Criterion of Bifurcation for Plastic Forming

摘要

The localisation into shear bands of a purely plastic, rate insensitive f.c.c. crystal has been analysed by methods developed for the solids of the continuum mechanics and specialised here to a crystalline material. Work hardening has been assumed uniform : hence, the yield locus remains the Bishop & Hill polyhedron throughout the deformation path. The knowledge of the rate constitutive law at each of the vertices of the polyhedron allows to calculate analytically the ratio between the microscopic work hardening modulus h and the common critical resolved shear stress τ_c as a function of the coefficients of the normal to the plane proposed for bifurcation ; the most favoured ratios and the corresponding shear planes and directions have been tabulated. The cubic symmetry implies that the vertices can be divided into types and the preferred bifurcation ratios take only four values ranging from 0.31 to 1.07. An application to the case of plane strain compression is proposed which shows that for the usual rolling texture components, the conditions in which intra granular shear banding becomes possible differ little from one grain to the other in spite of the range of crystallographic orientations.