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Weak Formulations and Solution Multiplicityof Equilibrium Configurations with Coulomb Friction

Published online by Cambridge University Press:  27 January 2009

M. Bostan  and
P. Hild
M. Bostan
Affiliation:
Laboratoire de Mathématiques, Université de Franche-Comté CNRS UMR 6623, 16 route de Gray, 25030 Besançon, France
P. Hild*
Affiliation:
Laboratoire de Mathématiques, Université de Franche-Comté CNRS UMR 6623, 16 route de Gray, 25030 Besançon, France
*
patrick.hild@univ-fcomte.fr
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Abstract

This work is concerned with the equilibrium configurations of elastic structures in contact with Coulomb friction. We obtain a variational formulation of this equilibrium problem. Then we propose sufficient conditions for the existence of an infinity of equilibrium configurations with arbitrary small friction coefficients. We illustrate the result in two space dimensions with a simple example.

Keywords

Coulomb frictionlinear elasticityequilibrium configurationsweak formulationscontinuum of solutions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 147 - 162
Copyright
© EDP Sciences, 2009

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References

Andersson, L.-E.. Existence results for quasistatic contact problems with Coulomb friction, Appl. Math. Optim., 42 (2000), 169–202. CrossRef
J.R. Barber, P. Hild. Non-uniqueness, eigenvalue solutions and wedged configurations involving Coulomb friction, Proceedings of the IJTC 2004, ASME/STLE International Joint Tribology Conference, Long Beach California, USA, 24-27 October 2004, Part A, 127–132.
Eck, C., Jarušek, J.. Existence results for the static contact problem with Coulomb friction, Math. Models Meth. Appl. Sci., 8 (1998), 445–468. CrossRef
C. Eck, J. Jarušek, M. Krbec. Unilateral contact problems: variational methods and existence theorems, Pure and Applied Mathematics 270, CRC Press, 2005.
W. Han, M. Sofonea. Quasistatic contact problems in viscoelasticity and viscoplasticity, American Mathematical Society, International Press, 2002.
J. Haslinger, I. Hlaváček, J. Nečas. Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Volume IV, Part 2, eds. P.G. Ciarlet and J. L. Lions, North Holland, 1996, pp. 313–485.
Hassani, R., Ionescu, I., Oudet, E.. Critical friction for wedged configurations, Int. J. Solids Structures, 44 (2007), 6187–6200. CrossRef
Hassani, R., Ionescu, I., Sakki, N.-D.. Unstable perturbation of the equilibrium under Coulomb friction. Nonlinear eigenvalue analysis, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2377–2389. CrossRef
Hild, P.. Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity, Q. Jl. Mech. Appl. Math., 57 (2004), 225–235. CrossRef
Hild, P.. Multiple solutions of stick and separation type in the Signorini model with Coulomb friction, Z. Angew. Math. Mech., 85 (2005), 673–680. CrossRef
Klarbring, A., Mikelíc, A., Shillor, M.. Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811–832. CrossRef
Klarbring, A., Mikelíc, A., Shillor, M.. On friction problems with normal compliance, Nonlinear Anal., 13 (1989), 935–955. CrossRef
Martins, J.A.C., Oden, J.T.. Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), 407–428. CrossRef
J.A.C. Martins, M.D.P. Monteiro Marques (Eds.) Contact Mechanics, Proceedings of the third Contact Mechanics International Symposium, Solid Mechanics and its Applications 103, Kluwer, 2002.
Naéjus, C., Cimetière, A.. Sur la formulation variationnelle du problème de Signorini avec frottement de Coulomb, C. R. Acad. Sci. Sér. I Math., 323 (1996), 307–312.
J. Nečas, J. Jarušek, J. Haslinger. On the solution of the variational inequality to the Signorini problem with small friction, Bolletino U.M.I., 17 (1980), No. 5, 796–811.
Oden, J.T., Martins, J.A.C.. Models and computational methods for dynamic friction phenomena, Comput. Methods. Appl. Mech. Engrg., 52 (1985), 527–634. CrossRef
Renard, Y.. A uniqueness criterion for the Signorini problem with Coulomb friction, SIAM J. Math. Anal., 38 (2006), 458–467. CrossRef
Rocca, R., Cocu, M.. Existence and approximation of a solution to quasistatic Signorini problem with local friction, Int. J. Engrg. Sci., 39 (2001), 1233–1255. CrossRef
M. Shillor (Ed.) Recent advances in contact mechanics, Mathl. Comput. Modelling, 28 (1998), No. 4–8, 1–534.
P. Wriggers, U. Nackenhorst (Eds.) Analysis and simulation of contact problems, Proceedings of the fourth Contact Mechanics International Symposium, Lecture Notes in Applied and Computational Mechanics 27, Springer, 2006.