  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 \cite{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 \cite{patsan}. The main
novelty here is that we allow dislocations to have different orientation and so
we have to deal with collisions of them.
