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
@article{CRMATH_2009__347_5-6_231_0,
     author = {Ariela Briani and R\'egis Monneau},
     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},
}
TY  - JOUR
AU  - Ariela Briani
AU  - Régis Monneau
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
DO  - 10.1016/j.crma.2009.01.006
LA  - en
ID  - CRMATH_2009__347_5-6_231_0
ER  - 
%0 Journal Article
%A Ariela Briani
%A Régis Monneau
%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
%R 10.1016/j.crma.2009.01.006
%G en
%F CRMATH_2009__347_5-6_231_0
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/

[1] E.C. Aifantis; D. Walgraef Dislocation patterning as the result of dynamical instabilities, J. Appl. Phys., Volume 58 (1985), pp. 688-691

[2] M.G. Crandall; H. Ishii; P.L. Lions User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), Volume 27 (1992) no. 1, pp. 1-67

[3] A. El Hajj; N. Forcadel A convergent scheme for a non-local coupled system modelling dislocations densities dynamics, Math. Comp., Volume 77 (2008) no. 262, pp. 789-812

[4] L.C. Evans The perturbed test function-method for viscosity solutions of nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, Volume 111 (1989), pp. 359-375

[5] I. Groma; P. Balogh Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater., Volume 47 (1999) no. 13, pp. 3647-3654

[6] H. Ishii; S. Koike Viscosity solutions for monotone systems of second order elliptic pdes, Comm. Partial Differential Equations, Volume 16 (1991) no. 6, 7, pp. 1095-1128

[7] L.P. Kubin; C. Fressengeas; G. Ananthakrishna Collective behaviour of dislocations in plasticity (F.R.N. Nabarro; M.S. Duesbery, eds.), Dislocation of Solids, vol. 11, Elsevier Science B.V., 2002, pp. 101-192

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

Cited by Sources:

Comments - Policy


Articles of potential interest

Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions

Hassan Ibrahim; Mustapha Jazar; Régis Monneau

C. R. Math (2008)


The velocity diagram for traveling waves

Mohammad Al Haj; Régis Monneau

C. R. Math (2023)


Résolution en temps court d'une équation de Hamilton–Jacobi non locale décrivant la dynamique d'une dislocation

Olivier Alvarez; Philippe Hoch; Yann Le Bouar; ...

C. R. Math (2004)