## 4 Simulation of the Crack Model  
As shown in Figure 7, a finite element model is built using the ANSYS 19.2 software to establish a more precise functional relationship between stress intensity factor range \(\Delta K\) and jacket defect depth \(a\). This model consists of six REBCO coils and six Nb\({}_{3}\)Sn coils. In the lower CS1U module which experiences high levels of stress due to vertical compression and hoop expansion, the detailed structure of three pancakes comprising REBCO cables, Nb\({}_{3}\)Sn cables, CHSN01 jackets, turn insulation, layer insulation, pancake insulation, and ground insulation is built. The innermost turns of both the REBCO coil and the Nb\({}_{3}\)Sn coil are configured with a semi-elliptical surface defect.  
In this model, frictional contact is considered between adjacent coils. At the bottom surface, axial displacement is constrained, while radial displacement remains free. Consequently, during cool-down and energization processes, the CS magnet is supported in its axial direction but can move radially. Mechanical analyses are performed under the following load conditions: (1) gravity, (2) preload at room temperature, (3) cooled down from room temperature to 4.2 K, and (4) the energization of the CS magnet at the start of waveform operation. After conducting the mesh independence analysis and accounting for a reasonable computational cost, the element size is set to 0.5 mm. And the numerical convergence threshold is set to 0.5%.  
Figure 6: Microstructure of the CHSN01 specimen’s fracture surface.  
Due to pulsed plasma operation during plasma scenarios, the fatigue assessment for the CS conductor jacket involves two stress cycles from cool down to the initial magnetization state. Figures 8 and 9 depict the maximum principal stress during the cool down and initial magnetization states for CS jackets, respectively. The maximum principal stress \(\mathsf{S_{cool\ down}}\) in the cool down state of the REBCO and \(\mathsf{Nb_{3}Sn}\) jackets is 47.0 MPa and 181.1 MPa, respectively. In the initial magnetization state, the maximum principal stress \(\mathsf{S_{initial\ magnetization}}\) is 621.2 MPa for the REBCO jacket and 742.4 MPa for the \(\mathsf{Nb_{3}Sn}\) jacket. The \(R\) ratios (\(\mathsf{S_{cool\ down}}/\mathsf{S_{initial\ magnetization}}\)) are calculated and are used in the Paris equation mentioned later.  
Figure 8: The maximum principal stress in the cool down state of the REBCO jacket (**a**) and the \(\mathsf{Nb_{3}Sn}\) jacket (**b**).  
Figure 7: The model for finite element analysis to establish a more precise functional relationship between stress intensity factor range \(\Delta K\) and jacket defect depth \(a\). The six modules of the CS magnet are numbered.  
The initial elliptical defect at various dimensions starts from depth \(a=0.25\) mm up to \(a=4\) mm. The data points of (depth \(a\), \(\Delta K\)) obtained from these models are fitted into \(\Delta K=f(a)\), which is then used in the Paris equation for LEFM. Figure 10 shows \(\Delta K\) as a function of defect size \(a\) in the REBCO and Nb\({}_{3}\)Sn jackets. The function \(\Delta K=f(a)\) is as follows:  
\[\text{REBCO}\qquad\Delta K=-3.1467a^{2}+20.4775a+11.3194 \tag{6}\] \[\text{Nb}_{3}\text{Sn}\qquad\Delta K=-3.3935a^{2}+20.4301a+14.6438 \tag{7}\]  
It can be observed from Figure 10 that the CS operational region of \(\Delta K\) is approximately between 15 and 43 MPa\(\cdot\cdot\cdot\)m\({}^{0.5}\), which is located in the blue line fitting curve (Equation (2)) in Figure 5. Therefore, only Equation (2) is considered in the model.  
Figure 10: Stress intensity factor range \(\Delta K\) as a function of defect size \(a\) in the REBCO and Nb\({}_{3}\)Sn jackets. The data points of (depth \(a\), \(\Delta K\)) are obtained from the finite element method.  
Figure 9: The maximum principal stress in the initial magnetization state of the REBCO jacket (**a**) and the Nb\({}_{3}\)Sn jacket (**b**).