## 4 Fatigue Life Estimation  
The fatigue life \(N_{f}\) is composed of the crack initiation life \(N_{i}\) and the crack propagation life \(N_{p}\) and can be calculated by Equation (3). The analytic formulation of the crack initiation life \(N_{i}\) was presented by [PERSON] and [PERSON] [9, 10] and was based on the dislocation dipole accumulation model. It was extended to estimate the crack initiation life \(N_{i}\) of the specimen with the different surface roughness by [PERSON] et al. [12]. It is [9, 10, 11, 12]:  
\[N_{i}=\frac{9\Delta K_{th}^{2}G}{E(S-S_{e})^{2}\pi(1-v)a_{0}} \tag{7}\]  
here \(\Delta K_{th}\) is the threshold of stress intensity factor, \(G\) is the shear modulus, \(E\) is the elastic modulus, \(S_{e}\) is the endurance limit, \(v\) is the Poisson's ratio, and \(a_{0}\) is the equivalent initial micro-defect size. \(a_{0}\) could be estimated as [12, 14]: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\gamma\) (\%) & R (\%) & 50 & 90 & 99 & 99.9 & Parameters \\ \hline
95 & \(34.0511\) & \(39.5910\) & \(44.8554\) & \(48.8012\) & \(a_{1}\) \\  & \(-11.4174\) & \(-13.8578\) & \(-16.1769\) & \(-17.9151\) & \(a_{2}\) \\ \hline
99 & \(34.8170\) & \(40.8567\) & \(46.8385\) & \(51.3489\) & \(a_{1}\) \\  & \(-11.7547\) & \(-14.4154\) & \(-17.0504\) & \(-19.0374\) & \(a_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of P-S-N curves with \(R_{\text{a}}\) = 1.6 \(\upmu\)m.  
Figure 8: The P-S-N curves with \(R_{\text{a}}\) = 1.6 \(\upmu\)m and confidence level **(a)**\(\gamma=95\%\); and **(b)**\(\gamma=99\%\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\gamma\) (\%) & R (\%) & 50 & 90 & 99 & 99.9 & Parameters \\ \hline
95 & \(28.6827\) & \(29.3268\) & \(29.9388\) & \(30.3976\) & \(a_{1}\) \\  & \(-9.4241\) & \(-10.1261\) & \(-10.7931\) & \(-11.2931\) & \(a_{2}\) \\ \hline
99 & \(28.7717\) & \(29.4739\) & \(30.1694\) & \(30.6938\) & \(a_{1}\) \\  & \(-9.5212\) & \(-10.2864\) & \(-11.0444\) & \(-11.6159\) & \(a_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of P-S-N with \(R_{\text{a}}\) = 0.8 \(\upmu\)m.  
\[a_{0}=2.97R_{\rm a} \tag{8}\]  
The number of cycles necessary to grow the crack from the initial micro-defect size to failure can be described by Paris' law as:  
\[\frac{\mathrm{d}a}{\mathrm{d}N}_{0}=C(\Delta K)^{n} \tag{9}\]  
where \(\Delta K\) is the amplitude of stress intensity factor, \(C\) the material constant, and \(a\) the crack size. The crack propagation life \(N_{p}\) can be calculated by the analytic solution of Paris' law integral as [12; 15; 16]:  
\[N_{p}=\frac{a_{0}^{(1-\frac{n}{2})}-a_{\rm sc}^{(1-\frac{n}{2})}}{CS^{n}\beta_{ 1}^{n}\pi^{\frac{n}{2}}\left(\frac{n}{2}-1\right)}+\frac{a_{\rm sc}^{(1-\frac{ n}{2})}-a_{\rm f}^{(1-\frac{n}{2})}}{CS^{n}\beta_{2}^{n}\pi^{\frac{n}{2}}\left( \frac{n}{2}-1\right)} \tag{10}\]  
where: \(a_{\rm sc}\) is the critical crack size for identifying long and short cracks, \(a_{\rm f}\) the crack size of final failure, \(\beta_{1}=0.5\sqrt{\pi}\) and \(\beta_{12}=1\). Generally, the number of crack growth cycles from \(a_{0}\) to \(a_{\rm sc}\) is much greater than the cycles needed to grow from \(a_{\rm sc}\) to \(a_{\rm f}\). Then \(N_{p}\) can be taken as [17]:  
\[N_{p}=\frac{a_{0}^{(1-\frac{n}{2})}}{CS^{n}\beta_{1}^{n}\pi^{\frac{n}{2}}\left( \frac{n}{2}-1\right)} \tag{11}\]  
It can be written by:  
\[S^{n}N_{p}=\mathrm{c} \tag{12}\]  
where:  
\[c=\frac{a_{0}^{(1-\frac{n}{2})}}{C\beta_{1}^{n}\pi^{\frac{n}{2}}\left(\frac{n }{2}-1\right)} \tag{13}\]  
By taking logarithms in Equation (12), then:  
\[\mathrm{lg}N_{p}=\mathrm{lg}c-n\mathrm{lg}S \tag{14}\]  
The material constant \(n\) in Equation (12) is the same as \(n\) in the S-N curves. The material constant \(C\) can be solved by Equation (13). \(n\) and \(c\) have been listed in Table 3 in Section 3.