Single-crystal plasticity was considered for the limiting case of infinite latent hardening; which signified that the crystal had to deform in single-slip at all points. This requirement introduced a non-convex constraint, and thereby induced the formation of fine-scale structures. Attention was restricted throughout to linearized kinematics and the deformation theory of plasticity, which was appropriate for monotonic proportional loading and conferred on the boundary-value problem of plasticity 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 beat 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 hence a length-scale, were included in the model. Different regimes led to several possible types of microstructural patterns. Constructions were presented which reproduced the various optimum scaling laws, and the relationship to experimentally known scalings such as the Hall-Petch law, was considered.

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