This chapter focuses on the numerical aspects of the characterization of friction-induced instabilities and their dynamic computation for linear and nonlinear problems. We begin by presenting briefly basic formulations and several computational methods for solving unilateral frictional contact problems, in quasi-statics and dynamics, and in elasticity and hyper-elasticity. The above specific dynamic formulations will be used to compute the flutter solutions presented in the last sections. Numerical schemes are then given for computing the various sufficient or necessary conditions for instability established together with Professor J.A.C. Martins Finally, the stability analysis and the computation of flutter solutions are carried out for two examples: the sliding of a Polyurethane block on a plane and the squeal of a rubber waist seal sliding on a car window.
In Section 1, a variational inequality formulation and numerical methods for solving quasistatic problems in elasticity are briefly recalled. Details can be found in a previous CISM course volume (see Raous (1999)).
This approach is extended to dynamic problems in Section 2. The formulation is written in terms of differential measures in order to deal with the non-smooth character of the solutions. It is an extension of those developed by J.J. Moreau and M. Jean.
In Section 3, the above formulations are extended to hyper-elastic problems and a method for computing directly the steady sliding solution is given.
Numerical analysis of the stability of quasistatic solutions in the context of linear elasticity is carried out in Section 4. The example of the sliding of a Polyurethane block is studied.
In Section 5, the stability analysis is carried out for a steady sliding solution in the context of hyper-elasticity and used to characterize the squeal of a waist seal sliding on a car window.
Friction Coefficient Complementarity Problem Dynamic Instability Generalize Eigenvalue Problem Unilateral Contact
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.
S. Barbarin (1997) Instabilité et frottement en élasticité linéaire. Application à un problème d’ondes de contrainte, Thesis, Université de Provence, Marseille.Google Scholar
G. Björkman, A. Klarbring, B. Sjödin, T. Larsson and M. Rönnqvist (1995). Sequential quadratic programming for non-linear elastic contact problems, Int. J. Numer. Meth. Engng., 38, 137–165.CrossRefMATHGoogle Scholar
P. Chabrand, F. Dubois and M. Raous (1998). Various numerical methods for solving unilateral contact problems with friction, Mathl. Comput. Modelling., 28, 97–108.CrossRefMATHGoogle Scholar
X. Chateau and Q.S. Nguyen (1991). Buckling of elastic structures in unilateral contact with or without friction, Eur. J. Mech., A/Solids, 10, 71–89.MATHMathSciNetGoogle Scholar
R.W. Clough and J. Penzien (1975). Dynamics of structures, Int. Stud Eds, Mc Graw Hill, Kogakuska.MATHGoogle Scholar
R. W. Cottle, J. S. Pang and R. Stone (1992). The linear complementarily problem, Computer Science and Scientific Computing, Academic press, New-York.Google Scholar
A. Curnier and P. Alart (1988). A generalized Newton method for contact problems with friction, J. Méca. Th. Appl., 7, 67–82.MATHMathSciNetGoogle Scholar
B. Feeny, A. Guran, N. Hinrichs and K. Popp (1998). A historical review on dry friction and stick-slip phenomena, Appl. Mech. Rev., 51, 321–341.ADSCrossRefGoogle Scholar
M. S. Gadala (1986). Numerical solutions of nonlinear problems of continua–II. A survey of incompressibility constraints and software aspects, Comput. Struct., 22, 841–855.CrossRefMATHMathSciNetGoogle Scholar
Q. C. He, J. J. Telega and A. Curnier (1996). Unilateral contact of two solids subject to large deformations: formulation and existence results, Proc. R. Soc. Lond. A, 452, 2691–2717.ADSCrossRefMATHMathSciNetGoogle Scholar
R.A. Ibrahim (1994). Friction-induced vibration, chatter, squeal and chaos: Part I–Mechanics of friction, Part II–Dynamics and modeling, Appl. Mech. Rev, 47, 209–253.ADSCrossRefGoogle Scholar
