An estimate of the linear strain rate dependence of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocineP. A. Conley and D. J. Benson Citation: Journal of Applied Physics 86, 6717 (1999); doi: 10.1063/1.371722 View online: http://dx.doi.org/10.1063/1.371722 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/86/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A molecular dynamics study of the early-time mechanical heating in shock-loaded octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine-based explosives J. Appl. Phys. 116, 033516 (2014); 10.1063/1.4890715 Initial chemical events in shocked octahydro-1,3,5,7-tetranitro-1,3,5,7- tetrazocine: A new initiationdecomposition mechanism J. Chem. Phys. 136, 044516 (2012); 10.1063/1.3679384 Flame spread through cracks of PBX 9501 (a composite octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine-basedexplosive) J. Appl. Phys. 99, 114901 (2006); 10.1063/1.2196219 Quantitative analysis of damage in an octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazonic-based composite explosivesubjected to a linear thermal gradient J. Appl. Phys. 97, 093507 (2005); 10.1063/1.1879072 Monte Carlo calculations of the hydrostatic compression of hexahydro-1,3,5-trinitro-1,3,5-triazine and β-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine J. Appl. Phys. 83, 4142 (1998); 10.1063/1.367168

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An estimate of the linear strain rate dependenceof octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine

P. A. Conleya) and D. J. BensonDivision of Mechanical Engineering, Department of Applied Mechanics and Engineering Sciences,University of California at San Diego, La Jolla, California 92093-0411

~Received 22 March 1999; accepted for publication 18 August 1999!

It is the long term goal of our work to elucidate the microscale mechanisms involved in the initiationof porous energetic materials~EMs! through direct numerical simulation of EM microstructuressubjected to dynamic loading. Through this effort it is hoped that we may suggest appropriatecontinuum level constitutive models~reactive and inert! for porous EMs. A major obstacle in thiseffort is the lack of reliable parameters for the constitutive models in current use in computation. Inthe current study, we examine the theoretical dependence of shock structure upon the viscous natureof the deformation. In particular, we attempt to demonstrate that a unique value of the viscousparameter exists for an elastoplastic strength model in concert with a simple Newtonian viscousshear rate dependence, which allows us to accurately simulate shock structures observed inexperimental studies of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine. In doing this, a reasonableestimate of the viscous parameter is suggested for use in future modeling efforts. ©1999American Institute of Physics.@S0021-8979~99!07522-2#

I. INTRODUCTION

Dynamic loading of heterogeneous condensed phase en-ergetic materials~EMs! is used to initiate large scale chemi-cal energy release in various industrial, aerospace and mili-tary applications. It has long been known that regions oflocalized deformation in the heterogeneous microstructure ofporous EMs lead to the formation of ‘‘hot spots’’ which mayresult in high order detonation through a shock-to-detonationtransition ~SDT! ~see, e.g., Ref. 1!.Over a large range ofinput shock strengths and porosity, the viscoplastic pore col-lapse mechanism has received the most attention. Notableare the works of Refs. 2–7.

For a rate dependent material, the viscous stress in thevicinity of a density discontinuity behind a shock or compac-tion wave is expected to be very high. Stress deviators localto the void surface will generally exceed the yield strengthvery near the wave front. The ensuing viscoplastic flow fillsthe void space and, in the process, generates a great deal ofentropic heating.

Shock initiation of reaction occurs in the followingsteps:

~1! localization of the average shock energy,E52 @(P1Po)/2# (V2Voo), at density discontinuities;

~2! a relatively fast phase change and/or chemical reactionof hot-spot regions;

~3! relatively slow growth of the reaction through thermaldiffusion over an induction period;

~4! coupling of the fast chemical energy release to the shockfront to form a self-sustaining detonation wave~in ex-plosives!.

Theoretical treatments of SDT~i.e., burn models! gener-ally simplify the details of the first and second steps by as-suming a certain mass fraction and temperature distributionof hot spots, and then assuming instant decomposition ofthese hot spots~see, e.g., Refs. 8 and 9!. The topology andthermal state of the decomposed hot spots are essentiallygiven as initial conditions to a thermal explosion initial-boundary value problem representing step~3! above. Al-though the nature of the burning in the growth stage is opento debate, the crucial input to these burn models is the heu-ristic approximation of the hot-spot distribution behind theshock and the dependence of this distribution upon shockstrength and duration, and upon various microstructural pa-rameters, including particle size distribution and porosity.

Based largely on the paucity of high strain rate experi-mental data, the strain rate dependence of common EMs isoften assumed to be linear~i.e., the flow stress is given as alinear function of the deviatoric strain rate where a constantviscosity coefficient represents the proportionality!. Whilesome have produced estimates of viscosity in various explo-sives, the range of reported values covers several orders ofmagnitude~see, e.g., Refs. 7,10 and 11!. The study of visco-plastic pore collapse mechanisms in SDT of heterogenousEMs would benefit from an improved estimate of the viscos-ity, or at least some corroboration of the various estimateswhich can be found in the prevailing literature. Due to thelarge scale use of the explosive component octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine~HMX !, we considerthis material in the current work.

Viscosity, as defined by the molecular kinetic theory ofcondensed matter, simply represents the irreversible diffu-sion of momentum along a velocity gradient. Although theviscosity is a concept which is germane to fluid flow, its usein solid mechanics is entirely justified, if not intuitive. Ina!Electronic mail: pconley@home.com

JOURNAL OF APPLIED PHYSICS VOLUME 86, NUMBER 12 15 DECEMBER 1999

67170021-8979/99/86(12)/6717/12/$15.00 © 1999 American Institute of Physics

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solids, the mechanisms available for momentum diffusionare more varied.

Due to the number and complexity of the various dissi-pation mechanisms, and due to the large range of relevantlength scales, constitutive models which address each dissi-pative mechanism individually are not currently justified dueto the relatively small amount of relevant experimental dataavailable for EMs. At this point, it is sufficient to define asingle viscosity parameter with the understanding that it rep-resents a convolution of the various dissipative mechanismsthat may exist below the spatial resolution of the model~i.e.,with length scales much smaller than a typical grain diam-eter!.

The approach taken in this work is semi-empirical. Withthe viscosity of HMX the principal unknown material param-eter, careful simulations of a series of recent shock loadingexperiments on coarse HMX powder are performed. Particlevelocity data from the experiments and simulations are com-pared to determine a consistent measure of the shock profilethat is relatively insensitive to anomalous transients. The vis-cosity of the HMX is estimated by determining the valuerequired in the calculations in order to yield similar averageparticle accelerations in both the calculated and observedshock fronts.

