Oblique penetration of a rigid projectile into an elastic-plastic target

Pergamon

Int. J. Impact En.qng Vol. 19, Nos. 9-10, pp. 769-795, 1997 ~? 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved P l h S0734-743X(97)00014-6 0734-743X/97 $17.00 + 0.00

OBLIQUE PENETRATION OF A RIGID PROJECTILE INTO AN ELASTIC-PLASTIC TARGET I. V. ROISMAN, A. L. YARIN* and M. B. RUBIN Faculty of Mechanical Engineering, Technion-lsrael Institute of Technology, 32000 Haifa, Israel

(Received 18 May 1996; in revisedJbrm 16 January 1996) Summary--The main objective of the present work is to develop an approximate solution of the problem of oblique penetration of a rigid projectile into an elastic-plastic target of finite thickness. This is accomplished by generalizing the work on normal penetration reported in [1]. Here, an irrotational isochoric velocity field is considered that consists of three parts, each of which together satisfy the condition of impenetrability at the projectile's surface. The first part is associated with the longitudinal motion of the projectile, the second part with the transverse motion, and the third part with the projectile rotation in the plane defined by the initial longitudinal projectile velocity and the normal to the target surface. The target material is assumed to be incompressible and the target region is subdivided into an elastic region ahead of the projectile, and a rigid-plastic region near the projectile. Using the above potential velocity field, inertia effects are included and the linear momentum equation is solved exactly in the elastic region. In the plastic region, the linear momentum equation is integrated numerically along the instantaneous streamlines to determine the pressure field on the projectile surface. Then the decelerating force and moment applied to the projectile are solved numerically. The model developed here predicts the residual velocity, the ballistic limit, as well as the residual angle of obliquity. Moreover, this model is able to describe the phenomenon of ricochet. It is shown that the agreement of the theory with experiments is good even though no adjustable parameters are used. Also, a user-friendly computer program has been developed that is available for distribution along with a Users' Manual. © 1997 Published by Elsevier Science Ltd.

1. INTRODUCTION Relatively few works are devoted to the problem of oblique penetration of projectiles in spite of its evident practical importance. However, some empirical correlations for the parameters of penetration have been proposed. By way of background, it is recalled that all points on the axis of symmetry of a rigid projectile have the same component U of velocity parallel to this axis. Then the ballistic limit Ub is defined as the minimum magnitude of the impact velocity Uo that causes perforation of the target, and the residual velocity Ur is defined as the magnitude of U after the projectile has perforated the target and lost contact with it. For metal plates made of steel and aluminum the ballistic limit Ub for oblique penetration has been found to be correlated with the ballistic limit Ubn for normal penetration and the initial angle of obliquity ~o [ - n/2 < C~o < n/2] by the relation: Ub = Ub, sec~o,

(1.1)

where Ubn is supposed to be known [2]. Some additional empirical correlations can be found in [3-5]. Due to the three-dimensional nature of oblique penetration, a fully numerical solution of the problem still poses extreme calculational difficulties. Examples of such numerical solutions that use finite difference hydrodcodes are given in [6, 7]. At present there seems to be two analytical approaches to the problem of oblique penetration. The first approach employs an energy and momentum balance to describe penetration. An example of this approach is given in [8] where normal and oblique perforation of a thin target with plug formation was considered. In that analysis it is

*On sabbatical leave at the Chemical Engineering Department, University of Wisconsin-Madison, Madison, WI 53706-1691, USA. 769

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I.v. Roisman et

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assumed that the plug possesses the same velocity and moves in the same direction as the projectile after perforation. Moreover, in [8], oblique penetration is assumed to be approximately equivalent to normal penetration as far as linear momentum in the direction of motion is considered. A similar point of view is adopted in [9] where the equation of motion of the projectile is a slightly modified version of that corresponding to normal penetration. The second approach focuses on the development of computer codes for oblique penetration by generalizing the ideas proposed in the analytical approaches for normal penetration. An example can be found in [10] where oblique penetration of a rigid projectile is described by means of auxiliary solutions of cavity expansion problems [1 1, 12] in order to relate target drag coefficients with target mechanical properties. The target drag imposed on a projectile and the moment of force about the projectile's center-of-mass are calculated in order to determine the projectile motion during the penetration process. In this approach, the determination of the target/projectile contact surface during oblique penetration is the most complex stage in the evaluation of drag and moment of force acting on the projectile. For modeling normal penetration it is well known that hydrodynamics of an inviscid fluid can be used to propose an approximate velocity field in the target. This method was successfully exploited using an integral work-rate balance to predict normal penetration of a rigid blunt cylindrical projectile in [13]. It has also been shown [14-:16] that this approach can drastically improve the characterization of the velocity field in the plastic target. The approach developed in the present work generalizes the analytic model for normal penetration developed in [1, 17] to consider the case of oblique penetration of a rigid projectile into an elastic-plastic target. The method of singularities is used to construct the flow field in the target corresponding to longitudinal and transverse motion of the projectile, as well as the projectile rotation in the plane of impact. The drag force and moment applied to the projectile are then calculated by integrating the stresses on its surface in contact with target material. In the present work, the projectile is considered to be an arbitrary rigid body of revolution. Initially, it is moving parallel to its axis of symmetry and it is not rotating. The impact plane contains the normal to the target surface and the axis of symmetry of the projectile (see plane SS in Fig. 1) so the velocity vector of the projectile's center-of-mass remains in this impact plane during penetration. The main objective of the present work is to develop an approximate solution of the problem of oblique penetration of a rigid projectile into an elastic-plastic target of finite thickness. In the following analysis, the location of the projectile's center-of-mass C relative to the fixed origin O is denoted by Xc and the angle of obliquity (Fig. 1) is denoted by ~ so

e~

er~\e~se~ e~'- t

'\.

plane .f~

Fig. 1. Sketch of the coordinate systemsassociatedwith the target and the projectile.

Oblique penetration of a rigid projectile into an elastic-plastic target

771

that the equations of motion of the projectile become MJtc = F,

Jc~ = Tc,

(1.2a, b)

where M is the constant mass of the projectile, Jc is the constant moment of inertia (about the point C) associated with rotations in the impact plane, F is the resultant force and Tc is the resultant moment about C applied to the projectile by the target. Throughout the text a superposed dot denotes material time differentiation which in Eqn (1.2) reduces to ordinary time differentiation. Equations (1.2) are solved using the initial values Uo and 7o at the moment of impact of the projectile velocity and the angle of obliquity, respectively. To calculate the stress field in the target and the resulting forces and moment applied to the projectile, the velocity field in the target is separated into three parts. The first part is due to the projectile's longitudinal motion, the second part due to its transverse motion, and the third part is due to its rotation in 'the SS plane. Following [1, 17], it is assumed that the velocity field is derived from a potential, which justifies the above separation. Next, the target material is assumed to be incompressible and the target region is divided into an elastic region ahead of the projectile where the strains remain infinitesimal, and a rigid-plastic region near the projectile where the strains can be arbitrarily large. Also, the rigid-plastic response is taken to be rate independent. Next, using the proposed potential velocity field, inertia effects are included and the linear momentum equation is solved exactly in the elastic region. However, in contrast with the velocity field for normal penetration I-1], the proposed velocity field for oblique penetration does not admit exact integration of the linear momentum equation in the plastic region. Consequently, an approximate solution is obtained by integrating the linear momentum equation along instantaneous streamlines in the target in the plastic region to determine the pressure acting on the projectile's surface. Furthermore, the effects of the free front and rear surfaces of the target are modeled in an approximate manner, and the decelerating force and rotating moment applied to the projectile by the target are calculated by numerically integrating over the relevant portion of the projectile's surface. A brief outline of the paper is as follows. Section 2 describes the coordinate systems that are used to formulate the problem. Section 3 discusses the analysis of the target. Section 4 determines the drag force and moment applied to the projectile. Section 5 proposes a simple engineering approximation of the solution and Section 6 presents a number of results and discussion. Conclusions are summarized in Section 7. Also, Appendices A and B present detailed formulas used to calculate the velocity and stress fields. Throughout the text, bold faced symbols are used to denote tensors, I denotes the unit tensor, the notation a-b denotes the usual scalar product between two vectors a, b, and the notation A. B = tr(AB T) denotes the inner product between two second-order tensors A, B. 2. C O O R D I N A T E SYSTEMS Consider a rectangular Cartesian coordinate system with coordinates {xl, Xz, x3} and base vectors {el, ez, e3}, with el and e 3 in the projectile's plane of symmetry SS. The fixed origin O is located at a point on the front surface of a target of thickness H (Fig. 1). Also, consider two other orthonormal coordinate systems: one with coordinates {x~, x~, x~} and base vectors {e'l, e~, e~} with e'l and e~ in the plane of symmetry SS; and the other with cylindrical polar coordinates {r, 0, 3} and base vectors {e,, eo, e~}; both with a moving origin B at a point on the projectile axis (see Fig. 1). The angle of obliquity ~ = or(t) is defined as an angle between projectile axis e~ and the vector e 3 normal to the target surface. The vector XB locates the point B and the vector x locates an arbitrary point P in the target, both relative to the fixed origin O. Also, the vector p locates the point P relative to B (Fig. 1) so that X =

xiei,

XB = x B l e l + XB3e3,

(2.1a, b)

p = x - XB,

(2.1C)

p = rer + ¢e¢ = (Xl - xB~)el + x2e: + (x3 - xB3)e3.

