Contact between representative rough surfaces

A numerical analysis of mechanical frictionless contact between rough self-affine elastic manifolds was carried out. It is shown that the lower cutoff wave number in surface spectra is a key parameter controlling the representativity of the numerical model. Using this notion we demonstrate that for representative surfaces the evolution of the real contact area with load is universal and independent of the Hurst roughness exponent. By introducing a universal law containing three constants, we extend the study of this evolution beyond the limit of infinitesimal area fractions.

Figure 1

Representativity of a rough self-affine surface can be estimated according to the normality of the height distribution. Here, the criterion of proximity is an error between the distribution of the surface's heights and the associated normal distribution; (a) this error is strongly dependent on the cutoff wave numbers $k_s$ and $k_l$. Every point on the graph represents an error averaged over 30 statistically equivalent surfaces. Examples of rough surfaces and the corresponding height distributions for different cutoffs $k_s=40$ and (b) $k_l=1$, (c) $k_l=4$, (d) $k_l=16$ are also depicted.

Figure 2

Evolution of the real-contact-area fraction $A/A_0$ with normalized external load $p_0/E^{*}$ is plotted for different cutoff wavelengths and Hurst exponents; the rms slope is kept constant for all surfaces $\sqrt{\langle |\nabla h{|}^2\rangle }=0.1$. The results for representative surfaces (${\lambda }_l=L/16$) are in a good agreement with the asymptotic limit of the Bush-Gibson-Thomas (BGT) theory [5] and are perfectly fitted by the suggested contact evolution law Eq. (2) ($\kappa =1.145\sqrt{2\pi }$, $\mu =0.55$, $\beta =0.21$). Moreover, if the surface is sufficiently smooth, ${\lambda }_s=L/32$, we do not observe any dependence of results on the Hurst exponent. The evolution of the real contact area predicted by Greenwood [18] and BGT [5] theories for spectrum bandwidth $m_0{m}_4/m_2^2=2$ (data from Ref. [21]) lie in the confidence interval of the results obtained for nonrepresentative surfaces (${\lambda }_l=L$) with high Hurst exponents $H⪆0.5$. The asymptotic limit of the Persson theory [23, 24] lies below all obtained results.

Figure 3

Evolution of $\kappa$ with the real-contact-area fraction $A/A_0$ is depicted for different Hurst roughness exponents and cutoff wavelengths ${\lambda }_l=L$, $L/8$, $L/16$, (a) ${\lambda }_s=L/32$, (b) ${\lambda }_s=L/128$; all results for representative surfaces ${\lambda }_l=L/16$ are fitted by the contact evolution law Eq. (2) with $\kappa =1.145\sqrt{2\pi }$, $\mu =0.55$, $\beta =0.21$ for (a) ${\lambda }_l=L/16$, ${\lambda }_s=L/32$, and with $\kappa =1.145\sqrt{2\pi }$, $\mu =0.56$, and $\beta \in [1.120,1.172]$ for (b) ${\lambda }_l=L/16$, ${\lambda }_s=L/128$. For comparison, the asymptotic limit ${\kappa }_{\mathrm{BGT}}=\sqrt{2\pi }$ from the Bush-Gibson-Thomas theory [5] is plotted, while the asymptotic limit of the Persson theory [23, 24] ${\kappa }_{\mathrm{P}}=\sqrt{8/\pi }\approx 1.60$ is out of the plot range.

Figure 4

The mean pressure derivative with respect to the external pressure $\partial \overline{p}/\partial p_0$ decreases nonlinearly with increasing fraction of the real contact area $A/A_0$. We plot its evolution for different Hurst roughness exponents and cutoff wavelengths ${\lambda }_l=L$, $L/8$, $L/16$ and (a) ${\lambda }_s=L/32$, (b) ${\lambda }_s=L/128$. For representative surfaces the results can be roughly approximated by a power law $\partial \overline{p}/\partial p_0=\beta {(A_0/A)}^{\mu }$; the solid line corresponds to parameters $\beta =0.21$, $\mu =0.55$, which fit well our results for representative ${\lambda }_l=L/16$ smooth surfaces.