Impact Initiation of Explosives and Propellants via Statistical Crack Mechanics [Sd]

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Journal of the Mechanics and Physics of Solids

54 (2006) 1237–1275

0022-5096/$ -

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Impact initiation of explosives and propellants viastatistical crack mechanics

J.K. Dienes, Q.H. Zuo�, J.D. Kershner

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 3 May 2005; received in revised form 13 October 2005; accepted 2 December 2005

Abstract

A statistical approach has been developed for modeling the dynamic response of brittle materials

by superimposing the effects of a myriad of microcracks, including opening, shear, growth and

coalescence, taking as a starting point the well-established theory of penny-shaped cracks. This paper

discusses the general approach, but in particular an application to the sensitivity of explosives and

propellants, which often contain brittle constituents. We examine the hypothesis that the intense

heating by frictional sliding between the faces of a closed crack during unstable growth can form a

hot spot, causing localized melting, ignition, and fast burn of the reactive material adjacent to the

crack. Opening and growth of a closed crack due to the pressure of burned gases inside the crack and

interactions of adjacent cracks can lead to violent reaction, with detonation as a possible

consequence.

This approach was used to model a multiple-shock experiment by Mulford et al. [1993. Initiation

of preshocked high explosives PBX-9404, PBX-9502, PBX-9501, monitored with in-material

magnetic gauging. In: Proceedings of the 10th International Detonation Symposium, pp. 459–467]

involving initiation and subsequent quenching of chemical reactions in a slab of PBX 9501 impacted

by a two-material flyer plate. We examine the effects of crack orientation and temperature

dependence of viscosity of the melt on the response. Numerical results confirm our theoretical finding

[Zuo, Q.H., Dienes, J.K., 2005. On the stability of penny-shaped cracks with friction: the five types of

brittle behavior. Int. J. Solids Struct. 42, 1309–1326] that crack orientation has a significant effect on

brittle behavior, especially under compressive loading where interfacial friction plays an important

role. With a reasonable choice of crack orientation and a temperature-dependent viscosity obtained

see front matter r 2006 Elsevier Ltd. All rights reserved.

.jmps.2005.12.001

nding author. Tel.: +1505 667 0377; fax: +1 505 665 5926.

dress: [email protected] (Q.H. Zuo).

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ARTICLE IN PRESSJ.K. Dienes et al. / J. Mech. Phys. Solids 54 (2006) 1237–12751238

from molecular dynamics calculations, the calculated particle velocities compare well with those

measured using embedded velocity gauges.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Penny-shaped cracks; Frictional heating; Melting and ignition; Initiation and quenching of reaction;

Explosion

1. Introduction

Explosives and propellants may react violently as a result of relatively mild stimuli forwhich the continuum heating is negligible (Jensen et al., 1981; Green et al., 1981; Idar et al.,1998). For example, in a test by Idar et al. (1998), an explosive (PBX 9501) shows evidenceof chemical reactions at impact speeds below 50m/s, and it undergoes violent reactions atsomewhat higher impact velocity. This anomalous behavior (XDT) has been of greatconcern to designers and users of explosives and rockets. To quantify this sensitivity andassess the risks associated with reactive materials, the actual mechanisms of initiation mustbe known in some detail. Since the temperature rise due to mechanical dissipation duringuniform continuum deformation is too low to initiate reactions, we can conclude that theinitiation is localized in small volumes (hot spots) where the heating is intense enough tolead to a vigorous reaction. Numerous localization processes have been proposed toaccount for the formation of hot spots that initiate reactions (e.g., Field et al., 1992;Bonnett and Butler, 1996), but for impacts at very low speeds it seems likely to us thatinterfacial friction in closed cracks is the dominant mechanism. For completeness the 10mechanisms noted by Field et al. are listed in Table 1, but crack friction is not amongthem. However, crack formation in propellant and explosives is often observed, bothmacroscopically and in micrographs (Howe et al., 1985; Skidmore et al., 1997). This mayoccur as a result of initial formulation or subsequent damage. One occasionally reads thatfracture does not lead to initiation (Chaudri, 1972; Balzer et al., 2002), but the evidenceconcerns open cracks, not closed cracks where friction plays a role. When cracks are closedwe find that the heat generated by interfacial friction can cause a significant reaction. The

Table 1

The hot-spot mechanisms cited by Field et al. (1992) (in condensed form)

1. Adiabatic compression of cavity gases

2. Heating of solid adjacent to collapsing cavity

3. Viscous heating of binder between grains

4. Friction between impacting surfaces

5. Localized adiabatic sheara

6. Heating at crack tips

7. Heating at dislocation pile ups

8. Spark discharge

9. Triboluminescent discharge

10. Decomposition followed by Joule heating of metallic elements

aAdiabatic shear is sometimes considered to be similar to shear cracking, but there is an essential difference in

the details, with the former usually involving an instability due to thermal softening and the latter a strictly

mechanical instability involving a competition of strain energy and surface energy, with very high crack speeds

possible.

ARTICLE IN PRESSJ.K. Dienes et al. / J. Mech. Phys. Solids 54 (2006) 1237–1275 1239

interfacial friction mechanism discussed here accounts for the XDT incidents reported byJensen et al., Green et al., and Idar et al. Moreover, the multiple-shock experiment ofMulford et al. (1993) that led to reactions following a second shock can be accuratelysimulated with this mechanism.

Our calculations represent a feasibility study to determine whether the proposed fractureand friction mechanisms would explain the observations, while we also considered whetherother mechanisms might not explain the observed details. Observations of particularconcern for XDT are: (a) that it occurs at very late times (hundreds of microseconds ratherthan just a few for shock-to-detonation transition (SDT); (b) it can be more violent thanSDT (greater air-blast pressures); (c) large samples are more sensitive than small ones (sizeeffect); and (d) violent reactions occurred in only 12 out of 50 shots in those first propellanttests of Jensen et al. (the remainder involved mild deflagrations).

All these features can be accounted for with the shear-crack heating mechanism andcrack statistics which are included in our material model (Statistical CRAck Mechanics,SCRAM, Dienes, 1978, 1985, 1996). The model accounts for the opening, shear, growth,and coalescence of an ensemble of penny-shaped cracks, as well as plastic flow and anonlinear equation of state. To characterize the response of reactive materials, the modelalso accounts for heating produced by interfacial friction on shear (closed) cracks, andpossible melting, ignition and fast burning of the material next to the crack surfaces. Inaddition to studying explosive sensitivity (Dienes, 1982, 1984, 1996), the model has beenused to study in situ retorting of oil shale and it was shown that the anisotropy of thebedded rock accounted for the formation of an aspirin-shaped cavity formed by a sphericalcharge (Dienes, 1981). It has also been used to model damage and failure of a ceramicarmor under ballistic impact (Meyer et al., 1999; Zuo et al., 2003).

Based on SCRAM, our colleagues at Los Alamos have developed two simplified modelsfor damage and failure of brittle materials. Addessio and Johnson (1990) proposed anisotropic damage model (ISO-SCRAM) for the dynamic response of brittle materialsunder nearly isotropic stress states by assuming that the crack distribution remainsisotropic during the deformation. Their calculations compared favorably with shockcompression and release experiments of plate impact for three ceramics. Based on SCRAMand ISO-SCRAM, Bennett et al. (1998) and Hackett and Bennett (2000) explored amechanical–thermal model (Visco-SCRAM) for ignition of PBX 9501 under non-shockimpacts. They applied it to simulate a non-shock ignition experiment by Asay et al. (1997)in which a small piece of confined PBX 9501 is impacted by a steel plunger and thedisplacements and temperatures on the surface are measured. The computed in-planesurface-displacement field matches the measured field reasonably well. Their results fullysupport the notion of Dienes (1982, 1984, 1996) that frictional heating can cause ignitionand initiation in propellants and explosives under weak stimuli. Those simplified models(ISO-SCRAM and Visco-SCRAM) assume that the crack distribution remains isotropicduring the deformation; consequently, they do not account for the anisotropic nature ofdamage. SCRAM accounts for material anisotropy, either originally present in thematerial (such as bedding cracks in oil shale) or induced by cracking, by tracking theevolution of crack sizes in various directions. Several other researchers have alsoconsidered anisotropic distribution of microcracks in their models for damage and failureof brittle materials (e.g., Curran et al., 1993; Espinosa, 1995; Gailly and Espinosa, 2002).Another feature of SCRAM, which is not included in ISO-SCRAM and Visco-SCRAM, isshear dilatancy. Shear dilatancy can result from opening of cracks with certain orientations


under shear, or from joint opening due to asperities and particles, and is an importantaspect of the dynamic response of brittle materials (Scholz, 2002). Joint or crack openingas a source of dilatancy has been incorporated into SCRAM and plays a role in theresponse of ceramic armor to impact (Zuo et al., 2003).In addition to accounting for anisotropic cracking and shear dilatancy, SCRAM

accounts for several thermal–mechanical–chemical effects which are important to themodeling of damage and initiation of reactive materials. (1) Crack coalescence isaccounted for by dividing the cracks into 2 types: active and inactive. Active (isolated)cracks can grow under moderate stresses, while inactive (connected) cracks do not growbecause of intersections with other cracks. The evolution of active and inactive crack sets isgiven by a Liouville equation which can be solved analytically when the initial distributionis exponential (Dienes, 1985). (2) The effect of heating by frictional sliding of crack faceson the thermal response (i.e., melting, ignition, burning, explosion) is accounted for by asubgrid model in which the flux due to frictional heating serves as a boundary condition toa one-dimensional heat equation with an Arrhenius source term representing chemicalreactions. The model accounts for melting of the material adjacent to a shear-crack surfaceby allowing transition of the mechanical heating mechanism from solid frictional toviscous shearing of the molten layer when the crack surface is completely melted. Theheating from viscous shearing can be substantial due to the high shear rate in the very thinmolten layer next to the crack surface. (3) The effects of latent heat of melting areaccounted for by allowing a region of mixed solid and liquid phases where the continuumtemperature remains at the melting point. The kinetics of phase transition is governed bythe ratio of volumetric heat generated and the latent heat. The latent heat affects thespread of the melting front from the crack surface into the bulk solid, and is accounted forin the model by solving the Stefan problem (Dienes et al., 2002). (4) The intense viscousheating in the molten layer can bring the local temperature to the ignition point. Theburning process is represented by the burn model of Ward et al. (1998) (WSB). The massflux predicted by WSB is used to calculate the pressure inside the burned gases, which is apart of the loading used in our crack dynamics calculation.One of the main differences between SCRAM and Visco-SCRAM is that SCRAM

models the complete process of initiation in which the late-stage fast burn followingignition plays an important role whereas Visco-SCRAM is only intended for modeling theresponse leading up to ignition. As such, many of the important effects discussed above areeither not considered (e.g., burning of cracks following ignition and the effects of gaspressure in the reaction products on crack responses), or accounted for by some roughapproximations (e.g., approximating the interfacial frictional heating on crack surfaces asa volumetric heating source, instead of a flux boundary condition as done in the currentmodel and consistent with physics).In this paper, we will show that in the multiple-shock experiment of Mulford et al.

