Restrictions on Quasi-Statically Moving Surfaces of Strong Discontinuity in Elastic-Plastic Solids

This paper educes the restrictions on a hypothesized quasi-statically propagating planar surface of strong discontinuity in an elastic-plastic solid. The derivation treats a solid under general three-dimensional conditions, employing a “small strain” formulation; the principal constitutive assumptions are that the elastic part of total strain be linearly related (although with arbitrary anisotropy) to the stress state, that the material possess a positive definite elastic strain energy function, and that inelastic deformation occur in accordance with the principle of maximum plastic resistance. It is proved that under these conditions all components of stress must be continuous across such a propagating surface. Further, the only components of strain which may suffer discontinuities across such a surface are shown to be the plastic components which do not deform elements in the plane of the surface, and these may be discontinuous only if the stress state at the moving surface meets certain specific conditions.