PHYSICS OF FLUIDS VOLUME 11, NUMBER 2 FEBRUARY 1999

Two-phase modeling of deflagration-to-detonation transitionin granular materials: A critical examination of modeling issues

J. B. Bdzil,a) R. Menikoff, and S. F. SonLos Alamos National Laboratory, Los Alamos, New Mexico 87545

A. K. KapilaRensselaer Polytechnic Institute, Troy, New York 12180

D. S. StewartUniversity of Illinois at Urbana-Champaign, Urbana, Illinois 61801

~Received 16 April 1997; accepted 23 October 1998!

The two-phase mixture model developed by Baer and Nunziato~BN! to study thedeflagration-to-detonation transition~DDT! in granular explosives is critically reviewed. Thecontinuum-mixture theory foundation of the model is examined, with particular attention paid to themanner in which its constitutive functions are formulated. Connections between the mechanical andenergetic phenomena occurring at the scales of the grains, and their manifestations on the continuumaveraged scale, are explored. The nature and extent of approximations inherent in formulating theconstitutive terms, and their domain of applicability, are clarified. Deficiencies and inconsistenciesin the derivation are cited, and improvements suggested. It is emphasized that the entropy inequalityconstrains but does not uniquely determine the phase interaction terms. The resulting flexibility isexploited to suggest improved forms for the phase interactions. These improved forms better treatthe energy associated with the dynamic compaction of the bed and the single-phase limits of themodel. Companion papers of this study@Kapila et al., Phys. Fluids9, 3885~1997!; Kapila et al., inpreparation; Sonet al., in preparation# examine simpler, reduced models, in which the fine scales ofvelocity and pressure disequilibrium between the phases allow the corresponding relaxation zonesto be treated as discontinuities that need not be resolved in a numerical computation.@S1070-6631~99!02002-4#

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I. INTRODUCTION

This paper is the first in a sequence of articles onmodeling, analysis and numerical simulation of deflagratito-detonation transition~DDT! in porous energetic materialsIt represents a group effort of the authors to critically evaate and carefully reformulate two-phase continuum modthat might have an extended, physically based predictivepability for DDT. The impetus for the larger study, whicincludes experiments and numerical implementation of mels in engineering design codes, stems from a program bat the Los Alamos National Laboratory~LANL ! and aimedat the development of advanced models to assess haassociated with the accidental initiation of detonation indamaged explosive device. When damage causes the esive to become cracked or pulverized, then it is more susctible to the accidental initiation of a violent reaction andetonation.

In this study we focus on various modeling issues tarise in the construction of a two-phase theory of DDT.be sure, several attempts aimed at the development of sutheory exist in the DDT literature, and we shall refer to soof these in due course. We emphasize, however, thatpaper is not an encyclopedic survey, nor even a retros

a!Corresponding author. Electronic mail: jbb@lanl.gov

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tive, of the field. Rather, we concentrate on what mayviewed as the current state of the art, and examine it ccally, with regard to its theoretical foundations~especiallythe manner in which principles of continuum mechanics athermodynamics are employed in its development!, the ap-propriateness of the modeling assumptionsvıs-a-vıs thephysical system at hand, and the way in which experimeinformation is incorporated into the theory. Such an examnation is warranted, and indeed essential, in a problemsuch complexity, so that any weaknesses, inconsistenand deficiencies that may exist are clearly identified. Fexample, while it is widely recognized that thermodynamconstrains the development of a constitutive theory, the pcise extent of the constraint, and more importantly, thegree of flexibility that it permits, appear not to have befully appreciated or exploited by the currently available theries. In this paper we address, explore, and clarify the issWe also identify other sources of constraints, such asrequirement that a two-phase description must have anpropriate single-phase limit, and that in partitioning the dsipation between the phases one must not lose sight offact. The mechanics and energetics of the process of cpaction are examined, with an eye on the extent and accuto which these aspects of the process are included intheory. This is an important modeling issue since a propotheory must not only predict the correct structure of comp

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379Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

tion waves but also the correct distribution of compactenergy. After all, it is the compaction energy that is respsible for the process of ignition.

Even though the ultimate, practical goal is the predictof DDT in damaged materials, in this study we concentron the behavior of simpler, well-characterizable explosivi.e., granular materials with well-controlled grain sizes. Twreasons lie behind this narrower focus. First, reliable expmental information on DDT in granular materials is at hanSecond~and more important!, it is crucial that the behavioof simpler explosives be thoroughly understood, and pretive models for them developed, before further complexitassociated with defining the amount and nature of accidedamage are inserted into the theory.

The experimental information to which we have just rferred comes from the so-called DDT-tube tests,1,2 wherein abed of granular explosive, such as 30% porous HMX is sjected to a low-velocity~approximately 100 m/s! impact. Ob-servations show that detonation does not occur immediaupon impact, but rather is the culmination of a successiondistinct events.2–4 Dominant identifiable features include~i!a lead compaction wave,~ii ! a burn front,~iii ! a high-densityregion called a plug,~iv! a secondary compaction wave,~v! ashock, and finally,~vi! a detonation front. Experiments5 fur-ther reveal that~i! length scales for such processes as copaction, energy release, and heat transfer can be of the oof the grain size,~ii ! the overall scale of the experimentlarge compared to the grain size, and~iii ! the most precisehigh-resolution measurements6 of such quantities as pressuand particle velocity, representing transverse ‘‘averageover 10 to 100 grains, are very reproducible. The implicatis that the granular explosive is well behaved, in the sethat its average response is insensitive to variations ingranular microstructure that must occur from one experimto the next. This argues for a ‘‘continuum’’ hydrodynammodel of DDT in granular explosives, which managesaverage out granularity and its fine-scale effects on the flA mathematical description at the level of the grains, eveit were feasible, would be an expensive overkill.

Within the continuum context itself there exist furthoptions. There is the minimalist approach reflected inrecent work of Stewartet al.,7 where the explosive is treateessentially as a single-phase material governed by the Eequations of hydrodynamics, with porosity and reactprogress introduced as independent thermodynamic vables. These added state variables obey new evolution etions along appropriate Lagrangian paths. With adequcalibration, this approach can reproduce the observedmary and secondary compaction wave, plug, and shoformation behavior seen in the tube test. But if suchproaches are to have a wider application to a broad clasexperiments beyond the DDT-tube experiment, additiocomplexity in the form of~i! enhanced constitutive theorand ~ii ! added state variables would need to be included

We opt for a two-phase framework, for which there aseveral motivating factors. First, while the single-phaminimalist description offers a simpler modeling structurehas insufficient degrees of freedom to describe the dynaeffects of temperature, pressure, and velocity disequilibriu

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The structure of two-phase models is better able to deal wsuch issues, accepting at the same time the only data thaavailable for the material response functions, namely, thfor the solid and gas phases in isolation. The porosity vable also enters naturally in such a description. Second, tphase models of DDT have an extensive literature~see be-low! and represent a substantial investment, so that a caconsideration of both~i! the applicability of such models an~ii ! the consequences resulting from their mathematstructure, is merited. Third, two-phase models encompagreat variety of complex wave phenomena. Therefore, sudescription ~possibly with additional variables to bettemodel energy localization, dissipation, and their effect onchemical reactions! can be regarded as a superset from whto rationally derive reduced models with fewer phenomeadmitted. Viewed in this way, there is a connection toduced models that corresponds to some enhanced unstanding of the physics.

The most common two-phase treatment of the explosas a mixture is explicitly formulated in terms of variables fits two separate constituents: a granular solid phase anseparate, combustion product gas phase. For a particulucid discussion of this topic see the article entitled ‘‘theory of multiphase mixtures,’’ by Passman, Nunziato, aWalsh, which is Sec.~5C.6! of Appendix 5C in a monographby Truesdell.8 An application of this class of theories tgranular explosives is exemplified by the work of Baer aNunziato;9 also see the contemporaneous studies of Buand Krier10 and Butler,11 the earlier work of Gokhale andKrier,12 and the more recent papers of Powers, Stewart,Krier.13,14Bibliographies of these papers provide a wealthadditional references.

In such an approach, each phase is identifiable andmixture assumed immiscible. The principle of phase sepation is invoked. Each phase is separately assumed to blocal thermodynamic equilibrium and described by a densspecific internal energy, and velocity. The phases arehowever, in equilibrium with each other. The volume fration of each phase is an additional variable requiredspecify the state of the mixture. The gas volume fractionreferred to as the porosity. Each phase is characterizedseparate equation of state~EOS!. To account for theconfiguration-dependent energy of the granular solid,EOS of the solid phase depends on the volume fractionwell as its density and energy. The dynamics is determiby a system of partial differential equations~PDEs!. Eachphase separately satisfies conservation of mass, momenand energy. The nonequilibrium interactions betweenphases are described by source terms for the exchangmass, momentum, and energy between the phases.

The evolution of the volume fraction variable requirthat an additional equation be specified to affect closurethe system. For a system in which both phases are compible, Passman, Nunziato, and Walsh introduced a mowhere closure is achieved with a PDE for the volume fration. They envision a thin, interfacial region, deemed to haa microinertia and a viscosity, which controls the evolutiof the volume fraction. This equation is an attempt to mothe way in which the microstructural forces at the interfac

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380 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

act to drive the volume fraction toward equilibrium.15 Thus,phenomena associated with the discrete, granular naturthe mixture are accounted for by the EOS of the granuphase, by the form of the volume fraction equation, andthe nature of the source terms therein. The dynamical beior of a granular solid can be very different from that of tfull-density solid phase.

Derivations of the interaction source terms in the contof explosives~and indeed, for multicomponent flows, in geeral! are usually phenomenological and heuristic. A commpractice is to postulate functional forms based on obsetions and experience with simpler problems. The fundamtal tenets of continuum theory provide some constraintsthe functional forms. Alternatively, guidance can be soufrom an averaging approach,16 wherein the source termhave a form relating them to exact solutions of the micrcale equations of motion. For a two-phase mixture ogranular solid and a gas, three types of source terms aThese correspond to~i! interactions at grain–grain boundaries,~ii ! interactions at interphase boundaries, and~iii ! fluc-tuations within each phase.16 An example of the first class ithe configuration pressure, an average of the localized sat the contacts between grains resulting from matestrength. The second class is exemplified by the energychange due to heat transfer between the phases. Drag ogas phase, caused by particle–velocity fluctuations duthe tortuous motion of the gas through the granular bed, iexample of the third class.

Computing power is reaching the point at which micrmechanical computations can be performed and the moing approximation tested with numerical experiments; sfor example, the simulation of shock compaction in copppowder by Benson.17 While serious questions remain cocerning numerical resolution and the adequacy of matemodels that are used, this development renders the averaapproach for developing continuum theory more attractthan it has been heretofore.

Regardless of the approach, specification of the inphase interactions remains the essence of multiphasetinuum modeling. Where the various models differ most, ofrom the other, is in their closure assumptions and souterms. The DDT application presents a special challengecause experimental information is seldom available ovesufficiently broad range. As a result, approximations andtrapolations abound. An example is the configuration prsure, typically calibrated to quasistatic experiments butplied to predict dynamical features such as the state behicompaction wave. Another example is the assumptionthe configuration pressurebs is a function only of the vol-ume fractionfs , notwithstanding the fact that above iyield strength the solid flows, and therefore cannot suppoconfiguration pressure. Both are examples of approximatthat, although simplistic, are forgiving; the first becausestiff EOS of the solid renders the gross mechanical featuof compaction waves insensitive to the approximation, athe second because wherebs is inaccurate it is dominated bthe bulk pressure in the solid, i.e.,bs /Ps!1. In other casessimplistic approximations lead to difficulties. We show, fexample, that in the limit as the gas mass fraction goe

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zero, insufficient attention to detail in the construction of texchange terms can have the gas play too significant a rothe energetics of the dominant solid phase. Our primaryin this effort is to clearly articulate the modeling issues in tDDT application, and, in particular, to clarify the natureassumptions and approximations, and, where feasible, tconsequences through an analysis of the model.

For the purpose of this paper we take the Baer–Nunz~BN! model as a starting point.9 It aims at describing agranular explosive in which a gas phase fills the interstipores between chemically reacting solid grains. It is badirectly on the previously cited work of Passman, Nunziaand Walsh8 on granular, multiphase mixtures. The velocittemperature, and pressure of the two phases are allowebe unequal. A dissipation inequality for the mixture is employed to formulate the source terms, requiring that at epoint the mixture entropy be nondecreasing with time.drag source in the momentum equations equilibrates velties, a heat-transfer term in the energy equations equilibrtemperatures, and a relaxation equation for the volume ftion serves to equilibrate pressures. The time rate of chaof the volume fraction is associated with the advection ofsolid phase, in keeping with BN’s association of granularwith the solid phase. Due to the low-pressure matestrength of the grains, modeled by the configuration pressthe solid and gas pressures are offset bybs at equilibrium.

In contrast to the BN model, many conventional twphase fluid models assume pressure equilibrium. The volufraction is then determined by an algebraic equation ratthan an evolution equation. In some situations, the constron the volume fraction causes the PDEs for the pressequilibrium models to become elliptic and ill posed, in thsense that the PDEs are not evolutionary, i.e., with arbitrinitial data the solution is not determined for all time. Thassumption of a dynamical compaction law in the BN modon the other hand, yields a system of hyperbolic PDThese PDEs have a well-defined wave structure, withpair of acoustic modes for each phase.18 A shock in a singlephase leads to a pressure imbalance behind the shock fAt low pressures, where reactivity is unimportant, the souterm in the compaction equation provides a relaxatmechanism to equilibrate the pressures. When the magniof the compaction source term is large, a compressive wapproaches pressure equilibrium within a narrow zonethe state behind the fully or partly dispersed wave rather tthe single phase shock is of physical interest.

