## 4 Comparison of Fatigue Limits with Proposed Model Estimation  
[PERSON] et al. [22; 23] proposed an estimation equation based on the rate process theory for the thermal activation of dislocations because ferritic/pearlitic steels are BCC lattice materials, and the strain-rate dependence is high. The proposed equations are shown in Equations (3) and (4).  
\[E=\alpha_{0}\text{exp}\left(\frac{\beta_{0}}{T_{0}\text{ln}\left(\frac{\ell_{ 0}}{f}\right)}\right) \tag{3}\]  
\[f_{0}=\frac{\dot{\varepsilon_{0}}}{2\Delta\varepsilon_{E}} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Loading Frequency _f,_ Hz & **Fatigue Limit \(\sigma_{\text{w}}\), MPa** \\ \hline JIS S10C & 70 & 235 \\ JIS S25C & 70 & 265 \\ JIS S35C & 7 & 285 \\ JIS S45C & 70 & 395 \\  & 70 & 415 \\ \hline \hline \end{tabular}
\end{table}
Table 3: List of rotating bending fatigue limits for each specimen.  
Figure 3: Results of estimating fatigue limit by using Vickers hardness.  
where \(\alpha_{0}\) and \(\beta_{0}\) are the material parameters for when the temperature \(T_{0}\) is 293 K, and \(f_{0}\) is the frequency coefficient linking to the frequency corresponding to \(\hat{\epsilon}_{0}\), while \(\Delta\varepsilon_{E}\) is the total strain range imposed by the fatigue test at the fatigue limit \(E\). As, in the case of this study, the materials used were low- and medium-carbon steels with ferrite/pearlite microstructures, we assumed that \(\Delta\varepsilon_{E}\) at the fatigue limit \(E\) was the same as that used by [PERSON] et al. (2022, 2023), which was \(\Delta\varepsilon_{E}=0.002\). The frequency coefficient \(\hat{\epsilon}_{0}\) was assumed to be \(\hat{\epsilon}_{0}=10^{7}\) s\({}^{-1}\) based on the work of [PERSON] et al. (2019) involving pure iron. By substituting these values into Equation (4), \(f_{0}=2.5\times 10^{9}\)s\({}^{-1}\) was calculated and applied to Equation (3).  
[PERSON] et al. (2016) analyzed the relationship between the fatigue limit and the loading frequency for low- and medium-carbon steels with ferrite/pearlite microstructures by grouping them according to carbon content and analyzing material parameters \(\alpha_{0}\) and \(\beta_{0}\) using the Pareto solution. As a result, the axial-loading model, in which the rate process theory is considered, was proposed, and the effect of loading frequency on the fatigue limit was different for each carbon content ([PERSON] et al., 2023). Figure 4 shows a plot of the rotating bending fatigue test results on the analytical model ([PERSON] et al., 2023). The solid line in the figure shows the model proposed by [PERSON] et al. The model is divided into three groups for each carbon content. When the rotating bending fatigue test results were classified to correspond to this model grouping, JIS S10C was compared with the model for the group with carbon contents between 0.08 and 0.11% (G1 Group), and JIS S35C and JIS S45C were compared with the model for the group with carbon contents between 0.33 and 0.46% (G3 Group). Although JIS S25C has no corresponding group in the proposed model, it was compared with models in the group with carbon contents between 0.12 and 0.22% due to its carbon content range being similar to that of the 0.12-0.22% group (G2 Group).  
The estimated fatigue limits were calculated for the two loading frequencies of 7 Hz and 70 Hz used in this fatigue test and compared with the results of rotating bending fatigue tests at the corresponding loading frequencies. Table 4 shows the comparison results. The relative error between the estimated fatigue limit \(E\) obtained using the model and the rotating bending fatigue limit \(\sigma_{\rm w