Problem on an inhomogeneity in a linearly elastic space or half-space

For a weakly contrasting anisotropic inhomogeneity in a linearly elastic homogeneous space or half-space, using the perturbation method, we obtain an approximate solution and estimate its accuracy. In the case of inhomogeneity of arbitrary contrast, we reduce the problem to a system of integral equations. In the general case, it is easy to compose the procedure for solving this problem approximately. In the special case of a homogeneous anisotropic ellipsoidal inhomogeneity in space, the strain state inside the inhomogeneity turns out to be homogeneous, and we thus obtain the exact solution of the problem.