# Design and Shear Analysis of an Angled Morphing Wing Skin Module
## 4 Results
### Effect of the Initial Angle  
Having demonstrated the effectiveness of the angled skin module design, in this section we investigate the effects of design parameters. First, consider the effect of the initial angle \(\theta_{0}\). The maximal and minimal critical wrinkling angles, \(\theta_{\text{max}}\) and \(\theta_{\text{min}}\), of the physical specimens M50-M90, are obtained from experiments and compared in Table 4. As expected, with the reduction in \(\theta_{0}\) from 90\({}^{\circ}\) to 50\({}^{\circ}\), \(\theta_{\text{max}}\) is generally increased whereas \(\theta_{\text{min}}\) is reduced, leading to a larger wrinkle-free deformation range \(\Delta\theta\) = \(\theta_{\text{max}}\) - \(\theta_{\text{min}}\). Notice that since the surface of the skin was scanned and examined during the experiment at the interval of 2\({}^{\circ}\) until wrinkles occured, the experimental critical wrinkling angles are only approximate values. To obtain the exact critical angles and to explore a wider range of \(\theta_{0}\), seven numerical models listed in Table 4, which have \(\theta_{0}\) from 30\({}^{\circ}\) to 90\({}^{\circ}\) at an interval of 10\({}^{\circ}\), were built and analyzed. All the other geometric and material parameters are identical to those of the physical specimens. The numerical results are presented in Table 4 and drawn in Figure 9a. Notice that the numerical \(\theta_{\text{max}}\) and \(\theta_{\text{min}}\) are determined by comparing the numerical principal stresses at the intersection of diagonals with the corresponding theoretical values calculated from Equation (1). When the numerical values deviate from the theoretical ones, the skin is considered to reach the critical wrinkling angle. It is found that the numerical results match the experimental ones reasonably well with differences less than 2\({}^{\circ}\), once again demonstrating the reliability of the numerical models.  
Moreover, the numerical principal stresses of all the seven models within the wrinkle-free deformation ranges are respectively drawn and compared in Figure 9b,c. It can be seen that the magnitude of the minimum principal stress \(\sigma_{2}\) is reduced as \(\theta_{0}\) becomes smaller at the same \(\theta\), and can even turn positive at a certain range when \(\theta_{0}\) is less than  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\theta_{0}\) (\({}^{\circ}\))} & \multicolumn{2}{c}{**Experiment**} & \multicolumn{2}{c}{**FEM**} \\ \cline{2-7}  & \(\theta_{\text{max}}\) (\({}^{\circ}\)) & \(\theta_{\text{min}}\) (\({}^{\circ}\)) & \(\Delta\theta\) (\({}^{\circ}\)) & \(\theta_{\text{max}}\) (\({}^{\circ}\)) & \(\theta_{\text{min}}\) (\({}^{\circ}\)) & \(\Delta\theta\) (\({}^{\circ}\)) \\ \hline
30 & / & / & / & 125.3 & 20.6 & 104.7 \\
40 & / & / & / & 131.5 & 29.1 & 102.4 \\
50 & 132.0 & 40.0 & 92.0 & 133.4 & 38.6 & 94.8 \\
60 & 128.0 & 50.0 & 78.0 & 127.5 & 48.2 & 79.3 \\
70 & 124.0 & 56.0 & 68.0 & 124.9 & 54.6 & 70.3 \\
80 & 122.0 & 62.0 & 60.0 & 123.1 & 60.4 & 62.7 \\
90 & 120.0 & 60.0 & 60.0 & 120.8 & 59.2 & 61.6 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of the experimental and numerical results in critical wrinkling angles.  
\(60^{\circ}\). In addition, the maximal principal stress \(\sigma_{1}\) increases substantially with the reduction in \(\theta_{0}\). These two factors in combination are accountable for the enlarged \(\Delta\theta\). However, as seen in Figure 9c, when \(\theta_{0}\) is too small, it can be seen from the result of \(\theta_{0}=30^{\circ}\) that the minimum principal stress develops quickly with the shear deformation due to the hardening phenomenon of the rubber material under large deformation, which will reduce \(\theta_{\text{max}}\). In addition, excessive tensile stress will increase the possibility of material failure and lead to a large driving force. Therefore, a too-small \(\theta_{0}\) is also inappropriate.