A critical and detailed examination of the BN modreveals certain deficiencies and inconsistencies. For examthe forms adopted for the interphase source termsheavily on the dissipation inequality for reactive, two-phamixtures. The BN source terms are only one realizationthe many possibilities that are compatible with the dissition inequality, and the BN derivation appears not to reconize this nonuniqueness. A part of the nonuniquenessvolves the distribution of energy or entropy productiobetween the phases for each dissipative process. We usenonuniqueness to advantage and show that it can be uscorrect inconsistencies and make improved modeling choin the original BN model related to compaction work and

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381Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

the single phase limit~porosity approaching 0 or 1!.Additionally, as Powerset al.13 have noted previously

the BN model does not treat in a thermodynamically content manner the assumed volume-fraction dependence osolid free energy in the development of the solid interenergy and/or entropy production. That is, the assumed ffor the differential of the solid entropy,

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with T, e, r, h, andP the temperature, specific internal eergy, density, entropy, and pressure, respectively, is incsistent with the pure phase equation of state,Ps

5Ps(rs ,es), that BN actually use for the solid. Importanconsequences follow inevitably from the inclusion offs as athermodynamic variable, namely,~i! volume-fraction depen-dence also needs to appear in the solid-phase internal en~ii ! quasistatic compaction is modeled as a reversible pcess, and~iii ! the intergranular stressbs emerges as a naturathermodynamic conjugate to the volume fraction. To haconfidence in the model over a wide range of conditions,important that the source terms are compatible with thermdynamics, and that each physical process is separately dpative. Then entropy is guaranteed to be nondecreasingevery flow. Moreover, the coefficients that characterize eprocess~such as heat conduction or drag! can be state dependent and set or empirically fit independently.

One of the weakest aspects of currently available twphase theories remains the manner in which they treat chcal reaction/combustion. This is an area requiring considable study, and although we identify fruitful directions ffurther work, no new burn model is suggested here. The bmodel of the BN theory has the notion of ‘‘hot spots’’ asmotivation~i.e., centers of rapid reaction forced by the locization of energy at many discrete locations in the granusolid!. But, in effect, it is calibrated to experiments in a limited dynamical range in the phase space. It does not limother hot-spot motivated burn models19 used to simulate HEinitiation at higher pressures and very low porosities thaPop plot data for the distance of run to detonation as a fution of pressure. The burn model does not explicitly incluthe effect of the dissipation and resulting localized grheating that occurs when the granular bed is compacteagrees with gas gun experiments on small HMX sample20

only for a very limited range of impact velocities. The dvelopment of a rate model that is predictive over a wrange of conditions is a key modeling issue in need of mfurther investigation.

In Sec. II we begin with the equations that compriseBaer–Nunziato model, and discuss the physical basis ofattributes of BN that distinguish it from conventional twphase models; nozzling and compaction. Also examithere are the fine-scale processes whose aggregate effreflected in the configuration pressurebs , the manner inwhich granularity and material strength enter the constitutheory, and the linkage of compaction-deposited energyignition. The dissipation inequality is the subject of Sec. IWe emphasize its role as a constraint on constitutive mo

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II. THE BAER–NUNZIATO MODEL

A. Postulates

The physical system envisioned in the BN model cosists of a granular bed of energetic material. The intergralar pores form an interconnected region that is occupiedthe gas phase. The model is postulated in accordancethe following principles of continuum-mixture theory.21

~i! Each phase is assigned a densityra , a specific vol-umeVa51/ra , particle velocityua , specific internal energyea , temperatureTa , and a volume fractionfa , where thesubscripta can be eithers or g, and denotes the solid and gaphase, respectively.

These variables represent the local, mesoscale matespecific average of the microscopic phase variables. Theume fractions satisfy the saturation condition,fs1fg51.The density and the volume fraction determine the mass ftion,

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where, clearly,ls1lg51.~ii ! The principle of phase separation8 holds.

We assume that on the mesoscale, each phase is inthermodynamic equilibrium and characterized by a thermdynamic potential. Phase separation, essentially the analothe axiom of equipresence8 for immiscible mixtures, thendictates that the averaged, material-specific variableseach phase~e.g., the Helmholtz free energyCa , the internalenergyea , the entropyha ,...) depend only on the independent variables of that phase~e.g., the densityra , the tem-peratureTa , shear strain, ...!. As a direct consequence of thdiscrete structure of a granular solid, we include the pobility of dependence upon volume fraction,fs , in the con-stitutive model of the solid phase.

~iii ! The properties of the mixture are weighted sumsthe properties of the constituents.

Quantities per unit volume, such as density~r! and pres-sure ~P!, are volume fraction-weighted sums of the indvidual phase quantities,

r5fsrs1fgrg , ~1!

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while quantities per unit mass, such as specific internalergy ~e! and specific entropy~h! are mass-weighted sums,

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The momentum density is volume weighted and is useddefine the mixture particle velocity,

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~iv! Material-frame indifference holds.

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382 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

The representation of the modeling PDEs and consttive functions must be independent of the observer. ForBN model this postulate is equivalent to Galilean invarian

~v! The motion of each constituent is described by bance laws for mass, momentum, and energy that are the sas those for single-phase materials.

These laws can be viewed as evolution equations forlocal, phase-averaged density, momentum, and energeach phase.16 Interaction among the constituents is describby source terms, which can depend on the independentables from both the phases.

~vi! The mixture satisfies the dissipation inequality.Each source term is constructed to model a single

change process and these processes are separately dtive.

~vii ! The modeling PDEs are evolutionary.We require that all modifications that the exchange ter

make to the basic single-phase balance laws lead to Pthat are hyperbolic, i.e., the eigenvalues for the lineariPDEs are real and the eigenvectors complete, so thatinitial-boundary value problems are well posed.22

~viii ! The volume fractionfs is carried with the solidphase.

The asymmetry between the material response ofsolid and gas phases motivates the advection of bothfs andfg512fs , referred to as the porosity, with the solid veloity. This postulate, as we shall see, leads to the appearannondissipative nozzling terms in the momentum and eneequations of each phase.

B. Governing equations

The balance laws for the model are expressed in terma system of seven PDEs,9 six arising from the conservatioof mass, momentum and energy for each phase, and theenth, an evolution equation for the volume fraction, providclosure. The governing equations are the following: Consvation of mass,

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compaction dynamics,

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2/2 is the total specific energyThe right-hand sides of Eqs.~7!–~12! are source terms thacharacterize phase interaction. These terms corresponmass~C !, momentum~M!, and energy~E! transfer betweenthe phases. The mass sourceC is negative when the solidburns and is converted to gas. The total mass, total momtum, and total energy are all conserved because the soterms for the gas and solid phases sum to zero. As we slater, the right-hand side of Eq.~13! describes the rate ocompaction. The eigenvalues of this system of equationsall real. Except for isolated surfaces in the state space,linearly nondegenerate characteristic velocities are uneand the eigenvectors are complete so that, effectively,PDEs are hyperbolic.18

Some general remarks about the governing equatioand the quantities appearing therein, are in order. In the ctinuum approximation the phases coexist at each point.individual phase variables can be viewed as local, mesosaverages over a subvolume that is large enough to contasufficient number of grains so that the local volume averais statistically meaningful. Underlying this view are two implicit assumptions. First, that self-equilibration of each phaoccurs at a rate much faster than that at which the phaseek equilibrium with each other. Second, that the ageforcing changes in the system, such as pressure gradithemselves vary on the mesoscale rather than the grain sThe formal averaging approach, long advanced by Drew16

provides some justification for this view, as we now arguLet ^ & l5*( )dV/Vl be the l-scale volume-averaging

symbol, wherel denotes the continuum or mesoscale. Givan indicator functionXa[$0,1%, whereXa51 in regions ofspace occupied by materiala and zero otherwise, then^Xa& l5fa . Upon averaging, the standard, microscale coservation equations yield their ‘‘averaged’’ or weightecounterparts that parallel the standard pure-phequations.16 Unlike the latter, however, the averaged equtions contain source terms that reflect the fact that for a nlinear function, ^ f (x,y,...)& lÞ f (^x& l ,^y& l ,...). Thesesources represent interactions both between the phaseswithin each phase, appearing as~a! forces and fluxes that acacross the interfacial boundaries and~b! averages of productsof fluctuations from the mean fields within each phase.

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383Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

example of the former are the forces established witgrains due to intergrain contacts and of the latter, intraphpressure fluctuations.

Here we focus on an issue not addressed by Drew; hthe nonlinear constitutive model forea comes through theaveraging when the forces driving changes on thel scale aresimple compressions. We consider that on the microscaleresponse of the solid,es(e% ,T;ak) is complex; acting ther-moelastically at low pressures and plastically at high prsures. In the above,e% is the strain tensor and theak areadded~hidden! internal state variables that are needed toscribe the transition from an elastic to inelastic respons23

Three time scales enter the problem: a short, fundameacoustic timetc based on the grain dimension, a longer evlution time t l@tc based on thel scale, and a process timscaleta@tc that characterizes the rates of change of the hden variablesak . Here we focus on the limitta.t l .

At first glance, the nonlinear dependence of the micrcale, pure-phase solid EOS functiones on e% , T, and ak

would result in ^Xsres(e% ,T;ak)& lÞ^Xsr& l es(^e% & l ,^T& l ;^ak& l). Given tc!t l and the smoothness of the applieforces on thel scale, the fluctuations in the strain and teperature fields within each grain are short lived, while theak

are frozen sinceta.t l . Therefore, the fields evolve quasisteadily on thel scale. When the grains are isolated, thleads to uniform fields and a bulk response within graisince the load they experience is hydrostatic. Th^Xsres(e% ,T;ak)& l5^Xsr& l es(^r& l ,^T& l ;^ak& l), where ^e% & l

5^r& l , since the average shear strain^S(e% )& l is zero for ahydrostatic load. When the grains are in contact, the fieldthe grains, although quasisteady, are nonuniform due tolocalized nature of the grain-to-grain contacts. Then theerage shear strainS(e% )& lÞ0, so that ^Xsres(e% ,T;ak)& l

Þ^Xsr& l es(^r& l ,^T& l ;^ak& l). Since^S(e% )& l is not a variablein this continuum theory, to close the model we approxim^Xsres(e% ,T;ak)& l simply as esp(^r& l ,^T& l ;^ak& l)1g(fs ,...;^ak& l), where esp is the mesoscale hydrostatcontribution, andg(fs ,...;^ak& l) represents, in an averagway, the effects on the mesoscale internal energy of thecroscale shear strainS(e% )& l ~which is related to the degreof compaction,fs). Similar arguments for the gas phayield ^Xgreg(r,T)& l5^Xgr& leg(^r& l ,^T& l). Thus, the mi-croscale material response to simple compressions is usthe mesoscale EOS, with an added term to reflect the effof the averaged localized strains.

Next, we turn to a related quantity that appears intheory as a result of the grain-to-grain interaction. Intergrcontacts produce strains that lead to contact stresses,played in Fig. 1, as gray shaded regions, with the darkgray indicating the highest stress. These stresses are main the BN theory as a continuum-level configuration presure, or intergranular stress,

bs~fs,•••;^ak& l !.

Dependence onfs is natural, as already remarked. The sophase experiences both the pressure carried by the gasnoted by arrows in Fig. 1, and the intergranular forces.mechanical equilibrium,Ps5Pg1bs.

nse

w

he

-

-

tal-

-

-

-

,,

inhev-

e

i-

asts

enis-stfest-

de-t

The configuration pressure andg(fs ,...;^ak& l) play asignificant role at low pressure when the solid phase is srelative to the gas. The pressure above which matestrength becomes unimportant is referred to as the cruspressure and is of the order of the yield strength; atcrush-up pressure the pores in a granular bed are nesqueezed out as the solid becomes fluid-like. A porous begranular HMX achieves 1% porosity forbs'5 kbar; we saythat the crush-up pressure for HMX is about 5 kbar.

We begin a discussion of the compaction equationobserving that in the BN model, the volume fractionfs

serves multiple purposes. Besides being a measure ofstrain induced in the grains, it is also the variable thatscribes the microstructure of the interface separatingphases. The theory of Passman, Nunziato, and Walsh,8 onwhich the BN model is based, postulates an evolution eqtion for the microstructure. Such concepts as ‘‘microstrutural inertia’’ and ‘‘microstructural force’’ are introduced tmodel the evolution offs with a second-order PDE of thform

~microinertia!•d2fs

dts2 1~viscosity!•

dfs

dts

5~microstructural forces!, ~14!

whered/dts is the convective derivative with respect to thsolid. In spirit, this equation is similar to the pore-collapmodel of Carroll and Holt,24 although its development wacouched in a ‘‘Rational Mechanics’’-type formalism.8,25 Intheir model, Baer and Nunziato,9 in effect, set the microin-ertia to zero and obtain closure by simply postulatingexistence of a constitutive expression relatingdfs /dts to theindependent variables of both phases,

dfs

dts5F ~rs ,rg ,Ps ,Pg ,fs ,...!, ~15!

FIG. 1. Schematic showing the forces acting on the HMX grains in a wcompaction wave. The variable gray shaded areas represent thestresses~Ref. 52!, while the arrows indicate the gas pressure. The unstresregions of the HMX crystals are shown as light gray. The solid gra‘‘feel’’ both of these forces.

alis

on

y

ouonin,e

o

rtl-

e

sf

ef

stb

rm

ilomq.or

ilib

laum-ili

ane

le,ac-

ofit,

ser-lidhe

N

rm,

s aflowurespo-

pac-t inof

forw.

eges

re

esn-

ture

en-

lidifi-

ro-vel.rker

384 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

where F is meant to mimic the effects of microstructurforces that, on the continuum level, describe the ‘‘restance’’ exhibited by the bed to changes in its configuratiThe form that BN derive is

F 5fsfg

mc~Ps2bs2Pg!, ~16!

where the parametermc is called the compaction viscositdue to the analogy with Eq.~14!.

Turning now to the source terms, we have separatedfrom the momentum and energy transfer rates, contributithat were developed by BN. These include the nozzlterms, proportional toPN , and the compaction-work termsproportional toPc . BN take the nozzling pressure to be thgas pressure, i.e.,

PN5Pg , ~17!

and the compaction pressure to be the difference of the spressure and the configuration pressure, i.e.,

Pc5Ps2bs. ~18!

