# Design and Shear Analysis of an Angled Morphing Wing Skin Module
## 3 Methods
### Validation  
The numerical results of specimen M50 were compared with the experimental results to validate the numerical model. As shown in Figure 5(a), four configurations near the critical wrinkling state of the skin with \(\theta\) of 130\({}^{\circ}\), 132\({}^{\circ}\), 136\({}^{\circ}\), and 140\({}^{\circ}\), from which it can be found that the same wrinkling mode was obtained with the experiment. Moreover, the differences in U3, which is the displacement of the skin module in the z-direction, of the numerical and experimental configurations at the four morphing angles, were calculated and normalized by the thickness of the module. The pie graphs of the errors shown in Figure 5(c) also indicate that the numerical and experimental results are very close. The principal stresses at the intersection of the diagonal lines of the module when the skin is in the ideal wrinkling-free state are calculated as follows based on the Yeoh hyperelastic constitutive model [33]:  
\[\sigma_{i}=\frac{\partial W}{\partial S}=\left(C_{10}+2C_{20}(I_{1}-3)+3C_{30 }(I_{1}-3)^{2}\right)\left(\lambda_{i}^{2}-\frac{1}{\lambda_{1}\lambda_{2}}\right) \tag{1}\]  
Figure 5: Finite element model of the skin module.  
Figure 6: Comparison of numerical and experimental results: (**a**) deformation process; (**b**) principal stress versus morphing angle curves; (**c**) pie graphs of the configuration errors.  
In the equation, \(W=C_{10}\left(I_{1}-3\right)+C_{20}\left(I_{1}-3\right)^{2}+C_{30}\left(I_{1}-3 \right)^{3}\) is the strain energy density function, \(C_{10}\), \(C_{20}\), and \(C_{30}\) are material constants listed in Table 2, \(I=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\) is the first strain invariant of the Cauchy-Green strain tensor \(S\), and \(\lambda_{i}\) (\(i\) = 1, 2, 3) is the principal extension ratio, which can be defined as:  
\[\left\{\begin{array}{l}\lambda_{1}=\frac{s}{s_{0}}=\frac{\sin\frac{\theta}{ \Omega}}{\sin\frac{\theta}{\Omega}}\\ \lambda_{2}=\frac{t}{t_{0}}=\frac{\cos\frac{\theta}{\Omega}}{\cos\frac{\theta }{\Omega}}\\ \lambda_{3}=\frac{1}{\lambda_{1}\lambda_{2}}=\frac{\sin\frac{\theta}{\Omega} \cos\frac{\theta}{\Omega}}{\sin\frac{\theta}{\Omega}\cos\frac{\theta}{\Omega}} \end{array}\right. \tag{2}\]  
where \(s_{0}\), \(t_{0}\), and \(s\), \(t\) are respectively the diagonal lengths before and after deformation, as shown in Figure 7.  
Finally, the maximal and minimal principal stress, \(\sigma_{1}\) and \(\sigma_{2}\), at the intersection of the diagonal lines of the module were obtained by substituting the experimental strain values into Equation (1), and are plotted versus the morphing angle \(\theta\) as shown in Figure 6b. It should be noted that the strain on the surface of the skin is the same everywhere before wrinkling. In general, a good agreement is obtained. The minor differences at the very large morphing angle are probably caused by fabrication defects of the specimens and the slight difference in boundary conditions. Since the focus of this section is on the critical wrinkling angle of the skins, these minor differences have little influence on the results, so the numerical model is considered acceptable.