Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics
Authors:
Patrick Hild and Yves Renard
Journal:
Quart. Appl. Math. 63 (2005), 553-573
MSC (2000):
Primary 74M10, 74G20; Secondary 35A07, 35J85, 65N30.
DOI:
https://doi.org/10.1090/S0033-569X-05-00974-0
Published electronically:
July 11, 2005
MathSciNet review:
2169034
Full-text PDF Free Access
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Abstract: This work is concerned with the frictional contact problem governed by the Signorini contact model and the Coulomb friction law in static linear elasticity. We consider a general finite-dimensional setting and we study local uniqueness and smooth or nonsmooth continuation of solutions by using a generalized version of the implicit function theorem involving Clarke’s gradient. We show that for any contact status there exists an eigenvalue problem and that the solutions are locally unique if the friction coefficient is not an eigenvalue. Finally we illustrate our general results with a simple example in which the bifurcation diagrams are exhibited and discussed.
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adams1975 R.A. Adams, Sobolev spaces, Academic Press, 1975.
al-cu1991 A. Alart, A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Engrg. 92 (1991), 353–375.
andersson2000 L.-E. Andersson, Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Optim. 42 (2000), 169–202.
br-fo1991 F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Vol. 15, Springer Series in Computational Mathematics, Springer-Verlag, 1991.
clarke1983 F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience Publications, 1983.
coulomb1785 C. A. de Coulomb, Théorie des machines simples, en ayant égard au frottement de leurs parties et à la roideur des cordages. Pièce qui a remporté le Prix de l’Académie des Sciences pour l’année 1781, Mémoire des Savants Etrangers, X, 1785, pp. 163–332. Reprinted by Bachelier, Paris 1821.
duvaut1971 G. Duvaut, Problèmes unilatéraux en mécanique des milieux continus, Actes du Congrès International des Mathématiciens (Nice 1970), Tome 3, Gauthier-Villars, Paris, 1971, pp. 71–77.
du-li1972 G. Duvaut, J.L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
ec-ja1998 C. Eck, J. Jaru$\check {\text {s}}$ek, Existence results for the static contact problem with Coulomb friction. Math. Models Methods Appl. Sci. 8 (1998), 445–468.
haslinger1983 J. Haslinger, Approximation of the Signorini problem with friction, obeying the Coulomb law. Math. Meth. Appl. Sci. 5 (1983), 422–437.
ha-so2002 W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, International Press, 2002.
ha-hi-io-sa2003 R. Hassani, P. Hild, I. Ionescu, N.-D. Sakki, A mixed finite element method and solution multiplicity for Coulomb frictional contact. Comput. Methods Appl. Mech. Engrg. 192 (2003), 4517–4531.
hild2002 P. Hild, On finite element uniqueness studies for Coulomb’s frictional contact model. Appl. Math. Comp. 12 (2002), 41–50.
hild2003 P. Hild, An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction. C. R. Acad. Sci. Paris 337 (2003), 685–688.
hild2004 P. Hild, Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity. Q. Jl. Mech. Appl. Math. 57 (2004), 225–235.
hiriart J.B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. of Operations Research 4 (1979), 79–87.
HL J.B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimisation Algorithms, Vol. 305, in Grundlehren der Mathematischen Wissenschaften, Springer–Verlag, 1993.
io-pa1994 I.R. Ionescu, J.-C. Paumier, On the contact problem with slip rate dependent friction in elastodynamics. Eur. J. Mech., A/Solids 4 (1994), 555–568.
jarusek1983 J. Jaru$\check {\text {s}}$ek, Contact problems with bounded friction. Coercive case. Czechoslovak Math. J. 33 (1983), 237–261.
kh-po-re2004 H. Khenous, J. Pommier, Y. Renard, Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of numerical solvers. To appear in Appl. Numer. Math.
ki-od1988 N. Kikuchi, J.T. Oden, Contact problems in elasticity, SIAM, Philadelphia, 1988.
la-re2003 P. Laborde, Y. Renard, Fixed points strategies for elastostatic frictional contact problems. Submitted, 2003.
li-ma1968 J.–L. Lions, E. Magenes, Problèmes aux limites non homogènes. Dunod, Paris, 1968.
ne-ja-ha1980 J. Ne$\check {\text {c}}$as, J. Jaru$\check {\text {s}}$ek, J. Haslinger, On the solution of variational inequality to the Signorini problem with small friction. Bolletino U.M.I. 5 (17-B) (1980), 796–811.
paumier1997 J.-C. Paumier, Bifurcations et méthodes numériques, Masson, Paris, 1997.
renard2000 Y. Renard, Singular perturbation approach to an elastic dry friction problem with a non-monotone friction coefficient. Quart. Appl. Math. 58 (2000), 303–324.
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Additional Information
Patrick Hild
Affiliation:
Laboratoire de Mathématiques de Besançon, CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France
Email:
hild@math.univ-fcomte.fr
Yves Renard
Affiliation:
MIP, CNRS UMR 5640, INSAT, Complexe scientifique de Rangueil, 31077 Toulouse, France
Email:
renard@insa-toulouse.fr
Keywords:
Coulomb friction,
unilateral contact,
local uniqueness,
bifurcation,
Clarke’s gradient.
Received by editor(s):
January 13, 2005
Published electronically:
July 11, 2005
Dedicated:
We dedicate this article to the memory of Jean-Claude Paumier.
Article copyright:
© Copyright 2005
Brown University
