Finite-size scaling in the interfacial stiffness of rough elastic contacts

The total elastic stiffness of two contacting bodies with a microscopically rough interface has an interfacial contribution $K$ that is entirely attributable to surface roughness. A quantitative understanding of $K$ is important because it can dominate the total mechanical response and because it is proportional to the interfacial contributions to electrical and thermal conductivity in continuum theory. Numerical simulations of the dependence of $K$ on the applied squeezing pressure $p$ are presented for nominally flat elastic solids with a range of surface roughnesses. Over a wide range of $p$, $K$ rises linearly with $p$. Sublinear power-law scaling is observed at small $p$, but the simulations reveal that this is a finite-size effect. We derive accurate, analytical expressions for the exponents and prefactors of this low-pressure scaling of $K$ by extending the contact mechanics theory of Persson to systems of finite size. In agreement with our simulations, these expressions show that the onset of the low-pressure scaling regime moves to lower pressure as the system size increases.

Figure 1

Log-log plot of the nondimensional contact stiffness $K{h}_{\mathrm{rms}}/E^{*}$ vs. nondimensional pressure $p/E^{*}$ for self-affine fractal surfaces with $H=0.7$ and rms slope $h_{\mathrm{rms}}^{\prime }=0.1$. In all cases, the surface is resolved with 8192 points in each direction, $L/{\lambda }_1=4096$, and the ratio of system size to roll-off wavelength, $L/{\lambda }_r$, is indicated. The (red) solid line is the prediction of Persson's theory while the dashed (red) line is the linear regime. Open squares (blue) show the interfacial stiffness obtained from a punch calculation with $L_{\mathrm{p}}/{\lambda }_1=1024$. Inset: Nondimensional pressure vs. separation for the same surfaces.

Figure 2

Dimensionless interfacial stiffness $K{h}_{\mathrm{rms}}/E^{*}$ as a function of pressure $p/E^{*}$ in the finite-size region for $L/{\lambda }_1=4096$ and different Hurst exponents $H$. All calculations are for $L/{\lambda }_r=1$. Solid lines show a fit to the first four data points.

Figure 3

Exponent $\alpha$ as a function of ${({\lambda }_1/{\lambda }_r)}^{0.77}$. Theoretically predicted value is denoted by $\blacksquare$. The inset shows selected numerical data for $K\left(p\right)$ from which $\alpha$ was deduced. Data for the following values of ${\lambda }_r/{\lambda }_1$ is presented: 4096 ($•$), 2048 ($⧫$), 768 ($*$), 512 ($▴$).

Figure 4

Power spectra for two surfaces without (a) and with (b) a roll-off at large wavelength as generated by a Fourier filtering algorithm. The solid lines show the prescribed power spectrum $C\left(q\right)$ and the dots the actual realization. Panel (b) indicates the wavevectors of the long-wavelength roll-off $q_r=2\pi /{\lambda }_r$ and the short-wavelength cut-off $q_1=2\pi /{\lambda }_1$. For $q<q_0=2\pi /L$, where $L$ is the linear system size, the surfaces have zero power. The noise at low $q$ is due to the fact that order $q^2$ Fourier components contribute to the power spectrum at wavevector $q$ of a realization of a surface.

Figure 5

Plot of the values of $1/{\theta }_0$, $1/{\theta }_1$, and $1/\theta =1/{\theta }_0+1/{\theta }_1$ as a function of Hurst exponent $H$. The quantities ${\theta }_0$ and ${\theta }_1$ are defined in the text.

Figure 6

The normal contact stiffness as a function of the applied nominal contact pressure for a silicon rubber block (cylinder shape with diameter $D=3\mathrm{cm}$ and height $d=1\mathrm{cm}$) squeezed against a road asphalt surface. The green and red lines are obtained from the measured $p\left(s\right)$ relation using Eq. (7) with $E^{\prime }=4.0$ and $4.2\mathrm{MPa}$ (see text), while the blue line is the theory prediction.