Multiscale problems in crystal plasticity

The macroscopic plasticity of ductile crystals is the net result of the collective motion of large numbers of crystal lattice defects, most notably, glissile dislocations, with structures forming at multiple length scales. These include the scale of the lattice, where dislocations may be regarded as discrete topological defects: the scale of the mean distance between dislocations, where the dynamics of the dislocation ensemble is of primary interest: and the sub-grain scale, where dislocations form characteristic patterns. The development of mathematical links between the behaviors at all scales, and the characterization of the effective macroscopic behavior of ductile crystals, remains a central and long-standing problem in physical metallurgy. Tools of the calculus of variations such as relaxation and Gamma convergence prove powerful and convenient in forging those links. I plan to review a number of mathematical problems that arise in that context, and some recent results pertaining to those problems, including: the formulation of a geometrical mechanics of discrete dislocations and the passage to the continuum: the effective energetics and dynamics of dislocation ensembles: and the relaxation and optimal scaling properties of single-crystals.