Effective equations for localization and shear band formation

We develop a quantitative criterion determining the onset of localization and shear band formation at high strain-rate deformations of metals. We introduce an asymptotic procedure motivated by the theory of relaxation and the Chapman-Enskog expansion and derive an effective equation for the evolution of the strain rate, consisting of a second order nonlinear diffusion regularized by fourth order effects and with parameters determined by the degree of thermal softening, strain hardening, and strain-rate sensitivity. The nonlinear diffusion equation changes type across a threshold in the parameter space from forward parabolic to backward parabolic, what highlights the stable and unstable parameter regimes. The fourth order effects play a regularizing role in the unstable region of the parameter range.