Local wave theory of elastic-body collision: Plane problem in the ideal-liquid approximation

Abstract

1. There are several well-known classical approaches to the investigation of elastic-body collision. In elementa~ impact theory, beginning with Newton's work, the basic parameter is the velocity recovery coefficient at impact, which is 1 for absolutely elastic bodies and 0 for absolutely plastic bodies. This approach cannot fully describe the internal state of the bodies. St Venant wave theory considers the longitudinal collision of elastic rods, i.e., one-dimensional objects; an advantage of the theory is the determination of the duration of collision, equal to twice the time for the compression wave to traverse the complete length of the rod. However, the one-dimensional stress----strain state determined in this way cannot be used for adc~iuatr description of the state of bodies of different shape. In addition, attempts to consider oblique or noneentral impact onthe basis of St Venant theory are evidently difficult. Classical local Hertz theory is based on the solution of a static contact problem regarding the action of a point force on the plane boundary of a halfspace, which is extended to the problem of the dynamic contact interaction of elastic bodies in direct central impact. Hertz theory, wifich is essentially quasi-static, is applicable when the contact time of the bodies in collision is considerably greater than the period of the lowest intrinsic oscillations of the bodies. Modification of the local theory by synthesis with St Venant theory has been proposed by Sears, Kil'chevskii, and others. Timoshenko has developed a theory of transverse impact at an elastic beam. Note that the assumption of elastic deformation of the colliding bodies limits the range of relative impact velocities that may be considered. At high velocities, the elastic limit of the material may be exceeded, and the behavior of the body must be described on the basis of more complex theological models, taking account of inelastic deformation. M. A. Lavrent'ev has proposed a hydrodynamic model of collision (interpenetration) of bodies at high velocity. It may be successfully used because the stress ellipsoid resembles a sphere in high-speed collision, i.e., the stress ellipsoid of a low-viscosity liquid. As a rule, numerical methods must be used to consider the collision of bodies with theologically complex behavior. In addition to these and several other approaches in modern continuum mechanics, for three-dimensional bounded by smooth surfaces, we may note an approach associated with the investigation of the collision (interpenetration) of bodies or the impact of a body on a medium in which the behavior of the body or medium is described by the equations of elasticity theory (or an ideal liquid in the case of collision at a liquid surface) and the collision problem is formulated, in the general case, as a nonsteady, often mixed, twoor three-dimensional problem, ff the motion of the bodies is specified only in the initial moment of collision, and their subsequent motion is determined in solving the problem, the boundary of the bodies' contact region is generally mobile and unknown in advance. This relatively complex formulation of the impact problem, which is free from the assumptions of classical appmacbes, has yielded a series of analytical (numerical--analytical) solutions by, among others, Grigolyuk and Gorshkov, Sagomonyan, Pomchikov, Galanov, Borodich, V. N. and V. V. Gavrilenko, Popov, and Tarlakovskii [ 1-3, 81. As a rule, the impact of a solid body or elastic shell on the surface of an infinite elastic or liquid medium is considered in this approach. Inthe present work, an approach to the collision of two elastic bodies such that at least their frontal surface is relatively smooth in the geometric sense (blunt bodies) is developed. The approach is based on research on the impact interaction of bodies of finite size with an infinite acoustic or elastic medium [3-8]. Its development takes into account that the depth of interpenetration of blunt bodies on impact is small in comparison with the clmracteristic dimension of the contact region and, consequently, the corresponding boundary conditions may be formulated at the unperturbed surface of the bodies. In addition, since the surface of a blunt body is only slightly distorted within the contact region, it may be regarded as plane. Finally, the