# Weak Formulations and Solution Multiplicity of Equilibrium Configurations with Coulomb Friction

Mathematical Modelling of Natural Phenomena (2009)

- Volume: 4, Issue: 1, page 147-162
- ISSN: 0973-5348

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topBostan, M., and Hild, P.. "Weak Formulations and Solution Multiplicity of Equilibrium Configurations with Coulomb Friction." Mathematical Modelling of Natural Phenomena 4.1 (2009): 147-162. <http://eudml.org/doc/222265>.

@article{Bostan2009,

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.},

author = {Bostan, M., Hild, P.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {Coulomb friction; linear elasticity; equilibrium configurations; weak
formulations; continuum of solutions; weak formulations},

language = {eng},

month = {1},

number = {1},

pages = {147-162},

publisher = {EDP Sciences},

title = {Weak Formulations and Solution Multiplicity of Equilibrium Configurations with Coulomb Friction},

url = {http://eudml.org/doc/222265},

volume = {4},

year = {2009},

}

TY - JOUR

AU - Bostan, M.

AU - Hild, P.

TI - Weak Formulations and Solution Multiplicity of Equilibrium Configurations with Coulomb Friction

JO - Mathematical Modelling of Natural Phenomena

DA - 2009/1//

PB - EDP Sciences

VL - 4

IS - 1

SP - 147

EP - 162

AB -
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.

LA - eng

KW - Coulomb friction; linear elasticity; equilibrium configurations; weak
formulations; continuum of solutions; weak formulations

UR - http://eudml.org/doc/222265

ER -

## References

top- L.-E. Andersson. Existence results for quasistatic contact problems with Coulomb friction, Appl. Math. Optim., 42 (2000), 169–202. Zbl0972.35058
- 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.
- C. Eck, J. Jarušek. Existence results for the static contact problem with Coulomb friction, Math. Models Meth. Appl. Sci., 8 (1998), 445–468. Zbl0907.73052
- C. Eck, J. Jarušek, M. Krbec. Unilateral contact problems: variational methods and existence theorems, Pure and Applied Mathematics 270, CRC Press, 2005. Zbl1079.74003
- W. Han, M. Sofonea. Quasistatic contact problems in viscoelasticity and viscoplasticity, American Mathematical Society, International Press, 2002. Zbl1013.74001
- 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. Zbl0873.73079
- R. Hassani, I. Ionescu, E. Oudet. Critical friction for wedged configurations, Int. J. Solids Structures, 44 (2007), 6187–6200. Zbl1159.74025
- R. Hassani, I. Ionescu, N.-D. Sakki. Unstable perturbation of the equilibrium under Coulomb friction. Nonlinear eigenvalue analysis, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2377–2389. Zbl1173.74343
- P. Hild. Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity, Q. Jl. Mech. Appl. Math., 57 (2004), 225–235. Zbl1059.74042
- P. Hild. Multiple solutions of stick and separation type in the Signorini model with Coulomb friction, Z. Angew. Math. Mech., 85 (2005), 673–680. Zbl1149.74338
- A. Klarbring, A. Mikelíc, M. Shillor. Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811–832. Zbl0662.73079
- A. Klarbring, A. Mikelíc, M. Shillor. On friction problems with normal compliance, Nonlinear Anal., 13 (1989), 935–955. Zbl0707.73068
- J.A.C. Martins, J.T. Oden. Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), 407–428. Zbl0672.73079
- 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.
- C. Naéjus, A. Cimetière. Sur la formulation variationnelle du problème de Signorini avec frottement de Coulomb, C. R. Acad. Sci. Sér. I Math., 323 (1996), 307–312. Zbl0856.73068
- 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. Zbl0445.49011
- J.T. Oden, J.A.C. Martins. Models and computational methods for dynamic friction phenomena, Comput. Methods. Appl. Mech. Engrg., 52 (1985), 527–634. Zbl0567.73122
- Y. Renard. A uniqueness criterion for the Signorini problem with Coulomb friction, SIAM J. Math. Anal., 38 (2006), 458–467. Zbl1155.35383
- R. Rocca, M. Cocu. Existence and approximation of a solution to quasistatic Signorini problem with local friction, Int. J. Engrg. Sci., 39 (2001), 1233–1255. Zbl1210.74126
- 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.

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