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

Nous établissons une inégalité de Korn non linéaire avec conditions au bord montrant que la distance dans H1 entre deux applications de ΩRn à Rn préservant l'orientation est majorée, à une constante multiplicative près, par la distance dans L2 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 Lp, 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 H1-distance between two mappings from ΩRn into Rn preserving orientation is bounded, up to a multiplicative constant, by the L2-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 Lp-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.

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DOI : 10.1016/j.crma.2011.01.011
Mardare, Cristinel 1

1 Université Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 75005 Paris, France
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Mardare, Cristinel. A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity. Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 229-232. doi : 10.1016/j.crma.2011.01.011. http://www.numdam.org/articles/10.1016/j.crma.2011.01.011/

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