Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body

Marius Cocou; Gilles Scarella

doi:10.1016/j.crma.2003.12.013

Mathematical Problems in Mechanics

Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body
[Existence d'une solution à un problème dynamique de contact unilatéral pour un milieu viscoélastique fissuré]

Présenté par : Philippe G. Ciarlet

Marius Cocou ¹ ; Gilles Scarella ²

¹ Laboratoire de mécanique et d'acoustique CNRS, 31, chemin Joseph-Aiguier, 13402 Marseille cedex 20, et Université de Provence, UFR MIM, Marseille, France
² INRIA Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 341-346.

Résumé
Abstract

Dans ce travail, on s'intéresse à un problème dynamique de contact unilatéral avec frottement non local pour un milieu viscoélastique fissuré, suivant une loi de comportement de Kelvin–Voigt. L'existence d'une solution du problème est obtenue par une méthode de pénalisation. Le passage à la limite est justifié par plusieurs estimations et par quelques résultats de compacité.

In this paper we study a dynamic unilateral contact problem with friction for a cracked viscoelastic body. The viscoelastic model is characterized by Kelvin–Voigt's law and a nonlocal friction law is investigated here. The existence of a solution to the problem is obtained by using a penalty method. Several estimates are obtained on the solution to the penalized problem, which enable us to pass to the limit by using compactness results.

Reçu le : 2003-08-11
Accepté le : 2003-12-01
Publié le : 2004-01-24

DOI : 10.1016/j.crma.2003.12.013

Affiliations des auteurs :

Marius Cocou ¹ ; Gilles Scarella ²

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Marius Cocou; Gilles Scarella. Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body. Comptes Rendus. Mathématique, Volume 338 (2004) no. 4, pp. 341-346. doi : 10.1016/j.crma.2003.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.013/

Bibliographie
Cité par

[1] P. Boieri; F. Gastaldi; D. Kinderlehrer Existence, uniqueness, and regularity results for the two-body contact problem, Appl. Math. Optim., Volume 15 (1987), pp. 251-277

[2] M. Cocou; J.M. Ricaud Analysis of a class of implicit evolution inequalities associated to viscoelastic dynamic contact problems with friction, Int. J. Engrg. Sci., Volume 38 (2000), pp. 1535-1552

[3] M. Cocou Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Z. Angew. Math. Phys. (ZAMP), Volume 53 (2002), pp. 1099-1109

[4] J. Jarušek Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Math. J., Volume 46 (1996) no. 121, pp. 475-487

[5] J.U. Kim A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, Volume 14 (1989), pp. 1011-1026

[6] K.L. Kuttler; M. Shillor Dynamic bilateral contact with discontinuous friction coefficient, Nonlinear Anal., Volume 45 (2001), pp. 309-327

[7] G. Lebeau; M. Schatzman A wave problem in a half-space with a unilateral constraint at the boundary, J. Differential Equations, Volume 53 (1984), pp. 309-361

[8] J. Muñoz-Rivera; R. Racke Multidimensional contact problems in thermoelasticity, SIAM J. Appl. Math., Volume 58 (1998), pp. 1307-1337

[9] J.-C. Paumier; Y. Renard Surface perturbation of an elastodynamic contact problem with friction, European J. Appl. Math., Volume 14 (2003), pp. 465-483

[10] A. Petrov; M. Schatzman One-dimensional viscoelastodynamics with Signorini boundary conditions, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 983-988

[11] J.M. Ricaud; E. Pratt Analysis of a time discretisation for an implicit variational inequality modelling dynamic contact problems with friction, Math. Methods Appl. Sci., Volume 24 (2001), pp. 491-511

[12] J. Simon Compact sets in the space L^p(0,T;B), Ann. Mat. Pura Appl., Volume 146 (1987), pp. 65-96

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