Noise Effect on Ice Surface Softening During Friction


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Here multiplier 2 is chosen for simplification of the corresponding Fokker–Planck equation (FPE).


\(\varepsilon\) :

Shear strain (dimensionless variable)

\(\sigma\) :

Shear stress (Pa)

\(\tau _\varepsilon\) :

Relaxation time of strain (s)

\(\eta _\varepsilon\) :

Effective shear viscosity (Pa s)

\(\tau _\sigma\) :

Relaxation time of stress (s)

G :

Non-relaxed shear modulus (Pa)

\(\eta\) :

Shear viscosity (Pa s)

\(G_\varepsilon\) :

Relaxed shear modulus (Pa)

\(G_0\) :

Typical shear modulus (Pa)

v :

Sliding velocity (m/s)

\(\omega\) :

Circular frequency (Hz)

T :

Temperature of ice surface (K)

\(T_\mathrm{c}\) :

Characteristic ice surface temperature (K)

Q :

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

\(\kappa\) :

Heat conductivity [1/(m s)]

l :

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

\(T_\mathrm{e}\) :

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

\(c_\mathrm{p}\) :

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

g :

Coefficient (dimensionless variable)

\(\tau _T\) :

Time of heat conductivity (s)

a :

Lattice constant or intermolecular distance (m)

c :

Sound velocity (m/s)

\(\rho\) :

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

V :

Synergetic potential (dimensionless variable)

\(T_\mathrm{c0}\) :

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)

D :

Integral of the correlation function (dimensionless variable)

I :

Effective noise intensity (dimensionless variable)

P :

Probability distribution (dimensionless variable)

U :

Effective potential (dimensionless variable)

dW :

Wiener process (s)

\(T_\mathrm{m}\) :

Maximal time (s)

N :

Number of time series members (dimensionless variable)

\(\mu ^2\) :

Dispersion (dimensionless variable)

\(r_{1,2}\) :

Pseudorandom numbers (dimensionless variable)

A :

Contact area (m\(^2\))

F :

Friction force (N)

\(S_\mathrm{p}\) :

Spectral power density (conventional units)

\(\nu\) :

Frequency (Hz)


  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)

    Article  Google 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)

    Article  Google 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

    Article  Google Scholar 

  4. 4.

    Blackford, J.R., Skouvaklis, G., Purser, M., Koutsos, V.: Friction on ice: stick and slip. Faraday Discuss. 156, 243–254 (2012)

    Article  Google 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)

    Article  Google 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

    Article  Google 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)

  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)

    Article  Google 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

    Article  Google 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

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google Scholar 

  24. 24.

    Khomenko, A.V., Yushchenko, O.V.: Solid–liquid transition of ultrathin lubricant film. Phys. Rev. E 68, 036110 (2003)

    Article  Google 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

    Article  Google Scholar 

  26. 26.

    Kietzig, A.M., Hatzikiriakos, S.G., Englezos, P.: Physics of ice friction. J. Appl. Phys. 107(8), 081101 (2010)

    Article  Google 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

    Article  Google 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

    Article  Google 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)

    Article  Google 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)

  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)

    Article  Google 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

    Article  Google 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)

    Article  Google Scholar 

  40. 40.

    Olemskoi, A.I., Khomenko, A.V.: The synergetic theory of the glass transition in liquids. Tech. Phys. 45, 677–682 (2000)

    Article  Google Scholar 

  41. 41.

    Olemskoi, A.I., Khomenko, A.V.: Synergetic theory for a jamming transition in traffic flow. Phys. Rev. E 63, 036116 (2001)

    Article  Google 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)

    Article  Google Scholar 

  43. 43.

    Persson, B.N.J.: Sliding Friction. Physical Principles and Applications, 2nd edn. Springer, Berlin (2000)

    Google 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)

    Google 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

    Article  Google Scholar 

  47. 47.

    Schulson, E.M., Fortt, A.L.: Friction of ice on ice. J. Geophys. Res. Solid Earth 117(B12), B12204 (2012)

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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 

Download references


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.

Author information



Corresponding author

Correspondence to Alexei Khomenko.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khomenko, A., Khomenko, M., Persson, B.N.J. et al. Noise Effect on Ice Surface Softening During Friction. Tribol Lett 65, 71 (2017).

Download citation


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