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Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem

Published online by Cambridge University Press:  15 March 2005

Patrick Ballard  and
Stéphanie Basseville
Patrick Ballard
Affiliation:
Laboratoire de Mécanique et d'Acoustique, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France. [email protected]
Stéphanie Basseville
Affiliation:
Laboratoire de Mécanique et d'Acoustique, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France. [email protected]
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Abstract

A simple dynamical problem involving unilateral contact and dry friction of Coulomb type is considered as an archetype. We are concerned with the existence and uniqueness of solutions of the system with Cauchy data. In the frictionless case, it is known [Schatzman, Nonlinear Anal. Theory, Methods Appl.2 (1978) 355–373] that pathologies of non-uniqueness can exist, even if all the data are of class C ∞. However, uniqueness is recovered provided that the data are analytic [Ballard, Arch. Rational Mech. Anal.154 (2000) 199–274]. Under this analyticity hypothesis, we prove theexistence and uniqueness of solutions for the dynamical problem with unilateral contact and Coulomb friction, extending [Ballard, Arch. Rational Mech. Anal.154 (2000) 199–274] to the case where Coulomb friction is added to unilateral contact.

Keywords

Unilateral dynamics with frictionexistence and uniqueness.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 59 - 77
Copyright
© EDP Sciences, SMAI, 2005

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References

Ballard, P., The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Rational Mech. Anal. 154 (2000) 199–274. CrossRef
H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Company (1973).
H. Cartan, Théorie Élémentaire des Fonctions Analytiques d'une ou plusieurs Variables Complexes. Hermann, Paris (1961).
A. Klarbring, 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. G. Del Piero and F. Maceri Eds., Springer-Verlag, Wien-New-York (1983) 173–221.
J.J. Moreau, Dynamique de systèmes liaisons unilatérales avec frottement sec éventuel : essais numériques. Note Technique No 85-1 (1985), LMGMC, Montpellier.
J.J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in Nonsmooth Mechanics and Applications, CISM Courses and Lectures No 302, J.J. Moreau and P.D. Panagiotopoulos Eds., Springer-Verlag, Wien-New-York (1988) 1–82.
J.J. Moreau, Bounded variation in time, in Topics in Non-smooth Mechanics, J.J. Moreau, P.D. Panagiotopoulos, G. Strang, Eds., Birkhaüser Verlag, Basel-Boston-Berlin (1988) 1–74.
Percivale, D., Uniqueness in the elastic bounce problem, I. J. Differ. Equations 56 (1985) 206–215. CrossRef
M. Schatzman, A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory, Methods Appl. 2 (1978) 355–373.
Schatzman, M., Uniqueness and continuous dependence on data for one dimensional impact problems. Math. Comput. Modelling 28 (1998) 1–18. CrossRef