Kinetic friction coefficient (stationary sliding) as a function of the logarithm (with 10 as basis) of the sliding velocity for a tread rubber on an asphalt road surface. The blue line, marked cold, is without the flash temperature. The red line, marked hot, is including the flash temperature. The black curves are showing the effective friction, experienced by a tread block, when it goes through the footprint, for the car velocity of 27 m/s and for several slip values (0.005, 0.0075, 0.01, 0.03, 0.05, 0.07, and 0.09). The experienced friction of the tread block follows first the cold rubber branch of the steady-state kinetic friction coefficient. And after, when the block has slipped a distance of order the diameter of the macroasperity contact region, and the flash temperature has fully evolved, the hot rubber branch is followed.

Kinetic friction coefficient (nonstationary sliding) as a function of the logarithm (with 10 as basis) of the sliding velocity for a sandpaper surface and a different rubber compound than in Fig. 1 for a car velocity of 16.66 m/s and for several slip values (0.07, 0.15, and 0.8). The experienced friction of the tread block follows first the cold rubber branch of the steady-state kinetic friction coefficient. Then, when the block has slipped a distance of order the diameter of the macroasperity contact region, the hot rubber branch is followed.

When the temperature increases, the tan δ = ImE/ReE spectrum shifts to higher frequencies. In general, this results in a decrease of the viscoelastic contribution to the rubber friction. It is assumed that the road asperities cause pulsating frequencies in the range between ω₀ and ω₁.

Energy dissipation per unit volume. It is highest in the smallest asperity contact regions.

Contact region between a tire and a road surface. At low magnification ζ < 1, it appears that the tire is in complete contact with the road. But increasing the magnification, the contact area decreases continuously, as indicated in the figure.

Scanning electron microscopy images of the surface region of a car tire tread block at low magnification (top) and at higher magnification (bottom). Acknowledgment is made to Marc Masen, Imperial College London, for providing the images.

Frictional shear stress acting on a tread block as a function of time for many slip values (0.005, 0.0075, 0.01, 0.03, 0.05, 0.07, 0.09, 0.12, 0.15, and 0.25). Car velocity 27 m/s and tire background temperature T₀ = 60°C. 1D tire model using the full friction model (green curves) and the cold-hot friction law (eq 1) (red curves). Passenger car tread compound.

μ-Slip curve for the 1D tire model using the full friction model (green curve) and the cold-hot friction law (eq 1) (red curve). Passenger car tread compound.

Logarithm of the real (red) and imaginary (blue) part of the viscoelastic modulus as a function of the logarithm of the frequency of the tread rubber compound at the reference temperature T₀ = 20°C. The square symbols are large strain or stress results obtained from strain-sweep data using the self-consistent stress procedure (eq 13 in [23]) for a substrate surface with the rms slope κ = 0.7.

Shift factor a_T as a function of the temperature T. The reference temperature T = 20°C. It is also used to shift the individual strain sweep measurements in Fig. 11.

Real part of E as a function of the applied strain during oscillation at fixed frequency of 1 Hz. The curves are obtained at different temperatures starting from 120 to −40°C.

Tan δ as a function of frequency, as obtained using experimental data for a tread rubber compound. The temperature T = 20°C and the red curve is for small strain (0.2%). The green squares are large strain or stress results obtained from strain-sweep data using the self-consistent stress procedure (eq 13 in [23]) for a substrate surface with the rms slope κ = 0.7.

Power spectrum for a used corundum P80 sandpaper, as a function of the wavevector q. The figure shows the top power spectrum on a log₁₀–log₁₀ scale.

1D model of a tire. A tread block is attached to a tire body block that is connected to the rim by viscoelastic springs. The springs have both elongation and bending elasticity (and damping) that are used in longitudinal (e.g., braking) or transverse (cornering) motion.

The μ-slip curve (where μ = F_x/F_N) for the 1D tire model compared with experimental data for the tire loads F_N = 3000, 5000, and 9000 N. The car velocity υ_car = 16.6 m/s at T₀ ≈ 37°C.

μ-Slipangle curve (where μ = F_y/F_N) for the 1D tire model compared with experimental data for the tire loads F_N = 3000 N (a), 5000 N (b), and 9000 N (c). The car velocity υ_car = 16.6 m/s.

(a) Measured μ-slipangle curves for the tire loads F_N = 3000, 5000, and 9000 N. (b) Tread surface temperature was measured at a center position of the tire after half a rotation. The slipangle has been changed with ±2°/s. It was first changed from 0 to 12° then to −12° and back to 0.

Lateral μ-slip curve for the 1D model compared with the experimental results for the tire load F_N = 5000 N. In the upper curve, the background temperature was fixed at 37.7°C. In the lower curve, the background temperature varied between 30 and 70°C (from Fig. 17b).

2D model of a tire (schematic). The car velocity υ_c points in another direction than the rolling direction, resulting in a nonzero cornering angle θ.