(1993) the particle velocities calculated by SCRAM agree well with the data for several(eleven) locations inside the PBX 9501 target where the measurements were taken. We willalso show that crack orientation and the temperature-dependent viscosity model haveimportant effects in the material response. We continue to seek out improvements tothe modeling of impact initiation, but selecting the most critical issue is difficult withoutgood experimental guidance. We will recommend an experiment involving a mixture ofinert and HMX (energetic) grains that would help clarify the mechanics and statistics ofhot spots.


The paper will proceed as follows. A fairly detailed description of the SCRAM modelincluding both mechanics and thermal–chemical analysis is given in Section 2. Two modelproblems will be provided in Section 3 as a verification of our theory and numericalimplementations. In Section 4, we will apply the model to PBX 9501, showing first a fit ofthe model to the stress–strain responses measured by Wiegand (1998a, b), then acomparison of the calculated initiation results with the multiple-shock experiments of PBX9501, using the material properties determined from Wiegand’s mechanical tests. InSection 5, we will discuss the effects of sample size and defect statistics on the safety ofexplosives and propellants, and recommend, for future work, a new test, intended toexamine microscopically the hot-spot mechanism used in the paper, namely, frictionalheating from sliding of shear cracks. A summary of the paper and some concludingremarks are given in Section 6.

1.1. Notation

For compactness, the following direct notation for vector and tensor operations (e.g.Gurtin, 1981) will be used in most of the paper:

i � dijei � ej; I � 12ðdikdjl þ dildjkÞei � ej � ek � el ,

u� v � uivjei � ej; A� B � AijBklei � ej � ek � el ,

Au � Aikukei; AB � AikBkjei � ej; CB � CijklBklei � ej ,

u � v � ukvk; A : B � trðATBÞ ¼ AikBik,

A^B � 12ðAikBjl þ AilBjkÞei � ej � ek � el ,

where i is the second-order identity; I, the fourth-order identity tensor; dij, the Kroneckerdelta; feigði ¼ 1; 2; 3Þ, an arbitrary basis; u, v, vectors; A, B, symmetric, second-ordertensors; C, fourth-order tensor.

2. Statistical crack mechanics model

It is generally known that materials can exhibit either ductile or brittle behavior, andthat the type of behavior depends on temperature, strain rate, and stress state. In SCRAM,we develop a general theory that subsumes both kinds of behavior by superimposing thestrain rates due to various physical processes. This is a generalization of the idea of Reussthat elastic and plastic strain rates should be superimposed, i.e.,

D ¼Xa

Da, (1a)

where D represents the symmetric part of the velocity gradient (the stretching, e.g., Gurtin,1981) and Da is the contribution of the physical mechanism of type a. The premise ofEq. (1a) is that deformation is often the consequence of numerous independent physicalprocesses which can be superimposed to obtain an overall deformation. This is illustratedschematically in Fig. 1. The difference in velocity between two points P1 and P2, can be

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Fig. 1. A conceptual diagram of the defects assumed in deriving the generalized superposition of strain rates

described by Eq. (1a). Detailed derivation is given by Dienes (1989, 1996, 2003).

J.K. Dienes et al. / J. Mech. Phys. Solids 54 (2006) 1237–12751242

written as

Du � u2 � u1 ¼X

Duc þX

Dud, (1b)

where the first sum is taken over the continuous regions between the defects, and thesecond sum is taken over the defects between continuous regions. The detailed derivationof Eq. (1a) from Eq. (1b), which is carried out for arbitrarily large deformation, has beengiven by Dienes (1989, 1996, 2003) and is somewhat involved. In its current form thesuperposition accounts for the opening, shear, growth and coalescence of penny-shapedcracks, and plastic flow

D ¼ Dm þDc þDg þDp, (2)

where Dm denotes the matrix (elastic) stretching (or the strain rate for small deformations);Dc, the contribution from opening and shearing of cracks (present for both stationary andgrowing cracks); Dg, the contribution due to crack growth; and Dp, the plastic stretching(strain rate). There is an extensive literature discussing the appropriateness of the additivedecomposition of the strain rate in the context of elasto-plasticity theory (e.g. Simo andHughes, 1998). We only note that for small elastic strain (compared to one), the additivesuperposition of strain rates Eq. (2) is consistent with the multiplicative decomposition ofthe deformation gradient (e.g. ABAQUS, 1998). For brittle materials considered in thispaper the elastic strains are always much smaller than one. Since the stress power is_e ¼ r : D, each term of Eq. (2) is associated with the energetics of a particular deformationmechanism. In a more general realization of the SCRAM algorithm, spherical voids,anisotropic high-pressure effects, and twinning, for example, could be added to the currentformulation.

2.1. Crack strain rate, Dc, and added compliance

Consider an ensemble of penny-shaped microcracks randomly distributed within astatistically homogeneous volume of a brittle material under multiaxial loading. Therequirements for a material volume to be statistically homogeneous are those discussed by


Krajcinovic (1998). The contribution to the stretching of a crack set, i.e., a homogeneousdistribution of similar penny-shaped cracks with the unit normal n and radius c, is (Dienes,1989, 1996)

DDcðc; nÞ ¼ bnðc; n; tÞc3ðdn þ dtÞDcDC, (3a)

dn ¼ ð2� nÞH½sn�ðn � rnÞn� n, (3b)

dt ¼ h1� ZiððrnÞ � nþ n� ðrnÞ � 2ðn � rnÞn� nÞ, (3c)

where r is the far-field (applied) Cauchy stress, and r represents the Green–Naghdi (polar)rate i.e., r � _rþ rX�Xr, with X representing the rate of material rotation (e.g., Dienes,1979, 1996; Belytschko et al., 2000). The elastic constant b � 8ð1� nÞ=½3Gð2� nÞ�, arisesfrom analytic solutions for open and closed penny-shaped cracks (Sack, 1946; Segedin,1950; Keer, 1966), with G and n the elastic shear modulus and Poisson’s ratio of theisotropic matrix (undamaged) material, respectively. nðc; n; tÞ is the number densitydefining the distribution of crack radii and orientations, which evolves with the time t.That is, nðc; n; tÞDcDC represents the number density of cracks (number of cracks per unitvolume) whose radii are between c and cþ Dc, and have a unit normal within a small solidangle DC around n (e.g., Oda, 1983; Dienes, 1985).

In Eq. (3b), sn � n � rn is the normal component of the remote (far-field) traction and H

is the Heaviside function (one for positive arguments and zero for negative arguments) sothat H½sn� ¼ 1 for an open crack and H½sn� ¼ 0 for a closed crack. The quantity Z in Eq.(3c), arising from the effects of interfacial friction between closed-crack faces (reducing theinterfacial sliding), is given by

Z �mh�sni

sn

, (3d)

where sn � ½n � r2n� ðn � rnÞ2�1=2 is the shear component of the remote traction and m thefriction coefficient (static or dynamic depending on whether the crack is sliding or at rest).The angled brackets in Eqs. (3c) and (3d) denote the Macaulay bracket, which takes thevalue of the argument when positive and is zero otherwise. The Macaulay bracket inEq. (3c) is used so that sliding of the crack faces (hence stretching due to crack shearing) isprohibited when the crack is locked by friction (�msn4sn, or Z41). This introduces thehysteretic effect of solid friction. It follows from the definition that Z ¼ 0 for an open crackðsn40Þ, and 0ph1� Zip1 for a closed crack. For certain orientations, closed cracks maybe friction-locked, i.e., h1� Zi ¼ 0; then both dn and dt vanish.

For closed (shear) cracks ðsnp0Þ, it follows from Eqs. (3a)–(3c) that

DDcðc; nÞ ¼ bnðc; n; tÞc3DcDCdt ¼ h1� Zibnðc; n; tÞc3DcDCcsr. (4a)

The shear fabric tensor (fourth-order) cs is (Oda et al., 1984)

cs � b� 2a, (4b)

where

a � n� n� n� n, (4c)

b � i^ðn� nÞ þ ðn� nÞ^i. (4d)


In Eq. (4d), the operator ‘‘^’’ is the cross-composition of two symmetric second-ordertensors defined in Notation.Similarly, for open cracks ðsn40Þ, Eqs. (3a)–(3c) reduce to

DDcðc; nÞ ¼ bnðc; n; tÞc3DcDCcor (5a)

with the open fabric tensor

co ¼ ð2� nÞaþ cs ¼ b� na. (5b)

The total stretching due to an ensemble of microcracks of all sizes and orientations (thecrack strain rate) is obtained by summing the contributions in Eq. (3a) over all crack sizesð0pcp1Þ and orientations (Dienes, 1989, 1996),

Dc ¼XC;c

DDcðc; nÞ ¼ bXC;c

nðc; n; tÞc3ðH½sn�co þ ð1�H½sn�Þh1� ZicsÞDcDCr. (6)

In terms of the usual polar coordinates y and f, the incremental solid angle is the area ofan element on the unit sphere, DC ¼ sinfDyDf. Cracks have symmetry such that areversal of 1801 leaves them unchanged. Thus, half the unit sphere is sufficient tocharacterize crack orientation; consequently, the integration limits for orientation are0pyp2p and 0pfpp=2.

2.1.1. Added compliance

It follows from Eq. (6) that the stretching due to an ensemble of cracks is the product ofthe added compliance and the stress rate:

Dc ¼ ðCoþ CsÞr. (7)

The added compliance due to open and closed (shear) cracks can be written as

Co¼ b

XC

H½sn�F ðn; tÞcoDC, (8a)

Cs¼ b

XC

ð1�H½sn�Þh1� ZiF ðn; tÞcsDC, (8b)

where the summation is taken over crack orientations. In SCRAM, the continuousdistribution of crack orientations n is approximated with a number of discrete orientations(crack bins). Crack orientations range over 2p, half a unit sphere, and one particularlyconvenient discretization scheme is to divide the hemisphere into N elements with equalareas so that DC ¼ 2p=N.In Eqs. (8a) and (8b), the function F ðn; tÞ is the third moment of the density-distribution

function nðc; n; tÞ:

F ðn; tÞ �

Z 10

nðc; n; tÞc3 dc. (9)

This function comes out of the analysis (Dienes, 1989) and was termed ‘‘effective crackvolume’’ previously (Dienes, 1996), though it is not the real volume of a crack, which iszero for cracks with zero thickness. Note that the sum over crack radii given by Eq. (6) hasbeen replaced by integrals over a continuous distribution of crack radii. Analyticexpressions for the integrals are developed in Section 2.4.