(2.1d)

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~v Fig. 2. Sketch of the projectile and its velocity and angular velocity.

By definition, the various base vectors are related by the equations el = cos ct e~ - sin ~ e3,

e~ = e2,

e, = cos 0 e~ + sin 0 e~,

e~ = e¢ = sin ~ el + COS ~t e3, e0 = -- sin 0 el + cos 0 e~,

(2.2a-c) (2.2d, e)

so that substitution of Eqns (2.2a-c) into (2.2d, e) yields the expressions e, = cos ~ cos 0 e~ + sin 0 e 2

-

-

(2.3a)

sin ~ cos 0 e3,

eo = -- cos ~ sin 0 el + cos 0 e2 + sin ~ sin 0 e 3 .

(2.3b)

Next, using Eqns (2.1d), (2.2a) and (2.3), the coordinates of an arbitrary point P become r = p. e, = (xl - xBl) cos ~ cos 0 + x2 sin 0 - (x3 - xB3) sin ~ cos 0, = p. e¢ = (xl - xBl) sin ct + (x3 - xB3) cos 0t, X'l = p "(1 = - ( x l

(2.4b)

- XB1) COS Ct -- (X3 -- XB3) sin ~,

0 = tan_~ (x.~) = tan_~ (

(2.4a)

(2.4c)

x2_ ). - (x3 - Xa3) sin 0~ - (xl - XB1) COS 0t

(2.4d)

Also, with the help of the above definitions, the vector Xc and its time derivative can be written as Xc = xB + ~ce~,

Xc = xB + ~c0te~,

(2.5a, b)

~B = -- Ve~ -- Ue~ = -- (V cos ~ + U sin or)el - ( - V sin ~ + U cos 00e 3.

(2.5c)

In E q n (2.5c) and t h r o u g h o u t the text, it is convenient to refer to the m o v i n g origin B because B is used to describe the shape of the projectile. Consequently, it is convenient to express the velocity of the point B in terms of its longitudinal velocity U in the ( - e ~ ) direction and its transverse velocity V in the ( - e~) direction (see Fig. 2). Then, the initial conditions on velocity used to solve E q n (1.2) become ~cl (0) = - Uo sin Cto,

~c3(0) = - Uo cos ~to,

~(0) = 0 ,

(2.6a-c)

where Uo( > 0) is the initial longitudinal velocity of the projectile (denoted as the impact velocity), and Cto is the initial angle of obliquity. Moreover, the initial transverse velocity is equal to zero and the initial position of the projectile is specified so that an extension of the projectile's axis of s y m m e t r y passes t h r o u g h the origin O and a point on its tip just makes contact with the target (the origin in Fig. 1 has been shifted d o w n w a r d for clarity in representing the vectors defined in the figure). 3. A N A L Y S I S O F T H E T A R G E T The target material is assumed to have a constant density p and to remain incompressible. This means that the conservation of mass and balance of linear m o m e n t u m equations

Oblique penetration of a rigid projectile into an elastic-plastictarget

773

have the forms V.v=0,

p ¼=-Vp+divo'.

(3.1a, b)

where the gradient operator V and the divergence operator div are defined with respect to the current position of a material point, v is the absolute velocity of a material point in the target, and the Cauchy stress o has been separated into a pressure p and its deviatoric part o', such that = - pl + a',

~ ' . I =0.

(3.2a, b)

In the following solution, a velocity field is proposed in the target which satisfies the continuity equation (3.1a) and the condition of impenetrability at the surface of the projectile. At the instant that the projectile touches the target's front surface the target material begins to deform elastically. At some point during the penetration process the target material begins to deform plastically and an elastic-plastic boundary propagates away from the projectile's tip. In the elastic region the strains remain small whereas in the plastic region the strains can be arbitrarily large. For simplicity, it is assumed that the material response in the plastic region is rate-insensitive and rigid-plastic so that the deviatoric stress can be approximated by

(D. D) 1/2 D,

(3.3)

where Y is the constant yield stress in uniaxial tension and D is the symmetric part of the velocity gradient. Furthermore, in the elastic region it is assumed that the material response is linear elastic and isotropic so that for isochoric motion the deviatoric stress is related to the linear deviatoric strain e by the Hook's law o ' = 2,u~:,

(3.4)

where/~ is the constant shear modulus.

3.1.

Velocity

and pressure fields

A projectile of length L is considered whose lateral surface is defined by a function r = R(~). Thus, at the surface of the projectile f = f(r,

~, t) = r -

R[~(t)] =0,

(3.5)

where the radius of the projectile asymptotically approaches a constant value R~o (Fig. 2). This surface will be material and the target material will not penetrate the projectile if f=0.

(3.6)

Consequently, with the help of (2.4) the expression (3.6) becomes dR • dR / = t: - ~ ~ = p.e, + p. 6, - ~ ([~. ee + p. 6~),

(3.7)

where the differentiation of Eqns (2.3a, b) and use of Eqns (2.2a, b) yields 6, = t)eo - ~ cos 0 e¢,

6~ = ~e'l.

(3.8a, b)

Moreover, with the help of Eqns (2.1c) and (2.5c) it can be shown that = i~ - x B ,

v = ~ = v r e , + Voeo +

veer,

(3.9a, b)

I.V. Roisman et al.

774

where vr, v¢ and Voare the components of the absolute velocity of a material point in the target material. Now, Eqns (2.1d) and (3.6)-(3.9) can be substituted into Eqn (3.5) to obtain the condition of impenetrability at the projectile surface in the form dR v~+ V c o s O - ¢ ~ c o s O = ~ - ~ ( v ¢ + U + R E c o s O ) .

(3.10)

Since the velocity field in the target is determined by a potential function 4(x, t) it follows that v = V4.

(3.11)

Thus, the continuity equation (3.1a), the balance of linear momentum (3.1b) and the impenetrability condition (3.10) reduce to the following forms: V24=0,

V

p-~+½pV4.V4+p

VcosO-~cosO=-d- ~

=dive'.

+ U + Ro~cosO

(3.12a, b)

.

(3.12c)

As mentioned in the introduction, the potential 4 can be represented as a sum of three potentials 4 (1), 4 (2) and 4 °) of flows characterizing a projectile moving with longitudinal velocity U, transverse velocity V, and rotating in the SS plane with an angular velocity a (see Fig. 2): (3.13)

4 = 4 (1) "q'-4 (2) + 4 (3).

The functional forms for these potentials can be deduced using the method of singularities 1-18-20] via the introduction of a distribution of sources and doublets along the axis of revolution. A distribution of sources 4 (1) generates 4 ") for the longitudinal motion, a distribution of doublets 4 (2) generates 4 (2) for the transverse motion, and a distribution of doublets 4 ~3~generates 4 (3) for the rotation. The resulting expressions for 4 (1), 4 ~2) and 4 (3) become

4 <'~= ~("URoo,

~)(')= _ ~, ~L[(~ _4.~(q)dq q)z + ~211/z'

4(2) = Zp,E,VRoo,

$,z, = f c o s 0 il [(~

4 o) = q~(3)~R2'

~(3) = f c o s 0

~-

(3.14a, b)

4t2)(q)d# Z-~+~]3/2' 4o)(#) dr7 .....

