Boundary value problems for elastic half-planes posed in terms of stress and displacement orientations

Abstract

This study investigates solvability of boundary value problems of plane elasticity formulated in terms of principal directions of the stress tensor and the orientations of the displacement vector. The analysis of solvability is performed by using the following approach. Firstly, boundary values of the complex potentials are represented by the Cauchy-type integrals with unknown density. Then a system of singular integral equations is obtained by satisfying particular boundary conditions. This system is further reduced to the system of the Riemann boundary value problems for the determination of sectionally holomorphic functions. Solvability of the Riemann problems is investigated by calculating their indexes. This allows one to determine the number of linearly independent solutions and hence the number of arbitrary parameters entering into the general solution. Two novel formulations have been investigated for the case of elastic half-planes. In both cases the initial system of equations has been reduced to the form that allow for successive solution of its equations.