## 2 Methods
### Fatigue Damage Model Parameters  
This study adopts the rail steel data from Figures 4 and 5 in [40], and follows the procedure to obtain damage model parameters in [32]. Although [40] presents quite old data, it contains \(\varepsilon\)-N (strain-life), K-da/dN (Paris law), and material properties data. This is beneficial, because it assures that the data are for the same material. Other fatigue data sources have been considered [41; 42; 43; 44; 45; 46], but either have only the S-N curves available, contain less data points, or do not have both \(\varepsilon\)-N and K-da/dN plots. The fatigue damage model parameters are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Value**} & \multicolumn{2}{c}{**X**} & \multicolumn{3}{c}{**Y**} \\ \cline{2-7}  & **FE, m** & **PD, m** & **Difference** & **FE, m** & **PD, m** & **Difference** \\ \hline Max & \(2.03\times 10^{-5}\) & \(1.90\times 10^{-5}\) & \(-6.95\%\) & \(2.35\times 10^{-6}\) & \(2.55\times 10^{-6}\) & \(7.92\%\) \\ Min & \(-6.59\times 10^{-7}\) & \(-8.30\times 10^{-7}\) & \(20.61\%\) & \(-4.69\times 10^{-5}\) & \(-5.07\times 10^{-5}\) & \(7.48\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum and minimum displacement values in the \(X\) and \(Y\) directions from the finite-element (FE) and Peridynamic (PD) simulations. Figure 4: The displacement in the cross-section of an undamaged model: (**a**) the Y displacement finite element (FE) model; (**b**) the Y displacement PD model; (**c**) the X displacement FE model; (**d**) the X displacement PD model. Deformations are increased 50 times. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Phase I** & **Phase II** \\ \hline
**A** & 426.00 & 25,237.48 \\
**m** & 2.77 & 4.00 \\ \(\varepsilon_{\infty}\) & 0.00186 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fatigue damage model parameters. The parameters for phase I (\(A_{I},\ m_{I},\varepsilon_{\infty}\)) are found from functions fitted to the \(\varepsilon\)-N plot (see Figure 5). The fitted power law function takes the same form, \(y=ax^{b}\), as the phase I damage model in (10). So, parameter \(m_{I}\) is the inverse of slope \(b\):  
\[b=-\frac{1}{m_{I}} \tag{18}\]  
and parameter \(A_{I}\) is calculated from the value of intercept:  
\[a=\frac{-logA_{I}}{m_{I}}\Rightarrow A_{I}=\frac{1}{a^{m_{I}}}. \tag{19}\]  
The fatigue limit of rail steel was determined from the function:  
\[\left(\frac{\Delta\varepsilon}{2}-\varepsilon_{\infty}\right)^{\xi}N=C, \tag{20}\]  
where \(\frac{\Delta\varepsilon}{2}\) is the strain amplitude, \(\varepsilon_{\infty}\) is the fatigue limit, \(N\) is the number of cycles, and \(\xi\), \(C\) are constants. A Paris law plot is required to find the parameters for phase II. In this study, \(R=0.05\), and moist air data from Figure 5 in [40] were used. The plot is replicated in Figure 6. The fatigue damage model in (11) has the same form as the Paris law for crack growth:  
\[\frac{da}{dN}=c\Delta K^{M}, \tag{21}\]  
where \(\frac{da}{dN}\) is the crack growth speed, \(c,\ M\) are constants, and \(\Delta K\) is the cyclic stress intensity factor. \(\Delta K\) is proportional to the bond strain at the crack tip (in [32] called \(\varepsilon_{core}\)); therefore, \(m_{II}=M\), so this parameter can be obtained directly from a Paris law plot. The remaining parameter \(A_{II}\), however, cannot. Instead, a simulation with some trial value \(A^{\prime}_{II}\) has to be run to obtain the trial crack growth speed \(\left(\frac{da}{dN}\right)^{\prime}\). Then, the real \(A_{II}\) value can be found from [32]:  
\[A_{II}=A^{\prime}_{II}\frac{\frac{da}{dN}}{\left(\frac{da}{dN}\right)}. \tag{22}\]  
Figure 5: The \(\varepsilon\)-N data, fitted functions, and damage model parameters for phase I. To find \(A_{II}\), a single edge notch (SEN) specimen with a pre-crack in uniaxial tension is simulated.