fan zhang, paul a. thibault and rick link- shock interaction with solid particles in condensed...

8/3/2019 Fan Zhang, Paul A. Thibault and Rick Link- Shock interaction with solid particles in condensed matter and related m

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doi: 10.1098/rspa.2002.1045, 705-7264592003Proc. R. Soc. Lond. A

Fan Zhang, Paul A. Thibault and Rick Linktransfercondensed matter and related momentumShock interaction with solid particles in

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10.1098/rspa.2002.1045

Shock interaction with solid particles in

condensed matter and related

momentum transfer

B y F an Z h a n g1, Paul A. Thibault2 a n d R i c k L i n k3

1Defence R&D Canada-Sueld, PO Box 4000, Stn Main,Medicine Hat, AB T1A 8K6, Canada ([email protected])

2Timescales-Scientic Ltd, 20 Bedford Street, Bedford, NS B4A 1W6, Canada3Combustion Dynamics Ltd, 1888 Brunswick Street, Halifax, NS B3J 3J8, Canada

Received 11 April 2002; accepted 9 July 2002; published online 20 January 2003

Detonation propagation in a condensed explosive with metal particles can resultin signicant momentum transfer between the explosive and the particles duringtheir crossing of the leading shock front. Consequently, the assumption of a `phase-interaction-frozen shock used in multiphase continuum models for detonation initi-ation and propagation may not be valid. This paper addresses this issue by perform-ing numerical and theoretical calculations in liquid explosives and RDX with variouscompressible metal particles under conditions of detonation pressure. The resultsshow that the momentum transferred to heavy-metal particles such as tungsten is

not signicant after the shock{particle interaction. However, light-metal particlesincluding aluminium, beryllium and magnesium rapidly accelerate during the shock{particle interaction. They reach a considerable speed immediately behind the shockfront, typically 60{100% of the ow speed of the explosive. It is important to takethis signicant momentum transfer at the shock front into account when modellingthe shock initiation and detonation structure for two-phase mixtures of condensedexplosive and light-metal particles.

Keywords: shock waves; detonations; multiphase; particle-uid ows

1. Introduction

For detonation propagation in a condensed explosive with metal particles, interactionbetween the leading shock front and the particles could result in a momentum lossto the explosive that inuences the detonation initiation and structure. Theoreticalmodels for detonation in a uid{solid particle system have mostly been based on two-phase uid dynamics models taking mass, momentum and heat transfer between thephases into consideration (Zeldovich & Kompaneets 1960). Dynamic compaction ofpacked solid particles has also been taken into account for modelling deagrationand detonation in granular materials (Baer & Nunziato 1986). In the two-phase uiddynamics models, a frozen shock{particle interaction is often assumed in which thesolid particles are not accelerated during crossing of the shock front. Behind theshock front, a drag force is assumed to determine the momentum transfer betweenthe uid phase and the particles. For detonation in two-phase mixtures of gas and

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706 F. Zhang and others

solid particles, the shock interaction time in which a particle crosses the shock frontis several orders of magnitude smaller than the velocity relaxation time related to thedrag. Thus, the particle crosses the shock front with negligible changes in its velocityand the assumption of frozen shock interaction has been proven adequate. Momentumloss behind the shock front plays a role in the detonation velocity decit for gas{

particle systems with considerable particle combustion beyond the sonic chokingpoint (Lee & Sichel 1986; Gelfand et al. 1991; Zhang et al. 1992; Zhang & Lee1994; Veyssiere & Ingignoli 2002). However, for detonation in condensed mattercontaining metal particles, the shock interaction time can be comparable with thevelocity relaxation time and therefore the assumption of frozen shock interactionmay fail. While the deformation of the metal particles and the multi-reected shockon their surface may cause hot spots in the explosive, the explosive loses momentumduring the shock{particle interaction if the reaction time of metal particles is largerthan the shock interaction time. Momentum transfer during the shock interactiontogether with that behind the shock front can both inuence the shock initiationand detonation structure. Detonation velocity decit was observed experimentally incondensed explosives with 0.1 mm aluminium particles (Baudin et al. 1998; Gogulyaet al. 1998; Haskins et al. 2002; Zhang et al. 2002).

