Comptes Rendus
Instability and friction
Comptes Rendus. Mécanique, Volume 331 (2003) no. 1, pp. 99-112.

A review on the stability analysis of solids in unilateral and frictional contact is given. The presentation is focussed on the stability of an equilibrium position of an elastic solid in frictional contact with a fixed or moving obstacle. The problem of divergence instability and the obtention of a criterion of static stability are discussed first for the case of a fixed obstacle. The possibility of flutter instability is then considered for a steady sliding equilibrium with a moving obstacle. The steady sliding solution is generically unstable by flutter and leads to a dynamic response which can be chaotic or periodic. This dynamic response leads to the generation of stick–slip–separation waves on the contact surface in a similar way as Schallamach waves in statics. Illustrating examples and principal results recently obtained in the literature are reported. Some problems of friction-induced vibration and noise emittence, such as brake squeal for example, can be interpreted in this spirit.

On présente dans cette Note une synthèse des résultats de la littérature sur l'analyse de stabilité des solides sous contact unilatéral avec frottement de Coulomb. Le problème de contact frottant d'un solide élastique avec un obstacle fixe ou mobile est examiné. La possibilité d'instabilité par divergence et la recherche d'un critère de stabilité statique d'un équilibre sont examinées dans le cas d'un obstacle fixe. La possibilité d'instabilité par flottement est ensuite discutée pour un équilibre résultant d'un glissement stationnaire avec un obstacle mobile. La solution de glissement stationnaire est dynamiquement instable par flottement et conduit à une réponse dynamique qui peut être chaotique ou périodique. En particulier, la réponse dynamique génère des ondes d'adhérence–glissement–séparation sur la surface de contact d'une façon comparable aux ondes de Schallamach en statique. Des exemples et des résultats récents de la littérature sont rapportés. Quelques problèmes de vibration et d'émission acoustique induites par le frottement, comme le crissement des freins par exemple, peuvent être interprétés dans cet esprit.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-0721(03)00020-2
Keywords: Friction, Unilateral contact, Instability, Stick–slip–separation waves
Mots-clés : Frottement, Contact unilatéral, Instabilité, Ondes adhérence–glissement–séparation

Quoc Son Nguyen 1

1 Laboratoire de mécanique des solides, CNRS-UMR7649, École polytechnique, 91128 Palaiseau cedex, France
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Quoc Son Nguyen. Instability and friction. Comptes Rendus. Mécanique, Volume 331 (2003) no. 1, pp. 99-112. doi : 10.1016/S1631-0721(03)00020-2. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00020-2/

[1] G. de Saxcé A generalization of Fenchel's inequality and its application to constitutive laws, C. R. Acad. Sci. Paris II, Volume 314 (1992), pp. 125-129

[2] J.J. Moreau On unilateral constraints, friction and plasticity, New Variational Techniques in Mathematical Physics, CIME Course, Springer-Verlag, 1974

[3] Q.S. Nguyen Stability and Nonlinear Solid Mechanics, Wiley, Chichester, 2000

[4] G. Duvaut; J.-L. Lions Les inéquations en mécanique et en physique, Dunod, Paris, 1972

[5] J.T. Oden; J.A.C. Martins Models and computational methods for dynamic friction phenomena, Comput. Methods Appl. Mech. Engrg., Volume 52 (1985), pp. 527-634

[6] N. Kikuchi; J.T. Oden Contact Problem in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988

[7] L.E. Andersson A quasistatic frictional problem with normal compliance, Nonlinear Anal., Volume 16 (1991), pp. 347-370

[8] L.E. Andersson; A. Klarbring Existence and uniqueness for quasistatic contact problems with friction (J.A.C. Martins; M.D.P. Montero Marques, eds.), Contact Mechanics, Kluwer Academic, London, 2001, pp. 245-260

[9] A. Klarbring; A. Mikelic; M. Shillor A global existence result for the quasistatic frictional contact problem with normal compliance (Del Piero; Maceri, eds.), Unilateral Problems in Structural Mechanics IV, Birkhäuser, Boston, 1991, pp. 85-111

[10] M. Cocu; E. Pratt; M. Raous Analysis of an incremental formulation for frictional contact problems, Contact Mechanics Marseille, 1995, Plenum Press, New York, 1996

[11] R. Hill A general theory of uniqueness and stability in elastic/plastic solids, J. Mech. Phys. Solids, Volume 6 (1958), pp. 236-249

[12] H. Petryk A consistent approach to defining stability of plastic deformed processes, IUTAM Symp. Stability in the Mechanics of Continua, Springer-Verlag, Berlin, 1982, pp. 262-272

[13] A. Klarbring Derivation and analysis of rate boundary value problems of frictional contact, Eur. J. Mech. A Solids, Volume 9 (1990), pp. 53-85

[14] A. Klarbring Stability and critical points in large displacement frictionless contact problems (J.A.C. Martins; M. Raous, eds.), Friction and Instabilities, Udine 2000, CISM Courses and Lectures, 457, Springer, Wien, 2002, pp. 39-64

[15] X. Chateau; Q.S. Nguyen Buckling of elastic structures in unilateral contact with or without friction, Eur. J. Mech. A Solids, Volume 10 (1991), pp. 71-89

[16] M.A. Biot Mechanics of Incremental Deformation, Wiley, New York, 1965

[17] Q.S. Nguyen Bifurcation and stability in dissipative media (plasticity, friction, fracture), Appl. Mech. Rev., Volume 47 (1994), pp. 1-31

