[Une approche lagrangienne de l'élasticité linéarisée intrinsèque]
On considère le problème en déplacement pur et le problème en traction pure de l'élasticité linéarisée tri-dimensionnelle. On montre que, dans chaque cas, l'approche intrinsèque conduit à un problème de minimisation quadratique avec des contraintes semblables à celles de Donati. Utilisant la condition inf–sup de Babuška–Brezzi, on montre ensuite que, dans chaque cas, le minimiseur du problème de minimisation avec contraintes trouvé dans une approche intrinsèque est le premier argument du point-selle d'un lagrangien approprié, ce qui fait que le second argument de ce point-selle est le multiplicateur de Lagrange associé aux contraintes correspondantes.
We consider the pure traction problem and the pure displacement problem of three-dimensional linearized elasticity. We show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations. Using the Babuška–Brezzi inf–sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints.
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@article{CRMATH_2010__348_9-10_587_0,
author = {Ciarlet, Philippe G. and Ciarlet, Patrick Jr. and Iosifescu, Oana and Sauter, Stefan and Zou, Jun},
title = {A {Lagrangian} approach to intrinsic linearized elasticity},
journal = {Comptes Rendus. Math\'ematique},
pages = {587--592},
publisher = {Elsevier},
volume = {348},
number = {9-10},
year = {2010},
doi = {10.1016/j.crma.2010.04.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.04.011/}
}
TY - JOUR AU - Ciarlet, Philippe G. AU - Ciarlet, Patrick Jr. AU - Iosifescu, Oana AU - Sauter, Stefan AU - Zou, Jun TI - A Lagrangian approach to intrinsic linearized elasticity JO - Comptes Rendus. Mathématique PY - 2010 SP - 587 EP - 592 VL - 348 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.04.011/ DO - 10.1016/j.crma.2010.04.011 LA - en ID - CRMATH_2010__348_9-10_587_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Ciarlet, Patrick Jr. %A Iosifescu, Oana %A Sauter, Stefan %A Zou, Jun %T A Lagrangian approach to intrinsic linearized elasticity %J Comptes Rendus. Mathématique %D 2010 %P 587-592 %V 348 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.04.011/ %R 10.1016/j.crma.2010.04.011 %G en %F CRMATH_2010__348_9-10_587_0
Ciarlet, Philippe G.; Ciarlet, Patrick Jr.; Iosifescu, Oana; Sauter, Stefan; Zou, Jun. A Lagrangian approach to intrinsic linearized elasticity. Comptes Rendus. Mathématique, Tome 348 (2010) no. 9-10, pp. 587-592. doi : 10.1016/j.crma.2010.04.011. http://www.numdam.org/articles/10.1016/j.crma.2010.04.011/
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