A spherical particle crossing a planar shock wave is a simple but good example toelucidate the suitability of the frozen shock interaction assumption. Considering aspherical particle of diameter d and density s moving in a continuum uid of density and neglecting any impressed force such as gravity, Newtons law of motion becomes

dus

dt=

u us

v; (1.1)

in which the velocity relaxation time of motion is

v =4d2s

3CdRe: (1.2)

Here, us stands for the particle velocity, u for the uid velocity and for the dynamicviscosity. The viscous drag coecient, Cd, dened in terms of the dynamic head ofthe relative ow and the frontal area of the particle, is a function of the Reynolds

number and the Mach number in a compressible uid based on Reynoldss principleof similarity (Schlichting 1979). On the other hand, the shock interaction time, ,can be dened as the time in which a particle crosses the shock front, which isapproximately

= d=D0; (1.3)

where D0 is the shock wave velocity. Thus, the ratio of the shock interaction timeover the velocity relaxation time is

v=

3CdRe

4dsD0: (1.4)

Substitution of typical numbers for a gaseous detonation into equation (1.4) showsthat, even for particles of 0.1{1 mm, the shock interaction time is about three to fourorders of magnitude smaller than the relaxation time. Thus, the particle crosses theshock front with negligible changes in its velocity and the assumption of frozen shock

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Shock interaction with solid particles in condensed matter 707

interaction is reasonable for detonation in two-phase mixtures of gas and solid par-ticles. However, for detonation in condensed matter such as in liquid explosive withaluminium particles of 0.1{1 mm, the shock interaction time is about the same orderof magnitude as, or one order smaller than, the relaxation time. Therefore, the parti-cle could be accelerated to a considerable velocity during passage of the shock front,

which makes the assumption of a frozen shock{particle interaction questionable.The objectives of the present paper are to calculate the momentum transfer duringthe shock interaction with compressible metal particles in host condensed matterand to understand the mechanism for the momentum transfer at the shock front ofa detonation wave. Numerical simulations were performed and analysed both for asingle particle and for multiple particles that cross a planar shock front. The studywas carried out for various material combinations of particles and host matter in thecondensed phase, and for shock strengths under conditions of detonation pressureswhich exceed the yield strength of solid particles. Quasi-one-dimensional analysiswas also conducted to obtain more insight into the mechanism of momentum transferwhen the particle crosses the shock front.

2. Shock interaction with a single particle

Classic detonation theories for homogeneous condensed explosives have been basedon the ZND model in which detonation occurs through a frozen shock transitionfollowed by an induction period, where the shocked temperature results in vibra-tional, rotational and electronic excitation followed by dissociation of the molecules(Zeldovich & Kompaneets 1960). During the frozen shock transition, explosives areassumed to be chemically unchanged. Various hot-spot mechanisms have been devel-oped to interpret the nature of the detonation initiation in solid heterogeneous explo-sives (Campbell et al. 1961; Dremin et al. 1970; Lee & Tarver 1980). The frozenshock transition, however, has remained a fundamental assumption following theZND model. Since the late 1970s, it has been speculated that detonation initiationcould start within the shock front due to the non-equilibrium kinetics (Dremin &Klimenko 1981; Walker 1988; Dremin 1992). Experimental proof of this postulatemust be inferred through observation inside the shock front and, therefore, remainsa dicult diagnostics challenge. In the present paper, only mechanical interactions

between the condensed explosive and the particles during passage of the shock frontwere addressed. Hence, it was assumed that decomposition of the condensed explo-sive would not start within the shock front. It was also assumed that the reactiontime-scale of solid particles is larger than the shock interaction time so that theparticles can be considered chemically frozen within the shock front.

Because of the assumption of the chemically frozen shock transition, the explosivematerial was described using the Murnaghan equation of state

p =K0

n

0n

1 + p0; (2.1)where p, , K0 and n are the pressure, density, bulk modulus and the pressurederivative of the bulk modulus, respectively; the subscript `0 denotes the initial state.Note that temperature does not appear as a variable in the Murnaghan equationsince it is mainly concerned with mechanical interactions. The response of the solid




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particle material during the shock interaction was described using the HOM equationof state (Mader 1998):

ps = (s=vs)(es eH) + pH (2.2)

with

pH

=c2s0(vs0 vs)

[vs0 ss0(vs0 vs)]2; e

H= 1

2pH

(vs0 v

s): (2.3)

The HOM equation of state is based on the shock Hugoniot and the Mie Gruneisenequation of state. In equations (2.2) and (2.3), e is the specic energy, v is the specicvolume, s is the Gruneisen coecient, and cs0 and ss0 are the coecient to the lineart of the shock and particle speed: Ds = cs0 +ss0us . The subscripts `s and ` H denotethe solid particle material and the Hugoniot state, respectively.

The Euler equations for mass and momentum were used to model the ow ofthe non-reacted explosive, and the governing equations for inviscid plastic ow wereemployed for modelling the dynamic response in metal particles. The interaction

between the explosive and the particles was solved by matching their boundary con-ditions of pressure and particle velocity. Numerical solution of these closure equationswere obtained using the IFSAS code (Thibault & Moen 1997), in which the Eulerequations were solved using a Lax{Wendro/FCT algorithm with sixth-order phaseerror correction and the plastic dynamic equations were solved using a nite-elementscheme (Boris 1976; Book & Fry 1984; Hallquist 1983). Deviatoric deformation of theparticle material that does not produce any volume change was also incorporated.The resolution for the calculations corresponded to 20 cells or elements for a particleradius, and a cylindrically axi-symmetrical mesh was used to represent a sphericalparticle immersed in the uid. As for initial and boundary conditions, a planar shockwave was applied as an in-ow boundary condition from the upstream of the parti-cle, whereas the downstream and lateral boundaries were treated as a free boundaryextrapolated with an expanding cell technique.

