Homogenization of the Peierls-Nabarro model for dislocation dynamics and the Orowan's law

Régis Monneau (1, 2) , Stefania Patrizi

1 CERMES - Centre d'enseignement et de recherche en mécanique des sols
2 navier umr 8205 - Laboratoire Navier

This paper is concerned with a result of homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic L\'{e}vy operator of order 1 and a potential depending periodically on $u/\ep$. The limit equation is a non-local Hamilton-Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane. In dimension 1, we are able to characterize the Hamiltonian of the limit equation close to the origin, recovering a property known in physics as the Orowan's law.

Domaines: Equations aux dérivées partielles [math.AP]

Régis Monneau, Stefania Patrizi. Homogenization of the Peierls-Nabarro model for dislocation dynamics and the Orowan's law. 2010. ⟨hal-00504598⟩