II. PLANE-WAVE SHOCK EXPERIMENTS

Experimental measurements of particle velocity,Up ,

and stress histories in ‘‘fine’’ grained (do;10 mm) and

‘‘coarse’’ grained (do;100 mm) samples of pressed HMXsubjected to flyer plate impacts of from 150 to 700 m/s havebeen performed independently at Los Alamos NationalLaboratory ~LANL ! and Sandia~SNL! National Laborato-ries. Details of the experimental procedure appear in the ap-propriate references~see, e.g., Refs. 12–14!. A summary ofthe procedure is given here for completeness.

Samples of granular HMX were pressed to 65% and74% theoretical maximum densities and confined in a Kel-Fcylinder with an inside diameter of 40.6 mm. The pressedthickness of the HMX samples was approximately 4 mm.The cylinder was closed with a Kel-F front plate and a backplug made from either 4-methyl-1-pentene~TPX! or polym-ethylmethacrylate~PMMA!. A Kel-F impactor on board aLexan projectile was fired from a gas gun. Particle velocitymeasurements were obtained at LANL using a 25mm thickFEP Teflon magnetic gauge with a 5mm stirrup gauge onboard. Stress measurements were obtained at SNL with poly-vinylidene difluoride~PVDF! gauges. The gauges were ep-oxied to the Kel-F front plate and the TPX/PMMA back plugat the HMX interface. The pressed thickness of the HMXsamples, i.e., the length of shock wave propagation, was ap-proximately 4 mm.

Analysis of previous experiments has demonstrated thatthe Kel-F front plate and the compacted HMX have verynearly the same shock impedance,13 obviating subsequentimpedance matching analysis. Therefore, the input wave pro-file at the front plate is expected to be representative of theactual profile transmitted into the HMX. At the back gauge,impedance mismatch at the HMX/TPX or HMX/PMMA in-

terface modifies the transmitted wave structure slightly. Risetimes are expected to be representative of the transmittedwave.14

The initiation of reaction in the HMX was clearly seenas an overshoot in stress and particle velocity histories at theback gauge. In the coarse samples some shots produced evi-dence of initiation at the front gauge as well. In general, thecoarse grained samples showed much greater sensitivity toinitiation, with evidence of reaction for shock pressuresgreater than 0.5 GPa compared to 0.9 GPa for the finegrained samples. In the fine HMX, no reaction-dependenttransients appear for impact velocities below 390 m/s. In thecoarse HMX, at particle velocities of 390 m/s, reaction tran-sients were observed. However, there is a very definite pla-teau after the inial compaction wave profile, indicating thatthe subsequent transients are not coupled to the leadingshock at the time they reach the back gauge. Therefore, theleading compaction wave profiles in these experiments areassumed to be determined only by the inert constitutive be-havior of the HMX, the initial microstructure and the impactvelocity. Only experiments with impact velocities at or be-low 390 m/s were used for the direct simulation and finalviscosity correlation.

It is initially unclear from the data if the transmittedwave profile has reached a stationary state by the time itarrives at the back gauge. However, it is well known that theachievement of a stationary profile relies on a state of equi-librium between the pressure gradients in the shock and thediffusion of momentum down those gradients through dissi-pative processes~see, e.g., Refs. 15 and 16!. For a conduct-ing solid exhibiting second order, nonlinear-elastic, rate de-pendent constitutive behavior, Bland17 showed that a step-wave particle velocity perturbation will evolve to astationary wave profile over a propagation distance

d58

3

Co

Se, ~2.1!

wherero is the initial density,Co is the bulk sound speedandS is the slope of the linearUs2Up relation for the solid.We may gain insights by extending this result as a heuristicapproximation of the ‘‘Bland length’’ for a more complexmaterial like HMX. For a homogeneous specimen of HMXwith a Co52900 m/s andS52.058, Eq.~2.1! predicts that astep-wave of amplitude;1.0 GPa producing a local strainrate of;13106 s21 will become stationary in a distance ofroughly ;3700mm. However, the addition of porosity intothe steady wave analysis decreases the effective sound speed.Benson18,19 has shown by direct simulation of metal powdercompaction that the slope of theUs2Up curve remains es-sentially constant as the initial porosity is increased. Basedon these findings, for initial porosities greater than only 5%,the effective sound speed is predicted to decrease to about25% of the sound speed in the homogeneous solid. The rep-resentative strain rate will increase significantly due to thelocalization of deformation in the vicinity of the collapsingpores. For initial diameters and shock strengths of interest,pore collapse calculations predict local strain rates at voidsurfaces of from 10 to 100 times those in the bulk material

6718 J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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~see, e.g., Ref. 20!. Taking these observations into consider-ation, an extension of the Bland criterion@Eq. ~2.1!# to aporous material appears to indicate a minimum propagationdistance for stationarity of no more than 25% of that in thehomogeneous material or, in the present case, approximately940 mm. If the increase in strain rate is taken into accountthis length may drop by another order of magnitude. Basedon this heuristic analysis, it seems likely that the transmittedwaves in these experiments are stationary after traversing theroughly 4.0 mm of pressed HMX microstructure.

In the first phase of the current study, it was necessary todevise a simple description of the experimentally measuredUp(t) wave profiles. In light of the very real uncertaintiesregarding the wave measurement—not least of which is theaveraging inherent in the measurement of the particle veloci-ties in the Lagrangian plane of the gauge—it was decided tofit a continuous trilinear function to the data. An optimal fitto the data was performed using a nonlinear least squaresoptimization algorithm.21 Figure 1 depicts an example of ameasured particle velocity history with its optimized trilinearprofile superimposed. This idealized shock profile embodiestwo independent parameters—the rise time and the averageparticle acceleration in the wave front~measured as the con-stant slope of the trilinear fit!. Alternatively, one of theseparameters may be replaced by the amplitude. This abstrac-tion provides a consistent basis for comparing the wave pro-files from multiple data sets. It is not intended to accuratelyrepresent, for example, the 5%–95% rise time commonlyused to describe a shock wave profile.

