## 3 Fatigue Tests at Low Temperature
### Data Processing  
The distribution of the fatigue life at a given stress range is often subject to a lognormal distribution. A distribution test of the logarithmic fatigue life was performed, and the \(P\)-value (\(P\)-value is the probability of an error when accepting the null hypothesis) of the logarithmic fatigue life under each stress range was greater than 0.05, indicating that the logarithmic fatigue life in this study satisfied the normal distribution. Due to space limitations, only one stress level of the normality test of T- and cruciform welded joints is shown in Figure 9a,b. Under the premise that the logarithmic fatigue life obeyed the normal distribution, the S-N curve with different failure probability and confidence based on the normal distribution model could be obtained [37].  
According to the guidance document given by the International Institute of Welding, \"Recommendations for Fatigue Design of Welded Joints and Components\" [36], if the number of specimens is less than ten or the test data were insufficient to obtain an accurate \(m\), \(m\) can be taken to have a fixed value of 3.  
The exponential form of the S-N curve is described by Equation (2):  
\[S^{m}N=A \tag{2}\]  
Figure 8: Position of strain gauges.  
Figure 9: The normality test of T- and cruciform welded joints at one stress level (percent is a normal distribution cumulative distribution function): (**a**) T-welded joints and (**b**) cruciform welded joints.  
Taking the logarithm on both sides of Equation (2), we obtain  
\[m\lg S+\lg N=\lg A \tag{3}\]  
where,  
\(S\) is the stress range,  
\(N\) is the number of cycles,  
\(m\), \(A\) is the material parameters related to fatigue.  
After \(m\) was determined, \(\lg A\) could be fitted using the maximum likelihood method. The standard deviation of the logarithmic fatigue life was assumed to be \(S_{i}\) under any stress range. The parameter \(\lg A\) could then be estimated from the test data.  
The maximum likelihood function is  
\[\begin{array}{l}L=\prod\limits_{i=1}^{n}\frac{1}{\sqrt{2\pi}\alpha_{\lg N}} \exp\left[-\frac{1}{2\alpha_{\lg N}}\left(\lg N_{i}-\lg A+m\lg S\right)^{2} \right]\\ =\left(\frac{1}{\sqrt{2\pi}\alpha_{\lg N}}\right)^{n}\exp\left[-\frac{1}{2 \alpha_{\lg N}}\sum\limits_{i=1}^{n}\left(\lg N_{i}-\lg A+m\lg S\right)^{2} \right]\end{array} \tag{4}\]  
The estimated value of the parameter \(\lg\hat{A}\) can be obtained using Equation (5):  
\[\lg\hat{A}=\frac{1}{n}\sum\limits_{i=1}^{n}\lg N_{i}-\frac{m}{n}\sum\limits_{ i=1}^{n}\lg S \tag{5}\]  
where  
\(\overline{\lg N}\) is the average logarithmic value of the fatigue life, \(\overline{\lg N}=\frac{1}{n}\sum\limits_{i=1}^{n}\lg N_{i}\).  
Taking \(\lg N_{P_{f}(\gamma)}\) as the logarithmic fatigue life with probability of failure \(P_{f}\) and confidence level \(\gamma\). Among them, the probability of failure \(P_{f}\) indicates the probability of failure when the life is \(N\) at a given stress level. The confidence level \(\gamma\) indicates the reliability of the probability of failure \(P_{f}\) at a given stress level with a fatigue life of \(N\). \(\lg N_{P_{f}(\gamma)}\) is obtained as shown in Equation (6).  
\[\lg N_{P_{f}(\gamma)}=\overline{\lg N}+k\alpha_{\lg N} \tag{6}\]  
where,  
\(\sigma_{\lg N}\) is the standard deviation of the logarithmic fatigue life,  
\(k\) is the unilateral tolerance coefficient, which can be determined from a probability book or related specifications [38], its value is related to the confidence level and failure probability.  
By replacing \(\overline{\lg N}\) with \(\lg N_{P_{f}(\gamma)}\) in Equation (5), the logarithmic fatigue life with a 50% and 95% confidence level under 10% failure probability could be obtained, as shown in Equation (6).