# Notch Effect on the Fatigue Behavior of a TC21 Titanium Alloy in Very High Cycle Regime
## 4 Discussion
### Fatigue Limit Prediction Based on the Theory of Critical Distance  
The theory of critical distance was assumed to estimate the fatigue limit based on the stress distribution around the notch. There are different manifestations of the TCD, such as the point method (PM) and line method (LM), which obtain a similar fatigue strength prediction result. As for the PM, the effective stress range \(\Delta\sigma_{\textit{eff}}\) at the critical distance around the notched specimens was equal to the stress range in smooth specimens, \(\Delta\sigma_{0}\), and the formula can be expressed as follows [20]:  
\[\Delta\sigma_{\textit{eff}}=\Delta\sigma_{y}(r=L_{PM},\theta=0)=\Delta\sigma_{0} \tag{4}\]  
where \(\theta\), \(r\) are the polar coordinates, \(\Delta\sigma_{y}\) is the axial stress range distribution along the x-axis and \(L_{PM}\) is the critical distance in the point method. According to TCD, the critical distance in the point method L\({}_{\rm PM}\) can be estimated by a\({}_{0}/2\), and a\({}_{0}\) is the El Haddad parameter defined as [18]:  
\[{\rm a}_{0}=\frac{1}{\pi}(\frac{\Delta K_{\rm th}}{Y\Delta\sigma_{0}})^{2} \tag{5}\]  
Figure 15: Relationship between the stress intensity factor range at the FGA and fatigue life. where Y is the geometry factor of the internal crack, and \(\Delta K_{\text{th}}\) is the threshold for the crack growth. The fatigue limit stress at the 10\({}^{9}\) cycles, \(\Delta\sigma_{0}\), was 430 MPa obtained by the smooth specimen fatigue test results, and the threshold value of the stress intensity factor range could be regarded as 4.0 MPam\({}^{1/2}\)[3]. The critical distance for the very high cycle fatigue \(L_{PM}=\text{a}_{0}/2\) was equal to 23.5 um for the TC21 titanium alloy. However, the VHCF crack was inclined to initiate in the subsurface of notch, and the effective stress range \(\Delta\sigma_{\textit{eff}}\) at the critical distance should include the distance from crack initiation site to notch root a. Thus, Equation (4) can be modified to become:  
\[\Delta\sigma_{eff}=\Delta\sigma_{y}(r=\text{a}+L_{PM},\theta=0)=\Delta\sigma_{0} \tag{6}\]  
Considering the effect of the stress triaxiality in the case of notched specimens, the fatigue damage depends on the shear stress amplitude, \(\tau_{\text{a}}\), and on the maximum normal stress, \(\sigma_{n,\text{max}}\), relative to the critical plane, which is defined as the plane experiencing the maximum shear stress amplitude [21]:  
\[\tau_{\text{max}}=\tau_{\text{a}}+\text{m}_{1}\frac{\sigma_{n,\text{max}}}{ \tau_{\text{a}}} \tag{7}\]  
where \(\tau_{\text{a}}\) is the shear stress amplitude relative to the materials plane experiencing the maximum shear stress amplitude, \(\sigma_{n,\text{max}}\) is the maximum stress perpendicular to this plane, and \(\text{m}_{1}\) is the material constant. Thus, Equation (6) can be modified as follows:  
\[\Delta\tau_{eff}=\Delta\tau_{\text{max}}(r=\text{a}+\text{L}_{\text{PM}})= \Delta\tau_{-1} \tag{8}\]  
In the above equation, the parameters \(\text{m}_{1}\) and \(\Delta\tau_{-1}\) can be calculated using the fatigue limits \(\Delta\sigma_{-1}\) and \(\Delta\sigma_{0}\) for R = \(-1\) and R = 0, respectively [21]:  
\[ \begin{array}{l}m_{1}=\frac{\Delta\sigma_{-1}-\Delta\sigma_{0}}{2}\\ \Delta\tau_{-1}=\Delta\sigma_{-1}-\frac{\Delta\sigma_{0}}{2}\end{array}  \tag{9}\]  
As for the TC21 alloy, \(\Delta\sigma_{0}\) can be estimated using the Goodman equation, and was about 358 MPa based on \(\Delta\sigma_{-1}\) = 430 MPa, and the parameters of \(m_{1}\) and \(\Delta\tau_{-1}\) were about 36 MPa and 251 MPa, respectively. The shear stress amplitude, \(\tau_{\text{a}}\), and the maximum normal stress, \(\sigma_{\text{n,max}}\), can be calculated using finite element methods for the notched specimens with K\({}_{\text{t}}\) = 2.67. Considering that the subsurface cracks may initiate at different sites, for example, the distance to the specimen surface was 0 um, 100 um, 200 um, and 400 um.