Investigation of Two-Dimensional Gravitational Stresses in Anisotropic Media Based on the Sherman-Type Integral Equations

We construct regularized Sherman-type integral equations for the anisotropic plane problem of the theory of elasticity. An integral representation of the general solution is obtained for a plane with holes in terms of the complex Lekhnitskii potentials with the help of the Cauchy theorem and, for a half plane, with additional application of the Green solutions. The properties of the constructed integral equations are investigated and their eigensolutions are found. On the basis of the Sherman approach, we introduce regularizing components, which enable us to find single-valued solution by numerical methods. By using the developed approach, we determine the elastic stresses formed near mine workings in rock massifs with anisotropic mechanical characteristics in the presence of gravitational forces. We perform the investigation of stresses near cylindrical cavities with circular and oval cross sections in isotropic materials and in aleurolite rock masses. The mutual influence of the cavities on the distribution of stresses is analyzed.