fragmentation of metal particles during...

Fragmentation of metal particles during heterogeneous explosion

R. C. Ripley · L. Donahue · F. Zhang

© Her Majesty the Queen in Right of Canada 2015

Abstract Heterogeneous explosives contain a mixture ofstandard explosive material and reactive metal particles.The inclusion of metal particles alters the energy densityand energy release timescales involved in the blast event.Available experimental evidence indicates that metal parti-cles may be damaged or fragmented during heterogeneousblast, altering the distribution of particle sizes from their ini-tial state. This paper discusses adaptation and applicationof fragmentation theory and physical models for particledamage during condensed matter detonation, aerodynamicbreakup of molten particles, and particle impact fragmen-tation with nearby structures. The shock compression andimpact fragmentation models are based on the energy meth-ods for dynamic fragmentation by Grady and Kipp, whileaerodynamic breakup is treated according to Weber numberstability criteria for droplets. These particle fragmentationmodels are validated against fundamental test cases from theliterature. The models are then applied to heterogeneous blastscenarios including free field and wall reflection in a semi-confined urban street. Comparison with experimental recordsof pressure shows good agreement despite challenges inher-ent in the complexity of heterogeneous blast measurementand multiphase simulation.

Communicated by L. Bauwens.

R. C. Ripley (B) · L. DonahueMartec Limited, 1888 Brunswick St., Suite 400, Halifax,NS B3J 3J8, Canadae-mail: [email protected]

L. Donahuee-mail: [email protected]

F. ZhangDefence Research and Development Canada,Station Main, PO Box 4000, Medicine Hat, AB T1A 8K6, Canadae-mail: [email protected]

Keywords Multiphase flow · Aerodynamic breakup ·Particle fragmentation · Heterogeneous explosion · Blastloading · Numerical modeling

1 Introduction

Heterogeneous blast explosives contain a mixture ofcondensed-phase explosive, binders, and metal particles.Most often aluminum particles are used for their high heat ofcombustion when reacted with atmospheric air. Micrometric-sized particles (typically 1–100 µm) are large enough thatthey do not react significantly within the detonation zone ofthe explosive itself, rather they are accelerated and heatedby shock compression and expanding detonation products,and are dispersed into the surrounding environment wheretheir reaction may augment the air blast [1–5]. Experimentalevidence indicates metal particles may be deformed or frag-mented during various stages of heterogeneous explosion,altering the particle size distribution (PSD) from the initialstate and affecting subsequent physical processes.

Some metal particles feature a thin passive oxide coating,which is presumed to be cracked or removed by the deto-nation shock. The high-strain-rate interaction with the det-onation shock may fragment the solid particle core. Yoshi-naka et al. [6] demonstrated that strong shocks in condensedmixtures can fracture aluminum particles with evidence thatthe particles did not melt within a microsecond timescale.Kim et al. [7] showed empirical evidence that modifyingthe initial PSD to approximate particle fragmentation canresult in closer agreement with experimental blast pressuredata.

The conditions within the condensed heterogeneous mat-ter detonation zone feature pressure levels typically above10 GPa. At the edge of the charge, the detonation shock

DRDC-RDDC-2015-P207

152 R. C. Ripley et al.

transitions into a very strong air shock that decays from10 GPa to 100 MPa within one charge radius, where high-temperature gas dissociation effects are significant. Measure-ments by Ogura et al. [8] and Goroshin et al. [9] indicate thatthe shocked air temperature may exceed 3,000 K by a fac-tor of 2–3. The heterogeneous blast pressure further decaysto just 1 MPa within ten charge radii and the shocked airtemperature falls below 1,000 K in the absence of particlereaction.

Within the transient region of rapid expansion and transi-tion from detonation pressure to air blast, the assumption ofconstant melting and evaporation temperatures, heat capac-ity, and surface tension for metal particles becomes inappro-priate and may affect the aerodynamic breakup and subse-quent combustion. Molten and liquid particles may becomeunstable under conditions of high relative flow (large slipvelocity) and may possibly fragment through aerodynamicbreakup. The melting behaviour of the particles under highpressure and the feasibility of subsequent breakup in the ini-tial expansion flow are critical for both dispersal range andefficient reaction of particles. Solid particles and stable liq-uid droplets may impact nearby target surfaces and subse-quently reflect inelastically or fragment. Frost et al. haveshown evidence that reactive metal particles can fragmentduring impact with target structures [10], and that the resul-tant reactive fragments can likely enhance reflected impulsefrom a heterogeneous explosive [11].

First-principles calculation of the wide range of lengthand time scales in heterogeneous explosion is infeasible,therefore macroscale simulation is a practical commonplacesolution. Numerical modeling of aluminized explosives nearstructures [12–15] has shown that aerodynamic breakup ofmolten particles and liquid metal droplets is one of the mostsignificant factors affecting the resulting heterogeneous blastloads. For the catastrophic droplet breakup mechanism inhigh relative flow conditions, the resulting fragment dropletsare typically ten times smaller in normalized diameter witha corresponding factor of 103 increase in number density.The smaller post-breakup size substantially increases thereaction rate in addition to the momentum and heat transferrates.

The present paper focuses on the establishment of thephysical models for use in hydrocodes to simulate vari-ous types of particle fragmentation and their effects duringheterogeneous explosions. The physical models consideredcover:

1. Particle damage or fragmentation during condensed mat-ter detonation.

2. Particle melting and aerodynamic breakup during earlyexpansion dispersal.

3. Particle and droplet impact fragmentation or coating dur-ing reflection on target surfaces.

2 Fragmentation theory

The mechanisms of dynamic fragmentation of metal particlesare distinguished by the metal state (brittle, ductile, or liquid)and velocity regime. Classical work on the dynamic fragmen-tation of metal particles includes the energy-based fragmen-tation theory developed by Grady and Kipp [16–19]. In gen-eral, the fragment size has been correlated to a representativediameter, dfrag = f

(1/ε̇2

)1/nwhere ε̇ is the strain rate and n

depends on the fragmentation mechanism. A fragmentationcriterion can be defined as σimpact ≥ pspall, where σimpact

is the impact stress and pspall is the spall pressure. Metalparticles relevant to heterogeneous blast explosives are usu-ally relatively ductile, such as high-purity aluminum, mag-nesium, and zirconium. Studies [4,14] have indicated thatimpact velocity during near-field blast interaction is belowthe hypervelocity regime. The following evaluation focuseson ductile and liquid fragmentation models of Grady [16] formetal particle impact events, and an aerodynamic fragmen-tation model for traveling molten particles.