Since the average shock velocity,Us , in the heteroge-neous microstructure will differ significantly from that of theHMX constituents, this piece of information is useful forvalidating the particular yield stress and shear modulus usedin the numerical simulation of the experiments. As an esti-mate of the shock velocity, a simple numerical difference inthe form of Us5DX/Dt was used. Here,DX was taken asthe initial thickness of the pressed HMX sample; this variedfrom experiment to experiment but was measured to thenearest micron. TheDt was taken as the difference betweenthe times at which the particle velocity in the equivalentideal profiles of both gauges reached one half of the best fitto the amplitude behind the shock. Table I lists the estimatedwave speeds, amplitudes and average particle accelerationsfor the ideal profiles of the measuredUp(t) histories used inthis study.

III. NUMERICAL METHODS

All numerical simulations were performed using thetwo-dimensional~2D! Eulerian hydrocode RAVEN devel-oped by Benson.22 RAVEN performs time-centered integra-tion of the governing rate equations of mass, momentum andenergy conservation. Each step of the temporal integration iscompleted in an operator split fashion,23 with the nodal dis-placements computed in the Lagrangian step followed by anEulerian advection step. The advection algorithm incorpo-rates a second order accurate van Leer MUSCL scheme.24

The multimaterial interface tracker is of Youngs type,25

modified by Johnson.26 Spatial integration is performed on alogically regular mesh of constant strain quadrilateral ele-ments. The conservation of energy formulation includes heattransfer effects and simple phase transformation involvingheats of melting or vaporization. The conservation of mo-mentum formulation incorporates a flux limited artificial vis-cosity in order to stably resolve shock discontinuities in aminimum number of computational cells.27,28 Interpolationof solution variables to Lagrangian tracer particles can beused to form virtual gauge records for the numerical simula-tion experiments.

IV. CONSTITUTIVE ASSUMPTIONS

Under ambient conditions, HMX is a polymorphic crys-talline solid. Quasistatic compaction experiments reveal abrittle character at low strain rates and pressures below a fewkbar.29 However, at the pressures and strain rates of interestin shock initiation scenarios, it is generally believed that

FIG. 1. Example of a trilinear fit to theUp(t) data~shot 994! at the backgauge. This piecewise continuous fit is referred to as the ‘‘ideal profile.’’

TABLE I. Ideal shock profile parameters for coarse HMXUp(t) data.

ShotNo.

Front gauge Back gauge

Amplitude~m/s!

t rise

(31029 s)U p

(3109 m/s2)Amplitude

~m/s!t rise

(31029 s)U p

(3109 m/s2)

912 228.9 65.0 3.522 207.4 446.6 0.464994 280.6 32.9 8.529 230.0 207.5 1.108995 196.1 39.9 4.915 155.8 400.2 0.389

6719J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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HMX exhibits a ductile flow behavior. Fieldet al.30 haveobserved distinct twinning striations in recovered, shockloaded HMX-based plastic bonded explosive. Based on thesefindings, we make the supposition that HMX undergoes ex-tensive plastic flow in the pressure regime of interest, seeTable II.

The Cauchy stress is given in component form by

s5sep8 1sv82~P1Q!I , ~4.1!

where the deviatoric components arise from materialstrength,sep8 and viscous stress,sv8 , and the spherical com-ponents are due to hydrostatic pressure,P, and shock viscos-ity, Q.

The elastoplastic stress rate is given by

sep8 52G~ e2 e p!, subject to J2~sep8!<Y, ~4.2!

whereG is the shear modulus,e is the total strain rate tensorand e p is the plastic strain rate tensor.J2 is the second in-variant of the elastoplastic stress andY is the yield strength.

The material strength model is necessarily simplistic dueto the fact that relatively little experimental data are availablein the high pressure, high strain rate regime. Steinberg andGuinan31 proposed a high strain rate material model for met-als. Based on limiting dislocation dynamics arguments, thismodel asserts that rate dependence of the yield strength van-ishes for strain rates above 105 s21. The major source ofentropic heating in heterogeneous microstructures resultsfrom local strain rates above 105 s21 near density disconti-nuities like voids, cracks and inclusions. Therefore, we usethis model as a basis for the strength calculations.

The unabridged strength model is given in Ref. 32 andasserts that both the shear modulus and yield strength arepressure and temperature dependent. However, due to a lackof relevant experimental data with which to estimate valuesfor these parameters, they are assumed to be zero. For similarreasons, the strain hardening parametersb andn are ignored,leaving only a melt factor to modify the ambient values ofGandY. With the melt factor a function of a single parameter,f, and the reference volume specific internal and melt ener-gies given byE andEmelt, respectively, the reduced strengthmodel is simply

Y5Yo expS 2 f E

~Emelt2E! D , ~4.3!

G5Go expS 2 f E

~Emelt2E! D . ~4.4!

The temperature,T, is not a state variable in the currentformulation. Instead it is calculated as

T5E2Ec~h!

rCp, ~4.5!

whereEc(h) is the cold compression energy and the com-pression,h, is defined asr/ro . The constant pressure spe-cific heat,Cp , is generally a function of temperature. How-ever, since no data are available forCp above 430 K,33 it isassumed to be constant given by its experimental value at thehighest available temperature.

For the solid material, a Mie-Gruneisen equation of stateis well characterized in HMX shock experiments.34,35There-fore, the pressure is computed from

P55roC2h@11~12go/2!h2ah2/2#

@12~S21!h#1~go1ah !E

~in compression!,

roCo2h1~go1ah !E ~in tension!,

~4.6!

where h5r/ro21, ro is the reference density,Co is thereference bulk sound speed and the parametersgo anda arethe Gruneisen parameter and the volume correction factor,respectively. The parameterS reflects the dependence of theshock velocity,Us , upon particle velocity,Up . A linearUs2Up relation,

Us5Co1SUp , ~4.7!

has been validated for HMX over the particle velocity rangeof interest.34,35

The cold compression energy in Eq.~4.5! is computed as

Ec~h!5E1

hP~h!

h2dh2rCpTref exp@a~12h21!#h (go2a),

~4.8!

where the integration occurs along the zero Kelvin isothermin P-r-E space. Instead of performing this integral multipletimes during a global integration step, the integral is evalu-ated at the beginning of the calculation and a ninth orderpolynomial function is fit to theEc(h) dependence.

The strain rate dependence of HMX is not well charac-terized. At present, the most common form of rate depen-dence is Bingham-type viscoplasticity represented by Eq.~4.1! ~see, e.g., Refs. 3,5 and 20!. The viscous stress,sv8 ,contributes to the stress deviator instead of only the shearterms, and is given by the linear strain rate expression,

sv852m~P,T!e8 ~4.9!

where the viscosity is proposed as depending on pressure andtemperature according to3

m~P,T!5H ms :T<Tmelt,

MIN @ms ,mo exp~P/Po!exp~Eo /T2Eo /To!#

:T.Tmelt,~4.10!

