Abstract
In this paper the modeling of a thin plate in unilateral contact with a rigid plane is properly justified. Starting from the three-dimensional nonlinear Signorini problem, by an asymptotic approach the convergence of the displacement field as the thickness of the plate goes to zero is studied. It is shown that the transverse mechanical displacement field decouples from the in-plane components and solves an obstacle problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Léger, A. and Miara, B., Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elasticity, 90, 2008, 241–257.
Léger, A. and Miara, B., The obstacle problem for shallow shells: A curvilinear approach, Intl. J. of Numerical Analysis and Modeling, Ser. B, 2, 2011, 1–26.
Ciarlet, P. G. and Destuynder, P., A justification of the two dimensional plate model, J. Mécanique, 18, 1979, 315–344.
Ciarlet, P. G., Mathematical Elasticity, Vol II, Theory of Plates, North-Holland, Amsterdam 1997.
Fichera, G., Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Mem. Accad. Naz. Lincei Ser., VIII 7, 1964, 91–140.
Duvaut, G. and Lions J.-L., Les Inéquations en Mécanique et en Physique, Dunod 1972.
Paumier, J. C., Modélisation asymptotique d’un problème de plaque mince en contact unilatéral avec frottement contre un obstacle rigide, Prépublication L.M.C., http://www-lmc.imag.fr/paumier/signoplaque.ps, 2002.
Lions, J.-L., Quelques Méthodes de Résolution des Problêmes aux Limites Non Linéaires, Dunod-Gauthier-Villars, Paris, 1969.
Acknowledgments
The author is greatly indebted to Professor Li Tatsien and Professor Zhou Yi for their instructive questions, corrections, encouragement, and help.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the Innovation Program of Shanghai Municipal Education Commission (No. 11YZ80) and the Program of Shanghai Normal University (No. SK201301).
Rights and permissions
About this article
Cite this article
Guan, Y. Mathematical justification of an obstacle problem in the case of a plate. Chin. Ann. Math. Ser. B 38, 1047–1058 (2017). https://doi.org/10.1007/s11401-017-1021-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-017-1021-9