We shall examine both of these terms in some detail shoThe residual momentum exchangeM and the residual energy exchangeE used by BN are

M5C us1~d1 12C !~ug2us!, ~19!

E5C ~es112us

2!1~d1 12C !us~ug2us!1H~Tg2Ts!.

~20!

Equation~19! is the BN form for the momentum exchangassociated with burning and drag, while Eq.~20! describesthe energy exchange due to burning, drag, and heat tranThe coefficientsd and H appearing in~19! and ~20! are,respectively, the interphase drag and heat transfer cocients.

In the remainder of this section we examine the contutive modeling of the phase-specific quantities. We startbriefly discussing the compaction law and the nozzling te

C. Compaction and nozzling

In contrast to typical two-phase fluid models,26 the BNmodel does not require the phases to be in pressure equrium. Instead, pressures are related dynamically via the cpaction law, Eq.~13!. The conservation of solid mass, E~7!, allows the compaction equation to be cast in a mstandard form as

dfs

dts2

C

rs52S ]rs

]t1

]~rsus!

]x D fs

rs5F . ~21!

Here, fs advects with the solid phase, and the sourceF

measures the rate of compaction. Without burn, the equrium value offs is given byF 50 and corresponds toPs

5Pg1bs. Thus, the compaction equation serves as a reation law that drives the phases toward pressure equilibriwith the compaction viscositymc characterizing the relaxation rate. A favorable consequence of pressure nonequ

-.

t,s

g

lid

y.

er.

fi-

i-y.

ib--

e

-

x-,

b-

rium is that the model equations are hyperbolic, rather thof the mixed hyperbolic–elliptic variety typical of two-phaspressure–equilibrium models.26

At low pressures the grains are nearly incompressibi.e., rs'const. Then, for a nonreactive system, the comption law reduces to

dfs

dts52fs

]us

]x, ~22!

and the solid pressure is determined by2fs]us /]x5F .Equation~22! also corresponds to the incompressible limitthe equation for mass conservation of the solid; in this limtherefore, compaction is simply a statement of mass convation, i.e., compaction is due to the relative solid-to-somotion which reduces the volume of the pores within tbed.

The asymmetric form of compaction dynamics in the Bmodel ~biased toward the solid!, along with the dissipationinequality, forces the appearance of the nozzling tePN]fs /]x, in the momentum and energy equations~see Sec.III !. The change in porosity on the continuum scale acts anozzle that can either accelerate or decelerate the gasand drive relative motion between the phases. At pressbelow the crush-up pressure of the granular bed, such arosity gradient can be set up, for example, across a comtion wave. Figure 2 shows the manner in which a gradienporosity ~produced by the gradient in the number densitygrains, with a higher grain density indicated as darker! actson the continuum scale like a nozzle. Nozzling accountsthe average effect of the solid interface on the gas floWhen the pressures are equal,PN5Pg5Ps , the nozzlingterm corresponds to thePdV work one phase does on thother when the internal phase interface varies and chanthe volume fraction.

Nozzling terms are included in some two-phase mixtumodels16 and discarded in others.13 They prevent the BNequations from being put in a conservative form. This donot cause any problems, provided the initial porosity is cotinuous, since the PDEs have a well-defined wave structhat maintains continuity offs .18 In Sec. III B we show thatthe nozzling term is required in order to guarantee thattropy is nondecreasing for all flows.

Nozzling plays a role at low pressure when the sophase is stiff relative to the gas, and for it to have a sign

FIG. 2. The variation in the local number density of grains on the micstructural level appears as a variation of porosity on the continuum leThe variation in porosity acts as a nozzle applied to the gas flow. Dashading indicates a higher density of particles.

ighmows

Dplve

tie-

gsithIn

o

cogof

accnslur

r.rehlieikedria

c

omiedol-on-for

sfy

to

port

ngheof aon-rge

s canieldro-ctsezednaling

irre-in

.esro-

ular

me-

i-wl-a

thein-iveicaledted. In

385Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

cant effect on the gas flow, drag must be small. At hpressure the solid becomes more fluid-like, drag becolarge, and nozzling diminishes in importance. Even at lpressures, when the ambient mass fraction of the gasmall, the solid EOS and the configuration pressurebs aresufficient to predict low-pressure compaction waves ingranular bed, that is, the lead compaction wave in a Dexperiment is insensitive to the nozzling terms. An examof the effect of nozzling on a piston-driven compaction wais shown in Ref. 27.

D. Equilibrium thermodynamic response

As discussed above, the BN theory9 treats the explosiveas a mixture of phase-separated substances.8 Constitutive re-lations must be provided for the phase-specific quantisuch as the Helmholtz free energyCa , and for the phaseinteraction or exchange terms such asC , M, andE . In thissection, we concentrate on expressions for the averaphase-specific, thermodynamic response. These expresdescribe how each phase interacts with itself, includingways in which granules interact across their interfaces.traphase response such as friction and viscosity could becluded in the interphase response, although the BN theomits such processes.

The material-specific phase response functions arestructed according to the postulates enunciated at the bening of this section, including, in particular, the principlesmaterial frame indifference and phase separation. Thus,Ca

is taken to depend on the independent variablesra , Ta ,fa ,..., where the dependence on the volume fractioncounts for the energy associated with the intergrain conta

Continuum mixture theory has been applied to matypes of materials,28,29 including solids for which the stresand strain tensors are important variables. The BN modeaimed at describing a granular bed. In contrast to a psolid, large shear strains do not necessarily give rise to lashear stresses, because grains can slide past each othemotivates replacing the stress tensor by a hydrostatic psure and the strain tensor by a density. Material strengtmodeled with the configuration pressure introduced earand discussed below. Thus, the BN model is a fluid-lmodel, and the set of independent variables is restricteSa[(ra ,Ta ,fa). Since all the variables are scalars, mateframe indifference reduces to Galilean invariance.

With the variable listSa , the derived thermodynamivariables are the entropy,

ha52S ]Ca

]TaD , ~23!

the internal energy,

ea5Ca1Taha , ~24!

the pressure,

Pa5ra2S ]Ca

]raD , ~25!

and the intergranular stress or configuration pressure,

es

is

aTe

s

edonse-

in-ry

n-in-

-ts.y

isregeThiss-isr

tol

ba5faraS ]Ca

]faD . ~26!

Since a gas has no material strength,bg is zero, i.e., the freeenergy of the gas has the pure single phase formCg(rg ,Tg).

We start by exploring the consequences of thefs depen-dence. The volume fraction is an internal degree of freedthat can be externally set only indirectly through an applpressure. It is natural to inquire about the equilibrium vume fraction for a porous solid according to the above cstitutive model, as determined by maximizing the entropya fixed solid mass, energy, and volume, i.e., fores and r5fsrs constant. The solid thermodynamic variables satithe differential relation

des5Ps

rs2 drs1Tsdhs1

bs

fsrsdfs . ~27!

Since the constraints imply thatdes50 and drs /rs

52dfs /fs , it follows that at equilibrium,Ps5bs . Thus, adirect consequence of thefs dependence ofCs is the emer-gence ofbs to describe the resistance of a granular bedcompaction under an applied pressurePs . Since the grainsare not bonded together, the granular bed cannot suptension and the configuration pressure must be positive.

We now turn to the question of consistently constitutithe functional form for the equilibrium response. Due to teffects associated with material strength, the responsegranular bed is complex. As already observed, the small ctact surfaces over which the grains interact can lead to lastress concentrations, even under moderate loads. Grainfracture, and when the local stresses exceed the ystrength, deform plastically. The resulting changes in micstructure affect the distribution of grain sizes and contaamong grains. As a result, small grains can then be squethrough pores between larger grains, giving rise to frictioheating. Thus, in addition to hydrostatic pressure arisfrom density changes, contact stresses may lead to~i! elasticdeformations,~ii ! fracture,~iii ! intergranular friction, and~iv!plastic deformation. Since some of these processes areversible, they lead to hysteresis effects or different pathsstate space when a granular bed is loaded and unloaded

The relative mix of reversible and irreversible processis material dependent. For example, compaction of ball ppellants is dominated by~i! rearrangement,~ii ! elastic load-ing, and, finally,~iii ! plastic deformation.30 Little or no frac-ture is observed. On the other hand, compaction of granHMX is dominated by fracture.31 This complexity of mate-rial response poses a significant challenge to the microchanical modeler.

In principle, given a sufficiently capable micromechancal code, physically accurate material models and a knoedge of the local microstructure, the fields derived frommicromechanical simulation, could be averaged to obtaincontinuum scale response function. Though valid in prciple, this approach is impractical, at least in a quantitatsense, given the present state of the art. A semiempirconstruction, guided by qualitative information gatherfrom averaging, and buttressed by a judiciously selecsuite of calibration experiments, is more likely to succeed

itdstrasla

n,isailic

ica

all-

het

ovof

lr

is

e

ten

theo.ten,nt

the

andatic,

atrnaless,ts.

en-nd

ange

the

solidre-

k to

N,

sti-of

tedex-

elyor ag

ofllmi-S,

386 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

either case, additional variables besidesfs are sure to beneeded to characterize the material state in all its complexalong with evolutionary equations for these variables. Moels more like those used to describe plastic or viscoelabehavior, that express stress rate in terms of the straincould lead to improved descriptions. Such extended contutive models are currently under development for granuexplosive materials; see Gonthieret al.32,33

Here we follow BN and continue with the prescriptiowhere fs is the only additional state variable in whatotherwise a fluid equation of state, keeping in mind thwhile such a continuum model cannot capture all the detaupon proper calibration it can describe states of mechanequilibrium and the gross energetics for a class of appltions. We seek to calibrateCs(rs ,Ts ,fs) to data on com-pressive loading of granular beds.34 In these experiments,uniform load is applied slowly to a thermally isolated, wecharacterized bed of granular HMX with little gas~ambientair!. This results in a uniform, quasistatic compaction of tbed, and serves to measure how the internal energy ofbed increases on compaction, as we now show.

The control variable in the experiments described abis the volume occupied by the porous HMX. A platen,areaA and initially a distanceL0 from the far side of thesample cell, is moved slowly into the bed so as to uniformcompress the bed. Since the pressure required to compthe bed is sufficiently low, the volumens of the solid grainsremains roughly constant andrs5(ms/ns)5rs

0, where the

solid volume fraction isfs5(ns/A•L). The bed remainsuniform due to the slowness of compaction, so that

dfs

dt52

Afs2•~us!L

ns, ~28!

where (us)L5dL/dt is the platen velocity. Since the solidthe only component, the solid energy equation, Eq.~11!, freeof source terms, can be integrated to get the change in enof the material,

E0

L ]~fsrsEs!

]tAdx1@fsus~rsEs1Ps!#LA50. ~29!

Bringing the time derivative through the integral

E0

L ]~fsrsEs!

]tAdx

5d

dt E0

L

~fsrsEs!Adx2us•~fsrsEs!LA,

and usingd/dt5(dfs /dt)d/dfs , leads to

d

dfsE

0

L

~fsrsEs!dx5ns•~Ps!L

Afs. ~30!

This can be integrated to obtain

es2es~rs0,fs

0!5Efs

0

fs ~Ps!L

fsrsdfs5B, ~31!

where the kinetic energy of compaction has been neglecSincers is essentially unchanged in the loading experime

y,-icte,ti-r

ts,al-

he

e

yess

rgy

d.t,

and (Ps)L is observed to principally depend onfs, it isreasonable to treatB as a function offs alone. We canidentify (Ps)L with bs, i.e.,

~Ps!L5bs5fsrs

dB~fs!

dfs.

The quantitybs is the intergranular stress, as measured inquasistatic compaction experiments of Elban and Chiarit34

~These experiments measure the applied force on the plaF5Afsbs , which needs to be reduced by 40% to accoufor the load carried by friction at the mold wall.! We excludethe possibility of a dependence ofbs on Ts ; more due to alack of information. Although, asTs is increased sufficiently,we expect that thermal softening can strongly influencemeasured (Ps)L .

We identify the change in energyB(fs), given in Eq.~31!, with the compaction energy,

B~fs!5Efs

0

fs bs

fsrsdfs . ~32!

With the Ts dependence ofbs neglected andrs'const,B isthe change in the internal energy of the bed on loading,describes phenomenologically the low-pressure, quasistcompressive loading of the bed.

Given this change in the energy of the bed, we areliberty as to how to represent this as a change in the inteenergy of the solid. Here, following the BN formalism, welect to associate all of this energy with a reversible procesuch as elastic deformation in the vicinity of grain contacThat would imply that neither the temperature nor the dsity of the solid grains had changed during compaction, athe internal energy would have the formes(rs ,Ts ,fs), withfs as an added internal-state variable to measure the chin energy. With this ansatz, the intergranular stress,bs has athermodynamic definition as the force conjugate tointernal-state variablefs . From Eq.~27! it follows that forthe assumed constant density process, the change inentropy for a quasistatic process would be zero with theversible ansatz.

With fs adopted as an internal-state variable, we seedevelop a consistent theory that treatsB as an approximationto the free energy of the bed for low-pressure loading. In Bthe intergranular stress derived fromB is the only way inwhich material strength and granularity enter into the contutive theory of the solid. The fits to the experimental dataElban and Chiarito for HMX are shown in Fig. 3. We nothat bs is relatively insensitive to the grain distribution anpacking of the bed, except near ambient pressure. Theperiments also indicate that a single function approximatdescribes the progressive crush-up of the bed, even floading cycle that involves unloading followed by reloadinduring the cycle.

