# Design and Shear Analysis of an Angled Morphing Wing Skin Module
## 5 Theoretical Analysis
### Theoretical Modeling
\tag{3}\]  
According to assumptions (iii) and (iv), the maximal and minimal principal stresses, \(\sigma_{1}\) and \(\sigma_{2}\), can be obtained as follows [37]:  
\[\left\{ \begin{array}{l}\sigma_{1}=\frac{E_{r}(\varepsilon_{1}+\mu_{r} \varepsilon_{2})}{1-\mu_{r}^{2}}\\ \sigma_{2}=\frac{E_{r}(\varepsilon_{2}+\mu_{r}\varepsilon_{1})}{1-\mu_{r}^{2 }}\end{array} \right. \tag{4}\]  
in which \(E_{r}\) and \(\mu_{r}\) are equivalent to the Young's modulus and [PERSON]'s ratio of the rubber. Then the stress components \(\sigma_{x}\), \(\sigma_{y}\), and \(\tau_{xy}\) can be worked out based on the theory of elasticity [37]:  
\[\left\{ \begin{array}{l}\sigma_{x}=\frac{\sigma_{1}+\sigma_{2}}{2}+\frac{ \sigma_{1}-\sigma_{2}}{2}\cos\theta\\ \sigma_{y}=\frac{\sigma_{2}+\sigma_{1}}{2}+\frac{\sigma_{2}-\sigma_{1}}{2} \cos\theta\\ \tau_{xy}=\frac{\sigma_{1}-\sigma_{2}}{2}\sin\theta\end{array} \right. \tag{5}\]  
Considering that the total cross-sectional areas of CFRP rods and KFRP fibers account for only a small portion (below 4.3%) of the skin cross-section, the rubber cross-sectional area is regarded to be the same as the cross-sectional area of the skin. Then, when the skin is sheared to \(\theta\), the in-plane load, and shear load, \(F_{x}^{\theta}\), \(F_{y}^{\theta}\) and \(F_{xy}^{\theta}\), can be expressed as:  
\[\left\{ \begin{array}{l}F_{x}^{\theta}=\sigma_{x}h\\ F_{y}^{\theta}=\sigma_{y}h\\ F_{xy}^{\theta}=\tau_{xy}h\end{array} \right. \tag{6}\]  
In which, \(h=H\)sin \(\theta_{0}/\sin\theta\) is the thickness after shearing based on the incompressibility of rubber. By substituting Equations (3)-(5) into Equation (6), \(F_{x}^{\theta}\), \(F_{y}^{\theta}\) and \(F_{xy}^{\theta}\) be expressed as:  
\[\left\{ \begin{array}{l}F_{x}^{\theta}=\frac{E_{r}H\sin\theta_{0}}{2\left(1 -\mu_{r}^{2}\right)\sin\theta}\biggl{[}(1+\mu_{r})\ln\biggl{(}\frac{\tan\frac{ \theta}{\varepsilon}}{\tan\frac{\theta}{\varepsilon}}\biggr{)}+(1-\mu_{r}) \cos\theta\ln\biggl{(}\frac{\tan\frac{\theta}{\varepsilon}}{\tan\frac{ \theta}{\varepsilon}}\biggr{)}\biggr{]}\\ F_{y}^{\theta}=\frac{E_{r}H\sin\theta_{0}}{2\left(1-\mu_{r}^{2}\right)\sin \theta}\biggl{[}(1+\mu_{r})\ln\biggl{(}\frac{\tan\frac{\theta}{\varepsilon}} {\tan\frac{\theta}{\varepsilon}}\biggr{)}-(1-\mu_{r})\cos\theta\ln\biggl{(} \frac{\tan\frac{\theta}{\varepsilon}}{\tan\frac{\theta}{\varepsilon}} \biggr{)}\biggr{]}\\ F_{xy}^{\theta}=\frac{E_{r}H\sin\theta_{0}}{2\left(1+\mu_{r}\right)}\ln \biggl{(}\frac{\tan\frac{\theta}{\varepsilon}}{\tan\frac{\theta}{\varepsilon }}\biggr{)}\end{array} \right. \tag{7}\]  
Figure 11: Deformation process of the skin. In addition, \(F_{x}^{90}\), \(F_{y}^{90}\), and \(F_{xy}^{90}\) at the end of the stage I can also be obtained by substituting \(\theta\) = 90\({}^{\circ}\) into Equation (7). At stage II, the critical wrinkling force of the skin is evaluated using the energy method. Since the skin is simply supported at the four edges, the double sinusoidal series [38] satisfying the boundary conditions are chosen as its displacement function \(w\), in which \(A_{mn}\) is the reciprocally independent coefficient, and \(m\) and \(n\) are the numbers of terms in the series.