Image Singularities of Green's Functions for an Isotropic Elastic Half-Plane Subjected to Forces and Dislocations

Abstract

Although the Green's function for an isotropic elastic half-space subjected to a line force or a line dislocation is well-known, the physical meaning of the solution is not clear. Green's functions for twodimensional plane-strain and plane-stress problems of an isotropic elastic half-space with a free or rigidly fixed surface subjected to line forces and line dislocations are reexamined in this study. The results are more explicit when compared with existing solutions in the literature. The Green's function for a half-space consists of four or five Green's functions for an infinite space, the number depending on the boundary condition at the half-space surface and the applied loading. One of the Green's functions in the infinite space has its singularity located in the half-space where the load is applied, and the other image singularities are located outside the half-space with the same distance from the surface as that of the applied load. The nature and magnitude of image singularities have been derived from the principle of superposition and classified according to different loads. The image singularities are found to possess some interesting properties. It is found that the fundamental solutions required to construct all the image singularities of applied forces and dislocations for the half-space are only forces and dislocations and their differentiations in the infinite space. Furthermore, the limiting case of the applied force or dislocation approaching the surface is also discussed in this study.