Comptes Rendus
Partial Differential Equations/Mathematical Problems in Mechanics
Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 231-236.

In this Note we are interested in the dynamics of dislocation densities in a material submitted to a time periodic stress. The dislocation densities solve a set of two coupled first order equations of Burgers' type. Our main aim is to give a description of the long time behaviour of those densities. By an homogenization procedure in the framework of viscosity solutions, we obtain that at the limit, the dislocation densities fulfills a single diffusion equation.

Dans cette Note, on s'intéresse à la dynamique de densités de dislocations dans un matériau soumis à un cisaillement périodique en temps. Ces densités sont solutions de deux équations couplées du premier ordre de type Burgers. Notre but est de décrire le comportement en temps long de ces densités. Nous développons une technique d'homogénéisation dans le cadre des solutions de viscosités, qui permet d'établir qu'à la limite les densités de dislocations sont solutions d'une seule équation de diffusion quasi-linéaire.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.01.006

Ariela Briani 1, 2; Régis Monneau 3

1 Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
2 ENSTA, 32, boulevard Victor, 75739 Paris cedex 15, France
3 CERMICS, Paris Est-ENPC, 6 and 8, avenue Blaise-Pascal, cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France
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Ariela Briani; Régis Monneau. Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 231-236. doi : 10.1016/j.crma.2009.01.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.006/

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[8] R. Monneau, P.E. Souganidis, Infinite Laplacian diffusion equations by stochastic time homogenization of a coupled system of first order equations, in preparation

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