73 2.86 2.98 3.18 1.85 1.97 2.12
2.06 1.80 2.71 2.80 3.89 3.42 7.36 10.09 12.53 13.96 17.90 1.51 2.52 3.59
8.74 8.28 8.35 8.43 9.00 8.52 9.12 9.60 9.55 10.06 10.14 8.19 8.43 8.63
N/Aa 3.31 3.92 3.70 N/Aa 4.08 4.88 5.23 N/Aa 5.27 5.50 N/A N/A N/A
WCE-12 WCE-2 WCE-1 WCE-13 WCE-3 WCE-10 WCE-4 WCE-11 WCE-5 WCE-18 WCE-19 WCE-7 a
Not measured.
as seen previously for ALOX2,7 and granular WC.11 Two examples are shown in Fig. 5. No elastic precursor is seen, but as the impact velocity increases the rise time of the wave decreases. In fact, if one compares the strain rate for these experiments with the Hugoniot stress, assuming the wave structures are steady, one finds that the strain rate is approximately proportional to stress to the fourth power. This is similar to the relationship observed for ALOX at 43% volume percent7 and for a variety of homogeneous metals and ceramics by Swegle and Grady.30 However, the relationship differs significantly from that observed for granular tungsten carbide and other granular materials,10,31 namely, that strain rate is approximately proportional to stress. Thus, the epoxy matrix appears to play a key role in determining the wave characteristics despite its much lower density than WC. Further, it suggests that significant voids between particles are necessary to obtain the linear relationship observed previously for granular ceramics. Following the main wave, the velocity shows a very gradual rise to a steady value. This gradual rise takes approximately 0.6 and 0.4 s for WCE-12 and WCE-13, respectively. A similar gradual increase in ve-
locity was also observed for ALOX.7,9 In general, the time to reach the equilibrium velocity decreases with increasing stress level. This gradual equilibration process may be due to reverberations of small amplitude waves within the microstructure of the shocked material. Velocity histories for two RI experiments 关see Fig. 2共b兲兴 are shown in Fig. 6. Following impact, a shock wave traverses the thin aluminum buffer and arrives at the monitored Al/LiF boundary. Once the wave arrives at this position, it quickly reaches a nearly constant velocity characteristic of the Hugoniot state. From the known behavior of the LiF window and the Al buffer and exploiting continuity of velocity and stress across the impact surface, the particle velocity u p and stress in the shock state are found. Equations 共1兲 and 共2兲 can then be used to find the shock velocity and density. Parameters for the shock state in these experiments are given in Table III. Because of the manner in which the shock parameters are determined for the RI experiments, the uncertainties in those values will be somewhat greater than for the TW configuration. The nearly constant interface velocity is maintained for 1 – 1.5 s, after which the records
up 0.2 (km/s)
arrival of release wave
WCE-13
Experiments Simulations
0.2
up (km/s) WCE-13
WCE-12
WCE-12
0.1
0.1
Experiments Mesoscale Simulations
0 1.25
1.5
1.75
t (µ µs)
2
FIG. 5. 共Color online兲 Interface velocity histories from experiments and mesoscale simulations illustrating steepening of the profiles with increasing impact velocity.
0
0
2
t (µ µs)
4
FIG. 6. 共Color online兲 Interface velocity histories from RI experiments and 11 realizations of the mesoscale model showing the wave form during unloading.
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2, 3, and 4 mm
up 0.2 (km/s)
0 0 1.2
3 mm 6 mm 9 mm
0.205 km/s
0.1
6 mm
0.329 km/s
0.449 km/s
3 mm 4.5 mm
0.4 1.6
0.8 2
1.2 2.4
1.6 2.8
t - arbitrary (µ µs)
FIG. 7. 共Color online兲 Interface velocity profiles for wave evolution experiment WCE-18, WCE-19, and WCE-7. Each experiment includes three sample thicknesses; the profiles have been shifted in time so all for a given experiment nearly overlay one another.