Each crack orientation (bin) na ða ¼ 1; . . . ;NÞ has its own number density distributionfunction, nðc; na; tÞ, and anisotropic damage is captured by tracking the evolution of thosedistribution functions with time. Cracks within the crack bin a are either open or closeddepending on the sign of sna ¼ na � rna for the bin; the third moment F ðn; tÞ in Eq. (9) forthe crack bin contributes either to Co [Eq. (8a)] or Cs [Eq. (8b)], accordingly. In general,whether a crack bin is open or closed depends on the stress state and the crack orientationof the bin, and some bins are open and some are closed. Under certain stress states,however, the status (open or closed) of a crack bin is independent of the bin orientation.For example, when the principal stresses are all positive (tensile), all crack bins are open.Consequently,

Co¼ b

2pN

Xa¼1;N

F ðna; tÞcoðnaÞ, (10a)

Cs¼ o. (10b)

Conversely, when the principal stresses are all negative (compressive), all crack bins areclosed, and

Co¼ o, (11a)

Cs¼ b

2pN

Xa¼1;N

h1� ZðnaÞiF ðna; tÞcsðnaÞ. (11b)

Furthermore, for certain stress states where the principal stresses are all compressive andclose to each other, cracks in all bins are closed and locked by friction; consequently, thereis no added compliance (or damage) due to cracks. This dependency of the materialcompliance on the stress state does not occur in a truly linear material model and impliesthat the material model is nonlinear, even though linear fracture mechanics is used toderive the crack opening and sliding. The range of the stress states for which all cracks arefriction-locked increases with the friction coefficient and has been given by Zuo and Dienes(2005) and Dienes et al. (2004).

2.2. Stretching due to crack growth, Dg

Cracks can contribute to the stretching by growing in size. The stretching due to the rateof growth of open cracks, _c, is

Dog ¼

_Cor, (12a)

_Co¼ b

XC

H½sn� _F ðn; tÞcoDC. (12b)

Similarly, for closed cracks,

Dsg ¼

_Csr, (13a)

_Cs¼ b

XC

ð1�H½sn�Þh1� Zi _F ðn; tÞcsDC. (13b)


It follows from Eq. (9) that the rate of the third moment F ðn; tÞ is given by

_F ðn; tÞ ¼

Z 10

qnðc; n; tÞ

qtc3 dc. (14)

The relationship between _F ðn; tÞ and _c is given in Section 2.4. The stretching due to thegrowth of open and closed cracks is then

Dg ¼ Dog þDs

g ¼ ð_Coþ _CsÞr. (15)

The rate of change of the added compliance (or material damage) is the sum ofcontributions of open and closed cracks

_Coþ _Cs

¼ bXC

_F ðn; tÞðH½sn�co þ ð1�H½sn�Þh1� ZicsÞDC. (16)

It is interesting to observe that the stretching due to crack growth is like a viscous strainrate in that it is proportional to the stress tensor, not its rate. Crack growth andcoalescence are discussed next.

2.3. Crack growth

The classic theory of crack instability and growth assumes that cracks grow at highspeed when the applied stress exceeds a critical level causing cracks to become unstable(Freund, 1990). Practical observations on a variety of materials reported by Stroh (1957)show that crack tip speeds can approach roughly a third the longitudinal wave speed, whileFreund (1990) argues on theoretical grounds that they approach the Rayleigh wave speedat high stress, somewhat faster than the Stroh result. On the other hand, it has beenobserved (Charles, 1958; Evans, 1974; Kanninen and Popelar, 1985) that, when the stress isbelow the critical level for instability, cracks may grow at low speeds. (This regime may begoverned by thermally activated diffusion rather than dynamic processes.) The empiricalresults can be expressed as

_c ¼ cRgðr; n; cÞ

g1

� �n

for gðr; n; cÞpgtr, (17a)

where gðr; n; cÞ is the energy-release rate for the crack with normal n and size c. Theterminal crack speed cR is the Rayleigh wave speed, and n is a model parameter typicallybetween 5 and 10. The constants g1 and gtr are related to the effective surface energy g andthe parameter n (Dienes and Kershner, 1998), and are given next.The above result applies when the stress is below a critical value. At high stress, when

cracks are unstable, Freund (1990) gives the result

_c ¼ cR 1�gc

gðr; n; cÞ

� �for gðr; n; cÞXgc, (17b)

where gc ¼ 2g is the critical energy-release rate, twice the effective surface energy g. Itfollows that for an unstable crack ðgðr; n; cÞXgcÞ, the crack growth speed asympto-tically approaches the terminal crack speed cR as the energy release rate increases. Theparameter gtr controls where the slow, diffusion-controlled crack growth transitions tofast, dynamic crack growth. g1 and gtr are found by requiring that the magnitude andslope of the curves defined by Eqs. (17a) and (17b) be continuous at the transition point


where gðr; n; cÞ ¼ gtr

gtr ¼ 1þ1

n

� �gc4gc, (18a)

g1 ¼ ðnþ 1Þ1=n 1þ1

n

� �gc4gtr. (18b)

For a typical value of n ¼ 6, the transition occurs at gtr ¼ ð7=6Þgc, slightly above thecritical energy-release rate, that is, the slow crack growth formulation extends into theunstable crack regime by a small amount. At the cross-over, gðr; n; cÞ ¼ gtr, the crackgrowth speed is _c ¼ cR=ðnþ 1Þ, which is ð1=7ÞcR for n ¼ 6. In this approach, we no longerdeal with crack instability, but with a transition from slow to fast crack growth. However,the growth is very slow for stable cracks ðgðr; n; cÞogcÞ. It follows from Eq. (17a) that for astable crack bin,

_cogc

g1

� �n

cR ¼nn

ðnþ 1Þnþ1cR. (19)

For n ¼ 6, the crack growth speed is _co0:057cR for stable cracks.The current formulation is a generalization of Dienes and Kershner (1998) in which the

stress intensity factor is used for calculating crack growth, rather than the energy-releaserate used here. When the crack is under mixed-mode loading (mode-I, and both mode-IIand mode-III for a penny-shaped crack), it is more convenient to use the currentformulation based on the energy-release rate. In the work of Dienes and Kershner (1998)and Bennett et al. (1998), since only closed cracks are considered, the two formulations areequivalent.

For a penny-shaped crack, the energy-release rate can be written as (Rice, 1984)

gðr; n; cÞ ¼4

pð1� nÞð2� nÞ

f ðr; nÞcG

. (20)

The expression for the stress function f ðr; nÞ depends on whether the crack is open (thenormal component of traction is tensile) or closed (the normal component is compressiveand controls the interfacial friction). For an open crack ðsn40Þ, both normal and shearstresses contribute to crack instability and the stress function f ðr; nÞ is (Keer, 1966)

f ðr; nÞ ¼ 1�n2

� �s2n þ s2n, (21a)

where sn and sn are the normal and shear components of the remote traction defined inconnection with Eq. (3d).

For a closed crack ðsnp0Þ, the friction on the crack surface is stabilizing. If theCoulomb friction law is assumed, then the stress function f ðr; nÞ is (Rice, 1984)

f ðr; nÞ ¼ hsn þ msni2. (21b)

When the crack is friction-locked ðsno� msnÞ, it follows from Eqs. (20) and (21b) thatf ðr; nÞ ¼ 0 and gðr; n; cÞ ¼ 0. Thus, the crack is friction-locked and remains stable andstationary ð_c ¼ 0Þ for any applied stress.


2.4. Crack coalescence

If cracks grew without coalescence the statistical problem would be trivial but, in fact,crack intersections are a vital part of material failure. Without coalescence the largestcrack would be the most unstable, and it would grow without bound, separating thematerial rapidly into 2 parts while the others are unaffected. This may be observed in somecircumstances, but SCRAM is concerned with problems where damage occurs as a resultof large numbers of microcracks. Failure in granular materials such as explosives,propellants, and ceramics is the result of microcrack growth and coalescence, and no singlemicrocrack dominates. This is consistent with the observation that the stress–strain curvesfor brittle materials often show a gradual increase of compliance as the stress increases.Furthermore, we observe crack networks with complex structure in micrographs ofdamaged materials. This growth and coalescence of cracks result, ultimately, infragmentation if the loading is of sufficient duration and intensity, notably in spall(Antoun et al., 2003).The first stage in formulating SCRAM crack statistics is to divide cracks into two

categories, active (isolated) and inactive (connected). Active cracks are capable of growthas a result of either the instability that occurs at stresses above the critical value (fastgrowth) or the thermally activated processes (slow growth). Inactive cracks are not capableof growth because they have intercepted a significant number (usually 3–4) of other cracks,thereby eliminating large portions of the regions of high stress at the crack edge. Theseinactive cracks are the ones we consider to form the faces of fragments. Of course, in realfragments the faces are polygons rather than circles, but we take the circles as a roughapproximation to the true area. Active and inactive cracks can be observed in mostgeological formations-active cracks are like the preceding dash and inactive cracks areobserved as something like a T or H on the face of an outcropping or a rock (Dienes,2005). Active cracks have a statistical distribution Lðc; n; tÞ representing the averagenumber of cracks per unit volume exceeding c in radius, while inactive cracks have adistribution Mðc; n; tÞ. The dependence on time, t, governs only during periods of crackgrowth.In terms of the distribution functions, Lðc; n; tÞ and Mðc; n; tÞ are

Lðc; n; tÞ ¼

Z 1c

nactðc; n; tÞdc, (22a)

Mðc; n; tÞ ¼

Z 1c

ninaðc; n; tÞdc. (22b)

Since a crack is taken to be either active or inactive, the total number of cracks is the sumof the cracks in the active and inactive sets:

nactðc; n; tÞ þ ninaðc; n; tÞ ¼ nðc; n; tÞ. (23c)

It is shown by Dienes (1978, 1985) that the distributions are related by a Liouville equation

qL

qtþ _c

qL

qc¼ �

qM

qt, (24)

which holds for each crack orientation (n) and does not involve other crack orientationsbecause cracks in various orientations are assumed to remain plane and to grow


independently. For compactness, the explicit dependence on crack orientation and time isdropped in this section when the context is appropriate. In the following, the crack speed _cis assumed constant in time and crack size (it still varies with the crack orientation), thoughit is straightforward to handle the case when it is variable (Dienes, 1985).