G. Isac (1992). Complementarity problems, Lecture notes in Mathematics, 1528, Springer Verlag.MATHGoogle Scholar
M. Jean and J.J. Moreau (1987). Dynamics in the presence of unilateral contact and dry friction: a numerical approach, in Del Piero and Maceri (Eds.), Unilateral problems in structural analysis -2, CISM lectures vol. 304, Springer Verlag, 151–196.Google Scholar
N. Josephy (1979). Newton’s method for generalized equations, Report TSR1965, Mathematics research center, University of Wisconsin.Google Scholar
A. Klarbring (1988). On discrete and discretized non-linear elastic structures in unilateral contact (stability, uniqueness and variational principles), Int. J. Solids Structures, 24, 459–479.CrossRefMATHMathSciNetGoogle Scholar
A. Klarbring (1990). Derivation and analysis of rate boundary-value problems of frictional contact, Eur. J. Mech., A/Solids, 9, 53–85.MATHMathSciNetGoogle Scholar
A. Klarbring and G. Björkmann (1992). Solution of large displacement contact problems with friction using Newton’s method for generalized equations, Int. J. Numer. Meth. Engng., 34, 249–269.CrossRefMATHGoogle Scholar
A. Klarbring (1997). Steady sliding and linear complementarity, in M. Ferris and J. C. Pang (Eds.), Complementarity and variational problems: state of the art, SIAM publication, Philadelphia, 132–147.Google Scholar
A. Klarbring (1999). Contact, friction, discrete mechanical structures and mathematical programming, in P. Wriggers–P. Panagiotopoulos (Eds.), New developments in contact problems, CISM Courses and Lectures, 384, Springer Verlag, 55–100.Google Scholar
A.M. Lang and D.A. Crolla (1991). Brake noise and vibration, the state of art, Vehicle Tribology, Tribologies series, 18, 165–173.CrossRefGoogle Scholar
C.E. Lemke (1980). A survey of complementarity theory, in Cottle-Gianessi-Lions (Eds), Variational Inequalities and Complementarily Problems, John Wiley, New York, 213–235.Google Scholar
C. Licht, E. Pratt and M. Raous (1991). Remarks on a numerical method for unilateral contact including friction, Int. Series Num. Math., 101, 129–144.MathSciNetGoogle Scholar
C. H. Liu, G. Hofstetter and H. A. Mang (1994). 3D finite element analysis of rubber-like materials at finite strains, Engng. comput., 11, 111–128.Google Scholar
J. A. C. Martins, J. T. Oden and F. M. F. Simbes (1990). A study of static and kinetic friction, Int. J. Engng. Sci., 28, 29–92.CrossRefMATHGoogle Scholar
J. A. C. Martins, S. Barbarin, M. Raous and A. Pinto da Costa (1999). Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction, Comput. Meth. Appl. Mech. Engng., 177, n 3–4, 289–328.ADSCrossRefMATHMathSciNetGoogle Scholar
J. A. C. Martins and A. Pinto da Costa (2000). Stability of finite dimensional systems with unilateral contact and friction: non-linear elastic behaviour and obstacle curvature, Int. Journal of Solids and Structures, 37, 2519–2564.CrossRefMATHMathSciNetGoogle Scholar
F. Moirot (1998). Etude de la stabilité d’un équilibre en présence de frottement de Coulomb, application au crissement des freins à disques, Ph.D. Thesis, Ecole Polytechnique, Paris, France.Google Scholar
J.-J. Moreau (1988a). Unilateral contact and dry friction in finite freedom dynamics, in Non Smooth Mechanics and Applications, CISM Courses and Lectures, 302, J.-J. Moreau, P.D. Panagiotopoulos (Eds), Springer-Verlag, Wien, 1–82.CrossRefGoogle Scholar
J.-J. Moreau (1988b). Bounded variation in time, In Moreau-Paniagiatopoulos-Strang (Eds), Topics in non-smooth mechanics, Birkhauser Verlag, 1–74.Google Scholar
J.-J. Moreau (1994). Some numerical methods in multibody dynamics: application to granular materials, Eur. J. Mech. A/Solids, 13 (4), 93–114.MATHMathSciNetGoogle Scholar
Q.S. NGuyen (1994). Bifurcation and stability in dissipative media (plasticity, friction, fracture), Appl. Mech. Rev., 47, 1–31.ADSCrossRefGoogle Scholar
