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

Patrick Ballard; Stéphanie Basseville

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 39.1 (2010): 59-77. <http://eudml.org/doc/194258>.

@article{Ballard2010,
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 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},
keywords = {Unilateral dynamics with friction; existence and uniqueness.; unilateral dynamics with friction; existence and uniqueness},
language = {eng},
month = {3},
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/194258},
volume = {39},
year = {2010},
}

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
DA - 2010/3//
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∞. 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.; unilateral dynamics with friction; existence and uniqueness
UR - http://eudml.org/doc/194258
ER -

References

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  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.  
  10. D. Percivale, Uniqueness in the elastic bounce problem, I. J. Differ. Equations56 (1985) 206–215.  
  11. M. Schatzman, A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory, Methods Appl.2 (1978) 355–373.  
  12. M. Schatzman, Uniqueness and continuous dependence on data for one dimensional impact problems. Math. Comput. Modelling28 (1998) 1–18.  

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