As the pressure increases beyond the yield strengthsolid HMX, then bs/Ps!1 and B represents only a smapart of the total energy. Then the material response is donated by the fluid-like, continuous pure-phase solid EOCsp(rs ,Ts). The composite EOS,

Cs~rs ,Ts ,fs!5Csp~rs ,Ts!1B~fs!, ~33!

anheith

panpa

bf1

tha

ela

acthre

M

rl

ar

th

incees,

ant

he

ssnoef-ge-ifi-

andtoentoisr-

m-

of

,en-

g

head-alliblergy

an

.gy

387Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

captures both the low- and high-pressure response of grlar HMX to compressive loading, and is consistent with tthermodynamic description given earlier in this section, w

es5esp1B, ~34!

hs5hsp~rs ,esp!, ~35!

Ps5Psp~rs ,esp!, ~36!

esp5Csp1Tshs . ~37!

In particular, in this description there is no additional entroassociated with the generation of grain surfaces, friction,plastic deformation that can accompany quasistatic comtion.

The total energy supplied to compact the bed cancomputed using the data of Fig. 3 and the expressionB(fs). For a typical quasistatic experiment it is of order3104 J/kg. Here we give some estimates of the energycould be associated with two processes described abovethat can be expected to occur during compaction~one revers-ible and the other irreversible!: ~i! elastic deformation and~ii ! fracture. Based on the low values of these energies rtive to the compaction energy, we argue that for HMXsignificant fraction of the compaction energy can becounted for by the intergranular friction that accompaniesrearrangement of the fractured grains. Given the highly irversible nature of this process, we examine data on Hgathered by Coyne, Elban, and Chiarito31 that shows the ex-tent of the irreversibility.

The elastic shear-strain energy for an isotropic lineaelastic material isY2/(6Gr0), whereY is the yield strengthandG is the shear modulus. For HMX, the maximum shestrain energy is 83102 J/kg, based onY50.3 GPa andG510 GPa. This is an order of magnitude smaller than

FIG. 3. Quasistatic compression data for a porous HMX bed by ElbanChiarito ~Ref. 34!. Plotted are the fits to their Exps.~8!, ~9!, ~11!, and~17!.Also shown is the fit to the analytic expressionbs52t•(fs2fs

0)• ln(fg)/fg , usingt51.2731022 GPa andfs

0'0.7.

u-

ydc-

eor

atnd

a-

-e-

X

y

-

e

compaction energy. Given that this is an overestimate, sonly a small fraction of each grain experiences these forcit is unlikely that shear stress is an energetically importmechanism.

The energy associated directly with fracture is of torder of 2 J/kg. This is based on the estimate of 0.05 J/m2 forsurface energy of HMX35 and the surface area per unit mafor grains 0.1 mm in diameter. Although fracture hasdirect influence on the HMX compaction energetics, thefects of intergranular friction that accompany the rearranment of fractured grains can impact the energetics signcantly. As mentioned above, the experiments of ElbanChiarito34 reveal that upward of 40% of the load appliedcompact a bed on the platen is carried by friction betwethe HMX grains and the smooth walls of the cylinder usedconfine the bed. All of the above argue that compactionhighly irreversible and that plastic deformation and integranular friction play important roles in the quasistatic copaction of granular HMX.

The experiments of Coyne, Elban and Chiarito31 show ahysteresis effect for the loading and unloading responseHMX. A representativefsbs vs 1/fs diagram is shown inFig. 4, for a sample initially atfs

050.73. In these variablesthe area under the curve corresponds to the compactionergy, Eq. ~32!. Also shown in Fig. 4 are two unloadincurves~corresponding to release isentropes! from states ofcompactionfs50.95 andfs50.85 and labeledA and B,respectively. A measure of the degree of reversibility of tcompaction process is the ratio of the area below the unloing curve to the area below the loading curve. The smratio implies that the energy dissipated as heat by irreversprocesses is a large fraction of the total compaction ene

dFIG. 4. The quasistatic compaction curve,bs52t•(fs2fs

0)• ln(fg)/fg fora granular HMX sample initially atfs

050.73, wheret512.7 MPa, isshown. Two release isentropes~unloading curves!, based on the data in Ref31 and denoted asA and B, are also shown. They reveal that the enerrecovered from the samples on unloading fromfs50.95 andfs50.85 isvery small~the shaded area!. The model treats all the energy under thefsbs

curve as being recoverable.

es

-paey

c.

yre

eti

t

e

llin

til

onngitn

acw

thofof tli

f tb

iket

oenbeanelytu

ith

rib

dur-itys-andter,

edside-gh-etyore,me-

in

erin amt the

nd

on-s,

med

de-ra-se..rnsofe-

l

ndem-

388 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

for HMX. These results are compatible with the low valuwe obtained for the energy estimates~given above!.

Alternatively, if we deem that compaction is fully irreversible, and that the entire energy associated with comtion is dissipated, then we would need to associate thisergy with a change inTs , and constitute the internal energas simplyes(rs ,Ts). We discussed this possibility in SeII B, with the introduction of the hidden EOS variablesak .From Eq. ~27!, it follows that the change in solid entropwould be positive if compaction were assumed to be irversible. This would correspond to the limitta,t l . Wecould, of course, assign a fraction ofB to reversible pro-cesses and the remainder to irreversible ones. Improvemin the modeling are needed to capture this added dissipaalong the lines of elastic–plastic theory.32,33

Because of its historical interest, we relate our resultsthose of the ‘‘P2a ’’ model of Herrmann36 and Carroll andHolt.37 When no gas is present, the pressure of the mixturthe P2a model is given by

P5fsPsp~r/fs ,esp!,

which is equivalent to Eq.~36!. However,es5esp in the P2a model, so the energy of quasistatic compaction is fudissipated. Sincefs is assumed to equilibrate instantlyresponse to the applied pressure@i.e., fs(P,rs5rs

0)#, com-paction waves are not dispersive, and no added dissipa~over and above that expected for a shock! enters the modedue to dynamic processes such as compaction waves.

For our modified BN model, the theoretical descriptiof B(fs) does not account for any of the dissipative heatito be sure. However, it is only the dissipation associated wquasisteady compaction that is absent. Since compactiomodeled as a dynamic process in this description, Eq.~21!,dissipation due to the nonequilibrium nature of the comption process is indeed allowed and can be significant, asshow in the following section.

E. Compaction energy and ignition

We now examine the energetics of compaction andmanner in which compaction is linked to the initiationcombustion in a loosely packed bed. During the passagecompaction wave, a substantial decrease in the volume obed results in work being done on the bed. With only a sopresent, this leads to an increase in the internal energy osolid grains that is significantly greater than that produceda shock of comparable strength in a full-density solid. Unlquasistatic compaction examined in the previous section,compaction wave is a dynamic, irreversible process, andalways generates entropy. For the low-velocity impactsgranular HMX explosives that are considered likely accidscenarios, the pressures induced in the bed are seldom athe crush-up pressure of the bed. Thus, although the incrin internal energy in the granular solid exceeds that obtaiin a full-density solid, the increase is still not particularlarge, and the corresponding rise in the bulk temperaalone is insufficient to initiate combustion. The increaseinternal energy also comes from a variety of sources, andputs an added burden on the modeler to faithfully desc

c-n-

-

ntson

o

in

y

on

,his

-e

e

f ahedhey

hesontovesed

renise

the various dissipation mechanisms that can be presenting the compaction of a granular bed. For the high-velocimpacts, more typical of prompt shock initiation, the presures and consequently the work done in compactingheating the bed are one to two orders of magnitude grealeading to the bulk solid temperature being many hundrof degrees higher. This leads to the onset of rapid and wspread combustion in the bulk of the solid grains. Such hivelocity impacts are not of interest for the explosive safproblem that motivates this study. One concludes, therefthat for granularity to play a role in low-speed impacts, soform of energy localization or ‘‘hot-spot’’ formation is required to reflect the combustion observed experimentallysuch situations.

To illustrate some of the issues involved, we considthe propagation of a one-dimensional compaction wavegas-free (rg50) granular bed. To have all the momentuand energy generated by the compaction wave directed asolid, we setPN5Pc5Pg50, d50, andC 50 in the model.In this gasless limit, the applied or mixture pressure amixture density are, respectively,

P5fsPs and r5fsrs .

The jump conditions across such a wave traveling at a cstant speedD are obtained from the conservation of masmomentum, and energy, as

r~us2D !52r0D,

P2P05r0Dus , ~38!

esp2esp0 1B2B05

1

2 S 1

r021

r D ~P1P0!.

In addition, the state behind the compaction wave is assuto be in mechanical equilibrium, i.e.,

Ps5bs .

Two constitutive expressions are needed to completelyfine the compaction wave: a prescription for the configution pressurebs , and an equation of state for the solid phaBoth are based on HMX data gathered from experiments

In order to quantify the compaction energetics, one tuto the Hugoniot condition for energy, the third equationset~38!. First, as is appropriate for weak waves, it is convnient to replaceesp2esp

0 by its linearization about the initiastate,

esp2esp0 5A•S Ps2Ps

0

Psc D 1B•S rs

0

rs21D 1¯ . ~39!

Here, pressure is scaled in units of the crush-up pressurePsc ,

and

A[Ps

c

rs0Gs

0 , B[~cs

0!2

Gs0 2

Ps0

rs0 ,

where Gs0 and cs

0 are the Gruneisen parameter and souspeed at ambient conditions, respectively. Second, weploy the decomposition,

go

lit

o

lmen

o

c-

ndf

e

udis

thopgeth

ith

ndionture

re..,

389Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

1

r0 S 12r0

r D5S 1

rs02

1

rsD 1

fs1

1

fs0rs

0 ~fs2fs0!

1

fs.

Then, with the initial pressure neglected, the energy Huniot in Eq. ~38! takes the form

A•S Ps

PscD 1B•S rs

0

rs21D 1B2B0

¯

5Ps

2rs0 S 12

rs0

rsD 1

Ps

2rs0 S fs

fs021D . ~40!

The work done on the solid by both compressing the sograins and compacting the bed are clearly separated onright-hand side, being represented by the first and secterms, respectively.

Our estimates on the response of HMX use the Heholtz free energy EOS for pure solid HMX that has becalibrated by Gustavsen and Sheffield.20 For this EOS, thecoefficientB is given by

B[CvsTs0Gs

01Ks

0

rs0Gs

0 ,

where the constants for HMX at the ambient temperatureTs

05300 K and Ps050 are the specific heat,Cvs51.05

31026 GPa m3/~kg K!, the isothermal bulk modulus,Ks0

512.9 GPa,rs0Gs

052090 kg/m3, andrs051900 kg/m3. With

Psc50.5 GPa, we obtainA52.431024 GPa m3/kg and B

56.531023 GPa m3/kg. Furthermore, forPs<Psc , B is, at

most, 8.2731026 GPa m3/kg and Ps /(2rs0) no larger than

1.3231024 GPa m3/kg. Thus, A, B, and Ps /(2rs0) are at

least an order of magnitude smaller thanB. Since Ps /Psc

and (fs2fs0) are both of order unity for the weak compa

tion waves under consideration, Eq.~40! implies that, toleading order,

12rs

0

rs5OS A

BD!1.

Consequently, below the crush-up pressure, the jump cotions for a compaction wavedo notdepend on the details othe energy equation~sincers'rs

0), and are given simply by

fs~us2D !52fs0D,

fsPs5fs0rs

0Dus , ~41!

Ps5bs~fs,rs5rs0!.

The important point here is that the lead compaction wavinsensitive to the energetics of the flow,38 any uncertaintywith regard to the appropriate form for the EOS for a granlar solid ~e.g., should an explicitfs dependence be includeor not! has little effect on the lead compaction wave. Thinsensitivity is displayed in theP2V Hugoniot diagram ofFig. 5~a!.

On the other hand, if we are interested in modelingignition of reaction in a granular bed, then the form we adfor the solid EOS will play an important role. Predictinignition, after all, is the goal of our program. In view of thorder-of-magnitude argument given above, it is clear thatwork associated with compaction of the bed,

-

dhend

-

f

i-

is

-

et

e

Ps

2rs0 S fs

fs021D ,

is typically substantially larger than the work associated wcompression,

Ps

2rs0 S 12

rs0

rsD .

Assuming that the work of compression is negligible, athat all the work associated with change of volume fractgoes to heat the solid, the increase in the solid temperadue to compaction would be

DTs5Ps

2rs0Cvs

S fs

fs021D ,

FIG. 5. Hugoniot loci~a! in theP–V plane and~b! in theTs2P plane. Thedotted line is the locus for a nonporous~pure! solid. The other three loci arefor a porous solid withfs

050.73. Whether compaction energy~the B inte-gral! is included in the jump condition has little influence on the pressuThe short dashed line neglects, in addition, variations in density, i.er5fsrs

0.

oeis

e

rg

ioth

momtoPe

tdu

-low.is-

o-enten-tial

ownot

itysing

e.isne

ate

thetion

e a

are

outm-edains,

onk

seionwe

390 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

which for granular HMX withfs050.73 andfs51, yields

DTs

Ps593 K/GPa. ~42!

Of course, the work associated with change in porosity dnot all go into heating. As discussed previously, in thmodel an amount of energyB(fs) is stored as recoverablcompaction energy, so that, at most, only the portion

Ps

2rsS fs

fs021D 2B~fs!

is available to heat the HMX. We call this dissipated enecompaction work. Figure 5~b! shows, on theT2P Hugoniotdiagram, the effect of including or ignoringB on heating.The ratio of compaction work toB as a function offs isdisplayed in Fig. 6, for two initial porosities (fs

050.65 andfs

050.73). Zero initially~i.e., whenfs5fs0), the ratio in-

creases withfs , becoming of the order of 2 or 3 whenfs

50.95. For strong waves~greater than 10 GPa pressure! theratio is large, reflecting the fact that dissipative compactwork dominates the static compaction energy. On the ohand, for waves of about 0.5 GPa,B is a significant fractionof the compaction work, and our assumption ofB as a re-versible energy with no dissipation to thermalize the copaction energy leads to a lower temperature behind a cpaction wave. One must emphasize, however, that thecompaction energy amounts to only 8.3 J/g for a 0.5 Gwave, and based on the specific heat, can affect the tempture by about 9 K. Since at lower pressures the amounenergy available is small, undercounting of the heating

FIG. 6. The ratio of the compaction work to theB integral ~i.e., @(Ps/2rs0)

3(fs /fs021)2B#/B) versus the inverse of the final state of compacti

(1/fs) is shown for two representative cases. The compaction worroughly two to three times theB integral atfs50.95. Since only the com-paction work measures the dissipated energy in this model, the premodel underestimates the total ‘‘frictional’’ heating available for reactignition by 25%–33% for these examples. Larger errors are made for lopressure waves.

s

y

ner

--

talara-ofe

to the irreversibility is important, particularly when the compaction energy is localized in hot spots, as discussed beThe dissipation that occurs within a compaction wave is dcussed further in Sec. IV B.

