LA-UR-10-01870-rev May 14, 2010Approved for public release;distribution is unlimited.

Title: Reactive Burn Models andIgnition & Growth Concept

Author(s): R. Menikoff, rtm@lanl.gov

M. Sam Shaw, mss@lanl.gov

Submitted to: 8th Biennial International Conference on

New Models and Hydrocodes for Shock Wave

Processes in Condensed Matter

Paris, FRANCE

24-28 May 2010

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Reactive Burn Models and Ignition & Growth Concept

Ralph Menikoff a and M. Sam Shaw b

Los Alamos National Laboratory, USA

Abstract. Plastic-bonded explosives are heterogeneous materials. Experimentally, shock initiationis sensitive to small amounts of porosity, due to the formation of hot spots (small localized regionsof high temperature). This leads to the Ignition & Growth concept, introduced by Lee and Tarver in1980, as the basis for reactive burn models. A homogenized burn rate needs to account for three meso-scale physical effects: (i) the density of burnt hot spots, which depends on the lead shock strength;(ii) the growth of the burn fronts triggered by hot spots, which depends on the local deflagrationspeed; (iii) a geometric factor that accounts for the overlap of deflagration wavelets from adjacenthot spots. These effects can be combined and the burn model defined by specifying the reactionprogress variable λ(t) as a function of a dimensionless reaction length rhs(t)/`hs, rather than byspecifying an explicit burn rate. The length scale `hs is the average distance between hot spots,which is proportional to [Nhs(Ps)]

−1/3, where Nhs is the number density of hot spots activated by

the lead shock. The reaction length rhs(t) =∫ t

0D(P (t′))dt′ is the distance the burn front propagates

from a single hot spot, where D is the deflagration speed and t is the time since the shock arrival.A key implementation issue is how to determine the lead shock strength in conjunction with a shockcapturing scheme. We have developed a robust algorithm for this purpose based on the Hugoniot jumpcondition for the energy. The algorithm utilizes the time dependence of density, pressure and energywithin each cell. The method is independent of the numerical dissipation used for shock capturing.It is local and can be used in one or more space dimensions. The burn model has a small number ofparameters which can be calibrated to fit velocity gauge data from shock initiation experiments.

1 Introduction

Plastic-bonded explosives (PBX) are heterogeneousmaterials consisting of explosive grains of various sizesand shapes, a polymeric binder to hold them togetherand a small amount of porosity; see figure 1. Due tothe heterogeneities, shock compression gives rise to lo-calized spatial regions of high temperature known ashot spots. Since chemical reaction rates are stronglytemperature dependent, reaction in and around hotspots dominate the burning in a shock-to-detonationtransition.

Pore collapse is a key hot-spot generation mech-anism for shock initiation. Experiments have shownthat initiation is sensitive to the meso-scale structureand in particular to porosity. The scale of the hetero-geneities in a PBX (tens of microns) is smaller thanthe cell size that can reasonably be used to simulate atypical high explosive (HE) application. Therefore, ahomogenized or volume averaged burn rate is neededto account for the sub-grid burn physics.

The underlying hot-spot reaction physics is bestdescribed by the Ignition & Growth concept reviewedin sec. 2. We then present a new formulation for a

a e-mail: rtm@lanl.govb e-mail: mss@lanl.gov

Fig. 1. Polarized light micrograph of PBX 9501; fromSkidmore et al. [1998, fig. 4].

burn rate in sec. 3 that better captures the under-lying physics of this concept. A key idea is to basethe burn model on the strength of the lead shock.In sec. 4 we present a robust algorithm based on theHugoniot jump condition for detecting the lead shock.

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A comparison of numerical results with velocity gaugedata for shock initiation experiments of PBX 9502 areshown in sec. 5. Final remarks and extensions of themodel formulation are briefly described in sec. 6.

2 Ignition & Growth Concept

Currently available burn models are empirical. Theywork well only for the limited regime in which theyare calibrated. The more physical hot-spot models arebased, at least heuristically, on the Ignition & Growthconcept introduced by Lee and Tarver in 1980. Theignition phase corresponds to the formation of hotspots, their reaction and thermal runaway on a fasttime scale. The growth phase corresponds to the sub-sequent burning from reactive wavelets triggered bythe burned hot spots. Typically, hot spots representa small fraction of the HE volume and most of theburning occurs in the growth phase.

