Comptes Rendus
Mathematical Problems in Mechanics
Some relaxation results for functionals depending on constrained strain and chemical composition
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 337-342.

We prove some relaxation results in the spirit of Anza Hafsa and Mandallena for integral functionals arising in the study of coherent thermochemical equilibria for multiphase solids. The energy density exhibits an explicit dependence on the deformation gradient and on a vector field representing the chemical composition. The deformation gradient satisfies a determinant type constraint and the chemical composition a constraint on the modulus.

On prouve quelques résultats de relaxation dans le même esprit que Anza Hafsa et Mandallena pour des fonctionnelles intégrales provenant de l'étude de l'équilibre thermochimique pour les solides multiphases. La densité d'énergie considérée dépend du gradient de la déformation ainsi que d'un champ de vecteurs représentant la composition chimique du solide. Le gradient de déformation satisfait une contrainte sur son déterminant et la composition chimique une contrainte sur son module.

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DOI: 10.1016/j.crma.2009.01.024
Elvira Zappale 1; Hamdi Zorgati 2

1 Università degli Studi di Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italy
2 Faculté des Sciences de Tunis, Campus Universitaire, 2092, Tunis, Tunisia
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Elvira Zappale; Hamdi Zorgati. Some relaxation results for functionals depending on constrained strain and chemical composition. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 337-342. doi : 10.1016/j.crma.2009.01.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.024/

[1] O. Anza Hafsa; J.-P. Mandallena Relaxation theorems in nonlinear elasticity, Ann. Inst. H. Poincaré, Volume 25 (2008), pp. 135-148

[2] B. Dacorogna Direct Methods in the Calculus of Variations, Springer, Berlin, 2008

[3] I. Ekeland; R. Temam Convex Analysis and Variational Problems, North-Holland Publishing Company, 1976

[4] I. Fonseca The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures Appl., Volume 67 (1988), pp. 175-195

[5] I. Fonseca; D. Kinderlehrer; P. Pedregal Energy functionals depending on elastic strain and chemical composition, Calc. Var. Partial Differential Equations, Volume 2 (1994), pp. 283-313

[6] H. Le Dret; A. Raoult Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results, Arch. Rational Mech. Anal., Volume 154 (2000), pp. 101-134

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