## 4 Numerical Simulation Analysis Methods and Models
### Numerical Calculation Model  
Based on the cross-scale mechanical theoretical framework, the macro-mechanical drive shaft models corresponding to Figure 6 were constructed, as shown in Figure 14a. The composite (SiC/TC\({}_{4}\)) material parameters, the fibre and matrix (SiC and Ti) material parameters, and the interface parameters (TiC) are presented in Table 2. For the SiC/TC\({}_{4}\) composite shaft, the critical failure loads of fibres and the matrix were predicted by applying progressive torque to the finite element model based entirely on the experimental load conditions.  
Based on the 3D Hashin failure criterion, the user-defined field (USDFLD) damage subroutine is prepared to simulate the damage of each component during the test. In the calculation process, the load is increased step by step, the stress state at the integral of each element is calculated, and the damage index is calculated according to the stress state of the element, which is stored in the state variable SDV. Therefore, we define SDV to be the maximum damage index (MDI). If MDI is 0, no damage occurs temporarily. The greater the MDI, the greater the degree of damage. When the MDI increases to 1, the element fails, resulting in stiffness degradation. Stress analysis is continued until the entire structure fails.  
The damage subroutine of the micro RVE model was employed to simulate the damage of the interface that occurs during the experiment [34]. Through macro-mechanical calculations, for the shaft, the actual stress of the dangerous position element was extracted, and the actual stress was applied to the RVE model after the macro-mechanics transformation. After loading, the response result of the interface could be obtained. The presence of damage was judged according to the stress state of the element, and MDI was calculated; when MDI reached the value of 1, the element at the integration failed, and stiffness degradation occurred. The stress analysis of the fibres and the matrix continued until the failure of the entire shaft [35].  
The RVE model was established using a fibre diameter of 100 \(\upmu\)m and a fibre volume fraction of 35%. Moreover, the quadrilateral representative volume element model, with a side length of 150 \(\upmu\)m, was established according to \(L=R\sqrt{\pi/V_{f}}\), as shown in Figure 14b;  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \multicolumn{2}{c}{**E/GPa**} & \multicolumn{2}{c}{\(\mu\)} & \multicolumn{2}{c}{**G/GPa**} \\ \hline \multirow{3}{*}{SiC/TC\({}_{4}\)} & E\({}_{1}\) & 234 & \(\mu_{1}\) & 0.23 & \(\mathrm{G_{1}}\) & 64.38 \\  & E\({}_{2}\) & 187 & \(\mu_{2}\) & 0.27 & \(\mathrm{G_{2}}\) & 71.69 \\  & E\({}_{3}\) & 187 & \(\mu_{3}\) & 0.23 & \(\mathrm{G_{3}}\) & 64.38 \\ \hline Material & E/GPa & \(\mu\) & \(\mathrm{X_{t}/MPa}\) & \(\mathrm{X_{c}/MPa}\) & \(\mathrm{S/MPa}\) \\ SiC & 400 & 0.17 & 1800 & 1800 & 900 \\ Ti & 110 & 0.30 & 950 & 1100 & 510 \\ \hline Material & E/GPa & \(\mu\) & \(\mathrm{X_{t}/MPa}\) & \(\mathrm{X_{c}/MPa}\) & \(\mathrm{g_{c}^{fail}/mm}\) \\ TiC & 330 & 0.3 & 98.5 & 98.5 & 0.0005 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters.  
Figure 14: Numerical calculation model: (**a**) the shaft model and (**b**) the RVE model.  
the C3D8R cell was selected for the fibre and matrix. The cohesive element (COH3D8) based on the bilinear cohesion model was also selected for the interface, periodic boundary conditions were applied, and meshing was performed.