## 2 Methods and Materials
### Vehicle Performance Assessment  
#### 2.4.1 Type Wear Quantity  
Type wear is defined according to the frictional or wear energy, \(E_{w}\)[Nm/m\({}^{2}\)], which is the energy due to slippage per unit area and can be evaluated as the product of the shear stress (\(\overrightarrow{\tau}\)) and slip (\(\overrightarrow{S}\)):  
\[E_{w}=\int_{0}^{b}\vec{\tau}d\overrightarrow{S} \tag{7}\]  
where \(b\) is the contact length; \(\tau_{x}\) and \(\tau_{y}\) are, respectively, shear stresses in \(x\)- and \(y\)-directions; and \(S_{x}\) and \(S_{y}\) are, respectively, slip distances in \(x\)- and \(y\)-directions. Wear energy during cornering: In order to calculate the wear energy (\(E_{w}\)) during cornering, a solid tyre is considered using a brush model (Figure 4a), while, for the wear  
Figure 3: (**a**) The modelling of the tread element as a cuboid and (**b**) the force deflection curves according to Equation (5) for pressures ranging [150:25:350] kPa. evaluation during cornering, a wear model is used (Figure 4b). In the considered wear model, parabolic pressure distribution (\(q_{z}\)) and small slip angles are assumed. The contact pressure is defined as:  
\[q_{z}=4p_{m}\frac{x}{b}\left(1-\frac{x}{b}\right) \tag{8}\]  
where \(p_{m}\) is the maximum pressure that occurs from the vertical load \(F_{z}\) and is equal to \(3F_{z}/2 ab\) if the contact patch is assumed as cuboid (Figure 3a). According to the wear model (Figure 4b), the lateral slip angle is \(\alpha\), while the lateral shear stress at a distance \(x\) measured from the the contact patch (0-\(b\)) is \(\tau_{y}\) and is expressed by Equation (9):  
\[\tau_{y}=C_{y}xtana \tag{9}\]  
where \(C_{y}\) [N/m] is the lateral shear elastic tread coefficient per unit area. The shear force \(\tau_{z}\) (=\(\mu_{s}q_{z}b_{h}\)) applied to the contact patch per unit area is equal to the lateral shear stress applied to the sliding point \(b_{h}\) (Equation (9)), i.e., the maximum frictional force per unit area (\(\tau_{ymax}\)). Therefore, combining Equations (8) and (9):  
\[\tau_{ymax}=C_{y}b_{h}tan\alpha=\mu_{s}q_{z}(b_{h})=4\mu_{s}p_{m}\frac{x}{b_{ h}}\left(1-\frac{x}{b_{h}}\right) \tag{10}\]  
where \(\mu_{s}\) is the static friction coefficient, while \(\zeta\) ( = \(\frac{C_{y}btana}{4\mu_{s}p_{m}}\)) is introduced as a new coefficient, to help in the simplification of the equations. Based on \(\zeta\), the sliding point (\(b_{h}\)) is defined as:  
\[ \begin{split} b_{h}=& b(1-\zeta_{h})\quad,0\leq \zeta_{h}\leq 1\\ b_{h}=& 0\quad\quad\quad\quad\quad,1>\zeta_{h} \end{split}  \tag{11}\]  
where the first part is when the tyre is in the adhesion region, while the second part considers that sliding occurs across the whole contact area when \(\zeta_{h}\) is greater than or equal to unity. Then, considering the force equilibrium in the sliding region (\(b_{h}\leq x\leq b\)), the sliding distance is defined according to Equation (12), where \(\mu_{d}\) is the kinematic friction coefficient:  
\[S_{y}=xtana-\frac{\mu_{d}q_{z}(x)}{C_{y}} \tag{12}\]  
Considering Equation (7), the wear energy per unit area is defined as:  
\[E_{y}{}^{w}=\int_{b_{h}}^{b}\mu_{d}q_{z}(x)dS_{y}=\int_{b_{h}}^{b}\mu_{d}q_{z }(x)\frac{dS_{y}}{dx}dx=\mu_{d}tan\alpha\int_{b_{h}}^{b}q_{z}(x)dx+\frac{1}{2} \frac{\mu_{d}^{2}}{C_{y}}q_{z}^{2}(b_{h}) \tag{13}\]  
where Equations (8) and (11) can be substituted, and the equation of wear energy can be derived at the two conditions explained in Equation (11). In addition, in this work, \(\mu_{d}\) and \(\mu_{s}\) are assumed to be equal to \(\mu\), while \(F_{z}\) is considered from the vehicle model:  
\[F_{z}=F_{Z_{0}}+C_{t}(\dot{z}_{t}-\dot{z}_{R})+K_{t}(z_{t}-z_{R}) \tag{14}\]  
where \(F_{Z_{0}}\) is the static load applied. Mass Loss: The wear energy per contact area described above can be related with the mass loss according to Equation (15), which was derived by [PERSON] et al. [17] from experiments:  
\[\Delta m=f_{1}E^{f_{2}} \tag{15}\]  
where \(f_{1}\) and \(f_{2}\) are constants at a given temperature.