## 4 Discussion
### Fatigue Life Prediction
measured the crack propagation rate of a Ti-22V-4 Al alloy subjected to heat treatment (solution treatment at 1023 K for 1 h and aging at 723 K for 8 h) under similar conditions as those used in this study. A linear regression of the data reported in their study was also plotted with a dashed line in Figure 15. Although the slope of the line was slightly smaller than that of the 0%, 70%, and 90% initiation life, the order of the crack propagation rate was consistent with that of this study. In addition, they reported that the crack initiation life was 70-80% of the experimental fatigue life [8; 25]. Figure 14: Fatigue life classification. Figure 15: Relationship between crack propagation rate and stress intensity factor range derived from the Tanaka-Akiniwa model in terms of fracture mode and crack initiation life. Based on this report and the fact that the dashed line was located around the 70% initiation life, the surface crack initiation life in this study was considered approximately 70% of the fatigue life. This indicates that the Tanaka-Akiniwa model is suitable for deriving the material constants of surface cracks with a particular crack initiation life in \(\beta\) titanium alloys. In addition, this indicates that the crack initiation life is not a negligible parameter in the surface fracture regime. The derived internal crack propagation rate has a similar order of magnitude with that of steel reported by many other researchers (\(\approx\)10\({}^{-11}\)-10\({}^{-13}\) m/cycles) [34; 35]. In addition, the lattice constant of the internal fracture regime of the \(\beta\) titanium alloys is listed in the diagram. Most of the derived internal crack propagation rate was below the lattice constant regardless of the crack initiation life. Previous studies have reported that the internal fatigue crack presumably exhibits an intermittent behavior. This indicates that it cannot propagate continuously, and its propagation rate is below the lattice constant [13; 14]. Therefore, the derived internal crack propagation rate can be considered reasonable. Furthermore, through X-ray CT observation of the internal fractures in Ti-6 Al-4V, previous studies have reported that the propagation rate of internal cracks reached 10\({}^{-14}\) m/cycles with two orders of variation at a lower \(\Delta K\) region around 2-3 MPa/m [36; 37]. According to Figure 15, the difference in the estimated crack propagation rate between the 90% and 0% initiation life was only one order of magnitude. Meanwhile, at a particular crack propagation rate, the deviation of the \(\Delta K\) between 90% and 0% initiation life was merely less than 0.2 MPa/m. This indicates that the deviation of the crack propagation rate due to various crack initiation life has no significant effect on the internal fracture. Namely, the crack initiation life is negligible to estimate the \(da\)/\(l\)N-\(\Delta K\) relation of internal cracks when the Tanaka-Akiniwa model is employed. Mathematically, the cliffy gradient in the \(da\)/\(l\)N-\(\Delta K\) curve (i.e., a large value of the material constant \(m\)) of the internal fracture also represents the different significance of the crack initiation life on the surface and internal fractures. As a conclusion, the crack initiation life has a significant effect on the derivation of the material constants in the surface fracture regime. However, it has a relatively negligible effect in the internal fracture regime.