Leading examples of this type of model, or modelsthat can be interpreted with the ignition and growthconcept, along with the form for the burn rate arelisted below:

i. Ignition & Growth model, Lee and Tarver [1980]Rate is a function of local pressure and reactionprogress variable. The functional dependence onthe reaction progress variable can be interpretedas proportional to the average burn front area dueto the hot-spot wavelets.

ii. DAGMAR, Wackerle et al. [1978] and Andersonet al. [1981]Rate is a function of lead shock pressure, equilib-rium mixture temperature and first order in reac-tion progress variable. Model parameters are cali-brated to embedded Manganin stress gauge data.

iii. JTF model, Johnson et al. [1985]Multi-step reaction with the rates a function of thelead shock pressure, equilibrium mixture pressureand first order in reaction progress variable. Note-worthy for comparison with embedded magneticvelocity gauge data; see Vorthman et al. [1985].

iv. WSD model, Wescott et al. [2005]This is a generalization of model i above. The shockdensity is used to switch between rates calibratedto the ignition and propagation regimes. In eachregime the rate depends on the local pressure andthe reaction progress variable.

v. CREST model, Handley [2006]Rate is a function of reactant entropy for two-phase model with pressure but not temperatureequilibrium. Two reaction progress variables areused to tune reaction profile based on analysis ofembedded magnetic velocity gauge data; see Jamesand Lambourn [2006].

At the time that the DAGMAR and JTF modelswere developed, artificial viscosity was used for shock

dissipation. The lead shock was determined by scan-ning the viscous pressure profile for the first local min-imum. This algorithm had difficulty when a compres-sive wave followed the lead shock. Due in part to thenumerical issue with robustness, these models nevergained favor.

The implementation of the WSD model in [Stewartet al., 2007, § 2.4] detects the time of arrival of the leadshock based on the maximum ∂tP for each cell. Then alevel set algorithm applied to the shock arrival time isused to determine the shock speed, which determinesthe shock state. This may not be too accurate butis good enough to use as a switch on the initiationregime or growth regime.

For the CREST model, the solid entropy can bethought of as a measure of the lead shock strengthsince the chemical energy released is apportioned tothe products. There are issues with shock capturing intwo-phase models since the component energy equa-tions are not in conservation form; see Petitpas et al.[2009] for details.

Burn rates based on the lead shock strength nat-urally account for the phenomena of shock desensi-tization. With the possible exception of the CRESTmodel, different rate parameters are needed to accountfor the initial temperature or initial porosity of a PBX.In effect, a PBX at a given initial condition is treatedas a distinct explosive. To overcome this and otherdeficiencies, a burn model needs to better account forthe sub-grid or meso-scale burn physics.

Based on the ignition and growth concept, threekey physical effects need to be incorporated in a burnmodel: (i) the density of burnt hot spots, which de-pends on the lead shock strength; (ii) the growth ofthe burn front triggered by the hot spots, which de-pends on the local deflagration speed; (iii) a geomet-ric factor that accounts for the overlap of deflagrationwavelets from adjacent hot spots.

3 New Formulation

The standard concept of a chemical reaction rate isappropriate for a uniform state. For a heterogeneousexplosive, burning is dominated by hot spots. The vol-ume average of the hot spot reaction can be accountedfor by directly specifying the reaction progress vari-able λ. The burn rate is determined implicitly as

R = (d/dt)λ . (1)

The key point is that hot spots are formed — andreaction begins — only after the lead shock pass overpotential sites for a hot spot; such as a void or anembedded high density particle. Moreover, the averageamount of burn depends on the statistical distributionof hot spots, how a hot spot evolves and how a hot spotinteract with its neighbors.

First, let us define the shock functionsts(x) = time the lead shock arrives at point x,Ps(x) = pressure of lead shock at point x.

New Models and Hydrocodes for Shock Wave Processes in Condensed Matter

We express the reaction progress variable as

λ(t, Ps, P ) =

{0 , for t ≤ ts;g(s(t− ts, Ps, P )

), for t > ts;

(2)

whereP (x, t) = local pressure,

g(s) = reaction scale function,s(t, Ps, P ) = scaled hot-spot reaction length

= rhs(t, P )/`hs(Ps),rhs(t, P ) = reaction length from single hot spot,`hs(Ps) = average distance between hot spots.

