Perturbation of the stress-strain state of an elastic half-space by the spherical inhomogeneity of elastic properties under shear in horizontal plane taking account of gravitational forces

The present paper is devoted to the solution of the problem related to the perturbation of the stress-strain state of an elastic isotropic half-space by the spherical inhomogeneity of elastic properties for the case of only gravity and the case of an additional superimposed pure shear on the half-space. The analytical solution is obtained for the displacement and volume strain components, expressed in terms of the potentials and quasipotentials of the sphere, for the case of only gravity forces acting on the half-spaces and the case of superposition of gravity and pure shear. Using the example of a specific seismic event, it was shown that the contribution of gravitational forces to the perturbation of the stress-strain state of a half-space by the spherical inhomogeneity of elastic properties (the distribution with respect to the volume deformation) is not small. Using the obtained analytical solution for the case of the action of gravity only, it is shown that the volume strain distribution is axisymmetric with respect to the axis. A spherical inhomogeneity of the shear modulus in an isotropic elastic half-space forms a zone of a relative compression located along the strike of the inhomogeneity, and zones of relative stretching above and below the inhomogeneity. A distinctive feature of the zones of relative stretching is their limiting size, while the size of the zone of a relative compression increases when the volume strain decreases. In this case, between these planes, the solution in the neighboring octants differs only in sign. When adding a term associated with the gravitational forces in the solution, the symmetry is not violated but the equality in magnitude of the volume deformations in neighboring octants is violated. Dimensions and geometry of isosurfaces of volumetric deformation varies significantly when gravity is taken into account. Depending on the direction of the axis of the maximum extension of the half-space with the inhomogeneity, the size of the zones of the relative compression increases and the corresponding reduction in the size of the zones of the relative stretching changes with the shape of both.