## 5 Discussion
### Examining the Effects of Different Loading Modes  
We examined the differences in values between the axial-loading model established in terms of the rate process theory [23] put forth by [PERSON] et al. in Section 4 and the rotating bending fatigue test results, using various data on the rotating bending fatigue limit, to see whether the difference was \"due to the effect of the difference in static strength\" or \"due to the effect of the difference in loading mode\". The axial-loading model established in terms of the rate process theory [23] was proposed based on the results of fatigue tests under axial loading fatigue with a stress ratio of \(R=-1\) on carbon steel with a duplex ferritic/pearlitic microstructure that has not undergone hardening and tempering. Since the number of data is insufficient for comparison with the fatigue test results in this study, the literature values were used for comparison with the data under axial-loading fatigue [23]. In comparing the axial-loading fatigue limit, the literature values were examined based on the following four conditions, except for the discrepancy caused by the loading mode. However, the loading frequency was limited to 120 Hz because the mechanism by which the rotating bending fatigue test is conducted does not allow fatigue tests in the high-frequency band as in the axial-loading fatigue test. 1. The microstructure of the carbon steel was predominantly duplex ferritic/pearlitic, except for strongly quenched steels. Moreover, cases involving ultrafine grain structures are outside the scope of the proposed model. 2. Fatigue tests were operated in air and at room temperature under rotating bending. The test environment in which the specimens were subjected to high temperatures was excluded. 3. The specimen shape inferred a stress concentration factor \(K_{t}\) less than 1.15. 4. The loading frequency of the fatigue test was within a wide range spanning from 1 to 120 Hz. In accordance with these criteria, a total of 23 fatigue limit data were collected. Details on the selected data are shown in Table 5 [32; 33; 34; 35; 36; 37; 38]. Figure 9: (**a**) A three-dimensional plot showing the results derived from the extended model; (**b**) a representation of the relative error for the selected axial-loading fatigue limit data [16]. The relationship between the axial-loading fatigue limit and the rotating bending fatigue limit was checked using three-dimensional plots. The results are shown in Figure 10. All plot points (the axial-loading fatigue limit and the rotating bending fatigue limit) were regressed using a quadratic shape regression surface. The correlation coefficient, \(r\), displays a strong correlation at 0.922. It can be expressed by one common surface despite the different loading modes. The relationship between the model and the rotating bending fatigue limit when using Equation (8) is shown in Figure 11a. For each selected rotating bending fatigue datum, the relative error of the fatigue limit estimation, provided by the \((\sigma_{\text{w}}-E)/E\) calculation, was then plotted in Figure 11b. The percentages of data within \(\pm 10\%\) and \(\pm 20\%\) of the  
Figure 10: Overview of the selected fatigue limit data, expressed alongside the Vickers hardness and loading frequency, including two different types: axial loading [16] and rotating bending. Second-order surface fitting is included for illustrative purposes. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Data No.** & **\%C** & \(H_{\text{V}}\) & \(f_{\text{,}}\) Hz** & \(\sigma_{\text{W,}}\)**Mpa** & **Ref.** \\ \hline
1 & 0.09 & 115 & 57 & 162 & [32] \\
2 & 0.1 & 115 & 70 & 235 & This study \\  
\hline \end{tabular}
\end{table} 
Table 5: General information on the selected fatigue data. relative error were 26% and 48%, respectively. In several datasets, the rotating bending fatigue limit was greater than that of the model based on Equation (8).