Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics

Cyril Imbert; Régis Monneau; Elisabeth Rouy

hal-00080397, version 2

Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics

Cyril Imbert () ¹, Régis Monneau ², Elisabeth Rouy ³

Résumé : This paper is concerned with a result of homogenization of a non-local first order Hamilton-Jacobi equations describing the dislocations dynamics. Our model for the interaction between dislocations involve both an integro-differential operator and a (local) Hamiltonian depending periodicly on $u/\eps$. The first two authors studied in a previous work homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator; the nonlinearity keeps memory of the short range interaction, while the Lévy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane.

1 : Institut de Mathématiques et de Modélisation de Montpellier (I3M)
CNRS : UMR5149 – Université Montpellier II - Sciences et techniques
2 : Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
INRIA – Ecole des Ponts ParisTech
3 : École Centrale de Lyon (ECL)
Ministère de l'Enseignement Supérieur et de la Recherche Scientifique

Collaboration : ACI JC 1025

hal-00080397, version 2
http://hal.archives-ouvertes.fr/hal-00080397
oai:hal.archives-ouvertes.fr:hal-00080397
Contributeur : Cyril Imbert <>
Soumis le : Vendredi 16 Février 2007, 14:13:11
Dernière modification le : Vendredi 16 Février 2007, 15:04:06

Voir la fiche détaillée

Documents associés

PDF :

Exporter

Bibtex EndNote TEI RefWorks

%% hal-00080397, version 2
%% http://hal.archives-ouvertes.fr/hal-00080397
@unpublished{imbert:hal-00080397,
    hal_id = {hal-00080397},
    url = {http://hal.archives-ouvertes.fr/hal-00080397},
    title = {{Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics}},
    author = {Imbert, Cyril and Monneau, R{\'e}gis and Rouy, Elisabeth},
    abstract = {{This paper is concerned with a result of homogenization of a non-local first order Hamilton-Jacobi equations describing the dislocations dynamics. Our model for the interaction between dislocations involve both an integro-differential operator and a (local) Hamiltonian depending periodicly on $u/\eps$. The first two authors studied in a previous work homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order L{\'e}vy operator; the nonlinearity keeps memory of the short range interaction, while the L{\'e}vy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane.}},
    keywords = {periodic homogenization, Hamilton-Jacobi equations, integro-differential operators, dislocations dynamics, non-periodic approximate correctors},
    language = {Anglais},
    affiliation = {Institut de Math{\'e}matiques et de Mod{\'e}lisation de Montpellier - I3M , Centre d'Enseignement et de Recherche en Math{\'e}matiques, Informatique et Calcul Scientifique - CERMICS , {\'E}cole Centrale de Lyon - ECL},
    note = {29 pages ACI JC 1025 },
    collaboration = {ACI JC 1025 },
    pdf = {http://hal.archives-ouvertes.fr/hal-00080397/PDF/homog-dislo.pdf},
}