# Study on Gage Widening Methods for Small-Radius Curves
## 4 Adaptability Analysis of Gage Widening Standards
### Influence of Gage Widening on Rail Wear  
The wheel-rail wear calculation model was established on the basis of [PERSON]'s material wear law [19], which is the most commonly used model for estimating the wear due to contact friction between objects. According to [PERSON]'s formula, given in Equation (16), the material loss due to wear is proportional to sliding distance and normal pressure as well as inversely proportional to material hardness. \[V_{w}=k_{w}\frac{N\cdot s}{H} \tag{16}\]  
In Equation (16), \(V_{w}\) represents the wear volume, \(N\) represents the normal force, \(s\) represents the sliding distance, \(H\) represents the material hardness, and \(k_{w}\) represents the wear coefficient, which is determined via laboratory tests. The wheel-rail wear calculation was performed on the basis of vehicle-track dynamics simulation. The Fastsim algorithm was needed to achieve some necessary information within the contact patch, such as the distribution of sliding and adhesive zones. Initially, the contact ellipse was discretized into \(m\times n\) elements. For any generic element with the size of \(\Delta x\times\Delta y\) (see Figure 11), the wear depth \(\Delta d\) was calculated as follows according to [PERSON]'s formula. \[\Delta d=\Delta V_{w}/\Delta x\cdot\Delta y=[k_{w}\frac{p(x,y)\cdot\Delta x \cdot\Delta y\cdot\Delta s}{H}]/\Delta x\cdot\Delta y=k_{w}\frac{p(x,y)\cdot \Delta s}{H} \tag{17}\]  
Here, \(\Delta V_{w}\), \(p(x,y)\), and \(\Delta s\) represent the wear volume, normal pressure, and sliding distance for this element, respectively. \(p(x,y)=\frac{3N}{2\pi\alpha b}\sqrt{1-(\frac{x}{a})^{2}-(\frac{y}{b})^{2}}\), where \((x,y)\) represents the Cartesian coordinate of this element in the contact patch reference system; \(N\) represents the normal wheel-rail contact force; and a and b represent the semi-major and semi-minor axes of the contact ellipse, respectively. If this element was in the adhesive zone, which could be identified by the Fastsim calculation, there would be no wear in it, as the sliding distance \(\Delta s\) would be zero. If this element was in the sliding zone, \(\Delta s\) was calculated using the following equation. \[\Delta s=\|\mathbf{v}\|\cdot\Delta x/v_{0} \tag{18}\]  
Figure 11: Wear calculation model. Here, \(\Delta x\) represents the longitudinal length of this element, \(v_{0}\) represents the running velocity of the vehicle, and \(\mathbf{v}\) represents the relative slip velocity between the wheel and rail surfaces, which was evaluated through the following formula. \[\mathbf{v}=\mathbf{v}_{r}-\mathbf{v}_{e}=\left[ \begin{array}{c}\xi_{x}-\phi y \\ \xi_{y}+\phi x\end{array} \right]\cdot v_{0}-\left[ \begin{array}{c}\frac{ \partial u_{x}}{\partial x}\\ \frac{\partial u_{y}}{\partial x}\end{array} \right]\cdot v_{0} \tag{19}\]  
In Equation (19), \(\xi_{x}\) and \(\xi_{y}\) represent the longitudinal and lateral creepages of the contact patch, respectively; \(\phi\) represents the spin creepage; and \(\mathbf{u}=(u_{x},u_{y})\) represents the elastic displacement. The elastic slip term \(\mathbf{v}_{e}\) is usually much smaller than the rigid slip part \(\mathbf{v}_{r}\); hence, \(\mathbf{v}_{e}\) was neglected in this calculation. The value of the wear coefficient \(k_{w}\) was determined through a series of tests conducted under dry and clean conditions. The \(k_{w}\) value changes depending on the slip velocity and normal pressure, as shown in Table 17[23]. Through the above calculation process, the wear depth for every element in the contact patch could be determined. Then, the depth corresponding to every longitudinal strip of the contact patch could be determined by adding the wear depths of the elements with the same y-coordinate, i.e., the wear depth distribution along the wheel or rail profile. Figure 12 shows the entire process of a wheel passing through a certain section of rail. It is shown that the section departure from the front end of the contact spot to the rear end of the contact spot is divided into k moments.