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/ / -SANDIA REPORTSAND94- 1490 UC -704Unlimited R eleasePrinted Octo ber 1994

Projecti le Penetration in toRepresentative Targets

George W. StonePrepared bySandla Natlonai LaboratoriesAlbuquerque, New Mexlco 87185 and L lvermore, California 94550for the Unlted States Departmentof Energyunder Contract DE-ACOe94AL85000

Approved for public re1 jbution i s unlimited.

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Issued by Sandia National Laboratories, operated for the United StatesDepartmen t of Energy by Sandia C orporation.NOTICE:This report was prepared as an account of work sponsored by anagency of the United State s Government. Neither th e United StatesGovern-men t nor an y agency thereof, nor any of their em ployees, nor any of theircontractors, subcontractors, or the ir em ployees, makes any warranty, expressor im lied, or assumes any legal liability or responsibility for th e accuracy,compPeteness, or usefulness of an y inform ation, appar atus , produc t, orprocess disclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product, process, orservice by trade name, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply ita endorsem ent, recommendation, or favoringby the United States Government, any agency thereof or any of theircontractors or subcontractors. The views and opinions expressed herein donot necessarily state or reflect those of the United States Government, anyagency thereof or an y of their contractors.Printed in the United States of America. This report has been reproduceddirectly from th e best available copy.Available to DOE nd DOE ontractors fromOffice of Scientific and Technical InformationPO Box 62Oak Ridge, T N 37831

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SAND94-1490Unlimited ReleasePrinted October 1994Distribution CategoryNumber: UC-704

Projectile Penetration into Representative Targets

George W. StoneSecurity and Survivability Department, 5822Sandia National LaboratoriesAlbuquerque, New Mexico 87 185

AbstractThe differential equation representing the penetration of a "hard" projectile into semi-infmite, homogeneous target materials is solved for several generic combinations of the targetmateriUprojectile characteristics. A "hard" projectile is defined as one that does not change sizeor shape and does not lose mass during the penetration process. The target materia ls evaluatedrange from the structurally "soft" materials (liquids) to structurally "hard" materials (armor plate)with viscous and fluid dynamic drag considered. The solutions to the differential equation(s) areexpanded in series form to demonstrate the underlying parameters governing projectile penetra-tion and the way they interact to limit penetration in a given target material. It is shown that thefundamental parameter governing projectile penetration into structurally "fm" aterials is theinitial kinetic energy of the projectile divided by the frontal area of the projectile and the inherentstructura l characteristic of the target. Experimental data on the penetration of steel spheres intoballistic gelatin and for armor piercing bullets into armor plate materials are used to verify thecharacteristics of the solu tions to the equation of motion for the projectile and to demonstratehow penetration can vary with projectile size and target characteristics. The penetration equationfor a s ingle "hard" target material isused to develop a solution for the penetration of multi-layered"hard" target materials.

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ContentsPreface ............................................................................................................................ 71. Introduction ............................................................................................................... 92. Analysis ...................................................................................................................... 9

A. Structurally "Hard" Materials.............................................................................. 12B. Structurally "Firm"Viscous Materials ................................................................. 3C. Structurally "Firm" Inviscid Materials ................................................................. 13D. Structurally "Finn" Materials with Viscosity and Fluid Drag ............................... 14 E. Structurally "Soft" Materials ............................................................................... 14

3. Experiments ............................................................................................................. 15A. "Soft" Target Materials....................................................................................... 53 . "Hard" Target Materials...................................................................................... 16

4.Multi-Layered "Hard" Materials ............................................................................... 225. Conclusions.............................................................................................................. 4References .................................................................................................................... 24

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Figures12

Penetration Schem atic and Nomenclature............................................................... 11The Variation of the Fluid Dynamic Drag Parameter, a, with Sphere Radiusfor Tests in 20%Gelatin......................................................................................... 17The Variation of the Fluid Friction Drag Parameter, b, with Sphere Radiusfor Tests in 20%Gelatin......................................................................................... 18The V ariation of the Target Structural Resistance Parameter c, withSphere Radius for Tests in 20% Gelatin................................................................... 19The Experimental Penetration Capability of Various Armor Piercing BulletsAgainst Rolled Homogeneous Armor as a Function of the Areal KineticEnergy of the Total Projectile ................................................................................. 20The Experim ental Penetration Capability of Various Armor Piercing BulletsAgainst Rolled Homogeneous Armor as a Function of the Areal KineticEnergy of Only the Hardened Projectile Core ......................................................... 20 The Experim ental Penetration Capability of Various Armor Piercing BulletsAgainst Aluminum Armor Plate as a Function of the Areal Kinetic Energyof the Total Projectile............................................................................................. 1The Experimental Penetration Capability of Various Armor Piercing BulletsAgainst Aluminum Armor Plate as a Function of the Areal Kinetic Energyof Only the Hardened Projectile Core .................................................................... 21