2.1 Impact fragmentation

During the high-velocity impact of a metal particle, rapiddeformation and expansion from unloading of the impactshock provide kinetic energy relative to the particle center ofmass. Grady formulated an energy-governed dynamic frag-mentation model for ductile materials, in which an inducedenergy must exceed strain deformation energy, W = Y εc.Here Y is the material flow stress in tension, and εc is the crit-ical strain at failure. Forgoing detailed derivation, the frag-ment size relation is [16]:

dfrag =(

8Y εc

ρdε̇2

) 12

for pspall = (2B0Y εc)12 (1)

where dfrag is the fragment size, ρd is the particle materialdensity, ε̇ is the representative strain rate, and B0 is the mate-rial bulk modulus. The flow stress may vary according toY = f (ε, ε̇, T ). The tensile spall strength, pspall, determinesa threshold for particle fragmentation.

Applying a similar energy balance approach during frag-mentation of liquids droplets, it is assumed the impact energyis transferred to surface energy of the resulting fragments.Grady also provides a strain-rate-dependent fragmentationmodel for liquid droplets [16]:

dfrag =(

48γd

ρdε̇2

) 13

for pspall =(

6ρ2d c3

0γdε̇) 1

3(2)

where γd is the liquid surface tension and c0 is the materialbulk sound speed.

Fragmentation of metal particles 153

2.2 Aerodynamic breakup

Aerodynamic breakup of liquid droplets is based on a forcebalance between flow shear and particle surface tension.Aerodynamic breakup has traditionally been correlated tothe non-dimensional Weber number, W e = ρgu2

slipdp0/γd,where ρg is the gas density, uslip = |ug − up| is the slipvelocity, and dp0 is the initial droplet diameter. For lowWeber numbers (i.e., W e < 350) a variety of mechanisms fordroplet breakup exist [20], each producing different fragmentsize outcomes. However during heterogeneous blast, highshear and shock interaction are expected in which the mag-nitude of aerodynamic loading is much greater. The explosivebreakup regime [20,21] of catastrophic failure is expected.This breakup regime is characterized by initial fragmentationfrom large-amplitude surface waves followed by the substan-tial stripping of small fragments from the droplet surface. Forthis type of droplet breakup, the maximum stable diameter,dfrag, at the conclusion of the droplet breakup process can bedetermined according to [20]:

dfrag = dp0We∗

We0(3)

where We∗ is the critical Weber number. Viscous liquids areable to maintain coherence under higher degrees of aero-dynamic loading, therefore the stability threshold is depen-dent on droplet viscosity through the correlation of Brodkey[20,22]:

We∗ = 12(

1 + 1.077On1.6)

; On = μd(ρddp0γd

)0.5(4)

where On is the Ohnesorge number andμd is the temperature-dependent viscosity.

Droplet breakup requires a finite amount of time to occurduring aerodynamic loading. While correlations for breakuptime have been extensively studied, by reviewing experimen-tal data Gelfand suggested that a dimensionless breakup time,τb, for droplets in shock waves is relatively consistent, fromwhich he determined an absolute breakup time, tb [21]:

tb = τbdp0

uslip

√ρd

ρg(5)

where τb = 4–5 is the droplet interaction timescale.

3 Macroscale physical models

To facilitate use of the fragmentation theory in a heteroge-neous explosion problem, it is necessary to introduce appro-priate physical models with fragmentation parameters eval-uated a priori. Special considerations are addressed due tothe unique class of blast problems, which involves solidductile metals (aluminum, magnesium, zirconium, etc.) and

molten/liquid metals when particles are melted, and particlesthat are typically micrometric (1–100 µm in diameter). Thefollowing physical models are organized by process evolu-tion in heterogeneous explosions: detonation, expansion dis-persal, and target interaction. Each physical model employsa fragment size model, a rate model, and failure criteria. Therange of applicability and influence of thermophysical para-meters are addressed throughout.

3.1 Shock compression fragmentation in detonation

For fragmentation of micrometric particles in explosive det-onation, no direct experimental data are available sincethe high-temperature gas combustion products that followbehind the detonation front would further influence the frag-mentation and reaction of damaged particles. Therefore, ashock compression particle fragmentation model is con-sidered for the detonation regime. It is presumed that themetal particles remain in the solid state during shock com-pression along the Hugoniot. Inviscid hydrodynamic cal-culation of packed aluminum particles saturated with liq-uid nitromethane explosive [3] showed deformation duringshock compression of 0.25 based on change in diameter in thedirection normal to the detonation front (i.e., strain definedas ε = 1 − �/dp0), and 0.18 based on change in volumeof the particle during compression by the detonation shock(i.e., ε = ρ/ρ0 − 1). In general, the deformation dependson the particle material density and that of the surround-ing explosive. Possible factors reducing particle deforma-tion may include the particle material strength, or the com-plex interactions with high-pressure detonation products andneighboring particles.

To represent the solid particle fragmentation effect withGrady’s dynamic fragmentation model (1), it is necessaryto estimate particle strain rate and dynamic flow strength.Considering that the majority of particle deformation in det-onation of a heterogeneous explosive occurs within the shockinteraction timescale, τS = dp0/D, where D is the detona-tion propagation velocity, a strain rate can then be defined asε̇det = ε/τS. In the shock frame of reference, particles can beviewed as impacting the detonation front where the densitydiscontinuity acts as an impact surface. A simple approxi-mation of the shock-induced particle strain rate, ε̇det, can begiven as:

ε̇det = χdet(D/dp0

) = χdet

(uf S + c0

dp0

)(6)

where the detonation velocity is analogous to an impactvelocity. A model rate coefficient, χdet, is used to relate theshock velocity to strain deformation during compression. Formodel implementation, the shocked-fluid material velocity,uf , is employed using the Hugoniot of the host fluid (S is the


Table 1 Shock Hugoniot parameters for host fluids

ρ0 (g/cc) c0 (mm/µs) S

Nitromethane 1.128 1.647 1.637

Heptane 0.684 1.803 1.402

slope and c0 is the intercept). Shock Hugoniot parameters forcommon liquids saturating particles are given in Table 1.