TABLE II. HMX material properties~taken from Ref. 33!.

Constitutive parameter Units Value

Sound speed,C m/s 2901Initial density,ro g/cm3 1.891Mie-Gruneisen coefficient,go 1.1S 2.058Initial shear modulus,Go MPa 2700Initial yield strength,So

y Mpa 48.3Melt parameter,f 0.45Ambient melt temperature,Tmo K 558

6720 J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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wherems is the ‘‘solid viscosity’’ associated with flow of theunmelted material,mo is the viscosity of the melt at ambientpressures, andPo , Eo andTo are material parameters whichdefine the continuous transition from the relatively high solidviscosity to the relatively low melt viscosity. A maximumbound on the melt viscosity is required since, in practice,temperature increases tend to lag behind pressure increasesin time, and the exponential pressure dependence was ob-served to lead to unrealistic increases in viscosity.

The dependence of hot-spot evolution near a collapsingvoid has been shown to be very sensitive toPo in shockinitiation calculations involving this constitutive model.36

However, since the greater portion of dissipative heatingtakes place prior to melting, we essentially are looking forthe solid viscosity in the current numerical study. A reason-able estimate of this material property is required for even azeroth order prediction of viscoplastic hot-spot mechanisms,and is the primary motivation of the current presentation.

To reiterate, this viscosity is not physically equivalent tothe shear viscosity as defined by molecular kinetic theory.Instead, thissolid viscosity is defined as an analog of theshear viscosity only. It implies the presumption of a lineardependence of flow stress upon adeviatoricstrain rate. How-ever, the mechanisms at work during large strain in solids areinherently nonlinear. In quasistatic compaction, the approxi-mation is expected to be reasonable since the amount of ran-dom motion behind the compaction zone is relativelysmall.37 This may not be true of the fully dynamic regimesince turbulent jetting increases the energy of random motion~microkinetic energy! and processes like self-impact actuallydissipate a significant amount of energy through inertial dis-persion instead of momentum diffusion. However, for explo-sives, the dynamic regime is dominated by fast chemicalreaction and the deviatoric constitutive assumptions are nolonger relevant, since a purely hydrodynamic phenomenol-ogy dominates the post-ignition flow.

In general, the melting temperature,Tmelt, is known todepend on the pressure~or volume compression! of the solid.For most materials, the melt temperature increases with bulkloading and a variety of melt laws have been proposed. Themelting factor,f, in Eq. ~4.3! is used in conjunction with amodified Lindemann melt law proposed by Steinberg andco-workers32 in the original presentations of the high strainrate strength model. Based on that model, the melt tempera-ture evolves according to

Tmelt5Tmelto exp~2a~121/h!!h2(go2a21/3), ~4.11!

and the melt energy evolve according to

Emelt5Ec~h!1rCpTmelt. ~4.12!

Melting introduces a latent heat of fusion,DHm , into thetemperature calculation so that the actual temperature iscomputed from

T55E2Ec~h!

rCp:E<Emelt,

Tmelt :Emelt,E,Emelt1DHm ,

E2Ec~h!2DHm

rCp:E>Emelt1DHm.

~4.13!

V. MODELING ISSUES

A. Boundary conditions

Figure 2 shows a subsection of the initial geometry usedto simulate shot 912~coarse HMX pressed tofo535%!.The left side of Fig. 2 shows the mesh resolution used in thecalculations. The entire microstructure uses over 230 par-ticles. For the simulations, the shock wave moves throughthe mesh from left to right. The top and bottom boundaryconditions are assumed to be periodic to minimize constrainteffects.

The shock is generated by a velocity boundary conditionon the left edge of the model using theUp(t) data takendirectly from front gauge records in each experiment. Theright boundary approximates a ‘‘silent’’ or ‘‘transmitting’’boundary condition to minimize reflections due to impedancemismatch there.

B. Virtual gauge

Three columns of tracer particles, 13 in each column, areembedded in the mesh, as is partially shown on the right-hand side of Fig. 2, and form virtual gauges. The first gaugeis placed 755mm from the left boundary and the othergauges are placed at 200mm increments~see Fig. 2!. TheUp(t) data calculated at each tracer particle are averagedover each column to produce three distinctUp(t) profiles foreach simulation. Each of these profiles is then fit with theidealized profile described in Sec. II. The slope, rise time andamplitude of the three idealized profiles are then averaged to

FIG. 2. Representative subsection of the initial simulation geometry for shot912.

6721J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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arrive at the final values predicted for each simulation. Theseare then compared against the idealized shock wave profilesof the experimental data.

C. Particle size distributions

It has been observed that the average particle size repre-sents a lower bound on the shock front thickness~see, e.g.,Ref. 37!. If we are to simulate experimental shock processes,then it is expected that the shock structure prediction maysuffer some systematic bias if we use, for instance, a mono-disperse particle distribution where the average particle sizediffers significantly from the real microstructure. To avoidsuch a bias, we went to great effort to reconstruct particlepopulations that bear some statistical resemblance to the trueexperimental populations.

The particle sizes and shapes in each simulation weregenerated using a Monte Carlo method with a suitably cho-sen probability distribution function to match the sieve datadescribed by Sheffieldet al.14 and measured38 for the par-ticular lot of coarse HMX used in the experiments describedhere. The particle distributions have a random character sub-ject to a rigorously defined distribution function. Each popu-lation of particles generated by this means is different fromthe other in detail. However, care was taken to ensure thatthe population constructed is statistically likely to be amem-ber of the much larger sieved population based on the sievedata.

As the particle size in the simulations decreases, thethermal gradients increase. In order to maintain a consistentspatial resolution of these gradients in the calculation, theratio of particle diameter to element size must be consistent.Since the element size determines the maximum stable timestep of the integration, the computational cost of the calcu-lations increases by at least an order of magnitude in the finegrained material. In addition, as the particle size decreases,the characteristic void size decreases while the surface-to-volume ratio increases and the number and area of inter-granular contacts increase. Therefore, in the fine grainedHMX, it is less certain that viscoplastic void collapse willdominate over, for instance, intergranular friction as a dissi-pation phenomenon. For these reasons, it was decided toperform the simulations for only the coarse grained material,where viscoplastic void collapse is expected to be the pre-dominant dissipative~shock spreading! mechanism. Thethree experiments chosen for correlation studies are shots912, 994 and 995~see, e.g., Ref. 14!. The defining charac-teristics of these experiments are summarized in Table III.

