# American Institute of Mathematical Sciences

April  2016, 9(2): 363-381. doi: 10.3934/dcdss.2016001

## Can the 'stick-slip' phenomenon be explained by a bifurcation in the steady sliding frictional contact problem?

 1 CNRS and UMPC Université Paris 06, Institut Jean le Rond d'Alembert, UMR 7190, 75005 Paris, France

Received  March 2015 Revised  October 2015 Published  March 2016

The stick-slip' phenomenon is the unsteady relative motion of two solids in frictional contact. Tentative explanations were given in the past by enriching the friction law (for example, introducing static and dynamic friction coefficients). In this article, we outline an approach for the analysis of the stick-slip' phenomenon within the simple framework of the coupling of linear elasticity with the Coulomb dry friction law. Simple examples, both discrete and continuous, show that the solutions of the steady sliding frictional contact problem may exhibit bifurcations (loss of uniqueness) when the friction coefficient is taken as a control parameter. It is argued that such a bifurcation could account, in some cases, for the `stick-slip' phenomenon. The situations of a single point particle, of a linear elastic bounded body with homogeneous friction coefficient and of the elastic half-space with both homogenous and piecewise constant friction coefficient are analysed and compared.
Citation: Patrick Ballard. Can the 'stick-slip' phenomenon be explained by a bifurcation in the steady sliding frictional contact problem?. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 363-381. doi: 10.3934/dcdss.2016001
##### References:
 [1] P. Ballard and J. Jarušek, Indentation of an elastic half-space by a rigid flat punch as a model problem for analyzing contact problems with coulomb friction, Journal of Elasticity, 103 (2011), 15-52. doi: 10.1007/s10659-010-9270-9. [2] P. Ballard, Steady sliding frictional contact problems in linear elasticity, Journal of Elasticity, 110 (2013), 33-61. doi: 10.1007/s10659-012-9381-6. [3] P. Ballard, Steady sliding frictional contact problem for an elastic half-space with a discontinuous friction coefficient and related stress singularities, to appear in the special issue of Journal of the Mechanics and Physics of Solids in honour of Pierre Suquet, (2015). [4] R. W. Cottle, J.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Society for Industrial and Applied Mathematics, Philadelphia, 2009. doi: 10.1137/1.9780898719000.ch1. [5] P. Hild, Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity, The Quarterly Journal of Mechanics and Applied Mathematics, 57 (2004), 225-235. doi: 10.1093/qjmam/57.2.225. [6] J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Volume 1, Dunod, Paris, 1968. [7] J. L. Lions and G. Stampacchia, Variational Inequalities, Communications on Pure and Applied Mathematics, 20 (1967), 493-519. doi: 10.1002/cpa.3160200302. [8] J. R. Rice and A. L. Ruina, Stability of steady frictional slipping, Journal of Applied Mechanics, 50 (1983), 343-349. doi: 10.1115/1.3167042. [9] F. G. Tricomi, Integral Equations, Interscience Publishers, New York, 1957.

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##### References:
 [1] P. Ballard and J. Jarušek, Indentation of an elastic half-space by a rigid flat punch as a model problem for analyzing contact problems with coulomb friction, Journal of Elasticity, 103 (2011), 15-52. doi: 10.1007/s10659-010-9270-9. [2] P. Ballard, Steady sliding frictional contact problems in linear elasticity, Journal of Elasticity, 110 (2013), 33-61. doi: 10.1007/s10659-012-9381-6. [3] P. Ballard, Steady sliding frictional contact problem for an elastic half-space with a discontinuous friction coefficient and related stress singularities, to appear in the special issue of Journal of the Mechanics and Physics of Solids in honour of Pierre Suquet, (2015). [4] R. W. Cottle, J.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Society for Industrial and Applied Mathematics, Philadelphia, 2009. doi: 10.1137/1.9780898719000.ch1. [5] P. Hild, Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity, The Quarterly Journal of Mechanics and Applied Mathematics, 57 (2004), 225-235. doi: 10.1093/qjmam/57.2.225. [6] J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Volume 1, Dunod, Paris, 1968. [7] J. L. Lions and G. Stampacchia, Variational Inequalities, Communications on Pure and Applied Mathematics, 20 (1967), 493-519. doi: 10.1002/cpa.3160200302. [8] J. R. Rice and A. L. Ruina, Stability of steady frictional slipping, Journal of Applied Mechanics, 50 (1983), 343-349. doi: 10.1115/1.3167042. [9] F. G. Tricomi, Integral Equations, Interscience Publishers, New York, 1957.
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