Existence and uniqueness of solutions to dynamical unilateral contact problems with coulomb friction: the case of a collection of points
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, p. 1-25

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.

DOI : https://doi.org/10.1051/m2an/2013092
Classification:  70F40,  49J52,  34A60
Keywords: unilateral dynamics with friction, frictional dynamical contact problems, existence and uniqueness
@article{M2AN_2014__48_1_1_0,
author = {Charles, Alexandre and Ballard, Patrick},
title = {Existence and uniqueness of solutions to dynamical unilateral contact problems with coulomb friction: the case of a collection of points},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {1},
year = {2014},
pages = {1-25},
doi = {10.1051/m2an/2013092},
mrnumber = {3177835},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_1_1_0}
}

Charles, Alexandre; Ballard, Patrick. Existence and uniqueness of solutions to dynamical unilateral contact problems with coulomb friction: the case of a collection of points. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 1-25. doi : 10.1051/m2an/2013092. http://www.numdam.org/item/M2AN_2014__48_1_1_0/

[1] L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives. Proc. Amer. Math. Soc. 108 (1990) 691-702. | MR 969514 | Zbl 0685.49027

[2] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154 (2000) 199-274. | MR 1785473 | Zbl 0965.70024

[3] P. Ballard and S. Basseville, Existence and uniqueness for dynamical unilateral contact with coulomb friction: a model problem, ESAIM: M2AN 39 (2005) 59-77. | Numdam | MR 2136200 | Zbl 1089.34010

[4] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Company (1973). | MR 348562 | Zbl 0252.47055

[5] 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). | MR 2128865 | Zbl 1079.74003

[6] A. Klarbring, Ingenieur-Archiv 60 (1990) 529-541.

[7] M.D.P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems. Birkhaüser, Basel-Boston-Berlin (1993). | MR 1231975 | Zbl 0802.73003

[8] 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. | Zbl 0619.73115

[9] J.J. Moreau, Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel: essais numériques, Note Technique No 85-1, LMGC, Montpellier (1985).

[10] 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. | Zbl 0703.73070

[11] 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. | MR 957087 | Zbl 0657.28008

[12] P. Painlevé, Sur les lois du frottement de glissement. C.R. Acad. Sci. (Paris) 121 (1895) 112-115. | JFM 26.0781.03

[13] D. Percivale, Uniqueness in the Elastic Bounce Problem, I, J. Diff. Eqs. 56 (1985) 206-215. | MR 774163 | Zbl 0521.73006

[14] M. Schatzman, A Class of Nonlinear Differential Equations of Second Order in Time, Nonlinear Analysis. Theory, Methods Appl. 2 (1978) 355-373. | MR 512664 | Zbl 0382.34003

[15] M. Schatzman, Uniqueness and continuous dependence on data for one dimensional impact problems. Math. Comput. Modell. 28 (1998) 1-18. | MR 1616372 | Zbl 1122.74473