[18] Z. Mroz Contact friction models and stability problems (J.A.C. Martins; M. Raous, eds.), Friction and Instabilities, Udine 2000, CISM Courses and Lectures, 457, Springer, Wien, 2002, pp. 179-232

[19] Q.S. Nguyen Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale, J. Méc. Théor. Appl., Volume 3 (1984), pp. 41-61

[20] J.A.C. Martins; A. Pinto da Costa; F.M.F. Simoes Some notes on friction and instabilities (J.A.C. Martins; M. Raous, eds.), Friction and Instabilities, Udine 2000, CISM Courses and Lectures, 457, Springer, Wien, 2002, pp. 65-136

[21] R. Cottle; J.S. Pang; R.E. Stone The Linear Complementarity Problem, Academic Press, New York, 1992

[22] A. Klarbring A mathematical programming approach to three-dimensional contact problems with friction, Comput. Methods Appl. Mech. Engrg., Volume 58 (1986), pp. 175-200

[23] G.G. Adams Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction, J. Appl. Mech., Volume 62 (1995), pp. 867-872

[24] J.A.C. Martins; J. Guimaraes; L.O. Faria Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions, J. Vib. Acoustics, Volume 117 (1995), pp. 445-451

[25] F. Moirot; Q.S. Nguyen Some problems of friction-induced vibrations and instabilities (J.A.C. Martins; M. Raous, eds.), Friction and Instabilities, Udine 2000, CISM Courses and Lectures, 457, Springer, Wien, 2002, pp. 137-178

[26] G.G. Adams Dynamic instabilities in the sliding of two layered elastic half-spaces, J. Tribology, Volume 120 (1998), pp. 289-295

[27] F. Moirot, Etude de la stabilité d'un équilibre en présence du frottement de Coulomb. Application au crissement des freins à disque, Thèse, École Polytechnique, Paris, 1998

[28] M. Renardy Ill-posedness at the boundary for elastic solids sliding under Coulomb friction, J. Elasticity, Volume 27 (1992), pp. 281-287

[29] K. Ranjith; J.R. Rice Slip dynamics at an interface between dissimilar materials, J. Mech. Phys. Solids, Volume 49 (2001), pp. 341-361

[30] H. Troger; A. Steindl Nonlinear Stability and Bifurcation Theory, Springer-Verlag, Wien, 1991

[31] F. Moirot; Q.S. Nguyen An example of stick–slip waves, C. R. Acad. Sci. Paris IIb, Volume 328 (2000), pp. 663-669

[32] A. Oueslati; L. Baillet; Q.S. Nguyen Transition vers une onde glissement-adhérence-décollement sous contact frottant de Coulomb, Journées Européennes sur les freins, Hermès, Paris, 2002, pp. 155-162

[33] M. Oestreich; N. Hinrichs; K. Popp Bifurcation and stability analysis for a non-smooth friction oscillator, Arch. Appl. Mech., Volume 66 (1996), pp. 301-314

[34] D. Vola; M. Raous; J.A.C. Martins Friction and instability of steady sliding squeal of a glass/rubber contact, Int. J. Numer. Methods Engrg., Volume 45 (1999), pp. 301-314

[35] N.J. Carpenter; R.L. Taylor; M.G. Katona Lagrange constraints for transient finite element surface contact, Int. J. Numer. Methods Engrg., Volume 32 (1991), pp. 103-128

[36] L. Baillet; Y. Berthier; O. Bontemps; M. Brunet Tribologie de l'interface fibre/matrice. Approche théorique et expérimentale, Rev. Composites et Matériaux Avancés, Volume 7 (1997), pp. 89-105

[37] B. van der Pol On relaxation-oscillations, Philos. Mag., Volume 7 (1926), pp. 978-992

[38] K. Popp; P. Stelter Stick–slip vibrations and chaos, Philos. Trans. Roy. Soc. London Ser. A, Volume 332 (1990), pp. 89-105

[39] G.G. Adams Steady sliding of two elastic half-spaces with friction reduction due to interface stick–slip, J. Appl. Mech., Volume 65 (1998), pp. 470-475

[40] O.Y. Zharii Frictional contact between the surface wave and a rigid strip, J. Appl. Mech., Volume 63 (1996), pp. 15-20

[41] J.A.C. Martins; S. Barbarin; M. Raous; A. Pinto da Costa Dynamic stability of finite dimensional linear elastic system with unilateral contact and Coulomb's friction, Comput. Methods Appl. Mech. Engrg., Volume 177 (1999), pp. 298-328

[42] D. Vola; E. Pratt; M. Jean; M. Raous Consistent time discretization for a dynamical frictional contact problem and complementary techniques, Rev. Eur. Elements Finis, Volume 7 (1998), pp. 149-162

[43] Y. Renard, Modélisation des instabilités liées au frottement sec des solides, aspects théoriques et numériques, Thèse de doctorat, Université de Grenoble, 1998

[44] M. Raous; S. Barbarin; D. Vola Numerical characterization and computation of dynamic instabilities for frictional contact problems (J.A.C. Martins; M. Raous, eds.), Friction and Instabilities, Udine 2000, CISM Courses and Lectures, 457, Springer, Wien, 2002, pp. 233-291

[45] D.A. Crolla; A.M. Lang Brake noise and vibration: state of art, Vehicle Tribologie, Volume 18 (1991), pp. 165-174

[46] M. Nakai; M. Yokoi Band brake squeal, J. Vib. Acoustics, Volume 118 (1996), pp. 187-197

[47] F. Moirot; Q.S. Nguyen Brake squeal: a problem of flutter instability of the steady sliding solution?, Arch. Mech., Volume 52 (2000), pp. 645-662

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