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

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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
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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
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Cité par 4 documents. Sources : zbMATH

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