×

A classification of thin shell theories. (English) Zbl 0531.73044

This paper gives a modern mathematical analysis of the relationships between several, different linear shell theories. It also discusses the asymptotic role played by membrane theory. It presents theorems on the existence and uniqueness of solutions of membrane equations depending on the concavity of the surface

MSC:

74K15 Membranes
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
Full Text: DOI

References:

[1] Adams, R.:Sobolev Spaces, Academic Press, New York, 1976. · Zbl 0339.46027
[2] Antman, S. S. and Osborn, J. E.: ’The principle of virtual work and integral laws of motion’,Arch. Rat. Mech. Anal. 69 (1979), 231–262. · Zbl 0403.73003 · doi:10.1007/BF00248135
[3] Bernadou, M. and Ciarlet, P. G.: ’Sur l’ellipticité du modèle linéaire de coques de W.T. Koiter’,Lecture Notes in Applied Sciences and Engineering, vol. 134, Springer-Verlag, Berlin, pp. 89–136.
[4] Brezzi, F.: ’On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers’,RAIRO R2, 1974, 129–151. · Zbl 0338.90047
[5] Budiansky, B. and Sanders, J. L.: ’On the best first order linear shell theory’,Progress in Applied Mechanics, W. Prager Anniversary Volume New York, MacMillan, 1967, pp. 129–140.
[6] Choquet-Bruhat, Y., De Witte-Morette, C. and Dillard-Bleick, M.:Analysis Manifolds and Physics, North Holland, Amsterdam, New York, Oxford, 1978. · Zbl 0385.58001
[7] Ciarlet, P. G. and Destuynder, Ph.: ’A justification of the two-dimensional linear plate model’,J. Mech. 18 (1979). · Zbl 0415.73072
[8] Destuynder, Ph.:Thèse d’état Université de P.M. Curie, Paris, 1980. · Zbl 0451.73041
[9] Destuynder, Ph.: ’Sur la propagation des fissures dans les plaques minces en flexion’,J. Méc. Théor. Appl. 13 (1982). · Zbl 0532.73088
[10] Dickey, R. W.: ’Bifurcation problems in nonlinear elasticity’,Research Notes in Mathematics, No. 3, Pitman, London, 1976. · Zbl 0335.73012
[11] Duvaut, G. and Lions, J. L.:Les inéquations en mécanique et en physique, Dunod, Paris, 1973.
[12] Friedrichs, K. O. and Dressler, R. F.: ’A boundary layer theory for elastic plates’,Comm. Pure Appl. Math. XIV (1961), 1, 33. · Zbl 0096.40001 · doi:10.1002/cpa.3160140102
[13] Gol’denveizer, A. L.,Theory of Elastic Thin Shells, Pergamon Press, Oxford, 1961.
[14] Koiter, W. T.: ’General theory of shell stability. Thin shell theory’,New Trends and Applications, C.I.S.M. Courses and Lectures 240, Springer-Verlag, Wien, New York, 1980, pp. 65–87.
[15] Lions, J. L.: Perturbations singulières dans les problèmes aux limites et en contrôle optimal’,Lectures Notes in Mathematics, vol. 323, Springer-Verlag, Berlin, 1973. · Zbl 0268.49001
[16] Lions, J. L. and Magenes, E.:Problèmes aux Limites non homogènes et applications Vol. 1, Dunod, Paris, 1968. · Zbl 0165.10801
[17] Love, A. E. H.:A Treatise on the Mathematical Theory of Elasticity, 4th edn., Dover Publications, 1927.
[18] Lukasiewicz, S.:Local Loads in Plates and Shells, P.W.N. Polish Scientific Publishers Warsaw, Noordhoff Int. Publ., Leyden, 1979. · Zbl 0415.73070
[19] Novozhilov, V. V.:Thin Shell Theory, Walters Noordhoff Publ., Groningen, 1959. · Zbl 0085.18803
[20] Rougée, P.: Thèse d’état, Paris, 1969.
[21] Valid, R.:La mécanique des milieux continus et le calcul des structures, Eyrolles, Paris, 1977. · Zbl 0454.73003
[22] Yosida, K.:Functional Analysis, 4th edn., Springer-Verlag, Berlin, 1975. · Zbl 0152.32102
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.