D. Artificial viscosity

In the absence of dissipation, the numerical solution ofthe hyberbolic Euler equations of motion admits the propa-gation of discontinuities in conserved variables. It is com-monplace to use some form of artificial dissipation in orderto resolve shocks within a discretized computational domain.However, for the viscoplastic constitutive equation given byEq. ~4.1!, a certain amount of physical dissipation is intro-duced by a rate dependent deviatoric strength model. It isinteresting to note that a plane wave, while exhibiting zeroshear strain rate, does exhibit nonzero deviatoric strain rates.Therefore, it was suspected that the addition of a viscousstrength model might allow us to forgo artificial dissipationin the form of classical shock viscosity pressure component,Q @see Eq.~4.1!#.

To evaluate the possibility of performing the current cal-culations with minimal artificial viscosity, investigative cal-culations were performed for a homogeneous slab of HMXsubjected to a velocity jump on one boundary. Three caseswere considered:~1! the coefficients of the artificial shockviscosity were set to zero,~2! no artificial viscosity waspresent, but a solid viscosity of 250 P was included in theflow stress model as per Eq.~4.9!, and, finally,~3! both theartificial and the solid viscosity were included.

In the absence of either artificial or physical viscosity~case 1!, numerical dispersion occurs in the shock front pro-ducing the characteristic Gibbs oscillations behind the front.However, when only solid viscosity is present, the dissipa-tion in the deviatoric stress component is enough to yield astable shock profile that is suitably resolved. The addition ofartificial viscosity has a negligible effect since the solid vis-cosity naturally diminishes the velocity gradients, thus di-minishing the magnitude of the viscous pressure,Q. There-fore, it appears that artificial viscosity may be replaced bysolid viscosity, even for calculations involving a plane wavemoving into a homogeneous material.

In viscous porous materials, however, the localization ofdeformation at the surface of collapsing voids provides adirect mechanism for dissipation. Therefore, the use of arti-ficial viscosity may actually be redundant in this instance.Moreover, since we are attempting to estimate a materialparameter, the presence of artificial viscosity only convolutesthe analysis. In preliminary simulations of the nonhomoge-neous initial geometries in the target experiments, it was de-termined that the artificial viscosity contribution has a non-negligible effect on the predicted shock profiles. Therefore,the artificial viscosity was not employed in any of the calcu-lations performed in this study.

VI. RESULTS

A. Rise time versus average particle acceleration

The most natural characteristic of the wave profile withwhich to correlate an appropriate value of viscosity is theshock rise time,t rise. As the viscosity increases in the cal-culation, the shock spreads out and the predicted rise timeincreases as well. However, it was observed that the rise timeof the calculatedUp(t) profiles tended to be sensitive to thepost-shock transients which were invariably present in the

TABLE III. Summary of the experiments of interest.

ShotNo.

Impactvelocity~m/s!

Initialporosity

~%!

Samplethickness

~mm!Backingmaterial

912 288.0 35% 3882 TPX994 390.0 25% 3782 PMMA995 270.0 25% 3830 PMMA

6722 J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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calculations. These transients are expected where the indi-vidual grain boundaries represent impedance mismatched in-terfaces and the underlying hyperbolic nature of the conser-vation equations admits multiple shock interactions, or‘‘ringing,’’ behind the shock. Also, a graded mesh@i.e., Dx5 f (x)] was used in some cases. The nonuniform mesh re-sults in a slight diffusive error in the discrete approximationof the governing differential equations~see, e.g. Ref. 39!.The nonuniform mesh region acts like a slightly ‘‘opaque’’boundary and small amplitude reflections spuriously appearthere.

The slope of the trilinear fit profiles was observed to beless sensitive to the postshock transients than either the am-plitude or rise time of the profile. In a physical sense, thisslope is simply the time-averaged particle acceleration in thewave front and is defined as

ap5E0

trise 1

t rise

]Up~X!

] td t. ~6.1!

Instead of using the rise time of the pressure wave, it wasdecided that all correlations should be in terms of the aver-age particle acceleration,ap .

B. Viscosity correlation

Simulations of experiments 912, 994 and 995 were per-formed with solid viscosity as a variable parameter. Therange of viscosities considered was from 100 to 500 P. Pre-liminary calculations for shot 912 indicated that this rangebrackets the most likely value of the viscosity since the av-erage simulated particle acceleration is far greater than themeasured value whenm5100 P was used, and far less thanthe measured value when 500 P was used. For values below100 P, the shock front began to show some spurious oscilla-tions due to the lack of dissipation. Therefore, we see that forviscosities below 100 P the rate of physical dissipation is notsufficient to provide a stationary solution to the shock changeequations.

For illustrative purposes, the ‘‘acceleration ratio’’ is de-fined here as the ratio of average particle acceleration calcu-lated in the simulations to that observed in the experimentaldata for each shot. Figure 3 summarizes the variation of theacceleration ratio as a function of the viscosity. An accelera-tion ratio of unity indicates that the simulated and experi-mentally obtained transmitted wave forms match and, there-fore, that the corresponding value of viscosity is estimated tobe representative of the actual material.

Based on this set of calculations, the solid viscosity ofHMX appears to be about 300–315 P. It is important to notethat this procedure yields the same predicted value for vis-cosity to within 15 P~or less than65%) when both theinput shock strength and the porosity are varied indepen-dently ~see Table III!.

C. Sensitivity to the elasticity parameters

To test the sensitivity of this result to various other pa-rameters in the deviatoric stress model, the baseline yieldstress and shear modulus were varied with the shear viscosity

held constant at 310 P. Figure 4 shows a plot of the accel-eration ratio and wave speed ratio for five different baselineyield stresses covering one decade variation above and be-low the value used in the estimation of viscosity for shot912. Thewave speed ratiois the ratio of the propagationspeed of the idealized trilinear wave in the simulations tothat estimated from the experimental data.

The yield stress has a significant effect on the predictedaveraged wave speed. At higher yield stresses the materialundergoes more elastic deformation before flow commencesand more elastic energy is stored in each particle. Therefore,the wave speed increases towards the elastic wave limit asthe yield stress increases relative to the shock amplitude. Itappears that the dependence of wave speed upon baselineyield stress becomes small for values below 100 MPa. Thisyield strength appears to mark a threshold ratio of shockpressure to yield stress, above which the dominating physicsat the shock front changes.