For a particle crossing a planar shock front, the inviscid governing equationsand the rate-independent material response models do not introduce any additionallength- or time-scales. Therefore, there exists a simple geometric scaling rule: at anygiven time, td2 , the surrounding ow eld and the dynamic response of a particlewith diameter d2 can be geometrically scaled to the same state as a particle withdiameter d1 at a time td1 :

td2=td1 = d2=d1: (2.4)Consequently, calculations need only to be conducted for one particle diameter, andthe results can be scaled to any other particle diameters. In the following calculations,the particle diameter was arbitrarily chosen as 10 mm. According to the geometryscaling equation (2.4), for example, the time for a 0.1 mm particle is scaled downto 1/100th of that for the 10 mm particle. Note that the simple geometry scalingwould no longer be valid for viscous ows since the additional length-scale associatedwith viscosity introduces the Reynolds number as an additional scaling parameter.Similarly, kinetic phenomena such as strain-rate eects or chemical reactions would

also introduce additional scaling parameters.

(a) Eects of a particles material density

Various metal particles were subjected to a 101.3 kbar shock in a liquid with thesame initial density and compressibility as water (0 = 1 g cm

3, c0 = 1477 m s1,




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3 ns 4 ns2 ns1 ns

Figure 1. Pressure contours for a magnesium particle subjected to a 101.3 kbar liquid shock.

3 ns 8 ns2 ns1 ns

Figure 2. Pressure contours for a tungsten particle subjected to a 101.3 kbar liquid shock.

K0 = 21:78 kbar and n = 7:15). The shocked ow properties of the liquid aredensity 1 = 1:639 g cm3, ow velocity u1 = 1987 m s

1 and shock velocityD0 = 5097 m s

1, where the values of ow velocity and shock velocity are relativeto a xed observer.

The particle motion, deformation and pressure distribution in the particle and thesurrounding uid are displayed as pressure contour plots in gures 1 and 2 for magne-sium and tungsten. Times of the plots refer to the beginning of the calculation whenthe incident shock is 2 mm before the leading edge of the particle. It can be seen thatthe shock interaction with the particle results in a reected shock into the liquid anda shock transmitted into the particle. The particle is severely deformed and distorteddue to the complex lateral expansion and the fact that the shock pressure exceedsthe yield stress of the material. The distortion is directly related to the relative veloc-ity between the leading and trailing edges of the particles, which are displayed ingure 3 for magnesium. Local velocities inside the particle are rst produced duringthe propagation of the transmitted shock. The velocity at the leading edge of the




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0 2 4 6 8

0

500

1000

1500

2000

-500

time (ns)

velocity(ms

-1)

Figure 3. Leading- and trailing-edge velocity histories for a magnesium particlesubjected to a 101.3 kbar liquid shock.

0 2 4 6 8

0

500

1000

1500

2000

time (ns)

us

(ms

-1)

10

W

Ni

Al

Mg

Figure 4. Particle mass-centre velocity histories in magnesium, aluminium,nickel and tungsten subjected to a 101.3 kbar liquid shock.

particle is very close to the analytical value of the one-dimensional wave transmis-sion, 1474 m s1 for light magnesium and 407 m s1 for heavy tungsten. The localvelocities decay behind the transmitted shock and along the propagation distancedue to the unloading eect of lateral expansion. Furthermore, the rarefaction that isreected o the trailing edge of the particle competes against lateral expansion andcauses the secondary acceleration.

Due to the non-uniform local velocities inside the particle, the particle mass-centrevelocity is calculated based on a mass average over the particle elements. Figure 4provides the histories of particle mass-centre velocity for four metals: magnesium,aluminium, nickel and tungsten with increasing initial density. The particles velocityhistories display two phases: a shock{particle interaction phase which occurs over atime = 1:96 ns from equation (1.3), followed by a drag phase. It can be seen fromthis gure that the particle acceleration decreases with the particle material den-




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Table 1. Velocity transmission factors

s0 cs0 p1material (g cm3) (m s1) (kbar)

liquid 0 = 1:0 g cm3

magnesium 1.770 4700 50.65 0.776101.3 0.790

202.6 0.803

beryllium 1.870 7975 101.3 0.781

aluminium 2.785 5350 101.3 0.600

nickel 8.860 4646 101.3 0.227

uranium 18.98 2540 101.3 0.108

tungsten 19.30 4060 101.3 0.108

202.6 0.114

RDX 0 = 1:4 g cm3

magnesium 1.770 4700 101.3 0.940aluminium 2.785 5350 101.3 0.754

RDX 0 = 1:8 g cm3

magnesium 1.770 4700 101.3 1.008

aluminium 2.785 5350 101.3 0.802

sity. While the heavy particles such as tungsten accelerate mildly, the light particlesaccelerate rapidly during the shock{particle interaction. For example, the magne-sium particle accelerates to a velocity of 1600 m s1 that corresponds to ca.80% ofthe surrounding shocked uid velocity of 1987 m s1.