Multilayered Target Schematic and Nomenclature ..................................................23

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PrefaceThe Nuclear Security Center of Sandia National Laboratories has been involved for years in thevulnerability of and in the ballistic protection for high value assets to small arms projectiles. Thiswork has included experimental vulnerability testing of complex structures and the developmentof sof t and hard armor blankets and protective structures. In the course of this work, the need forsome theoretical basis for analyses, interpolation, or extrapolation of the limited experimental testdata became apparent. This report presents some previously unpublished theoretical results andexperimental comparisons developed several years ago to provide a fm fou ndation for theexpanded application of existing and expensive test data.

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Projectile Penetration into Representative Targets1. Introduction

The penetration of projectiles into representative targets has been of interest to mankind eversince man ceased to rely on rocks alone and graduated to arrows and spears. Even without anyscientific background, it was soon observed that heavier, (i.e. larger) projectiles generallypenetrated deeper in to or through a given target material than smaller and/or lighter projectiles.With the advent of gunpowder, man came to possess the energy source necessary to deliverprojectiles to targets with ever increasing velocity. Soon, the field of ballistics reached the pointwhere projectile design and strength began to be a factor when the projectiles themselves began tofracture and breakup either when being launched or on impact with a target.The physics of launching a projectile from a tube or barrel, (interior ballistics), is extremelycomplex and, as a result, even today it is basically an empirical science based on a tremendousamount of experimental observations. The flight of a projectile from the barrel muzzle to thetarget, (exterior ballistics), is amenable to theoretical analysis and has been the subject of math-ematicians and engineers for decades an d today is rather well explained and understood. Thepenetration of a projectile into or through a target (terminal ballistics) is theoretically difficult indetail, but the bulk process is amenable to an extension of the techniques of the exterio r ballisti-cian where the target is now much more dense and has more inherent structural integrity than air.The continued development of special penetration ammunition (i.e. Saboted Light ArmorPenetrator [SLAP], Long Rod Penetrators, etc.) make the problem of penetration prediction andprotection more critical for security, safety and survivability. The purpose of this report is topresent som e previously developed theories for use in future analyses of potential penetrationvulnerabilities of high value assets.

2. AnalysisArthur Dzimian presents the results of some excellent penetration tests of steel spheres into 20%gelatin representing animal tissue and discusses the several proposed theories for predictingprojectile penetration in various target materials. The tests involved spheres with radii of 3/16,1/4,3/8,nd 7/16 inches and impact velocities up to 3300 feedsecond. By accuratelydifferentiating the experimental data, Dzimian found that the deceleration of the projectile duringpenetration was governed* by the differential equation:

MdV/dt=-[AV2 +BV +C]*Equation 1 was first proposed by W. A. Allen, E. B. Mayfield, and H. . Morrison, who thenargued that it was not applicable to experiments in sand.

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where,M =the projectile mass (assumed constant)V =he projectile velocityt =he independent variable timeA =a classical fluid dynamic drag parameterB =a deceleration parameter due to kinetic fict ion on the surface of the projectileC= he deceleration force due to the inherent structural characteristics of thetarget material.

By the nature of the experimental tests and the physics of the problem, Equation 1 is valid whenthe target material can be considered as a semi-infinite slab and boundary effects are negligible.As expected, Reference 1 found that when the impact velocity was small, a few tens of metersper second), the penetration was heavily influenced by boundary effects. In addition, Reference 1points out that Equation 1encompasses all of the other penetration theories proposed at that tim e.For Equation 1,nondimensional analysis of the fluid dynamic forces shows that the fluid dynamicdrag force is equal to the density of the target material, a constant depending on the projectileshape, (drag coefficient) , and the frontal area of the projectile, i.e.

where CD is independent of projectile size.Similiarly, the fric tion force can be shown to be equal to

where, however, CF hould depend on the projectile size (fluid Reynold's Num ber).The resistance force of the target material due to its inherent structural characteristics is difficultto describe analy tically because the penetration process necessarily occurs in the plastic regime ofmaterial characteristics but a consideration of the two special cases of a flat nosed projectile(punch press shear) and a conical projectile (pierce press with radial material flow) suggests thatthe constant C can be described in the form:

where Cm isdependent on the m aterial characteristics in the plastic regime and should bedependent on the projectile shape, R in Equations 2a through 2c is the cross-sectional radius ofthe projectile. A schem atic showing the geometry of the problem and illustrating the variables ispresented in Figure 1.

