  We consider a system of parallel straight edge dislocations and we analyse
its asymptotic behaviour in the limit of many dislocations. The dislocations
are represented by points in a plane, and they are arranged in vertical walls;
each wall is free to move in the horizontal direction. The system is described
by a discrete energy depending on the one-dimensional horizontal positions of
the n walls; the energy contains contributions from repulsive pairwise
interactions between all walls, a global shear stress forcing the walls to the
left, and a pinned wall at x=0 that prevents the walls from leaving through the
left boundary.
  We study the behaviour of the energy as the number n of walls tends to
infinity, and characterise this behaviour in terms of Gamma-convergence. There
are five different cases, depending on the asymptotic behaviour of a single
dimensionless parameter. As a consequence we obtain characterisations of the
limiting behaviour of stationary states in each of these five regimes.
  The results shed new light on the open problem of upscaling large numbers of
dislocations. We show how various existing upscaled models arise as special
cases of the theorems of this paper. The wide variety of behaviour suggests
that upscaled models should incorporate more information than just dislocation
densities. This additional information is encoded in the limit of the
dimensionless parameter.