A velocity transmission factor can be dened as the ratio of the particle mass-centre velocity us after the shock interaction time over the shocked uid velocityu1:

= us=u1: (2.5)

The velocity transmission factor varies between 0 for perfect reection o a rigid bodyto 1 for a perfect transmission into a particle with the same material properties as theunreacted uid. The computed velocity transmission factor is displayed in table 1

for the four metal particles.Similar to the velocity distribution, the pressure distribution inside the particleis initially highly non-uniform, as exhibited in gure 5, which displays pressure his-tories for elements at the leading edge, centre and trailing edge of the magnesiumand tungsten particles. The shock pressure at the leading edge of the particle is veryclose to the values from the one-dimensional wave transmission theory, 166.8 kbarfor magnesium and 358.3 kbar for tungsten. However, the shock pressure decays inthe particle due to the lateral expansion; the rate of decay is much faster in tung-sten than in magnesium due to the higher acoustic impedance of tungsten. Therelative pressure between the leading and tailing edge for both magnesium and tung-sten approaches zero shortly after the shock interaction. Thus, the particle pressure,computed on a mass average over the particle elements, rapidly reaches and oscil-lates around the initial liquid shock pressure of 101.3 kbar. Finally, given densityand pressure, we can also compute the particle temperature using the HOM equa-tions of state for temperature based on the Walsh and Christian method (Mader




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0 2 4 6 8

0

100

200

300

400

-100

time (ns)

staticpressure(103atm)

0 2 4 6 8

10

0

50

100

150

200

-50staticpress

ure(103atm)

(a)

(b)

Figure 5. Pressure history for leading edge, centre and trailing edge of a magnesium (a) anda tungsten (b) particle subjected to a 101.3 kbar liquid shock.

1998; Walsh & Christian 1955). Consequently, the temperature histories computedalso show a rapid equilibration to the Hugoniot temperature of the metal particle,i.e. 450 K, 371 K, 329 K and 314 K for magnesium, aluminium, nickel and tungsten,respectively.

Strictly speaking, the late-time calculations in the drag phase after the shock inter-action is not correct without using governing equations for viscous uid. However,some qualitative phenomena observed in the calculations for the shock interactionand shortly thereafter may bring attention to an appropriate description of the drag

under conditions of detonation pressures. Due to the severe particle deformation,particle acceleration in the drag phase can be relatively complex. For instance, theconsiderable increase in cross-sectional area associated with the above-mentionedlateral expansion and distortion can play a role in the drag force exerted on the par-ticle. The sharp edge produced at the trailing edge can also result in ow separationand a further increase in the drag coecient. Furthermore, the large deformationcan also lead to shedding of small particles at the sharp edge. Hence, the drag lawcannot be expressed simply in terms of the local relative ow properties, but mustalso account for the history-dependent particle shape.

(b) Eects of particle acoustic impedance

The dependence of the velocity transmission factor on a particles acousticimpedance, s0cs0 , was explored by performing calculations using materials withsimilar initial densities but with dierent sound speeds. Beryllium was compared




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4 ns2 ns1 ns

Figure 6. Pressure contours for a beryllium particle subjected to a 101.3 kbar liquid shock.

8 ns2 ns1 ns

Figure 7. Pressure contours for a uranium particle subjected to a 101.3 kbar liquid shock.

with magnesium for the low-density regime while uranium and tungsten were com-pared for the high-density regime. Figures 6 and 7 show the pressure contour plotsfor beryllium and uranium, respectively. It can be seen from a comparison of g-ure 6 with gure 1 that the transmitted shock propagates faster in beryllium thanin the softer magnesium, which also displays a slightly larger deformation. Similarly,comparison between gures 7 and 2 indicates that the transmitted shock propagatesfaster in tungsten than in the softer uranium. In spite of these dierences, the par-ticles mass-centre velocity imparted after the shock interaction remains almost thesame as shown in gure 8, thus resulting in a similar velocity transmission factor asdisplayed in table 1.

One noticeable dierence is the higher distortion produced for the lower-impedancematerials. In particular, much higher distortion is observed in the softer uranium thantungsten after the shock interaction process. The higher distortion causes a largerdrag force that causes the particle velocity for uranium to diverge from that fortungsten, as seen in gure 8.




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0 2 4 6 8

0

200

400

600

time (ns)

us

(ms

-1)

10

W

U

Be

Mg

0 2 4 6 8

0

500

1000

1500

2000

us

(ms

-1)

(a)

(b)

Figure 8. Particle mass-centre velocity histories for magnesium and beryllium (a) andfor tungsten and uranium (b).

