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Existence and uniqueness of solutions to dynamical unilateral contact problems with coulomb friction: the case of a collection of points

Published online by Cambridge University Press:  11 October 2013

Alexandre Charles  and
Patrick Ballard
Alexandre Charles
Affiliation:
Laboratoire de Mécanique et d’Acoustique, CNRS, 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, France. [email protected]
Patrick Ballard
Affiliation:
Laboratoire de Mécanique et d’Acoustique, CNRS, 31, chemin Joseph Aiguier, 13402 Marseille Cedex 20, France. [email protected]
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Abstract

This study deals with the existence and uniqueness of solutions to dynamical problems of finite freedom involving unilateral contact and Coulomb friction. In the frictionless case, it has been established [P. Ballard, Arch. Rational Mech. Anal. 154 (2000) 199–274] that the existence and uniqueness of a solution to the Cauchy problem can be proved under the assumption that the data are analytic, but not if they are assumed to be only of class C∞. Some years ago, this finding was extended [P. Ballard and S. Basseville, Math. Model. Numer. Anal. 39 (2005) 59–77] to the case where Coulomb friction is included in a model problem involving a single point particle. In the present paper, the existence and uniqueness of a solution to the Cauchy problem is proved in the case of a finite collection of particles in (possibly non-linear) interactions.

Keywords

Unilateral dynamics with frictionfrictional dynamical contact problemsexistence and uniqueness
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 1 - 25
Copyright
© EDP Sciences, SMAI, 2013

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References

Ambrosio, L. and Dal Maso, G., A general chain rule for distributional derivatives. Proc. Amer. Math. Soc. 108 (1990) 691–702. Google Scholar
Ballard, P., The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154 (2000) 199–274. Google Scholar
Ballard, P. and Basseville, S., Existence and uniqueness for dynamical unilateral contact with coulomb friction: a model problem, ESAIM: M2AN 39 (2005) 59–77. Google Scholar
H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Company (1973).
C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems in Mechanics. Variational Methods and Existence Theorems. Monographs & Textbooks in Pure & Appl. Math. No. 270 (ISBN 1-57444-629-0). Chapman & Hall/CRC, Boca Raton (2005).
Klarbring, A., Ingenieur-Archiv 60 (1990) 529–541.
M.D.P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems. Birkhaüser, Basel–Boston–Berlin (1993).
J.J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in Unilateral problems in structural analysis, edited by G. Del Piero and F. Maceri. Springer-Verlag, Wien–New-York (1983) 173–221.
J.J. Moreau, Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel: essais numériques, Note Technique No 85-1, LMGC, Montpellier (1985).
J.J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in Nonsmooth Mechanics and Applications, CISM Courses and Lectures No 302, edited by J.J. Moreau and P.D. Panagiotopoulos. Springer-Verlag, Wien–New-York (1988) 1–82.
J.J. Moreau, Bounded variation in time, in Topics in Non-smooth Mechanics, edited by J.J. Moreau, P.D. Panagiotopoulos and G. Strang. Birkhaüser Verlag, Basel-Boston-Berlin (1988) 1–74.
Painlevé, P., Sur les lois du frottement de glissement. C.R. Acad. Sci. (Paris) 121 (1895) 112–115. Google Scholar
Percivale, D., Uniqueness in the Elastic Bounce Problem, I, J. Diff. Eqs. 56 (1985) 206–215. Google Scholar
Schatzman, M., A Class of Nonlinear Differential Equations of Second Order in Time, Nonlinear Analysis. Theory, Methods Appl. 2 (1978) 355–373. Google Scholar
Schatzman, M., Uniqueness and continuous dependence on data for one dimensional impact problems. Math. Comput. Modell. 28 (1998) 1–18. Google Scholar