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 .
In this work we discuss, from a variational viewpoint, the equilibrium problem for a finite number of Volterra dislocations in a plane domain. For a given set of singularities at fixed locations, we characterize elastic equilibrium as the limit of the minimizers of a family of energy functionals, obtained by a finite-core regularization of the elastic-energy functional. We give a sharpasymptotic estimate of the minimum energy as the core radius tends to zero, which allows one to eliminate this internal length scale from the problem. The energy content of a set of dislocations is fully characterized by the regular part of the asymptotic expansion, the so-called renormalized energy, which contains all information regarding self- and mutual interactions between the defects. Thus our result may be considered as the analogue for dislocations of the classical result of Bethuel, Brezis and Helein for Ginzburg--Landau vortices. We view the renormalized energy as the basic tool for the study of the discrete-to-con...