Nonstationary Axisymmetric Motion of an Elastic Momentum Half-Space Under Nonstationary Normal Surface Displacements

Abstract

An elastic homogeneous isotropic half-space filled with the Cosserat medium has been considered. The deformed state is characterized by independent displacement and rotation vectors. At the initial time and at infinity, there are no perturbations.On the boundary of a half-space, normal displacements are given. All components of the stress–strain state are supposed to be limited. A cylindrical coordinate system with an axis directed inward to the half-space has been used. With axial symmetry, the resolving system of equations includes three hyperbolic equations with respect to the scalar potential and the nonzero components of the vector potential and the rotation vector. The components of displacement vectors, rotation angle, stress tensors, and stress moments are related to potentials by known relationships. The solution of the problem has been sought in the form of generalized convolutions of a given displacement with corresponding superficial influence functions. These functions have been constructed using a Hankel transform with respect to the radius and a Laplace transform with respect to time. All images have three terms. The first of these terms corresponds to the tension–compression wave, and the remaining two are determined by the associated shear and rotation waves. The originals of the first components have been found accurately through successive inversion of transforms. For the remaining terms, we have used expansion in power series in a small parameter characterizing the relation between shear and rotation waves. The images of the first two coefficients of these series have been found. The corresponding originals have been determined by successive inversion of transforms. Examples of calculations of the regular components of the influence function of a granular composite of aluminum shot in an epoxy matrix have been given.