[Le problème en déplacement pur en élasticité non linéaire tri-dimensionnelle : Formulation intrinsèque et théorèmes d'existence]
Dans cette Note, les équations de l'élasticité non linéaire tri-dimensionnelle correspondant au problème en déplacement pur sont ré-écrites, soit comme un problème aux limites, soit comme un problème de minimisation, l'inconnue étant dans les deux cas le tenseur des déformations de Cauchy–Green, au lieu de la déformation comme il est usuel. On montre ensuite que l'un et l'autre problème ont au moins une solution si les forces sont suffisamment petites et si la densité d'énergie satisfait certaines hypothèses naturelles. Le second problème constitue un exemple de problème de minimisation d'une fonctionnelle non coercive sur une variété de Banach.
In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold.
Publié le :
@article{CRMATH_2009__347_11-12_677_0,
author = {Philippe G. Ciarlet and Cristinel Mardare},
title = {The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems},
journal = {Comptes Rendus. Math\'ematique},
pages = {677--683},
publisher = {Elsevier},
volume = {347},
number = {11-12},
year = {2009},
doi = {10.1016/j.crma.2009.03.020},
language = {en},
}
TY - JOUR AU - Philippe G. Ciarlet AU - Cristinel Mardare TI - The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems JO - Comptes Rendus. Mathématique PY - 2009 SP - 677 EP - 683 VL - 347 IS - 11-12 PB - Elsevier DO - 10.1016/j.crma.2009.03.020 LA - en ID - CRMATH_2009__347_11-12_677_0 ER -
%0 Journal Article %A Philippe G. Ciarlet %A Cristinel Mardare %T The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems %J Comptes Rendus. Mathématique %D 2009 %P 677-683 %V 347 %N 11-12 %I Elsevier %R 10.1016/j.crma.2009.03.020 %G en %F CRMATH_2009__347_11-12_677_0
Philippe G. Ciarlet; Cristinel Mardare. The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 677-683. doi : 10.1016/j.crma.2009.03.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.020/
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