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Comptes Rendus Mathématique
Volume 355, n° 2
pages 232-242 (février 2017)
Doi : 10.1016/j.crma.2017.01.001
Received : 30 December 2016 ;  accepted : 30 December 2016
Une approche intrinsèque d'un modèle non linéaire de la théorie des coques
An intrinsic approach to a nonlinear model in shell theory

Philippe G. Ciarlet a , Oana Iosifescu b
a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 
b Départment de Mathématiques, université de Montpellier, place Eugène-Bataillon, 34095 Montpellier cedex 5, France 


On considère le modèle de coque non linéairement élastique « peu profonde » proposé par L.H. Donnell, V.Z. Vlasov, K.M. Mushtari & K.Z. Galimov et W.T. Koiter. On montre que les champs de tenseurs, linéarisés de changement de courbure et non linéaire des déformations, apparaissant dans l'énergie de ce modèle peuvent être pris comme les seules inconnues du problème, au lieu du champ des déplacements comme à l'accoutumée. Afin de justifier cette « approche intrinsèque » de ce modèle non linéaire de coques, on identifie des conditions de compatibilité non linéaires que ces nouvelles inconnues doivent satisfaire. Ces conditions sont du type de Donati, au sens qu'elles se présentent sous la forme de relations intégrales d'orthogonalité à des champs de tenseurs à divergence nulle.

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We consider the model of a nonlinearly elastic “shallow” shell proposed by L.H. Donnell, V.Z. Vlasov, K.M. Mushtari & K.Z. Galimov, and W.T. Koiter. We show that the linearized change of curvature and nonlinear strain tensor fields appearing in the energy of this model can be taken as the sole unknowns of the problem, instead of the displacement field as is customary. In order to justify this “intrinsic approach” to this nonlinear model, we identify nonlinear compatibility conditions that these new unknowns must satisfy. These conditions are of Donati type, in the sense that they take the form of integral orthogonality relations against divergence-free tensor fields.

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