# Notch Effect on the Fatigue Behavior of a TC21 Titanium Alloy in Very High Cycle Regime
## 4 Discussion  
### Fracture Mechanism of Notch Specimens  
Figure 12 shows the relationship between the depth of the crack initiation site and the number of cycles to failure for both the notched and smooth specimens. For notched specimens, the distance was scattered over a range between 80 and 220 \(\upmu\)m, and the fatigue life increased with the increase in crack initiation site depth. However, the depth of the crack initiation site was independent of the fatigue life for smooth specimens, indicating that the crack initiation site was randomly distributed under the uniform distribution of the axial stress. Figure 13 indicates that the fatigue life increased with the size of the FGA in both the smooth and notched specimens, ranging from 40 \(\upmu\)m to 68 \(\upmu\)m. As mentioned above, the fatigue life in the very high cycle regimes may have been mainly consumed in the wear process of the FGA. Based on the formula of the equivalent stress range defined by [PERSON] [17], the equivalent stress range of the subsurface crack in the range of blunt notch can be modified as:  
\[\Delta\sigma=\frac{\text{k}_{i}\Delta\sigma_{n}}{\sqrt{1+\frac{4.5(\alpha+d)} {\rho}}} \tag{2}\]  
Figure 11: Fracture surface for the notch specimens at \(\sigma=250\) MPa and N = 3.8 \(\times\) 10\({}^{5}\) cycles: (**a**) fatigue crack initiation site, and (**b**) crack initiation morphology. Figure 12: Relationship between the crack initiation site depth and the number of cycles to failure: (**a**) smooth specimens, and (**b**) notched specimens. where a is the distance from the subsurface crack initiation site to notch root surface, \(\rho\) is the radius of notch root, and d is the subsurface crack initiation size which can be obtained by fracture observation. According to Equation (2), the stress gradient of the blunt notch accelerated the fatigue crack initiation. The stress range was smaller at farther distances from the crack initiation site to notch root surface, thus resulting in the longer fatigue life. Based on the SEM observation, the equivalent stress at the subsurface crack initiation site can be calculated using Equation (2). S-N curves of smooth specimens and notched specimens for the equivalent stress are shown Figure 14. It is indicated that fatigue life of subsurface crack initiation was similar for both the smooth and notched specimens for equivalent stress. However, the subsurface initiation life of notched specimens was far longer than that of surface initiation for smooth specimens when both specimens were under the equivalent stress. According to the Murakami model [18], the stress intensity factor range at the FGA front can be calculated using:  
\[\Delta K_{FGA}=0.5\Delta\sigma\sqrt{\pi\sqrt{\mathrm{area}_{FAG}}} \tag{3}\]  
where \(\Delta\sigma\) is the stress amplitude range and \(\mathrm{area}_{FAG}\)is the area of the crack initiation. In ultrasonic fatigue with a mean load equal to zero (\(R=-1\)), \(\Delta\mathrm{KF}_{\mathrm{FGA}}\) is estimated by substituting the stress amplitude, \(\sigma_{\mathrm{a}}\), into Equation (3) instead of \(\Delta\sigma\) as only the tensile part of the cycle affects the fatigue crack propagation [19]. Figure 15 shows the both the smooth and notched specimens had the  
Figure 14: S–N curves of smooth specimens and notched specimens for equivalent stress. Figure 13: Relationship between the size of the FGA and the number of cycles to failure. scattering zone of \(\Delta K_{\rm FGA}\) value ranged from 3.8 MPa\(\cdot\)m\({}^{1/2}\) to 4.5 MPa\(\cdot\)m\({}^{1/2}\), which can be considered as the threshold for the crack growth \(\Delta K_{\rm th}\)[3]. It can be supposed that the VHCF crack initiation resulted from the interaction between the notch stress and heterogeneous microstructure. Under a relatively high stress amplitude, the stress that was introduced by the notch root was high enough to promote the formation of multiple cracks (Figure 8), which was similar to the characteristics of the low-cycle fatigue.