become more irregular. The cause of this irregularity is not known. When the wave in the WCE impactor reaches its back face, it reflects as a release wave that arrives at the interface at about 2 – 2.5 s. The arrival of this release wave at the window interface is then used to determine the unloading wave speed. Experiments conducted in the EW configuration shown in Fig. 2共c兲 allowed the evolution of the wave form to be observed at constant stress levels. Velocity histories for three such experiments are shown in Fig. 7 translated in time so that all histories for a given experiment nearly overlay one another. At the lowest impact velocity of 0.205 km/s 共WCE18兲, which corresponds to a stress of approximately 1.5 GPa, the wave profiles changed shape significantly as the propagation distance increased from 3 to 9 mm, with the rise time of the wave increasing with distance. The stress level of this experiment is comparable to that shown for ALOX in Fig. 8 of Setchell and Anderson,7 though the spreading of the waves is somewhat greater for WCE than for ALOX. As will be discussed in Sec. V, the attenuation does not appear to be due to an unloading wave from the back of the impactor; the real cause of the attenuation is not known. It is possible that the velocity probes for some of these samples were not po8
Us, CR (km/s)
CR - ALOX (Setchell)
6
CR - WC/Epoxy
sitioned in the center of the samples, which would lead to arrival of edge release waves earlier than anticipated. Despite these potential problems, we report these experiments to illustrate the manner in which waves of different stress levels evolve as they travel through the material. The attenuation probably affects the rise time of the waves somewhat but probably does not affect the overall trend that the rise time is increasing with sample thickness. At the middle velocity of 0.329 km/s 共WCE-19兲, which corresponds to a stress level of about 2.5 GPa, the wave becomes only slightly more spread out as propagation distance increases. Again, some attenuation in the wave amplitude is observed, though only for the thickest 共6 mm兲 sample thickness. The markedly different shape for this wave near the maximum amplitude suggests an experimental anomaly of some type. Finally, at the highest velocity of 0.449 km/s, which corresponds to a stress level of about 3.6 GPa, the wave form is essentially the same for the three samples of 2, 3, and 4 mm. In fact, the three records are remarkably close to one another considering the variation from sample to sample that might be expected for WCE. Parameters for the shock state calculated assuming the propagating waves are steady are given in Table III. Based on these experiments, the critical distance for a wave in WCE to become steady is less than 2 mm at stresses of 3.6 GPa and greater than 9 mm at 1.5 GPa. At 2.5 GPa, the critical distance appears to be on the order of 9 mm. Previous studies of wave form evolution30,32 have found a similar trend for metals such as aluminum. Based on these critical distances, the assumption of steady wave propagation underlying Eqs. 共1兲 and 共2兲 is valid for samples 3–6 mm thick at stresses above 3.6 GPa, reasonable at stresses of 2–3 GPa, and questionable for stresses below 2 GPa. The shock velocity, Us, and Eulerian release wave speed, CR in WCE are plotted as a function of particle velocity in Fig. 8. Uncertainties in the shock velocities for WCE are approximately the size of the symbols in the plot, but the uncertainties for the release wave speed are larger. Estimates for the uncertainties in these quantities are shown in the figure for selected data points. The uncertainty in CR arises from the uncertainties in Us, uncertainties in the compressed density in Eq. 共3兲, and, most importantly, from the difficulty in determining when the release wave arrives if the velocity record has noise or irregularities such as those in Fig. 6. Also shown for comparison are the results for ALOX of Setchell and Anderson.7 The shock velocities of both materials are represented well by a linear fit of the form Us = co + su p ,
4
where co and s are fitting constants given in Table IV. The release velocities are fit better by a quadratic function of the form
US - ALOX (Setchell)
2
US - WC/Epoxy
CR = CRo + a1u p + a2u2p ,
ultrasonic values for CB and CL
0
0
0.5
共4兲
1
up (km/s)
FIG. 8. 共Color online兲 Shock wave velocity and Eulerian release wave speed for WCE and ALOX as a function of particle velocity. Also shown are the ultrasonic values for longitudinal 共CL兲 and bulk 共CB兲 wave speeds.
共5兲
where CRo, a1, and a2 are fitting constants given in Table IV. Shock velocities are lower for WCE than for ALOX at a given particle velocity, as are the release wave speeds. However, the ratios of CR to Us are comparable for the two materials. Also shown on the plot are values for longitudinal and bulk sound speeds, CL and CB, from ultrasonic measure-
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TABLE IV. Coefficients for linear and quadratic fits to shock and release velocity data, respectively, for WCE and ALOX.
