Author(s)

A. N. Galybin & Sh. A. Mukhamediev

Abstract

This article addresses a new type of boundary condition in plane elastic boundary value problems. Principal directions are given on a contour separating interior and exterior domains; the stress vector is continuous across the contour. Solvability of this problem is investigated and the number of linearly independent solutions is determined. Some special cases in which the problem is underspecified have been reported. Keywords: plane elasticity, boundary value problems, principal directions, complex potentials. 1 Introduction Classical boundary value problems, BVP, of the plane elasticity require one of the following surface conditions to be known on the entire boundary of a domain (see Muskhelishvili [1]): (i) stress vector; (ii) displacement vector; or (iii) certain combinations of stress and displacement components (mixed problems). In these cases the BVP is well posed and possesses a unique solution. Galybin and Mukhamediev [2] and Galybin [3] considered different types of BVPs in which magnitudes of stresses, displacements or forces are not specified on the boundary. It has been shown that the BVPs of this type may have a finite number of solutions or be unsolvable. Solvability depends on the, so-called, index of singular integral equations (see, e.g., Gakhov [4]). It can be determined in every particular case from the analysis of principal directions (of the stress tensor), orientation of displacement or forces on the entire boundary of a considered domain.

Keywords

plane elasticity, boundary value problems, principal directions, complex potentials.