On evolution equations for impact deformation problems with consideration of plane discontinuity surfaces

The method of constructing approximate solutions of impact deformation problems for front areas of strong discontinuity surfaces is considered. It is shown that application of the method of matched asymptotic decompositions at certain distances from the shock wave front results in first order nonlinear wave equations, known as evolution equations. In the case of shear deformation, the evolution equation differs fundamentally from the volume wave equation (Hopf equation). Some variants of the solution of these equations and their application to definition of displacement field and deformations are offered. One of the variants is based on the additional parametric variable. The basic ideas of this method are illustrated by solving a number of one-dimensional problems on impact loading of a half-space occupied by a nonlinear elastic, isotropic compressible or incompressible medium. It is shown that the asymptotes obtained in this study can be used to develop numerical algorithms for solving the problems of impact deformation of a solid with consideration of discontinuity surfaces.