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Noise Effect on Ice Surface Softening During Friction

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Abstract

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.

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Notes

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

Abbreviations

\(\varepsilon\) :

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)

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Acknowledgements

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.

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  1. Sumy State University, Sumy, 40007, Ukraine

    Alexei Khomenko, Mariya Khomenko & Kateryna Khomenko

  2. Peter Grünberg Institut-1, Forschungszentrum-Jülich, 52425, Jülich, Germany

    Alexei Khomenko & Bo N. J. Persson

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  1. Alexei Khomenko
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Khomenko, A., Khomenko, M., Persson, B.N.J. et al. Noise Effect on Ice Surface Softening During Friction. Tribol Lett 65, 71 (2017). https://doi.org/10.1007/s11249-017-0853-5

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