Abstract : This work deals with the modeling, the analysis and the numerical analysis of the dislocation dynamics and with the very strong links which exists with mean curvature type motion. Dislocations are linear defects which move in crystals when those are subjected to exterior stress. More precisely, the dynamics of a dislocation line is described by an eikonal equation where the speed depends in a nonlocal way on the whole line. In the modeling, it is also possible to add a mean curvature term.
The first part of this work is devoted to the study of the qualitative properties of dynamics of a dislocation line (existence, uniqueness, asymptotic behaviour...). This study relies essentially on the theory of viscosity solutions. We also propose several numerical scheme for this dynamics and we show their convergence as well as error estimates.
In a second part, we establish the link between the dynamics of a finite number of dislocations and the dynamics of dislocation density by showing homogenization results. We also study, in a theoretical and numerical way, a model for the dynamics of dislocation density.