Three-dimensional Green's functions of steady-state motion in anisotropic half-spaces and bimaterials

Abstract

Three-dimensional Green's functions (GFs) of steady-state motion in linear anisotropic elastic half-space and bimaterials are derived within the framework of generalized Stroh formalism and two-dimensional Fourier transforms. The present study is limited to the subsonic case where the sextic equation has six complex eigenvalues. If the source and field points reside in the same material, the GF is expressed in two parts: a singular part that corresponds to the infinite-space GF, and a complementary part that corresponds to the reflective effects of the interface in the bimaterial case and of the free surface in the half-space case. The singular part in the physical domain is calculated analytically by applying the Radon transform and the residue theorem. If the source and field points reside in different materials (in the bimaterial case), the GF is a one-term solution. The physical counterparts of the complementary part in the half-space case and of the one-term solution in the bimaterial case are derived as a one-dimensional integral by analytically carrying out the integration along the radial direction in the Fourier-inverse transform. When the source and field points are both on the interface in the bimaterial case or on the surface in the half-space case, singularities appear in the Fourier-inverse transform of the GF. The singularities are treated explicitly using a method proposed recently by the authors. Numerical examples are presented to demonstrate the effects of wave velocity on the stress fields, which may be of interest in various engineering problems of steady-state motion. Furthermore, these GFs are required in the steady-state boundary-integral-equation formulation of anisotropic elasticity.