Analytical solutions for crack-tip sectors in perfectly plastic Mises materials under mixed in-plane and out-of-plane shear loading conditions

In this paper, analytical solutions for asymptotic crack-tip plastic sectors in perfectly plastic Mises materials are derived under mixed in-plane and out-of-plane shear loading conditions. Plastic strains in crack-tip plastic sectors are considered to be singular and non-singular. Sectors with singular plastic strains have the solution of centered fan type, and sectors with non-singular plastic strains have the solution of either centered fan or constant stress type. The requirement of stress continuity along the border between a constant stress and a centered fan sectors is then discussed. Discontinuities of the normal and out-of-plane shear stresses in the radial direction between two constant stress sectors are assumed in assembling the crack-tip fields under mixed mode II/III and I/III conditions. Crack-tip fields under mixed mode II/III and I/III conditions with small contributions of mode III are then presented to show the existence of asymptotic crack-tip fields for perfectly plastic materials under mixed in-plane and out-of-plane shear loading conditions. The trends of the angular variations of the mode III stresses under the mixed mode II/III and I/III conditions are generally in agreement with those of the available asymptotic and finite element analyses for low strain hardening materials.