As a further aid in understanding the role played by prosity in the heating of the solid grains, and the consequenhancement of ignition sensitivity, we estimate the depdence of the time to explosion on shock strength and iniporosity. We use a simple Arrhenius rate law,

Rate5~12ls!Z expS 2T‡

TsD ,

where Z is the exponential prefactor andT‡ the activationtemperature. The parameters for HMX areZ5531019s21

and T‡52.653104 K.39 For a given shock pressureP, theporosity is computed fromP5fsbs(fs,rs5rs

0), the densityrs from the energy Hugoniot and temperature,Ts from thethermal EOS, all using Sheffield’s calibration for HMX,

Ps~rs ,esp!5Ks

0

N H S F rs

rs0GN

21D2

Gs0

N21 S F rs

rs0GN21

21D 1Gs0S 12

rs0

rsD J

1Gs0rs

0Fesp2esp0 1CvsTs

0Gs0S 12

rs0

rsD G ,

~43!

Ts~rs ,esp!5Ts01

1

rs0CvsGs

0

3H Ps~rs ,esp!2Ks

0

N F S rs

rs0D N

21G J , ~44!

where N510.3. This EOS is reasonably accurate belabout 10 GPa. The temperature is an estimate and wasmeasured.

The initial rate corresponding toTs is plotted in Fig. 7 asa function of shock pressure for both porous and full-densmaterials. The rate is very sensitive to shock pressure, rifrom 1023 (s)21 at 2 GPa to 1~ms!21 at 5 GPa for theporous solid. Prompt initiation will occur in the latter casIn contrast, even at 10 GPa, the rate in full-density HMXless than 1 s21, and it would take some time for a detonatioto develop, if it occurred at all. This is consistent with thobservation that a single crystal of HMX does not detonin 50 mm size samples when shocked toO(10 GPa).40 Thevast difference in rate as porosity is varied is due toexponential temperature dependence of the rate and variaof shock heating with porosity. Thus, granularity can havdramatic affect on the chemistry.

The pressures achieved in a typical accident situationof the order of~0.2 GPa!. From Fig. 5~b!, the temperaturerise behind a compaction wave in a porous bed is only ab15 K. This is an inconsequential increase in the bulk teperature. If the compaction work available to heat the bcan be deemed to be deposited near the surface of the grmuch as the ‘‘elastic’’ energy component ofB(fs) is local-

is

nt

r-

thsethtotio-

rat

sa

thn

ovsue

tfe

lytioe

pap

isen

ornti

ix-ow

ruc-

-ots,to-

in

gas

temrge

gashysi-c-veor-s.–

ionngehenlat-

t:ate

ngeThes aro-

umate-

arethe

lledlim-

er,r a

ngesurealesex-gasno-s

y

nco

igh

391Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

ized in small volumes around the contact surfaces ofgrains, then the temperature increase would scale inverwith the fractional volume deemed to be associated withcompaction work. A fractional volume of 3% would leada local temperature increase of 450 K and an initial reacrate of 0.2~ms!21. A reaction rate of this magnitude corresponds to vigorous burning. A local region of high tempeture is commonly referred to as a ‘‘hot-spot.’’ If the ‘‘hospot’’ is large enough, then the energy generation duereaction can exceed the conductive losses, leading to anificant localized reaction that, in turn, could build up todetonation wave. Some complications can occur withsimple bookkeeping described above. HMX is known to udergo a solid-to-solid phase transformation at 145 K abroom temperature, which both consumes energy and rein a 10% volume increase in HMX. This can act to furthload a confined granular bed. Also, melting occurs at 200above room temperature and the latent heat correspondstemperature change of 216 K. These may significantly afthe temperature and hence the rate estimated above.

A leading deficiency in the present model is that onbulk temperatures are included and temperature localizais ignored. However, at least qualitatively, the model docapture the physical source of the heating, namely, comtion work. The dissipated energy could be used to develorational ‘‘hot-spot’’ reaction mechanism. Although thmechanism is probably the dominant one for ignition in mchanical loading of granular beds at low pressures, we doconsider the question of these reaction centers in this wInstead, we focus on examining the processes or exchaterms that describe the dissipation that generates the heaat least how they are mimicked in BN-type continuum mture models for materials with a microstructure, and h

FIG. 7. The initial reaction rate~computed using the Arrhenius form giveabove! as a function of shock temperature. The dotted and solid linesrespond to initial solid volume fractionfs

051 and 0.73, respectively. Theeffects of porosity are dramatic at higher pressures, leading to a much hbulk solid temperature and very rapid reaction.

elye

n

-

toig-

e-elts

rKo act

nsc-a

-otk.geng;

these exchange terms influence the equilibrium wave sttures supported by the model.

Embid, Majda, and Hunter41 have proposed an alternative wave resonance theory for the generation of hot spand imply that this could be the start of a deflagration-detonation transition. The theory is based on a singularitythe gas acoustic equations, noted by Embid and Baer,18 thatoccurs when the solid particle speed and either of theacoustic characteristics coalesce~i.e., when us5ug6cg).Then the set of eigenvectors is not complete and the sysof PDEs ceases to be hyperbolic. Given that the drag is lain low-porosity systems, and, therefore, the solid andparticle speeds are nearly equal, this resonance is an unpcal artifact of the model, and unlikely to occur in any pratical situation of interest. If the gas flow were sonic relatito the solid, the resulting drag force would increase enmously, leading to a rapid equilibration of the velocitieThis effect was specifically excluded from the EmbidMajda–Hunter theory.

In the next section we reexamine how the dissipatinequality for the mixture can be used to define the exchaterms. We demonstrate that the dissipation inequality, wso applied, permits considerably more freedom in formuing the exchange terms than BN would imply.

III. THE DISSIPATION INEQUALITY

The BN model requires two types of constitutive inpu~1! the EOS’ for the equilibrium response of the separcomponents discussed in the last section, and~2! the sourceterms for rates of mass, momentum, and energy exchabetween phases, which is the subject of this section.development of suitable models for both cases requireblend of analysis and observation. While an analysis pvides constraints imposed by the principles of continumechanics, observations provide data for the specific mrial~s! at hand.

The methods for developing single-phase EOS’rather standard and widely accepted. To a large extent,EOS experiments can be carried out on large, well-controsamples measuring response on a macroscopic scale to aited number of well-understood control variables. Howevobtaining a thermodynamically consistent EOS valid ovewide range may well be difficult.

Methods for measuring and parametrizing the exchaprocesses are not nearly so well developed. In large meathis is a consequence of the short spatial and temporal scon which these nonequilibrium processes can occur. Forample, the resistance of a low-porosity granular bed toflow is typically measured by slow-flow experiments omacroscopic samples.42 These measurements can only prvide limited information about the high relative-speed flowthat can occur over narrow, yet macroscopic~several grains!,regions.

On the analytical side, the principal tool~besides mate-rial frame indifference and phase separation! is the dissipa-tion ~entropy! inequality for mixtures proposed bTruesdell,8

r-

er

-nixtth

did

sfola

asuasio

le

na

-quh

-s

tiob

ndth

e,,to

fromor-

ter-g is

ofer-ro.withse

refer-n is

ix-

ctndis-

ex-ap-edeo-

ytheheuc-tedicalthe

meto

en-be

on-ngen-ter

nu-del.lid

ndss-

392 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

~hs2hg!C 1~fsrs!dhs

dts1~fgrg!

dhg

dtg>0, ~45!

whered/dta5] t1ua]x is the material derivative of an individual phase. Roughly, the inequality implies that the etropy of an adiabatically insulated sample of a granular mture cannot decrease spontaneously. Given thattemperature, pressure, and velocity of each phase togewith the volume fraction define the state of each phase,ferences in the values of these variables could be viewethe forces that drive changes in the state of the mixture.

A. Driving forces

To help motivate our choice of the set of driving forcewe first discuss the state of thermodynamic equilibriumthe mixture of granular solid and gas. This defines the retion between the pressure and temperature of the two phwhen complete thermodynamic equilibrium is achieved. Sficiently close to equilibrium, we can view compactionreversible, irrespective of whether we adopt the formulatCs(rs ,Tsfs) ~reversible! or Cs(rs ,Ts) ~irreversible!. In ei-ther case,bs should be viewed as representing the reversibelastic stress. We consider a nonreactive, quiescent~i.e., us

5ug50), adiabatically closed system whose total interenergy, e5lses1lgeg , total volume,V5ls /rs1lg /rg ,and mass fractions (ls ,lg) are constrained to remain constant. For such a mixture, the entropy of the system at elibrium is a maximum with respect to all variations, whicleavee and V unchanged.43 Using Eq.~27!, the thermody-namic identity for the granular solid,

des5Ps

rs2 drs1Tsdhs1

bs

fsrsdfs , ~46!

and the corresponding identity for the gas,

deg5Pg

rg2 drg1Tgdhg , ~47!

the variation of mixture entropy@defined by Eq.~4!#, nearequilibrium and withe, V, ls , andlg held constant, is

Tsdh51

r~Ps2bs2Pg!dfs1lg~Ts2Tg!dhg . ~48!

Since the variationsdfs and dhg can be completely arbitrary, it follows that at equilibrium, the following conditionmust both hold:

Ps2bs2Pg50, ~49!

Ts2Tg50. ~50!

Equations~49!–~50! are the statements of pressure~mechani-cal! and temperature~thermal! equilibrium, respectively, forthe granular mixture. Thus, we see that the configurapressure represents the equilibrium pressure differencetween the phases. With the definition of equilibrium at hawe now put forth two postulates about the departure ofsystem from equilibrium. First, even whenTsÞTg , Eq. ~49!will continue to define mechanical equilibrium. LikewisEq. ~50! will remain our definition of thermal equilibriumeven whenPs2bs2PgÞ0. The second postulate pertains

--heer

f-as

,r-es

f-

n

,

l

i-

ne-,e

the role of (Ts2Tg) and (Ps2bs2Pg) as independent driv-ing forces that serve as the agents that move the systemequilibrium. These forces are taken to be directly proptional to Ps2bs2Pg andTs2Tg .

Similarly, we assume that the departure of (us2ug)from zero is proportional to a force that acts to cause inphase motion. The notion here is that Stokes-type drataken to be the corresponding restoring force. The formsthese driving forces are certainly consistent with total thmodynamic equilibrium, when the driving forces are zeThey are also plausible, as we shall see, and consistentthe notion that the dissipation inequality is linear in theforces for small departures from equilibrium.

A stable equilibrium for such a sample would requithat dissipation be at least of second order in these difences at the equilibrium point. To ensure that the evolutiodissipative, source terms represented by the fluxesM, E ,andF are chosen such that the entropy change for the mture near equilibrium is the sum of positive-definite terms~infact, perfect squares!. Each term corresponds to a distinphysical process~drag, heat conduction, and compactiowork!, and these processes are required to be separatelysipative. This leads to relaxation-type expressions for thechange rates that are consistent with equilibrium. Thisproach is reminiscent of that of Prigogine, who employnonequilibrium thermodynamics to develop transport thries for simple mixtures.44

Baer and Nunziato9 imply that the exchange laws thederive in this way are, in fact, unique. Here we show thatgeneral procedure allows considerably more flexibility in tforms for exchange laws. In particular, the entropy prodtion for each physical process can be arbitrarily distribubetween the phases. Consideration of the underlying physprocess is needed to determine the free parameters inconstrained form of the source terms. In addition, when soexchange laws~such as the chemical conversion of solidgas, and to a lesser extent, compaction! are known from ex-periment to be irreversible, and to have a particular depdence on the state variables, the entropy inequality canused more creatively. Then, one can either place more cstraints on the form of the exchange law or take the excharate as given, which would then serve to more tightly costrain the remaining exchange laws. We utilize this idea lain this section.

B. Nonreactive system

We demonstrate this procedure and the issue of noniqueness by first considering a simpler nonreactive moThe differential thermodynamic relation for either the sophase, Eq.~27!, or the gas phase~with bg50) can be writtenas

Tadha5dea2Pa

ra2 dra2

ba

rafadfa . ~51!

On computing the material-specific time derivative aeliminatingdra /dta by means of the individual-phase maconservation, Eqs.~7!–~8!, one is led to the dissipation inequality for a nonreactive system,

loe

rcrg

the,maida,

-

roroo

v

pk

m

andider

the

asd-

ned

lity

qui-loc-

es,on

heua-sti-

ich

393Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

1

TsFfsrsS des

dts1

Ps

rs

]us

]x D1~Ps2bs!dfs

dtsG

11

TgFfgrgS deg

dtg1

Pg

rg

]ug

]x D1Pg

dfg

dtgG>0. ~52!

This expression assumes that the two-phase system is cand locally adiabatic, i.e., there is neither heat nor masschange with the surroundings. Upon introducing soutermsM andE into the single-phase momentum and eneequations@see Eqs.~9!–~12!#, and then eliminating the ki-netic energy from the latter, one finds that

S des

dts1

Ps

rs

]us

]x D51

rsfs~E2usM!,

S deg

dtg1

Pg

rg

]ug

]x D521

rgfg~E2ugM!, ~53!

whereE andM denote the flux of energy and momentumrespectively, that flow through the interface separatingtwo phases. In the absence of all interphase exchangE

5M50. Then, neither phase gains or loses energy andmentum across the interfaces, and the components are sbe phase isolated. Further, if no gasdynamic shockspresent, the changes in the overall entropy of the system

1

Ts~Ps2bs!

dfs

dts1

1

TgPg

dfg

dtg, ~54!

follow the changes in the volume fraction. On usingfg512fs and

dfg

dtg52

dfs

dts1~us2ug!

