On the equivalence and differences of three plane-stress yield criteria for modelling orthotropic sheet metals with the tension-compression strength differential effect

Department of Mechanical Engineering, Southern Methodist University, Dallas, Texas, USA

^* Corresponding author: wtong@smu.edu

Abstract

The plane-stress yield surface of a sheet metal exhibiting tension-compression asymmetry may be modelled by a single polynomial yield stress function consisting of both even and odd order stress terms. The simplest such polynomial yield stress function would be the one made of quadratic and linear polynomials of stresses. Historically, Hill's 1948 homogeneous quadratic polynomial yield stress function with kinematic hardening is a more widely used formulation. Recently, a yield stress function consisting of a square root of Hill's 1948 homogeneous quadratic polynomial plus a linear polynomial also appears in the literature. This theoretical study examines their equivalence and differences. As the set of six material constants in each formulation is non-linearly but uniquely related, the equivalence of their yield conditions was established directly per the imposed strict positivity and convexity conditions. Differences in their associated flow rules and equivalent plastic stress-strain relationships were pointed out for the first time. Possible generalizations of these three formulations to non-quadratic polynomial ones with some challenges were discussed at the end.