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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–Balogh in Acta Mater 47(13):3647–3654, 1999), a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of Patrizi and Sangsawang (Nonlinear Anal 202:112096, 2021). The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them. [ABSTRACT FROM AUTHOR]

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]