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Trends in Applications of Mathematics to Mechanics pp 159–177Cite as

Three Examples Concerning the Interaction of Dry Friction and Oscillations

Part of the Springer INdAM Series book series (SINDAMS,volume 27)

Abstract

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.

Partially supported by DFG via project B01 in SFB 1114.

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References

  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)

    CrossRef  Google Scholar 

  2. Brokate, M., Krejčí, P., Schnabel, H.: On uniqueness in evolution quasivariational inequalities. J. Convex Anal. 11, 111–130 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)

    CrossRef  Google Scholar 

  4. DeSimone, A., Gidoni, P., Noselli, G.: Liquid crystal elastomer strips as soft crawlers. J. Mech. Phys. Solids 84, 254–272 (2015)

    MathSciNet  CrossRef  Google Scholar 

  5. Gidoni, P., DeSimone, A.: On the genesis of directional friction through bristle-like mediating elements crawler. arXiv:1602.05611 (2016)

    Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  7. Gidoni, P., Noselli, G., DeSimone, A.: Crawling on directional surfaces. Int. J. Non-Linear Mech. 61, 65–73 (2014)

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  9. Heida, M., Mielke, A., Pipping, E.: Rate-and-state friction from a thermodynamical viewpoint. In preparation (2017)

    Google Scholar 

  10. Mielke, A.: Emergence of rate-independent dissipation from viscous systems with wiggly energies. Contin. Mech. Thermodyn. 24(4), 591–606 (2012)

    MathSciNet  CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  12. Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015)

    CrossRef  Google Scholar 

  13. Pfeiffer, F.: Mechanische Systeme mit unstetigen Übergängen. Ingenieur-Archiv 54, 232–240 (1984). (In German)

    CrossRef  Google Scholar 

  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. 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)

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  17. Popov, V.L.: Contact Mechanics and Friction. Springer, New York (2010)

    CrossRef  Google Scholar 

  18. Prandtl, L.: Gedankenmodel zur kinetischen Theorie der festen Körper. Z. Angew. Math. Mech. 8, 85–106 (1928)

    CrossRef  Google Scholar 

  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. 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)

    CrossRef  Google Scholar 

  21. Tomlinson, G.A.: A molecular theory of friction. Phil. Mag. 7, 905–939 (1929)

    CrossRef  Google Scholar 

  22. Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)

    CrossRef  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Alexander Mielke

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  1. Alexander Mielke
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Correspondence to Alexander Mielke .

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Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Prof. Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Prof. Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Prof. Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Prof. Augusto Visintin

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Mielke, A. (2018). Three Examples Concerning the Interaction of Dry Friction and Oscillations. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_8

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