Experiments by Yoshinaka et al. [6] observed recoveredsamples of aluminum powder subjected to high-pressure con-densed shock waves in an inert liquid. Their experimentalsetup used an explosively driven flyer plate to subject ValimetH30 spherical aluminum powder (dp0 of 36 ±16µm by mass)saturated with liquid heptane to a 12.9–29.3 GPa shock of1 µs duration. Since the sample was contained in an impactedcapsule, multiple shock reverberations are expected and thisis similar to the complex wave reflections in heterogeneousdetonation [3] although the frequency would be much differ-ent. Test results show particle damage characteristic of shearfailure and a shift in mean size towards smaller particles. Fig-ure 1 illustrates the change in PSD resulting from shock com-pression damage by counting the initial particles (N = 534)and fragments (N = 174) in the micrographs of [6]. Note thatthe particle sample sets are different between initial and testrecovery; as such, a batch distribution from Microtrac parti-cle size analysis is plotted for reference. From this analysis,damage from strong inert shocks appears to result in a shiftof approximately 25 % in mean particle diameter towardssmaller diameters. It is presumed that detonation interactionwill reduce the average particle size by a similar amount.

Rather than attempting to determine a physically basedstrain rate, the fundamental data of Yoshinaka et al. [6]were used to calibrate an analytical model for macroscopicshock compression fragmentation. The shocked-fluid mate-rial velocity based on the flyer plate impact experiment wasused to calculate the fragment diameter using (1) and (6).Initial particle diameters ranged from 1 to 80 µm with massweighting according to an idealized PSD of atomized alu-minum particles (type H30 made by Valimet Inc). Figure 2plots the fragment distribution curves against the idealizedinitial PSD. Through comparison of the observed (see Fig. 1)and analytical results, a rate coefficient for (6) of χdet = 0.06produces the resulting shift in average particle size in closestagreement with experimental data. Note that the symmetricalGaussian PSD may overpredict the appearance of large-sizedfragments since the probability of fragmentation of large par-ticles is high.

3.2 Aerodynamic breakup during expansion dispersal

After detonation, metal particles in the expanding blast floware dispersed while being subjected to extreme gas tempera-

tures and drag forces. As a result of these conditions, it is pos-sible that the particle temperature is beyond the melting point,Tm0, at lower pressures, and the resulting droplets becomeunstable and fragment in the gas flow. Any oxide coatinginitially present on the surface of the particles is assumed tohave broken off during the detonation shock compression; thedroplets are thus assumed to be pure materials. Table 2 sum-marizes typical parameters above Tm0, although it is notedthat these properties depend on the temperature of the liquid,composition of the surrounding gas (assumed to be air), anddegree of oxidization in the case of pure metal. The pres-

Particle Diameter (μm)

Rel

ativ

e M

ass

0 20 40 60 800%

5%

10%

15%

20%

25%

30%

35%

Initial Distribution (micrograph)Fragment Distribution (micrograph)Batch Distribution (microtrac)

Fig. 1 Shift in PSD from shock compression fragmentation. Measuredshock compression damage adapted from [6] using a histogram binwidth of 5 µm

Initial Diameter (μm)

Rel

ativ

eM

ass

0 20 40 60 800

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

χ = 0.06

χ = 0.15

χ = 0.10

InitialDistribution

Fig. 2 Analytical solutions to the detonation shock compression dam-age model for various χdet


Table 2 Droplet parameters (typical values at 1 atm.)

Material Temperature(K)

Surface tension,γ (N/m)

Viscosity, μ

(Ns/m2) × 103

Water 273 0.076 1.79

Magnesium [23] 924 0.56 1.25

Aluminum [24] 933 0.86 1.23

Titanium [23] 1,958 1.65 5.2

Zirconium [23] 2,128 1.48 8

x x

x

x

x

x

xxx

x

x

xx

x

xxx

Diameter (μm)

Max

imu

mW

eber

Num

ber

400 600 800 1000 1200 14000

10

20

30

40

50

FragmentStable

x

Fig. 3 Aerodynamic fragmentation threshold for aluminum/aluminaagglomerates in a convergent nozzle. Data reproduced from Cavenyand Gany [25]

sure dependence of the melting point will be described later.Water is included for reference.

For conventional hydrocarbon liquid droplets, the limitof stability is We∗ = 12–20 which indicates that the sur-face tension and droplet viscosity are insufficient to main-tain a coherent droplet shape and breakup occurs [20]. Eval-uation of the properties in Table 2 shows that the Ohne-sorge number and corresponding Brodkey correction (4)for viscous effects are insignificant for micrometric metaldroplets.

Caveny and Gany [25] provide one of the few availableexperimental works looking specifically at Weber thresholdfor breakup of reactive molten metal droplets. They stud-ied aluminum/alumina agglomerates in the combustion prod-ucts of a propellant. Observing aerodynamic breakup of theagglomerates passing through a convergent nozzle, Cavenyand Gany suggest that the stability threshold appears to beWe∗ = 20–30, which is slightly higher than values typi-cally observed for conventional liquids. Figure 3 plots theWeber number estimated in [25] from each observed agglom-

erate along with the corresponding aerodynamic breakupresult.

Despite these observations, Caveny and Gany cautionedthat their estimates may be systematically high as only cal-culated gas velocities could be used and direct observationof initial agglomerate diameters was challenged by strongluminescence from the reacting agglomerates [25]. Of addi-tional note, the authors used a surface tension value for liquidalumina of 0.69 N/m in their calculation of Weber number.Considering that the agglomerate surface composition couldpotentially contain liquid aluminum with a surface tensionof 0.86 N/m (see Table 2), the corresponding critical Webernumber would then be 15 % lower. Thus in the present work,We∗ = 25 was used for aluminum.

A final consideration for aerodynamic breakup of dropletsis accounting for the distribution of fragment sizes produced.According to the fact that the mass-median droplet fragmentsize is on the order of half of the stable droplet diameter [20],the model used in the numerical implementation is:

dfrag =max

⎛

⎜⎝

1

2We∗ γd

ρg

(u∗

slip

)2 ,dp0

102

⎞

⎟⎠ if tWe>We∗ > tb

(7)

and u∗slip is the fragment slip velocity at the conclusion of

the breakup process. The methods suggested by Pilch andErdman [22] and Kolev [20] determine u∗

slip using empiricalrelations for enhanced drag during the breakup timescale.Enhanced drag is thought to be a result of deformed non-spherical droplets, and interactions within the dense frag-ment cloud. As noted by Pilch and Erdman [22], the extentof enhanced drag increases with the extent of fragmenta-tion. For modeling of heterogeneous blast, where particlesare accelerated in dense gas, the relations for enhanced draghave been found to overpredict droplet acceleration. Instead,the concept of enhanced drag is used to place a limit on thesmallest fragment size. A lower bound of dfrag/dp0 ∼ 10−2

represents the limiting fragment size from the data of Pilchand Erdman [22], which occurs for We > 1,000. This modelis normally appropriate for breakup of water and hydrocar-bon droplets in air shocks, and has been adopted for moltenmetals in the present work.

