Shear band post-bifurcation analysis in a planar double slip single crystal model

Abstract

Abstract A shear band post-bifurcation analysis for a double slip ductile single crystal is presented. The crystal geometry is idealized in terms of a planar double slip model. The crystal is assumed to be rigid-plastic and rate-independent. The crystallographic slip is governed by Schmid's law. The strain hardening is expressed by a latent hardening matrix. By analysing kinematics and statics of shear banding, conditions for shear band bifurcation and post-bifurcation are obtained. The obtained bifurcation condition is a necessary and sufficient condition. The post-bifurcation condition allows us to determine the magnitudes of the crystallographic slips inside the band as well as those outside the band (matrix). The results show that shear band can form in a hardening single crystal without any defect, that its formation and its development are characterized by a rigid matrix during shear banding, and that they depend strongly on crystal slip geometry and on strain hardening properties, ruled by the “geometrical” softening due to the lattice rotation accompanied with shear banding. By analysis of shear banding kinematics we find that, when the slip geometry is such that the two slip directions make an angle greater than 90°, the localized shearing has a limit value; however, in the inverse case, the localized shearing has kinematically no limit. The static conditions allow us to determine in the latter case the saturation of the shear band. It is shown that shear banding can be described by a scalar parameter C. C = 0 corresponds to the bifurcation. If C after the bifurcation, the plastic flow will entirely be localized inside the band, while the matrix will be unloaded. If C becomes positive during shear banding, the matrix will restart to be deformed. Consequently, the following development processes of a shear band are predicted: bifurcation, large shear strain localization and saturation. These predictions are in agreement with experimental observations of shear band propagation mechanism on a sample scale.