Tracking of Shear Bands I. the One-dimensional Case

We develop a model for the dynamics of a fully developed shear band that allows eeective computation across several length scales. From a macroscopic point of view, a shear band is a discontinuity in tangential velocity that supports a shear stress. Numerical simulation of the full system of governing equations reveals that the internal structure of the band consists of a quasistatic core surrounded by a thermal layer. We show that the shear band can be modeled as a composite structure whose evolution is governed by an integral equation, coupled to the external ow through jump conditions. We establish the accuracy of the model equations by numerical experiments. 1. Introduction Metals that are subjected to large shear stresses develop highly localized regions of shear strain. These regions are called shear bands. As pointed out by Zener and Hollomon 10], shear bands form when thermal softening outweighs strain and strain rate hardening: plastic straining generates heat, which softens the metal, leading in turn to increased straining. Thus the strain is localized at points where the metal is weakest. Experimentally, it is found that shear bands are only a few tens of microns thick. Because shear bands are so thin, numerical simulation of ows in which they occur is very diicult. Our aim in this paper is to formulate the ow equations in a way that permits eeective computation of shear bands. We start from the conservative form of the ow equations for elasto-plastic ow in the Eulerian frame (Sec. 2). These conservation laws are supplemented by an equation of state, which determines the internal energy and stress, and by a ow rule, which speciies the plastic strain rate. Work-and strain-hardening are ignored here. For smooth solutions, the resulting set of equations are equivalent to those studied by several authors 1, 7, 6, 5]. In Sec. 3 we present numerical solutions of these equations. We use the numerical method of Walter 5]. The novel aspect of our work lies in the analysis of the numerical simulations, which brings out the solution features that form the basis of our model of shear bands. The basic idea of the model, developed in Sec. 4, is to treat the shear band as a boundary layer internal to the ow. Viewed on a large scale, the band appears as a discontinuity in the velocity, and it generates heat at a rate determined by the velocity jump …