We consider systems of @math parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modelled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations, and prove several convergence results in the limit @math . The main aim of the paper is to study the convergence of the evolution of the empirical measure as @math . We consider rate-independent, quasi-static evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasi-static evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slip-plane-confined. Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large self- or core energy. We prove Gamma-convergence of this renormalized energy, and we construct joint recovery sequences for which both the energies and the modified distances converge. With this augmented Gamma-convergence we prove the convergence of the quasi-static evolutions as @math .
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 G-limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form @PARASPLIT E = fO (W(se) + f (Curl se)) dx, @PARASPLIT where e represents the elastic part of the macroscopic strain, and Curl se represents the geometrically necessary dislocation density. The plastic energy density f 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. This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ( > 0 ) representing the lattice spacing of the crystal, we remove a disc of radius ( ) around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ( 0 ) , in terms of Γ-convergence. We focus on the self energy regime of order ( 1 ) ; we show that configurations with logarithmic diverging energy converge, up to a subsequence, to a finite number of multiple dislocations and we compute the corresponding Γ-limit.