Abstract
Plastically deformed crystals are often observed to develop intricate
dislocation patterns such as the labyrinth, mosaic, fence and carpet
structures. In this paper, such dislocation structures are given
an energetic interpretation with the aid of direct methods of the
calculus of variations. We formulate the theory in terms of deformation
fields and regard the dislocations as manifestations of the incompatibility
of the plastic deformation gradient field. Within this framework,
we show that the incremental displacements of inelastic solids follow
as minimizers of a suitably defined pseudoelastic energy function.
In crystals exhibiting latent hardening, the energy function 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 structures.
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
qualitative and quantitative agreement with observation.
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