Contact Problems Involving Friction

Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 593)


The Coulomb friction law is simple to apply in the formulation of elastic contact problems, but it is also a rich source of unexpected physical phenomena, including ranges of unstable dynamic response, history-dependence, ‘wedging’ and mathematical problems of existence and uniqueness of solution. We first explore the implications of the law in the context of simple discrete systems and demonstrate the importance of interaction [coupling] between the normal and tangential contact problems, particularly in problems of periodic loading. The discussion is then extended to problems of the elastic continuum, and to cases where elastodynamic effects must be included [for example, the interaction of a seismic disturbance with a frictional interface]. It is shown that finite element formulations of elastodynamic problems with Coulomb friction are inherently ill-posed and alternative friction laws that avoid this difficulty are discussed.


  1. Adams, G. G. (1995). Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction. ASME Journal of Applied Mechanics, 62, 867–872.CrossRefGoogle Scholar
  2. Adams, G. G. (1998). Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip. ASME Journal of Applied Mechanics, 65, 470–475.CrossRefGoogle Scholar
  3. Ahn, Y. J. (2010). Discontinuity of quasi-static solution in the two-node coulomb frictional system. International Journal of Solids and Structures, 47, 2866–2871.CrossRefGoogle Scholar
  4. Ahn, Y. J., & Barber, J. R. (2008). Response of frictional receding contact problems to cyclic loading. International Journal of Mechanical Sciences, 50, 1519–1525.CrossRefGoogle Scholar
  5. Ahn, Y. J., Bertocchi, E., & Barber, J. R. (2008). Shakedown of coupled two-dimensional discrete frictional systems. Journal of the Mechanics and Physics of Solids, 56, 3433–3440.CrossRefGoogle Scholar
  6. Andersson, L.-E., Barber, J. R., & Ahn, Y.-J. (2013). Attractors in frictional systems subjected to periodic loads. SIAM Journal of Applied Mathematics, 73, 1097–1116.MathSciNetCrossRefGoogle Scholar
  7. Andersson, L.-E., Barber, J. R., & Ponter, A. R. S. (2014). Existence and uniqueness of attractors in frictional systems with uncoupled tangential displacements and normal tractions. International Journal of Solids and Structures, 51, 3710–3714.CrossRefGoogle Scholar
  8. Archard, J. F. (1953). Contact and rubbing of flat surfaces. Journal of Applied Physics, 24, 981–988.CrossRefGoogle Scholar
  9. Archard, J. F. (1957). Elastic deformation and the laws of friction. Proceedings of the Royal Society of London, A, 243, 190–205.CrossRefGoogle Scholar
  10. Barber, J. R. (2013). Multiscale surfaces and Amontons’ law of friction. Tribology Letters, 49, 539–543.CrossRefGoogle Scholar
  11. Barber, J. R., Davies, M., & Hills, D. A. (2011). Frictional elastic contact with periodic loading. International Journal of Solids and Structures, 48, 2041–2047. Scholar
  12. Bowden, F. P., & Tabor, D. (1950). The friction and lubrication of solids. Oxford: Clarendon Press.zbMATHGoogle Scholar
  13. Cabboi, A., Putelat, T., & Woodhouse, J. (2016). The frequency response of dynamic friction: Enhanced rate-and-state models. Journal of the Mechanics and Physics of Solids, 92, 210–236.CrossRefGoogle Scholar
  14. Cattaneo, C. (1938). Sul contatto di due corpi elastici: Distribuzione locale degli sforzi. Rendiconti dell’Accademia Nazionale dei Lincei, 27, 342–348, 434–436, 474–478.Google Scholar
  15. Chez, E. L., Dundurs, J., & Comninou, M. (1978). Reflection and refraction of sh waves in presence of slip and friction. Bulletin of the Seismological Society of America, 68, 999–1011.zbMATHGoogle Scholar
  16. Cho, H., & Barber, J. R. (1998). Dynamic behavior and stability of simple frictional systems. Mathematical and Computer Modeling, 28, 37–53.CrossRefGoogle Scholar
  17. Ciavarella, M. (1998). The generalized Cattaneo partial slip plane contact problem. I—Theory, II—Examples. International Journal of Solids and Structures, 35, 2363–2378.MathSciNetCrossRefGoogle Scholar
  18. Comninou, M., & Dundurs, J. (1979). Interaction of elastic waves with a unilateral interface. Proceedings of the Royal Society of London, A, 368, 141–154. Scholar
  19. Dundurs, J. (1969). Discussion on edge bonded dissimilar orthogonal elastic wedges under normal and shear loading. ASME Journal of Applied Mechanics, 36, 650–652.CrossRefGoogle Scholar