Material WCE ALOX
co 共km/s兲 1.65 2.88
s
CoR 共km/s兲
a1
a2 共s/km兲
2.15 2.00
2.40 3.26
9.04 9.18
⫺6.72 ⫺4.63
ments for ALOX 共Ref. 7兲 and WCE. In general, CL ⬇ CRo and CB ⬇ co, though agreement is not as good for the latter relationship for ALOX. Finally, the complete set of parameters for the shock state as calculated using Eqs. 共1兲 and 共2兲 are given in Table III. Because of the uniaxial strain condition, the densities are related to the engineering strains through
⑀=1−
o ,
共6兲
where the strain ⑀ is positive in compression. The shock response for WCE is plotted in Fig. 9 along with the response previously found for ALOX.7 The two are very similar below about 10 GPa, suggesting that most of the deformation is accommodated by the softer epoxy matrix for both materials; a small difference in ceramic particle volume fraction 共45% WC versus 43% Al2O3兲 contributes to a slightly stiffer response for the WCE. While the ALOX results are represented well by a quadratic fit passing through the origin, the WCE data are better represented by a cubic fit. However, although the concave-down region 共to about 3 GPa兲 of the WCE fit suggests that propagating waves should separate into faster elastic and slower bulk components for stress levels below about 11 GPa, no such elastic precursor is seen experimentally 共see Figs. 5 and 7兲. This is probably because the ultrasonic sound speed CL = 2.04 km/ s is above the shock velocity Us for just four experiments 共and then only slightly so兲, making separation of an elastic precursor a very slow process. It should also be noted that the data shown in Fig. 9 include several experiments below 3 GPa for which the waves may not be steady for a significant portion of the sample. These data, therefore, should be regarded as somewhat less reliable than data for higher stress levels. 20 WCE Shock Experiments
(GPa)
ALOX Shock Experiments
WCE (cubic fit)
10 ALOX (quadratic fit)
0
0
0.1
0.2
B. Ramp loading experiments
Two experiments were conducted on the Veloce machine, as described in Sec. II C. In one experiment, WCEV4, samples 1.06 and 1.56 mm thick were loaded on opposite panels 共Table V兲. Velocity profiles for this experiment are shown in Fig. 10. The velocity profiles are characterized by a gradual foot that is due to the initially elastic response of the aluminum panels through which the WCE samples are loaded. After the foot, the velocity rises more abruptly, and the rise is somewhat steeper for the thicker sample due to nonlinearity in the material response. Significant attenuation in the peak velocity 共about 15%兲 occurs between the two sample thicknesses. The attenuation occurs because of the much higher velocity associated with unloading than with loading, analogous to the behavior seen under shock loading in Fig. 8. This has a significantly greater attenuation than has been seen in other studies of metals17,18,33 because the difference between the loading and unloading wave speeds is greater for ALOX than for typical metals. After the peak velocity is reached, the velocity decreases as the sample unloaded due to the decrease in current in the machine. The unloading history for the thicker sample appears erratic, perhaps due to edge release waves in the sample. The nature of the unloading wave in ramp experiments has been used previously17,18 to determine the strength of metals under isentropic loading conditions, but it would be ill-advised to attempt that here given the limited amount of data available. Through an iterative characteristics technique,34 the loading response of the WCE was determined from the velocity profiles shown in Fig. 10. An estimate of the uncertainty in the response is also shown. This analysis assumes a rate-independent response for WCE. Given its polymeric matrix, that assumption is at least somewhat inaccurate. Nevertheless, the response determined from the ramp loading experiment is extremely close to the cubic fit to the shock data, as shown in Fig. 11. Given that almost all of the shock data are for stresses above 2 GPa, this agreement should be regarded as fortuitous. The loading response shown in the figure also shows a concave-down regime below 1 GPa, just as TABLE V. Details of ramp loading experiments.
Expt.
Window
Diagnostic
Sample thickness
o 共g / cm3兲
WCE-V3
Sapphire
WCE-V4
LiF
Point VISAR Line-VISAR Point VISAR Point VISAR
1.532 2.073 1.061 1.560
7.58 7.51 8.21 8.15
FIG. 9. 共Color online兲 Longitudinal stress—engineering strain response of WCE and ALOX.
