International Journal of Impact Engineering 25 (2001) 187}191

E!ect of a spherical explosion upon the #ight path and spatialorientation of a projectile

G. Iosilevskii!,*, N. Farber"

!Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel"Wales Inc., Ramat Gan 52521, Israel

Received 17 December 1998; received in revised form 20 April 2000

Abstract

A possible defense of an armored vehicle against a high-kinetic-energy projectile is based on generatinga strong explosion in a close proximity of the projectile and at some distance from the vehicle. In thisexposition we suggest a simpli"ed analytical model that bounds the e!ect of an explosion upon the #ight pathand spatial orientation of the projectile. ( 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

It is well known that penetration depth of an armor by a high-kinetic-energy projectile is highlysensitive to the impact angle [1]. This observation suggests that a possible defense of a vehicleagainst such a projectile can, probably, be based on changing the spatial orientation of the latter soas to increase its impact angle. To this end, few kilograms of an explosive can, in principle, bedetonated in the close proximity of the projectile and at the distance of few tens of meters from thevehicle. A simpli"ed analytical analysis of the e!ect of such an explosion upon the impact angle ofthe projectile is the subject matter of this short exposition.

2. Strong explosion

To begin with, consider a spherical explosion of large magnitude* large enough to justify theassumption that the expanding spherical shock is in"nitely strong. For the sake of simplicity, the

*Corresponding author.

0734-743X/01/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 7 3 4 - 7 4 3 X ( 0 0 ) 0 0 0 3 2 - 4

gas, in which the explosion occurs, will be assumed perfect, with gas constants R and c remainingconstant through the shock. Under these assumptions, the density o

2, the pressure p

2, the

temperature ¹2

and the gas velocity v2

immediately behind the expanding shock, are completelydetermined by the density o

1of the quiescent gas before the shock and by the velocity v

4of the

shock front by the well-known relations [2]

o2"o

1

c#1c!1

, P2"o

1v2S

2c#1

, ¹2"

v2S

R2(c!1)(c#1)2

, v2"v

S

2c#1

. (1)

For typical values of c, the former of these relations implies that the density of the gasimmediately behind the shock is large as compared with the density of the gas in which the shockexpands. Therefore, we shall assume, subject to a posteriori veri"cation, that the gas behind theshock (which contains both the gases of the explosion and the air captured by the shock) isconcentrated in a thin layer adjacent to the shock front, with vacuum prevailing at the center of theexplosion.

Let us further assume that the gas properties (as velocity, temperature and density) are constantthrough this thin layer. As an immediate consequence, the kinetic energy E

,*/, total energy E, and

linear momentum I of the gas take the form

E"(m0#m

!)(c

7¹

2#1

2v22)"

4(m0#m

!)v2

S(c#1)2

, (2)

E,*/

"12(m

0#m

!)v2

2"

2(m0#m

!)v2

S(c#1)2

"12E, (3)

I"(m0#m

!)v

2"

2(m0#m

!)v

Sc#1

, (4)

by (1). Here, cv"R/(c!1) is an appropriate speci"c heat, m

0is the mass of the explosive (it is

tacitly assumed that the mass of the explosive equals the mass of the gases released by it), and m!is

the mass of the air captured by the explosion. With rS

being the instantaneous radius of the shockfront, m

!"4

3po1r3S

.Eqs. (1)}(4) may be somewhat simpli"ed by introducing

e,*/

"E,*/

/m0, (5)

the speci"c kinetic energy of an explosive, and

c"A3m

04po1B

1@3, (6)

the equivalent radius of an explosive mass. With these,

v2S"

e,*/

(c#1)22

c3r3S#c3

, (7)

by (2) and (3), and concurrently,

p2"o

1e,*/

(c#1)c3

r3S#c3

, ¹2"

e,*/

(c!1)R

c3r3S#c3

, v22"2e

,*/

c3r3S#c3

, (8)

188 G. Iosilevskii, N. Farber / International Journal of Impact Engineering 25 (2001) 187}191

Fig. 1. Estimated relative thickness d/rS

of the gas layer behind the shock.