For now we neglect the initial hot-spot volume, andexpress the reaction length from a single hot spot as

rhs(t, P ) =∫ t

ts

D(P (t′)

)dt′ , (3)

which depends on the local deflagration speed, D(P ).The average distance between hot spots can be written

`hs(Ps) = [Nhs(Ps)]−1/3, (4)

where Nhs(Ps) is the number density of burn centersor those hot spots activated by the lead shock thatburn on a fast time scale.

We expect the reaction scale function g to dependon the distribution of hot spots. It accounts for theinteraction of hot spots and the geometric factor forthe burn front area due to the overlap of burn waveletsfrom each hot spot. For a given PBX composition,g would depend on the structure of heterogeneities;grain size distribution and other factors such as grainmorphology, imperfections or impurities.

To summarize the burn model: it is determinedby specifying three functions g(s), Nhs(Ps) and D(P )that separate the sub-grid burning into distinct parts,each of which has a clear physical interpretation.

3.1 Fitting forms

For independent randomly distributed point hot spotsin three dimensions, the reaction scale function canbe shown to be g(s) = 1− exp(−s3); see Nichols andTarver [2002] and Hill et al. [2009]. This includes thegeometric effects of outward hot-spot burning and in-ward grain burning similar to the functional depen-dence of the burn rate on the reaction progress vari-able, λ2/3(1− λ)2/3, in the original ignition & growthmodel of Lee and Tarver [1980].

The geometry of burn wavelets triggered by thehot spots depends on the dissipative mechanism gen-erating the hot spots. We use a form corresponding tocylindrical hot spots

g(s) = 1− exp(−s2) . (5)

Measurements on the regression rate for propellantsand higher pressure measurements with diamond anvil

cells, see for example Esposito et al. [2003], suggestthat the deflagration speed is proportional to the pres-sure. We express it as

D(P ) = D(Ps) · (P/Ps)n . (6)

The sensitivity of distance-to-detonation with shockpressure suggests that the number of burn centers risesrapidly with shock strength. We use a shock strengthfactor of the form

f(Ps) = D(Ps) ∗ [Nhs(Ps)]1/3

= exp[A + B ∗min(Ps, Ps,max)/Pref

]t−1ref , (7)

where Pref and tref are units for pressure and time.The scaled reaction length is then expresses as

s = f(Ps) ∗∫ t

ts

[P (t′)/Ps]n dt′ . (8)

For these choices, the burn model has only 4 pa-rameters (A, B,Ps,max,n). The model can be calibratedto shock initiation experiments by fitting to embed-ded velocity gauge data; i.e., Lagrangian time histo-ries of the particle velocity for several points alonga shock-to-detonation trajectory; for details see Shawand Menikoff [2010].

We note that s = 1 corresponds to 1 e-fold of re-action. The time scale for this to occur is roughly τ =1/f(Ps), and the corresponding hot-spot length scale`hs = D(Ps)/f(Ps). For PBX 9502, used to illustratethe model in sec. 5, the reaction time scale for a shock-to-detonation transition varies between 10 ns and 1µs.Assuming D(Ps) is on the order of 0.1 mm/µs, the hot-spot length scale would be between 1 and 100microns.This is comparable to the grain size for the PBX.

The effective burn rate for the fitting form corre-sponding to Eq. (5) can be expressed as

R(λ,P, Ps) =d

dtλ =

d

dsg · d

dts

= 2[− ln(1− λ)]1/2(1− λ) · f(Ps) · [P/Ps]n (9)

since s2 = − ln(1−λ). We note that the λ dependenceof the rate is very similar to that in the CREST model;see [Handley and Lambourn, 2009, Eq. (2)]. Moreover,Eqs. (5) and (8) are similar to the history variablereactive burn model in the CTH code; see descriptionin [Starkenberg, 2002].

4 Lead Shock Detection Algorithm

A numerical shock profile mimics that of a steady-state solution to the Euler equations with viscosityand heat conduction. In the (V, P )-plane, the contin-uum profile lies between the Rayleigh line and theHugoniot locus, as shown in figure 2; see Gilbarg [1951].The distance between a point on a shock profile and

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Fig. 3. Evolution of pressure profiles (series of fixed times) and lead shock pressure (dashed red curve) from shock detectorfor cases with decaying (left) and strengthening (right) shocks.

the Hugoniot locus is roughly proportional to the Hugo-niot function,

h(V, e) = e− e0 − 12 (V0 − V )[P (V, e)− P0] , (10)

whose zero level set defines the Hugoniot locus. Thesubscript ‘0’ denotes the state ahead of the shock. Forthe lead shock, the ahead state is well defined; theambient or initial state of the HE.

