Three Examples Concerning the Interaction of Dry Friction and Oscillations

  • Alexander MielkeEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 27)


We discuss recent work concerning the interaction of dry friction, which is a rate independent effect, and temporal oscillations. First, we consider the temporal averaging of highly oscillatory friction coefficients. Here the effective dry friction is obtained as an infimal convolution. Second, we show that simple models with state-dependent friction may induce a Hopf bifurcation, where constant shear rates give rise to periodic behavior where sticking phases alternate with sliding motion. The essential feature here is the dependence of the friction coefficient on the internal state, which has an internal relaxation time. Finally, we present a simple model for rocking toy animal where walking is made possible by a periodic motion of the body that unloads the legs to be moved.



The authors is grateful to Martin Heida and Elias Pipping for stimulating discussions. The research was partially supported by DFG via the project B01 Fault networks and scaling properties of deformation accumulation within the SFB 1114 Scaling Cascades in Complex Systems.


  1. 1.
    Abe, Y., Kato, N.: Complex earthquake cycle simulations using a two-degree-of-freedom spring-block model with a rate- and state-friction law. Pure Appl. Geophys. 170(5), 745–765 (2013)CrossRefGoogle Scholar
  2. 2.
    Brokate, M., Krejčí, P., Schnabel, H.: On uniqueness in evolution quasivariational inequalities. J. Convex Anal. 11, 111–130 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)CrossRefGoogle Scholar
  4. 4.
    DeSimone, A., Gidoni, P., Noselli, G.: Liquid crystal elastomer strips as soft crawlers. J. Mech. Phys. Solids 84, 254–272 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gidoni, P., DeSimone, A.: On the genesis of directional friction through bristle-like mediating elements crawler. arXiv:1602.05611 (2016)Google Scholar
  6. 6.
    Gidoni, P., DeSimone, A.: Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler. Meccanica 52(3), 587–601 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gidoni, P., Noselli, G., DeSimone, A.: Crawling on directional surfaces. Int. J. Non-Linear Mech. 61, 65–73 (2014)CrossRefGoogle Scholar
  8. 8.
    Heida, M., Mielke, A.: Averaging of time-periodic dissipation potentials in rate-independent processes. Discr. Cont. Dynam. Syst. Ser. S 10(6), 1303–1327 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Heida, M., Mielke, A., Pipping, E.: Rate-and-state friction from a thermodynamical viewpoint. In preparation (2017)Google Scholar
  10. 10.
    Mielke, A.: Emergence of rate-independent dissipation from viscous systems with wiggly energies. Contin. Mech. Thermodyn. 24(4), 591–606 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mielke, A., Rossi, R.: Existence and uniqueness results for a class of rate-independent hysteresis problems. Math. Models Meth. Appl. Sci. 17(1), 81–123 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015)CrossRefGoogle Scholar
  13. 13.
    Pfeiffer, F.: Mechanische Systeme mit unstetigen Übergängen. Ingenieur-Archiv 54, 232–240 (1984). (In German)CrossRefGoogle Scholar
  14. 14.
    Pipping, E.: Existence of long-time solutions to dynamic problems of viscoelasticity with rate-and-state friction. arXiv:1703.04289v1 (2017)Google Scholar
  15. 15.
    Pipping, E., Kornhuber, R., Rosenau, M., Oncken, O.: On the efficient and reliable numerical solution of rate-and-state friction problems. Geophys. J. Int. 204(3), 1858–1866 (2016)CrossRefGoogle Scholar
  16. 16.
    Popov, V.L., Gray, J.A.T.: Prandtl-Tomlinson model: History and applications in friction, plasticity, and nanotechnologies. Z. Angew. Math. Mech. 92(9), 692–708 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Popov, V.L.: Contact Mechanics and Friction. Springer, New York (2010)CrossRefGoogle Scholar
  18. 18.
    Prandtl, L.: Gedankenmodel zur kinetischen Theorie der festen Körper. Z. Angew. Math. Mech. 8, 85–106 (1928)CrossRefGoogle Scholar
  19. 19.
    Radtke, M., Netz, R.R.: Shear-induced dynamics of polymeric globules at adsorbing homogeneous and inhomogeneous surfaces. Euro. Phys. J. E 37(20), 11 (2014)Google Scholar
  20. 20.
    Roubíček, T.: A note about the rate-and-state-dependent friction model in a thermodynamical framework of the biot-type equation. Geophys. J. Int. 199(1), 286–295 (2014)CrossRefGoogle Scholar
  21. 21.
    Tomlinson, G.A.: A molecular theory of friction. Phil. Mag. 7, 905–939 (1929)CrossRefGoogle Scholar
  22. 22.
    Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Weierstraß-Institut für Angewandte Analysis und StochastikBerlinGermany
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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