Lecture 11Penetration Mechanics II

Figure 6. Partom diagram for two layer target.

3. Layered TargetsFor relatively long projectiles, it is often very helpful to analyze impacts

in terms of a P-L diagram (Partom, [7]). In the Partom diagram, oneconsiders the penetration versus the remaining length of the projectile. Inthe example of Figure 3.5, the first target of thickness T1 is harder thanthe second block. Lo is the initial projectile length. If penetration ratewere uniform, the path would be along the dotted line.Important terms used in this field include: Penetration implies movement ofa projectile into a target. Perforation implies penetration all the waythrough a target. Limit velocity (also known as ballistic limit and V50) is thevelocity at which a projectile just perforates a target. Areal density isweight per unit presented area, and is the main measure of performance ofarmor materials. Mass efficiency is the ratio of the areal density of steel

to the areal density of a test armor for the same level of protection (so, forexample, an efficiency of 2 implies half the weight of steel for the sameprotection). RHA stands for "rolled homogeneous armor" which is the USstandard for steel armor; it is described by milspec 12560.

The transient effects in thick targets are associated with crater growthduring the impact stage. For relatively short projectiles, L

D<~

2 , this stage

dominates. The crater is excavated by the shock wave. Figure 7 showsexamples of impact craters. Usually the target flow in the crater iscompressive shear. It is approximately true that the kinetic energy of theprojectile equals the work done in forming the crater. If the target resistsdeformation with a stress equal to Y, then the work making a crater is simplyY times the crater volume.

Figure 8. Impact crater made by equal mass projectiles, aluminum strikingaluminum. (Gehring, [8])

By setting this equal to the projectile kinetic energy and replacing Y withan “effective strength” St, one derives the Charters Summers equation:

†

P =814p

rP

rt

ESt

Ê

Ë Á Á

ˆ

¯ ˜ ˜

1/ 3

, (12)

where E is projectile kinetic energy.The units of St are stress, or equivalently, energy per unit volume.

Notice that the penetration depth for cratering increases with impactvelocity as V2/3. In practice, the value of St is very close to orindistinguishable from the Tate target parameter R. Table 1 gives somemeasured values of cratering efficiency, or target resistance.

Table 1 Ranges of target resistance (GPa)Rt or St

Armor Steel 4.5 – 5.5Armor Titanium 3.5 – 5.0

Structural Aluminum 2.0 – 2.5Beryllium 4.0 – 4.5Alumina 6.0 – 7.0

Subramanian et al. [9] recently showed that a more precise conservationstatement is:

crater volume x target strength = impact kinetic energy – projectilevolume x projectile strength.

If a projectile is a sideways-flying cylinder, rather than a sphere, thesame line of reasoning leads to a somewhat reduced crater depth

†

P =DV2

rp

Rt(13)

4. Thin targetsWhen thin targets are struck by high-velocity projectiles, the impact

scenario is as depicted in Figure 8. A crater is produced in the target, whilethe back face is spalled. If the velocity is higher than the “limit velocity,”the projectile passes through the target, which now has a reducedthickness.

Figure 8. Crater from glass striking titanium at 5 km/s.

The thickness of a target that can be perforated by a projectile is lessthan the penetration of the same projectile in a thick plate. Figure 9, forexample, compares the protection thickness (the thickness of a target thatjust defeats the penetrator) to the semi-infinite penetration for steelplates struck by tungsten rods. A rule of thumb is that the protectionthickness will be about 1.5 cavity diameter less than the semi-infinitepenetration.

Figure 9 Protection thickness compared to penetration depth for L/Dtungsten rods striking titanium. (Gooch et al. [5])

When the target plate is oblique, the cavity will not be symmetric aroundthe penetration axis. The asymmetry in the crater creates an extremesensitivity to pitch when the pitch is the opposite sign of obliquity.

When mass is ejected from the rear of a target, but the projectileretains its original mass, then there is a well-known expression for residualvelocity:

VR =Vs

2 - VL2

1+ Ms / Mp

, (14)

where Vs is the striking velocity, VL is the limit velocity, Ms is the strikingmass, and Mp is the ejected mass.

