Analytic solution for the stress field in an orthotropic disc under diametral parabolic pressure

Abstract

A full-field analytic solution for the stress field developed in a circular disc made of an orthotropic material is obtained assuming that the disc is under diametral compression. The solution is achieved with the aid of Lekhnitskii's complex potentials technique for rectilinear anisotropic materials. The general formulae obtained for an orthotropic disc are particularized for a transversely isotropic one, since most rocks are described, in practice, as transversely isotropic materials. The innovation of the solution introduced, besides its applicability to both orthotropic and transverse isotropic materials, is the realistic simulation of the loading scheme at the disc's boundary. Indeed, the load applied is simulated by a parabolic distribution of radial stresses rather than by uniform pressure or diametral point forces. The solution is validated by considering special limiting cases both for the material and the loading scheme. The distribution of stresses along strategic loci is then considered, enlightening critical issues related to the severity of the stress field developed in the disc. Special attention is paid to the stress components at the disc's center, in an effort to assess the applicability of the Brazilian-disc test in determining the tensile fracture strength in case of materials for which the assumption of isotropy is not satisfactory. Before the development of the theoretical model proposed, a short chronological survey of the relative literature is presented, enlightening the path, full of thorns and difficulties that the pioneers of the relevant research have carved and travelled, in order to demonstrate the difficulties of the project and to pay tribute to their priceless contributions.