Nonconvex energy minimization and dislocation structures in ductile singlecrystals

Plastically deformed crystals are often observed to develop intricate dislo cation patterns such as the labyrinth, mosaic, fence and carpet structures. In this paper, such dislocation structures are given an energetic interpre tation with the aid of direct methods of the calculus of variations. We for mulate the theory in terms of deformation fields and regard the dislocation s as manifestations of the incompatibility of the plastic deformation gradi ent held. Within this framework, we show that the incremental displacements of inelastic solids follow as minimizers of a suitably defined pseudoelast ic energy function. In crystals exhibiting latent hardening, the energy fun ction is nonconvex and has wells corresponding to single-slip deformations. This favors microstructures consisting locally of single slip. Deformation microstructures constructed in accordance with this prescription are shown to be in correspondence with several commonly observed dislocation structu res. Finally, we show that a characteristic length scale can be built into the theory by taking into account the self energy of the dislocations. The extended theory leads to scaling laws which appear to be in good qualitativ e and quantitative agreement with observation. (C) 1999 Elsevier Science Lt d. All rights reserved.