I

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Figure 1 Penetration Schematic and Nomenclature11


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Dividing Equation 1 through by the projectile mass, M, and changing the independent variablefrom time to distance traveled into the target material, S, gives the following:

dS/dt =VVdVIdS =-(a@ +bV +c)

where,a =AIM, etc.

Equation 3 is variable separable and can be rearranged in the form:

VdV = - d SaV2 +bV +c

(3)

(4)

Solutions to Equation 5 can be easily obtained from any Table of Integrals for variouscombinations of parameters a, b, and c, (See. Reference 2. )A. Structurallv Hard Materials (armor plate, etc.), a =b =0.For a structurally hard target material where any fluid drag force can be neglected and the frictiondrag force on the surface of the projectile can be assumed small, the solution to Equation 4 is:

~2 =-2cs +constantIf we apply the boundary conditions that V =VOwhen S =0, and V =0 when S =P (totalpenetration), then we can write Equation 6a as:

P=Vo52c .Subsitution of Equations 2c and 4 into Equation 6b gives:

which states that the total projectile penetration is proportional to the initial kinetic energy of theprojectile per un it area divided by the target material structural parameters.

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B. Structurally Firm Viscous Materials (sand,* loose soils, etc.), a=O.When a =0, the solution to Equation 5 is given by,

b2S=cLn(c+bV) - bV +constant (7a)Again applying the boundary conditions to Equation 7a gives:

b2P=bVo - cLn(1+bVdc) . (7b)If the logarithmic function is expanded in series form, the result can be shown to be,

P =MJ02 1 {1 +....} (7c)2 c (1 +bVd2c)Again, the penetration is proportional to the initial kinetic energy of the projectile divided by thetarget material structural parameters but it is reduced somewhat by a term containing the ratio ofthe viscous parameter divided by the structural parameter in the denominator.C. Structurally Firm Inviscid Materials b =0. If b =0, the solution to Equation 5 is given by:

2 a ~-Ln(c +aV2) +constant (3a)which, when the boundary conditions are applied, can be written as:

2 a ~Ln(1 +aV&) . (8b)Expanding Equation 8b in series form gives the result:

P=M&)2 1 {1+ ...} (8c)2~ (1+av$/2c>Again, the total penetration is proportional to the initial kinetic energy of the projectile divided bythe target material structural parameter but, in this instance, it is reduced by a term in thedenominator containing the projectile fluid drag parameter divided by the ta rget structuralparameter.

W. A. Allen, E. B. Mayfield, and H. L. Morrison3 conducted experim ents of projectilepenetration in sand and concluded through some faulty reasoning and questionable mathematicsthat for sand c =0, (i.e. there was no structural integrity); at the Same time, Reference 3 discussesthe importance of crushing sand particles.

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P =MVn 1 {1+...}B ( l + a V @ )where, in Equation lOc, the penetration isgoverned by the initial impact momentum of theprojectile, rather than by the initial kinetic energy.

3. ExperimentsA. "Soft" Targe t MaterialsAs mentioned earlier in explaining the development of Equation 1,Arthur Dzimianl conducted anexcellent seriesof penetration tests of steel spheres into ballistic gelatin, (20%water). Bymeasuring the instantaneous penetration of a particular sphere as a function of time, Dzimian wasable to accurately differentiate the data and to obtain quantitative values for the parameters a, b,and c. For this particular target material, a partial justification of the form of Equation 1 can beobtained if the parameters a, b, and c do indeed vary with sphere diameter as suggested byEquation 2.For a sphere in viscous f l ow , (Stoke's Flow), the viscous drag coeffic ient in Equation 2b is givenby Reference 4 s:

and, since the m ass of a spherical projectile can be written as:

then, Equations 2,4, 11 and 12 can be combined to give, theoretically,

where ka, kb, and 5 are indeed constants of the target material and the material density of thespherical projectile.

