On an integral equation for axially-symmetric problems in the case of an elastic body containing an inclusion

Abstract

An approximate solution of the singular integral equation (SIE) which arises in spatial problems in the theory of elasticity with mixed conditions of one-sided detachment of inclusions under axially-symmetric torsion is considered. The singularity is taken into account using the exact solution of the equation which determines the conditions of the analogous detachment in a two-dimensional problem in the case of a sheet. It is proved that, subject to certain geometric constraints, the solution of the initial problem can be obtained by the method of successive approximations. The problem of the axially-symmetric torsion of a layer using a rigid circular disc embedded in this layer and fixed to it by one of its surfaces is treated as an example. The possiblity of the practical realization of this problem lies in the fact that a torsional moment which is applied to a rod which has been welded perpendicular to the centre of a disc can be transmitted through the disc to a layer and pierce a part of the layer.