J.T. Oden and N. Kikuchi (1982). Finite element methods for constrained problems in elasticity, Int. J. Num. Meth. Eng., 19, 701–725.CrossRefMathSciNetGoogle Scholar
P. Rabier, J. A. C. Martins, J. T. Oden and L. Campos (1986). Existence and local uniqueness of solutions for contact problems with non-linear friction laws, Int. J. Engng. Sci., 24, 1755–1768.CrossRefMATHMathSciNetGoogle Scholar
C. Rajakumar and C. Rogers (1991). The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem, Int. J. Numer. Meth. Engng., 32, 1009–1026.CrossRefMATHMathSciNetGoogle Scholar
A. Ralston and P. Rabinowitz (1978). A first course in numerical analysis, McGRAW-HILL, New-York.MATHGoogle Scholar
M. Raous (Ed) (1988). Numerical methods in mechanics of contact involving friction,J. de Mécanique Théorique et Appliquée, Special Issue, Supp. 1 to vol. 7, Gauthier-Villars.Google Scholar
M. Raous, P. Chabrand and F. Lebon (1988). Numerical methods for solving unilateral contact problems with friction, in [Raous (Ed.), 1988], 111–128.Google Scholar
M. Raous and S. Barbarin (1992). Conjugate Gradient for Frictional Contact, In Curnier A. (Ed), Proceedings of Contact Mechanics International Symposium, Presses Polytech. et Univ. Romandes, Lausanne, 423–432.Google Scholar
M. Raous, J.-J. Moreau and M. Jean (Eds) (1995). Contact Mechanics, Plenum Publisher, New York.Google Scholar
M. Raous, S. Barbarin, D. Vola and J.A.C. Martins (1995). Friction induced instabilities and sound generation, Proceed. ASME Design Engineering Technical Conference, 18–21 september 1995, Boston, USA, 799–802.Google Scholar
M. Raous and S. Barbarin (1996). Stress waves in a sliding contact. Part 2: modelling, in D. Dowson et al. (Eds.), Proceedings of the 22ndLeeds Lyon Symposium on Tribology, Elsevier Science, 39–44.Google Scholar
M. Raous (1999). Quasistatic Signorini problem with Coulomb friction and coupling to adhesion, in P. Wriggers, P. Panagiotopoulos (Eds.), New developments in contact problems, CISM Courses and Lectures, 384, Springer Verlag, 101–178.Google Scholar
M. Raous (2001), Constitutive models and numerical methods for frictional contact, In J.Lemaitre (Ed.), Handbook of material behavior–Non linear models and properties, Section 8.5, Academic Press, 777–786.Google Scholar
R. T. Rockafellar (1970). Convex analysis, Princeton University Press.Google Scholar
J. C. Simo and R. L. Taylor (1991). Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Comput. Meth. Appl. Mech. Engng., 85, 272–310.ADSCrossRefMathSciNetGoogle Scholar
T. Sussman and K.J. Bathe (1987). A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Comp. Stryct., 26, 1–2.CrossRefGoogle Scholar
D. Vola (1998). Frottement et instabilités en dynamique: bruit de crissement, thesis, Université de la Méditerranée, Marseille, France.Google Scholar
D. Vola, E. Pratt, M. Jean and M. Raous (1998). Consistent time discretization for a dynamical frictional contact problem and complementarity techniques, Revue Européenne des Eléments Finis, 7, 149–162.MATHGoogle Scholar
D. Vola, A. Pinto da Costa, S. Barbarin, J.A.C. Martins and M. Raous (1999). Bifurcations and instabilities in some finite dimensional frictional contact problems, in F. Pfeiffer, Ph. Glocker (Eds), Proceedings of 1998 IUTAM symposium: unilateral multibody dynamics, Kluwer, 179–190.Google Scholar
D. Vola, A. M. Raous and J.A.C. Martins (1999). Friction and instability of steady sliding: squeal of a rubber/glass contact, International Journal for Numerical Methods in Engineering, 46, 1699–1720.ADSCrossRefMATHMathSciNetGoogle Scholar
P. Wriggers, T. Vu Van and E. Stein (1990). Finite element formulation for large deformation impact-contact problems with friction, Comp. Struct., 37 (3), 319–331.CrossRefMATHGoogle Scholar
T. Zeghloul and B. Villechaise (1996). Stress waves in a sliding contact. Part 1: experimental study, in D. Dowson et al. (Eds.), Proceedings of the 22ndLeeds Lyon Symposium on Tribology, Elsevier Science, 33–37.Google Scholar