(3.14c, d)

,

(3.14e, f)

~, E(~ - ,7) ~ + e2] 3/~

r R~'

~ R~

?] R~'

~A

~L

(3.14g-k)

where 4(o(#) are the strengths of the singularities per unit length at the point ~- = q on the projectile axis, CA is the axial coordinate of the projectile's tip A, and ~r is that of the projectile's rear end. The expressions for the functions 4°~(#), 4~z)(#) and 4(3)(#) can be derived from the condition of impenetrability of the projectile surface by substituting Eqns (3.14) into Eqn (3.12c) and demanding that the coefficients of the terms U, V, and d vanish. These conditions give the following integral equations for determining 4(1)(#), 4(2)(#)and 4(3)(#)

Oblique penetration of a rigid projectile into an elastic-plastictarget

775

[18-20]: ~, [(~ _ #)2 + /~2] 1/2 q(~)(#) d# = 2

~~ ~'~(#) d#,

_

+

(3.15a)

1,

-rfiT

d}~ ~

¢'-~(3~(#)[(~-#)2-2/~Z]d# 4(3>(~)(~- 6) d# ~, [(~ _ 0)2 + /~215/2 + 3 - ~ /~ ~, [.( ~ _. 0 ) 2. +. ~ -. ~ 2

(3.15b)

i~d}~

~+

~-~,

(3.15c)

/~ =/#(~-) = __R(~) R='

(3.15d)

where/~(~) is the normalized function defining the projectile's surface. Also, it is noted that Eqn (3.15a) has been derived by multiplying the terms associated with the velocity U in Eqn (3.12c) [with Eqns (3.13) and (3.14a, b) used] by/#/U and integrating the result subject to the condition that R(~A) vanishes at the projectile's tip. In order to calculate the velocity field for a given projectile, it is necessary to solve these integral equations numerically. As a result, the distributions of sources and doublets corresponding to a given projectile shape are obtained. Such a solution for a projectile shaped as a semi-infinite ovoid of Rankine is shown in Fig. 3. The form of the ovoid of Rankine [1, 18, 19] is chosen for the calculations of the next sections because it has a simple analytical form that is close to the shape of real projectiles. The calculated distribution of sources ~(1)(#) (Fig. 3b) approximates the delta-function 6(#)/4, which is expected for a single source generating the ovoid for longitudinal motion. Also, for comparison purposes the distributions of sources and doublets corresponding to a conical shaped projectile are shown in Fig. 4.

(a) .

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Fig. 3. Distributionsof singularitiescorrespondingto a projectileshapedas an ovoidof Rankine.(a) the shape of the projectile'stip, (b) the distributionof sources4(", (c) the distributionof doublets~(2~, and (d) the distribution of doublets o~a~.

4

776

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. . . .

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-3,

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-4 6 ....

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Fig. 4. Distributions of singularities correspondingto a conical shaped projectile.(a) the shape of the projectile'stip, (b) the distribution of sources~1~,(c) the distribution of doublets Ot2~,and (d) the distribution of doublets ~t3~

Using Eqns (3.11) and (3.13) analytical expressions for the velocity field ¥ and the rate of deformation tensor D can be derived ¥ = V(]91 + V(~2 "31-V(~3 ,

(3.16a)

D = 2 (Vv + Vvt),

(3.16b)

in terms of the functions ~(1)(~), ~(2)(?~) and ~(3)(r3. These expressions are given in Appendix A. The distributions of sources and doublets obtained characterize the flow field in the target as well as on the projectile's surface. For example, for a projectile shaped as an ovoid of Rankine, the instantaneous streamlines in the plane of symmetry of the projectile and on its surface are shown for normal penetration in Fig. 5, for oblique penetration leading to embedding of the projectile in the target in Fig. 6, and for oblique penetration leading to ricochet in Fig. 7. In these figures the instantaneous streamlines relative to the target are associated with the absolute velocity field v and those relative to the projectile are associated with the relative velocity field [/~e~ where p = p~el]. For simplicity, the pressure is separated into two parts p = w + p(i),

(3.17)

where w and p") satisfy the equations Vw = V-a',

a++ l

p-~+~pV~b.V¢+p(O=O,

(3.18a, b)

and pt0 is seen to be the part of the pressure due to inertial terms that appear in the equation of linear momentum. Expressions for the function w must be found in both the elastic and the plastic regions separately and should satisfy matching and boundary conditions for

Oblique penetration of a rigid projectile into an elastic-plastic target

777

(b)

(a)

(c)

x

0.5"~/

/o -i

1

-0.5

0

0.5

1

X

Fig. 5. Instantaneous streamlines relative to the projectile (a) and relative to the target (b) in the plane of symmetry of the projectile, and instantaneous streamlines on the projectile's surface (c) for normal penetration of a projectile shaped as an ovoid of Rankine. U = 700 m/s, V = 0 m/s, ~R~ = 0 m/s, xB3 = - 10 mm, ~ = 0', H = 20 mm.

stress. F o r additional convenience, w is denoted by )'w (e) in the elastic region, w = [w(p ) in the plastic region.

(3.19)

F o r the case of n o r m a l penetration of an ovoid of Rankine it was shown [ 1, 17] that a scalar function w exists which satisfies E q n (3.18a) pointwise in both the elastic and plastic regions. However, since the functional form for the velocity field is specified in the target region and since the constitutive equation for the deviatoric stress o ' is determined by this velocity field, there is no guarantee that a function w satisfying Eqn (3.18a) even exists. In particular, for the case of oblique penetration being considered, it will be shown that w (eJ vanishes and that an exact form for w (pJ does n o t exist since the divergence of t~' is not the gradient of a scalar. Nevertheless, an a p p r o x i m a t e solution will be presented which determines the value of w (p) by integrating along instantaneous streamlines in the plastic region.

I.V. Roisman et al.

778

(b)

(a)

x

0.5~,~ 1

-i

-0.5

0 x

0.5

1

Fig. 6. Instantaneous streamlines relative to the projectile (a) and relative to the target (b) in the plane of symmetry of the projectile, and instantaneous streamlines on the projectile's surface (c) for oblique penetration leading to embedding of a projectile shaped as an ovoid of Rankine. U = 50 m/s, V = - 30 m/s, aR0o = - 25 m/s, xaa = - 9 m m , c( = -45% H =20 mm. U s i n g t h e p o t e n t i a l field (3.16a), t h e e x p r e s s i o n (3.18b) for pti) c a n be w r i t t e n in t h e f o r m P

-

2

pv¢(U + a r c o s O) - p ( V - ~&)(v, c o s O - vo sin O),

(3.20)

w h e r e v,, vo a n d v¢ a r e t h e c o m p o n e n t s o f t h e v e l o c i t y field.

3.2. Plastic zone o f the target I n t h e p l a s t i c z o n e e q u a t i o n (3.3) for t h e d e v i a t o r i c stress o ' c a n be r e w r i t t e n in t h e f o r m o ' = vD,

v --

(D . D ) 1/2'

(3.21a, b)

Oblique penetration of a rigid projectile into an elastic-plastic target

779

(b)

(a)

t (c)

x

0.5~// -i

i

-0.5

0 x

0.5

1

Fig. 7. Instantaneous streamlines relative to the projectile {a) and relative to the target (b) in the plane of symmetry of the projectile, and instantaneous streamlines on the projectile's surface (c) for oblique penetration leading to ricochet of a projectile shaped as an ovoid of Rankine. U = 700 m/s, V = 1 I0 m/s, ~R~ = - 20 m/s, xB3 = - 3 ram. • = - 6T, H =20 mm.

which, w i t h t h e h e l p o f t h e r e s u l t V . D = 0 d u e to i n c o m p r e s s i b i l i t y , yields

~r D,, + -~ D,¢ + -r -ffoD,o e, + -~r D,¢ + -~ D¢¢+ r "~ D~o e¢ + -~r Dro + -~ D¢o + r - ~ Doo eo,

(3.22)

in t e r m s of t h e c o m p o n e n t s o f D r e l a t i v e to t h e p o l a r c o o r d i n a t e basis. M o r e o v e r , f r o m E q n (3.21b) it c a n be s h o w n t h a t

Or = -

Y (D.D)3/2,

(3.23a)

I.V. Roisman et al.

