Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Self-Similar Mode of Ice Surface Softening During Friction

  • 156 Accesses

  • 1 Citations


Softening of ice surface under friction is explored in terms of the rheological model for viscoelastic matter approximation. The non-linear relaxation of strain and fractional feedbacks are allowed. Additive non-correlated noise associated with shear strain, stress as well as with temperature of ice surface layer, is introduced, and a phase diagram is built where the noise intensities of the stress and temperature define the domains of crystalline ice, softened ice, and two types of their mixture (stick-slip friction). Conditions are revealed under which crystalline ice and stick-slip friction proceed in the self-similar mode. Corresponding strain power-law distribution is provided by temperature fluctuations that are much larger than noise intensities of strain and stress. This behavior is fixed by homogenous probability density in which characteristic strain scale is absent. Since the power-type distribution is observed at minor strains, it meets self-similar crystalline ice surface. Constructed friction force time series are investigated for all rubbing regimes using fast Fourier transformation and autocorrelation function analysis. It is revealed that these series represent “pink”-colored noise and weak correlations are realized in the system. The conclusion is drawn that the results of tribological experiment can be forecasted.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    The coefficient 2 serves for simplification of the respective Fokker–Planck equation (FPE).

  2. 2.

    As stress noise \(I_\sigma\) decreases \(\varepsilon_0^m (T_{\rm{e}})\) dependence displays the similar pattern.

  3. 3.

    The \(I_T\) ascent enlarges the linear section insignificantly but it does not change the inclination angle.

  4. 4.

    It is assumed that friction force is caused both by elastic and viscous contributions.

  5. 5.

    The ice surface softens at large temperature \(T_{\rm{e}}\), but at small \(T_{\rm{e}}\) it solidifies.

  6. 6.

    It is assumed that the vibrations around zero are related to the accuracy of calculations of the time series. In this case, it is supposed that in the limiting case the ACF exponentially decreases to zero. Besides, it is noteworthy that according to Cheddok scale at \(0< A_{\rm{cf}} < 0.3\) the correlation between time series is very weak.


\(\varepsilon\) :

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)

\(T_{\rm{c}}\) :

Characteristic ice surface temperature (K)

a :

Fractional exponent (dimensionless variable)

Q :

Heat flow from the sliding block to the ice surface (K/m3)

\(\kappa\) :

Heat conductivity (1/(m s))

l :

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

\(T_{\rm{e}}\) :

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

\(c_{\rm{p}}\) :

Heat capacity (1/m3)

g :

Coefficient (dimensionless variable)

\(\tau_T\) :

Time of heat conductivity (s)

b :

Lattice constant or intermolecular distance (m)

c :

Sound velocity (m/s)

\(\rho\) :

Ice density (kg/m3)

V :

Synergetic potential (dimensionless variable)

\(T_{{\rm{c}}0}\) :

Critical thermostat temperature (K)

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

Intensities of strain, stress, and temperature noises (s−2, Pa3, K3)

\(\xi_i\) :

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

t :

Time (s)

\(f_a\) :

Deterministic force (dimensionless variable)

\(I_a\) :

Effective noise intensity (dimensionless variable)

\(P_a\) :

Probability distribution (dimensionless variable)

\(U_a\) :

Effective potential (dimensionless variable)

\(R^2\) :

Determination coefficient (dimensionless variable)

dW :

Wiener process (dimensionless variable)

\(T_{\rm{m}}\) :

Maximal time (s)

N :

Number of time series members (dimensionless variable)

\({\mu }^2\) :

Dispersion (dimensionless variable)

\(r_{1,2}\) :

Pseudo-random numbers (dimensionless variable)

A :

Contact area (m3)

F :

Friction force (N)

\(S_{\rm{p}}\) :

Spectral power density (conventional units)

\(\nu\) :

Frequency (Hz)

\(A_{\rm{cf}}\) :

Autocorrelation function (conventional units)

\(\gamma\) :

Mathematical expectation (conventional units)

\(\tau\) :

Lag or delay time (conventional units)


  1. 1.

    Bowden, F.P., Hughes, T.P.: The mechanism of sliding on ice and snow. Proc. R. Soc. Lond. A 172, 280–298 (1939)

  2. 2.

    Beeman, M., Durham, W.B., Kirby, S.H.: Friction of ice. J. Geophys. Res. Solid Earth 93(B7), 7625–7633 (1988). https://doi.org/10.1029/JB093iB07p07625

  3. 3.

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

  4. 4.

    Fortt, A., Schulson, E.: The resistance to sliding along coulombic shear faults in ice. Acta Mater. 55(7), 2253–2264 (2007). https://doi.org/10.1016/j.actamat.2006.11.022

  5. 5.

    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 116, C11012 (2011). https://doi.org/10.1029/2011JC006969

  6. 6.

