Comptes Rendus
Équations aux dérivées partielles, Mécanique
Remarks on homogenization and 3D-2D dimension reduction of unbounded energies on thin films
Omar Anza Hafsa1 ; Jean-Philippe Mandallena1 
1 Université de Nimes, Laboratoire MIPA, Site des Carmes, Place Gabriel Péri, 30021 Nîmes, France
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 903-910.

We study periodic homogenization and 3D-2D dimension reduction by Γ(π)-con-vergence of heterogeneous thin films whose the stored-energy densities have no polynomial growth. In particular, our results are consistent with one of the basic facts of nonlinear elasticity, namely the necessity of an infinite amount of energy to compress a finite volume of matter into zero volume. However, our results are not consistent with the noninterpenetration of the matter.

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DOI : 10.5802/crmath.454
Omar Anza Hafsa 1 ; Jean-Philippe Mandallena 1

1 Université de Nimes, Laboratoire MIPA, Site des Carmes, Place Gabriel Péri, 30021 Nîmes, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Comptes Rendus. Math\'ematique},
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     year = {2023},
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     language = {en},
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Omar Anza Hafsa; Jean-Philippe Mandallena. Remarks on homogenization and $3D$-$2D$ dimension reduction of unbounded energies on thin films. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 903-910. doi : 10.5802/crmath.454. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.454/

[1] Omar Anza Hafsa; Nicolas Clozeau; Jean-Philippe Mandallena Homogenization of nonconvex unbounded singular integrals, Ann. Math. Blaise Pascal, Volume 24 (2017) no. 2, pp. 135-193 | DOI | Numdam | Zbl

[2] Omar Anza Hafsa; Mohamed Lamine Leghmizi; Jean-Philippe Mandallena On a homogenization technique for singular integrals, Asymptotic Anal., Volume 74 (2011) no. 3-4, pp. 123-134 | DOI | Zbl

[3] Omar Anza Hafsa; Jean-Philippe Mandallena The nonlinear membrane energy: variational derivation under the constraint “det∇u≠0”, J. Math. Pures Appl., Volume 86 (2006) no. 2, pp. 100-115 | DOI | Zbl

[4] Omar Anza Hafsa; Jean-Philippe Mandallena The nonlinear membrane energy: variational derivation under the constraint “det∇u>0”, Bull. Sci. Math., Volume 132 (2008) no. 4, pp. 272-291 | DOI | Zbl

[5] Omar Anza Hafsa; Jean-Philippe Mandallena Relaxation et passage 3D-2D avec contraintes de type déterminant (2009) (https://arxiv.org/abs/0901.3688)

[6] Omar Anza Hafsa; Jean-Philippe Mandallena Relaxation and 3d-2d passage theorems in hyperelasticity, J. Convex Anal., Volume 19 (2012) no. 3, pp. 759-794 | Zbl

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[8] Hafedh Ben Belgacem Modélisation de structures minces en élasticité non linéaire, Ph. D. Thesis, Université Pierre et Marie Curie (1996)

[9] Hafedh Ben Belgacem Une méthode de Γ-convergence pour un modèle de membrane non linéaire, C. R. Acad. Sci. Paris, Volume 324 (1997) no. 7, pp. 845-849 | DOI | Zbl

[10] Hafedh Ben Belgacem Relaxation of singular functionals defined on Sobolev spaces, ESAIM, Control Optim. Calc. Var., Volume 5 (2000), pp. 71-85 | DOI | Numdam | Zbl

[11] Andrea Braides; Irene Fonseca; Gilles Francfort 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., Volume 49 (2000) no. 4, pp. 1367-1404 | Zbl

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[13] Hervé Le Dret; Annie Raoult Le modèle de membrane non linéaire comme limite variationnelle de l’élasticité non linéaire tridimensionnelle, C. R. Acad. Sci. Paris, Volume 317 (1993) no. 2, pp. 221-226 | Zbl

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[16] Y. C. Shu Heterogeneous thin films of martensitic materials, Arch. Ration. Mech. Anal., Volume 153 (2000) no. 1, pp. 39-90 | DOI | Zbl

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