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A justification of the von Kármán equations. (English) Zbl 0443.73034

74K20 Plates
35C20 Asymptotic expansions of solutions to PDEs
74B20 Nonlinear elasticity
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI
[1] Agmon, S., A. Douglis & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. XVII (1964), 35–92. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[2] Antman, S. S., Fundamental mathematical problems in the theory of non-linear elasticity, in Numerical Solution of Partial Differential Equations-III (B. Hubbard, Editor), pp. 35–54, Academic Press, 1976.
[3] Antman, S.S., The eversion of thick spherical shells, Arch. Rational Mech. Anal. 70 (1979/80), 113–123. · Zbl 0448.73088 · doi:10.1007/BF00250348
[4] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403. · Zbl 0368.73040 · doi:10.1007/BF00279992
[5] Berger, M.S., Nonlinearity and Functional Analysis, Academic Press, New York, 1977. · Zbl 0368.47001
[6] Ciarlet, P.G., Une justification des équations de von Kármán, C. R. Acad. Sci. Paris Sér. A 288 (1979), 469–472. · Zbl 0423.73015
[7] Ciarlet, P.G., Derivation of the von Kármán equations from three-dimensional elasticity in Proceedings of the Fourth Conference on Basic Problems in Numerical Analysis, Plzeň, September, 1978 pp. 37–49, 1979.
[8] Ciarlet, P.G., & P. Destuynder, A justification of the two-dimensional linear plate model, J. Mécanique 18 (1979), 315–344. · Zbl 0415.73072
[9] Ciarlet, P.G., & P. Destuynder, A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg. 17/18 (1979), 227–258. · Zbl 0405.73050 · doi:10.1016/0045-7825(79)90089-6
[10] Ciarlet, P.G., & S. Kesavan, Problèmes de valeurs propres pour les plaques: Comparaison entre les modèles tri- et bi-dimensionnels (to appear).
[11] Deny, J., & J.-L. Lions, Les espaces du type de Beppo Levi, Ann. Institut Fourier (Grenoble) V (1953–1954), 305–370. · Zbl 0065.09903
[12] Destuynder, P., Sur une Justification Mathématique des Théories de Plaques et de Coques en Elasticité Linéaire, Doctoral Dissertation, Université Pierre et Marie Curie, 1980.
[13] Duvaut, G., & J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. · Zbl 0298.73001
[14] Green, A. E., & W. Zerna, Theoretical Elasticity, University Press, Oxford, 1968.
[15] Hlaváček, I., & J. Naumann, Inhomogeneous boundary value problems for the von Kármán equations, I, Aplikace Matematiky 19 (1974), 253–269.
[16] Hlaváček, I., & J. Naumann, Inhomogeneous boundary value problems for the von Kármán equations. II, Aplikace Matematiky 20 (1975), 280–297.
[17] John, F., Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure and Appl. Math. 18 (1965), 235–267. · doi:10.1002/cpa.3160180120
[18] John, F., Refined interior equations for thin elastic shells, Comm. Pure and Appl. Math. 24 (1971), 583–615. · Zbl 0299.73037 · doi:10.1002/cpa.3160240502
[19] John, O., & J. Nečas, On the solvability of von Kármán equations, Aplikace Matematiky 20 (1975), 48–62.
[20] von Kármán, T., Festigkeitsprobleme im Maschinenbau, in Encyklopädie der Mathematischen Wissenschaften, Vol. IV/4, C, pp. 311–385, Leipzig, 1910.
[21] Koiter, W.T., On the foundations of the linear theory of thin elastic shells. I–II, Proc. Kon. Ned. Akad. Wetensch. B 73 (1970), 169–195. · Zbl 0213.27002
[22] Koiter, W.T., & J.G. Simmonds, Foundations of shell theory, in Proceedings of Thirteenth International Congress of Theoretical and Applied Mechanics (Moscow, 1972), pp. 150–176, Springer-Verlag, Berlin, 1973. · Zbl 0286.73068
[23] Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flows, Gordon & Breach, New York, 1969.
[24] Lions, J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
[25] Lions, J.-L., Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Mathematics, vol. 323, Springer-Verlag, Berlin, 1973. · Zbl 0268.49001
[26] Lions, J.-L., & E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968. · Zbl 0212.43801
[27] Marsden, J.E., & T.J.R. Hughes, Topics in the mathematical foundations of elasticity, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. II, pp. 30–285, Pitman, London, 1978.
[28] Morgenstern, D., & I. Szabò, Vorlesungen über Theoretische Mechanik, Springer-Verlag, Berlin, 1961. · Zbl 0097.16404
[29] Nečas, J., Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967.
[30] Novozhilov, V.V., Foundation of the Nonlinear Theory of Elasticity, Graylock, 1953.
[31] Oden, J.T., Existence theorems for a class of problems in nonlinear elasticity, J. Math. Anal. Appl. 69 (1979), 51–83. · Zbl 0413.73023 · doi:10.1016/0022-247X(79)90178-1
[32] Schwartz, L., Théorie des Distributions, Hermann, Paris, 1966.
[33] Stoker, J.J., Nonlinear Elasticity, Gordon and Breach, New York, 1968.
[34] Stoppelli, F., Un teorema di esistenza e di unicità relativo alle equazioni dell’elastostatica isoterma per deformazioni finite, Ricerche di Matematica 3 (1954), 247–267. · Zbl 0058.39701
[35] Témam, R., Navier-Stokes Equations, North-Holland, 1977.
[36] Timoshenko, S., & W. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959. · Zbl 0114.40801
[37] Truesdell, C., A First Course in Rational Continuum Mechanics, Volume 1 : General Concepts, Academic Press, 1976.
[38] Truesdell, C., Comments on Rational Continuum Mechanics, Three Lectures for the International Symposium on Continuum Mechanics and Partial Differential Equations, Instituto de Matemática, Universidade Federal do Rio de Janeiro, August 1–5, 1977, pp. 495–603 of Contemporary Developments in Continuum Mechanics and Partial Differential Equations, ed. G.M. de LaPenha & L.A. Medeiros, Amsterdam, North-Holland, 1978.
[39] Truesdell, C., & W. Noll, The Non-Linear Field Theories of Mechanics, Handbuch der Physik, vol. III/3. Springer, Berlin, 1965. · Zbl 0779.73004
[40] Valid, R., La Mécanique des Milieux Continus et le Calcul des Structures, Eyrolles, Paris, 1977.
[41] van Buren, M., On the Existence and Uniqueness of Solutions to Boundary Value Problems in Finite Elasticity, Thesis, Carnegie-Mellon University, 1968.
[42] Wang, C.-C, & C. Truesdell, Introduction to Rational Elasticity, Noordhoff, Groningen, 1973. · Zbl 0308.73001
[43] Washizu, K., Variational Methods in Elasticity and Plasticity, Second Edition, Pergamon, Oxford, 1975. · Zbl 0339.73035
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