    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)

  7. 7.

    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)

  8. 8.

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

  9. 9.

    Sukhorukov, S., Loset, S.: Friction of sea ice on sea ice. Cold Reg. Sci. Technol. 94, 1–12 (2013). https://doi.org/10.1016/j.coldregions.2013.06.005

  10. 10.

    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)

  11. 11.

    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). https://doi.org/10.1016/j.wear.2004.09.026

  12. 12.

    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). https://doi.org/10.1063/1.3173346

  13. 13.

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

  14. 14.

    Klapproth, C., Kessel, T., Wiese, K., Wies, B.: An advanced viscous model for rubber-ice-friction. Tribol. Int. 99, 169–181 (2016). https://doi.org/10.1016/j.triboint.2015.09.012

  15. 15.

    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). https://doi.org/10.1007/s11249-016-0665-z

  16. 16.

    Limmer, D.T., Chandler, D.: Premelting, fluctuations, and coarse-graining of water-ice interfaces. J. Chem. Phys. 141(18), 18 (2014)

  17. 17.

    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)

  18. 18.

    Persson, B.N.J.: Ice friction: role of non-uniform frictional heating and ice premelting. J. Chem. Phys. 143(22), 224701 (2015). https://doi.org/10.1063/1.4936299

  19. 19.

    Khomenko, A., Khomenko, M., Persson, B.N.J., Khomenko, K.: Noise effect on ice surface softening during friction. Tribol. Lett. 65(2), 71 (2017). https://doi.org/10.1007/s11249-017-0853-5

  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)

  21. 21.

    Akkok, M., Ettles, C.M.M., Calabrese, S.J.: Parameters affecting the kinetic friction of ice. ASME J. Tribol. 109, 552–559 (1987)

  22. 22.

    Eirich, F. (ed.): Rheology. Academic Press, New York (1960)

  23. 23.

    Khomenko, A.V.: Noise influence on solid-liquid transition of ultrathin lubricant film. Phys. Lett. A 329(1–2), 140–147 (2004)

  24. 24.

    Khomenko, A.V., Lyashenko, I.A.: Temperature dependence effect of viscosity on ultrathin lubricant film melting. Condens. Matter Phys. 9(4), 695–702 (2006)

  25. 25.

    Khomenko, A.V., Lyashenko, I.A.: Hysteresis phenomena during melting of an ultrathin lubricant film. Phys. Solid State 49(5), 936–940 (2007)

  26. 26.

    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. Usp. 55(10), 1008–1034 (2012). https://doi.org/10.3367/UFNe.0182.201210f.1081

  27. 27.

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

  28. 28.

    Pogrebnjak, A.D., Bondar, O.V., Abadias, G., Ivashchenko, V., Sobol, O.V., Jurga, S., Coy, E.: Structural and mechanical properties of NbN and Nb-Si-N films: experiment and molecular dynamics simulations. Ceram. Int. 42(10), 11743–11756 (2016). https://doi.org/10.1016/j.ceramint.2016.04.095

  29. 29.

    Goncharov, A.A., Konovalov, V.A., Volkova, G.K., Stupak, V.A.: Size effect on the structure of nanocrystalline and cluster films of hafnium diboride. Phys. Met. Metall. 108(4), 368 (2009). https://doi.org/10.1134/S0031918X0910007X

  30. 30.

    Jourani, A.: Effect of 3d fractal dimension on contact area and asperity interactions in elastoplastic contact. AIP Adv. 6(5), 055309 (2016). https://doi.org/10.1063/1.4949564

  31. 31.

    Persson, B.N.J.: On the fractal dimension of rough surfaces. Tribol. Lett. 54(1), 99–106 (2014). https://doi.org/10.1007/s11249-014-0313-4

  32. 32.

    Yang, H., Baudet, B.A., Yao, T.: Characterization of the surface roughness of sand particles using an advanced fractal approach. Proc. R. Soc. Lond. A 472, 2194 (2016). https://doi.org/10.1098/rspa.2016.0524

  33. 33.

    Gardiner, C.W.: Handbook of Stochastic Methods, 2nd edn. Springer, Berlin (1994)

  34. 34.

    Horstemke, V., Lefever, R.: Noise-Induced Transitions. Springer, Berlin (1984)

  35. 35.

    Olemskoi, A.I.: Theory of stochastic systems with singular multiplicative noise. Phys. Usp. 41(3), 269–301 (1998). https://doi.org/10.1070/PU1998v041n03ABEH000377

  36. 36.

    Risken, H.: The Fokker-Planck-Equation. Methods of Solution and Applications, 2nd edn. Springer, Berlin (1989)

  37. 37.

    Eisenberg, D.S., Kauzmann, W.: The Structure and Properties of Water, 1st edn. Oxford University Press, Oxford (2011)

  38. 38.

