From the Peierls–Nabarro model to the equation of motion of the dislocation continuum.

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 well known equation called by Head (1972) "the equation of motion of the dislocation continuum". The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan's law which states that dislocations move at a velocity proportional to the effective stress. [ABSTRACT FROM AUTHOR]