]fs

]x

~which assigns the porosity to the solid!, the above expression is rewritten as

S Ps2bs

Ts2

Pg

TgD dfs

dts1

Pg

Tg~us2ug!

]fs

]x. ~55!

To reiterate, we have considered changes in system entfor phases that are isolated both from each other and ftheir environment and interact only via changes in the vume fraction. Importantly, Eq.~55! can be of either sign, sothat the entropy is not guaranteed to be nondecreasing.

When the system is in mechanical, temperature andlocity equilibrium ~i.e., Ps5bs1Pg , Ts5Tg and us5ug),then changes in volume fraction produce no entropy.44 In thislimit, fs is not required in the modeling. The standard aproach for modeling condensed phase explosives mathese assumptions.45

To have the evolution of the system toward equilibriureflect that the three forces,

sedx-ey

,e

o-to

re

pyml-

e-

-es

~Ps2bs2Pg!, S 1

Ts2

1

TgD , and ~us2ug!,

can act independently, it is necessary that momentumenergy flow across the phase interfaces. We now conshow the forms for the fluxesM andE can be constituted toreflect our choice of the driving forces in such a way thatdissipation inequality is satisfied.

We begin by recognizing that the decomposition

S zs

Ts2

zg

TgD5S 1

Ts2

1

TgD @zs•b1zg•~12b!#

1~zs2zg!S 12b

Ts1

b

TgD ~56!

allows terms in the dissipation inequality to be expressedthe sum of terms proportional to the driving forces. By ajusting the parameterb between the limits 0<b<1, the en-ergy exchange due to compaction, say, can be partitioeither all to the solid (b50) or all to the gas (b51), oranywhere in between. We will take advantage of this abilater. Using Eq.~56!, we can rewrite Eq.~55!, to get

S 1

Ts2

1

TgD @~Ps2bs!•b1Pg•~12b!#

dfs

dts

1~Ps2bs2Pg!S 12b

Ts1

b

TgD dfs

dts1

Pg

Tg~us2ug!

]fs

]x.

~57!

This regrouping of terms in Eq.~55! highlights the threeagents that drive changes in the mixture entropy near elibrium; the differences in temperature, pressure, and veity at the interface.

The source term involving (us2ug) is unlike the otherterms. It can have either sign and unlike other sourccannot be removed with a constitutive statementdfs /dts . More than any of the other terms, it motivates tinclusion of source terms in the momentum and energy eqtions to ensure that dissipation inequality is satisfied. Subtution of Eqs.~53! into the dissipation inequality, Eq.~52!,yields

S 1

Ts2

1

TgD E2S us

Ts2

ug

TgD M

1S 1

Ts2

1

TgD @~Ps2bs!•b1Pg•~12b!#

dfs

dts

1~Ps2bs2Pg!S 12b

Ts1

b

TgD dfs

dts

1Pg

Tg~us2ug!

]fs

]x>0. ~58!

We now employ Eq.~56! to rewrite Eq. ~58!, obtainingthereby an expression for the dissipation inequality in whterms are grouped according to the three driving forces:

,

lat

ty

erivr

rs,a

seedlo

cee

niuent

be

Ap-er-

sealling

ta-thegh,pyra-x-

alarpel-

the

e

end

re-

tione

-y.

ac-icalid-idero-

ani-e

394 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

S 1

Ts2

1

TgD $E2@us•w1ug•~12w!#M

1@~Ps2bs!•b1Pg•~12b!#F %1~us2ug!

3S 12w

Ts1

w

TgD H Pg

Tg

]fs

]x S 12w

Ts1

w

TgD 21

2MJ1~Ps2bs2Pg!H S 12b

Ts1

b

TgD dfs

dtsJ >0. ~59!

The new parametersb and w are yet to be determinedthough it seems natural to restrict 0<w<1 and 0<b<1 inorder for the factors of temperature or pressure to interpobetween the values for the solid and gas phase. Definingthree terms in curly brackets to be (Tg2Ts)H, (us2ug)dand

1

mcS 12b

Ts1

b

TgD ~Ps2bs2Pg!,

respectively, whereH>0, d>0, and mc.0, ensures thepositivity of each of the terms in the dissipation inequaliWith this ansatz, the exchange terms are

M5Pg

TgS 12w

Ts1

w

TgD 21 ]fs

]x2d~us2ug!, ~60!

E5H•~Tg2Ts!2@~Ps2bs!•b1Pg•~12b!#F

1@us•w1ug•~12w!#M, ~61!

F 5dfs

dts5

fsfg

mc~Ps2bs2Pg!. ~62!

These forms for the exchange laws reflect the fact that nequilibrium the exchanges are taken to be linear in the ding forces. We note that BN actually use the following altenate form for the compaction source:

F 5H fsfg

mc~Ps2bs2Pg!, for Ps2bs.0;

2fsfg

mcPg , for Ps2bs<0;

~63!

which is an attempt to model compaction as being irreveible. Since the signs ofF andPs2b2Pg remain the samethe modified form is also consistent with the entropy inequity that we have presented in this section. The selectionthe BN form over the equilibrium form given by Eq.~62! isan example of how independent information can be ualong with the entropy inequality to deduce an improvform for an exchange term. In the next section we empthis idea more directly in treating reactivity.

We identify, as Baer and Nunziato did,H as a heat-transfer coefficient,d as a drag coefficient, andmc as a com-paction viscosity. These are the quantities, which, in conwith the prevailing levels of nonequilibrium, determine thrates of relaxation toward equilibrium. Their relative magtudes, state dependent, in general, will determine the seqtial order in which the system approaches velocity, mechacal, and thermal, equilibria. Since each process is separa

tehe

.

ar-

-

-

l-of

d

y

rt

-n-i-

ely

dissipative, the coefficientsH, d, andmc may be determinedand set independently. This allows the source terms tocalibrated and applied far from equilibrium.

The nonconservative term,

PN

]fs

]x[

Pg

TgS 12w

Ts1

w

TgD 21 ]fs

]x,

has already been identified as the nozzling term. In thependix we show that for the system of PDEs to be hypbolic, w51. This implies thatPN5Pg , the BN value. More-over, there is no entropy production in either phaassociated with nozzling. It is worth reiterating that nottwo-phase mixture models have a nozzling term. Eliminatnozzling by settingPN50, as suggested by Powerset al.,13

puts the modeling PDEs in conservation form; a computional advantage. However, the entropy production formixture is not guaranteed to be positive semidefinite, thouin many applications, nozzling is small and the total entromay well be nondecreasing. Here we follow the BN padigm, for which nozzling contributes no entropy to the miture.

With b5w51, we recover the BN exchange terms fornonreactive system. However, the selection of a particuform for the parameters should either be based on a comling physical argument or made with the realization thatchoice is arbitrary. The parameterb is associated with dis-tributing compaction work between the phases46 and is dis-cussed in more detail later in Sec. IV B.

Finally, we note that even though BN assumCs(rs ,Ts ,fs), which then implies the relation@see Ref. 18,Eq. ~2.11!#

Tsdhs5des2Ps

rs2 drs2

bs

fsrsdfs ,

they, in fact, use an EOS for the solid that does not depon fs @see Ref. 18, Eq.~2.9!, where es(rs ,Ps)#. This isinconsistent. To remain true to the assumption,es(rs ,Ps)would require the standard single-phase thermodynamiclation,

Tsdhs5des2Ps

rs2 drs ,

and would lead to some significant changes in the dissipainequality. This, in turn, would have implications for thform of the compaction sourceF . Importantly, if the com-paction energyB(fs) is not accounted for, then the configuration pressure,bs , would not appear naturally in the theorWe discuss some of these points later.

C. Reactive system

Progress was obtained relatively easily in the nonretive case,C 50, because the issues were mostly mechanand involved constitutive theory only via an EOS. In consering chemically reactive systems, we are forced to consthe very nonlinear constitutive functions typical of such prcesses. Also, instead of being symmetric as the ‘‘mechcal’’ processes were~e.g., heat can flow both to and from thsolid!, we expect the chemistry to be one sided, withC <0.

phao-ws

-

ifi

fos

n-

inem;

firstic-

lity

anrst

ofis-are

han

and

in-

n-rst

de-rmrce

395Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

Thus, we shall make a break with the traditional develoment here and include reactivity into the theory somewdifferently than the way in which we included the other prcesses. This apparent break from the development folloby BN is, in fact, in keeping with their approach, becauBN never use the expression forC derived from a consideration of the entropy inequality.

On including chemistry~i.e., mass transfer! in the elimi-nation of dra /dta from Eq. ~51!, the dissipation inequalityfor a nonreactive system, Eq.~52!, is transformed into thereactive-system version,

~hs2hg!C 2S Ps

Tsrs2

Pg

TgrgD C

11

TsFfsrsS des

dts1

Ps

rs

]us

]x D1~Ps2bs!dfs

dtsG

11

TgFfgrgS deg

dtg1

Pg

rg

]ug

]x D1Pg

dfg

dtgG>0. ~64!

As before, the momentum and energy exchange rates,M

andE , enter via the material-based rate of change of specinternal energy, given by

S des

dts1

Ps

rs

]us

]x D51

rsfs~E2usM!2

1

rsfsS es2

1

2us

2D C ,

~65!

S deg

dtg1

Pg

rg

]ug

]x D521

rgfg~E2ugM!1

1

rgfgS eg2

1

2ug

2D C .

~66!

Following the discussion surrounding Eq.~21!, we associatethe compaction rateF with

dfs

dts2

C

rs5F .

Substituting these expressions into Eq.~64! yields

S Gg

Tg2

Pg

rsTg2

Gs

Ts1

Ps2bs

rsTsD C 1

1

2 S us2

Ts2

ug2

TgD C

1S 1

Ts2

1

TgD E2S us

Ts2

ug

TgD M1

Pg

Tg~us2ug!

]fs

]x

1S Ps2bs

Ts2

Pg

TgD F >0, ~67!

where

Ga[Ca1Pa

ra5ea1

Pa

ra2haTa

is the Gibbs free energy. We note that with the allowancemass conversion, the Gibbs free energy now enters aadditional forcing term in the dissipation inequality.

In the limit of thermal equilibrium (T5Ts5Tg), veloc-ity equilibrium (us5ug) and mechanical equilibrium (Ps

5bs1Pg), porosity is an extraneous variable andF , M,and E play no role. Then the dissipation inequality is idetical to the form for a homogeneous system,

-t

ede

c

ran

S Gg2Gs

T D C >0. ~68!

In this case, it follows that the rate of mass conversion isthe direction to reduce the Gibbs free energy of the systi.e., C <0 whenGs.Gg . Typically, for explosives of inter-est, the change in the Gibbs free energy dominates theterm in Eq.~67! and the entropy inequality places no restrtion on C .

Here we assume that the mass conversion rateC isknown, and show that the previously demonstrated flexibiin choosing the other exchange rates~i.e., M andE! is suf-ficient to ensure that the mixture dissipation inequality cbe satisfied. In doing this, we focus our attention on the fiterm in Eq.~67! proportional toC , since it contains the newdriving force. The second term in Eq.~67! presents no newproblems since it can be lumped in with the coefficientsthe three driving forces for the nonreactive problem dcussed previously. The new modeling assumptions thatspecific to a reactive mixture are the following.

~i! Entropy: the entropy of the reacted gas is greater tthat of the solid, i.e., (hs2hg)<0.

~ii ! Irreversible reaction:C ,0.

The first term in Eq.~67! can be written as

S Gg

Tg2

Pg

rsTg2

Gs

Ts1

Ps2bs

rsTsD C 5S eg1Pg~Vg2Vs!

Tg2

es

TsD C

1S ~hs2hg!2bs

rsTsD C .

~69!

By the assumptions above, the second term on the right-hside is positive. Using Eq.~56!, we can rewrite the factor inthe first term on the right-hand side as

eg1Pg~Vg2Vs!

Tg2

es

Ts

5S 12n

Ts1

n

TgD @eg2es1Pg~Vg2Vs!#

2S 1

Ts2

1

TgD $nes1~12n!@eg1Pg~Vg2Vs!#%,

wheren is an adjustable parameter chosen to satisfy theequality

@eg2es1Pg~Vg2Vs!#S 12n

Ts1

n

TgD<S hg2hs1

bs

rsTsD .

~70!

With n selected so that Eq.~70! is satisfied, the contributionof the reactive term to the dissipation inequality is guarateed to be positive. The choice of decompositions for the fireactive term in the dissipation inequality, Eq.~67!, is some-what arbitrary. Other rational choices are possible. Thecomposition allows us to account for the remaining teproportional to the temperature difference in the souterms. Moreover, it reduces to the BN result whenn51.Typically, Tg.Ts , eg2es1Pg(Vg2Vs).0, andhg2hs isthe dominant term in inequality~70!. In this case, the choice

yra

ro-

m

orn

thusf

osho

cef

thm

toto

rag

that.e.,

by

ith

rial

ion

ot

ce

ter-

henthe

the

ane

-er toac-edasticitypa-en-nt

ersdis-pres-lau-

396 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

n51 is compatible with the principle of minimum entropproduction for a nonequilibrium process when the tempeture factor is assumed to interpolate betweenTs andTg .

We decompose the second reactive term in Eq.~67! ac-cording to

1

2 S us2

Ts2

ug2

TgD C 5

1

2 S 1

Ts2

1

TgD @us

2f 1ug2~12 f !#C

11

2~us2ug!~us1ug!S 12 f

Ts1

f

TgD C ,

~71!

wheref is a free parameter. The two forcing terms are pportional to differences inT andu as occurs in the nonreactive problem.

