Analytical Solutions for Nonlinear Strain-Gradient Softening and Localization

Abstract

Strain softening, no matter how small, is usually accompanied by localization of the softening deformation field. Inside the zone of localization, a decrease in stress is accompanied by an increase in strain; outside the zone the strain decreases. One way to describe this phenomenon is to use a nonlocal theory of plasticity in which the yield stress depends on an invariant of the gradient of plastic strain. In this paper analytical solutions to two problems are obtained for the case where the functional dependence on the invariant is quadratic. The solutions are interpreted and correlations are made between predictions of load softening and the size of the localization zone based on material parameters and experimental data. Consistency with thermodynamical restrictions is shown.