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The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc.
The description of the problem is indeed extremely complex in its generality.
In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations.
Under suitable scales we study the ``variational limit'' (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves.