Field-theoretical approach to estimate mean gap and gap distribution in randomly rough surface contact mechanics

Abstract

We extend the statistical field-theoretical framework of rough surface contact mechanics to characterize the interfacial gap between an elastic half-space and a randomly rough surface incorporating exponential repulsion. Building upon a cumulant expansion to second-order, we derive an explicit analytical relation between the mean gap and the applied normal pressure. This result provides a closed-form expression for the drift and diffusion coefficients in a convection-diffusion equation governing the scale-dependent evolution of the gap distribution. Solving this equation with appropriate initial and boundary conditions yields the gap distribution under varying external pressures. Both the mean gap and the gap distribution are found to be in good agreement with Green's function molecular dynamics (GFMD) simulations. Our results demonstrate that the field-theoretical approach enables quantitative predictions not only for contact stresses but also for interfacial gap in rough surface contacts.

Figures (6)

• Figure 1: (a) Schematic of frictionless contact between elastic half-space and rigid randomly rough surface. (b) Cross-section of deformed elastic half-space and rough surface; (c) A typical power spectral density (PSD) $C(q)$ of randomly rough surface.
• Figure 2: Reduced mean gap $g_0/\bar{h}$ as a function of reduced external pressure $\sigma_0/(E^*\bar{g})$ for interaction range $\rho/L$ equals to (a) $2.0 \times 10^{-3}$, (b) $1.0 \times 10^{-3}$ and (c) $5.0 \times 10^{-4}$ with Hurst exponent $H = 0.3$, where $L$ denotes the system size and $\bar{g}$ the root-mean-square gradient of height. Red dashed line represents the large pressure limit, blue dashed line denotes the small pressure limit, the solid line denotes the field approach and open circles the GFMD simulation results. Specifically, the longest wavelength $\lambda_{\mathrm l} / L = 0.25$, the shortest cut-off $\lambda_{\mathrm s} / L = 2.5 \times 10^{-3}$ and the root-mean-square gradient of surface height $\bar{g}$ is fixed to unity. The surface energy $\gamma_0 / (E^*L) = 1.0 \times 10^{-3}$.
• Figure 3: Reduced mean gap $g_0/\bar{h}$ as a function of reduced external pressure $\sigma_0/(E^*\bar{g})$ for interaction range $\rho/L$ equals to (a) $5.0 \times 10^{-3}$, (b) $2.0 \times 10^{-3}$ and (c) $1.0 \times 10^{-3}$ with Hurst exponent $H = 0.8$. Red dashed line represents the large pressure limit, blue dashed line denotes the small pressure limit, the solid line denotes the field approach and open circles the GFMD simulation results. Specifically, the longest wavelength $\lambda_{\mathrm l} / L = 0.25$, the shortest cut-off $\lambda_{\mathrm s} / L = 2.5 \times 10^{-3}$ and the root-mean-square gradient of surface height $\bar{g}$ is fixed to unity. The surface energy $\gamma_0 / (E^*L) = 1.0 \times 10^{-3}$.
• Figure 4: Reduced mean gap $g_0/\bar{h}$ as a function of Hurst exponent $H$ in the case of $\rho/L = 2.0\times 10^{-3}$ and $\sigma_0 / (E^* \bar{g}) = 0.01$. The solid line denotes the field approach and open circles the GFMD simulation results.
• Figure 5: Exactly the same as Fig. \ref{['fig:meangap-hurst-a']}, while $\sigma_0 / (E^* \bar{g}) = 0.1$.