## 2 Description of the Methodology
### Crack Initiation Model  
In this work, an energy dissipation method was selected to predict the eventual initiation of cracks. This kind of method is very useful and fast to calculate where the cracks are more probable to appear. However, this analysis does not provide any information about the characteristics of the predicted cracks. To take advantage of the ease and low time-consumption of the energy dissipation method, the crack characteristics will be predicted only when and where a crack is expected to appear. The crack growth model (see Section 2.4) implemented within the proposed methodology assesses the characteristics of the crack. The Damage Index Function developed by [PERSON] [14] was selected as the energy dissipation method to apply. This validated assessment, on Network Rail tracks, makes use of the wear number \(T\gamma\), which is the product of shear forces (tangential to the head of rails) and creepages (\(\gamma\)). The wear number is directly linked to shear force coefficient \(\mu\). The Damage Index is calculated as  
\[DI(x,y)=\sum_{\text{n}}m_{\text{n}}\sum_{\text{wh}}h_{\text{wh,n}}(x,y)d[\mu_{ \text{wh,n}}(x),T\gamma_{\text{wh,n}}(x)] \tag{2}\]  
where \((x,y)\) are the coordinates of any point of the rail, and the damage is calculated considering \(m\) vehicles of n different types with wh wheelsets on each vehicle. The parameter h describes a function with elliptic shape over the contact patch, with its maximum damage (\(d[\mu,T\gamma]\) value) at the centroid. The Damage Index Function (Figure 2) shows the influence of different ranges of \(T\gamma\) on the crack initiation behaviour with four different zones:  
1. For low values of \(T\gamma\), the rail surface is not affected because there is not enough work to cause incremental plastic straining of the material. 2. Afterwards, the risk of RCF cracks starts to increase linearly with energy dissipation. 3. Then this risk decreases when wear becomes the dominant form of surface damage. 4. Finally, high values of wear number prevent crack initiation. This occurs because the effect of wear damage is higher than the propagation of a theoretical crack, so this crack will never appear. According to the Damage Index Function, each contact patch on the rail accumulates some damage. Once the accumulated \(DI\) is higher than one, for an analysed point of the rail, a crack will appear in the rail; however, the damage function is not able to quantify the length, depth and even the propagation angle of these cracks. Thanks to the output parameters from MBS software, the wear number associated with the different contact patches of the calculation will be known, as well as the semi-axis of the contact ellipses. Therefore, the \(DI\) associated with each wheel-rail contact can be easily added to discretized sections (in both lateral and longitudinal directions) of the rails. If the total \(DI\) is higher than one in any of these sections, and a crack is expected to appear, the crack growth model will be applied (see Section 2.4). Even if the \(T\gamma\) model is generic for any point of the rail section, there are two variables of influence. One of them is that \(T\gamma\) value is affected by steering forces in a curve negotiation, where the contact point moves to the inner face of the high rail. Furthermore, the material properties defined by Young's modulus and Poisson's ratio are being considered for the calculation of contact patches. It is important to bear in mind that the damage index function is specific for the sites studied in [PERSON]'s report [14]. To assess different track sections, some modifications are required, as applied by [37; 38; 39]. This can be done recalculating the turning points of the damage function using the equation  
Figure 2: Rail rolling contact fatigue damage function for normal grade rails. Values are taken from [14].