3.3 Impact fragmentation

The premise that reactive particles may fragment on impactwith nearby structure surfaces, thereby increasing particlereactivity and influencing reflected blast wave loading, iswell supported by several previous studies [4,10,11,26]. Toapply Grady’s energy-based fragmentation theory, modelsfor impact stress and strain rate are required for fragmenta-tion threshold and fragment size. In the absence of simple


models for determining representative strain rate values foran impacting sphere on a flat plate, the following relation isused:

ε̇ = χimpact(up/dp

). (8)

Although simple, a similar relation was found to ade-quately represent strain rate by Champagne et al. [27] formodeling cold spray particle impact. To determine a valuefor the reduction factor, χimpact, comments by Grady [16] onthe plate impact experiments of Christman et al. [28] indicatesuitable values to be in the range of 0.25–0.50.

To obtain a velocity threshold for impact fragmentation,the spall strength criteria given by (1) and (2) can be evaluatedagainst impact stress. As material spall requires tensile load-ing, it is assumed that the tensile stress caused by unloadingof the impact shock is similar in magnitude to the initial com-pressive stress. For an upper-bound estimate on the impactshock, the Hugoniot pressure, pH, at the contact between theparticle (p) and target surface (s) can be used [19]:

pH = Zp Zs

Zp + ZsVi (9)

where Zm = ρmcm is the acoustic impedance, with ρm andcm as the density and sound speed of each material (m =s, p), and Vi is the interface velocity, which can be takenas the impact velocity. In a spherical particle, the transmit-ted internal shock is ellipsoidal and greatly diminishes instrength as it travels to the back of the particle [19]. Fromnumerical simulation, Grady and Kipp [29], and Dykhuizenet al. [30], suggest this pressure decrease is linear along theimpact axis. From this information a simple estimate of meanshock pressure can be made, p̄ = 0.5pH. A lower boundon particle impact stress, which is more appropriate for theimpact of molten particles, is the Bernoulli stagnation pres-sure, pstag = 0.5ρdV 2

i .

3.4 Heating and fusion during impact

In addition to fragment size, fragment temperature and veloc-ity may influence particle mixing and reaction rate afterimpact. Evaluation of previous modeling results [12,13] hasshown small diameter fragments quickly equilibrate withthe local fluid flow. In the present study fragment velocityis therefore approximated using a simple plastic reflectionmodel described in Sect. 3.5; future investigations could con-sider fragment spray velocity. To determine fragment temper-ature, plastic deformation heating and residual impact shockheating are both considered. Deformation energy is approxi-mated using the product of flow stress and strain at fragmen-tation, Y εc, while shock energy, eshock, is evaluated using thefollowing relation derived from Grady and Kipp [19]:

eshock = 1

2

(p2

H

ρpc2p

)

f (Vi/c0) . (10)

The non-linear function, f (Vi/c0) given in [19], accountsfor acoustic coupling of the shock into the particle andapproaches unity as Vi → 2c0. After unloading of the impactshock, Kipp et al. [31] estimated 10 % of total shock energy isretained as heat (i.e., fheat = 0.1). If it is assumed the addedheat is evenly distributed through the particle, it is possibleto estimate the bulk temperature increase as:

T = Y εc + fheateshock

ρdcv

(11)

where cv is the heat capacity at constant volume. Similarto fragment velocity, evaluation of this expression quicklyshows T to be small for the impact of metal particles below1,000 m/s.

Under certain conditions particles can fuse on impact withtarget surfaces, creating a coating on the target. This behav-iour is relevant to heterogeneous blast, as particles that coattarget surfaces may cool and quench, reducing overall parti-cle combustion. Overall the physical process by which con-tact bonding of solid particles occurs is not well established inthe literature, and simplified models for bonding criteria arenot readily available. The impact of molten particles presentsan alternate mechanism for target surface coating, as dropletscan cool and solidify on arbitrary surfaces. For low-speedmolten particle impact, Fauchais et al. [32] provide criteriato determine if a molten particle will coat the surface:

Kcoat = We1/2L Re1/4

L WeL = ρdV 2i dp

γdReL = ρdVidp

μd

(12)

where Kcoat is the coating parameter, WeL is a modifiedWeber number using the droplet density and impact veloc-ity, and ReL is a modified Reynolds number which uses thedroplet viscosity, in contrast with traditional definitions ofthese values that normally use gas properties. A transitionvalue of Kcoat = 70 is given in [32]; below this value coat-ing will occur, above this value the droplet will reflect orfragment on impact.

3.5 Reflection velocity of particles and fragments

Particle impacts, either with or without fragmentation, mayresult in inelastic reflection of the metallic particle at thetarget surface. High reflection velocities can reintroduce par-ticles into shock-heated regions and may increase overallmixing of reactive particles with oxidizers. Low reflectedparticle velocities may cause localized particle reactions, orconversely may reduce overall mixing and particle combus-tion. For these reasons, the use of a physical model to deter-mine the reflected particle velocity is a necessity.


Impact Velocity (m/s)

Res

titu

tio

n C

oef

fici

ent

100 101 102 1030

0.2

0.4

0.6

0.8

1

AluminumTitaniumZirconiumMagnesium

Fig. 4 Calculated restitution coefficients for ductile metal spheresimpacting a flat steel plate

The reflected particle velocity magnitude normal to thesurface is taken as Vr = eVi, in which e is the restitutioncoefficient and Vi is the incident particle velocity magnitudenormal to the reflection surface; the tangential velocity com-ponent is assumed to be perfectly conserved during reflec-tion. For ductile particle materials, the coefficient of resti-tution is a strong function of incident velocity and materialstrength properties. A simple model that accounts for a rangeof particle impact velocities is Johnson’s classical reflectionmodel [33] of a plastic sphere on a flat plate, which is inslightly adjusted form:

e = 1.718

((3Ymin)

5

E∗4ρp

)1/8

V −1/4i

E∗ =(

1 − ν2p

Ep+ 1 − ν2

s

Es

)−1

(13)

where Ymin is the lesser yield strength of the two materials,E∗ is the effective Young’s modulus, ν is Poisson’s ratio,and subscripts denote the particle (p) and surface (s) materi-als respectively. Figure 4 shows the inelastic reflection coef-ficient calculated using (13) for several commercially pureductile metal particles.

