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Nonconvex energy minimization and dislocation structures in ductile single crystals. (English) Zbl 0964.74012
From the summary: Plastically deformed crystals are often observed to develop intricate dislocation patterns such as the labyrinth, mosaic, fence and carpet structures. In this paper, we give an energetic interpretation of such dislocation structures with the aid of direct methods of the calculus of variations. We formulate the theory in terms of deformation fields and of the dislocations as manifestations of the incompability 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 addition, we show that a characteristic length scale can be built into the theory by taking into account the self-energy of dislocations. The extended theory leads to scaling laws which are in good qualitative and quantitative argreement with observations.

MSC:
74E15 Crystalline structure
74G65 Energy minimization in equilibrium problems in solid mechanics
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
82D25 Statistical mechanical studies of crystals
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