Remarks on homogenization and $3D$-$2D$ dimension reduction of unbounded energies on thin films

Omar Anza Hafsa; Jean-Philippe Mandallena

doi:10.5802/crmath.454

Équations aux dérivées partielles, Mécanique

Remarks on homogenization and

3 D

-

2 D

dimension reduction of unbounded energies on thin films

Omar Anza Hafsa ¹ ; Jean-Philippe Mandallena ¹

¹ Université de Nimes, Laboratoire MIPA, Site des Carmes, Place Gabriel Péri, 30021 Nîmes, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 903-910.

Résumé

We study periodic homogenization and $3 D$ - $2 D$ 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.

Reçu le : 2022-02-12
Accepté le : 2022-12-12
Publié le : 2023-07-18

DOI : 10.5802/crmath.454

Affiliations des auteurs :

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

@article{CRMATH_2023__361_G5_903_0,
     author = {Omar Anza Hafsa and Jean-Philippe Mandallena},
     title = {Remarks on homogenization and $3D$-$2D$ dimension reduction of unbounded energies on thin films},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {903--910},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.454},
     language = {en},
}

TY  - JOUR
AU  - Omar Anza Hafsa
AU  - Jean-Philippe Mandallena
TI  - Remarks on homogenization and $3D$-$2D$ dimension reduction of unbounded energies on thin films
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 903
EP  - 910
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.454
LA  - en
ID  - CRMATH_2023__361_G5_903_0
ER  -

%0 Journal Article
%A Omar Anza Hafsa
%A Jean-Philippe Mandallena
%T Remarks on homogenization and $3D$-$2D$ dimension reduction of unbounded energies on thin films
%J Comptes Rendus. Mathématique
%D 2023
%P 903-910
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.454
%G en
%F CRMATH_2023__361_G5_903_0

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/

Bibliographie
Cité par

[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 “ $det \nabla u \neq 0$ ”, 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 “ $det \nabla u > 0$ ”, 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

[7] Gabriele Anzellotti; Sisto Baldo; Danilo Percivale Dimension reduction in variational problems, asymptotic development in $Γ$ -convergence and thin structures in elasticity, Asymptotic Anal., Volume 9 (1994) no. 1, pp. 61-100 | DOI | Zbl

[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

[12] Gianni Dal Maso An introduction to $Γ$ -convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, 1993 | DOI

[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

[14] Hervé Le Dret; Annie Raoult The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., Volume 74 (1995) no. 6, pp. 549-578 | Zbl

[15] Danilo Percivale The variational method for tensile structures (1991) (Preprint 16, Dipartimento di Matematica Politecnico di Torino)

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

Cité par Sources :

Commentaires - Politique