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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 under study, 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-limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form