    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)

  39. 39.

    ter Haar, D. (ed.): Collected Papers of L.D. Landau. Pergamon Press, London (1965)

  40. 40.

    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)

  41. 41.

    Lifshits, E.M., Pitaevskii, L.P.: Course of Theoretical Physics. Physical Kinetics, vol. 10, 1st edn. Pergamon Press, Oxford (1981)

  42. 42.

    Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics. Theory of Elasticity, vol. 7, 3rd edn. Butterworth-Heinemann, Oxford (1986)

  43. 43.

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

  44. 44.

    Khomenko, A.V.: Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope. Condens. Matter Phys. 17(3), 33401:1–33401:10 (2014)

  45. 45.

    Haken, H.: Synergetics: An introduction: Nonequilibrium phase transitions and self-organization in physics, chemistry, and biology, 3rd edn. Springer, Berlin (1983)

  46. 46.

    Olemskoi, A.I., Khomenko, A.V.: Three-parameter kinetics of a phase transition. J. Exp. Theor. Phys. 83(6), 1180–1192 (1996)

  47. 47.

    Olemskoi, A.I., Khomenko, A.V.: Phenomenological equations of the glass transition in liquids. Tech. Phys. 45, 672–676 (2000)

  48. 48.

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

  49. 49.

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

  50. 50.

    Olemskoi, A.I., Khomenko, A.V., Kharchenko, D.O.: Self-organized criticality within fractional Lorenz scheme. Phys. A 323, 263–293 (2003)

  51. 51.

    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). https://doi.org/10.1007/s40544-013-0021-3

  52. 52.

    Khomenko, A., Lyashenko, I.: Stochastic theory of ultrathin lubricant film melting in the stick-slip regime. Tech. Phys. 50(11), 1408–1416 (2005). https://doi.org/10.1134/1.2131946

  53. 53.

    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). https://doi.org/10.1142/S0219477510000046

  54. 54.

    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)

  55. 55.

    Toropov, E., Kharchenko, D.: Influence of noise on the nature of synergetic systems. Russ. Phys. J. 39(4), 355–361 (1996). https://doi.org/10.1007/BF02068059

  56. 56.

    Khomenko, A., Lyashenko, I.: Melting of ultrathin lubricant film due to dissipative heating of friction surfaces. Tech. Phys. 52(9), 1239–1243 (2007). https://doi.org/10.1134/S1063784207090241

  57. 57.

    Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics. Statistical Physics, vol. 5. Butterworth, London (1999)

  58. 58.

    Sazaki, G., Zepeda, S., Nakatsubo, S., Yokomine, M., Furukawa, Y.: Quasi-liquid layers on ice crystal surfaces are made up of two different phases. Proc. Natl. Acad. Sci. 109(4), 1052–1055 (2012). https://doi.org/10.1073/pnas.1116685109

  59. 59.

    Amit, D.J.: Field Theory, the Renormalization Group, and Critical Phenomena. McGraw-Hill, New York (1978)

  60. 60.

    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)

  61. 61.

    Khomenko, A.V., Lyashenko, Y.A.: Periodic intermittent regime of a boundary flow. Tech. Phys. 55(1), 26–32 (2010). https://doi.org/10.1134/S1063784210010056

  62. 62.

    Sneddon, I.N.: Fourier Transforms. McGraw-Hill, New York (1951)

  63. 63.

    Kogan, S.: Electronic Noise and Fluctuations in Solids. Cambridge University Press, Cambridge (2008)

  64. 64.

    Wang, A., Muser, M.H.: On the usefulness of the height-difference-autocorrelation function for contact mechanics. Tribol. Int. 123, 224–233 (2018). https://doi.org/10.1016/j.triboint.2018.02.002

  65. 65.

    Box, G.E.P., Jenkins, G.M., Reinsel, G.C., Ljung, G.M.: Time Series Analysis: Forecasting and Control. Wiley Series in Probability and Statistics, 5th edn. Wiley, Hoboken (2015)

Download references


This work is supported by the Ministry of Education and Science of Ukraine (Project “Atomistic and statistical representation of formation and friction of nanodimensional systems,” No. 0118U003584) and visitor Grant of Forschungszentrum-J\(\ddot{u}\)lich, Germany. A.K. thanks Dr. Bo N.J. Persson for hospitality during his stay in Forschungszentrum-J\(\ddot{u}\)lich.

Author information

Correspondence to Alexei Khomenko.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khomenko, A. Self-Similar Mode of Ice Surface Softening During Friction. Tribol Lett 66, 82 (2018). https://doi.org/10.1007/s11249-018-1034-x

Download citation


  • Ice surface softening
  • Viscoelastic medium
  • Self-similarity
  • Fluctuations intensity
  • Phase diagram
  • Stick-slip friction