780

d~ -

Y (D. D) 3/2'

0-0= - \.~J

Y (D. D) 3/2'

(3.23b)

where the tensors R, S, and T are related to derivatives of the components of D and are defined in Appendix B. Then, Eqns (3.18a) and (3.22) yield the expressions dw tp)

3v

dv D

dr = -~r D , + -~

1 dv

,~ + -r -~-~D,o,

(3.24a)

dw {p) dv dv 1 c3v d~ - Or D,¢ + ~-~ Dec + r 0-0 D~o,

(3.24b)

dw ~") c3v dv dv dO - r -~r D,o + r -~ D:o + -~ Doo,

(3.24c)

which can be used to integrate the function w tp) along an instantaneous streamline x*(s) with arclength ds using the directional derivative

dx*(s) v ds = ivy,

dw{P) ds

-

[ dw tp) -~r

(3.25a)

10w tp)

e' + -r - - ~ e o

Ow{n) I

+ -~-

v

e¢ • ~-~ .

(3.25b)

3.3. Elastic zone of the target Next, a solution in the elastic zone of the target is determined where the strains remain small. To this end, it is assumed that the infinitesimal deviatoric strain e associated with the velocity field (3.16a) can be determined by integrating the equation de - - = D, dt

(3.26)

simultaneously with integration of the equation of motion of the projectile. The initial conditions are taken to be e = 0 and the material derivative is approximated by the partial derivative with respect to time because the strains and the velocity components are assumed to be small in the elastic region. Since V. ~' - 0 in the elastic region, Eqns (3.18a) and (3.19) require w re) to be a function of time only that will be determined by the boundary conditions. Then the expression for the stress tensor in the elastic region becomes 6 = - pI + o' = - [p") + wt~)(t)] I + 2 pc.

(3.27)

3.4. Boundary and matching conditions As in the model [1, 17] for normal penetration of a rigid projectile, it is not possible to satisfy the boundary conditions that the rear and front surfaces of the target are stress-free pointwise using the assumed velocity field. Also, it is not possible to satisfy the condition of continuity of the surface tractions pointwise on the elastic-plastic boundary. By way of background, it is recalled that for normal penetration [1, 17] these conditions were satisfied only along the line defined by an extension of the projectile's axis when the target's rear surface was in the elastic region. Alternatively, when the target's rear surface was in the

Oblique penetration of a rigid projectileinto an elastic-plastictarget

781

plastic region the free-surface condition was approximated by requiring the vanishing of the total force applied to the part of the rear surface in the plastic region. In both cases, continuity of the axial stress component was required at the intersection of projectile axis with the elastic-plastic boundary. This led to continuity of the pressure at that intersection point. Due to the three-dimensional nature of oblique penetration, it is necessary to consider different approximations of these boundary and matching conditions. Since the stresses in the elastic region of the target are much smaller than those in the plastic region it is assumed that it is sufficiently accurate to determine the value of w~e~(t) in the elastic region by requiring 6 to vanish as r ---,oo. With the help of Eqns (3.20) and (3.27) it then follows that

w~e~(t) =0,

(3.28)

in the elastic region. Next, the value of wtp~ is determined by a matching condition at the elastic-plastic boundary or by a boundary condition at the rear or front surface of the target. In this regard, it is noted that the elastic-plastic boundary in the target is a three-dimensional surface which is symmetric relative to the SS plane and determined by the values of x~, and x3 which cause the elastic solution to satisfy the von Mises yield condition [3~,.~,]1/2 _ y =/~[6t:.~]1/2 _ y =0.

(3.29)

Specifically, the value of wtp~is calculated by integrating (3.25) from a point on the surface of the projectile along the instantaneous streamline that intersects that point. If the streamline intersects the elastic-plastic boundary then the continuity of w is enforced so that wtp~ vanishes at that intersection point (see point Q3 in Fig. 8) w tp~ = w{~ = 0.

(3.30)

If the streamline intersects the rear (x3 = - H) or front surface (x3 = 0) of the target then the value of wtp~ is so determined that the normal component of the traction vector on that surface vanishes (see points Q1 and Q2 in Fig. 8) w~p~= e3. (er'e3) -

prO.

(3.31)

Furthermore, if the streamline returns to the projectile surface (point Q4 in Fig. 8) then it is assumed that the projectile loses contact with the target material so that w tp~ is determined from the condition of elimination of the normal component of the traction vector on the

H

Ql

Fig. 8. Approximationof the boundary and matching conditions.

782

I . V . Roisman et al.

projectile surface at that point w{p) = e.. (a'en) -- p(i).

(3.32)

Once wre) and w(p) are determined then the stress tensor a can be evaluated at any point in the target using the Eqns (3.2a), (3.17), (3.18b), (3.19) and (3.21). 4. DRAG FORCE AND M O M E N T A P P L I E D TO THE PROJECTILE To calculate the drag force and the moment applied to the projectile, the stresses are integrated over its surface. At any point on the projectile's surface the normal component of the deviatoric stress tensor is given by

ra~,, - 2(dR/d~)a'~ +

a'nn I, R

(dR/d~) 2 a ~ l

k

Jl,=.

=

(D.D) '/2

i ~ (--~/d~~

j[,=,

(4.1)

and the pressure p is given by Eqns (3.17)-(3.19) with w being determined as discussed in the last section. In the previous work on normal penetration [1, 17] it was shown that the value of the shear stress an, at the projectile surface [predicted by Eqns (3.2) and (3.21) with the rate of deformation tensor based on the potential flow generated by ~bl] is physically incorrect because it tends to accelerate the projectile instead of decelerate it. On the other hand, the normal stress a . . is predominantly determined by impenetrability of the target and projectile materials. The flow given by Eqn (3.16a) satisfies this impenetrability condition as well as the incompressbility conditions (3.1a). Moreover, the pressure is determined by integrating the balance of linear momentum along the instantaneous streamlines. Therefore, it is expected that an., given by Eqns (3.2a), (3.17), (3.18b), (3.19) and (4.1), will be reasonably accurate and that an, should be ignored. Consequently, the shearing component of stress vector on the projectile's surface is neglected in the calculation of the drag force and the moment applied to the projectile. Specifically, the components of the drag force and moment are obtained by integration of the normal stresses over a region A of the projectile surface F1 = F . e ~ = S an.en.e'l da,

F~=F~=F.e~=Sa.ne..e~da,

A

(4.2a, b)

A

Tc = ~ O'nn[{Re, + (4 -- ~c)e~} x e.] .e2 da,

(4.2c)

A

where the unit outer normal en and the element of area da are defined by (dR/d~)e¢ + e, [1 + (dR/d~)2] 1/2' -

en

=

(4.3a)

da = El + (dR/d~)2ql/2R d~ dO.

(4.3b)

The region A is that part of the projectile that is currently in contact with the target material. To determine this region A, it is convenient to introduce three coefficients k,, ko, and kv defined at an arbitrary point T on the projectile surface by the following expressions: if - H < x 3 _<~0 if - H < x 3 or 0 < x3

and an. < 0, or a . . > 0,

(4.4a)

if the point lies on the crater surface, otherwise,

(4.4b)

if Vp'en > 0, if Vp.en < 0,

(4.4c)

Oblique penetrationof a rigid projectileinto an elastic-plastictarget

783

where vp is the velocity of an arbitrary point on projectile's surface, which can be written in the form vp = [ - U - R(~)~ cos 0]e~ + [ - V cos 0 + ~ cos 0]e,. (4.5) The coefficient k, is used to express the sign of the normal stress an. acting on the projectile surface, kc determines positions of points relative to the crater in the target, and kv determines whether the projectile is tending to maintain contact with the target. The crater surface is determined by the collection of all the previous positions of the projectile surface inside the target. The coefficients k,, kc and kv are used to conveniently write expressions for the numerical calculation of the components of the drag force and the moment applied to the projectile such that 2n ~A+L

F; = ~ 0

~ k a . , R ( ~ ) c o s O d ~ dO,

(4.6a)

~A

2n ¢^+L k dR d F~ = F~ = - S ~ a,,,R(~) "-d-( ~ dO, o

Tc = ~ I ka,,R(~)cos0 0

(4.6b)

~A

R(~)+(~-~c)

d~d0,

(4.6c)

~A

k = k~k~k,, /73 = - F~ cos e + F~ sin e,

(4.6d)

F1 = F; sin e + F~ cos ~.

