## 3 Fatigue Test Results  
Figure 2a-d show the \(S\)-\(N\) diagrams obtained from the rotating bending fatigue tests involving JJS S10C, JIS S25C, JIS S35C, and JIS S45C, respectively, wherein the bilinear curves, represented by solid lines, were determined via a standard JSMS regression method [25]. The vertical axis, \(\sigma_{a}\), is the stress amplitude, and the horizontal axis, \(N_{\text{f}}\), is the number of cycles to failure. Figure 2 shows that the fatigue limit increases with increasing carbon content. The calculated fatigue limits are shown in Table 3. According to results reported in [26], the rotating bending fatigue limits of JIS S10C when annealed in a furnace at about 900 \({}^{\circ}\)C, and the limits of JIS S25C when annealed in a furnace at about 850 \({}^{\circ}\)C, are 135 to 245 MPa and 185 to 295 MPa, respectively. These fatigue test results are within a reasonable range. However, as reported in [26], the rotating bending fatigue limits of JIS S35C when annealed in a furnace at about 830 \({}^{\circ}\)C, as well as JIS S45C when annealed in a furnace at about 810 \({}^{\circ}\)C, are 195 to 295 MPa and 205 to 305 MPa, respectively. Although the heat treatment temperatures for JIS S10C and JIS S25C were different from those in the literature, the results were equivalent to the values described in the literature because the specimens were annealed. However, for JIS S35C and JIS S45C, hot-rolled bars were used as supplied. In general, annealing removes residual stresses and the hardening generated in the material during the material working process, resulting in lower fatigue limit values. The fatigue limit of JIS S45C is higher than that of materials with similar carbon content. This is considered to be due to the increase in static strength resulting from the presence of 0.5% or more of solid solution Mn [27; 28].  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Loading Frequency _f,_ Hz} \\ \cline{2-3}  & **7** & **70** \\ \hline JIS S10C & - & Conduct \\ JIS S25C & - & Conduct \\ JIS S35C & Conduct & Conduct \\ JIS S45C & Conduct & Conduct \\ \hline \hline \end{tabular}
\end{table}
Table 2: List of the loading frequencies for each specimen.  
When comparing the fatigue test results of medium-carbon steel at 7 Hz and 70 Hz, the results for both JIS S35C and JIS S45C showed an increase in the fatigue limit of about 3-5%. In a study by [PERSON] et al. [29], four low- and medium-carbon steels were tested in a rotating bending fatigue test at different loading frequencies ranging from 50 rpm (0.8 Hz) to 15,000 rpm (250 Hz), and the fatigue limit was reported to increase by about 7% at 0.4%C. Our fatigue test results are consistent with the results obtained by [PERSON] et al. [29]. However, the fatigue limit is deemed to fall within the _S-N_ curve's variation range, and it cannot be said that the fatigue limit's dependence on loading frequency has been observed.  
For carbon steel, a model for estimating the fatigue limit in rotating bending based on hardness in the range of Vickers hardness \(H_{\text{v}}\) below 400 HV was proposed by [PERSON] [3]:  
\[\sigma_{\text{w}}=1.6H_{\text{v}}\pm 0.1H_{\text{v}} \tag{2}\]  
where the unit for the fatigue limit \(\sigma_{\text{w}}\) is MPa. Figure 3 shows a comparison between Equation (2) and the fatigue test results. From the figure, the fatigue limit of each material was plotted to be higher than the estimated line of sight. It was confirmed that the fatigue test results were the same as those obtained using the conventional estimation method.