For a given cell in a numerical simulations, theshock state can be determined by calculating the timedependence of the Hugoniot function,

h̃(t) = h((V (t), e(t)

), (11)

and looking for either a zero crossing or a local min-imum. The former case occurs when the shock is fol-lowed by a compression wave and the latter whenthe shock is followed by an expansion wave. Follow-ing waves would cause the shock strength to changeand the wave front to accelerate.

The ability of the algorithm to detection the leadshock is shown in figure 3. In this example, a piston isused to drive a shock in non-reacting PBX 9502. Anaccelerating piston strengthens the lead shock, whilea deaccelerating piston causes the shock to decay. Thealgorithm works equally well for the case in which re-action accelerates the lead shock. Possibly the errorin one of the other shock jump conditions, such as,δ = (us − u0)2 − (Ps − P0)(V0 − Vs), can be used toestimate the error in the shock state.

V

PH

S

shockprofile

Fig. 2. Hugoniot locus and shock profile in (V, P )-plane.Also shown are Rayleigh line and isentrope thru shockedstate.

Three key features of the shock detection algo-rithm are: (i) It is independent of the numerical dis-sipation and can be used with a shock capturing al-gorithm based on either artificial viscosity or a Rie-mann solver, i.e., Godunov scheme. (ii) It is local to acell and can be uses in one or more space dimensions.(iii) It determines time of shock arrival along with theshock state. Consequently, the shock detection algo-rithm is well suited for our formulation of a reactiveburn model.

5 Numerical Results for PBX 9502

To illustrate the effectiveness of the burn model forshock initiation we compare simulated results withdata from gas gun experiments. A series of shock-to-detonation experiments were carried out on PBX 9502by Gustavsen et al. [2006] in which the initial shockpressure was varied. The gas gun drives a well sup-ported planar shock in the PBX. Hence the flow isone dimensional. Embedded magnetic velocity gaugeswere used to measure Lagrangian time histories for upto ten particles. This provides information on the flowbehind the lead shock as well as the trajectory of thelead wave.

Numerical simulations were performed utilizing theAmrita framework [Quirk, 1998a,b]. The solver algo-rithm is a Godunov scheme on a 1-D Lagrangian mesh.Both the projectile from the gas gun and the PBXwere included in the calculation. The measured pro-jectile velocity was used as the initial condition to setthe initial shock pressure.

Adaptive mesh refinement allowed shock fronts andthe reacting regions to be well resolved. A notewor-thy point of the shock detection algorithm is that theHugoniot function variable h̃ enabled the refinementcriterion to select the finest mesh for the lead shock,and to prevent reaction from occurring in the numer-ical shock profile. Coarsening the mesh by 1 level be-hind the lead shock is a smoothing operation thatdamps out numerical variations in the shock state dueto the discretization of the shock profile.

A comparison of the velocity gauge data for two ex-periments and numerical results are shown in figure 4.The initial shock pressure in shot 2s57 is 13.5GPa and

New Models and Hydrocodes for Shock Wave Processes in Condensed Matter

0 0.5 1 1.5time (µs)

0

0.5

1

1.5

2

2.5

velo

city

(km

/s)

0.00 mm 1.17 mm 1.96 mm 2.75 mm 3.54 mm 4.33 mm 5.11 mm 5.91 mm 6.70 mm 7.48 mm

Velocity Gauge Data(shot 2s57)

0 0.5 1 1.5 2 2.5time (µs)

0

0.5

1

1.5

2

2.5

velo

city

(km

/s)

0.00 mm 3.04 mm 4.13 mm 4.91 mm 5.70 mm 6.48 mm 7.28 mm 8.07 mm 8.86 mm 9.65 mm10.44 mm

Velocity Gauge Data(shot 2s68)

Fig. 4. Comparison of embedded velocity gauge data for shock initiation experiments in PBX 9502; shots 2s57 (13.5GPa)and 2s68 (11.5 GPa) from Gustavsen et al. [2006]. Solid lines are experimental data and dashed lines are simulations.Gague positions are shown in the legend of each plot.

in shot 2s68 is 11.5 GPa. We note that initiation issensitive to the shock pressure. The relatively smallpressure difference (less than 10 %), changes the tran-sition time from 1.3µs to greater than 2.0 µs (greaterthan 30 %). In addition, the timing of the jump offvelocity of the gauges depends on the shock velocity.Hence accurate equations of state are needed for thereactants and products of the PBX. The one we areusing is described in [Menikoff, 2009].