4 Numerical methods

The methods are implemented in the Chinook CFD codewhich utilizes a multiphase numerical framework [34]. Forapplication of particle fragmentation in heterogeneous blast,a two-phase macroscopic modeling approach is employed.

Lagrangian particles are used in a group method that repre-sents a range of particle diameters spanning two orders ofmagnitude in size [35]. Each Lagrange group contains thecollective mass of 102–104 physical metal particles, whichhave the same velocity, temperature, and size. An Eulerianfluid domain with an HLLC flux solver is used for capturingthe shock physics. The fluid solver utilizes a Mie–Grüneisenequation of state for condensed explosives, a reaction-ratemodel for the explosive detonation, and a JWL equation ofstate for the detonation products. Afterburning of detonationproducts and shocked air temperature are represented in areal gas equation of state [36].

The governing equations for three-dimensional flow, inwhich the gas (formerly subscript g) has been generalized asa fluid (subscript f) to include the condensed explosive, areas follows:

∂

∂t

⎡

⎢⎢⎢⎢⎣

σf

σf uf

σfvf

σfwf

σf Ef

⎤

⎥⎥⎥⎥⎦

+ ∇

⎡

⎢⎢⎢⎢⎣

σf Vn

σf uf Vn + φf pf nx

σfvf Vn + φf pf ny

σfwf Vn + φf pf nz

σf Ef Vn + φf pf Vn

⎤

⎥⎥⎥⎥⎦

= Ω−1

⎡

⎢⎢⎢⎢⎢⎢⎣

∑np( jp)∑np( jpup + fp,x )∑np( jpvp + fp,y)∑np( jpwp + fp,z)

∑np

(jpep + ∑

dirfp,i up,i + qp

)

⎤

⎥⎥⎥⎥⎥⎥⎦

(14)

φp = Ω−1∑

np

(π

6d3

p

)φf = 1 − φp (15)

where σf is the gas phase mass concentration, Vn is the gasnormal velocity, uf , vf , wf are Cartesian components of thegas velocity, φf pf is the gas phase partial pressure, nx , ny ,nz are the normal directions, and Ef is the sum of internaland kinetic gas phase energy. Ω represents the volume of thefluid cell. The volume fraction of particles, φp, in the Euleriangas phase solution is updated by summing individual particlevolumes to the computational mesh using optimized search-ing algorithms. Particle variables (subscript p) are describedbelow with their respective governing equations.

All particles within a group share the same characteristicsize, physical properties, and state. The governing differentialequations for each Lagrangian group are as follows:

∂

∂t

⎡

⎢⎢⎢⎢⎢⎢⎣

mp

up

vp

wp

ep

np

⎤

⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎣

jpfp,x/mp

fp,y/mp

fp,z/mp

qp/mp

Ψp

⎤

⎥⎥⎥⎥⎥⎥⎦

(16)

where mp is the mass of an individual particle within thegroup, jp is the reaction rate per particle, up, vp, wp are theCartesian particle velocity components, fp is the net force


acting on a single particle, ep is the internal energy of a par-ticle, qp is the thermal energy flux in J/s per particle, np isthe number of particles represented by the Lagrangian group,and Ψp is the fragmentation or agglomeration rate. Particlemass is initially determined according to particle diameter,which is chosen randomly and assigned to each group usinga weighting function designed to match the desired PSD.

The mass transfer source term, jp, is dictated by a hybridkinetic-diffusion reaction mechanism of Zhang et al. [37] andis employed for particle combustion. Condensed productsproduced by particle reactions (e.g., Al2O3) are treated as acondensed (cp = cv) ‘volumeless’ mass within the Euleriangas phase. In dilute flow, the particle force, fp, and heat trans-fer, qp, are from correlations for isolated spheres. Duringdense flow, shock compression acceleration in the detonationstage [3] and high-volume-fraction flow of particles duringearly dispersal [38] dominate the aerodynamic force on theparticles.

The fragmentation source term Ψp is developed in thepresent work. The aforementioned fragmentation theory andmacroscopic physical models are applied in the number den-sity governing equation:

Ψp = np0

t

[(dp0

dfrag

)3

− 1

]

(17)

where t is the numerical timestep. The above fragmentsize models are applied as an average size for a subgroupof the large number of Lagrange particle group sets andthe fragments resulting from fragmentation and aerodynamicbreakup therefore feature a transient size distribution of thetotal particles.

4.1 Equation of state for particles and drops

Analysis of the Biot number for micrometric metal parti-cles and droplets readily shows that a lumped capacitanceassumption is valid for most situations. Therefore, it isassumed a bulk particle can be adequately represented usinga single temperature, and phase change is entirely completeafter latent heat is overcome. These assumptions are includedin the following thermal equation of state for the particles:

Tp =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ep/cv ep < cvTm

Tm cvTm ≤ ep < cvTm + Lmep − Lm

cv

cvTm + Lm ≤ ep < cvTv + Lm

Tv cvTv + Lm ≤ ep < cvTv + Lm + Lvep − Lm − Lv

cv

cvTv + Lm + Lv ≤ ep

(18)

where ep is the internal energy, cv(T ) is the heat capacity,Tm(p) is the melting temperature, Lm is the latent heat of

melting, Tv(p) is the vaporization temperature, and Lv is thelatent heat of vaporization.

The melting and evaporation temperature of metal parti-cles increases with hydrostatic pressure. A model for meltingwas developed [15] for aluminum using fitting to experimen-tal data for low pressure and to theoretical data at high pres-sure:

Tm(p)

= Tm0+{

p/p∗ for p ≤ 8.16 GPa

a1 p3 + a2 p2 + a3 p + a4 8.16 < p < 150 GPa

(19)

where Tm0 = 933 K and p∗ = 0.015 GPa from linear fittingto the data of Lees and Williamson [39]; a1 = 0.0003, a2 =−0.16, a3 = 45, and a4 = 166 from polynomial fitting to thedata of Moriarty et al. [40]. In (19), p is the pressure in GPa.The upper limit of aluminum heating is dictated through ahigh-pressure evaporation curve by Gonor [41].

