Abstract
The method of asymptotic expansions, with the thickness as the parameter, is applied to the nonlinear, three-dimensional, equations for the equilibrium of a special class of elastic plates under suitable loads. It is shown that the leading term of the expansion is the solution of a system of equations equivalent to those of von Kármán. The existence of solutions of this system is established. It is also shown that the displacement and stress corresponding to the leading term of the expansion have the specific form generally assumed in the usual derivations of the von Kármán equations; in particular, the displacement field is of Kirchhoff-Love type. This approach also clarifies the nature of admissible boundary conditions for both the von Kármán equations and the three-dimensional model from which these equations are obtained. A careful discussion of the limitations of this approach is given in the conclusion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
Antman, S.S., The eversion of thick spherical shells, Arch. Rational Mech. Anal. 70 (1979/80), 113–123.
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403.
Berger, M.S., Nonlinearity and Functional Analysis, Academic Press, New York, 1977.
Ciarlet, P.G., Une justification des équations de von Kármán, C. R. Acad. Sci. Paris Sér. A 288 (1979), 469–472.
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.
Ciarlet, P.G., & P. Destuynder, A justification of the two-dimensional linear plate model, J. Mécanique 18 (1979), 315–344.
Ciarlet, P.G., & P. Destuynder, A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg. 17/18 (1979), 227–258.
Ciarlet, P.G., & S. Kesavan, Problèmes de valeurs propres pour les plaques: Comparaison entre les modèles tri- et bi-dimensionnels (to appear).
Deny, J., & J.-L. Lions, Les espaces du type de Beppo Levi, Ann. Institut Fourier (Grenoble) V (1953–1954), 305–370.
Destuynder, P., Sur une Justification Mathématique des Théories de Plaques et de Coques en Elasticité Linéaire, Doctoral Dissertation, Université Pierre et Marie Curie, 1980.
Duvaut, G., & J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
Green, A. E., & W. Zerna, Theoretical Elasticity, University Press, Oxford, 1968.
Hlaváček, I., & J. Naumann, Inhomogeneous boundary value problems for the von Kármán equations, I, Aplikace Matematiky 19 (1974), 253–269.
Hlaváček, I., & J. Naumann, Inhomogeneous boundary value problems for the von Kármán equations. II, Aplikace Matematiky 20 (1975), 280–297.
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.
John, F., Refined interior equations for thin elastic shells, Comm. Pure and Appl. Math. 24 (1971), 583–615.
John, O., & J. Nečas, On the solvability of von Kármán equations, Aplikace Matematiky 20 (1975), 48–62.
von Kármán, T., Festigkeitsprobleme im Maschinenbau, in Encyklopädie der Mathematischen Wissenschaften, Vol. IV/4, C, pp. 311–385, Leipzig, 1910.
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.
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.
Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flows, Gordon & Breach, New York, 1969.
Lions, J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
Lions, J.-L., Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Mathematics, vol. 323, Springer-Verlag, Berlin, 1973.
Lions, J.-L., & E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968.
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.
Morgenstern, D., & I. Szabò, Vorlesungen über Theoretische Mechanik, Springer-Verlag, Berlin, 1961.
Nečas, J., Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967.
Novozhilov, V.V., Foundation of the Nonlinear Theory of Elasticity, Graylock, 1953.
Oden, J.T., Existence theorems for a class of problems in nonlinear elasticity, J. Math. Anal. Appl. 69 (1979), 51–83.
Schwartz, L., Théorie des Distributions, Hermann, Paris, 1966.
Stoker, J.J., Nonlinear Elasticity, Gordon and Breach, New York, 1968.
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.
Témam, R., Navier-Stokes Equations, North-Holland, 1977.
Timoshenko, S., & W. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959.
Truesdell, C., A First Course in Rational Continuum Mechanics, Volume 1 : General Concepts, Academic Press, 1976.
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.
Truesdell, C., & W. Noll, The Non-Linear Field Theories of Mechanics, Handbuch der Physik, vol. III/3. Springer, Berlin, 1965.
Valid, R., La Mécanique des Milieux Continus et le Calcul des Structures, Eyrolles, Paris, 1977.
van Buren, M., On the Existence and Uniqueness of Solutions to Boundary Value Problems in Finite Elasticity, Thesis, Carnegie-Mellon University, 1968.
Wang, C.-C, & C. Truesdell, Introduction to Rational Elasticity, Noordhoff, Groningen, 1973.
Washizu, K., Variational Methods in Elasticity and Plasticity, Second Edition, Pergamon, Oxford, 1975.
Author information
Authors and Affiliations
Additional information
Communicated by S. Antman & J. L. Lions
Rights and permissions
About this article
Cite this article
Ciarlet, P.G. A justification of the von Kármán equations. Arch. Rational Mech. Anal. 73, 349–389 (1980). https://doi.org/10.1007/BF00247674
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00247674