(4.6e, f)

The component F2 of the drag force normal to the penetration plane SS in Fig. 1 is equal to zero because the problem is symmetric relative to this plane. Also, Tc is the only nonzero component of the moment. It is further noted that the present approach allows a rather straightforward generalization for the case when points on the projectile are not confined to move parallel to the plane of the projectile's axis of symmetry and the normal to the target surface. 5. AN ENGINEERING APPROXIMATION The analysis of normal penetration of a rigid projectile [1, 171 allows one to conclude that the component F~ of the drag force in the e~ direction is relatively independent of the projectile longitudinal velocity, and therefore can be written approximately in the form F~= rtYR~2 [~5 In(4) + In ~ ] , (5.1a) - 2 = F 2d ~l 2/3 (3/~)

d (4Y']"2,

(5.1b, c)

where ~2 is the nondimensional axis coordinate of the elastic-plastic boundary, and d is the instantaneous depth of penetration of the projectile tip into the target. Moreover, the calculations in [1, 17"1(e.g. Figs 4 and 10 in [1"1)indicated that the magnitude of this force depends only slightly on the projectile shape, which is justified by experimental results (e.g. [21]). Thus, for the engineering approximation it is assumed that the normal stress a.. on the projectile's surface is equal to the average stress given by ann ~

~ F¢

- Y[~ln(4) + ln~22].

(5.2)

Moreover, when Eqn (5.2) is used for the case of oblique penetration it is tacitly assumed that cr.. is independent of the transverse velocity and the rotation of the projectile. Also, the depth of penetration d is approximated by the amount of penetration of the tip of the projectile d =

+

where XA~ and XA3 are the coordinates of tip A of the projectile.

(5.3)

784

I.V. Roisman et

al.

Thus, the expressions (5.2) and (5.3) can be substituted into Eqn (4.6) to calculate the force and m o m e n t applied to the projectile of an arbitrary shape. Note that in spite of the fact that ann as given by Eqn (5.2) is axisymmetric, the contact area of an axisymmetric projectile with the target is not axisymmetric during oblique penetration. Consequently, the calculated resultant m o m e n t is nonzcro and the projcctilc can rotate in the impact plane during oblique penetration. 6. R E S U L T S A N D D I S C U S S I O N A user-friendly computer program was developed based on the algorithm outlined in the previous sections and is available for distribution. This program calculates the drag force and the m o m c n t (4.6) applied to the moving projectile and solves the equations of motion (1.2) and (2.5). The model predicts the rcsidual velocity, the ballistic limit, as well as the residual angle of obliquity aftcr targct penetration. It is also possible to predict the location of the projectile [Fig. 9(a)] and the shape of the plastic region and the crater in the target [Fig. 9(b)] at different instants during penetration. Moreover, the model is able to predict the phenomenon of richochet (Fig. 10).

(a) = 0.8 x

10-6 sec

= 19.6 x 10 -6 scc

t = 2.3 x

10 -6 sec

t = 48.3 × 10-6 scc

(b)

Fig. 9. Oblique perforation of a plate by a rigid projectile shaped as an ovoid of Rankine: (a) the shape of the plastic zone at different instants of time shown in the figure;(b) the initial location of the projectile and its location at these instants of time. Simulation Data: Target: steel plate with H =20mm, Y =490.4 MPa; Projectile: 2R~ =6.2ram, L =22.6mm, M =5.2g, Uo =820 m/s, ~to = - 30~', residual velocity U, = 225.8 m/s, residual angle at, = - 32.Y'.

O b l i q u e p e n e t r a t i o n of a rigid projectile into an e l a s t i c - p l a s t i c target

785

(a) = 0 . S x 10-6 sec

/

t = 6.4 x 10.6 see

t = 27.8 x 10 .6 sec

t = 43.7 × 10 .6 sec

I

/ /

(h)

Fig. 10. Ricochet of a rigid projectile s h a p e d as a n ovoid of Rankine: (a) the s h a p e of the plastic zone at different i n s t a n t s of time s h o w n in the figure; (b) the initial location of the projectile a n d its location at these i n s t a n t s of time. S i m u l a t i o n Data: Target: steel plate with H = 2 0 m m , Y = 490.4 M P a ; Projectile: 2 R~ = 6.2 mm, L = 22.6 mm, M = 5.2 g, Uo = 820 m/s, ~o = - 6 5 , residual velocity U, = 439.1 m/s, residual angle ~r = - 115.0.

In order to examine the accuracy of the numerical solution of the equations of motion of the projectile computations have been compared with a number of experiments. A list of these experiments, together with the associated references and relevant material and geometric properties is recorded in Table 1. The value of the dynamic yield stress Y differs from the static one, and for the material used in EXP1 the value of Y was calculated by the empirical equation given in [23] Y I-MPa] = 3.484 × (BHN - I1.24),

(6.1)

where BHN is the Brinell Hardness Number. For aluminum AI 2024-T3 the value was taken to be BHN = 125 as in [23]. Since published values of the Brinell Hardness Number or of the dynamic yield stress for the materials used in EXP2 and EXP3 have not been found, the values of Y for the simulations associated with these experiments were determined by fitting the computed residual velocity with experimental data for normal penetration. Consequently, the values of Y used in the calculations for EXP2 and EXP3 depend on the accuracy of the normal penetration simulation. However, this procedure is justified by the fact that the method of I-1-1 and its modification used here to calculate normal penetration were compared with numerous experimental data and shown to be rather accurate. Note also that for EXP1 the

I. V. Roisman

786

et

al.

Table 1. A list of experiments, references, and relevant material and geometrical properties Projectile EXP

1 2 3

Target

Ref.

Shape

M (g)

L (mm)

Ro~ (mm)

Material

10 10 22

Conet Conet Ovoid*

13.0 13.0 5.2

28.5 28.5 22.6

4.750 4.750 3.100

AI 2024-T3 Cu

p (kg/m 3)

Y (MPa)

p (GPa)

2800 8440 7800

396 370 490

28 37 83

Mild steel

*The shape of the projectile is close to the ovoid of Rankine. tThe shape of the projectile is conical with the half angle of the cone being 30'.

Table 2. Comparison of the theoretical predictions with the experimental values of the residual velocity obtained for different initial angles of obliquity and target thicknesses EXP

Uo (m/s)

1 1 I 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

600.0 953.0 971.0 913.0 819.9 821.4 828.9 809.0 815.3 811.7 817.7 806.1 819.2 840.2 817.5 797.3 812.8 842.3 799.5 819.0

~o (deg) 0 - 50 - 60 0 - 60 -62 -63 - 57 - 59 -60 - 30 - 45 -51 - 55 - 30 - 45 - 51 0 - 15 - 50

H (mm)

U, (predicted) (m/s)

U, (experimental) (m/s)

12.67 12.67 12.67 9.56 10.0 10.0 10.0 12.0 12.0 12.0 16.0 16.0 16.0 16.0 20.0 20.0 20.0 25.0 25.0 25.0

432.1 768.0 674.9 840.0 339.5 179.2 0.0 193.5 0.0 0.0 465.5 226.0 0.0 0.0 256.3 0.0 0.0 256.9 0.0 0.0

465.0 757.0 541.0 848.0 493.7 0.0 0.0" 368.9 0.0 0.0" 496.3 0.0 0.0 0.0 151.9 0.0 0.0 107.6 0.0 0.0

The cases when the target was not penetrated correspond to U, = 0.0 and the symbol * indicates that ricochet occurred.

value of Y was not determined by matching experimental data for normal penetration and the comparison with the experiments for this case (see Table 2) is satisfactory. Furthermore, it is mentioned that the present model for oblique penetration and the engineering approximation are based on the velocity field past a projectile moving through an infinite target and are valid for thick targets where the influence of the target free surfaces on this velocity field is not large. Consequently, the phenomena of bulging and spallation of the rear surface of the target are not modeled. Moreover, the fitted value of the yield stress Y for simulating EXP2 (where relatively thin targets were used) contains some error corresponding to the influences of those phenomena and may not be appropriate for numerical simulations of oblique penetration of rigid projectiles into thick targets. As mentioned previously, the boundary conditions and matching conditions used to determine the functions w ¢c) and W (p) in this oblique penetration problem are different from those used for normal penetration l-l, 17]. Consequently, it is desirable to first examine the influence of these approximations for the normal penetration case. To this end, the computer program for oblique penetration was used to predict the normal penetration ease (~o = 0) and the results were compared with predictions of the previous model [-1, 17] for the material data of EXP1 in Table 1.

