On the ellipticity of linear shell models. (English) Zbl 0765.73046

The refined shell theory approximating the exact three-dimensional stress state of thin shells introduced by W. T. Koiter is analyzed. A new proof of the ellipticity of the Koiter’s two-dimensional strain energy is given. The proof is based on the lemma of J. L. Lions (used originally for the establishing of the generalized Korn’s inequality) and on the “rigid displacement lemma” due to M. Bernadou and the first author [Lect. Notes Econ. Math. Syst. 134, 89-136 (1976; Zbl 0356.73066), C. R. Acad. Sci., Paris, Ser. A 282, 793-795 (1976; Zbl 0331.73091)].


74K15 Membranes
35Q72 Other PDE from mechanics (MSC2000)
Full Text: DOI


[1] Amrouche, C. and Girault, V.,Propriétés fonctionnelles d’opérateurs, applications au problème de Stokes en dimension quelconque (1991, to appear).
[2] Bernadou, M. and Ciarlet, P. G.,Sur l’ellipticité du modèle linéaire de coques de W. T. Koiter, in Computing Methods in Applied Sciences and Engineering, Lect. Notes in Economies & Math. Systems, Vol. 134, pp. 89-136, Springer-Verlag, Heidelberg 1976. · Zbl 0356.73066
[3] Bernadou, M., Ciarlet, P. G. and Miara, B.,On the ellipticity of two-dimensional linearly elastic shell models (1991, to appear).
[4] Borchers, W. and Sohr, H.,On the equations rot v=g and div u=f with zero boundary conditions, Hokkaido Math. J.19, 67-87 (1990). · Zbl 0719.35014
[5] Ciarlet, P. G. and Miara, B.,Justification d’un modèle bidimensionnel de coque ?peu profonde? en élasticité linéarisée, C. R. Acad. Sci. Paris, Sér. I,311, 571-574 (1990). · Zbl 0707.73044
[6] Ciarlet, P. G. and Miara, B.,Une démonstration simple de l’ellipticité des modèles de coques de W. T. Koiter et de P. M. Naghdi, C. R. Acad. Sci. Paris, Sér. I,312, 411-415 (1991a). · Zbl 0762.73050
[7] Ciarlet, P. G. and Miara, B.,Justification of the two dimensional equations of a linearly elastic shallow shell (to appear in Comm. Pure Applied Math., 1991b).
[8] Coutris, N.,Flexions élastique et élastoplastique d’une coque mince, J. de Mécanique12, 463-475 (1973). · Zbl 0271.73065
[9] Coutris, N.,Théorème d’existence et d’unicité pour un problème de coque élastique dans le cas d’un modèle linéaire de P. M. Naghdi, RAIRO Analyse Numérique12, 51-57 (1978). · Zbl 0413.73082
[10] Destuynder, P.,Sur une Justification des Modèles de Plaques et de Coques par les Méthodes Asymptotiques, Doctoral Dissertation, Université Pier?e et Marie Curie, Paris, 1980.
[11] Destuynder, P.,A classification of thin shell theories, Acta Appl. Math.4, 15-63 (1985). · Zbl 0561.73062 · doi:10.1007/BF02293490
[12] Duvaut, G. and Lions, J. L.,Les Inèquations en Mécanique et en Physique, Dunod, Paris, 1972. · Zbl 0298.73001
[13] Gordeziani, D. G.,On the solvability of some boundary value problems for a variant of the theory of thin shells, Dokl. Akad. Nauk SSSR215 (1974). (English translation: Soviet Math. Dokl.15, 677-680 (1974)). · Zbl 0321.73062
[14] John, F.,Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure Appl. Math.18, 235-267 (1965). · doi:10.1002/cpa.3160180120
[15] John, F.,Refined interior equations for thin elastic shells, Comm. Pure Appl. Math.24, 583-615 (1971). · Zbl 0299.73037 · doi:10.1002/cpa.3160240502
[16] Koiter, W. T.,On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. B69, 1-54 (1966).
[17] Koiter, W. T.,On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. B73, 169-195 (1970). · Zbl 0213.27002
[18] Magenes, E. and Stampacchia, G.,I problemi al contorno per le equazioni differenziali di tipo ellitico, Ann. Scuola Norm. Sup. Pisa12, 247-358 (1958). · Zbl 0082.09601
[19] Marsden, J. E. and Hughes, T. J. R.,Mathematical Foundations of Elasticity, Prentice-Hall, Engiewood Cliffs 1983. · Zbl 0545.73031
[20] Miara, B., to appear.
[21] Naghdi, P. M.,Foundations of elastic shell theory, inProgress in Solid Mechanics, Vol. 4, pp. 1-90, North-Holland, Amsterdam 1963.
[22] Naghdi, P. M.,The theory of shells and plates, inHandbuch der Physik, Vol. VI a-2, pp. 425-640, Springer-Verlag, Berlin 1972.
[23] Ne?as, J.,Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris 1967.
[24] Novozhilov, V. V.,Thin Shell Theory, Wolters-Noordhoff Publ., Groningen 1970. · Zbl 0135.43602
[25] Rougée, P.,Equilibre des Coques Elastiques Minces Inhomogènes en Théorie non Linéaire, Doctoral dissertation, Université de Paris, 1969.
[26] Sanchez-Palencia, E.,Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée, C. R. Acad. Sci. Paris, Sér. I309, 411-417 (1989a). · Zbl 0697.73051
[27] Sanchez-Palencia, E.,Statique et dynamique des coques minces. II. Cas de flexion pure inhibée-Approximation membranaire, C. R. Acad. Sci. Paris, Sér. I309, 531-537 (1989b). · Zbl 0712.73056
[28] Sanchez-Palencia, E.,Passage à la limite de l’elasticité tridimensionnelle à la théorie asymptotique des coques minces, C. R. Acad. Sci. Paris, Sér. II311, 909-916 (1990). · Zbl 0701.73080
[29] Shoikhet, B. A.,On existence theorems in linear shell theory, PMM38, 567-571 (English translation: J. Appl. Math. Mech.38, 527-531 (1974).
[30] Valiron, G.,Equations Fonctionnelles, Applications, Masson, Paris, (2nd Ed.) 1950. · Zbl 0061.16607
[31] Vekua, I. N.,Theory of thin shallow shells of variable thickness, Akad. Nauk Gruzin. SSR Trudy Tbilissi Mat. Inst. Razmadze30, 3-103 (1965) (Russian). · Zbl 0166.20904
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.