Self-Similar Mode of Ice Surface Softening During Friction


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.

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  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)


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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.

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Khomenko, A. Self-Similar Mode of Ice Surface Softening During Friction. Tribol Lett 66, 82 (2018).

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