An interesting trend is seen in the particle accelerationratio. Instead of steepening, the waves appear be become

FIG. 3. Variation of the dimensionless particle acceleration~ratio! withsolid viscosity. A ratio value of 1.0 indicates the appropriate value of theviscosity.

FIG. 4. Sensitivity of wave speed and particle acceleration predictions to theassumed baseline yield strength.

6723J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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more diffuse as the yield stress is increased. It is presumedthat the higher yield stress causes a significant dispersiveeffect at the leading edge of the shock front due to the factthat the leading elastic precursors are able to propagate stressbridges through the Hertzian contacts at the particle inter-faces. If this hypothesis is correct, one would expect that thebottom of theUp(t) traces should be spread out relative tothe top of the traces since elastic precursors of low amplitudemove out ahead of the slower plastic stress waves. Furtherinvestigation of the simulation results confirmed that, indeed,this is the case. Therefore, as the yield stress increases rela-tive to the shock strength we appear to move into a regimewhere dispersive~as opposed to diffusive! processes providethe necessary mechanism for momentum transport down thedriving pressure gradients.

Neither the wave speed nor the average particle accel-eration is predicted to depend significantly upon the shearmodulus. This result is expected in light of the fact that weare using a Gruneisen equation of state. As the material inthe shock front is compressed, the elastic shear modulusstays essentially constant while the instantaneous bulk modu-lus,K, increases exponentially. Since the sound speed is pro-

portional toAK1 43G, we expect that the instantaneous bulk

modulus will have a much greater effect than the shearmodulus for the large pressures considered. Also, in the pres-ence of plastic flow, the wave speed decreases since 2G isreplaced by the much lower hardening modulus,h, in thesound speed computation.

VII. DISCUSSION

A. Stationary wave assumption revisited

With the parameter estimation complete, what remainsto be done is to analyze the possible systematic bias in thesemi-empirical approach used here. We have already dis-cussed the importance of comparing only measured and cal-culated wave profiles or, equivalently, amplitude histories,which represent stationary, time invariant wave propagation.Comparisons of nonstationary waves are not reliable sincethe shock structure~i.e., the rise time and particle accelera-tion history through the shock front! is changing with timeand, implicitly, with propagation distance.

From the simple analysis provided in Sec. II, the mini-mum required propagation length for a stationary wave wasestimated to be somewhat less than 1000mm. With the av-erage particle size in the coarse microstructures at 80– 100mm, this is equivalent to a wave propagating through about10 particles.

Corroboration of this estimate can be found in simula-tions of the compaction of monosize copper and aluminapowders, where Benson and co-workers37 have found thatcompaction waves in those material become stationary in apropagation distance of roughly 8–10 particle diameters.

A criterion for stationarity may be extracted from theclassical Hugoniot relations which cast the fundamental con-servation equations into a convenient form. A stationary en-ergy balance requires

E2Eo5 12 Up

2, ~7.1!

whereUp is the final particle velocity, andE andEo repre-sent the final and initial energy, respectively. Therefore, be-hind a stationary shock wave, the energy is~on average!equally partitioned between internal potential energy and ki-netic energy.

A test calculation was performed using the initial geom-etry and input gauge data corresponding to the shot 912 ex-periment. The new viscosity estimate of 310 P was used.Figure 5 shows the calculated internal and kinetic energyprofiles in the mesh at 2.2ms. The internal energy is highlyoscillatory in and behind the shock front due to the intra-granular pressure wave reflections from the particle bound-aries. This ringing is an unavoidable consequence of the hy-perbolic nature of the governing equations, and are indeedpresent to some degree in the actual microstructure. Gaugedata are inherently integral making direct observation ofringing impossible.

While the traces appear disparate near the shock front, itis evident that average differences behind the shock transi-tion zone~shock front! are not significant and, therefore, thewave appears to be stationary. Furthermore, since the speci-mens impacted in the LANL experiments were nearly 4 mmin length, this result supports the notion that the transmittedwave is stationary when it reaches the back gauge in theexperiments.

B. Analytical viscosity estimate

Based on the assumption that total deviatoric stress maybe decomposed into elastic and viscous stress contributions,and on the assumption that viscous flow provides the major-ity of the dissipation resisting the formation of the shock, thecurvature of the Hugoniot is directly related to the materialviscosity. The specific Rayleigh line pertaining to the steadyshock event fixes the first pressure derivative of the bulkmodulus, which is equivalent to the slope of the presumedlinear Us2Up relation. This derivative represents a dimen-sionless pressure associated with the steepening of the waveprofile as the shock develops. The dimensionless stress(2m/t)/Ds is an estimate of the resisting force, i.e., thegeneralized shock spreading force, whereDs represents theshock strength andt represents the shock rise time. Grady40

FIG. 5. Comparison of predicted internal and kinetic energy profiles for ashock which has traversed roughly 1000mm in the initial microstructuregenerated to model shot 912.

6724 J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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used this heuristic approach to estimate the viscosity from2m.Sst/4. Using the experimental data from shot 994,Ds.0.4 GPa, t.0.32ms and the estimateS52.058 forHMX, this yields a viscosity estimate of approximately 329P. Clearly, this is within the error bars for the 310 P estimateobtained here, giving a loose theoretical corroboration of theestimate.

C. Uniqueness of the viscosity fit

We have performed a single parameter fit based on theassumption that other material parameters in the model arereasonably representative of reality. For typical shock load-ing of EMs the macroscopic behavior is sensitive to the volu-metric ~or ‘‘bulk’’ ! response of the constituents and the po-rosity of the aggregate. The approach outlined in this articlerelies heavily on use of an appropriate equation of state~EOS! for the constituent phases, as well as reliable data forthat EOS. For the case of unreacted HMX, much work hasrelied on the use of the Gruneisen EOS. In addition, Hugo-niot data for HMX have been obtained experimentally33 andare in wide use today in the experimental, theoretical andmodeling communities. It is believed that these parametersare known within reasonable accuracy.

The yield strength of solid HMX, however, is not knownwith the same degree of accuracy. Due to the reactive natureof explosives, standard strength measurements are impracti-cal. The dependence of the viscosity estimate upon the yieldstrength of HMX appears to be significant. Therefore, uncer-tainty in the yield stress reflects directly on the uncertainty ofthe viscosity estimation. If we imagine that the effects ofyield stress and viscosity are separable, then Fig. 4 tells usthat if, in the worst case scenario, the true yield stress differsfrom the value used in the correlations by an order of mag-nitude, the viscosity estimate is likely to change by630%.

