Strain localization in ductile single crystals

This paper is concerned with an analysis of strain localization in ductile crystals deforming by single slip. The plastic flow is modelled as rate-insensitive, and localization, viewed as a bifurcation from a homogeneous deformation mode to one which is concentrated in a narrow ‘shear band’, is found to be possible only when the plastic hardening modulus for the slip system has fallen to a certain critical value hcr, sensitive to the precise form of the constitutive law governing incremental shear. We develop the general form of this constitutive law, incorporating within it the possibility of deviations from the Schmid rule of a critical resolved shear stress, and we show that hcr may in fact be positive when there are deviations from the Schmid rule. It is suggested that micromechanical processes such as ‘cross-slip’ in crystals provide specific cases for which stresses other than the Schmid stress may influence plastic response and, further, there is an experimental association of localization with the onset of large amounts of cross-slip. Thus, we give the specific form of hcr for a constitutive model that corresponds to non-Schmid effects in cross-slip, and we develop a dislocation model of the process from which we estimate the magnitude of the parameters involved. The work supports the notion that localization can occur with positive strain-hardening, hcr > 0, and the often invoked notions of the attainment of an ideally-plastic or strain-softening state for localization may be unnecessary.