# A New Method for Fatigue Evaluation of Titanium Alloy Welded Structures
## 2 The Principle of Notch Equivalent Stress and the Proposal of a New Method for Life Prediction
### Proposition of Augmented-Reverse Notch Equivalent Stress Method
#### 2.3.1 The Method of Proposing the New Three-Parameter Power Function Model  
The known expression of the existing three-parameter power function S-N curve is as follows:  
\[\left(S-S_{f}\right)^{m}\cdot N=C \tag{6}\]  
where \(m\) and \(C\) are material parameters, \(S_{f}\) is the fatigue limit, \(S\) is the stress range, and \(N\) is the fatigue life.  
When the traditional three-parameter power function S-N curve is used to evaluate the fatigue life of metal structures, although its accuracy is higher than that of the exponential, power function, Zheng formula, and other S-N curves, its fitting degree to the test data is not greatly improved. Therefore, on the basis of the existing three-parameter power function S-N curve, it is improved, and a new three-parameter power function S-N curve is proposed. The proposed process of the new three-parameter power function S-N curve is as follows:  
1. The fatigue test is carried out on \(n\) welded joints loaded with the same stress level, and the \(n\) groups of test life under the stress level are obtained as (\(N_{11}\),... \(N_{1i}\),... \(N_{1n}\));
2. Perform random sampling (\(m\cdot k\)) times on the above \(n\) fatigue test data (\(m\) and \(k\) are both positive integers), and divide the sample data after sampling into \(m\) groups, with \(k\) in each group;  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{_as/mm_} \\ \cline{2-4}  & **1–10** & **10–100** & **100–1000** \\ \hline \(A/\text{mm}^{-1}\) & 0.52067 & 0.80875 & 2.0598 \\ \(B\) & 0.13915 & 0.0565 & 0.00928 \\ C/mm\({}^{-1}\) & \(-0.01092\) & \(-8.41088\times 10^{-4}\) & \(-1.97016\times 10^{-5}\) \\ \(D/\text{mm}^{-2}\) & \(6.35004\times 10^{-4}\) & \(7.23825\times 10^{-6}\) & \(1.90644\times 10^{-8}\) \\ \(E/\text{mm}^{-3}\) & \(-1.60256\times 10^{-5}\) & \(-2.48689\times 10^{-8}\) & \(-6.86043\times 10^{-12}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Piecewise function parameters of notch stress equivalent value point [23].  
Calculate the corresponding mean value of each group of data. The calculation formula of the life mean value of group \(j\) is as follows (\(1\leq j\leq m\)): \[\overline{N_{j}}=\frac{\sum\limits_{i=1}^{k}N_{ji}}{k}\] (7) After \(m\) times of calculation, \(m\) mean values can be obtained: \(\overline{N_{1}}\),..., \(\overline{N_{j}}\),..., \(\overline{N_{m}}\).
4. Count the \(m\) mean values obtained in step (3) and determine the distribution law. If it is a normal distribution, the fatigue life \(N\) under the stress level in the new three-parameter power function S-N curve is the normal distribution mean value. If it is a skewed distribution, the fatigue life \(N\) under the stress level in the new three-parameter power function S-N curve is the median value of the skewed distribution;
5. Repeat steps (1-4) to determine the fatigue life \(\overline{N}\) under other stress levels in the new three-parameter power function S-N curve, then the new three-parameter power function S-N curve calculation model is as follows: \[\left(S-S_{f}\right)^{m}\cdot\overline{N}=C\] (8)