## 5 Discussion
### Studying Model Using Vickers Hardness  
The confirmed relationship between carbon content and Vickers hardness in the data from [PERSON] et al. (2018) was used to investigate factors that influence fatigue limit other than loading frequency. Figure 6 shows the results. Since Vickers hardness is not commonly defined in the literature, the following empirical equation was used to evaluate the relationship between tensile strength \(\sigma_{\text{B}}\) [MPa] and Vickers hardness. \[\sigma_{\text{B}}\cong\frac{1}{3}(9.8\times H_{\text{v}})\cong 3.27\times H_{ \text{v}} \tag{5}\]  
Figure 6 shows that Vickers hardness increases with increasing carbon content, indicating a positive correlation between the two. The least-square method was used for linear regression, with each plotted point shown by the solid line. The Fe-C phase diagram demonstrates that carbon steels with ferrite/pearlite microstructures experience an increase in pearlite volume fraction as ferrite volume fraction decreases. [PERSON] et al. (2018) established a proportional relationship between tensile strength and pearlite volume fraction by decomposing tensile strength into strengthening mechanisms based on the Hall-Petch rule and relating them to microstructural parameters. The Fe-C phase diagram shows that the volume fraction of pearlite increases as the volume fraction of ferrite decreases in carbon steels with ferrite/pearlite microstructures. The trend shown in Figure 6 is due to the increase in the volume fraction of pearlite. Below, Figure 7 shows the relationship between the axial-loading fatigue limit and the Vickers hardness. The dotted line in the figure is an empirical relationship equation between the Vickers hardness \(H_{\text{v}}\) and the axial-load fatigue limit \(\sigma_{\text{w}}\) in a double-ended tension-compression fatigue test for steel materials with a Vickers hardness \(H_{\text{v}}\) of 400 HV or less, as shown in Equation (6) [4]. \[\sigma_{\text{w}}=1.47H_{\text{v}} \tag{6}\]  
The fatigue limit data utilized in this study were derived from the work of [PERSON] et al. (2018). Based on Figure 7, when the loading frequency is less than 100 Hz, the relationship between the fatigue limit and Vickers hardness can be described by Equation (6). However, at frequencies exceeding 100 Hz, the correlation between the axial-load fatigue limit and Vickers hardness diverges from the conditions of Equation (6). This finding is significant for materials with Vickers hardness values below 150 HV. Therefore, a three-dimensional plot was made to evaluate the relationship between Vickers hardness, loading frequency, and fatigue limit. The results are shown in Figure 8. All plot points were regressed using a quadratic shape regression surface. The correlation coefficient, \(r\), displays a strong correlation at 0.944. Figures 7 and 8 show that Vickers hardness and loading frequency are both factors that affect the fatigue limit. Figure 6: Relationship between carbon contents and Vickers hardness. Based on Equation (3), the smaller the value of the material parameter \(\beta_{0}\), the smaller the effect of the loading frequency on the fatigue limit. The effect of the loading frequency on the fatigue limit tends to decrease as the Vickers hardness value increases, as shown in Figure 8. Based on the above, we proposed Equation (7) by reflecting these contents in the material parameters \(\alpha_{0}\) and \(\beta_{0}\) in Equation (3). \[E=AH_{\text{v}}\text{exp}\left(\frac{B/H_{\text{v}}}{T_{0}\text{ln}\left(\frac {f_{0}}{f}\right)}\right) \tag{7}\]  
where \(A\) and \(B\) are material parameters. \(A\) and \(B\) were estimated from axial-loading fatigue data [23]. \(E/H_{\text{v}}\) was used for the vertical axis, and \(\text{exp}((1/H_{\text{v}})/T_{0}\text{ln}(f_{0}/f))\) was used for the horizonal axis. Each axis was converted to a logarithmic axis and subjected to linear regression using the least-square method.