At the continuum level, homogeneous material yield is afairly unambiguous concept. However, the appropriate yieldstress to be used in the modeling of porous granular EMmust be determined implicitly based on an integral responseof the material during compaction. The fundamental com-paction processes for quasistatic compaction will, in general,be different from those under fully dynamic compaction. Inthe classic study of Elban and Chiarito,29 quasistatic compac-tion of very coarse HMX was observed for a range of po-rosities. The changes in porosity with applied stress weretracked and it was observed that compaction began at appliedstresses;O(1) MPa. However, micrographic evidence ofgrain fracture was also observed for compaction up to aboutan 80% theoretical maximum density~TMD!. The appliedstresses at 80% TMD are;O(10) MPa in their results.Since fracture was observed to become secondary for furthercompaction, it seems reasonable that plastic flow should be-gin to dominate at applied stresses in excess of 10 MPa.More recently, Dick and co-workers35 performed a battery ofplate impact experiments with PBX-9501 and its constituents~HMX, estane and nitroplasticizer!. Single crystals ofb2HMX 3 mm thick were grown and mounted on Kel-Fdisks such that the plate impacts were aligned with two dif-ferent crystallographic axes. VISAR laser interferometric

data revealed a Hugoniot elastic limit~HEL! for both orien-tations that is approximately 475 MPa. Therefore, one couldjustify using a yield stress which is;O (100) MPa.

Both of the works just described represent extremes inthe experimental configuration from which yield magnitudesmay be estimated. In the case of quasistatic compaction, thelarge initial porosities and the large average grain size~900mm) lend themselves to almost purely brittle deformationmodes. The heterogeneous grain structure is expected to con-tain a high density of dislocations in the form of pileups oreven microcracking. Therefore, the integral experimentaldata are likely to predict a lower bound of the actual materialyield strength. On the other extreme, the single crystal con-figuration has a very low dislocation density by design. Itsstrength is expected to be much closer to the theoreticalstrength of the material. While this mode of measurement isvaluable, the yield stress obtained from the HEL of the intactcrystal is not necessarily representative ofin situ granularHMX.

In the end, the ‘‘true’’ material yield stress and the valuemost suitable for the model may not be the same value. Forour part, it was decided that taking an integral approach todeformation at the grain boundaries is justified on the basisthat tractable constitutive theories do not yet exist for mod-eling fracture at the mesoscale~i.e., a scale too small forstandard brittle fracture models, and too large for consideringfractures individually!. Therefore, a yield stress whose mag-nitude is between 10 and 100 MPa seems to be a reasonableapproximation for the modeling effort put forth here. Vary-ing the yield stress over this range and again assuming aseparable yield stress and viscosity contribution to the shockstructure, Fig. 4 suggests that a reasonable error bar on theviscosity estimate is closer to675 P. In light of the widerange of viscosity values in the literature on modeling, webelieve this level of confidence represents a meaningful im-provement.

D. Melting

Melting has the potential to pollute our estimate of vis-cosity since the melting reduces the effective viscosity in thecompacting granular bed. Therefore, if a significant massfraction of solid melts, the effective viscosity will be lowersince the melt viscosity is generally several orders of mag-nitude smaller.41 Based on the Frey melt model3 and theLindeman melt law31 description for the dependence of melttemperature on pressure, no significant melting was pre-dicted in any of simulations of the experiments used in theviscosity correlation. Therefore, melting is not much of anissue.

E. Intergranular friction

In the current model, the sources of dissipation whichresist shock formation are irreversible inviscid and viscousflow at the stresses,sep8 and sv8 , respectively. In general,however, the relative motion at particle-particle interfacesadds another source of dissipation through intergranular fric-tion. This is particularly relevant to the current discussionsince we have implicitly assumed that the dissipation is pre-

6725J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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dominantly due to viscous flow. If friction were to play amajor role in the shock spreading, then the viscosity estimatederived from the data would be too high.

Friction is not explicitly included in the present model.The finite element solution of the governing equations isbased on an element averaged stress,sE. Also, the velocitygradients applied to the individual materials within an ele-ment are determined from nodal velocities. The Eulerian for-mulation therefore requires a homogenization model, ormix-ture theory, to relate the ‘‘element’’ stresses and strains tothe individual ‘‘material’’ stresses and strains. The tradi-tional set of assumptions required for this process includes ano-slip assumption for material interfaces within an element.Currently, the alternative is a multiphase flow formulationwhere the trade-off is an inability to resolve particle inter-faces at all~see, e.g., Refs. 42 and 43!.

The incorporation of Coulombic friction into an Eulerianformulation has not been a topic of much research until re-cently. Notably, Benson44 has proposed an appropriate con-tact mixture algorithm for Eulerian hydrocodes. However,the resulting set of constraint equations is nearly singular andhighly nonlinear, representing a formidable challenge from anumerical methods standpoint. At this time, the contact mix-ture algorithm is still a topic of research.

There appears to be some experimental justification forignoring intergranular friction in the current compactionmode. Linse45 performed a series of shock compaction ex-periments in titanium alloy. The recovered samples whereobserved with optical and scanning electron microscopy~SEM!. At shock strengths below those required for fullcompaction, deformations at particle-particle contact sur-faces appear to be isotropic. Had friction played a major role,these contact surfaces would be elliptic instead of circular.At higher shock strengths, elliptic interfaces are indeed ob-served, accompanied by evidence of local melting/bonding.However, the micrographs also reveal a heavily striated layerjust below the amorphous melt layer. In contrast to earlierinterpretations of these data, we interpret this as evidencethat the interparticle motion induced substantial materialflow belowthe contact interfaces. Therefore, a perfect bond-ing model like that implicitly used in the current calculationsseems reasonable, if not appropriate.

In an effort to help quantify the relative contribution toshock dissipation coming from friction, a special modifica-tion was made to the strength model and a calculation whichis otherwise identical to the simulation for shot 912 wasperformed. The strength modification simply sets the yieldstrength,Y, to zero in all multimaterial Eulerian computa-tional cells for each material present. In this way, particlecontact regions are not allowed to support any deviatoricstresses and the viscoplastic work in those regions is con-strained to zero. This situation approximates a frictionlessparticle bed.

Figure 6 shows a virtual gauge result for the particlevelocity history approximately 1 mm into the microstructure.One can see that, while the elimination of all contact surfacedissipation has a marginal effect on the amplitude of thepropagated wave, the shape of the wave is not affected. Therate of change of the particle velocity at the virtual gauge

~i.e., the average particle acceleration! appears to be essen-tially identical in both cases.

The propagation velocity predicted for the ‘‘friction-less’’ case is somewhat slower than that in the unmodifiedcalculation. This is expected if we consider that the overallpropagation velocity is controlled by a weighted average ofthe local sound speeds in the microstructure. The local soundspeed is proportional to the instantaneous stiffness of thematerial and inversely proportional to the local density. Inthe calculations, for the mass fraction of material in cells atthe particle surfaces~i.e., multimaterial cells!, the inability tosupport deviatoric stresses decreases the effective stiffnessthere and the local sound speed is diminished, which tends toslow the overall wave propagation slightly. The lack of stiff-ness in the outer layer of the particles also accounts for theabsence of entropic work there and the overall decrease indissipation at the particle interface zones accounts for theextra particle velocity amplitude observed for the frictionlesscase. Since the average particle acceleration is used for esti-mating the viscosity, it does not appear that explicit interpar-ticle friction has much bearing on the estimate.

F. Rate dependent viscosity

While three different experiments were used to correlatethe final estimate of viscosity, the range of strain rates forthese experiments is limited. Higher strain rate experimentsproduced significant reaction which rendered theUp(t) dataunreliable since the transmitted wave profiles become aug-mented through chemical energy release and the rise timesand particle accelerations are no longer characteristic ofcompaction dissipation only. On the other hand, lower am-plitude loadings than those considered here result in incom-plete compaction and dramatically alter the important phys-ics from diffusive ~e.g., viscous flow! to dispersive~e.g.,ringing!. It appears that for relatively sensitive EMs likeHMX the high strain rate limitation is intrinsic.

We have estimated a single viscosity value which hasbeen implicitly assumed to be constant. In fact, it is likelythat the viscosity is not constant, but is itself rate dependent.For instance, we have pointed out that the nebulous defini-tion of viscosity used here contains contributions from anumber of different dissipative micromechanical processes

FIG. 6. Effect of eliminating deviatoric stresses in multimaterial cells in thesimulation of shot 912.

6726 J. Appl. Phys., Vol. 86, No. 12, 15 December 1999 P. A. Conley and D. J. Benson

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including microcracking, fracture and friction. Each of theseprocesses will in general exhibit its own characteristic lengthand time scales. Therefore, it is likely that, as the prevailingstrain rate changes, so will the prevailing dissipative re-sponse leading to variations in the effective viscosity.

Experimental evidence in metals points to a ‘‘rate depen-dent rate dependence.’’ Swegle and Grady15 have observed afourth power dependence of characteristic strain rates uponshock amplitude. This observation appears to apply for alarge number of materials and its consistency points to theexistence of a fundamental thermodynamic relationship be-tween shock structure and material microstructure. Partom46

has derived a possible connection between the fourth powerlaw and a quadratic viscous flow law~i.e., e8;bs82). Inhigh strength brittle solids, Grady40 described a rate depen-dence of viscosity based on fracture kinetics. Basically, asthe strain rates increase, the characteristic time for fracturebecomes large in comparison with that of dislocation dynam-ics and a quasi-ductile mode of deformation prevails, withrate dependence eventually vanishing due to dislocationdrag.

Finally, shock induced phase transitions also provide animplicit rate dependence of the viscosity. Melting, for in-stance, has a dramatic effect on the strength and rate depen-dence of a material. In turn, the thermal energy localized inhot spots is a function of the local strain rate history andthermal diffusion properties. Therefore, melting builds in animplicit upper bound of the viscous stresses by decreasingthe viscosity by several orders of magnitude once the melttemperature is reached. This coupling effectively results in arate dependent viscosity which approaches a negligible con-stant value. This behavior, while not based on the theory ofdislocation dynamics, is intuitively appealing and has beenexploited to some advantage in pore collapse models.3,7

VIII. CONCLUSIONS

In order to gain new insights into the role that variousmicrostructural phenomena play in the shock initiation ofcondensed phase EMs, it is necessary to formulate materialmodels which contain the relevant physical mechanisms. Inshock loading of porous heterogeneous microstructures, theviscoplastic collapse of voids has been widely studied as themajor dissipative mechanism leading to thermal activation ofphase transformations and/or large scale chemical energy re-lease. However, to date, very little reference has been madeto the appropriate numerical values for the ‘‘solid’’ viscosityof the materials of interest.

In the current work we have proposed a method for es-timating the solid viscosity of condensed phase EMs. Themethod uses computer simulations of controlled dynamic ex-periments performed at Los Alamos and Sandia NationalLaboratories on well characterized samples of pressed HMX.The range of peak strain rates in the experiments of interestcovers approximately the decade from 13106 to 13107 s21. The range of initial porosity considered covers25%–34%. The methodology described here may be appliedto determine physically meaningful values for the rate de-

pendence of a number of commonly used energetic constitu-ents once more data become available.

The major observations from this work are the follow-ing:

~1! The mass of hot-spot material expected to exist behind acompaction wave in HMX depends upon the flow vis-cosity of the solid. Therefore, the value of viscosity usedin pore collapse models has direct bearing on the ignitionparameters assumed for burn models which are based onignition and growth scenarios.

~2! Based on the equation of state parameters that arethought to be representative, the viscosity of HMX isestimated to be 310 P.

~3! The value of viscosity estimated here is not unique, ex-hibiting a significant dependence upon the yield strengthassumed for HMX. However, based on experimentaldata for the compaction behavior of HMX, it is thoughtthat this value of viscosity is correct to within675 P.

~4! The viscosity obtained in this study is not limited toclassical fluid dynamics definitions. It includes a host ofmolecular dissipation phenomena as well as various me-soscale acoustic, fracture and tribological mechanisms.The dissipative component due to intergranular frictionwas considered in a qualitative sense and determined tohave a negligible bearing on the viscosity estimate ob-tained here.

~5! The addition of physical dissipation may take the placeof numerical dissipation and, thus, render steady shockprofiles in a monotonic fashion without artificial viscos-ity.

ACKNOWLEDGMENTS

The authors wish to thank Rick Gustavsen and SteveSheffield for their efforts in supplying them with the all im-portant experimental stress/particle velocity histories in po-rous HMX. Several discussions with them regarding meth-odology and interpretation were integral to our effort.

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