## 5 Result and Discussion  
In the Paris equation, as \(C\) and \(m\) are treated as constants, they fail to capture the inherent uncertainty associated with fatigue crack propagation. To address this limitation, this article proposes a modification to the Paris equation by introducing a lognormal random variable to Equation (2), which is then incorporated into Equation (8). The probability distribution of FCGR with the random variable is calculated using the MLEM to accurately represent the uncertainty in fatigue crack propagation. \[da/dN=C(R)(\Delta K)^{m}\times exp\Big{[}N\Big{(}0,b^{2}\sigma^{2}\Big{)} \Big{]} \tag{8}\]  
where  
\[b=\frac{n\Sigma_{i=1}^{n}\ x_{i}y_{i}-\Sigma_{i=1}^{n}\ x_{i}\Sigma_{i=1}^{n}y _{i}}{n\Sigma_{i=1}^{n}\ x_{i}^{2}-\sum_{i=1}^{n}x_{i}\Sigma_{i=1}^{n}\ x_{i}}\]  
\[b^{2}\sigma^{2}=\frac{1}{n}\underset{i=1}{\overset{n}{\underset{i=1}{\sum}}}\ (y_{i} -a-bx_{i})^{2}\]  
\[a=\frac{1}{n}\underset{i=1}{\overset{n}{\underset{i=1}{\sum}}}y_{i}-b\frac{1 }{n}\underset{i=1}{\overset{n}{\underset{i=1}{\sum}}}\ x_{i}\]  
\[x_{i}=ln(\Delta K)_{i},\ y_{i}=ln(da/dN)_{i}\]  
The uncertain FCGR equations of the REBCO jacket and the Nb\({}_{3}\)Sn jacket are calculated using Equation (8). \[da/dN=8.7944\times 10^{-12}\times(\Delta K)^{4.1935}\times exp\Big{[}N \Big{(}0,0.05644^{2}\Big{)}\Big{]} \tag{9}\]  
\[\text{Nb}_{3}\text{Sn}\qquad da/dN=8.7944\times 10^{-12}\times(\Delta K)^{4.19 35}\times exp\Big{[}N\Big{(}0,0.06354^{2}\Big{)}\Big{]} \tag{10}\]  
Assuming that the lower quantile of the standard normal distribution with a probability \(P\) is denoted as \(\mu_{P}\), the FCGR equation for achieving a failure probability of \(1-P\) (reliability \(P\)) can be derived as follows:  
\[\big{(}da/dN\big{)}_{P}=C(R)(\Delta K)^{m}exp(\mu_{P}b\sigma) \tag{11}\]  
When the probability \(P\) is set to 99%, \(\mu_{P}\approx 2.33\) can be obtained from the standard normal distribution. The FCGR uncertainty equation with a probability of \(P=99\%\) can be derived using Equation (11):  
\[\text{REBCO}\qquad\ \ \ \ \Big{(}da/dN\big{)}_{0.99}=1.0030\times 10^{-11} \times(\Delta K)^{4.1935} \tag{12}\]  
\[\text{Nb}_{3}\text{Sn}\qquad\ \ \ \Big{(}da/dN\big{)}_{0.99}=1.0198\times 10^{-11} \times(\Delta K)^{4.1935} \tag{13}\]  
The FCGR curve in Figure 11, with a probability of 99% (failure probability of 1%), provides a more comprehensive characterization of the dispersion law observed in the experimental data and effectively captures the influence of probability on the uncertainty associated with the FCGR curve. The derived uncertainty equation for crack propagation aligns well with the experimental results. Using Equations (2)-(5), Equations (12)-(13), and the \(R\) value derived from the actual cyclic load during CS magnet operation, the \(C(R)\) for the CHSN01 jackets is obtained, where the \(C(R)\) of the REBCO jacket is 1.1158 \(\times\) 10\({}^{-11}\) and the \(C(R)\) of the Nb\({}_{3}\)Sn jacket is 1.6851 \(\times\) 10\({}^{-11}\). Substitute Equations (6) and (7) into Equations (12) and (13) and perform the following integration:  
\[b=\frac{n\Sigma_{i=1}^{n}\ x_{i}y_{i}-\Sigma_{i=1}^{n}\ x_{i}\Sigma_{i=1}^{n}y _{i}}{n\Sigma_{i=1}^{n}\ x_{i}^{2}-\sum_{i=1}^{n}x_{i}\Sigma_{i=1}^{n}\ x_{i}}\]0  
where \(a_{i}\) is the initial defect depth and \(a_{f}\) is the critical defect depth, which is the jacket thickness here. \(N_{if}\) is the number of cycles required for crack growth from \(a_{i}\) to \(a_{f}\). Figure 12 shows the relationship between the number of cycles and the maximum tolerable initial defect area. The results indicate that for a wall thickness of 4.3 mm, the maximum allowable initial defect areas to sustain 60,000 plasma pulses are 6.18 mm\({}^{2}\) and 4.07 mm\({}^{2}\) for the REBCO and Nb\({}_{3}\)Sn jackets, respectively. Due to its larger \(\Delta K=f(a)\) and \(R\) value compared to the REBCO jacket, the Nb\({}_{3}\)Sn jacket has a much smaller initial defect area. Table 4 presents the maximum acceptable defects for different cycles.