On the theory of perfectly plastic anti-plane straining

A general formulation of two-dimensional elastic-perfectly-plastic anti-plane straining is presented for materials with arbitrary anisotropic convex yield surfaces. Stress and strain distributions in plastic regions adjoining portions of the boundary are obtained directly in terms of the yield surface geometry. When specialized to the classical torsion problem, results lead directly to a generalization of the well-known plastic roof construction for limit loads. Examples of the determination of fully plastic stress distributions and corresponding limit torques are given for circular and rectangular shafts with various yield conditions. Another specialization is made to the contained plastic deformation created by longitudinal shearing of a body containing a sharp edge notch. Here the determination of the elastic-plastic boundary and strain distribution is reduced to a potential theory problem for a region in the stress plane bounded by straight line segments and a portion of the yield surface, and a membrane analogy is presented which allows effective visualization of the solution. A solution valid for small scale yielding near a crack is given in terms of a conformal transformation of the yield surface to a unit circle, and some specific examples are worked. Particular attention is given to single crystal type yield surfaces made up of straight line segments corresponding to discrete slip planes.