5 Fundamental validation

Validation of fragmentation of metal particles in heteroge-neous explosions is challenged by a lack of fundamental testdata. Very limited data are available for detonation damageto particles, aerodynamic breakup of melted particles, andhydrodynamic fragmentation of molten metal particles inheterogeneous explosions. As an example, experiments ofFauchais et al. [32] on the impact of molten metal dropletsshow complex fragmentation behavior involving liquid jet-ting and streams of fragments where the observed fragmentsize can be orders of magnitude smaller than the originaldroplet. For impact fragmentation, the majority of experi-mental data of Grady and Kipp [16,19,29] focus on veryhigh velocity impacts (>2,500 m/s) of millimeter-scale par-ticles with thin plate targets. In the context of heterogeneousblast, impact velocities on the order of 1,000 m/s, micron-to-millimeter-scale particles, and solid target surfaces areof interest. Therefore more relevant experiments have beensought, including Van Steenkiste et al. [42] on the spray coat-ing of aluminum particles, and Janidlo [43] on the impactfragmentation of various metals. Aerodynamic breakup ofwater drops in shock waves in air is used for validation; sim-ilar analyses are common for hydrocarbon droplets.

5.1 Water drops in normal shock waves

A series of simple calculations to validate the droplet breakupmodel for conventional liquids was performed using waterdroplets and compared with the data collected by Pilch andErdman [22]. The numerical model featured a single waterdroplet in a shock tube; droplet diameters of 25 and 100 µm


Time (μs)

Web

er N

um

ber

dfr

ag /

dp

0

-20 0 20 40100

101

102

103

104

10-3

10-2

10-1

100

101

Wedfrag / dp0

tb

Fig. 5 History of a 100-µm-diameter water droplet broken up by aMach 2.5 normal shock

Weber Number

No

rmal

ized

Fra

gmen

tSiz

e,d

frag

/dp0

101 102 103 104 105 10610-3

10-2

10-1

100

ExperimentalModel Result

Fig. 6 Comparison of numerical results with experimental data [22]for water droplet aerodynamic fragmentation subjected to normal shockwaves (W e∗ = 12)

were subjected to normal shocks of Mach 1.5, 2.5, 4.0, and6.0. Figure 5 plots the loading history from the 100-µm-diameter water droplet subjected to a Mach 2.5 shock. Thenumerical results show the Weber loading of approximately2,500, and a breakup timescale of approximately 14 µs,which is consistent with (5). In the numerical model thebreakup is treated as instantaneous once the droplet breakuptimescale is exceeded. In Fig. 6, calculated fragment diam-eter is plotted against maximum Weber number, along withthe collected experimental data from [22].

Calculated fragment sizes compare reasonably well tocollected data. During calculations of high We flows, theimposed fragment size limit results in fragments that


Fra

gm

entD

iam

eter

,dfr

ag(μ

m)

0 500 1000 1500 20000

10

20

30

40

50

60

70

80

90

dp0 = 80 μmdp0 = 50 μm

Observed onset of ductile fracture

Fig. 7 Comparison of observed [20] and numerical onset of ductilefracture during impact of aluminum particles using χimpact = 0.5

Table 3 Model parameters for the impact fragmentation of variousmetals

ρ0 (kg/m3) c0 (m/s) B0 (GPa)[16]

Y (MPa)[16]

εc[16]

Aluminum† 2,710 5,266 72.9 30 0.42

Titanium 4,510 4,937 105 900 0.23

Zirconium† 6,570 3,891 89.8 250 0.40∗

Brass 8,450 3,660 105 – –

Steel 7,890 4,507 168 344 0.3

∗ Estimated value† High-purity metals

exceeded the maximum stable diameter criteria; however,the simulated fragments quickly equilibrated with the gasflow without undergoing additional fragmentation.

5.2 Metal particle impact fragmentation on a plate

In the experiments by Van Steenkiste et al. [42], aluminumparticles were sprayed at velocities ranging from 400 to500 m/s onto a brass surface. Examination of post-impactparticles showed evidence of ductile fracture for impactvelocities above 450 m/s. Figure 7 plots the calculated(1), (8), and (9) fragment diameter of solid aluminum parti-cles, against the observed onset of ductile fracture in sprayedparticles [42]. Table 3 provides model parameters used in thecalculations.

Observations by Janidlo [43] also provide relevant data;the experiments utilized a gas gun to accelerate 6.35-mm-diameter particles (Al, Ti, and Zr) up to 900 m/s for impactwith a thick steel plate. For aluminum particles the observedfragmentation threshold was between 600 and 800 m/s. Thisis considerably higher than the observed value of 450 m/s in[34]. The high fragmentation threshold may be attributed to



Num

ber

ofF

ragm

ents

,Nfr

ag

500 600 700 800 9001

2

3

4

5

6

7

Observed

Analytical

Fig. 8 Comparison of experimental [42] and numerical results for zir-conium particle impact fragmentation using χimpact = 0.7

high thermal conductance in aluminum, which may inhibitthe fracture mechanism of adiabatic shear banding [43]. It isnoteworthy that the experiments conducted by Janidlo [43]utilized 6.35 mm particles, while the experiments by VanSteenkiste et al. [42] utilized 50–80 µm particles. Accord-ing to (8), this constitutes a two orders-of-magnitude differ-ence in strain rate, which could potentially account for dis-crepancies in the observed threshold fragmentation behavior.Grady and Kipp [19] also impacted 6.33 mm diameter par-ticles, but at a much higher velocity of 3,920 m/s and witha thin PMMA target. Using radiography, they observed analuminum fragment debris cloud with a 0.4–1.5 mm sizedistribution. Equations (1) and (8) with the parameters inTable 3 give dfrag/dp0 = 0.1, which is consistent with themean fragment size found by Grady and Kipp.

To compare the analytical model with the zirconium par-ticle experiments by Janidlo [43], Fig. 8 plots the calculatednumber of fragments against observed values. Table 3 pro-vides model parameters used in the calculations; as noted forzirconium, no values for dynamic failure strain were foundin the literature, and an assumed value of εc = 0.40 was usedto be consistent with observed values for other ductile mate-rials. Calibration with the data required a higher rate coeffi-cient, χimpact = 0.70, with (8) indicating that rate-dependentmaterial properties may be required.

