Mean-field theory of the Stribeck effect

Abstract

We present a theoretical analysis of frictional transitions along the Stribeck curve for rough elastic contacts lubricated by a Newtonian fluid. Building on the mean-field framework of Persson and Scaraggi (J. Phys.: Condens. Matter 21 (2009) 185002), we formulate a minimal elastohydrodynamic model that couples contact mechanics and lubrication through a homogenized pressure decomposition. Dimensional analysis reveals three independent dimensionless parameters governing the frictional response, which correspond to a dimensionless speed, normal load, and surface roughness. Using asymptotic expansions, we first characterize the boundary and hydrodynamic lubrication regimes, which arise naturally as the quasistatic and smooth-surface limits of the model. In both limits, the contact morphology converges toward Hertzian contact in the regime of large elastic deformation, with boundary layers regularizing the separation profile at the edge of the contact zone. We then analyze the mixed lubrication regime and derive asymptotic expressions for the friction coefficient in the low- and high-speed limits. At high speeds, friction decomposes into a viscous contribution and a residual contact term, leading to a roughness- and load-dependent criterion for the transition to hydrodynamic lubrication that departs from constant-Λ ratio theories. At low speeds, friction reduction results from the progressive redistribution of the applied load between asperity contact and hydrodynamic pressure, yielding a characteristic transition speed from boundary to mixed lubrication. These results are summarized in a phase diagram that generalizes the classical Stribeck curve to a multidimensional parameter space.

Figures (12)

• Figure 1: Stribeck curve schematic. Friction coefficient, ratio between shear force $F$ over the normal force per unit length $F_N$, versus the Hersey Number $\eta U/F_N$, where $\eta$ and $U$ are the lubricant viscosity and sliding speed respectively. Three regimes of boundary, mixed and hydrodynamic lubrication are encountered as the Hersey number increases.
• Figure 2: Schematic of the homogenization method. A rigid rough cylinder is sliding along a soft solid immersed in a viscous liquid. We model the contact area at the mesoscopic scale using a mean-field approach (see red zoomed region), where the asperities (see green zoomed region) are averaged spatially to a contact pressure.
• Figure 3: Numerical Stribeck curve and phase diagram. (a) Friction coefficient versus the Hersey number for $\bar{h}_\mathrm{rms} = 10^{-4}$, $\bar{F}_N = 0.1$ and $\mu_0 = 1$. (b) Schematic phase diagram of the lubrication problem in the three-dimensional parameter space defined by the dimensionless speed $\lambda$, load $\bar{F}_N$ and roughness $h_\mathrm{rms}$. The diagram highlights key asymptotic regimes: the smooth limit ($\bar{h}_\mathrm{rms}=0$, blue plane), corresponding to classical elastohydrodynamic lubrication; the quasistatic limit ($\lambda = 0$, pink plane), where the friction is purely contact-dominated; and their intersection (green line), representing the Hertzian indentation problem for dry contacts, as schemed in the green box.
• Figure 4: Separation distance profiles in the smooth limit. Dimensionless separation distance as a function of the lateral position for various values of $\lambda$ (indicated with colors) both in the $\lambda \ll 1$ (a) and $\lambda \gg 1$ (b) limits. The inset in (a) displays a zoom in the outlet region, highlighting the $\lambda$ dependence of the separation distance in the central region. In the panels (c) and (d), we plot the same data as in (a)-(b) but with the appropriate $\lambda$ rescaling to highlight the asymptotic solutions. The schematics in (e) and (f) highlight the structure of the lubrication film. The pink dot shows the location of the Hertz contact radius.
• Figure 5: Central film thickness and hydrodynamic lubrication friction in the smooth limit. (a) Dimensionless central film thickness $h(x=0)/u_\mathrm{Hz}$ versus $\lambda$, where the blue and green dashed lines show \ref{['eq:central_thickness_large']} and \ref{['eq:film_thickness_small_deformation']} respectively. (b) Dimensionless hydrodynamic friction coefficient versus the dimensionless speed $\lambda$ in logarithmic scales, where the circles show the numerical integration of \ref{['eq:hydrofriction_full']} and the solid lines the interpolation function \ref{['eq:interpolated_hydrodynamic_friction']}. The blue and green dashed lines display the asymptotic law \ref{['eq:hydrofriction_large']} and \ref{['eq:hydrofriction_small']} respectively.