Stress-strain state of a cracked elastic wedge under anti-plane deformation with mixed boundary conditions on its faces

Abstract

A problem about the stress-strain state of an elastic wedge with an arbitrary angel of opening, when on its bisector there is a system of a finite number collinear cracks, is studied by the means of the theory of elasticity. The anti-symmetric mixed boundary conditions given on both wedge-faces, together with the forces applied to the cracks' surfaces are provoking the anti-plane deformation of the wedge. The displacement components are given for the same group of nonintersecting intervals on each wedge-face and the stress components are given on the rest of the faces. The problem is formulated as a known mixed boundary problem of the theory of harmonic functions for a half-wedge because of the wedge symmetry relative to its bisector. The solution of this mixed boundary problem is derived in the closed form by using the Mellin integral transformation in combination with the methods of singular integral equations. Based on this the density of displacements' dislocations on the cracks' surfaces, the stress intensity factors, the stresses on those intervals on the wedge-faces, where the displacements are given, and other characteristics of the investigating problem are determined by explicit analytical formulas. Particular cases are discussed as well.