(c) Eects of shock strength

To determine the dependence of the velocity transmission factor on the incidentshock strength, additional calculations were performed with incident shock pressuresof 50.7 kbar and 202.6 kbar. The pressure contour plots for these calculations arepresented in gure 9 for tungsten. An increase in shock strength does result in a fasterparticle distortion rate which plays a role in calculating the drag force. However,within the shock interaction process, the increase in velocity transmission factor withshock strength is relatively small as seen in table 1. The velocity transmission factorsin table 1 are calculated based on the particles mass-centre velocity at the shockinteraction time , which is 2.5 ns, 1.96 ns and 1.48 ns for the 50.7 kbar, 101.3 kbarand 202.6 kbar shocks, respectively.




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3 ns 6 ns2 ns1 ns

Figure 9. Pressure contours for a 50.7 kbar (a) and 202.6 kbar (b) liquid shockinteracting with a tungsten particle.

(d) Eects of host matterThe studies presented above have dealt with particles in a liquid matter. To exam-

ine the inuence of dierent condensed matter on the velocity transmission factor, asolid explosive, RDX, is chosen to replace the liquid. RDX is commonly consideredas a heterogeneous explosive with a grain size distribution and cannot be rigorouslydescribed by a homogeneous uid model. Moreover, mixing of metal particles withRDX introduces the further complexity of heterogeneity. However, the bulk responseof unreacted RDX to shock waves is still worthwhile studying in respect of the inu-ence of shock Hugoniots and material densities of host matter on the shock{particle

interaction process. Thus, curve ts of the Murnaghan equation of state were con-ducted to RDX shock Hugoniots (Dremin et al. 1970; Marsh 1980) for initial densitiesof 1.44 g cm3 and 1.6{1.8 g cm3. This yields the following parameters of the equa-tion of state: K0 = 52:77 kbar and n = 4:006 for 1.44 g cm

3 and K0 = 109:8 kbarand n = 7:294 for 1.6{1.8 g cm3, respectively.




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0.5

50003000

D0

(m s-1)

9000700060004000 100008000

0.6

1.1

0.7

0.8

0.9

1.0

a

Figure 10. Velocity transmission factors for magnesium and aluminium particles subjected toRDX shocks. , Mg, RDX (1.8 g cm3); N, Al, RDX (1.4 g cm3); O, Al, RDX (1.8 g cm3).

50003000

D0 (m s

-1)

9000700060004000 100008000

ps

(kbar)

0

200

400

600

Figure 11. Mass-averaged particle pressures at the shock interaction time for magnesium andaluminium subjected to RDX shocks. , Mg, RDX (1.8 g cm3); N, Al, RDX (1.4 g cm3); O,Al, RDX (1.8 g cm3).

The velocity transmission factor was computed for magnesium and aluminiumparticles at various RDX shock strengths up to a shock velocity ofD0 = 8930 m s1,which corresponds to the Chapman{Jouguet detonation of RDX at an initial densityof 1.8 g cm3 with a shock interaction time of = 1:12 ns. It is expected that theparticle distortion rate is increased with shock pressure on the high-density uid.However, as displayed in gure 10, the variation of the velocity transmission fac-




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a

0p

0 /ps0

0.2

0.4

0.8

0.6

1.0

0.2 0.4 0.80.6 1.0

liquid

RDX1.4

RDX1.8

curve fit

1D theory

Figure 12. Velocity transmission factors for various particle materialssubjected to a 101.3 kbar shock in a liquid and RDX.

tor after the shock interaction is within 10% for a range of RDX shock velocityfrom 4000 m s1 to 9000 m s1. It is noticeable that the velocity transmission factorbecomes larger than 1 for a magnesium particle immersed in 1.8 g cm3 RDX. Thisis due to the transmitted expansion caused by the smaller density of magnesium

(1.77 g cm3

) than that for RDX. At the shock interaction time, the mass-averagedparticle pressure rapidly reaches the initial RDX shock pressures over the shock speedrange between 4000 m s1 and 9000 m s1 (see gure 11).

The velocity transmission factor computed for various metal particles immersedin dierent condensed matters at several shock pressures are summarized in table 1.Comparison of the results for shocks in liquid and RDX with various metal parti-cles (gure 12) reveals that the velocity transmission factor depends on the initialdensity ratio of host condensed matter over particle, 0=s0, rather than on the hostcondensed matter or the particle alone. Among the material properties and shockstrength investigated, is most dependent on this initial density ratio.

3. Shock interaction with multiple particles

(a) A linear cluster of particles

To understand the complex shock interaction processes for multiple particles, one-dimensional calculations were performed for a linear cluster of aluminium particlessubjected to a 101.3 kbar liquid shock that served as an in-ow boundary conditionfrom the upstream of the particles. The linear cluster was composed of twelve 10 mmthick aluminium slugs with a 10 mm gap lled with liquid between slugs.

