Green's Functions for Transversely Isotropic Elastic Half Space

Abstract

This paper presents a comprehensive analytical treatment of the three‐dimensional response of a transversely isotropic elastic half space subjected to time‐harmonic excitations. General solutions for equations of equilibrium expressed in terms of displacements are derived by applying Fourier expansion with respect to the circumferential coordinate and Hankel integral transforms with respect to the radial coordinate. The general solutions are used to derive the explicit solutions for Green's functions (displacements and stresses) corresponding to a set of time‐harmonic circular ring loads acting inside a half space. The circumferential variation of the ring loads are assumed to be cosmθ for loadings in the vertical and radial directions and sinmθ for the loading in the circumferential direction. These Green's functions can be used as the kernel functions of the boundary‐integral‐equation method and in the development of solutions for a variety of elastodynamic boundary value problems. Comparisons with existing numerical solutions for an isotropic half space are presented to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses are presented to portray the dependence of the response of the half space on the frequency of excitation and the degree of anisotropy of the medium.