Oblique penetration of a rigid projectile into an elastic-plastic target

787

Figure 11 confines attention to normal penetration and shows predictions of the depth of penetralion P [Fig. 1 l(a)] and the residual velocity U, [Fig. 1 l(b)], as well as the ballistic limit Ub [Fig. 1 l(c)] for both the new model for oblique penetration (when it is applied to normal penetration with 0~= 0) and the model of [1, 17-]. These figures indicate that the modified boundary and matching conditions of the new model cause the target to offer more resistance to penetration than those of the model of [1, 17]. This is because the normal stress is only required to vanish far away from the projectile so that the normal stress applied to the target's rear surface can be compressive when the target's rear surface is in the elastic range. On the other hand, it is seen from Fig. 1 l(c) that there is a relative decrease in resistance to penetration as the target gets thinner because the new model attempts to enforce zero normal stress pointwise on the part of the target's rear surface in the plastic zone instead of only in an integral sense. Consequently, the new model can be expected to be less accurate than the model of [1, 17-1 when the target's rear surface is in the elastic region and more accurate when the rear surface is in the plastic region.

3O

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....

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30

35

....

I ' ' ' ' 1

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....

45

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....

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Fig. 11. Investigation of the influence of the modified boundary and matching conditions of the new oblique penetration model. Predictions of the ballistic limit for normal penetration of a rigid projectile shaped as an ovoid of Rankine using the new model, the model for normal penetration presented in [1, 17], and the alternative model which includes the influence of the free rear target surface. Simulation Data: Target: steel plate with Y =490.4 MPa; Projectile: 2 R ~ = 6 . 2 r a m , L = 22.6 mm, M = 5.2 g, ~o = 0'.

I.V. Roisman et al.

788

To further explore the influence of the free rear target surface, an alternative procedure for determining the value of w(e) was considered. In this alternative procedure, the point E is defined as the intersection point of the rear target surface and the extension of the projectile's axis (see Fig. 8). When the point E is in the elastic region the value of w(e) is determined by requiring the normal component of the stress vector applied to the rear target surface to vanish at the point E. This yields an equation similar to Eqn (3.31) with wcp) replaced by wteJ.The results of this alternative procedure are shown as the small dashed line in Fig. 1l(c). This alternative procedure yields results closer to those of the model in I-1, 17] for the full range of target thicknesses. However, in the following simulations the simpler model with vanishing w(~) will be used. Figure 12 shows a comparison between the results of a number of computations (which are denoted by symbols) of the ballistic limit and the empirical formula (1.1). The agreement of the empirical formula with the solution presented here is good even though the ballistic limit appears to be very sensitive to parameters of the projectile and target. Furthermore, numerical simulations of oblique penetrations of rigid projectiles corresponding to the experiments given in Table 1 were performed. In Table 2 the results of the numerical predictions of residual velocity Ur are compared with the experimental data. These results indicate that the present model for oblique penetration accurately predicts the impact angles at which projectile embedding occurs and the residual velocity after perforation. However, some discrepancies between the predicted and experimental residual velocities may be attributed to such phenomena as: projectile failure or breakup during penetration; inaccuracy of the reported obliquity angles; and the necessity to estimate the target yield stress due to lack of data. Also, note that for EXP3 the impact angles at which ricochet of the projectile takes place are eo = - 63 ° and - 6 0 °, for target thicknesses of 10 and 12 mm, respectively (as is shown in Table 2). The calculated values of these angles of ricochet for the same impact velocities as in the corresponding experiments are eo = - 66 ° and - 64.5°, respectively. Figure 13 shows the predictions of the residual velocity for the case of normal penetration computed using the complete model as it is outlined in the previous sections, and the engineering approximation as given in Section 5. The predictions based on the complete model agree better with the experimental data than those of the engineering approximation which uses stresses at the projectile surface corresponding to a semi-infinite target. Figure 14 compares four versions of the model: (1) the complete model; (2) a version in which the projectile rotation in the SS plane is neglected ['(~(3) in Eqn (3.13) was taken to be zero]; (3) a version which totally neglects transverse motion and rotation of the projectile and uses the model of normal penetration [1, 17] of target with a modified thickness H/cos(cto); and (4) the engineering approximation. Figure 14 shows that in comparison with the experiments the first three versions of the model give better results than the engineering

1.2

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~

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Fig. 12. Comparison of the empirical formula (1.1) [Ub./Ub = COSatO] for the normalized ballistic limit with the numerical predictions for two targets of different thicknesses. The target and projectile parameters correspond to EXP 3 in Table I.

Oblique penetration of a rigid projectile into an elastic-plastic target 80O

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. . . .

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. . . .

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(ram)

Fig. 13. Normal perforation of a finite target. Comparison of the predictions of the residual velocity with experimental data for a range of target thicknesses. The solid line presents results of the complete model and the dashed line presents results of the engineering approximation given in Section 5. The initial velocity is the same as that in the experiments and varies in the range 779-842 m/s.

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60

70

Fig. 14. Comparison of the predictions of the residual velocity with the experimental data for different target thicknesses. The initial velocity is the same as that in the experiments and varies in the range 779-842 m/s. The four different curves denote: (1) predictions based on the complete model; (2) predictions obtained neglecting the projectile rotation in the symmetry plane; (3) results of calculations of normal perforation of a target with the thickness H/cos ~o; and (4) predictions of the engineering approximation given in Section 5.

approximation. Moreover, the agreement of models l, 2 and 3 with the experiments is good for impact angles I0tol less than about 45 °. Also, it is noted that both the complete model and the engineering approximation predict such phenomena as embedding and ricochet of the projectile. The experiments of [10] for oblique penetration with the impact velocity and the initial obliquity angle Uo = 914 m/s and ~o = - 60 ° were also investigated. In these experiments,

790

I.v. Roisman et al.

three c o p p e r targets of different thicknesses were used. All the o t h e r p a r a m e t e r s c o r r e s p o n d to E X P 2 in T a b l e 1. In the first case (H = 6.5 mm) the t a r g e t was perforated. In the s e c o n d case (H = 9.7 mm) the projectile was e m b e d d e d a n d the target was a l m o s t perforated, a n d in the third case (H = 12.9 m m ) r i c o c h e t occurred. In the n u m e r i c a l s i m u l a t i o n s b a s e d on the engineering a p p r o x i m a t i o n the s a m e q u a l i t a t i v e results as in the e x p e r i m e n t s were o b t a i n e d at the i m p a c t angle eo = - 6 0 ° for the first case, ~o = - 6 2 . 5 ° for the s e c o n d case, a n d eo = - 63° for the t h i r d case. T h e results of the two s i m u l a t i o n s in w h i c h the projectile e m b e d d i n g a n d ricochet o c c u r r e d are s h o w n in Fig. 15. T h e p r e d i c t e d residual angle of o b l i q u i t y ~r as a function of the initial angle of o b l i q u i t y Co, the yield stress Y of the t a r g e t a n d the t a r g e t thickness is s h o w n in F i g s 16-18, respectively. T h e residual angle c o r r e s p o n d s to the i n s t a n t when the rear edge of the projectile leaves the target, or the i n s t a n t when the projectile s t o p s in the t a r g e t when the ballistic limit is n o t exceeded.

(a)

(b)

Fig. 15. Locations of a rigid conical projectile during oblique penetration at different instants of time: (a) H = 9.7 ram, Uo = 914 m/s, ~o = - 62.5"; (b) H = 12.9 mm, Uo = 914 m/s, So = - 63'. All the other parameters correspond to those of EXP 2 in Table 1.

i

80

,

,

.

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

Ballistic limit 60 ¸

40 ,,

..

20 ¸

0

'

0

'

'

'

I

10

'

'

'

'

I

20

'

'

'

'

I

'

'

30

'

'

I

40

. . . .

'I

50

'

'

'

'

60

o~o ( d e g )

Fig. 16. Residual angle of obliquity ~tr of the projectile as a function of the initial angle of obliquity • o. H = 16 ram, Uo = 800 m/s, the other parameters correspond to EXP 3 in Table 1.

Oblique penetration of a rigid projectile into an elastic plastic target 52

.

.

.

.

.

.

.

.

.

I

.

.

.

.

.

.

.

.

.

i

.

.

.

.

.

.

.

.

,

791

6 0 0

51 50 x....

" " xx

-400 , ~

49-~

-%

-200

48-. 47 46 300 .

\ .

.

.

.

.

.

.

.

i

.

.

400

.

.

.

.