Figure 13 displays an oscillatory structure for the mass-centre velocity history ofthe leading particle of the linear cluster. The rst sharp velocity rise correspondsto the shock transmitted into the metal and agrees with the analytical value of1110 m s1. This pulse is followed by a second pulse which is due to the rarefactionwave produced when the shock reaches the metal{liquid interface at the trailing edge




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0 2 4 6 8

time (ns)

10

0

500

1000

1500

2000

-500

velocity

(ms

-1)

12 14

Figure 13. Velocity history of the leading particle in a one-dimensional linear cluster of12 aluminium particles with a 10 mm gap, subjected to a 101.3 kbar liquid shock.

3 ns 4 ns2 ns1 ns 5 ns

Figure 14. Pressure contours for two cylindrical aluminium particles with a 5 mm gap subjectedto a 101.3 kbar liquid shock. There is a 2.5 mm gap between the particles and the side wall.

of the particle. A third weak velocity jump also occurs when the rarefaction wavereects o the metal{liquid interface at the leading edge of the particle. Finally, a

sharp velocity drop takes place when the particle is subjected to the wave reected othe second particle. The process is similar for successive particles with the exceptionthat the mean particle velocity slightly increases as the shock wave penetrates thecluster. When the gap between the particles is reduced, the rarefaction wave does nothave sucient time to propagate in the particle before the arrival of the wave thatis reected o the downstream particle. In this case, oscillations become asymmetricand the mean particle velocity is reduced and becomes closer to that behind thetransmitted shock.

(b) A matrix cluster of particlesThe complexity of the shock{particle interaction process is further increased by

additional waves produced by neighbouring particles in the lateral direction. Two-dimensional calculations were conducted for a matrix cluster of 10 mm cylindricalaluminium particles subjected to a 101.3 kbar liquid shock. Although the results of




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0 1 2 3 4

time (ns)

0

500

1000

1500

2000

-500

veloci

ty(ms

-1)

5

Figure 15. Velocity histories for the leading, centre and trailing edges of the leading particle oftwo cylindrical aluminium particles subjected to a 101.3 kbar liquid shock.

the cylindrical particle calculations cannot be directly applied to spherical particles,they do provide a qualitative trend for the eect of particle volume fraction onparticle velocity. Figure 14 displays pressure contours for two particles with a 5 mmgap in the direction of motion. The eect of neighbouring particles in the lateraldirection is simulated by introducing a wall at the plane of symmetry between theparticles and the lateral particles with a 5 mm gap, that is, a 2.5 mm gap betweenthe particles and the sidewall. It can be seen that, apart from the wave interactiondescribed in the calculations for a linear cluster of particles, there are also additional

shock waves reected o the lateral particles around the leading particle. Theselateral shocks compete against the lateral expansion and prevent the local velocitiesfrom decaying inside the particle, as seen in gure 15. Collision of these lateral shocksproduces a secondary pressure pulse to the leading particle, thus causing a gradualacceleration which is less signicant than the deceleration eect from the wave thatis reected o the downstream particle in the direction of motion.

Figure 16 displays a high-loading matrix of cylindrical aluminium particles com-pacted by a 101.3 kbar liquid shock into a tight cluster of particles. The particlesare largely deformed and coalesce due to the time lag between the acceleration ofthe leading and trailing particles. As shown in gure 17, due to the inuence of theneighbouring particles (particularly the downstream particles), the time in whichthe initially non-uniform velocity inside the particle reaches and oscillates around anaverage value behind the shock front is almost twice as much as the time required forthe single particle case (see gure 3). It should be noted that geometric arrangementof the particles for a given particle volume fraction may result in dierent inuences.However, resonant eects that can be created in ordered systems may not appearin more realistic stochastic distributional ensemble systems (Baer 2002). Thus, thetime required for achieving a relatively uniform velocity within a particle in a realisticsystem would be expected to be not much more than that for an ordered system.

Considering the eects on the shock interaction from the neighbouring particles,the particle mass-centre velocity must be computed at a time longer than the shockinteraction time for a single particle. To include the eect of the downstreamparticle, the particle mass-centre velocity for multiple particles, computed at a timeequal to twice the shock interaction time , is summarized in gure 18. The results




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3 ns 7 ns 11 ns

Figure 16. Pressure contours for a matrix of cylindrical aluminium particlessubjected to a 101.3 kbar liquid shock.

0 2 4 6 8

time (ns)

10

0

500

1000

1500

2000

-500

velocity(ms-

1)

12 14

Figure 17. Velocity histories for the leading, centre and trailing edges of the central leading

particle in a matrix of cylindrical aluminium particles subjected to a 101.3 kbar liquid shock.

indicate that the transmitted particle velocity decreases with an increase in particlevolume fraction .