  20. Greenwood, J. A., & Williamson, J. B. P. (1966). The contact of nominally flat surfaces. Proceedings of the Royal Society (London), A 295, 300–319.Google Scholar
  21. Haslinger, J., & Nedlec, J. C. (1983). Approximation of the Signorini problem with friction, obeying the Coulomb law. Mathematical Methods in the Applied Sciences, 5, 422–437.MathSciNetCrossRefGoogle Scholar
  22. Hassani, R., Hild, P., Ionescu, I. R., & Sakki, N. D. (2003). A mixed finite element method and solution multiplicity for Coulomb frictional contact. Computer Methods in Applied Mechanics and Engineering, 192, 4517–4531.MathSciNetCrossRefGoogle Scholar
  23. Hild, P. (2004). Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity. Quarterly Journal of Mechanics and Applied Mathematics, 57, 225–235.MathSciNetCrossRefGoogle Scholar
  24. Hu, Z., Lu, W., Thouless, M. D., & Barber, J. R. (2016). Effect of plastic deformation on the evolution of wear and local stress fields in fretting. International Journal of Solids and Structures, 82, 1–8.CrossRefGoogle Scholar
  25. Iwan, W. D. (1967). On a class of models for the yielding behaviour of continuous and composite systems. ASME Journal of Applied Mechanics, 34, 612–617.CrossRefGoogle Scholar
  26. Jager, J. (1998). A new principle in contact mechanics. ASME Journal of Tribology, 120, 677–684.CrossRefGoogle Scholar
  27. Kikuchi, N., & Oden, J. T. (1988). Contact problems in elasticity: A study of variational inequalities and finite element methods. Philadelphia: SIAM.CrossRefGoogle Scholar
  28. Klarbring, A. (1990). Examples of non-uniqueness and non-existence of solutions to quasi-static contact problems with friction. Ingenieur-Archiv, 60, 529–541.Google Scholar
  29. Klarbring, A. (1999). Contact, friction, discrete mechanical structures and discrete frictional systems and mathematical programming. In P. Wriggers & P. Panagiotopoulos (Eds.), New developments in contact problems (pp. 55–100). Wien: Springer.zbMATHGoogle Scholar
  30. Klarbring, A., Ciavarella, M., & Barber, J. R. (2007). Shakedown in elastic contact problems with Coulomb friction. International Journal of Solids and Structures, 44, 8355–8365.CrossRefGoogle Scholar
  31. Martins, J. A. C., Montiero Marques, M. D. P., Gastaldi, F., Simoes, F. M. F. (1992). A two degree-of-freedom “quasistatic” frictional contact problem with instantaneous jumps. In Contact Mechanics International SymposiumGoogle Scholar
  32. Melan, E. (1936). Theorie statisch unbestimmter Systeme aus ideal-plastischen Baustoff. Sitzungsberichte der Akademie der Wissenschaften in Wien, 145, 195–218.zbMATHGoogle Scholar
  33. Mindlin, R. D. (1949). Compliance of elastic bodies in contact. ASME Journal of Applied Mechanics, 16, 259–268.MathSciNetzbMATHGoogle Scholar
  34. Munisamy, R. L., Hills, D. A., & Nowell, D. (1994). Static axisymmetrical hertzian contacts subject to shearing forces. ASME Journal of Applied Mechanics, 61, 278–283.CrossRefGoogle Scholar
  35. Persson, B. N. J. (2001). Theory of rubber friction and contact mechanics. Journal of Chemical Physics, 115, 3840–3861.CrossRefGoogle Scholar
  36. Prakash, V. (1998). Frictional response of sliding interfaces subjected to time varying normal pressures. ASME Journal of Tribology, 120, 97–102.CrossRefGoogle Scholar
  37. Rabinowicz, E. (1951). The nature of the static and kinetic coefficients of friction. Journal of Applied Physics, 22, 1373–1379.CrossRefGoogle Scholar
  38. Ranjith, K., & Rice, J. R. (2001). Slip dynamics at an interface between dissimilar materials. Journal of the Mechanics and Physics of Solids, 49, 341–361.CrossRefGoogle Scholar
  39. Spence, D. A. (1975). The Hertz problem with finite friction. Journal of Elasticity, 5, 297–319. Scholar
  40. Storakers, B., & Elaguine, D. (2005). Hertz contact at finite friction and arbitrary profiles. Journal of the Mechanics and Physics of Solids, 53, 1422–1447.MathSciNetCrossRefGoogle Scholar
  41. Thaitirarot, A., Ahn, Y. J., Hills, D. A., Jang, Y. H., & Barber, J. R. (2014). The use of static reduction in the finite element solution of two-dimensional frictional contact problems. Journal of Mechanical Engineering Science, 228, 1474–1487.CrossRefGoogle Scholar
  42. Turner, J. R. (1979). Frictional unloading problem on a linear elastic half-space. Journal of the Institute of Mathematics and its Applications, 24, 439–469. Scholar

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© CISM International Centre for Mechanical Sciences 2020

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MichiganAnn ArborUSA

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