Three different models of “rough” surfaces. (a) All the asperities are equally high and have identical radius of curvature. (b) Introducing asperities with a random height distribution gives the GW approach toward contact mechanics. (c) A real, randomly rough surface is shown, where the asperities are of different heights and curvature radii.

Elastic block (dotted area) in adhesive contact with a rigid rough substrate (dashed area). The substrate has roughness on many different length scales, and the block makes partial contact with the substrate on all length scales. When a contact area is studied, at low magnification it appears as if complete contact occurs, but when the magnification is increased it is observed that in reality only partial contact exists.

Area of real contact, as a function of the nominal contact pressure (in units of E), between an elastic solid with the Young's modulus E and Poisson's ratio ν = 0.5, and an asphalt road surface that is considered rigid.

Surface roughness power spectrum used in the adhesion and contact mechanics calculations.

Surface height probability distribution for five different asphalt road surfaces. The average of the skewness over all five surfaces is SK ≈ −0.21.

Surface height probability distribution for three mathematically generated (self-affine fractal) surfaces with the same surface roughness power spectra.

Normalized contact area as a function of the squeezing pressure [in units of the low frequency reduced modulus E_r0 = E₀/(1 − ν²)] as obtained from numerical calculations. Results are shown for three different surfaces with the same surface roughness power spectrum, but different skewness SK. The sliding speed is (a) v = 1 mm/s and (b) v = 2.336 m/s.

Viscoelastic contribution to the friction coefficient as a function of the squeezing pressure [in units of the low frequency reduced modulus E_r0 = E₀/(1 − ν²)] as obtained from numerical calculations. Results are shown for three different surfaces with the same surface roughness power spectrum but different skewness SK. The sliding speed v = 2.336 m/s.

Area of real contact between an elastic solid with the Young's modulus E and Poisson's ratio ν = 0.5 and a road surface. The asphalt road surface has the power spectrum shown in Fig. 3 and is assumed rigid. The red line is without adhesion, and the other lines with adhesion using the work of adhesion w (also denoted by Δγ) between flat surfaces w = 0.05 J/m², as is typical for the (adiabatic) work of adhesion between rubber and many solids (in this case the physical origin of w is usually due to the weak van der Waals interaction).

Effective interfacial binding energy γ_eff as a function of the magnification ζ = q/q₀. For an elastic solid with the Young's modulus E = 10 MPa and Poisson ratio ν = 0.5 and for the nominal contact pressure p = 1 MPa. The asphalt road surface has the power spectrum shown in Fig. 3 and is assumed rigid. The work of adhesion between flat surfaces γ(ζ₁) = w = 0.05 J/m², as is typical for the (adiabatic) work of adhesion between rubber and many solids (in this case the physical origin of w is usually due to the weak van der Waals interaction).

Simple friction tester (schematic) used for obtaining the friction coefficient μ = M′/M as a function of the sliding speed. The sliding distance is measured using a distance sensor and the sliding velocity obtained by dividing the sliding distance with the sliding time. This setup can only measure the friction coefficient on the branch of the μ(v)-curve where the friction coefficient increases with increasing sliding speed v.

Measured and calculated friction coefficient as a function of sliding speed for a rubber compound on concrete surface. The upper green lines are the total calculated rubber friction coefficient and the lower green lines are the viscoelastic contribution. The solid lines are without flash temperature and the dashed lines with the flash temperature. The background temperature is T₀ = 8°C and the nominal contact pressure p = 0.05 MPa (red squares), p = 0.3 MPa (black squares), and p = 0.1 MPa (blue squares and green +).

Measured friction coefficient as a function of sliding speed for a rubber compound on an asphalt road surface. The background temperature is T₀ = 20°C and for the nominal contact pressure is indicated in the figure.

Elastic block (e.g., rubber) sliding on a substrate. For a randomly rough surface the concentration of macroasperity contact regions increases proportional to the normal force. This will affect the lateral elastic coupling between the macroasperity contact regions, and results in a friction force that depends nonlinearly on the load.

Schematic illustrating that an increase in the temperature shifts both μ_cont (v) and μ_visc (v) toward higher sliding speeds.

Measured temperature profile before run-in, after sliding s = 3.5 m. The rubber block is L = 2.5 cm long in the sliding direction and 0.7 cm high. The color (mainly green) on the road surface behind the rubber block results from the temperature increase on the road surface from the interaction with the rubber block.

Measured temperature profile T(x) after run-in. Run-in consists of 10 repetitions, each involving sliding of s = 3.5 m, that is, a 35 m total distance. Between each run is 10 s of waiting time. The temperature profile is after the last repetition.

Measured profile of the rubber block (a) before run-in and (b) after run-in (10 repetitions, each involving sliding s = 3.5 m).

Measured friction coefficients as a function of the number of repetitions during run-in. Each repetition involves sliding 3.5 m at the sliding speeds (a) v = 0.1, (b) 0.5, and (c) 1 m/s. Results are shown for the nominal contact pressures 0.05 (blue lines), 0.15 (red), and 0.3 MPa (green).

Measured maximal temperature on the rubber side wall (red data points) and the calculated average (over the y-direction) rubber temperature at x = L/2 at rubber–road interface (green symbols and lines). Results are shown when the initial temperature equals T = 20 and 40°C, and for the contact pressures (a) p = 0.05 and (b) 0.15 MPa. The block is L = 2.5 cm long in the sliding direction and the temperature refers to a sliding distance s = 3.5 m.

Calculated temperature T(x) as a function of the position x along the sliding direction with x = 0 at the leading edge and x = L = 2.5 cm at the trailing edge. The temperature profile is after sliding s = 3.5 m at the nominal contact pressure p = 0.15 MPa and with the initial temperature T = 20°C. The red line is the temperature in the macroasperity contact regions and the green line the temperature averaged over the y-direction orthogonal to the sliding direction.

Red and blue lines: the kinetic friction coefficient (stationary sliding) as a function of the logarithm (with 10 as basis) of the sliding velocity. The blue line denoted “cold” is without the flash temperature, whereas the red line denoted “hot” is with the flash temperature. Black curves: the effective friction experienced by a tread block as it goes through the footprint. For the car velocity 27 m/s and for several slip values 0.005, 0.0075, 0.01, 0.03, 0.05, 0.07, and 0.09. Note that the friction experienced by the tread block first follows the cold rubber branch of the steady state kinetic friction coefficient and then, when the block has slip a distance of order the diameter of the macroasperity contact region, it follows the hot rubber branch.

μ-Slip angle curves for the elliptic footprint with the tire loads F_N = 3000, 5000, and 7000 N, and the footprint pressure p = 0.3 MPa. For the rubber background temperature, T₀ = 80°C and the car velocity, 27 m/s.