Displacements and stresses induced by a point source across a plane interface separating two elastic semi-infinite spaces: An analytical solution

The problem of computing the static deformations and stresses produced by a point source in a homogeneous infinite medium was solved by Volterra [1907] in a closed analytical form at the beginning of this century. The similar problem of computing fields generated by point sources in a homogeneous half-space bounded by a free surface was later studied by Steketee [1958a,b] and several others [see Okada, 1985, 1992], who focused on point as well as on rectangular fault sources of interest in seismology. Here the model taken into account consists of two elastic half-spaces characterized by different elastic properties (rigidity modulus μ and Poisson coefficient ν) and separated by a planar interface: assuming that a point source is active in one half-space, static deformations and stresses generated by the source in the whole space are computed. The similar problem of two half-spaces welded together was solved by Heaton and Heaton [1989], but they imposed the simplifying constraint that both materials are Poissonian (i.e., both have the same Poisson coefficient ν=0.25). The present approach, which is based on the Galerkin vector method, is general and applicable to an arbitrary point source. In this paper the computations have been carried out explicitly only for the special case of a dislocation source having the form of a strike-slip double couple. The solution is provided in a closed analytical form by means of expressions involving the source descriptors (position and intensity) as well as the elastic parameters of the heterogeneous medium. The solutions have been illustrated and discussed with special attention given to the dependence of the displacement and stress components on the elastic parameters of the model. One interesting finding concerns the limiting case when the rigidity modulus of the half-space not containing the point source is equalized to zero. Although the solution in this half-space no longer makes sense, the solution in the other reduces exactly to the one computed for a half-space with a free surface, that is, to the solutions computed by Steketee [1958a] and Okada [1985] following an alternative approach.