Tribology Letters

, 65:71 | Cite as

Noise Effect on Ice Surface Softening During Friction

  • Alexei Khomenko
  • Mariya Khomenko
  • Bo N. J. Persson
  • Kateryna Khomenko
Original Paper


The ice surface softening by friction is investigated considering the additive non-correlated fluctuations of the shear strain and stress, and the temperature. The premelting is construed by the Kelvin–Voigt equation for shear strain and by the relaxation equations of Landau–Khalatnikov type for shear stress and temperature. Taking into account the noises in these equations, the Langevin and Fokker–Planck equations are derived. Their analysis is based on the investigation of extrema of the distribution function, i.e., steady-state values of the shear strain using the Stratonovich interpretation. The phase diagrams are constructed, where the noises intensities and thermostat temperature determine the regions of ice, softened ice and their mixture (stick–slip rubbing). We present that domain of ice friction is bounded by relatively small background sliding block temperatures and fluctuation intensities of the stress and temperature. The ice film softens with growth of the stress noise intensity even at small thermostat temperatures. The friction force time series for all rubbing modes are calculated and compared with experimentally observed ones.


Ice surface softening Viscoelastic medium Plasticity Fluctuations intensity Phase diagram Stick–slip friction 

List of symbols


Shear strain (dimensionless variable)


Shear stress (Pa)

\(\tau _\varepsilon\)

Relaxation time of strain (s)

\(\eta _\varepsilon\)

Effective shear viscosity (Pa s)

\(\tau _\sigma\)

Relaxation time of stress (s)


Non-relaxed shear modulus (Pa)


Shear viscosity (Pa s)


Relaxed shear modulus (Pa)


Typical shear modulus (Pa)


Sliding velocity (m/s)


Circular frequency (Hz)


Temperature of ice surface (K)


Characteristic ice surface temperature (K)


Heat flow from the sliding block to the ice surface (K/m\(^3\))


Heat conductivity [1/(m s)]


Distance into which heat penetrates ice (scale of heat conductivity) (m)


Thermostat temperature (temperature far away from rubbing surfaces) (K)


Heat capacity (1/m\(^3\))


Coefficient (dimensionless variable)

\(\tau _T\)

Time of heat conductivity (s)


Lattice constant or intermolecular distance (m)


Sound velocity (m/s)


Ice density (kg/m\(^3\))


Synergetic potential (dimensionless variable)


Critical thermostat temperature (K)

\(I_{\varepsilon ,\sigma ,T}\)

Intensities of strain, stress and temperature noises (s\(^{-2}\), Pa\(^2\), K\(^2\))

\(\xi _i\)

\(\delta\)-correlated Gaussian source (white noise) (dimensionless variable)


Integral of the correlation function (dimensionless variable)


Effective noise intensity (dimensionless variable)


Probability distribution (dimensionless variable)


Effective potential (dimensionless variable)


Wiener process (s)


Maximal time (s)


Number of time series members (dimensionless variable)

\(\mu ^2\)

Dispersion (dimensionless variable)


Pseudorandom numbers (dimensionless variable)


Contact area (m\(^2\))


Friction force (N)


Spectral power density (conventional units)


Frequency (Hz)



This work is supported by the Ministry of Education and Science of Ukraine (Project “Nonequilibrium thermodynamics of metals fragmentation and friction of spatially nonhomogeneous boundary lubricants between surfaces with nanodimensional irregularities,” No. 0115U000692) and visitor Grant of Forschungszentrum-Jülich, Germany. A.K. is grateful to Dr. Bo N.J. Persson for hospitality during his stay in Forschungszentrum-Jülich. We thank Daria Troshchenko for attentive reading and correction of the manuscript.


  1. 1.
    Akkok, M., Ettles, C.M.M., Calabrese, S.J.: Parameters affecting the kinetic friction of ice. ASME J. Tribol. 109, 552–559 (1987)CrossRefGoogle Scholar
  2. 2.
    Baurle, L., Kaempfer, T.U., Szabo, D., Spencer, N.D.: Sliding friction of polyethylene on snow and ice: contact area and modeling. Cold Reg. Sci. Technol. 47(3), 276–289 (2007)CrossRefGoogle Scholar
  3. 3.
    Beeman, M., Durham, W.B., Kirby, S.H.: Friction of ice. J. Geophys. Res. Solid Earth 93(B7), 7625–7633 (1988). doi: 10.1029/JB093iB07p07625 CrossRefGoogle Scholar
  4. 4.
    Blackford, J.R., Skouvaklis, G., Purser, M., Koutsos, V.: Friction on ice: stick and slip. Faraday Discuss. 156, 243–254 (2012)CrossRefGoogle Scholar
  5. 5.
    Ducret, S., Zahouani, H., Midol, A., Lanteri, P., Mathia, T.: Friction and abrasive wear of UHWMPE sliding on ice. Wear 258(14), 26–31 (2005). doi: 10.1016/j.wear.2004.09.026. (Second International Conference on Erosive and Abrasive Wear) CrossRefGoogle Scholar
  6. 6.
    Eirich, F. (ed.): Rheology. Academic Press, New York (1960)Google Scholar
  7. 7.
    Eisenberg, D.S., Kauzmann, W.: The Structure and Properties of Water, 1st edn. Oxford University Press, Oxford (2011)Google Scholar
  8. 8.
    Fortt, A., Schulson, E.: The resistance to sliding along Coulombic shear faults in ice. Acta Mater. 55(7), 2253–2264 (2007). doi: 10.1016/j.actamat.2006.11.022 CrossRefGoogle Scholar
  9. 9.
    Fortt, A.L., Schulson, E.M.: Frictional sliding across coulombic faults in first-year sea ice: a comparison with freshwater ice. J. Geophys. Res. Oceans (2011). doi: 10.1029/2011JC006969 Google Scholar
  10. 10.
    Gardiner, C.W.: Handbook of Stochastic Methods, 2nd edn. Springer, Berlin (1994)Google Scholar
  11. 11.
    Ghrib, T. (ed.): New Tribological Ways, Adhesion Theory for Low Friction on Ice by Katsutoshi Tusima, 1st edn. InTech, University of Toyama, Toyama (2011)Google Scholar
  12. 12.
    ter Haar, D. (ed.): Collected Papers of L.D. Landau. Pergamon Press, London (1965)Google Scholar
  13. 13.
    Haken, H.: Synergetics. An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology, 3rd edn. Springer, Berlin (1983)Google Scholar
  14. 14.
    Horstemke, V., Lefever, R.: Noise-Induced Transitions. Springer, Berlin (1984)Google Scholar
  15. 15.
    Kennedy, F.E., Schulson, E.M., Jones, D.E.: The friction of ice on ice at low sliding velocities. Philos. Mag. A 80(5), 1093–1110 (2000)CrossRefGoogle Scholar
  16. 16.
    Khomenko, A., Lyashenko, I.: Stochastic theory of ultrathin lubricant film melting in the stick–slip regime. Tech. Phys. 50(11), 1408–1416 (2005). doi: 10.1134/1.2131946 CrossRefGoogle Scholar
  17. 17.
    Khomenko, A., Lyashenko, I.: Melting of ultrathin lubricant film due to dissipative heating of friction surfaces. Tech. Phys. 52(9), 1239–1243 (2007). doi: 10.1134/S1063784207090241 CrossRefGoogle Scholar
  18. 18.
    Khomenko, A.V.: Noise influence on solid–liquid transition of ultrathin lubricant film. Phys. Lett. A 329(1–2), 140–147 (2004)CrossRefGoogle Scholar
  19. 19.
    Khomenko, A.V.: Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope. Condens. Matter Phys. 17(3), 33401 (2014)CrossRefGoogle Scholar
