Applications of an Existence Result for the Coulomb Friction Problem

  • Vincent AcaryEmail author
  • Florent Cadoux
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 56)


In a recent paper [2], we prove an abstract existence result for the Coulomb friction problem in discrete time. This problem must be solved at each time step when performing a simulation of the dynamics of a mechanical system involving unilateral contact and Coulomb friction (expressed here at the level of velocities). In this paper, we only recall this result and the gist of its proof and then give an overview of its range of applicability to show the power of our existence criterion. By considering several mechanical systems (Painlevé’s example, granular material on a plan or in a drum) and several particular cases (cases with no moving external objects, cases without friction), we demonstrate the broad range of use-cases to which the criterion can be applied by pure abstract reasoning, without any computations. We also show counter-examples where the criterion does not apply. We then turn to more complicated situations where the existence result cannot be used trivially, and discuss the computational methods that are available to check the criterion in practice using optimization software. It turns out that in suffices to solve a linear program (LP) when the problem is bi-dimensional, and a second order cone program (SOCP) when the problem is tri-dimensional.


Contact Force Existence Result Contact Problem Coulomb Friction External Object 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. LNACM, vol. 35. Springer (2008)Google Scholar
  2. 2.
    Acary, V., Cadoux, F., Lemaréchal, C., Malick, J.: A formulation of the linear discrete Coulomb friction problem via convex optimization. Zeitschrift für Angewandte Mathematik und Mechanik 91, 155–175 (2011)zbMATHCrossRefGoogle Scholar
  3. 3.
    Al-Fahed, A.M., Stavroulakis, G.E., Panagiotopulos, P.D.: Hard and soft fingered robot grippers. The linear complementarity approach. Zeitschrift für Angewandte Mathematik und Mechanik 71, 257–265 (1991)Google Scholar
  4. 4.
    Alart, P., Curnier, A.: A mixed formulation for frictional contact problems prone to Newton like solution method. Computer Methods in Applied Mechanics and Engineering 92(3), 353–375 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for the three dimensional rigid body dynamics. Computer Methods in Applied Mechanics and Engineering 177, 183–197 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Christensen, P., Klarbring, A., Pang, J., Stromberg, N.: Formulation and comparison of algorithms for frictional contact problems. International Journal for Numerical Methods in Engineering 42, 145–172 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Curnier, A., Alart, P.: A generalized Newton method for contact problems with friction. Journal de Mécanique Théorique et Appliquée (suppl. 1-7), 67–82 (1988)Google Scholar
  8. 8.
    De Saxcé, G., Feng, Z.Q.: New inequality and functional for contact with friction: The implicit standard material approach. Mech. Struct. & Mach. 19, 301–325 (1991)CrossRefGoogle Scholar
  9. 9.
    De Saxcé, G., Feng, Z.Q.: The bipotential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms. Math. Comput. Modelling 28(4-8), 225–245 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Duvaut, G., Lions, J.L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972)Google Scholar
  11. 11.
    Feng, Z.Q.: 2D and 3D frictional contact algorithms and applications in a large deformation context. Commnications in Numerical Methods in Engineering 11, 409–416 (1995)zbMATHCrossRefGoogle Scholar
  12. 12.
    Haslinger, J.: Approximation of the Signorini problem with friction, obeying the Coulomb law. Mathematical Methods in the Applied Sciences 5, 422–437 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Haslinger, J.: Least square method for solving contact problems with friction obeying Coulomb’s law. Applications of mathematics 29(3), 212–224 (1984), MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jean, M., Moreau, J.J.: Unilaterality and dry friction in the dynamics of rigid bodies collections. In: Curnier, A. (ed.) Proc. of Contact Mech. Int. Symp., vol. 1, pp. 31–48. Presses Polytechniques et Universitaires Romandes (1992)Google Scholar
  15. 15.
    Jean, M., Touzot, G.: Implementation of unilateral contact and dry friction in computer codes dealing with large deformations problems. J. Méc. Théor. Appl. 7(1), 145–160 (1988)zbMATHGoogle Scholar
  16. 16.
    Jourdan, F., Alart, P., Jean, M.: A Gauss Seidel like algorithm to solve frictional contact problems. Computer Methods in Applied Mechanics and Engineering 155(1), 31–47 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Klarbring, A.: A mathematical programming approach to three-dimensional contact problems with friction. Computer Methods in Applied Mechanics and Engineering 58, 175–200 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Klarbring, A., Björkman, G.: A mathematical programming approach to contact problems with friction and varying contact surface. Computers & Structures 30(5), 1185–1198 (1988)zbMATHCrossRefGoogle Scholar
  19. 19.
    Klarbring, A., Pang, J.S.: Existence of solutions to discrete semicoercive frictional contact problems. SIAM Journal on Optimization 8(2), 414–442 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Leung, A.Y.T., Guoqing, C., Wanji, C.: Smoothing Newton method for solving two– and three–dimensional frictional contact problems. International Journal for Numerical Methods in Engineering 41, 1001–1027 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Mitsopoulou, E.N., Doudoumis, I.N.: A contribution to the analysis of unilateral contact problems with friction. Solid Mechanics Archives 12(3), 165–186 (1987)zbMATHGoogle Scholar
  22. 22.
    Mitsopoulou, E.N., Doudoumis, I.N.: On the solution of the unilateral contact frictional problem for general static loading conditions. Computers & Structures 30(5), 1111–1126 (1988)zbMATHCrossRefGoogle Scholar
  23. 23.
    Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. In: Progress in Nonlinear Differential Equations and their Applications, vol. 9. Birkhauser, Basel (1993)Google Scholar
  24. 24.
    Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D. (eds.) Nonsmooth Mechanics and Applications. CISM, Courses and lectures, vol. 302, pp. 1–82. Spinger, Wien- New York (1988)Google Scholar
  25. 25.
    Pang, J.S., Stewart, D.E.: A unified approach to frictional contact problem. International Journal of Engineering Science 37, 1747–1768 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Pang, J.S., Trinkle, J.C.: Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction. Mathematical Programming 73, 199–226 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Park, J.K., Kwak, B.M.: Three dimensional frictional contact analysis using the homotopy method. Journal of Applied Mechanics, Transactions of A.S.M.E 61, 703–709 (1994)zbMATHCrossRefGoogle Scholar
  28. 28.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. In: Non-linear Dynamics. John Wiley & Sons (1996)Google Scholar
  29. 29.
    Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. International Journal for Numerical Methods in Engineering 39(15) (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.INRIA Rhone-AlpesGrenobleFrance

Personalised recommendations