.

.

.

Y (MPa)

I

.

.

.

.

.

.

.

.

.

0

500

600

Fig. 17. Residual angle of obliquity ct, (curve 1) and the residual velocity Uo (curve 2) as functions of the yield stress Y of the target. H = 16 mm, Uo = 800 m/s, ~o = 45, the other parameters correspond to EXP 3 in Table 1. 53

,

525150-

,

,

z

,

.

,

i

.

,

,

" "'''',

2

/~400 i

~

,

,

~

5 0 0

/

~-300 ~

I

I

g

49-

47 12

14

16 H (an)

18

20

Fig. 18. Residual angle of obliquity ~t, (curve 1) and the residual velocity Uo (curve 2) as functions of the target thickness H. Uo = 800 m/s, ~o = 45', the other parameters correspond to EXP 3 in Table 1. The projectile perforates the target for values of H less than the ballistic limit H = 17.2 mm while for greater values of H the projectile is embedded in the target. The results shown in Fig. 14 indicate that the formulation based on normal penetration with a modified thickness H/cos(~o) predicts the value of residual velocity fairly well. However, such a simple model c a n n o t predict the p h e n o m e n a of ricochet nor can it predict changes in the angle of obliquity due to penetration, both of which can be predicted by the present model. In particular, the results shown in Fig. 16 indicate that the residual angle of obliquity can differ from the initial angle of obliquity by as much as 10%.

7. C O N C L U S I O N S The model developed in the present w o r k characterizes oblique penetration of a rigid projectile into an elastic-plastic target of finite thickness. D u r i n g penetration, the velocity vector of the projectile's center-of-mass remains in the impact plane defined by the axis of symmetry of the projectile and the normal to the target surface. As a result, the trajectory of the projectile's center-of-mass is a planar curve. However, the velocity field in the deformable target is m o d e l e d in a fully three-dimensional manner. Generalization of the present a p p r o a c h for the general case of oblique penetration leading to three-dimensional m o t i o n of the projectile's center-of-mass is algebraically c u m b e r s o m e but is straightforward. The target material is assumed to deform in a rigid-plastic m a n n e r near the projectile and remains elastic far from it. The position and shape of the elastic-plastic b o u n d a r y are determined using the von Mises yield condition. The influences of the free surfaces of the target on the penetration process are accounted for in an approximate manner. However,

792

I.v. Roisman et al.

the p h e n o m e n a of bulging and spalation of the rear surface of the target are neglected, which makes the approximations valid only for rather thick plates. A plausible kinematically admissible velocity field is p r o p o s e d in the target. This field satisfies the conditions of incompressibility and impenetrability of the target and projectile materials. The velocity field allows the projectile to translate and rotate in the impact plane containing the curved m o t i o n of the projectile's center-of-mass. Moreover, this velocity determines the m o s t i m p o r t a n t n o r m a l c o m p o n e n t of the traction acting on the projectile's surface, which is used to calculate the total force and m o m e n t resisting the projectile's penetration. The model depends only on well-defined material and geometric characteristics of the target and projectile. Consequently, it does not depend on any empirical constants. It is capable of predicting a n u m b e r of i m p o r t a n t features of the penetration process which include: the projectile's residual velocity, the ballistic limit, the residual angle of obliquity, and the threshold angle for ricochet. Theoretical predictions have been c o m p a r e d with a n u m b e r of experimental data found in the literature. This c o m p a r i s o n is fairly g o o d even for such characteristics as the ballistic limit and the ricochet angle, which are both sensitive to parameters of the target and projectile. Some discrepancies between the predicted and experimental residual velocities m a y be attributed to such p h e n o m e n a as: projectile failure or breakup during penetration; inaccuracy of the reported obliquity angles; and the necessity to estimate the target yield stress due to lack of data. Also, a simple engineering a p p r o x i m a t i o n of the model was proposed. However, the results of the complete model were shown to be superior, especially at large angle of obliquity. The complete theoretical model has been p r o g r a m m e d into a user friendly c o m p u t e r code that is available for distribution along with its operation manual. Acknowledgement-- This research was partially supported by MAFAT, State of Israel.

REFERENCES 1. A. L. Yarin, M. B. Rubin and I. V. Roisman, Penetration of a rigid projectile into an elastic-plastic target of finite thickness. Int. J. Impact Engng 16, 801-831 (1995). 2. J. A. Zukas, High Velocity Impact Dynamics, John Wiley & Sons, New York (1990). 3. M. E. Backman and W. Goldsmith, The mechanics of penetration of projectiles into targets. Int. J. Engn9 Sci. 16, 1-99 (1978). 4. S.J. Bless, J. P. Barber, R. S. Bertke and H. F. Swift, Penetration mechanics of yawed rods. Int. J. Engng Sci. 16, 829-834 (1978). 5. T. W. Bjerke, G. F. Silsby, D. R, Schefflerand R. M. Mudd, Yawed long-rod armour penetration. Int. J. Impact Engng 12, 281-292 (1992). 6. G R. Johnson, R. A. Stryk, T. J. Holmquist and O. A. Souka, Recent EPIC code developments for high velocity impact. 3D element arrangements and 2D fragment distributions. Int. J. Impact Engng 10, 281-294 ( 1990). 7. J. J. Pyun, C. M. Kennedy and D. Hruska, A new slideline/eroding algorithm for EPIC2. Int. J. Impact Engng 10, 473-482 (1990). 8. R. F. Recht and T. W. Ipson, Ballistic perforation dynamics, J. Appl. Mech. 30, 384-390 (1963). 9. J. Awerbuch and S. R. Bodner, An investigation of oblique perforation of metallic plates by projectiles. Exp. Mech. 17, 147-153 (1977). 10. J. Falcovitz, M. Mayseless, Z. Taubes, D. Keck, R. Kennedy, K. Oftedahl and P. Singh, A computer model for oblique impact of a rigid projectile at ductile layered targets, l l t h Int. Syrup. on Ballistics, pp 1-11, Brussels, 1989. 11. R. F. Bishop, R. Hill and N. F. Mott, The theory of indentation and hardness tests. Proc. Phys. Soc. London 57, 147-159 (1945). 12. M.J. Forrestal, K. Okajima and V. K. Luk, Penetration of 6061-T651 aluminum targets with rigid long rods. J. Appl. Mech. 55, 755-760 (1988). 13. M. Ravid and S. R. Bodner, Dynamic perforation of viscoplastic plates by rigid projectiles. Int. J. Engng Sci. 21,577-591 (1983). 14. A. Tate, A simple hydrodynamic model for the strain field produced in a target by the penetration of a high speed long rod projectile. Int. J. Engng Sci. 16, 845-858 (1978). 15. A. Tate, Long rod penetration models--Part I. A flow field model for high speed long rod penetration. Int. J, Mech. ScL 28, 535-548 (1986).

Oblique penetration of a rigid projectile into an elastic-plastic target

793

16. A. Tate, Long rod penetration models--Part II. Extensions to the hydrodynamic theory of penetration. Int. J. M ~ h . Sci. 28, 599-612 (1986). 17. 1. V. Roisman, Description of projectile penetration with small Deborah number. M.Sc. Thesis, Technion, Haifa, 1994. 18. N. E. Kochin, I. A. Kibel and N. V. Roze, Theoretical Hydrodynamics. Interscience Pub., New York (1964). 19. G. K. Batchelor, An Introduction to Fluid Mechanics. Cambridge Univ. Press, Cambridge (1967). 20. H. Rouse, Advanced Mechanics of Fluids. Wiley, New York (1964). 21. J. Awerbuch and S. R. Bodner, Analysis of the mechanics of perforation of projectiles in metallic plates, Int. J. Solids Struct. 10, 671-684 (1974). 22. N. K. Gupta and V. Madhu, Normal and oblique impact of a kinetic energy projectile on mild steel plates. Int. J. Impact Engnq 12, 333-343 (1992). 23. C. E. Anderson Jr, B. L. Morris and D. L. Littlefield, A penetration mechanics database. Southwest Research Institute Report 3593/001, 1992.