A noteworthy feature of the data is displayed bythe first gauge which is at the PBX interface. After theshock, the particle velocity is nearly constant. This iscompatible with a small initial hot-spot volume andburning resulting from the subsequent growth of thehot-spot reaction length; i.e., rhs(t, P ). This is quali-tatively different than the original Ignition & Growthmodel in which the burn rate is a function of the localpressure. We expect that the different behavior for therate behind the lead shock would have a large effecton short-shock initiation experiments.

Further details on the burn model and compar-isons with additional experiments can be found in acompanion paper of ours [Shaw and Menikoff, 2010].

6 Final Remarks

We have focused on shock initiation. But a burn modelneeds to deal with two additional detonation phe-nomenon: (i) The curvature effect for propagating det-onation waves. (ii) Change of the ignition sensitivitywith the initial state of a PBX. We conclude witha brief comments on incorporating these propertieswithin the context of a hot-spot model.

The detonation speed and state of an underdrivenplanar detonation wave is uniquely determined by theCJ condition and the equation of state of the HE. Ingeneral, however, the detonation speed and state de-pends on the local curvature of the detonation front.This is known as the curvature effect. Due to the jumpconditions for a quasi-steady detonation [Menikoff et al.,1996], the curvature effect is sensitive to the reaction

zone width. This can lead numerical simulations tohave a mesh dependent curvature effect. Typically, thereaction zone width decreases with a finer mesh andresults in a smaller numerical curvature effect.

Alternatively, the reaction rate of a burn modelcan be adjusted to tune the reaction zone width inorder to fit the curvature effect. For a propagatingdetonation wave, the reaction zone depends largely onthe rate for pressures greater than PCJ . Since shockinitiation depends on the rate for pressures less thanPCJ , a model should be able to be calibrated for bothinitiation and propagation.

In fact, Wescott et al. [2005] claim to fit bothregimes for PBX 9502 using a variation of the originalLee & Tarver model. They did this by adding termsto the rate to switch from the initiation regime to thepropagation regime based on the shock density; see[Wescott et al., 2005, Eqs. (44) and (45)] and [Stewartet al., 2007, § 2.4 & § 2.5.2]. This method of fitting bothinitiation and propagation regimes requires the reac-tion zone to be resolved. Very likely adaptive meshrefinement of the detonation front would be neededfor efficiency. For our formulation of a hot-spot burnmodel, the two regimes can be taken into account withthe hot-spot reaction length by modifying the defla-gration speed; i.e., D(P ).

Initiation sensitivity of a PBX depends on its ini-tial state. Lower initial density increases the shocksensitivity. This is compatible with a higher porosityresulting in an increased number of potential activa-tion sites for hot spots. In addition, a PBX is moresensitive when hot then when cold. This can be at-tributed to varying the number of hot spots activatedby a shock.

Dissipative mechanisms that localize energy in ashocked PBX give rise to a temperature distribution;see for example Menikoff [2004]. Hot spots would cor-respond to the high temperature tail of the distribu-tion. Due to the temperature sensitivity of a chemicalrate, active hot spots (in which thermal runaway oc-curs on a fast time scale) can be approximated as thoseabove a cutoff temperature. A higher initial temper-

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ature would shift the temperature distribution result-ing in a greater density of active hot spots. For ourformulation of a hot-spot burn model, the change insensitivity could be taken into account by modifyingthe hot-spot length scale to depend of the initial stateas well as the shock pressure; i.e., f(Ps, ρ0, T0).

Finally, we note that the burn models mentionedin sec. 2 all have been calibrated to fit some exper-imental data. To assess the relative merits of differ-ent models one needs to compare each model with alarge number of experiments covering a wide range ofdetonation phenomenon. We think that a burn modelthat incorporates the underlying physics of hot spotsis likely to do better than ad hoc fitting forms for thereaction rate.

Acknowledgments

This work was carried out under the auspices of theU. S. Dept. of Energy at LANL under contract DE-AC52-06NA25396. We would like to thank Rick Gus-tavsen for providing the gauge data from his shockinitiation experiments in electronic form.

References

A. B. Anderson, M. J. Ginsberg, W. L. Seitz, andJ. Wackerle. Shock initiation of porous TATB. InSeventh Symposium (International) on Detonation,pages 385–393, 1981.

