## 4 Discussion
### Fatigue Life Prediction  
Fatigue life prediction was carried out to evaluate the validity of the obtained results from another point of view. By the reverse calculation of Equation (4), the fatigue life prediction equation was derived, as shown in Equation (7):  
\[N_{p}\ =\ \frac{2\sqrt{area_{ini}}}{\left(\Delta K_{ini}\right)^{m}C(m\ -\ 2)} \tag{7}\]  
where \(N_{p}\) is the predicted fatigue life, \(C\) and \(m\) are the material constants derived from the \(\Delta K_{ini}-N_{e}/\sqrt{area_{ini}}\) diagram in Figure 12. To evaluate the accuracy of the prediction, the ratio of the predicted fatigue life to the experimental fatigue life (\(N_{p}/N_{e}\)) was calculated. The closer this ratio is to 1, the better the accuracy of prediction obtained. Figure 13a,b show the prediction results using the total facet area and the major facet(s) area, respectively. The stress ratio had no significant effect on the fatigue properties. In addition, a similar feature to that of the \(\Delta K_{ini}-N_{e}/\sqrt{area_{ini}}\) diagrams in Section 4.1 was observed. For the surface fracture regime, the total facet area and major facet(s) area exhibited great prediction accuracy. In contrast, the \(N_{p}/N_{e}\) range of the internal fracture regime decreased sharply when the major facet area was used than when the total facet area was used. These results also confirm the reliability of the two-step facet initiation and propagation process. Effect of the Crack Initiation Life and Facet Formation Life on the Derivation Process of the Material Constants  
In general, fatigue life can be divided into the crack initiation life and crack propagation life. In facet-originated fractures, the crack initiation life represents the number of loading cycles before the first facet formation. The crack propagation life can be sub-divided into the facet formation life and the loading cycles consumed after the facet formation, respectively. Considering the two-step facet growth behavior, the schematic of the fatigue process in the \(\beta\) titanium alloy is shown in Figure 14. In Section 4.1, the experimental fatigue life was applied to derive the material constants (i.e., the crack initiation life and facet formation life were not considered in the calculation). [PERSON] et al. conducted X-ray computed tomography (CT) to observe the internal fractures in an \(\alpha+\beta\) titanium alloy Ti-6 Al-4V. They reported that the crack initiation life varied from 20% to 80% of the total fatigue life, and that the first facet was formed instantaneously [33]. This indicates that the first facet formation life (i.e., fatigue life consumed by the major facet formation in the present study) is negligible. However, the crack initiation life should be considered for  
Figure 13: Relationship between the predicted and experimental fatigue life (\(N_{p}/N_{e}\)). (**a**) Using total facets area; (**b**) Using major facet area. the estimation of the crack propagation rate. Therefore, in this section, we paid particular attention to the effect of the crack initiation life on the derivation process of the material constants based on the Tanaka-Akiniwa model. Figure 15 shows the relationship between the crack propagation rate and stress intensity factor range (_da/IN-AK_ curves) obtained using the material constants from the major facet(s). To illustrate the effect of the crack initiation life, three cases were chosen as examples. The first case was obtaining the _da/IN-AK_ curve without considering the initiation life. In the second and third cases, the crack initiation life occupied 70% and 90% of the fatigue life, respectively (i.e., 30% experimental fatigue life (\(N_{c}\)) and 10% \(N_{e}\) were applied to Equation (4), respectively, instead of \(N_{c}\)). With an increase in the crack initiation life, the derived crack propagation rate increased. For the surface fracture regime, [PERSON] et al.