A P P E N D I X A: C O M P O N E N T S

OF THE VELOCITY FIELD

Equations (3.14) and (3.16a) yield the components of the velocity field for an arbitrary distribution of singularities ~(~)(0), 4t2)(~), t](3)(~): V -----V(1) 7!- V(2'3) = (r~ l) + Vr . (1)

(~(~(1) =

v,

Or

)(2,3)

l~r -~

- 0

l)~ 1)

= rI3,

0~b(2)

0(/)(3)

=

(~(])(1)

~

(Ala)

(2,3)~^

~t-

1

( A l b , c)

= 13,

(Ald)

COS O ( J 0 - - 3 r 2 J ° ) ,

v~2'3) - Oq~(2--)+ OcP'3--)= -- 3 f c o s 0 j1,

(Ale)

v~2,3 , = a~b(2__ ) + O~b(3__ ) = _ sin 0 jo, 00 ~0

(Alf)

where the functions I,.(~, " - ~) are defined as

~,(+, ¢) =

"

(~--O"j(l__'(0_)d,~

(A2)

7 E(~- 0) ~ + ~2)-~,

+,

and the functions J,?,(~, " - r-) are defined as J~(~, i ) = V ~ q(2)(q)(~ - ~)" dr7 q(3)(rT)(~ - 0)" d6 +, e2] + ~R+ -~ E(~". . . . 0) 2 "l-"~-]-~ - I(~---,7) ~ + =/------~

(A3) "

Also, the identities I2+2(~, ~) = I°(~ -, f ) -

f 2 i o,,+ 2t¢, r= 7),

(A4a)

J.~+ 2(~, ~) = jo(L ~) _ ~2jOr.+2,., '~ ~),

(A4b)

are used in order to minimize the n u m b e r of functions that are required to express the resulting solution. The rate of deformation tensor D as defined by Eqn (3.16b) can be then written in terms of its components Dij relative to the polar coordinate basis in the form D = "~(:) (2,3))(er ® e~) + ~¢¢ i/-)(:) + r',.(2,3),~+" [i~rr -'1- Drr tp¢¢ Ite¢ ~) e¢) .~(1)

r..(2,3)x.

+ (voo + voo

~,eo ®

---(1)

n(2,3)xl~

eo) + (D~, + ~,~ )t,+¢® e, + e, ® e¢) (2,3)

+ (D(,o2'3))(eo® e, + e, ® %) + (D¢o)(eo ® e~ + e+ ® eo),

(A5)

794

1. V. Roisman et al.

where the components corresponding to the longitudinal motion of the projectile are 1

o

~rr/-)(1) ~---~--£ (I3

Dcx) ~ o0 =

io

2

-- 3r2 I°),

~¢¢r)(1)= - - - I °~oR

+ ~

3

r2I°'

r~O) 3 fI~ ~¢r = -- R---~ "

,

(A6a, b) (A6c, d)

Here, and throughout the text, superscript (1) corresponds to the longitudinal motion of the projectile, and superscripts (2, 3) correspond to the overall effect of the transverse and rotational motions. Also, due to the symmetry about the ~-axis D(,lo) = Dtol,) = Oto~) = O~lo' = 0.

(A7)

Next, the components of the part of D corresponding to the transverse and rotational motions of the projectile are given by 3

D(,2,"3' -

fcos O(3J ° - 5f2 J°),

R~

(A8a)

3 D~{' 3) = _ _ f cos 0(4 do _ 5~2 jo),

(Agb)

R~ F)(2,3)__

~oo

3 fcos0jO, R~

/')(2,3) = 3 fsin ~o R~

3

F1(2'3) = -~o

O J O,

3 cos0(Jg--5tv2dl), R~

/.)(2,3)__

~ sin 0 J~.

(A8c, d) (A8e, f)

APPENDIX B: C O M P O N E N T S OF THE TENSORS R, S, AND T Tensors R, S and T used in Eqns (3.23) can be written in the convenient forms R = R (~) + R (2'3),

S = S (1) -~ S (2'3),

T = T (1) + T (2'3),

(Bla-c)

where the components of these tensors are obtained by differentiating the corresponding components of D which are given in Appendix A. Specifically, using the definitions of the functions 1~,(~, ~) and Jm(~, " - r) - of Appendix A, the components of the tensor R Cu are defined by R,,,=

10_rno)l=__~.~f(31o_5:2io),

R(1)

1 L [-/)(1)1 =

(B2a)

f(4i o _ 5~2Io),

3

(B2b)

R(1)= 1 O rnml 3 flO oo Ro~ ~f L~°° a = ---gTRo~

(B2c)

R(1)= 1 c~ ___(1)1 3 ~' Roo &: LO¢,, = -~-~ (I~ - 5f214),

(B2d)

Ro) rO

(B2e)

~,(u = u(u "'Or -'~ "'0~

= R(~)_0, "'~0

and those of S (1) a r e defined by S(1) --rr

1 ~

R~o

t3

(u

1

/~oo

"~

~r ! ~"

--¢,1,

1 c3---(l)q ¢¢ -- R~ O-~Lo~¢a = R ~ [2I~ - 5f 2 I~],

S(1)_

°°0 = R~ 0-~ I-D°° ] = -- R TIs'..~

(B3a) (B3b) (B3c)

Oblique penetration of a rigid projectile into an elastic-plastic target

S(1)

0 (1)-1 = 1 _O rl)(t)-1 ~(1) o~[D.-, £ 0:~. " = ''"'

1 = ~

(I)

if(1)

r0 = ~0r

(B3d)

~,(I) -----~,(1) = 0 .

= '-'0~

795

(B3e)

~j0

Also, due to the s y m m e t r y the tensor T (1) vanishes (B4)

T (1) - 0. T h e c o m p o n e n t s of the tensor R (2"3) are defined by

R(r;, 3) .

.]

0. r/)[2, . .3)] J

3 cos 013J ° - 30~2j ° + 35F4J°], -fly R~

R ~ OFL--"

R[2,3) = 1 0 (2,3) ~¢ R ~ 0---r[D~¢ ]. = ~(2,3) = 1 0 rn(2 .3)_I "oo R ~ Ofu~°° • R(2,3) = ~r

*'re

3 cos O(J ° - 5Fza°), R2

1 63 FD(2,3).1 = ~15 f c o s O(3J~ J R~o

-

goo OF L u ~ r

R(2,3) = - -1

(3 rD[2 ,3).1

R~o ~FL~'°

R~,3)=

a=

7r2j1),

(B5d) (B5e)

i5 sin0J~',

J

(B5b) (B5c)

~ 3 sin O(J~ - 5 f2d°),

I A.~,2,3,.1_ R o~ 0f L~¢o

° + 35F4J°],

3 cos014:-35:2j

(B5a)

(B5f)

Rg~

those of S (2' 3) are defined by S(2,3)

1 ~FD(2,3)]=

"

I/~

S(2,3) __ 1

¢~

S~2,3)

~

0

15

] = - -m-R~ecos

I ----=0i_r~(2,3)1 = 15 = R-'-~0~ ,-~oo J -fiT'R. f c o s 0

i~(2,3)

'

~ ="~'

O(4J4 -

7e2j~),

1

(B6a) (B6b) (B6c)

J7,

S(2,3) ~--- 1 0 rD(2,3).1 _ 1 __0 i_i,1(2,3)_1 (2,3) ¢" R~o 0q L ~ ¢ r J R~ 0/7 L~¢~j J = Ree ,

(B6d)

S(2,3) ~--- 1 ~ 1_D(2,3)_1 ,o R~o0~u ,0 J = - ~

15 fsin0Jv 1 R~o

(BBe)

32 sin 0(4J ° - 5 f2jo). R~

(B6f)

~o K~ o¢ and those of

[2,3)

R---~0~ I-D~

1 0 rn(2,3).1

R-Z o-~ ~ '

'"

[ D ~ 3)3 -

T (2' 3) are defined by 0 rD[2,3)l = 3 Ty'3' = 0--0 ~-" ~ k--~ ¢sin

T(2,3) ~¢

0(3: -5:J°),

63 3 f s i n O(4J ° = ~-0 [D~#' 3)] = _ r-~

T (2,3) -oo =

r/3(2'3)-1= - k~oo J R

-

Fsin OJ °,

T~(~'3) = ~0 r/)(2'3).1L~r J = -~sinO(J~--5fzJ~),

Tf2o ,3) = T t2,3)

¢o

O0

rn(2,3)3 - ~ o J = - ~3c o s 0 J ° , R:o

= ~ ,0- ~rn(2,3). o _1 , = - - 3~ c o s 0 J ~ . R~

5tzzJ°),

(B7a) (B7b) (BYe) (B7d) (B7e) (B7f)