[Une inégalité de Korn non linéaire et son relation à l'existence de minimiseurs en elasticité non linéaire]
Nous établissons une inégalité de Korn non linéaire avec conditions au bord montrant que la distance dans entre deux applications de à préservant l'orientation est majorée, à une constante multiplicative près, par la distance dans 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 , 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 -distance between two mappings from into preserving orientation is bounded, up to a multiplicative constant, by the -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 -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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@article{CRMATH_2011__349_3-4_229_0,
author = {Mardare, Cristinel},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.01.011/}
}
TY - JOUR AU - Mardare, Cristinel 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2011.01.011/ DO - 10.1016/j.crma.2011.01.011 LA - en ID - CRMATH_2011__349_3-4_229_0 ER -
%0 Journal Article %A Mardare, Cristinel %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 %U http://www.numdam.org/articles/10.1016/j.crma.2011.01.011/ %R 10.1016/j.crma.2011.01.011 %G en %F CRMATH_2011__349_3-4_229_0
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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