Based on experimental observations (Seaman et al., 1976; Curran et al., 1987; Curranand Seaman, 1996; Scholz, 2002), it is assumed here that the initial distribution isexponential in crack size (Dienes, 1985)

Lðc; n; 0Þ ¼ L0ðnÞ expð�c=c0ðnÞÞ, (25)

where c0ðnÞ and L0ðnÞ are the mean initial crack size (radius) and total number of cracksper unit volume per 2p, respectively. It is typically assumed that the crack number densityand mean crack size are isotropic, initially, though the theory allows for an initiallyanisotropic distribution of cracks. For oil shale an additional set of large cracks in thebedding planes was generated (Dienes, 1981).

If the crack density is not too large, the rate at which cracks coalesce is proportional toLðc; n; tÞ,

qM

qt¼ kL, (26a)

where k is a constant that depends on the initial distribution of cracks. An analysis of themean free path given by Dienes (1989) leads to the expression for the coalescence constant:

k ¼ 2p4L0c20 _c=a. (26b)

The constant a is the number of intersections required to convert an active crack to onethat is inactive (termination of crack growth). It can be estimated on intuitive grounds tolie between 3 and 4 [analysis of one series of experiments leads to 3.4 (Dienes, 2005)]. Since,in general, L0, c0, _c depend on the crack orientation n, the constant k is in generalorientation dependent, k ¼ kðnÞ, though in practice it has been assumed constant.

With the initial distribution of cracks given by Eq. (25), the Liouville equation has theanalytical solution:

L ¼ 0; M ¼1

b½ð_c=c0Þe

�kc=_c � ke�c=c0 � be�kt�L0 for co_ct, (27a)

L ¼ L0e�ðc�_ctÞ=c0�kt; M ¼

kbðebt � 1Þe�c=c0L0 for c4_ct (27b)

with b � _c=c0 � k. The total number of cracks should, and does, remain equal to L0. Tosee this note that the number of active cracks with radius exceeding zero is the same as thenumber exceeding _ct, L ¼ L0e

�kt. The number of inactive cracks reduces, for c ¼ 0, toM ¼ L0ð1� e�ktÞ, so that the sum is indeed L0. Recall that the time t here is the durationof the crack growth, which can be significantly less than the total elapsed time.

Let FLðn; tÞ and FM ðn; tÞ be the contributions from active and inactive cracks to the thirdmoment

FLðn; tÞ ¼ �

Z 10

Lcðn; c; tÞc3 dc, (28a)


F Mðn; tÞ ¼ �

Z 10

Mcðn; c; tÞc3 dc, (28b)

where the subscript c denotes the derivative with respect to crack size. Analyticalexpressions for F Lðn; tÞ and FM ðn; tÞ have been found: for active cracks,

F Lðn; tÞ ¼ L0 expð�ktÞðc3 þ 3c0c2 þ 6c20cþ 6c30Þ, (29a)

where

c � _ct ¼ cðn; tÞ � c0 (29b)

is the extent of crack growth for the crack orientation n, and cðn; tÞ is the current meancrack size for each orientation. For inactive cracks,

F Mðn; tÞ ¼L0

c0b½6ð~c4 � c40Þ þ expð�ktÞH�, (30a)

where ~c � _c=k and

H � c3ðc0 � ~cÞ þ 3c2ðc20 � ~c2Þ þ 6cðc30 � ~c

3Þ þ 6ðc40 � ~c4Þ. (30b)

The corresponding rates of change of the third moments are

_F Lðn; tÞ ¼ L0 expð�ktÞ½ð3c2 þ 6c0cþ 6c20Þ_c� kðc3 þ 3c0c2 þ 6c20cþ 6c30Þ�, (31a)

_F Mðn; tÞ ¼L0kc0b

expð�ktÞ½�kH þ ð3c2ðc0 � ~cÞ þ 6cðc20 � ~c2Þ þ 6ðc30 � ~c

3ÞÞ_c�. (31b)

The sums of the contributions from active and inactive cracks are the third momentsneeded for calculating the added compliance and the stretching due to crack growth

F ðn; tÞ ¼ F Lðn; tÞ þ FM ðn; tÞ, (32a)

_F ðn; tÞ ¼ _F Lðn; tÞ þ _FM ðn; tÞ. (32b)

When the number density of cracks becomes large it can no longer be assumed that therate of coalescence is constant and equal to the initial value. In that case it is still possibleto make progress toward an analytical solution, but the analysis is much more complex, asreported by Dienes (1985, 1989). The general solution is not used in the current SCRAMprogram, though the asymptotic solution is; the number of inactive cracks at late times isused in the computer algorithm to limit crack growth.For crack bin a ða ¼ 1; . . . ;NÞ, the crack growth is computed using Eq. (17) with n ¼ na

and c ¼ cðna; tÞ, the mean crack radius for the bin. The growth rate of the mean crackradius is

_cðna; tÞ ¼ _cðr; na; cðna; tÞÞ; a ¼ 1; . . . ;N, (33)

where _cðr; na; cðna; tÞÞ is given by either Eq. (17a) or Eq. (17b), depending on the energy-release rate, g. Since the energy-release rate depends on the mean crack radius andorientation of the bin, the crack growth rates are different for different bins, resulting inanisotropic damage. Similarly, Eqs. (32a) and (32b) for the third moments apply to eachcrack bin a with constants k, b, and c, ~c for the bin.It is assumed that for each crack bin, the growth rate for cracks of all sizes is the same as

that for the crack with mean crack radius of the bin, _c ¼ _cðna; tÞ. This approximation is


justified by noting that the crack distribution affects the overall behavior only through thecompliance, which involves integrals of the type (Eq. (28))

�

Z 10

Lcðc; n; tÞc3 dc. (34)

Since Lcðc; n; tÞ is nearly exponential in crack size c, with mean cðn; tÞ, the integrand has amaximum for c ¼ 3cðn; tÞ. Thus, it is only important, to first order, to represent theintegrand correctly near c ¼ 3cðn; tÞ. In particular, the distribution for cocðn; tÞ does notcontribute significantly to the overall behavior.

2.5. Stress rate-stretching relationship

The matrix stretching is related to the stress rate by

Dm ¼ Cmr, (35)

where Cm is the compliance of the matrix material. For an isotropic, linear elastic matrixmaterial, Cm

¼ ð1=ð3KÞ � 1=ð2GÞÞi� i=3þ 1=ð2GÞI where K � 2Gð1þ nÞ=ð3ð1� 2nÞÞ is thebulk modulus, and I is the fourth-order identity tensor defined in Notation. SubstitutingEqs. (7) and (35) into Eq. (2) gives

D ¼ ðCmþ Co

þ CsÞrþDg þDp. (36)

It follows that the elastic stretching is

De ¼ Dm þDo þDs ¼ Cr, (37)

where C is the overall compliance of the material containing an ensemble of cracks:

C ¼ Cmþ Co

þ Cs. (38)

The inelastic stretching is the sum of contributions from crack growth, Dg, andplastic flow Dp. In a computer program, the total stretching D and rate of materialrotation X are obtained from the momentum equation and one needs to find the stressrate. To that end, the constitutive law of Eq. (36) can be inverted to give the polar rate ofstress as

r ¼ C�1De ¼ C�1ðD�Dg �DpÞ. (39)

It is assumed here that the plastic response of the material can be represented bykinematic hardening. Furthermore, it is assumed that the yield surface is of the form

fðr;aÞ � 12ðrd � aÞ : ðrd � aÞ � Y 2 ¼ 0, (40)

where rd � r� ðtrrÞi=3 is the stress deviator; a, the ‘‘back stress’’ (center of the yieldsurface); ‘‘:’’, the scalar product defined earlier; and Y, the flow stress in simple shear.Consideration of hardening and hysteresis suggests that the center of the yield surface instress space evolves linearly with the plastic stretching (Prager, 1955),

a ¼ bDp (41)

with b the kinematic hardening modulus. The plastic stretching is assumed to havethe form

Dp ¼_lðrd � aÞ, (42)


where _l is the plastic multiplier, which can be found by requiring that the stress stateremain on the yield surface during plastic deformation (the consistency condition),fðr;aÞ ¼ 0:

_l ¼~r : C�1ðD�DgÞ

2bY 2þ ~r : C�1 ~r

, (43)

where ~r � rd � a. With the plastic multiplier found, the plastic stretching is then obtainedby substituting Eq. (43) into Eq. (42). With the total stretching D prescribed, the plasticstretching Dp can be determined by using the total compliance C and the stretching due tocrack growth Dg given by Eqs. (38) and (15), respectively.Finally, the stress rate is

_r ¼ C�1ðD�Dg �DpÞ þXr� rX. (44)

Eq. (44) can be used to update the stress. A simple kinematic hardening model is used here,but it is straightforward to incorporate other plasticity models when the experimental datawarrant them.

2.6. Frictional crack heating and melting

Consider the effects of heating due to crack friction on the thermo-chemical response ofa reactive material. In SCRAM, we model this heating by means of a subgrid calculation inwhich the frictional heating involves only a few microns normal to the crack surface. It isassumed that the heated zone is thin compared to the crack radius, so that the critical heatconduction can be taken to be one dimensional and normal to the crack plane. The heatingper unit area (flux) due to interfacial friction is

_q ¼ mh�sniv, (45)

where m is the solid friction coefficient defined earlier, and v is the interfacial slidingvelocity. This heating is taken as a boundary condition for the modified Frank–Kame-netzky equation for one-dimensional heat conduction (Frank-Kamenetzky, 1942; Mader,1979)

rCv_T ¼ kTxx þ r _Q, (46)

where T denotes the absolute temperature, r mass density, Cv heat capacity at constantvolume, k thermal conductivity. The coordinate axis ðxÞ is along the crack normal with theorigin at the center of the crack. The rate of heating per unit mass, _Q, for a reactivematerial, is the sum of chemical and mechanical sources:

_Q ¼ QrZ expð�EA=RTÞ þ1

rðð1� f ÞY þ f m_eÞ_e. (47)

The first term on the right accounts for chemical reactions and the second for mechanicaldissipation, which is the sum of plastic work in the solid phase and viscous heating of themelted liquid. Qr is the heat of reaction per unit mass for chemical decomposition;Z expð�EA=RTÞ Arrhenius reaction rate; Z is the pre-exponential (frequency) factor; EA

activation energy; R the universal gas constant; f ð0pfp1Þ, fraction of the phase transition(solid to liquid) completed (f ¼ 0 below the melting point and f ¼ 1 for complete melting);Y flow stress of the solid phase as before; m the viscosity of the melt; _e the shear strain rate


in the melting or molten material adjacent to the crack surface. The thermal boundarycondition at the crack surface ðx ¼ 0Þ is

�kTx ¼ _q. (48)

When the material adjacent to the crack surface is melting, the interface sliding velocity v

of Eq. (45) used to compute _q is not appropriate, but, rather, the sliding velocity should bea fraction of the crack-sliding velocity vc, which decreases as the fraction increases,

v ¼ ð1� f Þvc. (49)

Since we do not know the exact relationship between the total crack sliding velocity ðvcÞand that associated with the solid phase, a simple linear interpolation is used here. Theremainder of the crack velocity is associated with local shearing in the melting and moltenmaterial adjacent to the crack. Thus,

_e ¼ fvc=l, (50)

where l is the width of the region at or above the melting point. (A typo in Dienes, 1996says ‘‘below’’ rather than ‘‘above’’.) The crack sliding velocity vc is computed as the centralvelocity of a penny-shaped shear (closed) crack:

vc ¼3b2p½h_sn þ m _snicþ hsn þ msni_c�, (51)

where the elastic constant b has been defined in Eq. (3a), and c is the crack radius. The firstterm is due to the change of traction on the crack surface, and the second due to crackgrowth ð_c40Þ. The second term normally dominates when the crack is unstable.

