## 3 Numerical Analysis  
The present study employed a robust methodology to investigate the bending process of a composite material helical spring cylindrical shell using SolidWorks and ABAQUS finite element software. Specifically, a non-linear finite element analysis was carried out by directly importing the SolidWorks model into the ABAQUS software to simulate the bending process of the composite material helical spring cylindrical shell. Furthermore, a reverse bending process simulation was performed by creating a finite element model with shell element S4R5 and employing ABAQUS software. To ensure that the space deployable mechanism model achieves a purely bending state during actual operational processes, MPC constraints were used to establish nodes on both ends of the model, with the centroid of the end face serving as the control node, and coupled them as MPC constraints. In the ABAQUS finite element software, boundary conditions were applied at the end face's centroid to simulate the bending moment's loading effect. The automatic meshing feature of ABAQUS was used with an approximate global size of 0.5 mm and a maximum offset factor of 0.1 for the curvature control. Additionally, the minimum size control was set as a proportion of the global size, with a factor of 0.1. The finite element analysis process employed a large non-linear deformation (NLGEOM) incremental analysis using a solution \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Critical Bending** & **Steady-State Bending** & **Unit Mass Critical** & **Unit Mass** \\  & **Moment (Nmm)** & **Moment (Nmm/g)** & **Bending Moment** & **Steady-State Bending** \\  & & & **(Nmm/g)** & **Moment (Nmm/g)** \\ \hline Normal tape spring & 6664.5 & 956.5 & 166.2 & 17.7 \\ Composite tape spring & 2249.0 & 625.8 & 166.6 & 46.4 \\ Auxetic re-entrant & 2998.7 & 277.4 & 269.8 & 25.3 \\ honeycomb & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Different material structure spring mechanical properties comparison. Figure 9: Different material structure spring mechanical properties comparison. method based on. Moreover, the automatic stabilization coefficient STABILZE was set to 0.0002. The boundary conditions with applied displacement constraints were established as follows: among the six degrees of freedom on one end face of the model, the X-direction displacement, Y- and Z-direction displacements, and rotations were constrained, and an X-direction rotation of 1.57 radians was applied, i.e., an X-direction rotation of 1.57 radians was applied while the other five degrees of freedom were constrained. For the other end section, the X- and Y-direction displacements were constrained, the Y- and Z-direction rotations were constrained, and an X-direction rotation of \(-1.57\) radians was applied. This application ensured that the model bent into a U-shape, which is consistent with the actual working state of the space unfolding mechanism. The parameters used in the analysis are given in Tables 1 and 2, the boundary conditions are shown in Table 6. And the ply angles are shown in Figure 10. Figure 11 presents the FEM model in detail. Figure 12 shows the stress contour plot of the composite tape spring after folding and bending. The figure shows that the stress distribution is similar to a rectangle, and the middle part of the cylindrical shell becomes flat, which is in line with the variation of cross-sectional curvature in the theoretical model. The maximum stress values for the four layers are 214.0 Mpa, 145.8 Mpa, 153.7 Mpa, and 238.8 Mpa, respectively. The maximum stress value occurs in the fourth layer, and the stress values \(\sigma\)1, \(\sigma\)2, and \(\tau\)12 in the principal axis direction of the material at the node with the maximum stress value of \(-255\) Mpa, \(-30.9\) Mpa, and \(-13.6\) Mpa are extracted. The material strength parameters are shown in Table 7. Calculations are performed based on the Tsai-Hill failure criteria.