Scale effects in plasticity of random media: status and challenges

When the separation of scales in random media does not hold, the representative volume element (RVE) of classical continuum mechanics does not exist in the conventional sense, and various new approaches are needed. This subject is discussed here in the context of plasticity of random, microheterogeneous media. The first principal topic considered is that of hierarchies of mesoscale bounds, set up over a statistical volume element (SVE), for elastic-plastic-hardening microstructures; these bounds, with growing mesoscale, tend to converge to RVE responses. Following a formulation of the said hierarchies from variational principles and their illustration on two specific examples of power-law hardening materials, we turn to rigid-perfectly-plastic materials. The latter are illustrated by simulations in the setting of a planar random chessboard. The second principal topic is the analysis of spatially non-uniform response patterns of randomly heterogeneous plastic materials. We focus here on the geodesic properties of shear-band patterns, and then on the correlation of strain fields to the underlying microstructures. In the case of perfectly-plastic materials, shear-bands become slip-lines, but their spatial disorder is still present, and is described in ensemble sense by wedges of randomly scattered characteristics.