A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity

Cristinel Mardare

doi:10.1016/j.crma.2011.01.011

Mathematical Problems in Mechanics

A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity
[Une inégalité de Korn non linéaire et son relation à l'existence de minimiseurs en elasticité non linéaire]

Présenté par : Philippe G. Ciarlet

Cristinel Mardare ¹

¹ Université Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 229-232.

Résumé
Abstract

Nous établissons une inégalité de Korn non linéaire avec conditions au bord montrant que la distance dans $H^{1}$ entre deux applications de $Ω \subset R^{n}$ à $R^{n}$ préservant l'orientation est majorée, à une constante multiplicative près, par la distance dans $L^{2}$ entre leurs métriques. Cette inégalité est ensuite utilisée pour montrer l'existence d'un minimiseur unique de l'énergie totale d'un corps hyperélastique, sous les hypothèses que la norme de la densité des forces appliquées est suffisamment petite en norme $L^{p}$ , et la densité d'énergie de déformation est minorée par une fonction quadratique du tenseur de Green–Saint Venant.

We establish a nonlinear Korn inequality with boundary conditions showing that the $H^{1}$ -distance between two mappings from $Ω \subset R^{n}$ into $R^{n}$ preserving orientation is bounded, up to a multiplicative constant, by the $L^{2}$ -distance between their metrics. This inequality is then used to show the existence of a unique minimizer to the total energy of a hyperelastic body, under the assumptions that the $L^{p}$ -norm of the density of the applied forces is small enough and the stored energy function is bounded from below by a positive definite quadratic function of the Green–Saint Venant strain tensor.

Reçu le : 2010-11-12
Accepté le : 2011-01-07
Publié le : 2011-01-20

DOI : 10.1016/j.crma.2011.01.011

Affiliations des auteurs :

Cristinel Mardare ¹

¹ Université Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 75005 Paris, France

@article{CRMATH_2011__349_3-4_229_0,
     author = {Cristinel Mardare},
     title = {A nonlinear {Korn} inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {229--232},
     publisher = {Elsevier},
     volume = {349},
     number = {3-4},
     year = {2011},
     doi = {10.1016/j.crma.2011.01.011},
     language = {en},
}

TY  - JOUR
AU  - Cristinel Mardare
TI  - A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 229
EP  - 232
VL  - 349
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2011.01.011
LA  - en
ID  - CRMATH_2011__349_3-4_229_0
ER  -

%0 Journal Article
%A Cristinel Mardare
%T A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity
%J Comptes Rendus. Mathématique
%D 2011
%P 229-232
%V 349
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2011.01.011
%G en
%F CRMATH_2011__349_3-4_229_0

Cristinel Mardare. A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 229-232. doi : 10.1016/j.crma.2011.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.011/

Bibliographie
Cité par

[1] J. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337-403

[2] D. Blanchard, Personal communication.

[3] P.G. Ciarlet Mathematical Elasticity, vol. I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988

[4] P.G. Ciarlet; C. Mardare Continuity of a deformation in $H^{1}$ as a function of its Cauchy–Green tensor in $L^{1}$ , J. Nonlinear Sci., Volume 14 (2004), pp. 415-427

[5] P.G. Ciarlet; C. Mardare Existence theorems in intrinsic nonlinear elasticity, J. Math. Pures Appl., Volume 94 (2010), pp. 229-243

[6] G. Friesecke; R.D. James; S. Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., Volume 55 (2002), pp. 1461-1506

[7] P. Quintela-Estevez Critical points in the energy of hyperelastic materials, RAIRO Model. Math. Anal. Num., Volume 24 (1990), pp. 103-132

[8] K. Zhang Energy minimizers in nonlinear elastostatics and the implicit function theorem, Arch. Ration. Mech. Anal., Volume 114 (1991), pp. 95-117

Patrick E. Farrell; Luis F. Gatica; Bishnu P. Lamichhane; Ricardo Oyarzúa; Ricardo Ruiz-Baier Mixed Kirchhoff stress-displacement-pressure formulations for incompressible hyperelasticity, Computer Methods in Applied Mechanics and Engineering, Volume 374 (2021), p. 24 (Id/No 113562) | DOI:10.1016/j.cma.2020.113562 | Zbl:1506.74050
Alessandro Musesti A nonlinear Korn inequality based on the Green-Saint Venant strain tensor, Journal of Elasticity, Volume 126 (2017) no. 1, pp. 129-134 | DOI:10.1007/s10659-016-9582-5 | Zbl:1362.74011
Hardik Kabaria; Adrian J. Lew; Bernardo Cockburn A hybridizable discontinuous Galerkin formulation for non-linear elasticity, Computer Methods in Applied Mechanics and Engineering, Volume 283 (2015), pp. 303-329 | DOI:10.1016/j.cma.2014.08.012 | Zbl:1423.74895
Patrizio Neff; Dirk Pauly; Karl-Josef Witsch Poincaré meets Korn via Maxwell: extending Korn's first inequality to incompatible tensor fields, Journal of Differential Equations, Volume 258 (2015) no. 4, pp. 1267-1302 | DOI:10.1016/j.jde.2014.10.019 | Zbl:1310.35012

Cité par 4 documents. Sources : zbMATH

Commentaires - Politique