# The Effect of Inflatable Pressure on the Strain Deformation of Flexible Wing Skin Film
## 3 Inflation Deformation Modeling and Bearing Deformation Simulation
### Inflatable Flexible Deformation Modeling
For the inflated wing under pressure stabilization, accounting for stress, geometric parameters, aerodynamic pressure load, and external loads, the following relationship among these variables can be derived [7; 8]:  
\[(p_{in}-p_{E}-p_{a}-\rho g)=\frac{(V_{f}\sigma_{xf}+V_{m}\sigma_{xm})t}{R_{x}} \frac{2\sin\frac{\alpha}{2}}{\alpha}+\frac{(V_{f}\sigma_{yf}+V_{m}\sigma_{ym}) t}{R_{y}}\frac{2\sin\frac{\beta}{2}}{\beta} \tag{2}\]  
Taking into account the volume stress of the fiber and the matrix, the relationship between the biaxial stress of the flexible film material and the deformation curvature in the \(x\) direction and the \(y\) direction at the time of formation is as follows [7]:  
\[\frac{(V_{f}\sigma_{xf}+V_{m}\sigma_{xm})}{k_{x}}=\frac{(V_{f}\sigma_{yf}+V_{ m}\sigma_{ym})}{k_{y}},\;\text{among them}:R_{x}=\frac{1}{k_{x}},\quad R_{y}=\frac{1}{k_{y}} \tag{3}\]  
Here, \(K_{x}\) and \(K_{y}\) (the reciprocals of \(R_{x}\) and \(R_{y}\)) represent the biaxial curvatures. Usually, the matrix in flexible inflatable membrane material is an isotropic material, and the fiber yarn bundle is an anisotropic material of laminated reinforced warp and weft plain weave. Therefore, in the modeling process, it is necessary to further consider the elastic modulus of the flexible inflatable membrane material along the \(x\) direction and the \(y\) direction. The biaxial elastic modulus is expressed as \(E_{x}\) and \(E_{y}\), respectively. When the flexible wing is pressurized, the pressure in the film is equal everywhere, and the change in the inflation radius in the biaxial direction is equal to \(r\). The relationship between the biaxial strain and the elastic modulus \(E_{x}\) and \(E_{y}\) in both directions can be further obtained to characterize the relationship between the volume fraction of the matrix and the fiber yarn in the flexible inflatable film material and the component stress. So, the flexible skin film and the strain-stress relationships between the x-direction and y-direction could be further identified as follows:  
\[\epsilon_{x}=\frac{(V_{f}\sigma_{xf}+V_{m}\sigma_{xm})}{E_{x}}-\mu\frac{(V_{f} \sigma_{yf}+V_{m}\sigma_{ym})}{E_{y}},\quad\epsilon_{y}=\frac{(V_{f}\sigma_{ yf}+V_{m}\sigma_{ym})}{E_{y}}-\mu\frac{(V_{f}\sigma_{xf}+V_{m}\sigma_{xm})}{E_{x}} \tag{4}\]  
Figure 2: Deformation and bearing relationship model of flexible film RVE. By further simplifying the biaxial strain deformation expression of the flexible inflatable wing, the relative radius of curvature of the biaxial deformation of the matrix and the fiber bundle can be set as follows:  
\[R_{y}=nR_{x} \tag{5}\]  
Therefore, the above biaxial stress expression (4) can be further simplified. Since the biaxial strain deformation of the flexible inflatable wing film is related to the biaxial stress, and the tensile curvature of the matrix and the fiber bundle along the two biaxial directions and the biaxial material property deformation law can be obtained, the biaxial strain deformation of the flexible inflatable wing film material is as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{x}=\frac{(p_{in}-p_{E}-p_{a}-\rho g)}{t/R _{y}}\left[\frac{1}{(E_{y}\frac{1+\mu\mu}{n+\mu}+E_{x}n)}-\frac{\mu}{(E_{x} \frac{x^{2}+2m}{1+\mu\mu}+E_{y})}\right]\\ \varepsilon_{y}=\frac{(p_{in}-p_{E}-p_{a}-\rho g)}{t/R_{y}}\left[\frac{1}{(E_{ x}\frac{x^{2}+\mu\mu}{1+\mu\mu}+E_{y})}-\frac{\mu}{(E_{y}\frac{1+\mu\mu}{n+\mu}+E_{x}n)} \right]\end{array} \right. \tag{6}\]  
The flexible inflatable wing skin film RVE is obtained in the bidirectional plane stress state. According to the material mechanical deformation theory, the physical relationship could be further studied between the inflation overpressure deformation characteristics of the flexible skin film material and the internal inflation pressure. Furthermore, the deformation stiffness characteristics of the flexible film can be analyzed according to the deformation conditions.