A notion of polyconvex function on a surface suggested by nonlinear shell theory

Philippe G. Ciarlet; Radu Gogu; Cristinel Mardare

doi:10.1016/j.crma.2011.10.002

Mathematical Problems in Mechanics

A notion of polyconvex function on a surface suggested by nonlinear shell theory
[Une notion de fonction polyconvexe sur une surface suggérée par la théorie des coques non linéaires]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Radu Gogu ¹ ; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Université Pierre et Marie Curie, laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1207-1211.

Résumé
Abstract

Combinant les définitions proposées par J. Ball en 1977 et par J. Ball, J.C. Currie, et P.J. Olver en 1981, nous proposons dans cette Note une définition de « fonction polyconvexe sur une surface ». Quand la surface est vue comme la surface moyenne dʼune coque non linéairement élastique et la fonction comme sa densité dʼénergie, on montre quʼil est loisible de supposer en plus que cette fonction est coercive pour des normes de Sobolev convenables et quʼelle satisfait des conditions de croissance particulières qui empêchent les vecteurs des bases covariantes le long de la surface déformée de devenir linéairement dépendants, une condition qui est « lʼanalogue pour une surface » de la condition de préservation de lʼorientation de J. Ball. On montre ensuite quʼune fonctionnelle avec un tel intégrande polyconvexe est faiblement semi-continue inférieurement, ce qui finalement permet dʼétablir lʼexistence de minimiseurs.

Combining the definitions set forth by J. Ball in 1977 and by J. Ball, J.C. Currie, and P.J. Olver in 1981, we propose in this Note a definition of a “polyconvex function on a surface”. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, which eventually allows to establish the existence of minimizers.

Reçu le : 2011-09-23
Publié le : 2011-10-18

DOI : 10.1016/j.crma.2011.10.002

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Radu Gogu ¹ ; Cristinel Mardare ²

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     title = {A notion of polyconvex function on a surface suggested by nonlinear shell theory},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1207--1211},
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     doi = {10.1016/j.crma.2011.10.002},
     language = {en},
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Philippe G. Ciarlet; Radu Gogu; Cristinel Mardare. A notion of polyconvex function on a surface suggested by nonlinear shell theory. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1207-1211. doi : 10.1016/j.crma.2011.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.002/

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