Steady State Deformations of a Rigid Perfectly Plastic Rod Striking a Rigid Cavity

The axisymmetric steady state deformations of an infinite cylindrical rod made of a rigid/perfectly plastic material and striking a known cavity in a rigid target are analyzed by the finite element method. The contact between the deforming rod and the target surface is assumed to be smooth. It is found that the axial force experienced by the rod depends strongly upon the square of its speed. Results computed and presented graphically include the velocity field in the deforming region, the dependence of the shape of the upset head of the striker upon its speed, and the distribution of normal tractions upon the cavity wall. INTRODUCTION Penetration of metal targets by projectiles is influenced by such variables as material properties, impact velocity, projectile shape, target support position, and relative dimensions of the target and the projectile. Recently, emphasis has been placed on kinetic energy penetrators, which for terminal ballistic purposes may be considered as long metal rods traveling at high speeds. Wright [l], in his survey article on long rod penetrators, ellucidated vividly some of the problems with the existing penetration models. In another extensive review article, Backman and Goldsmith [2] discussed superbly the work done in penetration mechanics until 1977. Jonas and Zukas [3] reviewed various analytical methods for the study of kinetic energy projectile-armor interaction at ordance velocities and placed particular emphasis on three-dimensional numerical simulation of perforation. Anderson and Bodner [4] have recently reviewed the status of the ballistic impact modeling. A penetration model that is not too difficult to use has been proposed by Ravid and Bodner [5]. They studied the penetration problem by presuming a kinematically admissible flow field in the target and found the unknown parameters by utilizing an upper bound theorem of plasticity modified to include dynamics effects. In an attempt to shed some light on questions raised by Wright [l], Batra and Wright [6] recently studied an idealized penetration problem that simulates the following situation. Suppose that the penetrator is in the intermediate stages of penetration so that the active target/penetrator interface is at least one or two penetrator diameters away from either target face, and the remaining penetrator is much longer than several diameters and is still traveling at a speed close to its striking velocity. This situation has been idealized as follows. It is assumed that the rod is semi-infinite in length, the target is infinite with a semi-infinite hole, the rate of penetration and all flow fields are steady as seen from the nose of the penetrator, and that no shear stress can be transmitted across the target/penetrator interface. This last assumption is justified on the grounds that a thin layer of material at the interface either melts or is severely degraded by adiabatic shear. These idealizations make it possible to decompose the penetration problem into two parts in which either a rigid rod penetrates a deformable target or a deformable rod is upset at the bottom of a hole in a rigid target. Of course, in the combined case the contour of the hole is unknown, but if it can be chosen so that normal tractions match in the two cases along the entire boundary between penetrator and target, then the complete solution is known irrespective of the relative motion at the boundary. Even without matching the normal tractions, it would seem that valuable qualitative information about the flow field and distribution of stresses can be gained if the chosen contour is reasonably close to those that are found in experiments. Whereas Batra and Wright [6] studied the problem of the deforming target and a rigid penetrator, we analyze herein the companion problem of a deformable, semi-infinite and cylindrical penetrator striking a known semi-infinite cavity in an infinite and rigid target. Only the axisymmetric and steady state problem in which the penetrator material is rigid/perfectly 183 184 R. C. BATRA and PEI-RONG LIN plastic has been studied. This problem is more challenging than the one studied earlier by Batra and Wright [6] because of the presence in it of free surfaces whose shapes are not known a priori. We hope that the kinematic and stress fields found in this study would help in devising and/or checking results from simpler engineering theories of penetration. FORMULATION OF THE PROBLEM We describe the deformations of the cylindrical rod upset at the bottom of a semi-infinite cavity in an infinite rigid target with respect to a cylindrical coordinate system with origin at the center of the hole and z-axis pointing into the rod. Equations governing the steady state axisymmetric deformations of the rod are div v = 0, (1) div Q = pi’, (2.1) = p(v grad)v. (2.2) Here v is the velocity of a rod particle, p is the mass density and u is the Cauchy stress tensor. Equation (1) implies that the deformations of the rod are isochoric, and eqn (2) expresses the balance of linear momentum. The operators grad and div signify the gradient and divergence operators on fields defined in the present configuration. We neglect the elastic deformations of the rod and assume that it is made of a homogeneous and isotropic material that obeys the Von-Mises yield criterion and the associated flow rule. Thus we take the following constitutive relation for o, e.g. see Prager and Hodge [7].