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

Previous Article
Parametric nonlinear PDEs with multiple solutions: A PGD approach
DCDS-S Home
This Issue
Next Article
Preface

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?

Patrick Ballard ^1,

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

Abstract
Related Papers

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.

Keywords: stick-slip phenomenon, Elasticity, bifurcation, variational inequality., friction.

Mathematics Subject Classification: 74B05, 74M10, 74M1.

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.

show all references

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.

[1]	Stanislaw Migórski. A class of hemivariational inequalities for electroelastic contact problems with slip dependent friction. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 117-126. doi: 10.3934/dcdss.2008.1.117
[2]	Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[3]	Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058
[4]	S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial and Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155
[5]	Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
[6]	Thanyarat JItpeera, Tamaki Tanaka, Poom Kumam. Triple-hierarchical problems with variational inequality. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021038
[7]	Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[8]	Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621
[9]	Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[10]	T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675
[11]	Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial and Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673
[12]	Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645
[13]	Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545
[14]	Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261
[15]	Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012
[16]	Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[17]	Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331
[18]	G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure and Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583
[19]	Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[20]	Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Patrick Ballard
on Google Scholar
Patrick Ballard

[Back to Top]