It should be noted that when equations 13 are substituted back into the complete equation ofmotion o r any solutions containing the viscous parameter, the dynamics of the pro jectile donot scale linearly with projectile size as several investigations have expected or tried toexplain.

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Experimental results for the parameters a, b, and c , were taken from Reference 1for the varioussizes of spherical projectiles and are shown versus sphere radius in Figures 2 to4, respectively.As can be seen, the variation of these experimental results does indeed follow the predictedvariation given by Equations 13 surprisingly well.B. "Hard" Target MaterialsExperimental data for the penetration of armor piercing bullets into typical armor plate materialswere taken from Reference 5 and were com pared to the predicted behavior for "hard" targetmaterials given by Equation 6c. The armor piercing bullets selected have a rather so ft jacketmaterial surrounding a hardened steel core with a soft lead filler.The f m t comparison of experimental data** with Equation 6c is shown in Figure 5for "rolledhomogeneous armor plate" where the initial kinetic energy of the projectiles is based on the totalmass of the resw ctive projectiles. While it can be seen in Figure 5that the penetra tion of theindividual projectiles does indeed follow a linear variation with initial kinetic energy divided by theprojectile diameter squared, as theoretically predicted, the three curves shown do not collapse intoa single line representing a single target material constant. When the construction of the bullets isconsidered , it appears reasonable to expect the soft bullet jacket material and the lead filler to beeasily stripped away by the hard target material and a better comparison might be to use the initialkinetic energy of the bullet core only. Such a comparison is shown in Figure 6where it can beseen that using only the kinetic energy of the core does collapse the three curves in Figure 5into asingle line as the theory predicts.One feature of Figures 5and 6 (that is not theoretically predicted) is that the data do not appearto pass through the origin of the figures. There are two phenomenon that the theory does notaccount for in its present form. One mentioned earlier is boundary effec ts on the front and backfaces of the target material. For very small penetrations of thick targets, the cratering effec t onthe front surface may have a significant influence on the penetration process. In addition,experiments have shown that complete penetrations of finite thickness targets frequently result ina material "spall" of the rear surface. If the rear face of the target "spalls ," the "penetration" willbe greater than predicted.The second phenomenon that has not been considered is that, at low initialprojectile kineticenergy, the deformations of the target material will occur in the "elastic regime" while at higherenergies, the deformation will certainly be in the "plastic regime" assumed by the theory. Thisdramatic difference in target material characteristics for low initial kinetic energy impacts probablywill cause the experimental data to bend toward the origin of the figures. There are someexperimental indications of this behavior at low impact kinetic energies but it is for softer "ball"ammunition and the results cannot be considered conclusive.**Traditionally, ballisticians have worked with projectile calibers, i.e. diameters, as anindependent variable. The comparisons in Figures 5through 8 use the traditional projectileparameters Eo/D2, rather than Eo/ ICR2.

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Figure 2. The Variation of the Fluid Dynamic Drag Parameter, a, withSphere Radius for Tests in 20% Gelatin

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1000800

600

400lsecl

I 200c00

1oc

- Reference 1- lv . - _ -1- = a V z + b V + cdtI n2 1 \Theoreticallv b (x --R '

I I I I I I I l l I I I I I I I L0.01 0.10R, inches

1 o

Figure 3. The Variation of the Fluid Frict ion Drag Parameter, b, withSphere Radius for Tests in 20% Gelatin


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100

c 10-3FT/sec 2

10 I I I I-0.01

1

REFERENCE 1- dV- aV* + b V + cdt RTheoretically c a -

0.10R, inches

1 o

Figure 4. The Variation of the Target Structu ral Resistanc e Parameter, c, withSphere Radius for Tests in 20% Gelatin


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0 50 CALIBER MPAPA 30 CALIBER M P A P0 7.62CALlBER M61AP1.5

3fI0 1.05F3Wn

1.5

8 1.05FaanI-W

0.5

Eo/ *x I O 4 Ft Ibs I in2Figure5. The Experim entalPenetrationCapabilityof Various Armor Piercing Bullets AgainstRolled Homogeneous Armor as a Func tionof the Areal Kinetic Energy ofthe Total Projectile

