[Homogénéisation en temps d'un système du premier ordre intervenant dans la modélisation de la dynamique de densités de dislocations]
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.
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.
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@article{CRMATH_2009__347_5-6_231_0,
author = {Briani, Ariela and Monneau, R\'egis},
title = {Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities},
journal = {Comptes Rendus. Math\'ematique},
pages = {231--236},
publisher = {Elsevier},
volume = {347},
number = {5-6},
year = {2009},
doi = {10.1016/j.crma.2009.01.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.01.006/}
}
TY - JOUR AU - Briani, Ariela AU - Monneau, Régis TI - Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities JO - Comptes Rendus. Mathématique PY - 2009 SP - 231 EP - 236 VL - 347 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.01.006/ DO - 10.1016/j.crma.2009.01.006 LA - en ID - CRMATH_2009__347_5-6_231_0 ER -
%0 Journal Article %A Briani, Ariela %A Monneau, Régis %T Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities %J Comptes Rendus. Mathématique %D 2009 %P 231-236 %V 347 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.01.006/ %R 10.1016/j.crma.2009.01.006 %G en %F CRMATH_2009__347_5-6_231_0
Briani, Ariela; Monneau, Régis. Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities. Comptes Rendus. Mathématique, Tome 347 (2009) no. 5-6, pp. 231-236. doi : 10.1016/j.crma.2009.01.006. http://www.numdam.org/articles/10.1016/j.crma.2009.01.006/
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