It is known that the melting point plays an important role in the sensitivity of explosives(Bowden and Yoffe, 1952). At low temperatures ðToTmÞ the crack faces are solid, butwhen the temperature at the crack interface reaches the melting point, a phase transition isinitiated. During the melting process ð0ofo1Þ, the temperature remains at the meltingpoint ðT ¼ TmÞ, and a part of the heat generated from chemical reaction and mechanicaldissipation is consumed in the phase transition. The rate of melting can be calculated fromthe rate of heat generation:

_f ¼ _Q=L, (52)

where L is the latent heat of melting, and _Q has been given by Eq. (47). The velocity of themelting front moving into the solid phase, _lðtÞ, is given by the Stefan condition (Crank,1984)

rL_lðtÞ ¼ DðkTxÞ, (53)

with DðkTxÞ the discontinuity in the heat flux across the melting front.Evidence for melting in shear cracks is scant, but it has been observed in pseudotachyltes

(Scholz, 2002) and in shocked materials (Grady, 1988; Schmitt et al., 1989) and in animpacted TNT (Howe et al., 1985). It is appropriate to mention here that there is adistinction between shear cracks and shear bands as traditionally idealized, but thisdistinction may disappear as it becomes recognized that these are two extreme cases of ageneral kind of localization that involves plastic flow at one extreme and brittle failure atthe other. Shear bands in reactive materials are traditionally modeled with a one-dimensional nonlinear equation representing softening behavior at high temperature(Dienes, 1986), while shear cracks are typically represented with three-dimensional linear


elasticity with interfacial friction (e.g., Rice, 1984, 2000; Zuo and Dienes, 2005). Thisfriction is uniform over most of the crack, but nonlinear in the sense that it changesdiscontinuously when the sign of the local velocity changes. The ultimate effect of theresulting behavior, which can be a stick–slip instability, can be quite complex (e.g., Heaton,1990; Rice, 2000).

2.7. Ignition and burning of reactive materials

When melting is completed ðf ¼ 1Þ, the heat flux ð _qÞ from solid friction and thevolumetric plastic dissipation vanish. At this point the only mechanical dissipationmechanism is viscous shearing. Normally viscous heating (m_�2 in Eq. (47)) is small forliquids, but it is significant in the molten layer next to a shear-crack surface because thestrain rate is very high (the thickness of the layer l is on the order of a micron in theexamples studied), especially when the crack is unstable (high vc). Our previous analyses ofa solid propellant showed that the cracks had grown to centimeters in length at ignitiontime (Dienes, 1996). This intense viscous heating causes the temperature in the thin moltenlayer to rise rapidly. Ignition occurs when the peak temperature reaches a critical value Tc

above which the Arrhenius source term in Eq. (47) becomes important, causing a rapid riseof temperature. According to a theory of Linan and Williams (1971), the criticaltemperature (ignition point) for a reactive material under constant energy flux _q is

T cðT i; _qÞ ¼EA

R

1

lnðrQrZkT i= _q2Þ, (54)

where T i is the initial temperature, and the other variables have already been defined.Though the energy flux due to frictional heating normally changes with time, the criticaltemperature given above does provide a useful check on our numerical scheme. The criticaltemperatures for five explosives (HMX, TNT, PBX 9501, PBX 9502, PBX 9504) for_q ¼ 3060 cal=cm2=s and T i ¼ 300K have been calculated by Dienes (1995). The resultsrange from 755K for HMX to 1179K for TNT.Following ignition, there is a rapid (sub-nanosecond) transition to burning, so that burn

can be considered to begin immediately when the ignition point is reached ðT4T cÞ. Thisprocess can be represented by the burn model (WSB) of Ward et al. (1998). This is a two-step model with high activation energy in the condensed phase followed by a gas-phasereaction with vanishing small activation energy ðEg=RT51;Eg the activation energy of thegas phase). Compared with the burn models assuming high activation energy for the gasphase, the WSB model provides a better match of both the temperature profile and theburn rate behavior for several explosives measured experimentally (Ward et al., 1998).For our application, WSB can be briefly summarized as a set of three coupled equations

for mass flux (m), surface temperature ðT sÞ of the condensed phase, and the flame thicknessðxgÞ. Specifically:

mðT sÞ ¼AcT

2s e�Ec=T s

EcðT s � T0 �Qc=2Þ

� �1=2, (55)

T sðm;xgÞ ¼ T0 þQc þQg

xgmþ 1; xgðmÞ ¼

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ 4Dg

p�m

, (56a,b)


which are Eqs. (13)–(15) of WSB. In the above equations, we have followed the notation ofWSB, in which all the variables are dimensionless, and the subscripts c and g denote thecondensed and gas phases, respectively. Ac is the reaction rate prefactor (the frequencyfactor); Ec, the activation energy; Qc, the chemical heat release; and T0, the initialtemperature for the condensed phase. Qg is the chemical heat release in the gas-phasereaction. The parameter Dg is the Damkohler number for the gas phase,

DgðpgÞ ¼kgBgp

2gMW2

cpðmrRÞ2

(57)

with kg the thermal conductivity; Bg, the reaction rate prefactor; pg, the gas pressure insidethe crack; MW, the molecular weight; cp, the specific heat at constant pressure; and mr, areference mass flux for the gas phase. The nonlinear coupled equations are solved by ageneralized Newton’s method (Dienes and Kershner, 2001). The model constants providedby WSB are used in our calculations, without modifying the units (MKS). The initialpressure at ignition is given by the ideal gas law

pi ¼ rg0RT f=MW, (58)

where rg0 is the initial density of the reactive product (gas); and T f , the flame temperature.The subsequent pressure is calculated assuming that the burn products following ignitionobey an adiabatic gas law

pg ¼ piðrg=rg0Þg (59)

with rg the running (current) density of the burned gas and g � cp=cv, the ratio of thespecific heats at constant pressure and volume. Consider a mass of reacted material, whichis assumed to fill a cylinder of crack radius c and height of crack roughness d0 at the time ofignition. The initial amount of mass per unit crack area at ignition is then

mg0 ¼ rg0d0. (60)

As discussed by Dienes and Kershner (2001), the initial gas density at ignition, rg0, isdifficult to estimate accurately due to the complexity of the process involved during high-rate shearing and liquefaction of irregularities of crack interfaces (in geophysics suchsliding interfaces are typically separated by a layer of fine particles, Sammis and Biegel,1989). In the current calculations, rg0 is taken as a fraction (1/10) of the solid density andthe crack roughness is assumed to be 1 mm. During the burn, the pressure in the burnedgases opens the crack forming an oblate ellipsoid whose volume on one side of the crack is2pc2ðdþ rbÞ=3 with d the current opening of the crack and rb, the distance from the cracksurface to the gas-condensed phase interface. The current density is then (Dienes andKershner, 2001)

rg ¼mg

23ðdþ rbÞ

. (61)

The amount of the burned gas per unit crack area inside the cavity and the distance rb ofthe interface during the burning are given by the mass flux

mg ¼ mg0 þ

Z t

tc

_M dt, (62a)

ARTICLE IN PRESS

Fig. 2. An overview of the stages involved in initiation according to SCRAM. (a) Shear cracks form under shock;

(b) cracks grow easily inside HMX crystals, but are inhibited by binder; (c) cracks grind and generate frictional

heat; (d) HMX reaches ignition point ðT4T cÞ; (e) burned gases open cracks creating high-pressure zone. Opening

is inhibited by inertia, stiffness, and damping; (f) burning increases general pressure, accelerating burn in

adjoining hot spots and (g) cracks coalesce when percolation threshold is exceeded, causing general explosion.


rb ¼

Z t

tc

_M

rgdt, (62b)

where tc is the time of ignition, and _M � mmr is the dimensional mass flux in the notationof WSB. The gas pressure pg given by Eq. (59) is then used as a part of the loading in crackdynamics calculations (Dienes, 2001; Dienes and Kershner, 2001),

reff ¼ rþ pgn� n, (63)

where reff is the effective stress, r and n, as before, the externally applied stress and thenormal of the crack, respectively.This process continues till a critical condition when there is a transition to rapid burn, as

described by Dienes and Kershner (2001). The entire scenario is summarized in Fig. 2,from initial shock to violent reaction.

3. Verification and validation

The SCRAM theory discussed above has been implemented into three explicit, three-dimensional finite-element codes: PRONTO (Taylor and Flanagan, 1987), DYNA(Hallquist and Whirley, 1989), and EPIC (Johnson et al., 2001). The details ofimplementing a new constitutive model into the codes can be found in the users’ manuals


provided by the code developers. We have paid particular attention to verifying andvalidating the programming and physical concepts that are combined in SCRAM. This isconsidered crucial at this point because the introduction of new physics such as the set-sizetheory for percolation processes and a random crack size generator makes the damagebehavior very complex, and thus it is difficult to confirm the calculations. This is especiallythe case where three-dimensional behavior is involved, and three-dimensions are requiredto represent impacts that involve radial cracking and extensive damage. Radial cracking isnot a deterministic process; the cracks form at roughly random angles and may numberfrom 3 to more than 20. As a part of the initial condition for the material, a randomnumber generator is used to generate the number of cracks in each cell with eachorientation, but the statistics of crack size for each cell and orientation is based on theexponential distribution which is discussed by Korvin (1989) and for which new evidence isgiven by Antoun et al. (2003) and Scholz (2002). With this random cracking stronglyaffecting the details of impact response, it is difficult to completely verify the coding, andthis is the primary reason for the needed emphasis on validation at this time.

