Abstract
In this paper we present and analyze a geometrically nonlinear formulation that models the unsteady motion of two and three-dimensional elastic structures in the case of large displacements-small strains. In a first part (Section 2) we derive the equations describing the motion of the body. In a second part (Section 3), existence of a weak solution is proved using a Galerkin method. We also prove that the solution is unique.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I.: Mathematical methods of classical mechanics, translated from the Russian. In: Vogtmann, K., Weinstein, A. (eds.) Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, Berlin Heidelberg New York (1989)
Ciarlet, P.G.: Mathematical elasticity. vol. I. North-Holland, Amsterdam (1988)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Cohen-Tannoudji, C., Diu, B., Laloe, F.: Mécanique quantique. Vol. 2. Hermann, Paris (1973)
Duvaut, G., Lions, J.-L.: Les inéquations en mécanique et en physique. Dunod, Paris (1972)
Farhat, C., Pierson, K., Lesoinne, M.: The second generation of FETI methods and their application to the parallel solution of large-scale linear and geometrically nonlinear structural analysis problems. Comput. Methods Appl. Mech. Eng. 184 (2000)
Felippa, C.A.: A systematic approach to element-independent corotational dynamics of finite elements. Report CU-CAS-00-03, center for aerospace structures, University of Colorado, Boulder (2000)
Fraeijs De Veubeke, B.: The dynamics of flexible bodies. Int. J. Eng. Sci. 14 (1976)
Grandmont, C., Maday, Y.: Some remarks on fluid-structure interaction problems in case of rigid body plus small perturbations. CFSWA proceedings, Springer series Notes on numerical fluid mechanics (2003)
Grandmont, C., Maday, Y., Métier, P.: Existence of a solution for an unsteady elasticity problem in large displacement and small perturbation. C. R. Acad. Sci., Ser. 1 Math. 334 (2002)
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod, Gauthier-Villars, Paris (1969)
Lions, J.L., Magenes, E.: Problèmes aux limites non-homogènes et applications., vol. 1. Dunod, Paris (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grandmont, C., Maday, Y. & Métier, P. Modeling and Analysis of an Elastic Problem with Large Displacements and Small Strains. J Elasticity 87, 29–72 (2007). https://doi.org/10.1007/s10659-006-9097-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-006-9097-6