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1.061 mm
up (km/s)
0.1
u (km/s)
1.560 mm WCE-4V
0.05
p
0.1
2.073 mm
0.05
0 1,200
1.532 mm
800
y (µm)
WCE-3V
0 0.5
1
2
1.5
t (µ µs)
FIG. 10. 共Color online兲 Interface velocity profiles for Veloce experiment WCE-V4 and WCE-V3. The velocity record for the thicker sample of WCE-V3 comes from the spatial average of the line-VISAR record.
the fit does. As mentioned previously, no elastic precursor is distinguishable despite the concavity of the response. A second experiment, WCE-V3, was performed on two samples, 1.53 and 2.07 mm thick, with c-axis sapphire windows 共Table V兲. One sample utilized the same velocity interferometry system as on all the other experiments, while a spatially resolved interferometry system known as lineVISAR 共Ref. 35兲 was used on the other. Sapphire windows were used because they have been shown to give much better results for line-VISAR experiments in ramp loading.36 The spatially resolved velocity history extracted from the streak image using a technique similar to that of Celliers et al.37 is shown in Fig. 12. One interesting feature of the velocity history is the existence of two peaks in velocity, one for Y ⬇ 200 m and the other closer to 800 m, due to local variations in the material response. When the response shown in Fig. 12 is averaged in the spatial dimension, the average velocity, ¯u p, is close to the conventional VISAR measurement made on the thinner opposite sample as shown in Fig. 10. The measured velocities are significantly lower for WCE-V3 than for V4 because of the higher impedance sapphire windows used in the former. At any given time, the velocity at different positions varies significantly, as can be seen in Fig. 13. Also shown in the 4
400 0 1.2
1.4
1.6
t (µs)
FIG. 12. 共Color online兲 Spatially resolved interface velocity history for Veloce experiment WCE-V3.
figure is the standard deviation of the velocity at all positions for a given time, S关u p共y , t兲兴 共scaled by a factor of 5 for clarity兲. The variation is greatest during the part of the ramp where the velocity changes most rapidly and then is essentially constant as observed previously for a tungsten alloy38 and granular sugar.39 The ratio of S to ¯u p after the peak is 5%–7%, which is comparable to the ratio observed in previous studies.38,39 V. MODELING RESULTS A. Simulations of plate impact experiments
In the mesoscale simulations, an initial impact velocity was prescribed for the WCE impactor. An image of waves propagating in a simulation of the TW experiment WCE-13 is shown in Fig. 14. Two wave fronts are propagating simultaneously to the right in the sample and to the left into the impactor. Both fronts have significant width, and their fronts are irregular. The irregularities seen in these simulations are much less dramatic than those observed in similar mesoscale simulations on granular WC 共Ref. 11兲 due to the smoothing effect of the epoxy matrix. In WCE, few particles are touching one another even after they have reached the maximum stress state, and the force chains observed in simulations of granular WC 共Ref. 11兲 are not observed for WCE. By averaging the velocity histories of ten tracer points distributed vertically through the model at multiple horizontal locations, the velocity of the compression wave can be determined. Average particle velocity profiles for four posi-
WCE Shock Experiments
(GPa) 3
0.1
ALOX Shock Experiments WCE (cubic fit)
2 WCE response from ramp loading
ALOX (quadratic fit)
up (km/s)
up (t)
0.05
5 x S(u (y,t)) p
1 0 1.2
0 0
0.04
1.4
1.6
1.8
t (µs)
0.08
FIG. 11. 共Color online兲 Loading response from Veloce experiment WCE-V4 compared to plate impact results.
FIG. 13. 共Color online兲 Interface velocity history from Fig. 12 for WCE-V3 showing the average velocity ¯u p, the range of velocities 共colored band兲, and the standard deviation in the velocities S 共blue, magnified 5⫻兲.
