Elastic orthotropic or transversely-isotropic half-plane under the action of normal unsteady surface perturbations

Abstract

Abstract The unsteady plane problem of the action of different normal perturbations on an elastic orthotropic or transversely-isotropic uniform half-plane is considered. Its solution is represented in the form of convolutions of the perturbations with surface influence functions. Explicit expressions for these functions are found using a Laplace transform with respect to time and a Fourier transform with respect to a spatial coordinate. The corresponding preimages are determined using their analytical representations. Without major changes, this method enables us to obtain explicit formulae for the remaining influence functions (such as, for example, those corresponding to tangential perturbations). The reliability of the method was evaluated by passing to the limit of an isotropic medium, as a result of which the well-known formulae for plane Lamb problem were obtained. The special features of the behaviour of influence functions are explained. The influence functions found allow us to investigate the stress-strain state of a half-plane for any normal perturbation using quadratures.