Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model
Abstract
We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations groma, a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of patsan. The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them.