Longitudinal and transverse tire vibrational modes of an unloaded tire with fixed rim.

Rubber block sheet of square form (side b) and thickness d exposed to a uniform stress σ = F/db will elongate a distance u.

Calculated and experimental results for the longitudinal stiffness K_L for the tire load F_N = 5000 N and drive velocity υ = 2.2 mm/s.

Calculated and experimental results for the lateral stiffness K_T for the tire load F_N = 5000 N and drive velocity υ = 6.45 mm/s.

Calculated results for the stiffness parameters K_L and K_T for three different loads and the corresponding measured data for two loads.

Tire footprints pressure distribution for the normal loads F_N = 3000 N (a), 5000 N (b), and 9000 N (c).

Longitudinal μ-slip curve for the 1D and 2D models compared with the experimental results for the tire load F_N = 5000 N.

Self-aligning moment as a function of the slipangle. The measured result is given by the solid line and the prediction of the 2D tire model by the dashed line. Tire load F_N = 3000 N (a), 5000 N (b), and 9000 N (c).

Variation of the transverse force on the tire (in units of the normal force F_N) as a function of time when the cornering angle increases linear with time between t₀ = 0 s and t₁ = 0.02 s, from θ₀ = 0° to θ₁ = 1, 3, 7, and 12°, for F_N = 5000 N.

Snapshots of the tire-body deformations for the normal load F_N = 3000 N, 5000 N, and 9000 N. In all cases, the slip s = 0.05 and the cornering angle θ = 0. The short vertical lines indicate the displacement of the tire body from the undeformed state. The maximum tire-body displacements are 0.92, 1.39, and 1.84 cm for the tire loads F_N = 3000 N, 5000 N, and 9000 N, respectively. Rubber background temperature T₀ = 80°C and car velocity 16.6 m/s.

Snapshots of the tire-body deformations for the normal load F_N = 3000 N, 5000 N, and 9000 N. In all cases, the slip s = 0 and the cornering angle θ = 5°. The short horizontal lines indicate the displacement of the tire body from the undeformed state. The maximum tire-body displacements are 1.20, 1.81, and 2.31 cm for the tire loads F_N = 3000 N, 5000 N, and 9000 N, respectively. Rubber background temperature T₀ = 80° C and car velocity 16.6 m/s.

Uniform transverse force acting on the tire tread area deforms the tire body as indicated in the figure.

The μ-slip curve in dependency of the tire inflation pressure. The tire body stiffness parameters {k} and the footprints have been modified to correspond to the inflation pressures 0.2 and 0.3 MPa.

μ-Slipangle curve in dependency of the tire inflation pressure. The tire body stiffness parameters {k} and the footprints have been modified to correspond to the inflation pressures 0.2 and 0.3 MPa.

When a rubber block slides on a rough surface, the heat produced in the asperity contact regions will result in hot tracks (dotted area) on the rubber surface. When an asperity contact region moves into the hot track resulting from another asperity contact region in front of it (in the sliding direction), it will experience a rubber temperature higher than the background temperature T₀. This “thermal interaction” between hot spots becomes important if the slip distance is larger than the average separation between the (macro) asperity contact regions.

μ-Slip curve for car velocity v_c = 27 m/s. The maximum of the μ-slip curve occurs for the slip s = s* = 0.07. The ABS control algorithm should increase the braking torque when s < s* and reduce the braking torque when s > s*.

(a) Car velocity υ_c and the rolling velocity υ_R as a function of time t. The slip (b) and the braking moment (c) as a function of time t. For ABS braking, using algorithms a (see text for details).

(a) Car velocity υ_c and the rolling velocity υ_R as a function of time t. The slip (b) and the braking moment (c) as a function of time t. For ABS braking, using algorithms b (see text for details).

μ-Slip curves for ABS braking using algorithms a (top) and b (bottom). The green curve is the steady-state μ-slip curve for car velocity υ_c = 27 m/s, whereas the red curve shows the instantaneous effective friction coefficient.

Car velocity υ_c as a function of time t during ABS braking using two algorithms a (with s* = 0.05) and b. The procedures a and b result in nearly the same time, ≈2 s, for reducing the car velocity from 27 to 10 m/s. The effective friction μ ≈ 0.87 is smaller than the maximum kinetic friction (≈1.1).

Car velocity υ_c as a function of time t during ABS braking using algorithm a with s* = 0.05 and 0.07. Both cases result in nearly the same time for reducing the car velocity from 27 to 10 m/s. The effective friction μ ≈ 0.87 is smaller than the maximum kinetic friction (≈1.1).

Dynamical μ-slip curves for ABS braking using two different chosen s*-slip values for control algorithm a. The blue curve is the steady-state μ-slip curve.

Car velocity υ_c and the rolling velocity υ_R as a function of time t for ABS braking with two different chosen s*-slip values using algorithm a.

Dynamical μ-slip curves for ABS braking using two different chosen s*-slip values using algorithm a. The blue curve is the steady-state μ-slip curve.