Bifurcation into shear bands on the Bishop and Hill polyhedron. Part I: General analysis

Abstract

The present paper is the first of a series of three papers devoted to the micromechanical conditions which render possible the appearance of shear bands in crystalline materials. The phenomenon is analysed as a bifurcation from an initially homogeneous mode of deformation. Following a previous work by Hill and Hutchinson, the criterion of bifurcation is seen as the compatibility between equilibrium, the particular form of the shear velocity field and the state of the material, expressed by its rate constitutive equation. The analysis is restricted to the case of rigid-plastic crystals with uniform strain hardening whose flow surface is the Bishop and Hill polyhedron. The paper discusses the form of the criterion according to the state of deviatoric stress on the yield surface, which determines the various geometries of the slip and the form of the rate law of behaviour. It shows that when only two or three independent slip systems are available, only coplanar and codirectional slip systems currently originate shear banding. With a higher number of slip systems, the conditions required for the bifurcation are different, as will be studied in the subsequent papers.