## 2 Description of the Methodology
### Crack Growth Model  
The crack growth model allows predicting the future characteristics of an already existent crack under some conditions. The developed methodology is based on the work of the Korea Railroad Research Institute (KRRI) [21] to calculate the minimum crack size for growth under rolling contact between wheel and rail. Owing to the crack truncation rate, a minimum crack size is needed to allow propagation. KRRI model considers physically small cracks that eventually become long cracks. The purpose of their work is not the investigation of the growth of microstructurally or mechanically small cracks. According to KRRI work, our methodology uses a minimum crack size value derived from the traction coefficient and creepages in the wheel-rail contact point. Then, for an existing new crack, whose length has already been predicted, the crack growth rate will be a function of the crack depth. This dependence will be again related to the traction coefficient and creepages in the wheel-rail contact point.  
This work from KRRI employs [PERSON] and [PERSON]'s \"2.5D\" fatigue crack growth model [41], which performs calculations with the SIF procedure [42]. With the use of [PERSON]'s functions, it is possible to determine the crack propagation as a function of the crack depth for a known traction coefficient and creepages. In addition, this model takes into account the wear rate using the popular Archard's model. Therefore, the calculated crack will be the net crack (Figure 5) considering  
\[\frac{da_{\text{net}}}{dN}=\frac{da}{dN}-\frac{1}{\sin\left(90-\theta\right)} \times\frac{dw}{dN} \tag{7}\]  
Cracks are initiated while [PERSON]'s energy model accumulates \(DI\), which consider wear and crack propagation as well. Once \(DI\) is equal to one, the minimum crack size determined is long enough for the plastic zone surrounding the crack tip to reach an almost steady state in terms of shape, although the magnitude of plastic strain increases with the number of cycles. Thus, restrictions regarding plastically deformed material can be avoided [43]. For the rail points where the accumulated \(DI\) is higher than one, the crack growth and evolution are analyzed. With the aim of improving the computational cost of the methodology a database is implemented which stores the minimum crack lengths and the coefficients of fourth-degree polynomials \(p_{\text{q}}\) for different values of creepages \(\gamma\) and  
Figure 4: Location of _Total DI_ for both left and right rails.  
traction coefficients \(tc\), as shown by the examples in Figure 6 and Table 1. These coefficients allow calculating the net crack growth rate \(da_{\text{net}}\) as a function of the crack depth \(a\):  
\[\frac{da_{\text{net}}}{dN}=p_{4}.a^{4}+p_{3}.a^{3}+p_{2}.a^{2}+p_{1}.a+\ p_{0} \tag{8}\]  
Although KRRI work is based on [PERSON]'s functions used also in the 2.5D model to enable rapid calculation, the use of this kind of database allows introducing future crack growth calculations by FEM for ad hoc solutions. When cracks are explicitly modeled by FEM, crack growth calculations are very time-consuming because these models make use of rigorous fracture mechanics concepts.  
Regarding the propagation angle, there is not a simple relationship. Near the contact surface, the angle is heavily dependent on plastic damage, which depends on the material response to applied surface tractions. Afterwards, crack tip growth direction and angle influences the propagation angle. As a first approximation, the propagation angle \(\theta\) is calculated by means of the normal and shear forces. Then, the characteristics of the predicted crack are saved according to its location on the rails. In the next calculations, if the _Total_\(D1\) is higher than one, the methodology verifies the existence of a previous crack to apply the crack growth model.