Image singularities of Green’s functions for isotropic elastic bimaterials subjected to concentrated forces and dislocations

Abstract

Green’s functions for isotropic materials in the two-dimensional problem for elastic bimaterials with perfectly bonded interface are reexamined in the present study. Although the Green’s function for an isotropic elastic bimaterial subjected to a line force or a line dislocation has been discussed by many authors, the physical meaning and the structure of the solution are not clear. In this investigation, the Green’s function for an elastic bimaterial is shown to consist of eight Green’s functions for a homogeneous infinite plane. One of the novel features is that Green’s functions for bimaterials can be expressed directly by knowing Green’s functions for the infinite plane. If the applied load is located in material 1, the solution for the half-plane of material 1 is constructed with the help of five Green’s functions corresponding to the infinite plane. However, the solution for the half-plane of material 2 only consists of three Green’s functions for the infinite plane. One of the five Green’s functions of material 1 and all the three Green’s functions of material 2 have their singularities located in the half-plane where the load is applied, and the other four image singularities of material 1 are located outside the half-plane at the same distance from the interface as that of the applied load. The nature and magnitude of the image singularities for both materials are presented explicitly from the principle of superposition, and classified according to different loads. It is known that for the problem of anisotropic bimaterials subjected to concentrated forces and dislocations, the image singularities are simply concentrated forces and dislocations with the stress singularity of order O(1/ r). However, higher orders (O(1/ r 2) and O(1/ r 3)) of stress singularities are found to exist in this study for isotropic bimaterials. The highest order of the stress singularity is O(1/ r 3) for the image singularities of material 1, and is O(1/ r 2) for material 2. Using the present solution, Green’s functions associated with the problems of elastic half-plane with free and rigidly fixed boundaries, for homogeneous isotropic elastic solid, are obtained as special cases.