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

Patrick Ballard; Stéphanie Basseville

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 1, page 59-77
  • ISSN: 0764-583X

Abstract

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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 the existence 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.

How to cite

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Ballard, Patrick, and Basseville, Stéphanie. "Existence and uniqueness for dynamical unilateral contact with Coulomb friction : a model problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.1 (2005): 59-77. <http://eudml.org/doc/244894>.

@article{Ballard2005,
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^\infty $. 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 the existence 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.},
author = {Ballard, Patrick, Basseville, Stéphanie},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {unilateral dynamics with friction; existence and uniqueness},
language = {eng},
number = {1},
pages = {59-77},
publisher = {EDP-Sciences},
title = {Existence and uniqueness for dynamical unilateral contact with Coulomb friction : a model problem},
url = {http://eudml.org/doc/244894},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Ballard, Patrick
AU - Basseville, Stéphanie
TI - Existence and uniqueness for dynamical unilateral contact with Coulomb friction : a model problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 1
SP - 59
EP - 77
AB - 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^\infty $. 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 the existence 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.
LA - eng
KW - unilateral dynamics with friction; existence and uniqueness
UR - http://eudml.org/doc/244894
ER -

References

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  1. [1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Rational Mech. Anal. 154 (2000) 199–274. Zbl0965.70024
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  8. [8] 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. Zbl0703.73070
  9. [9] 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. Zbl0657.28008
  10. [10] D. Percivale, Uniqueness in the elastic bounce problem, I. J. Differ. Equations 56 (1985) 206–215. Zbl0521.73006
  11. [11] M. Schatzman, A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory, Methods Appl. 2 (1978) 355–373. Zbl0382.34003
  12. [12] M. Schatzman, Uniqueness and continuous dependence on data for one dimensional impact problems. Math. Comput. Modelling 28 (1998) 1–18. Zbl1122.74473

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