One kind of test problem concerns the response of a thick uniform ring to suddeninternal pressure. We have developed an analytic solution to the linear problem bystandard methods, and confirmed that the finite-element solution is indeed uniform andconsistent with the analytic solution. The real purpose of ring calculations, however, isverification that the damage is homogeneous around the ring when SCRAM is activated.This is far from trivial for a number of reasons. First, the finite-element programs arenoisy. Second, there may be bugs and/or noise in the SCRAM algorithm or itsimplementation. Third, configurations with only a few crack orientations are anisotropicand this anisotropy leads to inhomogeneous behavior in a geometry that is nominallysymmetric. Calculations of explosions in oil shale demonstrated the utility of this approachto verification (Dienes, 1981). Calculations with large numbers of orientations test for bugsbut also determine how many orientations are needed to get accurate damage response. Wehave found that 3 orientations gives very erratic results, 9 orientations is not enough, 30 isadequate, and 480 gives good uniformity. This conclusion, which is illustrated in Fig. 3,requires some elaboration, however. (Note that work on related approaches typicallyinvolves 5 or fewer orientations). The average behavior is good even when the number ofcrack orientations is small. The problem is that fluctuations about the average can be largewhen the number of orientations is small, say 9. These calculations allow us to examine theeffect of radial cracks, which we believe is very large, and may dominate material response.However, radial crack formation is not directly relevant to sensitivity, which depends onthe response of closed cracks. An important reason for carrying out this verification andvalidation at this point is discussed in what follows.

An essential difficulty with the original version of SCRAM is that there was noprovision for displaying explicitly the large cracks that are visible to the naked eye; nor didthe large cracks formed by coalescence dominate failure, as they do in real catastrophes.The concern was to account for the growth and coalescence of microcracks in a statisticalmanner. This approach was adequate at modest levels of damage, and we computedcoalescence in order to account for the limited damage that occurs when microcracksintersect and is a feature of brittle failure. More extensive damage (rupture) frequentlyoccurs when macroscopic cracks form from microcracks via a percolation process.(Cleavage is a competing mechanism.) In a sense the percolation phenomenon was implicitin the original SCRAM, but there was no mechanism for determining the set size that is a

ARTICLE IN PRESS

Fig. 3. A comparison of the response of a thick ring to internal pressure for 4 levels of resolution of crack

orientation. With only 3 orientations the anisotropy is very large, while with 480 the anisotropy is small and the

uniformity of the response is good. For many purposes 30 orientations would give satisfactory results. The colors

denote the specific energy consumed in brittle fracture ranging from 0 (blue) to 6� 105 erg=g (red). Lengths are in

centimeters. The internal pressure was ramped to 25 bar in 30ms, after which it remained constant. The response

does not change after the time shown, 200ms.


consequence of many intersections. An approach to computing the set size and thepercolation threshold has been under development for several years, but this process israther subtle and we were reluctant to incorporate it into SCRAM, which is alreadycomplex, without extensive testing. Hence the emphasis on verification and validationdiscussed above.


3.1. Validation of the melting and ignition algorithm

In the first analyses of crack heating as an initiation mechanism, it was assumed thatignition could be modeled by solving the heat equation with an Arrhenius source term anda boundary condition representing the heat generation due to interfacial friction in closedpenny-shaped cracks. As the theory was refined there was some concern that melting mightplay a role in reducing the heat generated when the crack surface reaches the melting point.On further analysis it appeared that the molten boundary layer at the crack surface is sothin that significant heat is produced by shearing of the viscous melt, especially when thecrack is unstable, but the latent heat of melting was accounted for only at the cracksurface. More recently, we have refined the treatment of latent heat. To investigate itseffect, we have developed a one-dimensional finite-difference ignition model containing acell between the molten and solid regions that accounts for discontinuous heat flux.Motion of this special cell accounts for the latent heat of melting. The algorithm has beenvalidated in several ways. First, when the heat flux is held constant at the boundary thetemperature rises according to the square root of time, as predicted by a well-knownanalytic solution. Second, the time to ignition predicted by the current model in a semi-infinite medium has been verified by comparison with results of Cook (1958), Linan andWilliams (1971), and Dienes (1995). Our prediction is within a few percent of the timepredicted using the theory of Linan and Williams. Third, the energy balance accounting forheat input, viscous dissipation, reaction, and melting, is good to a few percent. Fourth, theresults are insensitive to changes in resolution when Dx2/Dt is fixed. For the semi-infiniteproblem when the temperature at a boundary is fixed (above the melting point),the numerical solution is compared with the analytic (similarity) solution available in theliterature (Carslaw and Jaeger, 1959). For this problem, the numerical solution shows themelting front moving away from the boundary with distance increasing with the squareroot of time, as predicted by the analytic solution. The numerical and analytic solutionsare compared in Fig. 4, showing agreement to within a few percent at a typical time.The precision could be increased by refining the numerical algorithm, but this does notseem justified at this point since few of the required material properties are known to betterthan a few percent accuracy, and they are not, in fact, constant. Furthermore, theassumption of one-dimensional heat flow due to friction in a penny-shaped crack issomewhat idealized.

In a typical initiation calculation the effect of latent heat was to increase the pre-ignitiontime 34%, from 139 to 187ms. This is the time when the Arrhenius source term hascontributed an amount of heat equal to 1% of the total heating. The actual latent heat wastaken as 2:081� 109 erg=g, while the lowered (and negligible) value was 2:08� 108 erg=g.In that calculation, the rate of heating was _q ¼ 1:0� 1011 erg=cm2=s until the melting pointðTm ¼ 519KÞ was reached. After melting, viscous heating due to an assumed slidingvelocity of 0:35 cm=ms produced a fairly intense source term because the molten layer wasvery thin (7:94mm at ignition for the estimated latent heat, 9:03mm for the lowered value).

A similar result may be obtained for the problem with the fixed temperature at aboundary. In that case, when the surface temperature is fixed at 700K the melt layer moves0:42mm with latent heat accounted for, while it moves 0:53mm (26% more) when the latentheat is reduced tenfold, at a time earlier than that for Fig. 4.

The treatment appears to be flexible and sufficiently accurate. Thus, a seconddiscontinuity representing the beta–delta transition can be implemented when sufficient

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300

350

400

450

500

550

600

650

700

0 2 4 6 8 10 12 14 16 18 20

Distance (Microns)

Tem

pera

ture

(de

gree

Kel

vin)

2.08E9

Latent heat = 2.08E8

Fig. 4. A comparison of the effect of latent heat on thermal profiles normal to a shear crack surface according to

analytic and numerical solutions with a fixed temperature (700K) on the crack surface (left boundary). The melt

front ðT ¼ 519KÞ moves from left to right, its distance from the crack surface increasing with the square root of

time. The analytic solutions lie slightly above the numerical ones. The actual latent heat of PBX 9501 is

2:08� 109 erg=g.


data become available to represent the chemical-source term and the physical properties inthe two phases separately.

4. Initiation of PBX 9501

A class of high explosives (HE) known as Plastic Bonded Explosives (PBX) is widelyused in both conventional and defense applications, due to their reliable performance andmaximum safety envelopes (Thompson et al., 2002). We are concerned with modeling theinitiation of PBXs, in particular, PBX 9501, under low-amplitude mechanical impacts. Asdiscussed earlier, this explosive can initiate at impact velocities below 50m/s (Idar et al.,1998). This particular explosive is a heterogeneous material of 95% (weight percent)energetic HMX (High Melting point eXplosive octahydro-1,3,5,7-tetranitro-1,2,3,5-tetrazocine) grains embedded in a polymer binder, largely estane (a polyurethane), anda small amount ð�0:1%Þ of stabilizer. The HMX is very brittle with very low specificsurface energy (about 50 erg=cm2). The polymer binder is viscous and much tougher. Thesize of the HMX grains is random and roughly follows a bi-modal distribution with 3:1ratio of coarse grains (average size of �200mm) to fine ones (average size of �1mm)(Clements and Mas, 2004). The details of the compositions of PBX 9501, such as thecomposition of the binder and the size distribution of the HMX grains, are available in theliterature (e.g., Thompson et al., 2002). Fig. 5a shows the heterogeneous microstructure(HMX grains and polymer binder) and microcracks present in a pristine sample of PBX

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Fig. 5. Micrographs showing the microstructure (HMX grains and polymer binder) and cracks in a PBX 9501

(Courtesy of Skidmore and Idar of Los Alamos National Laboratory). (a) a pristine sample of PBX 9501 pressed

to a density of 1:8 g=cm3 showing distribution of microcracks prior to testing; (b) a disk of explosive (with 500

diameter) impacted by a hemispherical-nosed steel projectile at low-speed (a modified Steven test by Idar et al.,

1998), illustrating brittle fracture with both radial and circumferential cracking (the projectile is slightly off

center). The cover plate has been removed to view the explosive; (c) a fragment recovered from a quenched test

showing cracks, melt, probable reaction, and bubbles of gas products and (d) a grain of HMX sheared during a

quenched test.

J.K. Dienes et al. / J. Mech. Phys. Solids 54 (2006) 1237–1275 1261

9501 pressed to a density of 1:8 g=cm3 (Skidmore et al., 1997), as well as (5b) themacroscopic radial and circumferential cracks caused by a mechanical impact (Idar et al.,1998). It also shows (5c) cracks, melt, probable reaction, and bubbles of gas products in afragment recovered from a quenched burn test, and (5d) a grain of HMX sheared during aquenched test. Due to the complex microstructure, the thermo-mechanical and chemicalbehaviors of this explosive are very complicated (e.g., Dienes and Kershner, 1998, 2001;Bennett et al., 1998; Hackett and Bennett, 2000; Knauss and Sundaram, 2004).