4. Simple models

(a) Single particles

Under the assumption that the condensed explosive and the solid particles are chem-ically frozen during the particle crossing the shock front, the numerical studies fora single particle suggest that the velocity transmission factor , dened in equa-tion (2.5), strongly depends on the initial density ratio of host condensed matterto particle. The inuence on from other parameters, such as particle acousticimpedance, shock strength and unreacted explosives shock Hugoniots, is relatively




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a

0

0.6

0.7

0.9

0.8

1.0

0.2 0.4 0.80.6 1.0

linear model

2 cylinders

11 cylinders

f

0.5

m

Figure 18. Particle mass-centre velocities for various matrices of aluminium particles.

host matter (H) particle (S)

1

0 s0

2

s3

s4

Figure 19. Wave diagram of the one-dimensional model.

weak or could be implicitly included in the eect of the initial density ratio. A curvet of the numerical values of suggests the following correlation,

=0s0

a + b

0s0

1

a + b; (4.1)

where a = 3:947 and b = 1:951 (see gure 12).To get more insight into the mechanism for shock interaction, a one-dimensional

analytical model based on the wave dynamics is considered. For simplicity, it isassumed that the impedance matching of the host condensed matter and the particlematerial results in a reected shock in the host matter and a transmitted shock inthe particle as shown in gure 19. For the solid particle, the mass and momentum

conservation equation across the shock, along with the linear c{s Hugoniot relation,Ds0 us0 = cs0 + ss0(us us0); (4.2)

yield a relation between transmitted pressure ps3 and ow velocity us3,

ps3 ps0 = s0(us3 us0)[cs0 + ss0(us3 us0)]: (4.3)




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In equations (4.2) and (4.3), the plus sign is for a right-crossing shock and the minussign for a left-crossing shock. The variables Ds, us , ps , and s are the shock velocity,ow velocity, pressure and density for the particle. The subscript `0 represents theparticle initial state and `3 is for the transmitted shock state.

For the host condensed matter, the mass, momentum and energy conservation

equations crossing the reected shock are used, and they result inu2 = u1

p(p2 p1)(1=1 1=2); (4.4)

e2 = e1 +12

(p2 + p1)(1=1 1=2); (4.5)

where the plus sign is for a right-crossing shock and the minus sign for a left-crossingshock. The variables u, p, and e denote the ow velocity, pressure, density and thespecic internal energy for the host matter. The subscript `2 denotes the reectedshock state and `1 is for the incident shock state. The Murnaghan equation of state(2.1) must also be referred to the incident shock state, i.e.

p =K1n1

1

n1

1

+ p1 (4.6)

with

K1 = 1c21 = 1c

20(1=0)

(n01); (4.7)

n1 = 1 + 2ln(c1=c0)

ln(1=0): (4.8)

From (4.6), the specic internal energy can be obtained:

e =p (n1p1 1c

21)

(n1 1): (4.9)

Applying equation (4.9) to (4.5) results in a Hugoniot = (p), which is substitutedinto (4.4) to yield a relation u2 = u(p2). Solving the host matter relation u2 = u(p2)and that for the particle (4.3) along with the interfacial conditions

ps3 = p2; us3 = u2 (4.10)

results in the transmitted velocity

s =us3u1

= 1 ps3 p1q

1u21[12

(n1 + 1)ps3 +12

(n1 1)p1 (n1p1 1c21)]; (4.11)

in whichps3 = ps0 + s0(us3 us0)(cs0 + ss0(us3 us0)): (4.12)

As the shock in the surrounding condensed matter with a velocity D0 reaches theparticle trailing edge, the transmitted shock with a velocity Ds0 still propagates inthe particle if Ds0 < D0, or it is already reected o the particle trailing edge andthe reected rarefaction runs back into the particle ifDs0 > D0. Taking into accountthe shock velocity dierence, a momentum balance rule is assumed at the shockinteraction time, i.e.

susd = s3us3d3; if Ds0 6 D0; (4.13)

susd = s3us3(d d4) + s4us4d4; if Ds0 > D0; (4.14)




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D0

D0

D0

D0

Ds0

Ds0

D

0

d4

d

us4u

s3

-

Figure 20. One-dimensional model.

where the geometric quantities are illustrated in gure 20. The subscript `s denotesthe mass-averaged state at the shock interaction time , the subscripts `3 and `4represent the transmitted shock and the reected rarefaction state, respectively. Thedistances d3 and d4 are dened by the shock interaction time

=d

D0=

d3Ds0

=d

Ds0+

d4cs3 us3

: (4.15)

The ow velocity us4 behind the rarefaction wave can be analytically obtained. How-

ever, for simplicity, us4 2us3, s s3 s4 and also cs3 cs0 are assumed.Substitution of equation (4.15) into equations (4.13) and (4.14) yields

=Ds0D0

s ; for Ds0 6 D0; (4.16)