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2, 3, and 4 mm
up 0.2 (km/s)
FIG. 14. 共Color online兲 Contours of pressure in a propagating wave from a mesoscale simulation of experiment WCE-13 at 0.50 s after impact. The domain shown is 1 mm tall. 0
tions in the simulation are shown in Fig. 15. Although some variations in the wave front are observed at the different positions, the wave front appears to be fluctuating around a steady state. Thus, for this stress level of about 4 GPa, 0.5 mm appears to be a sufficient propagation distance for the wave to reach a steady state. Also shown in Fig. 15 is the average wave profile at the interface between the 0.5 mm thick aluminum buffer backing the WCE and the LiF window. The structure of this wave is similar to that observed in the experiment, as shown in Fig. 5, though the wave in the simulation is somewhat more spread out and arrives slightly later than in the experiment. It also appears that an elastic precursor is propagating in the simulation. Agreement between the simulated and experimental velocity histories is also good for WCE-12, as shown in Fig. 5, though the simulated wave again arrives late and has a noticeable elastic precursor. The velocities from the simulations display a gradual rise in the velocity near the plateau level that is similar to that seen in the experiments. Close examination of the simulations reveals that many small waves continue to reverberate through the microstructure long after the main wave has passed. Ultimately, these small reverberations attenuate and the steady state velocity is reached. To assess the effect of material variability on the interface velocity history, ten additional realizations of the simulation were conducted. Each realization had a different random arrangement of WC particles but was otherwise identical to the others. Velocity histories for all 11 realiza-
0.2
0
0
0.5
buffer-window interface (11 realizations)
2.0 mm
1.5 mm
0.1
1.0 mm
X = 0.5 mm
up (km/s)
1
1.5
2
t (µ µs)
FIG. 15. 共Color online兲 Average velocity profiles for tracer particles at Lagrangian positions of 0.5, 1.0, 1.5, and 2.0 mm from the impact plane for the mesoscale simulation of WCE-13. Also shown is the average velocity history for the buffer/window interface 共X = 3.5 mm兲 for 11 different realizations of the mesoscale model.
0
0.4
3 mm 6 mm 9 mm
0.205 km/s
0.1
0.329 km/s
0.449 km/s
3, 4.5, and 6 mm
0.8
1.2
1.6
t - arbitrary (µ µs)
FIG. 16. 共Color online兲 Velocity histories from mesoscale simulations of wave evolution experiments WCE-18, WCE-19, and WCE-7. Profiles for the different thicknesses have been shifted in time so all for a given experiment nearly overlay one another.
tions are shown for the buffer-window interface in Fig. 15. The velocity histories for these 11 realizations fall into a band only slightly wider than the width of a line in the plot, indicating that the response depends only slightly on the specific arrangement of particles. Of course, the velocity history considered is after the wave has traveled through 0.5 mm of aluminum, which will smooth out local heterogeneities, and the 1 mm lateral dimension of the simulation is larger than the region monitored by a point VISAR. Also, variations in the WC volume fraction similar to those seen in the various samples are expected to create larger differences in the velocity history than those shown here. Simulations of RI experiments also show good agreement with measured velocity histories, as shown in Fig. 6. Again, 11 distinct realizations of the WC particle arrangement are considered. The constant velocity level is correctly simulated for both experiments as is the approximate arrival of the release waves. However, the shape of the profile in the unloading regime differs significantly from the experiments. Additional study is needed to determine the cause of this discrepancy. Significantly greater variations between the different realizations are observed for the RI experiments than for the TW experiment examined previously. In general, the variations decrease in amplitude with time following the impact, and they increase in amplitude with impact velocity. The irregularities observed in the experimental velocity histories 1.5– 2 s after impact do not appear in the simulations, so, no real insight into their origin can be gained from the simulations. Finally, the variations appear to decrease during the unloading portion of the experiment. The EW experiments were also simulated, as shown in Fig. 16. Velocity profiles for the highest velocity, 0.449 km/s, are very similar to the experimental ones shown in Fig. 7. No significant difference is seen for the different thicknesses, though the simulated histories are slightly more spread out in time than the experimental ones. At the middle velocity, 0.329 km/s, the waves change only slightly for the different thicknesses, but a distinct elastic precursor can be seen. Finally, for the lowest velocity, 0.205 km/s, significant spreading of the waves is observed, though somewhat less than in the experiments. Again, a distinct elastic precursor is ob-
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8
up (t)
Us, CR (km/s)
(simulation)
6
CR - WC/Epoxy
4
US - ALOX (Setchell)
up (km/s)
CR - ALOX (Setchell)
0.05
up (t)
(experiment)
5 x S(u (y,t)) p
US - WC/Epoxy
2
0 1.2
scatter in realizations of the mesoscale model
0
0
0.5
1.4
1.6
1.8
t (µs) 1
up (km/s)
FIG. 17. 共Color online兲 Shock wave and release wave speeds for WCE from experiments 共symbols兲 and mesoscale simulations 共dashed lines兲. Scatter bars show range of values obtained for Us and CR of WCE from ten realizations of the microstructure.