I"43po1c3A2e

,*/

r3S#c3c3 B

1@2, (9)

by (1) and (4).Consistent with the above, the thickness d of the gas layer behind the shock can be estimated by

d"m

0#m

!4po

2r2S

"rS

c!13(c#1)

r3S#c3r3S

. (10)

It turns out to be of the order of 0.1rS

for c"1.4 and all practical combinations of rS

and m0

(seeFig. 1) * small enough to render valid the assumption stated in the paragraph immediatelyfollowing Eq. (1).

3. Interaction between the explosion and a projectile

Let the speci"c kinetic energy of an explosive be a typical 2 MJ/kg [3] and the gas in which theexplosion occurs have all the properties of air. Under these circumstances, Eq. (8) implies than anexplosion of 2 kg should yield v

2&1000m/s at r

S"1 m (see Fig. 2), i.e. v

2should be comparable

with the velocity < of the projectile itself. Accordingly, the interaction between the gases of theexplosion and the projectile can, probably, be analyzed in the framework of a cross-#ow analysis.Moreover, the typical interaction time

q&dv2

&

c!13(c#1)A

(r3S#c3)3

2e,*/

c3r4SB

1@2(11)

(see Eqs. (8) and (10)), turns out to be extremely short * of the order of 100ls for all practicalcombinations of r

4and m

0. All these observations imply that the maximum possible e!ect of an

explosion on a projectile (other than damaging it) can be estimated by considering an ideal

G. Iosilevskii, N. Farber / International Journal of Impact Engineering 25 (2001) 187}191 189

Fig. 2. Gas velocity v2

behind the shock.

interaction in which all linear momentum of the gas impinging the projectile is instantaneouslytransferred to the projectile's geometrical center.

Let, therefore, i be the linear momentum of the gas per unit area of the shock front; noting (9), ittakes the form

i"I

4pr2S

"o1

c3

3r2SA2e

,*/

r3S#c3c3 B

1@2. (12)

Let, also, S be the area of the side projection of the projectile, and h the distance between theprojectile's center of gravity and its geometrical center. Thus, the maximum possible values of thelinear and angular moments received by the projectile upon the interaction are, respectively,

I1&iS and H

1&iSh. (13)

4. E4ect of the explosion at the impact

Now, let the distance r between the explosion and the defending vehicle be small as comparedwith the shortest wave length of the projectile's rigid-body modes (which is known to be of theorder of a hundred meters). In this case, if no damage has been in#icted upon the projectile by theexplosion, the above estimates can be readily translated into the transverse *x and angular *adeviations of the projectile from its original impact point and impact angle, respectively. Thus, withm and J being the mass and the pertinent moment of inertia of the projectile,

*x&I1r

m<&r

o1Sc3

3mr2SA2e

,*/V2

r3S#c3c3 B

1@2, (14)

*a&H

1r

J<&

mh*xJ

, (15)

by (12) and (13).

190 G. Iosilevskii, N. Farber / International Journal of Impact Engineering 25 (2001) 187}191

Fig. 3. Maximal transverse *x and angular *a deviations of a typical projectile at the distance of 15m from theexplosion. Here, the shock radius is equivalent with the lateral seperation between the center of the explosion and theprojectile.

Given typical values m"4 kg, S"0.0175m2, h"7.2 cm, J"0.11 kgm2 and <"1500 m/s, thepertinent values of *x and *a are shown in Fig. 3 at r"15m. The e!ect of the explosion on the#ight path seems to be insigni"cant, but the e!ect on the impact angle seems to be signi"cantenough [1] to warrant a detailed investigation.

References

[1] Bless SJ, Barber JP, Bertke RS, Swift HS. Penetration mechanics of yawed rods. Int. J. Eng. Sci. 1978;16:829}33.[2] Landau LD, Lifshitz EM. Fluid mechanics. Oxford: Pergamon Press, 1987. p. 404.[3] Federo! B, She$eld OE, Kaye SM, editors. Encyclopedia of explosives and related items. PATR 2400 Vol. 4.

Picatinny Arsenal, Dover, NJ, 1969. p. D463.

G. Iosilevskii, N. Farber / International Journal of Impact Engineering 25 (2001) 187}191 191