=

1 +

cs0 us3Ds0

Ds0D0

1

s ; for Ds0 > D0; (4.17)

where is dened in equation (2.5) and s is given in equation (4.11).The theoretical velocity transmission factor obtained from equations (4.16) and

(4.17) lies between state `s3 behind the transmitted shock and state `s4 behind therarefaction. In gure 12, comparison of the theoretical results from equations (4.16)and (4.17) with the numerical calculations subjected to a 101.3 kbar shock shows afairly good agreement except that the theoretical values are generally larger thanthe numerical ones, since the one-dimensional theory does not consider the loss to




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0

9

5

6

7

8

Al mass % in RDX/Al

detonationvelocity(kms

-1)

10 20 30 40 50

RDX

RDX/inert Al

RDX/react. Al

Figure 21. Equilibrium Chapman{Jouguet detonation velocities forRDX with aluminium additive.

be caused by lateral deformation and expansion. Note that under conditions of verystrong shocks, the shocked host matter density can become larger than the particlematerial density. In this case, the one-dimensional theory does not take the lateralcompression into account and can therefore underpredict the numerical results.

(b) Multiple particles

For shock interaction with multiple particles immersed in a condensed matter,the particle mass-centre velocity calculated varies with the selection of the time forthe shock interaction over multiple particles. It may also vary with the geometricarrangement of the particles for a given particle volume fraction. For complex casesinvolved in multiple particles, it may simply assume a linear model for the velocitytransmission factor between the value for a single particle (i.e. the particle volumefraction = 0) and the value for the transmitted shock in the solid s (i.e. = 1),i.e.

m = (1 ) + s : (4.18)

In equation (4.18), can be obtained from equation (4.1) or (4.16) and (4.17), ands can be obtained from equation (4.11). The results predicted using equation (4.18)for multiple aluminium particles are displayed in gure 18 and their variation trendin terms of particle volume fraction is in good agreement with the numerical calcu-lations.

5. Final remarksFor a charge of condensed explosive with metal particles, the present study indicatesthat the momentum transfer from condensed explosive to heavy-metal particles suchas tungsten and uranium is insignicant during the particle crossing of the shockfront. However, the momentum transferred to light-metal particles is signicant, and




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the particle velocity for aluminium, beryllium and magnesium achieves 60{100% ofthe value of the shocked explosives velocity. The particle velocity after the shockinteraction strongly depends on the initial density ratio of explosive to metal. Theinuence of other parameters, such as particle acoustic impedance, shock strengthand bulk unreacted explosives shock Hugoniot, can be implicitly included in the

eect of the initial density ratio. The transmitted particle velocity decreases withincrease in the particle volume fraction. A theoretical model is also established togain the insights into the main shock interactions that dominate the momentumtransfer.

The signicant momentum transfer during a particle crossing the shock front mustbe taken into account when modelling the shock initiation and detonation structurefor two-phase mixtures of condensed explosive and light-metal particles. Its impor-tance can be illustrated with the equilibrium Chapman{Jouguet detonation for RDXwith molecular aluminium with a mixture density of 1.66 g cm3 using the Cheetah

code (Friedet al

. 1998), and the results are shown in gure 21. Detonation veloc-ity decit is generally predicted to increase with the mass of the particle additive,regardless of the reactive or chemically frozen nature of the particles. For chemicallyfrozen particles, the considerable velocity decit is originated in the momentum andheat transferred to the particles up to their equilibrated state. For reactive particles,apart from a negative oxygen balance and changes in the explosives detonation prod-ucts, the inuence of the momentum and heat transferred to the rest of the particlesand the particle combustion products still remains. Unlike the metallic molecules,metallic particles may not be fully equilibrated with the detonation products of theexplosive within the detonation wave. However, for light small particles, both the sig-

nicant momentum transfer during the shock{particle interaction and that occurringbehind the shock front play an important role in the detonation velocity decit.

This work was partly supported under the auspices of the DND contract W7708-8-R748. Theauthors thank Dr Alexander Gonor for his valuable suggestions and comments.

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http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/0040-6031^28^29384L.351[aid=4127344]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/0040-6031^28^29384L.351[aid=4127344]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/0040-6031^28^29384L.351[aid=4127344]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/0040-6031^28^29384L.351[aid=4127344]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/1364-503X^28^29339L.355[aid=4127347]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/1364-503X^28^29339L.355[aid=4127347]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/1364-503X^28^29339L.355[aid=4127347]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/1364-503X^28^29339L.355[aid=4127347]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/1364-503X^28^29339L.355[aid=4127347]http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/0040-6031^28^29384L.351[aid=4127344]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/1364-503X^28^29339L.355[aid=4127347]http://pippo.ingentaselect.com/nw=1/rpsv/cgi-bin/linker?ext=a&reqidx=/0040-6031^28^29384L.351[aid=4127344]


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