## 4 Discussion  
### Derivation of the Material Constants in Paris Law Based on the Tanaka-Akinitwa Model  
To verify the accuracy of the facet initiation and propagation of the facet patterns established in Section 3, the Paris law was employed, as shown in Equation (2) [30]. \[\frac{da}{dN}=C(\Delta K)^{m} \tag{2}\]  
where \(C\) and \(m\) are material constants used to describe crack behavior in terms of the crack propagation rate. By integrating the Paris law from the initial crack length to infinity, a linear relationship between the initial stress intensity factor range (\(\Delta K_{ini}\)) and the value of crack propagation life over initial crack length (\(N_{propa}/a_{ini}\)) can be exhibited in a double logarithmic chart. This relationship shown in Equation (3) is called the Tanaka-Akinitwa model or the \(\Delta K_{ini}\ -\ N_{propa}/a_{ini}\) diagram [12; 31]. \[\log(\Delta K_{ini})=\ -\ \frac{1}{m}\log\left(\frac{N_{propa}}{a_{ini}}\right)+ \frac{1}{m}\log\left[\frac{2}{C(m\ -\ 2)}\right] \tag{3}\]  
The gradient of the \(\Delta K_{ini}\ -\ N_{propa}/a_{ini}\) diagram was used to obtain the exponential material constant, \(m\), and the coefficient material constant, \(C\), was calculated by substituting \(m\) into the intercept. Considering that the fatigue crack in the \(\beta\) titanium alloys was initiated from the facet, \(a_{ini}\) was replaced by the root of the facet area projected on the fracture surface,  
Figure 11: Example of the combination type. These facets are the same facets shown in Figure 7b (\(\sigma_{max}\ =625\) MPa, \(R\ =-1\), \(N_{t}\ =2.0\ \times\ 10^{7}\)). **(a)** 3D-SEM image; **(b)** 3D-model schematic image. Figure 10: Example of the peak type. These facets are the same facets shown in Figure 6c (\(\sigma_{max}\ =1000\) MPa, \(R\ =0.1\), \(N_{t}\ =3.0\ \times\ 10^{5}\)). **(a)** 3D-SEM image; **(b)** 3D-model schematic image. \(\sqrt{area_{ini}}\), which can be measured by fractography. In addition, the experimental fatigue life, \(N_{e}\), was substituted for \(N_{propq}\). This indicates that the crack physically propagated from a chosen facet throughout the entire fatigue life. In other words, the crack initiation life and facet formation life were omitted. The effect of the crack initiation life and facet formation life on this calculation will be discussed in Section 4.3. Using the above steps, Equation (3) evolved into Equation (4), which is appropriate for the present work:  
\[\log(\Delta K_{ini})=\ -\ \frac{1}{m}\ \log\biggl{(}\frac{N_{e}}{\sqrt{area_{ini} }}\biggr{)}+\frac{1}{m}\ \log\biggl{[}\frac{2}{C(m\ -\ 2)}\biggr{]} \tag{4}\]  
where \(\Delta K_{ini}\) can be calculated using the following Murakami formulas [32] in Equations (5) and (6):  
\[\text{Surface crack}:\ \Delta K_{ini}=\ 0.65\Delta\sigma\sqrt{\pi\sqrt{area_{ini}}} \tag{5}\]  
\[\text{Internal crack}:\ \Delta K_{ini}=\ 0.5\Delta\sigma\sqrt{\pi\sqrt{area_{ini}}} \tag{6}\]  
where \(\Delta\sigma\) is the range of the applied stress. For the negative stress ratios, \(\Delta\sigma\) represents the positive parts of the applied stress (i.e., from zero to the maximum applied tensile stress). In previous studies, the multiple facets were not properly distinguished [15]. Therefore, in this study, as a no-treatment control group, not only the major facet(s) area, but also the total facet area (i.e., the whole area of multiple facets) was applied to determine the crack initiation site area, (\(\sqrt{area_{ini}}\)). Figure 12a,b shows the \(\Delta K_{ini}-\ N_{e}/\sqrt{area_{ini}}\), in which the total facet area and major facet(s) area were applied, respectively. The \(N_{e}\) values were obtained from the \(S\)-\(N\) curve in Figure 2. The linear relationship was calculated using the least squares method. The data were plotted separately in terms of fracture mode (i.e., surface fracture and internal fracture) and can be clearly distinguished.