We consider systems of $n$ 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 $n\to\infty$.
The main aim of the paper is to study the convergence of the evolution of the
empirical measure as $n\to\infty$. 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 $n\to\infty$.