A. P. Esposito, D. L. Farber, J. E. Reaugh, and J. M.Zaug. Reaction propagation rates in HMX at highpressure. Propellants, Explosives, Pyrotechnics, 28:83–88, 2003.

D. Gilbarg. The existence and limit behavior of theone-dimensional shock layer. Am. J. Math., 73:256–274, 1951.

R. L. Gustavsen, S. A. Sheffield, and R. R. Alcon.Measurements of shock initiation in the tri-amino-tri-nito-benzene based explosive PBX 9502: Waveforms from embedded gauges and comparison offour different material lots. J. Appl. Phys., 99, 2006.

C. A. Handley. The CREST reactive burn model. InThirteenth Symposium (International) on Detona-tion, pages 864–880, 2006.

C. A. Handley and B. D. Lambourn. Predicting theeffect of porosity on th eshock sensitivity of explo-sives. In Shock Compression of Condesnsed Matter,pages 221–224, 2009.

L. G. Hill, B. Zimmermann, and A. Nichols. On theburn topology of hot-spot initiated reactions. InShock Compression of Condesnsed Matter, pages432–435, 2009.

H. R. James and B. D. Lambourn. On the system-atics of particle velocity histories in the shock-to-detonation transition regime. J. Appl. Phys., 100,2006.

J. N. Johnson, P. K. Tang, and C. A. Forest. Shock-wave initiation of heterogeneous reactive solids. J.Appl. Phys., 57:4323–4334, 1985.

E. L. Lee and C. M. Tarver. Phenomenologicalmodel of shock initiation in heterogeneous explo-sives. Phys. Fluids, 23:2362–272, 1980.

R. Menikoff. Pore collapse and hot spots in HMX. InShock Compression of Condensed Matter – 2003,pages 393–396, 2004.

R. Menikoff. Complete EOS for PBX 9502. Techni-cal Report LA-UR-09-06529, Los Alamos NationalLaboratory, 2009.

R. Menikoff, K. S. Lackner, and B. G. Bukiet. Model-ing flows with curved detonation waves. Combustionand Flame, 104:219–240, 1996.

A. Nichols and C. Tarver. A statictical hot spot reac-tive flow model for shock initiation and detonationof solid HE. In Twelveth Symposium (International)on Detonation, pages 489–496, 2002.

F. Petitpas, R. Saurel, E. Franquet, and A. Chin-nayya. Modelling detonation waves in condensedenergetic materials: Multiphase CJ conditions andmultidimensional computations. Shock Waves, 19:377–401, 2009.

J. J. Quirk. Amrita - a computational facility for CFDmodelling. In 29th Computational Fluid Dynamics,VKI Lecture Series, chapter 4. von Karmen Insti-tute, 1998a.

J. J. Quirk. AMR sol: Design principles and practice.In 29th Computational Fluid Dynamics, VKI Lec-ture Series, chapter 5. von Karmen Institute, 1998b.

M. S. Shaw and R. Menikoff. Reactive burn modelfor shock initiation in a PBX: Scaling and separa-bility based on the hot spot concept. In FourteenthSymposium (International) on Detonation, 2010.

C. B. Skidmore, D. S. Phillips, J. T. Mang, and J. A.Romero. The evolution of microstructual changesin pressed HMX explosives. In Eleventh Sympo-sium (International) on Detonation, pages 556–564,1998.

J. Starkenberg. Modeling detonation propagation andfailure using explosive initiation models in a conven-tional hydrocode. In Twelveth Symposium (Interna-tional) on Detonation, pages 1001–1007, 2002.

D. S. Stewart, S. Yoo, and B. L. Wescott. High-ordernumerical simulation and modelling of the interac-tion of energetic and inert materials. CombustionTheory and Modelling, 11:305–332, 2007.

J. Vorthman, G. Andrews, and J. Wackerle. Reactionrates from electromagnetic gauge data. In EigthSymposium (International) on Detonation, pages99–110, 1985.

J. Wackerle, R. L. Rabie, M. J. Ginsberg, and A. B.Anderson. A shock initiation study of PBX-9404.In Actes du Symposium International sur le Com-portement des Milieux Denses sour Hautes PressionDynamiques, pages 127–138, 1978.

B. L. Wescott, D. S. Stewart, and W. C. Davis. Equa-tion of state and reaction rate for condensed-phaseexplosives. J. Appl. Phys., 98, 2005.