6 Application to heterogeneous explosion

To evaluate the physical models for particle fragmentationduring heterogeneous explosion, simulation of detonation,particle dispersal, blast expansion, and multiphase reflectionare considered. A fundamental explosive system of spher-ical aluminum particles saturated with nitromethane (NM)

Position (mm)

Pre

ssur

e(G

Pa)

dfr

ag/d

p0

10 20 30 40 50 600

5

10

15

20

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Flow PressureFragment Size

Fig. 9 Results for detonation flow pressure with particle fragmentationduring NM/Al detonation

is presented for detonation and free-field dispersal. A finaldemonstration is conducted using a cylindrical charge in asemi-confined urban street setting.

6.1 Heterogeneous detonation

Fragmentation of metal particles begins at the detonationstage of a heterogeneous explosion. No direct experimen-tal data for particle size are available for this regime underconditions of explosive detonation and reacting metal parti-cles. Figure 9 demonstrates the present physical models fordetonation of NM containing Valimet H30 aluminum parti-cles. In this macroscale calculation using a 0.2 mm cell size,the interaction of the NM shock and reaction zone promptlyfragments the aluminum particles resulting in a shift in thePSD towards smaller sizes, which is illustrated in Fig. 10.

6.2 Free-field heterogeneous blast

To validate the physical models during multiphase disper-sal, a comparison with experimental measurements for blastpressure and impulse was conducted for a spherical explosivecharge containing packed aluminum powder saturated withNM. The charge is 12.3 cm in diameter and approximately1.8 kg in mass, and contains aluminum powder correspond-ing to a measured size distribution of 54 ± 21 µm (ValimetH50). Additional details on the experimental setup can befound in [5,11].

The configuration is idealized using a 1D sphericaldomain. Figure 11 provides a comparison with experimen-tal data for peak overpressure and maximum positive-phaseimpulse. The numerical results overpredict the experimentalblast pressure in the very near field, but otherwise are within


the range of the experimental results indicated by the errorbars. Blast impulse results agree quite well at all ranges.

The numerical results may be analyzed to illustrate thecomplexity of the blast field. The initial particle distribution isshown in Fig. 12 (left) along with the numerically-predictedfragment distribution at 0.8 ms, which becomes bimodal dueto detonation damage and aerodynamic breakup. Figure 12(right) shows the two-phase radial velocities at the same time,which was chosen to correspond with particle dispersal tothe radius where maximum impulse occurred. The smallestparticle and droplet fragments are decelerated through the


Par

ticle

Mas

s

0 20 40 60 800%

5%

10%

15%

20%

25%

Initial DistributionFragment Distribution

Fig. 10 Shift in PSD during NM/Al detonation

rarefaction wave into higher density, lower velocity gas witha large number of particles nearly in equilibrium. The largestparticles do not melt and are slow to relax; a number of largerparticles and fragments have overtaken the shock at 1.02 mbut will later fall behind due to drag.

6.3 Near-field heterogeneous blast

To further highlight the importance of aerodynamic breakupof melted particles, an essentially uncased cylindrical chargecontaining a heterogeneous explosive mixture with a highmass fraction of aluminum particles was simulated. Thecharge features a tertiary mixture of aluminum particles andexplosives in a coaxial configuration, surrounding a cylindri-cal high explosive booster at the center that is initiated fromthe top. The total charge mass is 2.2 kg, including the boosterand the mixture. The charge configuration is described in [4].

The physical models and corresponding parameters devel-oped in this paper are applicable to this complicated alu-minized explosive mixture. As before, a breakup criterionbased on Weber number (We∗ > 25) was utilized for par-ticles that have overcome their latent heat of fusion andremained melted. Constant surface tension (γd = 0.86 N/m)was assumed while the high-pressure melting curve for alu-minum was used. Figure 13 illustrates the extent of particlefragmentation at very early times by plotting the number ofparticle fragments, Nfrag, normalized by the initial number ofparticle groups, Np0. There is damage to the particles duringthe detonation shock compression that shifts the mean PSDtowards smaller diameters as the detonation wave passes.Further behind the detonation shock, aerodynamic breakupand reaction of the particles begin, most prominently along

Radius (m)

Pea

k O

ver-

Pre

ssu

re (

MP

a)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

ExperimentalNumerical

Radius (m)

Max

imu

m Im

pu

lse

(kP

a-m

s)

0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

300

ExperimentalNumerical

Fig. 11 Comparison of experimental [5] and numerical results for pressure (left) and impulse (right) for free-field explosion of a 1.8 kg NM-H50charge



Par

ticle

Mas

s(k

g)

0 20 40 60 80 100 120 140 1600

0.02

0.04

0.06

0.08

0.1

Initial Distribution

Fragment Distribution (t = 0.80 ms)

Radius (m)

Rad

ial V

elo

city

(m

/s)

0 0.2 0.4 0.6 0.8 1 1.20

200

400

600

800

1000

1200

Particle VelocityGas Velocity

Fig. 12 Numerical results during near-field dispersal at t = 0.8 ms showing (left) bimodal PSD and (right) complex interaction of gas and particlevelocity field due to the range in particle fragment sizes. Symbol size is proportional to dp

Fig. 13 Two-dimensional axi-symmetric model of multiphase detonation and near-field dispersal from a metalized explosive. The particle stateis identified by colored points; lines denote flow pressure on a log scale. Left results during detonation (t = 10 µs). Right results during earlyexpansion to approximately six charge diameters (t = 100 µs)

the interface with the high explosive booster and at the freeedges of the charge as the detonation products break out. Dur-ing the very early expansion (Fig. 13, right) there is extensiveaerodynamic breakup of the dispersed particles resulting ina significantly higher number of fragments with correspond-ingly smaller mean droplet sizes. The reaction of particles ishighlighted by the reduction in particle diameter relative tothe post-fragment size, dp0 = dfrag.

