Abstract. 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 formwhere β e represents the elastic part of the macroscopic strain, and Curl β e represents the geometrically necessary dislocation density. The plastic energy density ϕ 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 aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework. © 2014 Springer-Verlag Berlin Heidelberg

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We prove that the classical line-tension approximation for dislocations in crystals, that is, the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ -limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length ψ(b, t), which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length ψ 0 (b, t) obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its H 1 -elliptic envelope.

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