Single-point blowup for nonlocal parabolic problems

Abstract

The formation of shear bands in materials has important implications to a variety of physical processes. These bands are observed in very thin zones and are generally regarded as a precursor to material failure. Shear band formation is caused by the heat generated in regions with highest strain rate. With insufficient time for diffusion of this heat, a localized thermal softening of the material occurs which enhances plastic flow in a thin zone. This adiabatic strain localization can be modelled as nonlinear thermally-activated reaction–diffusion equations. This leads to a class of nonlocal parabolic problems and their associated time-independent steady-state counterparts. A discussion of the models and an analysis of the problems will be presented.