Figure 14 provides a comparison of reaction histories withand without aerodynamic breakup of melted particles for two

particle melting criteria. Figure 14 (left) gives the results dur-ing the timescale of detonation and early dispersal, whichshows significantly increased consumption of aluminum dueto extensive breakup and corresponding increased reactionrates of smaller droplets. On the timescale of 100 µs, thealuminum reaction is almost exclusively with the detona-tion products gases. The early-time aerodynamic breakupand increased reaction in the detonation products result inmuch fewer particles escaping the fireball and reacting withoxygen in the air. Figure 14 (right) shows the PSD following


Time (ms)

Tot

alR

esid

ualM

ass

(%)

0 0.02 0.04 0.06 0.08 0.150

60

70

80

90

100

No Aero BreakupTm = f(P)Tm = 933 K


Mas

s (k

g)

100 101 102

0

0.01

0.02

0.03

0.04

0.05

0.06

No Aero BreakupTm = f(P)Tm = 933 K

Fig. 14 The effect of aerodynamic breakup of melted particles on the global reaction history and distribution of particle sizes. Left detonationand early expansion timeframe (t < 100 µs). Right PSD following detonation of the explosive charge (t = 25 µs). Dashed line shows the initialcondition for reference

Fig. 15 Photograph of the DRDC Suffield straight urban street

the detonation stage (t = 25 µs). The results with aero-dynamic breakup illustrate an order-of-magnitude shift inparticle sizes as the smaller droplet fragments are created.The results with the high-pressure aluminum melting curvereduced the amount of aerodynamic breakup compared tothe fixed melting temperature for standard atmospheric pres-sure. Consequently fewer small fragments were available toincrease the degree of particle reaction resulting in a slightlylarger residual aluminum mass fraction. Overall the high-pressure melting curve had a marginal impact on the com-bustion of aluminum particles in the very early time for theinitial size of particles considered. Beyond a timescale of100 µs, the flow pressure does not significantly affect themelting temperature of aluminum.

Fig. 16 Schematic of gauge locations in the straight urban street

6.4 Semi-confined heterogeneous blast

The blast field containing high-speed reactive particles mayinteract with nearby structures. A two-wall street environ-ment is used to study such a semi-confined heterogeneousblast; the experimental configuration is described in [4].Briefly, the test facility features parallel steel walls 6.1 mlong by 3.66 m tall, each 16,300 kg in mass, with the chargelocated halfway between the walls and 1.93 m above theground (see Fig. 15). With the open ends and top of thestreet, the semi-confined blast falls between the limits of free-field and closed-volume explosion. Pressure transducers arelocated on the wall to measure blast loads (see Fig. 16).

For the two large steel walls placed 1.8 m apart in a streetconfiguration, the particle interaction with the rigid wall andthe rigid ground begins to occur after 250 µs. Figure 17 illus-trates the particle fragmentation outcome, where the met-alized explosive charge described in Sect. 6.3 was initially


centered at (x , y, z) = (0.9, 4.53, 1.93 m). A significant num-ber of normal impact fragmentation events are in the solidphase, whereas hydrodynamic fragmentation is predominantat the foot of the wall. The particles impacting near the walltop are due to the influence of the rarefaction from the topopening. Figure 18 presents the particle fragment size alongthe street length and as a function of the incident normalvelocity. Below 400 m/s, the impact fragmentation is mainlyhydrodynamic; between 100 and 300 m/s, the fragments aremostly from aerodynamic breakup of small droplets. Above400 m/s incident velocity, the larger fragments are from solid

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X

X

X

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Distance Along Wall, y (m)

Hei

gh

t,z

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0 1 2 3 4 5 6

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2

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ground

wall

X

Fig. 17 Wall impact fragmentation outcome results. Impact eventswith Vi < 10 m/s not plotted. Charge location normal to gauge 3Cat (y, z) = (4.53, 1.93) m

impact whereas the smaller fragments are from impact oflarger droplets.

The near-field and semi-confined heterogeneous blastregime is the most challenging for numerical models, andcomparison with experimental gauges is often difficult. Fig-ure 19 shows a comparison of pressure measurements on thewall of the 2.4 m wide street at various distances surround-ing the explosive charge using the models described in thispaper (see Fig. 16 for the gauge layout). The normal reflectiongauge (3C) and a very near-field gauge (4A) demonstrate anexcellent comparison of blast pressure history and reflectedimpulse. Further down the street, the numerical and exper-imental results for pressure and impulse are in good agree-ment. In general, the mid-field (gauges 5A–9C) experimentalimpulse is under predicted by the model. However, the arrivaltimes of the incident shock, complex blast effects, and streetreflections match well, thus indicating that the correct shockstrength is calculated.

To assess the grid dependency of these second-order accu-rate results, the calculation was performed on two 3D mesheswith resolutions of 25 and 50 mm, respectively. Across 36numerical gauge locations on the wall, the sensitivity in shockarrival time was 0.6 % and the change in maximum impulsewas less than 4.6 %. Further rigorous grid independence stud-ies would need to consider the number of Lagrangian particlegroups.

7 Conclusions

Metal particles can be damaged or fragmented during explo-sion of metalized explosives, altering the distribution of par-

Fig. 18 Non-dimensional fragment size versus distance along the street wall (left) and as a function of incident normal velocity (right). Symbolsize is proportional to dp0


Fig. 19 Comparison of numerical and experimental [4] pressure gauge results along the two-wall street. 3C : (y, z) = (4.53, 1.93) m; 4A : (y, z) =(3.79, 0.09) m; 5A : (y, z) = (3.05, 0.09) m; 6A : (y, z) = (2.32, 0.09) m; 8B : (y, z) = (0.85, 1.01) m; and, 9C : (y, z) = (0.114, 1.93) m


ticle sizes and influencing subsequent dispersal and com-bustion phenomena. In order to incorporate fragmentationbehavior into a macroscopic multiphase CFD code for het-erogeneous blast events, physical models for various frag-mentation mechanisms have been collected and adapted towork within a group Lagrange particle solver. Energy-baseddynamic fragmentation models have been applied to modelparticle damage during detonation and high-speed parti-cle impact fragmentation. During particle dispersal, aerody-namic breakup of molten particles is modeled according todroplet stability criteria.

The particle fragmentation models have been validatedby limited fundamental test cases for shock compressiondamage during detonation, liquid droplet fragmentation inair shocks, and single particle impact scenarios. The resultsof these basic tests indicate that the models are capableof representing the relevant physical behaviors expected inheterogeneous explosion. In some circumstances, such aswith the aerodynamic breakup model, sufficient experimentalresources for conditions of reacting high-speed liquid metalparticles are not yet available in the literature.

To evaluate the fragmentation models in the context ofheterogeneous explosion, applied calculations of multiphasedetonation, free-field dispersal, and reflected heterogeneousblast were performed. Calculation results show moderateamounts of particle fragmentation during the detonationstage that shifts the PSD towards smaller particles. During thefree-field blast calculation, the molten droplet aerodynamicbreakup model predicted the extent of particle fragmentationas indicated by the associated reaction rate and blast enhance-ment from an aluminized explosive. Blast loads in the freefield and on reflecting surfaces in confinement are in goodagreement with experimental data.

Acknowledgments The authors greatly appreciate the contributionsof J. Leadbetter.

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