The majority of the SCRAM model constants for PBX 9501 were obtained by fitting themodel to the uniaxial compression test obtained data by Wiegand (Wiegand, 1998a, b;Aidun, 1998). The data are taken at a strain rate of 0.01/s and at room temperature. Usingthe same set of model constants, we predicted the stress–strain responses at a higher strainrate (1000/s) and compared the predictions with the available experimental data (Dobratzand Crawford, 1974; Funk et al., 1996). A comparison of the predicted and measuredresponses is shown in Fig. 6 (the predicted response at 0.022/s is not shown because it isessentially the same as 0.01/s). The agreement is fairly good. The stress–strain data involve3 main features: peak stress, the corresponding strain, and the stress at 4%. By matchingthese we have been able to estimate 3 microstructural material parameters: effective surface

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Fig. 6. A comparison of the stress–strain responses calculated by SCRAM at two strain rates with data obtained

from Wiegand (1998a), Funk et al. (1996), and Dobratz and Crawford (1974). The calculated response at 0.022/s

(not shown here) is essentially the same as 0.01/s.


energy ðgÞ, number density of cracks L0, and the initial average size of cracks ðc0Þ. It seemsremarkable that the process of fitting the macroscopic data leads to a unique and plausibleset of parameters characterizing the microstructures. If this approach can be confirmed,this would be a very useful means of characterizing the defects and failure behavior. InFig. 6, some adjustment of the other properties such as crack speed and elastic modulus ismade to optimize the fit.The multiple-shock experiment by Mulford et al. (1993), which involves the initiation

and quenching of the hot spots in a PBX 9501 slab, was selected for validation of both themechanics and initiation and burn algorithms in SCRAM. The experiment involved a two-material flyer plate, with Kel-F (Plexiglas) striking first, followed by Vistal (Al2O3),impacting a PBX 9501 target plate at 0:0911 cm=ms and resulting in two shocks; only thesecond one is strong enough to initiate a reaction causing an observable increase in particlevelocity. The experimental setup is shown schematically in Fig. 7a. The thicknesses ofVistal, Kel-F, and PBX 9501 are 1.1, 0.08 and 1.0 cm, respectively. Waves generated areone-dimensional for about 4ms. We use HYDROX (Shaw and Straub, 1981), a one-dimensional shock physics code, with SCRAM embedded, to simulate the experiment. TheVistal and Kel-F in the flyer plate are represented using the standard HYDROX materialmodel, while the PBX 9501 target is modeled with SCRAM. Some model constants for the3 materials are listed in Table 2. The properties of PBX 9501 are the same used in fittingthe uniaxial stress–strain data shown in Fig. 6. In this calculation, the number of discretecrack orientations (bins) N is chosen to be 9. The first three have normals in the impactdirection and perpendicular to it, forming an orthogonal basis. The remaining six arealigned with the face-diagonals of a unit cube having the orthogonal basis as edges. Thus,under compressive shock loading, four crack bins that are at 451 to the impact directionare under pressure and shear. Our calculation indicates that those shear cracks becomeunstable and grow in size following the first shock, becoming hot spots. The heat fluxgenerated by the interfacial friction on these shear cracks continues to cook the explosive,

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Fig. 7. A comparison of the measured and calculated particle velocities for the multiple-shock impact experiment

reported by Mulford et al. (1993) in which a slab of PBX 9501 is impacted by a two-material (Kel-F and Vistal)

flyer plate at 0:0911 cm=ms. The experimental setup and a time-position diagram showing intersection of shock

waves are shown in (a). The velocity plots shown in (b) are measured with 11 velocity gauges embedded in the

PBX 9501. The plots shown in (c) are calculated by HYDROX-SCRAM. The reactive heating is based on cracks

at 451 to the impact direction. The first shock is caused the Kel-F (Plexiglas) while the second shock is due to

reflection by the high-density Vistal sheet bonded to Kel-F flyer. The second shock leads to frictional heating in

shear cracks sufficient to raise the local temperature and pressure, initiate reaction, and thereby increase the

material velocity.


raising its temperature with time, in turn increasing the rate of chemical reaction. Thenshortly after the second shock these reacting shear cracks begin to burn as a result ofinterfacial friction and heating. During burning, the gaseous product is first highly

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Table 2

Material constants (cgs units, except as noted; blank entries indicate data not needed in the calculation)

Vistal Kel-F PBX 9501

r, mass density 4.103 2.12 1.86

K, bulk modulus 4:64� 1012 80:6� 109 93:85� 109

S, slope of us � up line in EOSa 2.0 1.87 2.26

G, Gruneisen’s ratio in EOSa 1.50

G, shear modulus 29:0� 109

n, Poisson’s ratio 0.36

c0, initial crack size 32� 10�4

L0, crack number density per 2p 45� 103

g, specific surface energy 50.0

ms, static coefficient of friction 0.8

md, dynamic coefficient of friction 0.2

n, power-law parameter for slow crack growth 6.0

a, coalescence parameter 3.4

Y , yield stress in pure shear 4:0� 109

b, kinematic hardening modulus 0:2� 109

Cv, heat capacity 9:96� 106

k, thermal conductivity (cal cm�1s�1K�1) 9:7� 10�4

Tm, melting point (K) 519

T i, initial temperature (K) 300

L, latent heat of melting 2:081� 109

EA, activation energy in Arrhenius kinetics (cal/mole) 5:27� 104

Qr, heat of reaction (cal/g) 500

Z, pre-exponential factor in Arrhenius kinetics 5:9� 1019

g, ratio of specific heats for the reactive product 1.3

aS and G are the material constants used in a Mie–Gruneisen equation of state (EOS), which relates pressure to

the density and internal energy (Dienes, 1985). A nonlinear EOS is needed to capture formation of shock waves at

high pressure.


compressed inside a shear crack, developing high internal pressure which gives rise to theobserved increase in particle velocities. For an individual crack, opening and fast growth(at roughly the sound speed) reduce the gas pressure so quickly that the crack becomesstable, quenching the burn. We believe that this kind of behavior is responsible for theobserved initiation and subsequent quenching of burn and reactions in PBX 9501.A comparison of the velocity histories of particles at eleven locations inside the target

calculated by HYDROX-SCRAM and measured by embedded gauges (Mulford et al.,1993) is shown in Figs. 7b (measured) and 7c (calculated). The first shock (jump in particlevelocities) is caused by the Kel-F while the second shock is due to reflection by the high-density Vistal sheet. The second shock leads to frictional heating in shear cracks sufficientto raise the local temperature and pressure, initiate reaction, and thereby increase theparticle velocities. The decrease in particle velocities at late times is the result of quenching.The agreement between the calculation and data is reasonably good, but improvement canbe hoped for, especially at the late times when our modeling of the gaseous reactionproducts can be improved.In this calculation isolated unstable cracks are ultimately quenched; however, in other

applications, interactions between cracks may exceed a percolation threshold generating a


dynamic instability, which can lead to a violent explosion. This behavior is too complex fordetailed modeling at this point because of the multitude of physical processes involved andthe difficulty of dealing with high-speed crack growth numerically. (Still, it may beapproximated in numerical codes by cell interactions.) Dienes and Kershner (2001) modelthe response of a burning crack to an oscillating pressure field that idealizes the effect ofneighboring cracks. It would be very useful to test the mechanisms governing the latestages of reaction; one approach is described in Section 5.

4.1. Role of crack orientation

In a recent APS symposium it was shown that the orientation of shear cracks is crucial(Dienes et al., 2004); thus, we regard with some skepticism brittle-failure models in whichcrack orientation is not carefully accounted for. In fact, new theoretical work has shownthat friction plays a major role in fracture, and determines the stress states that lead toinstability in penny-shaped cracks (Zuo and Dienes, 2005). For materials with highcoefficients of friction, the range of unstable crack orientations in compression turns out tobe rather narrow. The fundamental theory implicit in our computational model has notneeded modification—we simply had not examined the consequences of the computationalmodel in detail until recently.

In the analysis of the experiment of Mulford et al. discussed above, for which the resultsare given in Fig. 7, cracks in four bins (out of nine total) that had normals oriented at451 to the impact direction were found to form frictional hot spots; the remaining fivecrack bins had normals either parallel or perpendicular to the impact direction and hencedid not shear. In the light of our recent studies of the effect of friction which suggest thatcracks more nearly aligned with the direction of impact are more critical, we repeated theHYDROX-SCRAM calculation of the experiment using several different crack orienta-tions for the four crack bins which formed hot spots; the orientations of the remaining fivebins were unchanged. The effect of crack orientation is shown in Fig. 8, where the angle isthat between the crack normals and the impact direction and was kept the same for all fourcrack bins during the calculations. We conclude that the most critical crack has its normalnear 551 to the impact direction (an angle of 351 between the crack plane and the impactdirection), consistent with our theoretical prediction.

4.2. Effects of the viscosity

As with most modern materials models, a variety of material properties are required tocarry out realistic SCRAM calculations. Some can be obtained by molecular dynamics(MD) simulation, but properties that depend on the history of the material sample and itsresident defects, such as porosity and ultimate strength, can only be obtained from directmeasurements. When properties depend only on molecular structure and can be obtainedby MD, we have an advantage in that the effects of temperature and pressure can beaccounted for. In this paper only one example will be considered, the viscosity of HMX,for which calculations have been carried out by Bedrov et al. (2000) (BSS). A strongsensitivity to temperature was found. The values of viscosity first used in our calculationsresult in relatively modest heating in the shear layer, but the values in BSS allow ourcalculated velocities to agree reasonably well with the very precise multiple shockexperiment of Mulford et al., as shown in Fig. 7 and discussed above.

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Fig. 8. The effect of crack orientation on the computed response of PBX 9501 in the experiment of Mulford et al.

(1993) described in the text and in Fig. 7. Time is in microseconds, and velocity is in cm=ms. (a) 451 (same as Fig.

7c); (b) 551; (c) 701; (d) 801. The angle is that between the crack normal and the direction of impact. Only crack

orientations in a small range lead to significant reactions. (The velocity remains high at late times because we have

not allowed for a transition from solid to burn products in the solid EOS.)


The heating due to shear in HMX has been considered by Winter and Field (1975), Frey(1981), and Dienes and Kershner (1998). Values of viscosity based on the reports of theseauthors vary from below 0.001 to 0.0139 Pa s (MKS units are used throughout this sectionto maintain consistency with BSS). BSS have actually calculated the viscosity by means ofMD, and obtained the result:

mðTÞ ¼ m0 expðEvis=ðRTÞÞ (64)

with parameters Evis, m0 determined by fitting the formula with the viscosities at 800 and750K calculated by MD: m0 ¼ 3:45� 10�7 Pa s;Evis=R ¼ 7744K (a misprint in BSS saysmðTÞ ¼ m0 expð�Evis=ðRTÞÞ). The viscosity used in the original calculations of Dienes andKershner (0.00726) corresponds to Eq. (64) at 778K. However, at temperatures near themelting point (519K) the formula gives much higher viscosity.Plots of the computed particle velocities for the experiment by Mulford et al. (1993) are

compared in Fig. 9 for various viscosities. With the constant value of 0.0139, the upper

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Fig. 9. The effect of melt viscosity: (a) m ¼ 0:0139Pa s; (b) m ¼ 1:042 and (c) mðTÞ ¼ 3:45� 10�7 expð7744=TÞ.



limit of the reported values, there is no ignition (Dienes, 1996), as shown in Fig. 9a. Takingthe viscosity at 600K from Eq. (64), 0.139, there is still no significant reaction (not shown).With the constant viscosity at 550K (the lower limit of the calculations of BSS), 0.45, thereis some reaction, but the peak velocity is well below the measured value. Raising theviscosity to 1.042, the value at the melting point obtained from the BSS formula (whichextrapolates the BSS calculations) increases the peak velocity, but it is still somewhatbelow the measured value, as shown in Fig. 9b. The calculation using the temperature-dependent viscosity given by Eq. (64) gives the best result (Fig. 9c). It is emphasized that inaddition to the temperature-dependency, the values of viscosity calculated from Eq. (64)are significantly higher than the values used in previous studies (Winter and Field, 1975;Frey, 1981; Dienes and Kershner, 1998), especially at temperatures near the melting point.This comparison illustrates the importance of good estimates of the temperature

dependence of material properties, particularly if we are to compute the effects oflocalization and its effect on explosive hot spots. A particular challenge is the coupling ofchemical, thermal and mechanical properties in hot spots as they transition towardexplosive reactions. This requires experimental guidance because of the complexity of thecombined chemical, physical, and mechanical processes, as discussed in the section thatfollows.

