# Design and Shear Analysis of an Angled Morphing Wing Skin Module
## 5 Theoretical Analysis  
### Theoretical Modeling  
From the design point of view, a mathematical formula is desired to estimate the critical wrinkling angles of the skin concerning the specified geometry and material. However, the strong geometry and material nonlinearity involved in the deformation process make it very difficult to build a rigorous analytical model. Therefore, a simplified theoretical analysis is performed based on the following assumptions:  
1. The skin is considered as a constructional orthogonal anisotropic plate [36] since the CFRP rods and KKRP fibers are evenly spaced in the rubber. The four edges of the skin are considered to be simply supported. This is close to the realistic boundary condition as the frame provides a limited constraint on the rotation of the skin edge. 2. The shear deformation process of the skin is divided into two stages as shown in Figure 11. At stage I, the skin is sheared from the initial angle to \(90^{\circ}\). It is known from the previous results that the skin is guaranteed to be wrinkle-free at the end of this stage and is subjected to uniform biaxial in-plane loads and a shear load, denoted as \(F_{x}^{00}\), \(F_{y}^{90}\), and \(F_{xy}^{90}\) as shown in Figure 11. Note that the dimension of the three loads is force per unit length along the edges of the plate. At stage II, the skin is further sheared in either direction until it wrinkles. At this stage, we consider the skin as a pre-loaded plate and calculate the critical shear force \(F_{xy}^{cr}\) under \(F_{x}^{90}\) and \(F_{y}^{90}\). Then \(F_{xy}^{cr}+F_{xy}^{90}\) and \(F_{xy}^{cr}-F_{xy}^{90}\) are critical wrinkling forces of the skin in two directions. 3. The stresses in the skin are generated only by the rubber since the CFRP rods and KKRP fibers are found from previous results to undergo nearly no deformation before wrinkling. 4. The rubber, CFRP rods, and KKPR fibers here are all assumed to be linear elastic materials. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Group** & \(E_{c}\) **(GPa)** & \(d_{1}\) **(mm)** & \(I_{1}\) **(mm)** & \(L\) **(mm)** & \(E_{c}d_{1}^{\textbf{\text{a}}}/I_{1}\textbf{L}^{\textbf{\text{2}}}\) & \(\theta_{\textbf{max}}\) **(c)** & \(\theta_{\textbf{min}}\) **(c)** & \(\Delta\theta\) **(c)** \\ \hline \multirow{4}{*}{A} & 675 & 1.5 & 7 & 210 & & 142.7 & 36.0 & 106.7 \\  & 300 & 1.84 & 7 & 210 & & 142.4 & 36.2 & 106.2 \\  & 300 & 1.5 & 3.11 & 210 & & 143.0 & 35.7 & 107.3 \\  & 300 & 1.5 & 7 & 140 & & 142.8 & 36.3 & 106.5 \\ \hline \multirow{4}{*}{B} & 533 & 1.5 & 7 & 280 & & 135.9 & 40.3 & 95.6 \\  & 300 & 1.73 & 7 & 280 & & 136.0 & 39.8 & 96.2 \\  & 300 & 1.5 & 3.94 & 280 & & 135.1 & 39.3 & 95.8 \\  & 300 & 1.5 & 7 & 210 & & 135.4 & 40.4 & 95.0 \\ \hline \multirow{4}{*}{C} & 168 & 1.5 & 7 & 210 & & 129.8 & 43.5 & 86.8 \\  & 300 & 1.30 & 7 & 210 & & 129.6 & 43.2 & 86.4 \\  & 300 & 1.5 & 12.44 & 210 & & 130.1 & 42.9 & 87.2 \\  & 300 & 1.5 & 7 & 280 & & 129.9 & 43.7 & 86.2 \\ \hline \multirow{4}{*}{D} & 108 & 1.5 & 7 & 210 & & 124.8 & 46.5 & 78.3 \\  & 300 & 1.16 & 7 & 210 & & 124.9 & 46.2 & 78.7 \\ \cline{1-1}  & 300 & 1.5 & 19.44 & 210 & & 124.5 & 46.7 & 77.8 \\ \cline{1-1}  & 300 & 1.5 & 7 & 350 & & 125.1 & 46.1 & 79.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Maximal and minimal critical wrinkling angle of the skins with various CFRP reinforcement parameters. With the above assumptions, a simplified theoretical model can be derived. First, consider stage I. When an angled model is sheared to the configuration of \(\theta\), as shown in Figure 7, the maximal and minimal principal strains, \(\varepsilon_{1}\) and \(\varepsilon_{2}\), can be calculated by the following equation under pure shear deformation:  
\[\left\{ \begin{array}{l}\varepsilon_{1}=\ln\biggl{(}\frac{\sin\frac{\theta}{ \varepsilon}}{\sin\frac{\theta}{\varepsilon}}\biggr{)}\\ \varepsilon_{2}=\ln\biggl{(}\frac{\cos\frac{\theta}{\varepsilon}}{\cos\frac{ \theta}{\varepsilon}}\biggr{)}\end{array} \right.