served with a wave speed of about 2.09 km/s. No attenuation of the waves occurs in the simulations, supporting the assessment that the attenuation observed experimentally is not due to release from the back of the impactor. Of course, any attenuation due to release waves from the edge of the sample will not be reflected in the mesoscale simulations because they rely upon an RVE with uniaxial strain boundary conditions. Shock and release wave velocities were determined from simulations in the RI configuration, as described previously, and these velocities are shown in Fig. 17 along with the experimentally measured values. Agreement for the shock velocities is excellent. Calculated Eulerian release wave velocities agree well with experimental values, though the form of the data appears somewhat incorrect for ALOX. In general, the model performs very well at predicting the impact loading behavior of WCE and ALOX. Multiple realizations of the model are shown as scatter bars for Us and CR of WCE. The 11 realizations have very nearly the same Us, but more significant differences are seen in CR. These larger differences in CR arise from the fluctuations in the plateau velocity that make it difficult to determine the precise time of arrival for the release waves. The scatter bars for CR are comparable in magnitude to the estimated experimental uncertainties shown in Fig. 8. B. Simulations of ramp loading experiments
Only a limited number of simulations were run for ramp loading experiments on Veloce. Because the input for the geometry of panels and charge voltage used has not been calibrated, the simulations were not considered to be very accurate in representing the experiments. Also, the presence of an elastic precursor in the simulations, but not in the experiments, greatly affects the velocity history in ramp loading simulations. Nevertheless, the model was used to explore the variability in the velocity response measured using lineVISAR 共see Figs. 12 and 13兲. To do this, a simulation was run with the same geometry as in experiment WCE-3V and 50 Lagrangian tracers placed along the interface between the WCE sample and the sapphire window, mimicking the type
FIG. 18. 共Color online兲 Interface velocity history from a mesoscale simulation of WCE-V3 showing the average velocity ¯u p, the range of velocities 共colored band兲, and the standard deviation in the velocities S 共blue, magnified 5⫻兲. Also shown is ¯u p for the experiment.
of measurement made with the line-VISAR. For this case, the actual density of the experiment, 7.55 g / cm3, was used. An energy input used previously14 was scaled in amplitude to give results that were close to the velocity histories from the experiments, but no significant effort was made to match the simulations. Results from the simulation are shown in Fig. 18 in the same manner as the experimental data. As seen in the figure, the average velocity ¯u p from the simulation is reasonably close to that from the experiment. The band of velocities is similar, though the simulation band is narrower at low velocities than that from the experiment. The standard deviation S of the velocity is also generally somewhat lower than in the experiment. In addition to inaccuracies in the constitutive model and the unrealistic particle shapes, the mesoscale simulations are 2D while the experiments are, of course, three-dimensional. Also, errors associated with recording the velocity history on the streak camera, including distortion in the streak camera itself, may play a significant role in the velocity variation. Additional study is needed on both the experimental and modeling sides in order to understand the origins and sensitivities of the velocity variations. VI. DISCUSSION
In this study, the dynamic behavior of a WCE under planar loading has been examined under a variety of loading conditions to a stress level of nearly 20 GPa. This material is very similar to ALOX formulations that have been the subject of several previous studies2,7,9 except for the different ceramic particles used. As such, it complements those earlier studies by demonstrating the role of particle properties in the overall behavior of the composite. Qualitatively, WCE behaves in a very similar manner to ALOX in that propagating waves develop finite rise times due to local heterogeneity and the viscoelastic response of the epoxy matrix. Sharp shocks generated by planar impact developed finite rise times whose rise time decreases with increasing stress. On the other hand, when relatively low amplitude ramp waves were introduced into the sample, they tended to steepen with propagating distance. The stress-strain responses of WCE and ALOX are very similar, suggesting that the response of the composite is very close to rigid particles in a compliant epoxy matrix. The two materials also display similar behav-
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ior in their release wave speeds; in both cases they are significantly faster than one would expect for homogeneous materials. The ratios of CR to Us at a given u p are similar for the two materials. WCE provides an interesting contrast to the granular WC material studied previously.10 Not surprisingly, the WCE composite is significantly less compressible than granular WC, even though the composite’s density was lower. Although both WCE and granular WC displayed structured waves with finite rise times, the strain rate of WCE scaled with stress to the fourth power, similar to ALOX 共Ref. 7兲 and many other fully dense metals and ceramics.30 In contrast, the strain rate in granular WC scaled linearly with stress, a scaling also observed for granular TiO2 共Ref. 10兲 and sand.31 Thus, the epoxy matrix plays a very important role in determining the characteristics of propagating waves, even though it is about 13 times less