On certain problems of deformation-induced material instabilities

Deformation-induced material instabilities may be of elastic (buckling, martensitic transformations) or plastic (necking/shear banding, dislocation patterning) type. In plasticity, the emergence of material instabilities is mainly associated with the properties of the underlying microstructure such as the motion, interaction and production/annihilation of dislocations: the carriers of plastic deformation. In any case, a common mathematical basis for describing the emergence and evolution of pattern-forming material instabilities in a large class of physical problems may be sought on non-monotonous equations of state, in conjunction with the introduction of suitable time and space derivatives in the state variables. This was an approach elaborated upon by Aifantis and co-workers in the early eighties for plastic instabilities. In about the same time, a similar and perhaps broader approach, based on a non-convex energy functional for various physical systems and the introduction of higher-order field variables to stabilize the behavior in the non-convex regime, was independently proposed by Berdichevsky and co-workers. This paper, written on the occasion of his 65th birthday, provides an updated discussion of dislocation patterning, shear banding and strain instabilities during epitaxial film growth.