Elastic Displacement in a Half-Space Under the Action of a Tensor Force. General Solution for the Half-Space with Point Forces

The elastic displacement in an isotropic elastic half-space with free surface is calculated for a point tensor force which may arise from the seismic moment of seismic sources concentrated at an inner point of the half-space. The starting point of the calculation is the decomposition of the displacement by means of the Helmholtz potentials and a simplified version of the Grodskii-Neuber-Papkovitch procedure. The calculations are carried out by using generalized Poisson equations and in-plane Fourier transforms, which are convenient for treating boundary conditions. As a general result we compute the displacement in the isotropic elastic half-space with free surface caused by point forces with arbitrary structure and orientation, localized either beneath the surface (generalized Mindlin problem) or on the surface (generalized Boussinesq-Cerruti problems). The inverse Fourier transforms are carried out by means of Sommerfeld-type integrals. For forces buried in the half-space explicit results are given for the surface displacement, which may exhibit finite values at the origin, or at distances on the surface of the order of the depth of the source. The problem presented here may be viewed as an addition to the well-known static problems of elastic equilibrium of a half-space under the action of concentrated loads. The application of the method to similar problems and another approach to the starting point of the general solution are discussed.