dense than the WC particles. The rich data set reported here for WCE, as well as the previous data for ALOX, provide an excellent opportunity to exercise 2D mesoscale models previously used for granular materials11–13 and for mixtures of different materials and epoxy.23,24 Although very simple models are used to describe the behavior of the WC and epoxy constituents, the simulations capture much of the complex behavior observed in the experiments, suggesting that many aspects of the composite’s behavior are due strictly to the two disparate phases. Because of the multiple types of experiments included in this investigation 共TW, RI, EW, and ramp loading兲, many aspects of the model are tested. The simulations do a very good job of predicting the overall shock and release wave behavior for both WCE and ALOX. In addition, they capture many of the important features of the structured waves that propagate in WCE for relatively low stress levels. The main shortcoming of the simulations is the propagation of an elastic precursor ahead of the main wave at low stress levels. No evidence of precursors has been observed in WCE or previous studies of ALOX. It is possible that refinements in the constitutive model for epoxy that account for the viscoelasticity and pressure dependence of epoxy would correct this problem. Ideally, one would like to calibrate a constitutive model for the epoxy based on experiments on neat epoxy before utilizing the model to simulate the behavior of WCE or ALOX. Limited simulations of ramp loading experiments were conducted and compared with the line-VISAR measurements made on experiment WCE-V3. Although the spatially averaged velocity ¯u p did not agree particularly well with that from the experiment due to inaccuracies in the material model as well as the unknown loading input, the simulation results displayed spatial variations in the velocity that were similar to the experimental variations. Further investigation of the origin of these variations and the material parameters on which they depend could lead, ultimately, to a more rigorous means by which to probe material behavior and validate mesoscale models. While the mesoscale model predicts many aspects of behavior well, it does not capture all of the behavior observed experimentally. For example, the jaggedness of the velocity late in time for some of the RI experiments shown in Fig. 6 are not reproduced in the simulations, and the predicted un-
J. Appl. Phys. 107, 043520 共2010兲
loading behavior is somewhat inaccurate. Similarly, the predicted spreading of the waves with thickness shown in Fig. 16 is significantly less than that observed experimentally for the 0.205 km/s impact velocity. Finally, the predicted spatial variability S under ramp loading is qualitatively different from that observed experimentally. In general, the variability in the simulations is less than in the experiments. In part, this is likely due to “noise” in the measurements of velocity, which obviously is not present in the simulations. However, it is quite possible that there is more variability 共e.g., matrixrich regions兲 in the real material than in the simulations. Such variability could arise due to the preparation of the material despite the care taken to produce a uniform material. There may also be mechanisms that play a role in the material behavior that are not accounted for in the simulations. For example, debonding of the particles from the matrix could occur, or the material could form localized shear bands; neither of those potential effects would be captured by the mesoscale model. In many ways, the shortcomings of the model are the most interesting aspects since they suggest the areas where our understanding of material behavior is weakest. Although the mesoscale model was evaluated against a wide range of data, the model itself was not optimized. Nor were parametric studies performed to understand the sensitivities of the model as was done for granular WC.13 In general, the overall shock response is probably sensitive to the particle volume fraction but insensitive to most other parameters, including the strength of the ceramic particles. On the other hand, the structure of propagating waves will likely be influenced by a wider range of parameters including particle size 共as observed experimentally for ALOX兲9 and the constitutive behavior of the epoxy matrix. The simulations also provide some insight into the interplay of material viscosity and heterogeneity in determining wave structure. In general, both aspects play a role, with two-phase makeup of the material appearing to play a primary role and viscoelastic behavior in the matrix playing a secondary role. The mesoscale model demonstrates that, even in the absence of viscosity in the matrix, the composite can display significant apparent viscosity as heterogeneous behavior spreads out the wave front. Although the mesoscale models used here require only modest computational resources, on the order of 10’s of processors for a day, they are not suitable for continuum level simulations in which WCE or ALOX are used as encapsulants. For such applications, a continuum or multiscale approach will be required. The continuum model of Drumheller4 seems to be reasonable for ALOX, but its use for WCE would require calibration of the model. The experiments reported here could, of course, be used to perform such a calibration. However, such a model would not be expected to be predictive of new formulations such as WCE with a lower volume fraction of particles or a mixture of WC and Al2O3 in an epoxy matrix. There is reason to believe that the mesoscale model would be predictive for such materials, so a multiscale model incorporating it would be a valuable tool for simulating the behavior of ceramic/epoxy mixtures at the continuum level.
