## 5 Discussion
### Proposing the Model in Terms of the Rate Process Theory  
In terms of rate process theory proposed by [PERSON] et al. [22; 23], the estimation equation on which our model was based is based on an Arrhenius-type equation. The material parameters \(\alpha_{0}\) and \(\beta_{0}\) in the estimation equation in terms of the rate process theory shown in Equation (3) are considered to contain information on the mode of loading. Equation (9), which shows the stress dependence of the strain rate during steady-state creep deformation, is also an Arrhenius-type equation [39]. \[\dot{\epsilon}=\dot{\epsilon_{0}}\Big{(}\frac{\sigma}{G}\Big{)}^{n}\text{exp} \bigg{(}-\frac{U_{\text{a}}}{RT}\bigg{)} \tag{9}\]  
where \(n\) is the stress index, \(\sigma\) is the loading stress, \(G\) is the shear modulus, \(U_{\text{a}}\) is the activation energy, \(R\) is the gas constant, and \(T\) is the absolute temperature. Equation (9) was transformed logarithmically into the relationship between applied stress and strain rate shown in Equation (10):  
\[\sigma=\text{Gexp}(n)\text{exp}\big{(}nU_{\text{a}}/\big{(}RT\text{ln}\big{(} \dot{\epsilon}/\dot{\epsilon_{0}}\big{)}\big{)}\big{)}+G\big{(}\dot{\epsilon} /\dot{\epsilon_{0}}\big{)} \tag{10}\]  
Equation (10) shows that the strain rate at applied stress varies with the stress index \(n\). Similar to the rate process theory [22; 23], Equation (10) relates the loading frequency at the fatigue limit stress, \(E\), at a room temperature of 293K. By substituting Equation (4) into Equation (10), we obtained Equation (11):  
\[E=\text{Gexp}(n)\text{exp}(nU_{\text{a}}/(RT\text{ln}(f/f_{0})))+G(f/f_{0}) \tag{11}\]  
Comparing Equations (7) and (11), focusing on the relationship between loading frequency and stress, the material parameters \(A\) and \(B\) in Equation (7) can be expressed as \(A=\text{exp}(n)\ A^{\prime}\) and \(B=nB^{\prime}\). Substituting these relationships into Equation (7), we obtain Equation (12):  
\[E=A^{\prime}\text{exp}(n)H_{\text{v}}\text{exp}\left(\frac{B^{\prime}\frac{n} {H_{\text{v}}}}{T_{0}\text{ln}\Big{(}\frac{f_{0}}{f}\Big{)}}\right) \tag{12}\]  
Figure 11: (**a**) A three-dimensional plot of the results derived from the extended model; (**b**) a representation of the relative error for every selected datum. where \(A^{\prime}\) and \(B^{\prime}\) are the material parameters. The stress index \(n\) varies depending on the activation volume [40]. Since the activation volume is known to be subject to change in axial-loading fatigue tests and rotating bending fatigue tests, the stress index was applied to the model as a difference in loading mode. Here, axial-loading fatigue, which can be tested up to a high-frequency band, is considered as a standard. We calculated the material parameters \(A^{\prime}\) and \(B^{\prime}\) in Equation (11) by assuming that the stress index in the axial-loading fatigue was \(n=1\). The material parameters \(A^{\prime}\) and \(B^{\prime}\) were estimated from axial-loading fatigue data [9]. \(E/(\exp(n)H_{\mathrm{V}})\) was used for the vertical axis, and \(\exp((n/H_{\mathrm{V}})/T_{0}\mathrm{ln}(f_{0}/f))\) was used for the horizonal axis. Each axis was converted to a logarithmic axis and subjected to linear regression using the least-square method. The intercept and slope obtained through linear regression were utilized as the material parameters \(A^{\prime}\) and \(B^{\prime}\), which are shown in Equation (13):  
\[E=0.334\mathrm{exp}(n)H_{\mathrm{V}}\mathrm{exp}\left(\frac{3.68\times 10^{5} \times\frac{n}{H_{\mathrm{V}}}}{T_{0}\mathrm{ln}\left(\frac{f_{0}}{f}\right)}\right) \tag{13}\]  
Based on the fatigue limit estimation equations using the Vickers hardness shown in Equations (2) and (6), the coefficients used in the estimation depend on the loading mode, even for the same Vickers hardness.