Steady-state shear-bands in thermo-plasticity—I. Vanishing yield stress

Abstract

This paper concerns the construction and stability properties of steady-state solutions of a system of partial differential equations that model simple shearing of a slab of thermo-plastic material. The class of constitutive laws that give rise to a variational formulation of the steady-state problem is identified and a phase-plane argument is used to construct time-independent solutions that may be interpreted as steady-state shear-bands. Our variational framework captures several commonly adopted constitutive laws. Techniques from bifurcation theory for variational problems arc applied to classify stable and unstable solutions merely in terms of the shape of the solution branch in the distinguished bifurcation diagram that arises when average strain-rate is plotted against shearing force. There are two novel features to our approach. First, the two problems in which loading is imposed by either stress boundary conditions or velocity boundary conditions are treated by one analysis and the differing stability properties of solutions are explained naturally. Second, the stability analysis is based upon a symmetric eigenvalue problem arising from the appropriate second variation. The link with dynamic behavior is made through a Lyapunov functional and the linearized dynamics are not considered directly. Provided the proper existence theorems for the time-dependent problem can be proven or are assumed, our Lyapunov approach yields the appropriate nonlinear dynamic stability properties of steady-state solutions. In this paper we shall consider the case in which vanishing strain-rate implies zero stress, i.e. there is no residual or yield stress present in our model, but our analysis can be extended to encompass constitutive laws modelling nonzero yield stress.