Single-crystal plasticity was considered for the limiting case of infinite latent hardening. This required that the crystal should deform via single slip at all points. The requirement introduced a non-convex constraint, and thus induced the formation of finely-scaled structures. Attention was restricted throughout to linearized kinematics and the deformation theory of plasticity. This was suitable for monotonic proportional loading and conferred, upon the plasticity boundary-value problem, a well-defined variational structure which was analogous to elasticity. A scale-invariant (local) problem was first studied. It was shown that, by developing microstructures in the form of sequential laminates of finite depth, crystals could overcome the single-slip constraint. That is, the macroscopic (relaxed) constitutive behavior was indistinguishable from multi-slip ideal plasticity. In a second step, dislocation line energies, and thus a length-scale, were included. The different regimes led to several possible types of microstructural pattern. Constructions were described which produced the various optimal scaling laws, and their relationship to experimentally found scalings such as the Hall-Petch law.

Dislocation Microstructures and the Effective Behavior of Single Crystals. S.Conti, M.Ortiz: Archive for Rational Mechanics and Analysis, 2005, 176[1], 103-47