Intricate dislocation patterns (labyrinth, mosaic, fence, carpet) were interpreted energetically by using direct methods from the calculus of variations. The theory was formulated in terms of deformation fields, and the dislocations were regarded as being manifestations of an incompatibility of the plastic deformation gradient field. It was shown that the incremental displacements of inelastic solids served to minimize a suitably defined pseudo-elastic energy function. In crystals which exhibited latent hardening, the energy function was non-convex and had wells which corresponded to single-slip deformation. This favored microstructures which consisted locally of single slip. Deformation microstructures which were constructed in accord with this scheme were found to be in agreement with several commonly observed dislocation structures. It was shown that a characteristic length-scale could be built into the theory by taking account of the self-energy of the dislocations. The extended theory led to scaling laws which appeared to be in good qualitative and quantitative agreement with observations.

Non-Convex Energy Minimization and Dislocation Structures in Ductile Single Crystals. M.Ortiz, E.A.Repetto: Journal of the Mechanics and Physics of Solids, 1999, 47[2], 397-462