## 2 Background
### Road-Rubber Friction
#### 2.2.2 Tyre Friction Force as a Function of Stress Frequency  
The friction force between the tyre and the road is generated as a result of the relative slippage between the elastomer and the road surface. If there is no relative slippage then there is no tyre force. Two stress mechanisms intervene during the generation of tyre forces: molecular adhesion and indentation. * Molecular adhesion: The first mechanism is the adhesion [57]. The grip derived from the adhesion between the rubber and the road is the result of the Van der Waals bonding phenomena. The rubber's molecular chains form, stretch and break, following a cycle of stretching and breaking, and generating visco-elastic work. This adhesion mechanism occurs in a range of stress frequencies between \(10^{6}\) Hz and \(10^{9}\) Hz, and requires the separation distance between the road and the rubber to be below \(10^{-6}\) mm [2]. The bonding phenomena can be explained in a simplified manner by three steps, Figure 5b. In the first step the bond is created. After that,  
Figure 4: The influence of stress frequency on the tyre as a viscoelastic material. The modulus of elasticity becomes maximum above a stress frequency threshold. Hysteresis (energy loss) is maximum at this threshold and reduces at frequencies different to that [2]. in step 2, the molecular chain is stretched, and a friction force which opposes the tyre skidding is generated. Finally, in the last step, the bond breaks and new bonds form again successively. * Road roughness effect: The road roughness effect (also denoted as indentation) is primarily caused by the road irregularities [58; 59] and the hysteresis of the rubber [2]. The road texture (with rough spots that vary from 1 centimetre to 1 micron) induce a high-frequency excitation on the rubber (with frequencies ranging from \(10^{2}\) to \(10^{6}\) Hz [2]), which is distorted and undergoes several compression-relaxation cycles. As the rubber presents an inherent hysteresis, the rubber does not return immediately to its initial position, but exhibits an asymmetrical movement (and therefore an energy loss). Such asymmetrical movement of the rubber block around the rough spot results in a force field, with a tangential component which opposes the slippage and is seen as the tyre force [2]. The proportion in which each friction generation mechanism contributes to the total amount of grip available between the tyre and the road depends on the road irregularities and the road contamination, the latter factor having the highest influence. Regarding the road roughness, an analytical expression for the kinetic coefficient of friction has been proposed by [PERSON] [60] as a sum of a set of hysteresis forces caused by the multiple scales of the road surface texture,  
\[\mu =\frac{1}{2}\int_{q_{L}}^{q_{1}}q^{3}C(q)P(q)dq\int_{0}^{2\pi} \cos(\phi)Im\bigg{(}\frac{E(qv\cos(\phi))}{(1-\
u^{2})\sigma_{0}}\bigg{)}d\phi \tag{9}\] \[P(q) =\frac{2}{\pi}\int_{0}^{\inf}\frac{\sin(x)}{x}\exp(-x^{2}G(q))dx =\text{erf}\bigg{(}\frac{1}{2\sqrt{G(q)}}\bigg{)}\] (10) \[G(q) =\frac{1}{8}\int_{q_{L}}^{q}q^{3}C(q)dq\int_{0}^{2\pi}\bigg{|} \frac{E(qv\cos(\phi))}{(1-\
u)^{2}\sigma_{0}}\bigg{|}^{2}d\phi\] (11) \[\xi =\frac{2\pi}{q}\] (12) \[q =(q_{x},q_{y})=(q\cos(\phi),q\sin(\phi)) \tag{13}\]  
where \(q\) is the spatial angular frequency or magnitude of the wave vector corresponding to the wavelength \(\xi\), \(q_{L}\) is the lower integration limit (where \(L\) depends on the length of a tread block), \(q_{1}\) is the upper cut-off frequency, \(E\) is the complex viscoelastic modulus of the rubber, \(v\) is the sliding velocity, \(v\) is rubber's Poisson ratio, \(\phi\) is the direction of the wave vector in relation to sliding and \(P(q)\) is the relative contact area. For dry friction, the short wavelength limit is 1 micrometre, as mentioned in previous works [61]. Figure 5: (**a**) Road roughness (indentation) friction mechanism; (**b**) Molecular adhesion friction mechanism.