# Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train
## 3 Simulation Model
### Archard Wear Model  
The Archard wear model is used to calculate the wear loss in the contact patch; the wear loss is written as:  
\[V_{wear}=k_{w}\frac{Nd}{H} \tag{9}\]  
where \(V_{wear}\) is the wear loss in the contact patch, \(N\) is the normal contact force, \(d\) is the sliding distance, \(H\) is the hardness, and \(k_{w}\) is the friction coefficient. The contact patch can be divided into many elements of equal size (Figure 11), and the length and width of the element are, respectively, \(\Delta x\) and \(\Delta y\). \(p_{z}\) is the normal force on the element, \(S_{x}\) is the lengthwise sliding velocity, \(S_{y}\) is the lateral sliding velocity, and \(a\) and \(b\) are the semi major axis and the semi minor axis of the contact patch, respectively. Figure 10: Solving method of \(\phi_{f}\). For an element of the contact patch, the wear loss is:  
\[\Delta z=k_{w}\frac{p_{z}\Delta_{d}}{H} \tag{10}\]  
The normal force on the element is:  
\[p_{z}=p_{z}(x,y)=\frac{3N}{2\pi ab}\sqrt{1-\left(\frac{x}{a}\right)^{2}-\left( \frac{y}{b}\right)^{2}} \tag{11}\]  
The sliding displacement is:  
\[\Delta d=|s|\Delta t=\sqrt{{S_{x}}^{2}+{S_{y}}^{2}}\frac{\Delta_{x}}{v_{c}} \tag{12}\]  
where \(v_{c}\) denotes the velocity of the contact patch. Combining Equations (10)-(12), the wear loss of any elements of the contact patch can be written as:  
\[\Delta z(x,y)=\frac{3 Nk_{w}}{2\pi abH}\sqrt{1-\left(\frac{x}{a}\right)^{2}- \left(\frac{y}{b}\right)^{2}}\sqrt{{S_{x}}^{2}+{S_{y}}^{2}}\frac{\Delta_{x}}{ v_{c}} \tag{13}\]  
where \((x,y)\) is the position of the element of the contact patch. If the velocity component caused by the flexible deformation is ignored, Equation (13) is expressed as:  
\[\Delta z(x,y)=\frac{3 Nk_{w}\Delta_{x}}{2\pi abH}\sqrt{1-\left(\frac{x}{a} \right)^{2}-\left(\frac{y}{b}\right)^{2}}\sqrt{\left(\xi-\phi x\right)^{2}+ \left(\eta+\phi y\right)^{2}}\frac{v_{x}}{v_{c}} \tag{14}\]  
If the rolling contact patch is assumed to have a slight local slip, it implies \(v_{c}\approx v_{x}\), and thus, Equation (14) is expressed as:  
\[\Delta z(x,y)=\frac{3 Nk_{w}\Delta_{x}}{2\pi abH}\sqrt{1-\left(\frac{x}{a} \right)^{2}-\left(\frac{y}{b}\right)^{2}}\sqrt{\left(\xi-\phi x\right)^{2}+ \left(\eta+\phi y\right)^{2}} \tag{15}\]  
where \(\xi\) is the longitudinal creepage, \(\eta\) is the lateral creepage, and \(\phi\) is the spin creepage. These parameters can be calculated using the vehicle-track rigid-flexible coupling dynamical model. Equation (15) indicates that the wear loss of the contact patch depends on the normal force and creepages. When the distribution of these parameters along the wheel circumference fluctuates, the wear loss distribution along the wheel circumference also fluctuates, and it can easily lead to polygonal wear under certain conditions. Selecting the wear coefficient (\(k_{w}\)) is important for accurately calculating the wear that occurs on the wheel-rail because of their interaction. In [24], the authors constructed the friction coefficient based on the normal contact compressive stress and sliding velocity between the wheel and the rail obtained through tests. The wear coefficient within the range of 0.0001 to 0.001 means slight wear. In this paper, \(k_{w}\) is 0.0004. Using the model discussed above to calculate polygonal wear, for simplicity, the contact points are considered to be distributed along the rolling circle of the wheel between the two adjacent contact points discretely. At these contact points, the normal load and slips of wheel-rail are calculated using the rigid-flexible coupling dynamical model of vehicle-track, discussed above. At each contact point,  
Figure 11: Forces on elements of contact patch. the wear depth of elements is calculated in the contact patch using the known normal load and slips of the wheel-rail. Then, the wear depths along the rolling circle of the wheel are summed. This process is repeated 50 times.