Elastic-Plastic Plane-Strain Solutions With Separable Stress Fields

Abstract

A stress field σij (χ, t) depending on position χ and time t will be called separable if the time-dependence enters only through a scalar multiplier; i. e., if σij(χ, t)=s(χ, t)σ̂ij(χ). It is shown here that elastic-plastic plane-strain solutions in the infinitesimal (flow) theory of plasticity satisfying Tresca’s yield criterion and associated flow rule with linear isotropic hardening, based on either equivalent plastic strain or total plastic work, can occur with separable stress fields only in the following instances: (a) solutions with uniaxial stress fields, as in bending, (b) solutions with stress fields such that the entire domain changes from elastic to plastic at the same time, and (c) solutions with stress fields for which the elastic-plastic boundary coincides with a principal shear stress trajectory. Whether or not a plasticity solution has a separable stress field can be determined a priori by examining the corresponding elasticity solution.