  20. 20.
    Khomenko, A.V., Khomenko, K.P., Falko, V.V.: Nonlinear model of ice surface softening during friction. Condens. Matter Phys. 19(3), 33002 (2016)CrossRefGoogle Scholar
  21. 21.
    Khomenko, A.V., Lyashenko, I.A.: Statistical theory of the boundary friction of atomically flat solid surfaces in the presence of a lubricant layer. Phys. Uspekhi 55(10), 1008–1034 (2012). doi: 10.3367/UFNe.0182.201210f.1081 CrossRefGoogle Scholar
  22. 22.
    Khomenko, A.V., Lyashenko, I.A., Borisyuk, V.N.: Multifractal analysis of stress time series during ultrathin lubricant film melting. Fluct. Noise Lett. 09(01), 19–35 (2010). doi: 10.1142/S0219477510000046 CrossRefGoogle Scholar
  23. 23.
    Khomenko, A.V., Lyashenko, Y.A.: Periodic intermittent regime of a boundary flow. Tech. Phys. 55(1), 26–32 (2010). doi: 10.1134/S1063784210010056 CrossRefGoogle Scholar
  24. 24.
    Khomenko, A.V., Yushchenko, O.V.: Solid–liquid transition of ultrathin lubricant film. Phys. Rev. E 68, 036110 (2003)CrossRefGoogle Scholar
  25. 25.
    Kietzig, A.M., Hatzikiriakos, S.G., Englezos, P.: Ice friction: the effects of surface roughness, structure, and hydrophobicity. J. Appl. Phys. 106(2), 024303 (2009). doi: 10.1063/1.3173346 CrossRefGoogle Scholar
  26. 26.
    Kietzig, A.M., Hatzikiriakos, S.G., Englezos, P.: Physics of ice friction. J. Appl. Phys. 107(8), 081101 (2010)CrossRefGoogle Scholar
  27. 27.
    Klapproth, C., Kessel, T., Wiese, K., Wies, B.: An advanced viscous model for rubber-ice-friction. Tribol. Int. 99, 169–181 (2016). doi: 10.1016/j.triboint.2015.09.012 CrossRefGoogle Scholar
  28. 28.
    Kozin, V., Zhestkaja, V., Pogorelova, A., Chizhiumov, S., Dzhabrailov, M., Morozov, V., Kustov, A.: Applied problems of ice cover dynamics. Natural Sciences Academy Publishing, Moscow (2008). (in Russian)Google Scholar
  29. 29.
    Lahayne, O., Pichler, B., Reihsner, R., Eberhardsteiner, J., Suh, J., Kim, D., Nam, S., Paek, H., Lorenz, B., Persson, B.N.J.: Rubber friction on ice: experiments and modeling. Tribol. Lett. 62(2), 1–19 (2016). doi: 10.1007/s11249-016-0665-z CrossRefGoogle Scholar
  30. 30.
    Landau, L.D., Khalatnikov, I.M.: On the anomalous absorption of sound near a second-order phase transition point. Dokl. Akad. Nauk SSSR 96, 469–472 (1954)Google Scholar
  31. 31.
    Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, Vol.7: Theory of Elasticity, 3rd edn. Butterworth-Heinemann, Oxford (1986)Google Scholar
  32. 32.
    Landau, L.D.: Course of Theoretical Physics, Vol.5: Statistical Physics. Butterworth, London (1999)Google Scholar
  33. 33.
    Lifshits, E.M., Pitaevskii, L.P.: Course of Theoretical Physics, Vol.10: Physical Kinetics, 1st edn. Pergamon Press, Oxford (1981)Google Scholar
  34. 34.
    Limmer, D.T., Chandler, D.: Premelting, fluctuations, and coarse-graining of water–ice interfaces. J. Chem. Phys. 141(18), 18 (2014)CrossRefGoogle Scholar
  35. 35.
    Lishman, B., Sammonds, P., Feltham, D., Wilchinsky, A.: The rate- and state- dependence of sea ice friction. In: Proceedings of the 20th International Conference on Port and Ocean Engineering under Arctic Conditions, pp. POAC09–66 (2009)Google Scholar
