A crack at the interface between an elastic-perfectly plastic solid and a rigid substrate

A crack at the interface between an incompressible, elastic-perfectly plastic solid and a rigid substrate is studied for the stationary and quasi-statically growing cases under the condition of plane strain. A one-parameter family of asymptotic near-tip stress fields is obtained for the case of a stationary crack and compared with the finite element results given by Shih and Asaro for a crack at the interface of a power-law hardening material and a rigid substrate. Shih and Asaro's finite element results for a deformable material of low hardening strongly suggest this one-parameter family of stress distributions as appropriate for an interface crack between an elastic-perfectly plastic solid and a rigid substrate. The one-parameter family of solutions presented here for an interface crack is analogous to Shih's family of solutions at a mixed mode homogeneous crack tip with mixities that are arbitrary as far as the asymptotic analysis is concerned, but determinate from a full-field analysis. A complete asymptotic near-tip stress and deformation field based on the Prandtl-Reuss flow rule and the Huber-Mises yield criterion is obtained for a crack growing quasi-statically along the interface of an incompressible, elastic-perfectly plastic solid and a rigid substrate. Similar to a mode II growing plane stress or strain crack in a homogeneous medium, the near-tip field consists of three different types of angular sectors: a "centered fan" plastically deforming sector near the interface, a "constant stress" plastically deforming sector near the crack free surface and an elastically deforming sector in between. The velocity components are fully continuous, and the strains are found to be singular like (ln r)² as r → 0, where r is distance from the crack tip.