# Wrinkling Patterns and Stress Analysis of Tensile Membrane with Rigid Elements
## 2 Non-Uniform Stress Field  
To explain the change in the wrinkling pattern of the tension membrane with the position of the square rigid element, in this paper, we propose a calculation method for the non-uniform stress field based on the Eshelby elastic inclusion theory, whose heterogeneous membrane structure is shown in Figure 1. When the rigid element is bonded with the membrane, since the thickness of the membrane is much smaller than the thickness of the rigid element, the strain distribution of the membrane structure can be expressed as:  
\[\sigma_{ij}=\sigma_{ij}^{0}+\sigma_{ij}^{\prime} \tag{1}\]  
where \(\sigma_{ij}^{\prime}\) is the perturbation stress caused by rigid elements; \(\sigma_{ij}^{0}\) is the stress distribution of the membrane without rigid elements. In order to obtain the stress distribution of a uniform membrane, its stress function can be expressed as:  
\[\Phi=Apsint\theta \tag{2}\]  
Through the compatibility equation \(\
abla^{4}\Phi=0\), the stress component in the polar coordinate system can be obtained as:  
\[\left\{ \begin{array}{c}\sigma_{\rho}=\frac{2}{\rho}\ A\rho sin\theta\\ \sigma_{\theta}=0\\ \tau_{\theta\rho}=0\end{array} \right. \tag{3}\]  
Since there is no surface force on the membrane boundary, the stress boundary condition of the membrane can be expressed as:  
\[(\sigma_{\theta})_{\theta=\pm\alpha,\ \rho\
eq 0}=0,\ (\tau_{\theta})_{\theta=\pm \alpha,\ \rho\
eq 0}=0 \tag{4}\]  
Figure 1: Schematic diagram of heterogeneous film structure. From the stress boundary conditions, the equilibrium equation can be obtained as:  
\[\int_{-\infty}^{\infty}\sigma_{\rho}\rho d\theta cos\theta=T \tag{5}\]  
By substituting Equation (3) into Equation (5), and after integration, we can obtain:  
\[A=\frac{2 Tcos\theta}{\pi+2} \tag{6}\]  
Considering the thickness \(h_{m}\) of the membrane, the radial stress of the uniform membrane in the polar coordinate system can be obtained by substituting Equation (6) into Equation (3):  
\[\sigma_{\rho}=\frac{4 Tcos\theta}{\rho h_{m}(\pi+2)} \tag{7}\]  
According to the transformation formula from polar coordinates to rectangular coordinates, the radial stress given by Formula (6) is decomposed into the \(x\) and \(y\) directions:  
\[\left\{ \begin{array}{c}\sigma_{x}=\sigma_{p}sin^{2}\theta+\sigma_{\theta} cos^{2}\theta-2\tau_{\theta,\theta}sin\theta cos\theta\\ \sigma_{y}=\sigma_{p}cos^{2}\theta+\sigma_{\theta}sin^{2}\theta+2\tau_{ \theta,\theta}sin\theta cos\theta\\ \tau_{xy}=\left(\sigma_{\rho}-\sigma_{\theta}\right)sin\theta cos\theta+\tau _{\theta}\left(sin^{2}\theta-cos^{2}\theta\right)\end{array} \right. \tag{8}\]  
where \(\theta\) and \(\rho\) can be expressed in Cartesian coordinates as:  
\[cos\theta=\frac{R-y}{\rho},\;sin\theta=\frac{x}{\rho} \tag{9}\]  
Substituting Equations (7) and (9) into Equation (8), we can obtain:  
\[\left\{ \begin{array}{c}\sigma_{x}=\frac{4 Tx^{2}(R-y)}{\rho^{4}h_{m}(\pi+2)} \\ \sigma_{y}=\frac{4T(R-y)^{3}}{\rho^{4}h_{m}(\pi+2)}\\ \tau_{xy}=\frac{4T(R-y)^{2}}{\rho^{4}h_{m}(\pi+2)}\end{array} \right. \tag{10}\]  
The boundary stress at the top of the membrane due to the edge effect can be expressed as [16]:  
\[T_{c}=-\frac{4T}{2\pi R} \tag{11}\]  
According to the principle of stress superposition, the stress distribution \(\sigma_{ij}^{0}\) of the membrane can be obtained by superimposing the tensile force at the four vertices of the membrane and the boundary stress:  
\[\sigma_{ij}^{0}\left\{ \begin{array}{c}\sigma_{x}=-\frac{4T}{2\pi R}+\sum \limits_{i=1}^{i=4}\frac{4T(xcos\theta-ysin\theta)^{2}[R-(xcos\theta-ysin \theta)]}{\rho^{4}h_{m}(\pi+2)}\\ \sigma_{y}=-\frac{4T}{2\pi R}+\sum\limits_{i=1}^{i=4}\frac{4T[R-(xcos\theta- ysin\theta)]^{3}}{\rho^{4}h_{m}(\pi+2)}\\ \tau_{xy}=-\frac{4T}{2\pi R}+\sum\limits_{i=1}^{i=4}\frac{4T(xcos\theta-ysin \theta)[R-(xcos\theta-ysin\theta)]^{2}}{\rho^{4}h_{m}(\pi+2)}\end{array} \right. \tag{12}\]  
where \(T\) is the tension, \(R\) is the radius of the circumscribed circle of the membrane, and \(h_{m}\) is the thickness of the film.