Asymptotics of shell eigenvalue problems

Lorenzo Beirão da Veiga; Carlo Lovadina

doi:10.1016/j.crma.2006.03.004

Mathematical Problems in Mechanics

Asymptotics of shell eigenvalue problems

Presented by: Philippe G. Ciarlet

Lorenzo Beirão da Veiga ¹; Carlo Lovadina ²

¹ Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
² Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 707-710.

Abstract
Résumé

The asymptotic behaviour of the smallest eigenvalue in linear shell problems is studied, as the thickness parameter tends to zero. When pure bending is not inhibited, such a behaviour has been essentially studied by Sanchez-Palencia. When pure bending is inhibited, the situation is more complex and some information can be obtained by using the Real Interpolation Theory. In order to cover the widest range of mid-surface geometry and boundary conditions, an abstract approach has been followed. A result concerning the ratio between the bending and the total elastic energy is also announced.

Dans cette Note, on étudie le comportement asymptotique de la plus petite valeur propre pour un modèle de coque linéaire, lorsque l'épaisseur de la coque tend vers zéro. Pour les situations de flexion pure non-inhibée, ce comportement a déjà été largement étudié par Sanchez-Palencia. Dans un cas de flexion pure inhibée le problème est plus complexe et on peut employer la théorie de l'interpolation réelle pour l'aborder. Nous proposons une méthodologie abstraite qui vise à couvrir un éventail de configurations le plus large possible en matière de géométrie de la surface moyenne et de conditions aux limites. On annonce également un résultat portant sur le ratio entre énergie de flexion et énergie totale.

Received: 2006-01-25
Accepted: 2006-02-22
Published online: 2006-03-29

DOI: 10.1016/j.crma.2006.03.004

Author's affiliations:

Lorenzo Beirão da Veiga ¹; Carlo Lovadina ²

¹ Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
² Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

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     title = {Asymptotics of shell eigenvalue problems},
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     pages = {707--710},
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Lorenzo Beirão da Veiga; Carlo Lovadina. Asymptotics of shell eigenvalue problems. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 707-710. doi : 10.1016/j.crma.2006.03.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.004/

References
Cited by

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[2] L. Beirão da Veiga, H. Hakula, J. Pitkäranta, An analysis of the eigenvalue problem of clamped cylindrical shells, in preparation

[3] L. Beirão da Veiga, C. Lovadina, An interpolation theory approach to shell eigenvalue problems, in preparation

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[8] E. Sanchez-Palencia Statique et dynamique des coques minces. II. Cas de flexion pure inhibée – Approximation membranaire, C. R. Acad. Sci. Paris, Ser. I, Volume 309 (1989), pp. 531-537

[9] E. Sanchez-Palencia Asymptotic and spectral properties of a class of singular-stiff problems, J. Math. Pures Appl., Volume 71 (1992), pp. 379-406

[10] K. Yosida Functional Analysis, Springer-Verlag, 1980

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