0 =CALIBER M2 AP

I I I IO O 10 20 30 40

~ I f l x l O ~l b s I l n 2Figure6. The ExperimentalPenetration Capabili ty of Various Armor Piercing BuiietsAgainst RoiledHomogeneousArmor as a Function of the Areal Kinetic Energy of only the HardenedProjectileCore

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10 20 30 40 5006 D2

Armor Plate as a Functionof the Areal Klnet lc Energy of the Total Projecti leFT b d n2

Flgure 7. The ExperimentalPeneiratlon Capabliity of Varlous Armor Plerclng Bul lets Agalnsl Aiurnlnum 2024-T4

M7ABWBB17.12

0 50 CALIBER M2 A PA 30 CALIBER M2 A Pa 7.62 CALIBER M61 A P

I I I I10 20 30 40 50E d 104 FT- *din 2

F ~ U N. The Exp.rimmt.l Penetnt lonC a p a b ~ ~f Various A m o rPiercing Bullets Agah.t Aludn um Annor Platen aFunctiond he Areal Kinetic Energy of Only the HardendPmjectu. CON

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A second comparison of the predictions of Equation 6c with experimental data for softeraluminum armor plate is shown in Figure 7 where it can be seen that a similarseparation of theindividual curves is again evident when the total mass of the projectile is used. Figure 8 shows thesame comparison when only the mass of the hardened core of the projectile is considered in theinitialkinetic energy of impact. Using only the kinetic energy of the core results in the correlationof the penetration data on three different armor piercing bullets as predicted by the theory. Again,the fact that the experimental data do not appear to pass through the origin of the figures, asdiscussed above, is again evident.4. Multi-Layered "Hard" M aterials

Since the experimental data c o n f i i the theoretical predictions of Equation 6c or a single "ha rdtarget material, Equation 6a can be used to develop a penetration theory for a m ulti-layered* hardtarget material illustrated in Figure 9. It is assumed that the projectile possesses sufficient energyto completely penetrate k ayers of the target and is stopped in layer k+l. Applying the newboundary conditions to Equation 6a that V =Vo when S i =0, and V =V1 when S I =TI: V =V Iwhen S2 =0 and V =V2 when S2=T2: etc, a series of equations are developed as follows:

etc., with the final equation for layer k+ l as

where the projectile stops in layer k+l with Vk+l =0, when

Combining Equation 14 a, b, c, d gives the result:

The total penetration is given by:

An "effective" structural constant, c*, is defined as:P=v02//2c* .

Each layer is assumed to be a "hard" material or an airgap.

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. Figure 9 Mult i layered Target Schematlc and Nom enclature.(Each L ayer Is assumed to be a "Hard" M aterial oran Airgap)-

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Equation 17 can now be used with experimental data for a layered "hard" target to determine thevalue of the effec tive structural constant,c*, which then can be used to estimate the performanceof other projectiles and/or the effect of different initial impact velocities on the Same laveredQas

5. ConclusionsA family of solutions to the general equation motion for a "hard" projectile penetrating sem i-infinite generic target materials has been developed and compared to existing experimental datafor two types of targets. The comparison showed that, for these two target materials, the theoryadequately predicts the general characteristics of the penetration process with the exception ofvery small penetrations at low initial impact velocities. The solution for a "hard'projectilepenetrating a single layer of target material was used to develop an equation for estimating thepenetration behavior for a "layered" target using experimental data to define an effective value forthe target structu ral constant. The solutions should be very useful in predicting the performanceof various projectiles against a particular target material or conversely, the protection provided byvarious target materials against a particular projectile.

References

1. Arthur J.Dzimian, The Penetration of Steel Spheres into Tissue Mo dels, U.S. Army,Chemical Warfare Laboratories, Technical Report, CWLR 2226, August 1956.2. H. B. Dwight, Tables of ntegrals and Other Mathematical Da ta, Fourth Edition,The MacMillan Co., 1961.3. W. A. Allen, E. B. Mayfield, and H. L. Morrison, ,Dynamicsofa Projectile Penetratingsand, Journal of Applied Phvsics, Volume 28, Number 3, pp 370 o 376, March, 1957.4. H. Schlicting, , oundary Layer Theory, McGraw-Hill Book Company., 1955, pp 84-85.5. F. S . Masciancia, Ballistic Technology ofLightweight Armor-1981,AMMRC TR 8 1-20,U.S.Army Materials and Mechanics Research Center, May 1981.


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