Projectile exit through a finite target is often accompanied bydelamination and plugging. Delamination refers to a tensile failure parallel tothe target rear surface and is often initiated by spall. However, it can also

occur in quite thick targets, where it seems to be caused by shear bandsaround the projectile head that can propagate as cracks near the exitsurface. Plugging refers to the ejection of target material that fails inshear ahead of the projectile. The work required to eject a plug isapproximately equal to the circumferential area of the plug times the targetshear strength. In the case in which plugging is the dominate perforationmechanism, then the projectile energy at the limit velocity will be aboutequal to the energy to eject a plug.

Energy to eject plug = 14

p YDxdx = 18o

T

Ú pYDT 2 (15)

The above equation implies that the limit velocity is proportional to T/D(target thickness/projectile diameter). This linear relationship usually holdsup until a velocity or a target thickness at which the competing process ofductile hole growth comes to dominate. In that case, penetration occurs bypushing the target aside, rather than forward, and the energy to penetrateis proportional to the crater volume, D2T. The limit velocity is proportionalto ÷(T/D), which is characteristic of higher velocity impacts.

Damage to a long projectile that traverses thin plates takes place in twostages. The first is within the target, and is equivalent to erosion as takesplace in a thin target. Cavity expansion stress, R, is reduced in thin targets,so this effect is diminished at low values of Tt. Projectiles can also continueto shed material after emergence from a target plate. This effect isenhanced at high values of Tp.

Very thin plates referred to as “meteor bumpers” or “Whipple Shields”can be used to protect against very high velocity particles such as meteorsand orbital debris. The thickness of the plates is typically one quarter ofthe diameter of a (spherical) projectile. This thickness is about optimum toinduce destructive fragmentation of incident high velocity projectiles that inturn is stopped by a second plate, often the hull. Figure 11 shows early high-speed photos of a single Whipple Shield. Recent, more complex compositesystems have replaced the simple bumper sketched in Figure 10. [SeeHypervelocity Impact Symposia for overviews of current shieldtechnologies.]

Figure 10. Particle striking bumper and producing a debris cloud whichsubsequently perforates a hull plate. (Swift, [10])

[1]. Tate, A., “Further Results in the Theory of Long Rod Penetration,”Journal of the Mechanics and Physics of Solids, Vol. 17, No. 3, 1969, pg. 141[2]. S.J. Bless, D. Jurick, and B. Yoon, "Secondary Penetration of RodProjectiles," Proc. 9th Int'l. Symp. Ballistics, May 1986.[3]. Forrestal, M. J., Piekutowski, A. J., “Penetration Experiments with 6061-T6511 aluminum targets and spherical-nose steel projectiles at strikingvelocities between 0.5 and 3.0 km/s,” International Journal of ImpactEngineering, Vol. 24, No. 1, 2000, pp. 57-67.[4]. Subramanian R, Bless SJ. Reference correlations for tungsten long rodsstriking semi-infinite steel targets. 19th Int. Symp. Ballistics, Interlaken,Switzerland, 7-11 May 2001.

[5]. Gooch, W. A., Burkins, M. S., Frank, K., “Ballistic Performance ofTitanium Against Laboratory Penetrators,” Proceedings of the FirstAustralasian Congress on Applied Mechanics: ACAM-96, edited by R. H.Grzebieta, Institution of Engineers, Australia, Melbourne, Australia, 1996,pp. 21 - 23.[6]. Anderson, C. E., Jr., Morris, B. L., Littlefield, D. L., “A PenetrationMechanics Database,” Southwest Research Institute, SwRI Report3593/001, San Antonio, TX, January 1992. (Updated versions available ondisk from Southwest Research Institute).[7]. Partom, Y., and Littlefield, D. L., “Validation of the Cavity ExpansionResistance Model for Ceramic Targets with Computer Simulations,” Institutefor Advanced Technology, IAT Technical Report No. 25, Austin, TX, May1993. Also available as ADA313988.[8]. Gehring, J. W., “Engineering Considerations in Hypervelocity Impact,”High-Velocity Impact Phenomena, edited by R. Kinslow, Academic Press, NewYork, NY, 1970, Chap. 9.[9]. Subramanian, R., Satapathy, S., and Littlefield, D. L., “Observations onthe Ratio of Impact Energy to Crater Volume (E/V) in Semi-InfiniteTargets,” 19th International Symposium on Ballistics, edited by Iris RoseCrewther, 2001, Vol. 3, pp. 1273-1280.[10]. Swift, H. F., private communication, 1980.