Work hardening

Definition

Consider a plasticity model at a material point with yield function $f(\sigma, \kappa)$. Here $\sigma$ is the stress and $\kappa \ge 0$ is a hardening variable that measures accumulated plastic strain. A common choice is $\kappa(t) = \int_0^t \sqrt{\tfrac23 \dot\varepsilon^p : \dot\varepsilon^p} \, ds$. In isotropic hardening, $f(\sigma, \kappa) = \bar\sigma(\sigma) - \sigma_y(\kappa)$, where $\bar\sigma$ is an equivalent stress and $\sigma_y$ is the current yield stress. The material shows work hardening if $\sigma_y$ is increasing in $\kappa$, so that the elastic domain $\set{\sigma : f(\sigma, \kappa) < 0}$ grows as plastic deformation accumulates. More generally, it shows work hardening if continued plastic flow requires an increasing stress. That is, the plastic modulus $H$ in $d\bar\sigma = H \, d\kappa$ during plastic loading is positive.