5. Discussions and recommendations

Nonlinear behavior of some materials can be idealized with plasticity theory or somegeneralization thereof, but explosives and propellants exhibit much more complexbehavior which should be accounted for in assessing the risk of accidental explosions.In particular, most ductile materials such as metals do not show significant size effects, butexplosives are typically sensitive to the scale of the mass being investigated, as are mostsolids in a brittle condition. In particular, larger samples generally have lower strengthbecause they contain larger defects, but also a higher probability that smaller defects willcombine (Bazant and Planas, 1998; Persson et al., 1994). A related complication is thatstrength may fluctuate between samples, again due to the presence of defects. Failure ofbrittle materials may be catastrophic, with the speed of crack propagation near or evenexceeding the shear wave speed in special cases (Rosakis et al., 1999; Rosakis, 2002). Whilebrittle materials may behave stably in compression and appear to have a certain ductility,they may exhibit sudden failure. The materials used in explosives and propellants oftenhave brittle constituents, though they appear to be plastic when taken in a large mass dueto a soft binder material. The test results shown in Fig. 6 shows that significant failure mayoccur at only 2% strain in one common explosive, and incipient failure occurs at an evenlower strain. SCRAM theory provides a framework for dealing with some of thesecomplexities, but it needs to be combined with appropriate experiments to make reliablepredictions of risk possible. Experiments are necessary because the defect structuredepends on the actual history of material which is rarely known and, consequently,requires experiments to characterize it.To illustrate, consider the XDT data cited in the Introduction, where 12 in 50 of the

impact tests of propellant cylinders resulted in a violent reaction, while the others exhibiteda mild deflagration. Impact analysis with SCRAM has shown that the region sufferingsignificant frictional heating lies in a rectangular torus with a 1� 1mm2 section and amean radius of 2mm, leading to a critical volume of 0:012 cm3. The number density of


grains was measured at 0:85� 106=cm3 and the average grain size was 100mm. We estimatethe typical number density of large cracks (about 1% of the number density of grains) as1:0� 104=cm3 and the average radius of defects, when idealized as penny-shaped cracks, as10mm. With an exponential distribution of crack sizes we can conclude that the numberdensity of cracks exceeding 50mm in radius is 67=cm3. We assume that 1/4 of these have anorientation that would make them unstable in compression. Then the expected number ofdangerous cracks per shot would be 0:012� 67=4 ¼ 0:2, i.e., roughly 1 in 5 explodes, asobserved. This estimate cannot be taken too seriously because the details of the reactiveprocess are not fully established; the point of the calculation is to illustrate that combiningdefect behavior with statistics can lead to useful estimates of risk, and that sample size iscrucial.

It has long been known that the sensitivity of explosives is affected by their porosity,even when below 1%, but the actual hot spot mechanism has never been confirmedexperimentally. At high pressures some investigators have attributed the sensitivity to thecollapse of spherical hot spots, but this mechanism does not explain initiation at lowimpact speeds such as those studied by Jensen et al. ð�300m=sÞ or Idar et al. ð�50m=sÞ.More specifically, it has not been established whether hot spots are initiated within thebinder, at its interface with an explosive grain, or within the explosive grain itself. Asdiscussed in Section 4, HMX is very brittle (with specific surface energy about 50 erg=cm2),hence we take intragranular fracture to be the dominant mechanism. This view issupported by recent experimental evidence concluding that explosive sensitivity is a resultof intragranular, not extragranular, voids (Borne and Beaucamp, 2002). This does not ofitself confirm that shear cracks dominate initiation at low speeds, but it is supportive.

Though cracks may be initially open, they can be quite thin, so that a modest pressurecould close them and the subsequent behavior would be that of a shear crack. Recentstudies show that unstable shear cracks can propagate at above sonic speeds (Rosakis,2002; Abraham, 2001; Gao et al., 2001). Melting may well occur due to interfacial frictionin such cracks since the interfacial sliding velocity is related to the crack growth rate; theformation of pseudotachyltes, which is being actively explored in the geophysics literature(Scholz, 2002; Di Toro et al., 2005), is evidence that melting occurs in faults. Explosiveshave a much lower melting point than rocks so they can melt easily under intense frictionalheating. In many explosives melting is accompanied by reactions that form gaseousproducts that could open the shear cracks, possibly inducing further instability. Crackinteractions result from both stress waves (Dienes and Kershner, 2001) and intersections.(Permeability due to crack intersections can be important in explosives because it wouldallow hot reaction products to be more readily transported within the solid than bydiffusion.) Furthermore, the interaction of hot spots is probably crucial to initiation, but isdifficult to assess. However, experimental evidence that hot-spots will die out withoutinteractions was given recently by Proud et al. (2002, 2004) and Field et al. (2004).

The need for better insights is clear from the results of drop tests and such, where anumber of repetitions are needed and an average is noted, though the scatter may be verylarge. As pointed out by Chakravarty et al. (2002), ordering of explosives by sensitivitycannot be done. Different tests provide different ordering. Furthermore, standard tests donot show by what process porosity, grain size, and method of fabrication influencesensitivity. As restrictions on testing increase for reasons of safety and economics, itbecomes more attractive to make use of theoretical simulations, but the details of damage,initiation, and variability need to be better understood before we can rely on computer


simulations. Though we have made idealized calculations of shear crack heating, dynamicthree-dimensional shear crack behavior may be much more complex, and the critical cracksize and orientation for a specific scenario may be different from our predictions.In the SCRAM formalisms, we have attempted to quantify these issues in the simplest

way possible, retaining the mechanisms that dominate brittle behavior. In this paper, weemphasize the behavior of reactive materials such as explosives and propellants, but theformalism has been used to study other kinds of brittle behavior as well, which has providedadditional validation (Meyer et al., 1999; Zuo et al., 2003). Unfortunately, there have beenno experimental programs designed to provide detailed information on sensitivity and crackstatistics in the same material, allowing us to confirm that the shear-crack mechanism ofinitiation dominates sensitivity; we can only claim that SCRAM results are consistent withobservations. In particular, only a few mechanical properties of explosive materials arenormally reported, and such data as the mean crack size, number density of cracks, frictioncoefficient, fracture toughness, and shear modulus must be inferred.How can assessments of risk be confirmed? Even if extensive defect and reaction data

were available, there could remain some concerns that the mechanisms for initiation mightbe overly simplified. A natural validation procedure would be to examine samples ofvarious sizes to see that the trend is as predicted. This can be expensive, and such programshave had limited success in the field of rock mechanics. An alternative would be to create amixture of explosive and sugar (or other inert that simulates explosive grains, what we calla PBMIX) and examine the damage and sensitivity to impact when the fraction ofexplosive is increased, as illustrated in Fig. 10. The extent of fracture, reaction, and phasechanges in the explosive grains could be safely examined following an impact near criticalconditions, while test samples on a realistic explosive device near threshold conditionscannot be safely recovered and examined. (The tests of Howe et al. (1985) with polishing ofdamaged TNT could probably not be repeated today.)Specific results from tests of a PBMIX that would not be available from standard

experiments are listed below:

�

Fig

rea

sur

Severely damaged HMX grains can be recovered and examined as part of a nearly intactsample.
� The location, orientation, and shape of fractures and hot spots can be determined. � Large pieces can be conveniently sectioned, polished, etched, and examined for phasechanges and reactions at grain and crack surfaces.
. 10. Conceptual diagram of the expected response from testing a PBMIX. The red region represents largely

cted material, the dark blue dots denote partially reacted material (the fizz zone), pink denotes unreacted crack

face, and green denotes binder.


�
The importance of hot-spot interactions can be assessed. (Are adjacent hot spots hotterand more extensively reacted than isolated ones?) � Crack and hot-spot statistics can be obtained and correlated. The number and size ofdefects that are stable, exhibit phase changes, and show evidence of reaction can becompared. � Physically motivated models for the initiation of burning and hot-spot interactions canbe validated.
6. Summary and conclusions

We have presented a statistical theory (SCRAM) for modeling impact initiation ofreactive materials (explosives and propellants) which contain brittle constituents. Thetheory has been implemented and was used to model a multiple-shock experimenton a typical explosive (PBX 9501). The calculated particle velocities compare reasonablywell with those measured using gauges embedded inside the explosive. The effects ofcrack orientation and temperature dependence of viscosity of the melt on the responsehave also been examined. It is shown that crack orientation has a significant effect on theresponse, especially under compressive loading where interfacial friction plays animportant role.

Our hypothesis for the initiation mechanism is that intense frictional heating in a shearcrack during unstable growth forms a hot spot, and that the reactive material adjacent to ahot spot can undergo localized melting, ignition, and fast burn, which can lead to violentexplosions following relatively mild impacts. The evidence for this hypothesis has beendiscussed and it appears to be a viable explanation for a variety of events. Idealization ofhot spots as penny-shaped cracks makes it possible to model individual crack behavior, aswell as ensembles of cracks, theoretically. The behavior of open cracks and closed crackswith friction are quite different, but the model lends itself to detailed analysis of both. Inaddition it is feasible to examine the interactions of defects in the form of circular discs andto explain how ensembles of defects may form violent reactions, such as XDT. We haveshown (Dienes, 2005) that an ensemble of circular cracks has a percolation threshold suchthat a network of connected cracks forms above that threshold.

Shear cracks are hard to observe in any detail because there is so little of them, and theheating they produce is hard to observe because it is so transient. Means for observing thenumber and interactions of shear cracks and hot spots in more detail should be developedin order to confirm quantitative estimates of risk and size effects. Tests of a mixture ofreactive and inert materials, a PBMIX, would make it possible to quantify the evolution ofindividual hot spots in an ambiance of reduced violence.

Acknowledgments

This work was supported by the Joint US Department of Energy (DOE) and USDepartment of Defense (DoD) Munitions Technology Development Program, the DOEAdvanced Simulation and Computing (ASC) Program. We are indebted to D.A. Wiegandand J.B. Aidun for providing the experimental stress–strain data for PBX 9501, which wereused to determine the model constants. We thank J. Middleditch for his contributions tothe model implementation and calculations, and C.A. Bronkhorst, R.M. Hackett, C. Liu


for technical discussions. We are particularly grateful to W.G. Proud for his thoroughreading of an earlier version of the manuscript and providing many constructivecomments.

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