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ACKNOWLEDGMENTS
The authors wish to thank the teams at the STAR and DICE facilities for the execution of the gas gun and Veloce experiments. They would also like to thank Tom Ao for the analysis of the line-VISAR experiment presented herein. This work was supported by the Joint DoD/DOE Munitions Technology Development Program. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Co., for the National Nuclear Security Administration of the United States Department of Energy under Contract No. DE-AC04-94AL85000. W. Mock and W. H. Holt, J. Appl. Phys. 49, 1156 共1978兲. D. E. Munson, R. R. Boade, and K. W. Schuler, J. Appl. Phys. 49, 4797 共1978兲. 3 L. C. Chhabildas and J. W. Swegle, J. Appl. Phys. 53, 954 共1982兲. 4 D. S. Drumheller, J. Appl. Phys. 53, 957 共1982兲. 5 M. D. Furnish, J. Robbins, W. M. Trott, L. C. Chhabildas, R. J. Lawrence, and S. T. Montgomery, in Shock Compression of Condensed Matter— 2001, edited by M. D. Furnish, N. N. Thadhani, and Y. Horie 共American Institute of Physics, New York, 2002兲, pp. 205–208. 6 J. C. F. Millett, N. K. Bourne, and D. Deas, J. Phys. D: Appl. Phys. 38, 930 共2005兲. 7 R. E. Setchell and M. U. Anderson, J. Appl. Phys. 97, 083518 共2005兲. 8 J. C. F. Millett, D. Deas, N. K. Bourne, and S. T. Montgomery, J. Appl. Phys. 102, 063518 共2007兲. 9 R. E. Setchell, M. U. Anderson, and S. T. Montgomery, J. Appl. Phys. 101, 083527 共2007兲. 10 T. J. Vogler, M. Y. Lee, and D. E. Grady, Int. J. Solids Struct. 44, 636 共2007兲. 11 J. P. Borg and T. J. Vogler, Int. J. Solids Struct. 45, 1676 共2008兲. 12 T. J. Vogler and J. P. Borg, Shock Compression of Condensed Matter— 2007, edited by M. Elert, M. D. Furnish, R. Chau, N. Holmes, and J. Nguyen 共American Institute of Physics, Melville, NY, 2007兲, pp. 227–230. 13 J. P. Borg and T. J. Vogler, Modell. Simul. Mater. Sci. Eng. 17, 045003 共2009兲. 14 T. Ao, J. R. Asay, S. Chantrenne, M. R. Baer, and C. A. Hall, Rev. Sci. Instrum. 79, 013903 共2008兲. 15 E. Lassner and W.-D. Schubert, Tungsten: Properties, Chemistry, Technology of the Element, Alloys, and Chemical Compounds 共Kluwer/Plenum, Dordrecht/New York, 1999兲. 16 L. M. Barker and R. E. Hollenbach, J. Appl. Phys. 43, 4669 共1972兲. 17 J. R. Asay, T. Ao, J.-P. Davis, C. A. Hall, T. J. Vogler, and G. T. Gray, J. Appl. Phys. 103, 083514 共2008兲. 18 T. J. Vogler, T. Ao, and J. R. Asay, Int. J. Plast. 25, 671 共2009兲. 1 2
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