## 3 FCGR Experiment  
The sample size is shown in Figure 4; its thickness is 4 mm, and the initial slit width introduced into the CT specimen is 0.2 mm. The FCGR test was carried out by a servo hydraulic power testing machine (Instron Model 8802, Illinois Tool Works Inc., Norwood, MA, USA) with a capacity of 100 kN. In the first step, the fatigue crack was prefabricated at room temperature, and in the second step, the FCGR test was performed at 4.2 K. In order to obtain a low-temperature environment of 4.2 K, the fixture and sample were first put into a low-temperature Dewar filled with liquid nitrogen for pre-cooling. After a period of time, the liquid nitrogen was poured out, and liquid helium was transferred into the low-temperature Dewar to cool the fixture and sample. When the sample temperature reached 4.2 K, the transfer of liquid helium was stopped, and the FCGR experiment was started. The relevant experiments for pre-cracked CT specimens followed the procedures described in the ASTM E647 standard [33], employing a constant load control mode with a load ratio \(R\) of 0.1 and a constant frequency of 15 Hz to obtain classic FCGR behavior for the CHSN01 material. An Elipon clamping crack opening displacement COD (3541-005M-025M-LHT) meter was used to monitor the load displacement data, and the crack length was measured using the compliance method. Tests were performed with an initial stress intensity factor of around 10 MPa-m\({}^{0.5}\), which increased to approximately 80 MPa-m\({}^{0.5}\) by the end. Figure 5 illustrates FCGR versus the stress intensity factor range for two types of developed CHSN01 jackets. Both the REBCO and Nb\({}_{3}\)Sn jackets must undergo cold working, and the Nb\({}_{3}\)Sn conductor requires additional 650 \({}^{\circ}\)C heat treatment aging after cold working. It can be found that no significant degradation in FCGR was observed in the CHSN01 jacket after the aging process. Consequently, fitting one curve represents data from both the REBCO and Nb\({}_{3}\)Sn jackets within the Paris region. The FCGR test data presented in Figure 5 were obtained through experimental measurements conducted on over 10 independent production batches. To ensure a conservative assessment of FCGR performance, the maximum values observed across all batches were adopted as the results in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Process**} & **Temperature** & **YS** & **UTS** & **EL** & **KIC** \\  & **(K)** & **(MPa)** & **(MPa)** & **(\%)** & **(MPa m\({}^{0.5}\))** \\ \hline \multirow{2}{*}{4\%CW} & 300 & 602 \(\pm\) 12 & 805 \(\pm\) 15 & 45 \(\pm\) 2 & 316 \(\pm\) 7 \\  & 4.2 & 1640 \(\pm\) 28 & 1980 \(\pm\) 32 & 34 \(\pm\) 1 & 181 \(\pm\) 4 \\ \hline \multirow{2}{*}{4\%CW+HT} & 300 & 596 \(\pm\) 10 & 795 \(\pm\) 14 & 40 \(\pm\) 2 & 285 \(\pm\) 6 \\  & 4.2 & 1600 \(\pm\) 27 & 1940 \(\pm\) 30 & 30 \(\pm\) 1 & 163 \(\pm\) 4 \\ \hline \end{tabular}
\end{table} 
Table 3: Mechanical properties of the CHSN01 jacket. Figure 3: The cross sections of the REBCO jacket (**a**) and the Nb\({}_{3}\)Sn jacket (**b**) used in the CS magnet. Both jackets are made from the CHSN01 material. Generally, the fatigue crack growth behavior of any material is represented by the well-known Paris equation as follows:  
\[da/dN=C(R)\times\left(\Delta K\right)^{m} \tag{2}\]  
where \(da/dN\) is the fatigue crack growth rate, \(\Delta K\) is the stress intensity factor range (\(\Delta K=K_{max}-K_{min}\)), \(K_{max}\) and \(K_{min}\) are the maximum and minimum stress intensity factors, respectively, the exponent m is constant which depends on the material, and \(C(R)\) is the value of \(C\) at a certain \(R\) ratio. The coefficient \(C(R)\) is determined based on the following set of equations:  
\[C(R)=C_{0}\times(1-R)^{-n} \tag{3}\]  
\[n=-m\times(m_{w}-1) \tag{4}\]  
\[C_{0.1}=C_{0}\times 0.9^{m(m_{w}-1)} \tag{5}\]  
where \(C_{0}\) is the \(C\) value of the material at \(R=0\) and \(m_{w}\) is the Walker coefficient which is calibrated to 0.5 by experiments mentioned later.