Abstract
This paper deals with the asymptotic formulation and justification of a mechanical model for a shallow shell in frictionless unilateral contact with an obstacle. The first three parts of the paper concern the formulation of the equilibrium problem. Special attention is paid to the contact conditions, which are usual within two or three dimensional elasticity, but which are not so usual in shell theories. Lastly the limit problem is formulated in the main part of the paper and a convergence result is presented. Two points are worth stressing here. First, we point out that unlike classical bilateral shell models justifications, the functional framework of the present analysis involves cones. Secondly, while the cones result from a positivity condition on the boundary as long as the thickness parameter is finite, leading to a Signorini problem in the Sobolev space H 1, the cone results from a positivity condition in the domain, giving rise to a so-called obstacle problem in the Sobolev space H 2 at the limit.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Adams, R.A.: Sobolev Spaces. Academic, New York (1975)
Anicic, S.: From the exact Kirchhoff–Love model to a thin shell model and a folded shell model. PhD, Grenoble University (2001)
Blouza, A., Le Dret, H.: Existence et unicité pour le modèle de Koiter pour une coque peu régulière. C.R. Acad. Sci., I t.319, 1127–1132 (1994)
Berger, M.S.: On von Kármán’s equations and the buckling of a thin elastic plate. Comm. Pure Appl. Math. 20, 687–719 (1967)
Berger, M.S.: On the existence of equilibrium states of thin elastic shells (I). Indiana Univ. Math. J. 20(7), 591–602 (1971)
Ciarlet, P.G.: Plates and junctions in elastic multi-structures. An asymptotic analysis. Masson, Paris (1990)
Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. J. Elast. 78–79, 1–215 (2005)
Ciarlet, P.G., Miara, B.: Justification of the two-dimensional equations of a linearly elastic shallow shell. Comm. Pure Appl. Math. XLV, 327–360 (1992)
Ciarlet, P.G., Paumier, J.-C.: A justification of the Marguerre-von Kármán equations. Comput. Mech. 1, 177–202 (1986).
Cimetière, A.: Un problème de flambement unilatéral en théorie des plaques. J. de Mécanique 19, 182–202 (1980)
Cimetière, A., Léger, A.: Un résultat de différentiabilité dans le problème d’obstacle pour des poutres en flexion. C.R. Acad. Sci., I t.316, 749–754 (1994)
Conrad, F., Herbin, R., Mittelmann, H.D.: Approximation of obstacle problems by continuation methods. SIAM. Numer. Anal. 25, 1409–1431 (1988)
Duvaut, G., Lions, J.L.: Les inéquations en Mécanique et en Physique. Dunod, Paris (1972)
Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. XX, 493–519 (1967)
Lions, J.L.: Quelques méthodes de résolution des problêmes aux limites non linéaires. Dunod – Gauthier-Villars, Paris (1969)
Lions, J.L.: Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, 323. Springer (1973)
Paumier, J.-C.: Le problème de Signorini dans la théorie des plaques minces de Kirchhoff–Love. C.R. Acad. Sci., I t.335, 567–570 (2002)
Schaeffer, D.G.: A stability theorem for the obstacle problem. Adv. Math. 16, 34–47 (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article can be found online at http://dx.doi.org/10.1007/s10659-009-9230-4.
Rights and permissions
About this article
Cite this article
Léger, A., Miara, B. Mathematical Justification of the Obstacle Problem in the Case of a Shallow Shell. J Elasticity 90, 241–257 (2008). https://doi.org/10.1007/s10659-007-9141-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-007-9141-1