Grouping the terms in Eq.~67! according to the forcesrepresented by the differences (1/Ts21/Tg), (us2ug), and(Ps2bs2Pg), the source terms can be chosen to form suof squares, thus ensuring positivity. The result is

M5PN

]fs

]x2d~us2ug!1

1

2 S 12 f

Ts1

f

TgDTg~us1ug!C ,

~72!

E5H~Tg2Ts!2PcF 1usM2 12@us

2f 1ug2~12 f !#C

1$nes1~12n!@eg1Pg~Vg2Vs!#%C , ~73!

F 5dfs

dts2

C

rs5

fsfg

mc~Ps2bs2Pg!, ~74!

where

PN5Pg and Pc5~Ps2bs!b1Pg~12b!. ~75!

With n51, all the energy released by the reaction is depited in the gaseous reaction products. Typically, when buing occurs,Tg.Ts and eg2es1Pg(Vg2Vs).0. Then n.1 corresponds to distributing some of the energy ofreaction to the solid phase reactants. In principle, this cobe accounted for by heat conduction, i.e., in the heat trancoefficient H. Similarly, n,1 would withdraw additionalenergy from the solid and transfer it to the gas. This is psible when the reaction corresponds to a process sucevaporation but is not physically plausible for a highly exthermic irreversible reaction.

In addition to satisfying the entropy inequality the sourterms must be compatible with Galilean invariance. The lehand sides of the internal energy equations, Eqs.~65!–~66!,are Galilean invariant. Therefore, the source term onright-hand side must be Galilean invariant as well. The terinvolving the velocities are given by

1

2~us

22ug2!~12 f !

C

fsrs, for the solid;

Tg

2Ts~us

22ug2!~12 f !

C

fgrg2

d

fgrg~us2ug!2, for the gas.

In both cases Galilean invariance requiresf 51.

-

-

s

s--

elder

-as

-

t-

es

D. Additional dissipation terms

There are two additional degrees of freedom relatedthe distribution of energy from drag. First, adding a termthe energy source,

E→E1ad~ug2us!2,

changes the total dissipation due to drag fromd(us

2ug)2/Tg to

dS a

Ts1

12a

TgD ~us2ug!2.

This is positive for 0<a<1. The choicea50 in the BNmodel corresponds to assigning all the dissipation from dto the gas phase.

The second degree of freedom is related to the factthe particle velocity changes when the material burns; ius→ug . Adding a drag term to the sourcesM→M

2a(ug2us)C and E→E2 12a(ug1us)(ug2us)C in-

creases the entropy production of the mixture2 1

2aC (ug2us)2. For an irreversible reaction,a>0 would

satisfy the entropy inequality and also is compatible wGalilean invariance. The choicea50 in the BN model mini-mizes the entropy increase due to drag when the mateburns.

We now summarize the development of the interactsource terms.

~i! The dissipation inequality constrains but does nuniquely determine the source terms.

~ii ! The physical requirements of Galilean invarianand a well-posed initial value problem~hyperbolic system ofPDEs! provides additional constraints (w51 and f 51).

~iii ! The remaining degrees of freedom must be demined based on application-dependent considerations.

~iv! The BN model is recovered whenb5n51 anda50. Settingn51 gives a termes in the energy source termand seems natural for the energy balance equations. Wthe change in Gibbs free energy dominates the burn,entropy inequality would not restrictn. The parameterb isrelated to the distribution of compaction energy betweenphases.

~v! When the solid-phase EOS is constituted asEOS for a granular phase~i.e., volume-fraction dependencis included!, the configuration pressurebs appears ‘‘natu-rally’’ in the compaction law. However, this formalism models quasistatic compaction as a reversible process. In ordreflect the irreversibility seen during the quasistatic comption of HMX, additional variables and source terms will neto be introduced to model such processes as fracture, pldeformation, and frictional heating. The entropy inequalwill continue to be satisfied if these were separately dissitive, and each added a positive definite contribution totropy production.47 Such an approach is the subject of recework by Gonthieret al.32

In the following sections, we show that the parametare related to how the dissipation from each process istributed between the phases. We suggest a number of exsions for the exchange terms that are physically more psible than those used by BN.

ndiceW

thw

b

p

alseth

ni

ioed

tfo

-enhthi

,re

isgni-es-asAs

e

getionon-uchif

de-reular

on-ch

s inle-the

ithetheon

con-gy,pac-

n.s-in

to a

or-

the

di-

enthe

-ns,theag

of

397Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

IV. DISTRIBUTION OF DISSIPATION BETWEENPHASES: MODIFIED INTERACTION SOURCE TERMS

In their derivation of a two-phase model, Baer aNunziato9 used the entropy inequality to motivate the choof the source terms for the interaction between phases.have shown that many different choices are possiblesatisfy the overall entropy inequality, and suggest the folloing modifications to the BN source terms:

E5H~Tg2Ts!1usM1ad~us2ug!2

1$2 12us

21nes1~12n!@eg1Pg~Vg2Vs!#%C , ~76!

where 0<a<1 andn satisfies Eq.~70!, and

Pc5~Ps2bs!b1Pg~12b!, 0<b<1. ~77!

In this section, we argue on physical grounds for reasonachoices for the parametersa, b, andn. For this purpose, itshelpful to record the individual phase energy and entroequations, where we have takenf 5w51 in Eqs.~65!–~66!.We get for the solid,

fsrsS des

dts1Ps

dVs

dtsD

5H~Tg2Ts!1bsF 1~12b!~Ps2bs2Pg!F

1ad~us2ug!22~12n!@es2eg2Pg~Vg2Vs!#C

5fsrsTs

dhs

dts1bs

dfs

dts, ~78!

where we have used Eq.~27!, and for the gas,

fgrgS deg

dtg1Pg

dVg

dtgD

52H~Tg2Ts!1b~Ps2bs2Pg!F

1~12a!d~us2ug!22n@es2eg2Pg~Vg2Vs!#C

5fgrgTg

dhg

dtg. ~79!

We have rewritten the left-hand side of each individuphase energy equation so as to highlight the material-baisolated-phase, first-law form of the equation. This haseffect of moving the compaction-related part of]ua /]x tothe right-hand side as a source term. Thus, the apportioof dissipation between phases is controlled for drag bya, forcompaction byb, and for burn byn. Also, Eqs.~78!–~79!exhibit the property of the model that quasistatic compact~i.e., Ps5bs1Pg) does not generate any entropy in eithphase, which is a consequence of our thermodynamicscription ofB(fs) as a potential energy.

The assumption that the solid has a stiff EOS relativethe gas leads to the formulation of an advection equationcompaction dynamics, Eq.~21!, along solid particle trajectories. Consistent with this idea is the consignment of thetire drag dissipation to the gas. Thus, at low pressures wthe compaction dynamics is physically reasonable,choicea50 is appropriate. At higher pressures the dragnot significant compared to other dissipative processesparticular burn, so the partition for drag is unimportant the

eat-

le

y

-d,e

ng

nre-

or

-enesin.

Usually the empirical form used for the reaction ratetaken to be a strong function of gas pressure once the ition criteria for the reaction is exceeded. Thus, at low prsures takingn51 and sending the reaction energy to the gwould magnify the reactive response of the granular bed.the pressures increase andPs'Pg , the reactive responswould be insensitive to the value ofn.

In actual fact, the forms presented for the exchanterms should be thought of as a first guess. More informaon the microstructural physics needs to be developed to cstitute the continuum-scale exchange terms used in stheories as BN with improved realism. Nevertheless,physically plausible exchange terms of the sort we haverived are used intelligently, we should be able to explowhether the phenomena observed in experiments on granexplosives can be sensibly mimicked by multiphase ctinuum mixture models such as a modified BN. Using sumodified source terms, we now show how inconsistenciethe BN model related to compaction work and the singphase limit can be corrected by appropriately modifyingsource terms.

A. Compaction work

Here we discuss the dissipative terms associated wcompaction in this theory, called ‘‘compaction work.’’ Thesdissipative terms contribute compaction related energy tomixture, in addition to the previously discussed contributito the energy of a granular solid,B(fs). We argued thatB(fs) represented an energy that was localized near thetact surfaces of the solid grains. This compaction enerdiscussed in Sec. II E, is measured in the quasistatic comtion experiments of Elban and Chiarito34 in which a granularmaterial within a cylinder is slowly compressed by a pistoThe ‘‘compaction work’’ terms play no role there; being asociated with a dynamic process, they play a role onlydynamic compaction experiments, and, in general, leadrate dependence of the achieved end state.

Recall that compaction dynamics involves terms proptional toF . In the original BN model, whereb51, the com-paction process is dissipative for the gas phase sinceentropy change of the gas is given by

~Ps2bs2Pg!F }~Ps2bs2Pg!2>0,

whereas the entropy change for the solid is zero. Our mofication to the exchange terms includes a parameter 0<b<1. It allows us to distribute the compaction work betwethe phases in any fashion, with the contribution to bothsolid and gas entropy equations, Eqs.~78!–~79!, being posi-tive,

~12b!~Ps2bs2Pg!F >0, for solid,

b~Ps2bs2Pg!F >0, for gas.

The termb(Ps2bs2Pg)F , which appears with equal magnitude but opposite sign in the individual phase equatiosets the partition of the energy of compaction betweensolid and the gas. This is a similar functionality to the drterms that we considered earlier. Whenb takes on the BNvalue of 1, the gas would gain the maximum amount

m

f tn-w

-n

-in

omEqer

ioth

tiv

n

siain

uThinex

lidolynbl

go

douis

n

thethanl,hen

thats

theum,u-

heidsis-

qua-In

ex-s in

ot

aretum

398 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

energy allowed during compaction, while none of the copaction energy would go to the gas whenb50. The termbsF on the right-hand side of Eq.~78! consistently repre-sents the increase in the quasistatic compaction energy osolid and (Ps2bs2Pg)F represents the increase in the iternal energy of the solid due to compaction work, asnow show.

We integrate the solid energy equation, Eq.~78!, underthe following assumptions:~i! rs5rs

0, because as we discussed in Sec. II E the density of the solid phase doeschange much on compaction.~ii ! Compaction provides sufficient dissipation to lead to a fully dispersed steady travelwave profile.~iii ! With a negligible gas mass fraction,Pg

'0. Then, for a compaction wave, we obtain

Des'E1

2 dB~fs!

dfsdfs1E

1

2 ~12b!~Ps2bs!F

rs0fs

dts ,

~80!

where the subscripts 1 and 2 denote the initial and final cpaction states. The first term on the right-hand side in~80! is the quasistatic compaction energy from the intgranular stress, denotedB in Eq. ~32!. The second term is thedissipative compaction work due to the dynamic compactprocess. The sum of the two terms is determined byHugoniot jump relation, Eq.~40! given in Sec. II E,

Des'Ps

2rs0 S fs

fs021D ,

where we have takenrs'rs0. As we showed in Sec. II E, the

quasistatic compaction energy is a third of the dissipacompaction work when material initially atfs

050.65 is dy-namically compressed to 5% porosity.

The parameterb distributes the compaction dissipatiobetween the solid and gas phases. The choiceb51 in theoriginal BN model corresponds to all the compaction dispation occurring in the gas. In this case, when the gas mfraction is small~as occurs for the lead compaction wavea DDT-tube experiment!, attributing all the dissipation to thegas leads to an unphysically high gas temperature, mhigher than the temperature behind a detonation wave.is because the fixed energy from compaction is placedan arbitrarily small mass of gas, as we show in the nsection.

A better choice for the parameter isb50. This corre-sponds to attributing the compaction dissipation to the soAs the next section also shows, this choice is needed totain fully dispersed weak compaction waves. Alternativeb5lg or Pg /P would distribute the dissipative compactiowork between the solid and gas in a physically plausimanner.

B. Single-phase limit

As the mass fraction of the gas becomes vanishinsmall, it is natural to expect that the influence of the gasthe behavior of the system should decline accordingly, anthe limit the model should reduce to that for a gasless porsolid. We now explore whether this single-phase limitachieved in a consistent way.

-

he

e

ot

g

-.

-

ne

e

-ss

chis

tot

.b-,

e

lynins

In order to derive the limit, we start with the assumptiothat the gas has a small mass fraction, i.e.,

fgrg!fsrs . ~81!

Below crush-up the porosity may be moderate and thenabove condition holds because the gas is far less densethe solid (rg!rs). Above crush-up the porosity is smalfg!1, causing the above condition to be satisfied even wthe gas has been compressed to a density comparable toof the solid@rg5O(rs)#. We also assume that the velocitieus , ug and the specific internal energieses , eg are of orderunity. Sincepa /ra is a measure ofea , the condition on theinternal energies implies thatpa5O(ra) for both phases aswell. It then follows that

fgPg!fsPs and fgrgug!fsrsus . ~82!

For the mixture variables, Eqs.~81! and ~82! imply that

r[fsrs1fgrg'fsrs ,

P[fsPs1fgPg'fsPs ,

e[~fsrses1fgrgeg!/r'es ,

u[~fsrsus1fgrgug!/r'us .

As a result, the gas contributions can be neglected inmixture equations for the conservation of mass, momentand energy@obtained by adding together respective contribtions from each phase in Eqs.~7!–~12!#, reducing them to thestandard, single-phase fluid equations,

]r

]t1

]ru

]x50,

]~ru!

]t1

]~ru21P!

]x50, ~83!

]~rE!

]t1

]@~rE1P!u#

]x50,

]fs

]t1u

]fs

]x5F . ~84!

Although derived above from the conservation laws for tmixture, Eqs.~83! are simply the balance laws for the solphase alone, with the exchange terms removed. For contency, therefore, the exchange terms in the solid-phase etions themselves must vanish in the single-phase limit.order to ensure the satisfaction of this requirement, weamine the momentum and energy exchange processeturn.

