Elasticity of thin dilatational disk in an isotropic half-space


In this article, the solution to the elasticity boundary-value problem for an infinitesimally thin dilatational disk (ITDD) embedded in an elastically isotropic half-space is presented. The plane of the disk is parallel to the free surface. To solve the boundary value problem, the method of virtual defects is used. The image (mirror) ITDD and radial Somigliana dislocation loops (RSDLs), distributed continuously over the free surface coaxially with the ITDD, are chosen as virtual defects. The ITDD displacements, strains, and stresses are represented in the form of the sums of the Lipschitz-Hankel integrals. It is shown that the elastic field of the ITDD disk is distorted near the free surface and possesses nonzero dilatation and hydrostatic stress. Numerical estimates of dilatation are given and its influence on some characteristics of semiconductors is briefly discussed.