An elastic-plastic problem under conditions of antiplanar deformation

Abstract

By antiplanar deformation is meant the state of stress in an infinitely long cylinder subjected to the action of loading which is applied in the direction of the generators and which is constant along them. The elastic-plastic problem under the condition of antiplanar deformation has already been considered in the works of Trefftz [1], Hult and McClintock [2], Neuber [3]. Paper [1] gives an exact solution of the elastic-plastic problem on the antiplanar deformation of an angle section with right-angle opening, and also of the analogous problem for a region exterior to a circular hole. The elastic-plastic problem for a half-plane with a sharp notch has been solved for small values of the loading parameter in [2]. Neuber [3] considered a strip with two symmetric sharp notches, and, moreover, for an arbitrary single-valued relation between the stresses and the strains, the solution of the problem was reduced to a system of two ordinary differential equations and for a specially selected law the solution was obtained in closed form. Below, a treatment will be given of the solution in quadratures of the static elastic-plastic problem for the exterior of an arbitrary contour wholly enclosed by the plastic zone and loaded arbitrarily (Section 2); an exact solution of the problem for the exterior of a contour consisting of segments of straight and curved lines in the case when the straight sections are free of stresses and the parts of the curved arcs, which are arbitrarily loaded, are wholly contained in the plastic zone (Section 4). The solutions of the problems of Section 4 are based mainly on the solution of a certain nonlinear boundary value problem (Section 3). Throughout this article the Prandtl diagram has been taken as the relation between the stresses and the strains.