Single-force point-source static fields: An exact solution for two elastic half-spaces

Static fields produced by point sources in elastic media are important for several disciplines and are of special interest in earth sciences, where distributions of displacements, deformations and stresses determined by sources inside the Earth are often needed: consider, for example, permanent deformations generated by earthquakes, or surface displacements produced by magma intrusions in volcanic areas, and consider, moreover, that extended sources can very frequently be treated by superposing fields associated with proper elementary point sources, since linear theory almost always holds in geophysical applications. This paper examines the problem of computing static fields caused by point sources consisting of single forces in an elastic medium comprising two half-spaces that are characterized by different elastic parameters and are separated by a plane interface. The problem is posed in the framework of Volterra's (1907) theory and is solved analytically in both half-spaces for the two canonic cases of a force normal and parallel to the interface. It is stressed that the solutions deduced here can be used as the basis for the calculation of fields produced by an arbitrary point source of geophysical interest: indeed, forces pointing in arbitrary directions can be dealth with through linear combinations of the fundamental solutions, whereas single-couple or double-couple point sources can be handled by means of appropriate differentiations and subsequent linear combinations of them. It is emphasized that the solution is provided in a closed analytical form. The solution formulas are discussed, paying special attention to the dependence of the resulting static fields on the elastic parameters of the model, namely the Poisson coefficients and the rigidity moduli of the two half-spaces. One interesting result concerns the limiting case when the rigidity modulus of the half-space not containing the point-source is equated to zero: although the solution in this half-space no longer makes sense, the solution in the half-space containing the source is identical to the solution for the case of a homogeneous half-space delimited by a free surface, that is a surface on which no external traction is acting, as is assumed in several geophysical applications for the surface of the Earth.