*Published Paper*

**Inserted:** 25 jun 2008

**Last Updated:** 27 jul 2011

**Journal:** J. Eur. Math. Soc.

**Volume:** 12

**Number:** 5

**Pages:** 1231-1266

**Year:** 2010

**Abstract:**

In this paper, we deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations.

We restrict our analysis to the case of a cylindrical symmetry for the crystal in exam, so that the mathematical formulation will involve a two dimensional variational problem.

The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so called core region. We show that the Gamma limit as the core radius tends to zero and the number of dislocations tends to infinity of this energy (suitably rescaled), is given by the sum of an elastic term, depending on the strain, and a plastic term, depending on the dislocation density.

The plastic energy density is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively $1$-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.

**Keywords:**
relaxation, dislocations, strain gradient theories, plasticity

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