It is natural to take the mass exchange rateC 50 in thesingle-phase limit; otherwise, the gas mass fraction will nstay small. In the momentum exchange termM, the onlycontributions that remain in the absence of chemistrythose due to nozzling and drag, so that the solid momenequation~9! reduces to

]

]t~fsrsus!1

]

]x~fsrsus

21fsPs!5d~ug2us!2Pg

]fg

]x.

noaner

t

is

sthbahthe

e

rm

gle

arkfone

,

rorita

ica-eemors

ive

oforeiatom-T-

elythe

acttde

m-tedponb-italns,ionsn its

o

theula-ederi-aveal-

heual-eBN.

i-lown,ub-dis-ase

eic-sents aac-tionbyrs-ivethem-inbeerial

e ofhottheare

ion

399Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

The aforementioned consistency requirement forces thezling and drag combination on the right-hand side to be vishingly small. Now the nozzling term limits to zero eithbecausePg!1 or becausefg!1. The drag termd(us

2ug) must then also vanish on its own, i.e.,

d~us2ug!!1.

In order to assess the consistency of the energetics insingle-phase limit, we turn to the solid energy equation~11!.With mass exchangeC set to zero and drag neglected, thequation reads as

]

]t FfsrsS es11

2us

2D G1]

]x H fsusFrsS es11

2us

2D1PsG J52Pgus

]fg

]x2@b~Ps2bs!1~12b!Pg#F .

Here we have ignored heat transfer between the phasewell, to ensure that the single-phase limit exists even inworst case, when the gas is unable to transfer any heatto the solid. As argued above, the nozzling term on the righand side of the above equation is negligible, so thatcompaction term must be vanishingly small on its own. Blow crush-up, whenPg is small butfg ~and henceF ! can bemoderate, this requires that the partition coefficientb mustbe set to zero. Above crush-up, the compaction term is nligible because porosity~and henceF ! is small.

To complete the argument we examine the entropy foof the gas energy equation,

fgrgS deg

dt2

Pg

rg2

drg

dt D 5b~Ps2bs2Pg!F ,

obtained from Eq.~79!, where once again the chemical, draand heat-transfer contributions have been neglected. Thehand side vanishes in the limit, and the selectionb50 madeabove causes the right-hand side to vanish as well.

Thus, we see that the one-phase limit is consistentlytained. Additionally, the deposition of all compaction wointo the solid provides a natural dissipation mechanismweak compaction waves to be fully dispersed, a phenomeobserved in practice. The above analysis corrects a dciency in the BN model, which, having made the choiceb51 and thereby assigning all compaction work to the gasunable to limit properly.

V. FINAL REMARKS

Modeling the behavior of damaged explosives and ppellants is important for safety studies. The wide dispabetween the particulate scale of the damaged materialthe scale of the device would make a micromechanmodel impractical. Additionally, reproducibility of DDT experiments on granular HMX implies a well-behaved rsponse, in the sense that the ‘‘averaged’’ behavior is inssitive to variations in the microstructure that occur froexperiment to experiment. These observations argue fcontinuum model, one that would employ a two-phaframework to account for the solid grains of the explosand the gaseous products of combustion.

z--

he

aseckt-e-

g-

,ft-

t-

ronfi-

is

-yndl

-n-

ae

Various versions of a two-phase, continuum theoryDDT have been proposed. Among the latest, and mprominent, examples of this approach is the Baer–Nunzmodel.9 This model has been tested by comparing its coputational results to experimental observations from DDtube experiments with granular explosives. Appropriatcalibrated, the model can reproduce quite well a subset ofavailable experimental observations, but not all of the impdata as the projectile velocity varies.6,5 To evaluate accidenscenarios, it is important for a model to apply over the wirange of conditions that may occur.

In this paper we have critically examined the continuumechanical underpinnings of the BN model; both the staphysical approximations and the tacit assumptions uwhich it is based. Given the complexity of the physical prolem being modeled, such an examination is absolutely vso that the strengths of the model, as well as its limitatioare clearly identified and understood before any extensare undertaken or generalizations attempted to broaderange of validity.

The BN model requires constitutive information on twlevels: ~1! an EOS for each of its two constituents, and~2!rate expressions for the exchange terms that act acrossparticulate/gas interfaces. We have reexamined the formtion of the material-specific constitutive terms and identifitheir physical basis; a feature not present in the original dvation. Inconsistencies in the EOS for the granular solid hbeen corrected. BN relies heavily on the dissipation inequity for the development of constitutive expressions for tinterfacial exchange rates. We demonstrate that this ineqity allows considerably more flexibility in constraining thinterphase exchange terms than has been suggested byWe identify and exploit this flexibility to construct physcally improved exchange terms. These improvements alone to remove inconsistencies from the original formulatioespecially those connected to single-phase limits and ssonic dispersed compaction waves. Both require that thesipative compaction work be apportioned to the solid phrather than the gas phase, as in the original model.

In order to extend the domain of applicability of thmodel and to increase its utility as a tool for safety predtions, several improvements remain to be made. The preformalism, wherein the solid-phase internal energy havolume-fraction-dependent component, treats the comption energy as a potential and, hence, quasistatic compacas reversible. Further work, along the lines indicatedGonthieret al.,32 is needed to capture the observed irreveible aspects of slow compaction. With the simple constitutmodel of compaction energetics and configuration stress,fluid-like BN model, as it stands, can describe subsonic copaction waves and DDT. To have an extended validitysome multidimensional situations, the model will need toextended to treat other properties dependent on the matstrength of the grains, such as mesoscale shear.

Burn models, at the moment, are the weakest featurthe theory. Sufficiently strong compaction waves lead tospots, i.e., regions where energy is localized. Owing tosensitivity of the reaction to temperature, these hot spotsthe sites of ignition, and dominate the rate of combust

ndpth

on

tsoteNa

of

nlressud

olede

aheep

ituen

pre

l

-

o

vi-u

veus

redsilythfoor

pu-

-us

eNo.e

af-enaryi-

weandg,

en-

con-

en-nt

gas

n be

400 Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

prior to the onset of detonation. Ignition at hot spots depemost strongly on the size and temperature of the hot sitself, and not on the bulk temperatures that appear incontinuum theory. While the~experimentally derived! com-paction work describes the net energy available for ignitithe theory is silent on how this energy is partitioned~local-ized!. It needs constitutive information, from experimenand, possibly, detailed micromechanical calculations, abthe phenomenon of localization and about the way it initiaignition. Additional degrees of freedom are required in Bto treat such issues. Without such a feature, DDT modelsnot likely to be applicable over a sufficiently wide rangeconditions.

The model, in its present form, is designed to deal owith a granular material that is well characterized. What vaables are needed to characterize damage in an explosivmains an important but open question that must be addresuccessfully if the model is to be used for engineering sties of explosive safety.

Whatever constitutive expressions are in place, they ctain within them information about the length and time scaof the underlying microstructure. These scales, in turn,termine the spatial and temporal extents of the zoneswhich the exchange processes strive to move the systemward equilibrium. Accuracy requires that in a numericcomputation, these zones be adequately resolved. If tcharacteristic scales are thin compared to the physical dimsions of the system being examined, the associated comtational burden could be prohibitively excessive. Such a sation is ripe for exploring asymptotic reductions of thmodel, in which the thin zones would be shock-like discotinuities that need not be resolved, and across which appriate variables would undergo predetermined jumps. Forample, experiments on permeation42 as well as numericasimulations with the BN model46 show that at low porosity~relevant to the situation at hand! the drag is high,d'200 kg/~m3 ms!, and the time scale for velocity equilibration is thus short,

tu;fgrg /d50.1ms,

leading to a length scale for velocity equilibration basedthe solid sound speed,cs of tucs;0.3 mm. This scale iscomparable to the grain size, and thus very small relativethe scales at which the continuum model holds. Thus, omajor portions of the DDT event, carrying two velocity varables is an unnecessary complication because the soterms associated with drag are stiff. In addition, the twolocities lead to difficulties at the interface with a nonporoinert material that has only one velocity.~Witness the pres-sure oscillations at the interface in Fig. 5 of Ref. 48, wheBN-based computations of impact experiments are report!This suggests a natural reduction of the model, over anificant portion of the DDT event, to one that carries onone velocity to a first approximation. At pressures aboveyield strength of the explosive grains, the length scalepressure equilibration is found to be small as well, thus mtivating a further reduction to a single pressure. These

sote

,

uts

re

yi-re-ed-

n-s-

into-lirn-u--

-o-x-

n

toer

rce-

e.

g-

er-

e-

duced models, economical from both analytical and comtational viewpoints, are derived49 and studied27 in twoforthcoming papers in this series.

ACKNOWLEDGMENTS

We thank Larry Hill of Los Alamos for helpful discussions and Don Drew and Steve Passman for sharing witha manuscript of their book~in preparation! on continuum-mixture theory.50 This work has been supported by thUnited States Department of Energy, under ContractW-7405-ENG-36. AKK was also supported partially by thNational Science Foundation.

APPENDIX: HYPERBOLICITY

The modified source terms derived in Sec. III B canfect the mathematical structure of the modeling PDEs. Givour postulate that the modeling PDEs should be evolution~hyperbolic!, in this appendix we examine whether the modfications to the nozzling term~i.e., allowingwÞ1) can leadto PDEs that are not hyperbolic. To simplify the analysis,useP, u, andh as the dependent variables for each phaseset all algebraic source terms~such as heat transfer, dracompaction, and chemical reaction! to zero.

The evolution equations for the solid and gas phasetropy are

fsrsTs

dhs

dts1~12w!~us2ug!PN

]fs

]x50, ~A1!

fgrgTg

dhg

dtg2~12w!~us2ug!PN

]fs

]x50. ~A2!

The solid and gas mass equations can be put in a morevenient form by eliminatingVa using the thermodynamicidentities

VsdPs52gsPsdVs1GsTsdhs ,

VgdPg52ggPgdVg1GgTgdhg ,

whereGa is the Gruneisen parameter. In deriving these idtities we have used the definition of the adiabatic expone

ga52S ] ln Pa

] ln raD

ha ,fa

5raca

2

Pa,

whereca is the sound speed. The source-free solid andphase mass equations thus become

fs

dPs

dts1fsrscs

2 ]us

]x2GsfsrsTs

dhs

dts50, ~A3!

fg

dPg

dtg1fgrgcg

2 ]ug

]x2GgfgrgTg

dhg

dtg

1rgcg2~us2ug!

]fs

]x50. ~A4!

The source-free solid and gas momentum equations cawritten as

a-

is

401Phys. Fluids, Vol. 11, No. 2, February 1999 Bdzil et al.

fscs

]Ps

]x1fsrscs

dus

dts1~Ps2PN!cs

]fs

]x50, ~A5!

fgcg

]Pg

]x1fgrgcg

dug

dtg2~Pg2PN!cg

]fs

]x50. ~A6!

This system of PDEs, Eqs.~A1!–~A6! plus Eq.~21!, canbe written as a matrix equation,

H] tqW 1F]xqW 5source,

where

qW T5~qW sT ,fs ,qW g

T!, ~A7!

and the single-phase variables areqW aT5(Pa ,ua ,ha). By tak-

ing linear combinations of the rows, we find

te

b

vedin

rao

eio

em

F[H21F5S Fs SW 0

0W T us 0W T

0 GW Fg

D , ~A8!

whereFa is the flux matrix for the single-phase fluid equtions,

Fa5S ua raca2 0

Va ua 0

0 0 ua

D , ~A9!

and the coupling of the phases with the volume fractiongiven by the vectors

SW 5S Gs~12w!PN

fs~us2ug!,

Ps2PN

fsrs,

~us2ug!~12w!PN

fsrsTsD T

,

GW 5S 2

FGg~12w!Tg

TsPN2rgcg

2Gfg

~us2ug!,PN2Pg

fgrg, 2

~us2ug!~12w!PN

fgrgTs

D T

.

ron

l

rgyon-oesasible

e

ous-

to-

.es

ofplo-

o-f

pp.

nndn

n-

The simple form of the matrixF allows the eigenvaluesand eigenvectors to be calculated analytically. The characistic velocities are simply those for the solid phaseus2cs ,us , us1cs , the gas phaseug2cg , ug , ug1cg , and an ad-ditional characteristic velocityus from the compaction dy-namics equation.

We note that the characteristic velocityus is two-folddegenerate. For the BN model~i.e., w51) this does not leadto any problems, since the right eigenvectors are found tocomplete~except at isolated surfaces in the state space!. Thesituation is different for the modified BN model. The aboanalysis shows that the modified nozzling terms can leathe model not being hyperbolic. We investigate this ponext.

The right eigenvector corresponding to the degeneeigenvalueus requires special consideration. Let the compnents of the eigenvector be denoted byqi . The third compo-nent of the eigenvalue equationFqW 5usqW is

usq31~us2ug!~12w!PN

fsrsTsq45usq3 .

This implies thatq450 unless eitherw51, PN50 ~nonozzling!, or us5ug ~single velocity!. If q450 then there isonly one right eigenvector~0, 0, 1, 0, 0, 0, 0!, correspondingto the two-fold degenerate eigenvalueus . A second linearlyindependent eigenvector for the two-fold degenerate eigvalueus can be found in terms of a time-dependent solutof the form

~b1 ,b2 ,t,b4 ,b5 ,b6 ,b7!, ~A10!

where bi are calculable constants. However, thet depen-dence in Eq.~A10! renders the associated linearized probl

r-

e

tot

te-

n-n

ill posed.51 Any initial data that select this right-eigenvectowould lead to an unbounded, linear growth of the solutiwith time.

If w51, then the additional right eigenvector is@(PN

2Pg)/fs,0,0,1,0,0,0#. Hence, for the two-phase modeequations to be hyperbolic we requirew51. Thus,PN5Pg

and the velocity associated with the nozzling in the enesource term must be the solid particle velocity used to cvect the volume fraction. As a consequence, nozzling dnot contribute to the dissipation of either the solid or the gphase. Since the Hugoniot jump equations are compatwith fs continuous across a shock,18 the nonconservativenozzling term,]fs /]x, does not cause a problem for thwave structure.

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