  36. 36.
    Marmo, B.A., Blackford, J.R., Jeffree, C.E.: Ice friction, wear features and their dependence on sliding velocity and temperature. J. Glaciol. 51(174), 391–398 (2005)CrossRefGoogle Scholar
  37. 37.
    Olemskoi, A.I.: Theory of stochastic systems with singular multiplicative noise. Phys. Uspekhi 41(3), 269–301 (1998). doi: 10.1070/PU1998v041n03ABEH000377 CrossRefGoogle Scholar
  38. 38.
    Olemskoi, A.I., Khomenko, A.V.: Three-parameter kinetics of a phase transition. J. Exp. Theor. Phys. 83(6), 1180–1192 (1996)Google Scholar
  39. 39.
    Olemskoi, A.I., Khomenko, A.V.: Phenomenological equations of the glass transition in liquids. Tech. Phys. 45, 672–676 (2000)CrossRefGoogle Scholar
  40. 40.
    Olemskoi, A.I., Khomenko, A.V.: The synergetic theory of the glass transition in liquids. Tech. Phys. 45, 677–682 (2000)CrossRefGoogle Scholar
  41. 41.
    Olemskoi, A.I., Khomenko, A.V.: Synergetic theory for a jamming transition in traffic flow. Phys. Rev. E 63, 036116 (2001)CrossRefGoogle Scholar
  42. 42.
    Olemskoi, A.I., Khomenko, A.V., Kharchenko, D.O.: Self-organized criticality within fractional Lorenz scheme. Phys. A 323, 263–293 (2003)CrossRefGoogle Scholar
  43. 43.
    Persson, B.N.J.: Sliding Friction. Physical Principles and Applications, 2nd edn. Springer, Berlin (2000)CrossRefGoogle Scholar
  44. 44.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007)Google Scholar
  45. 45.
    Risken, H.: The Fokker-Planck-Equation. Methods of Solution and Applications, 2nd edn. Springer, Berlin (1989)CrossRefGoogle Scholar
  46. 46.
    Samadashvili, N., Reischl, B., Hynninen, T., Ala-Nissilä, T., Foster, A.: Atomistic simulations of friction at an ice–ice interface. Friction 1(3), 242–251 (2013). doi: 10.1007/s40544-013-0021-3 CrossRefGoogle Scholar
  47. 47.
    Schulson, E.M., Fortt, A.L.: Friction of ice on ice. J. Geophys. Res. Solid Earth 117(B12), B12204 (2012)CrossRefGoogle Scholar
  48. 48.
    Skokov, V.N., Koverda, V.N., Skripov, V.P.: A critical nonequilibrium phase transition and 1/f noise in a current-carrying thin HTSC film-boiling nitrogen system. Cryogenics 37(5), 263–265 (1997). doi: 10.1016/S0011-2275(97)00001-5 CrossRefGoogle Scholar
  49. 49.
    Sukhorukov, S., Loset, S.: Friction of sea ice on sea ice. Cold Regions Sci. Technol. 94, 1–12 (2013). doi: 10.1016/j.coldregions.2013.06.005 CrossRefGoogle Scholar
  50. 50.
    Toropov, E., Kharchenko, D.: Influence of noise on the nature of synergetic systems. Russ. Phys. J. 39(4), 355–361 (1996). doi: 10.1007/BF02068059 CrossRefGoogle Scholar
  51. 51.
    Wiese, K., Kessel, T.M., Mundl, R., Wies, B.: An analytical thermodynamic approach to friction of rubber on ice. Tire Sci. Technol. 40(2), 124–150 (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Alexei Khomenko
    • 1
    • 2
  • Mariya Khomenko
    • 1
  • Bo N. J. Persson
    • 2
  • Kateryna Khomenko
    • 1
  1. 1.Sumy State UniversitySumyUkraine
  2. 2.Peter Grünberg Institut-1Forschungszentrum-JülichJülichGermany

Personalised recommendations