# Wrinkling Patterns and Stress Analysis of Tensile Membrane with Rigid Elements
## 2 Non-Uniform Stress Field
According to the Eshelby elastic inclusion method, the stress field within rigid element can be expressed as:  
\[C_{jkl}^{1}\left(\varepsilon_{ij}^{0}+S_{ijkl}\varepsilon_{ij}^{*}\right)=C_{ jkl}\left(\varepsilon_{ij}^{0}+S_{ijkl}\varepsilon_{ij}^{*}-\varepsilon_{ij}^{*}\right) \tag{13}\]where \(C_{ijkl}\) and \(C_{ijkl}^{1}\) are the elastic stiffness matrices of the membrane and the rigid element, respectively. \(\epsilon_{ij}^{0}\) is the strain field of the membrane without rigid components, which can be obtained by [PERSON]'s law from Equation (12). Here, \(\epsilon_{ij}^{\prime}\) is the disturbance strain caused by the rigid element. The disturbance strain of the rigid element is regarded as caused by the eigenstrain \(\epsilon_{kl}^{*}\) generated in the inclusion, where \(\epsilon_{ij}^{\prime}\) can be expressed as:  
\[\epsilon_{ij}^{\prime}=\left\{ \begin{array}{l}S_{ijkl}\epsilon_{kl}^{*}\\ D_{ijkl}\epsilon_{kl}^{*}\end{array} \right. \tag{14}\]  
\(S_{ijkl}\) is the Eshelby tensor of the rigid element, which can be obtained from the Somigliana equation. The Eshelby tensor in a cylindrical inclusion under plane stress can be calculated as [17]:  
\[ \begin{array}{l}S_{ijkl}(\eta)=\frac{1}{8\pi(1-\
u)}\left\{\delta_{ij} \delta_{kl}\left[2\
u I_{i}(\eta)-I_{i}(\eta)+a^{2}I_{kl}(\eta)\right]\right. \\ \left.+\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)\left[a^{2} I_{ij}(\eta)-I_{j}(\eta)+(1-\
u)(I_{k}(\eta)+I_{m}(\eta))\right]\right.\end{array}  \tag{15}\]  
where \(\
u\) is the [PERSON]'s ratio of the membrane, \(\delta_{ij}\) is the unit second-order tensor, and \(I(\eta)\) is the ellipsoid integral, which can be expressed as:  
\[I_{i}(\eta)=2\pi a_{1}a_{2}a_{3}\int_{\eta}^{\infty}\frac{ds}{\left(a_{i}^{2} +s\right)\sqrt{\left(a_{1}^{2}+s\right)\left(a_{2}^{2}+s\right)\left(a_{3}^{ 2}+s\right)}} \tag{16}\]  
\[I_{ij}(\eta)=2\pi a_{1}a_{2}a_{3}\int_{\eta}^{\infty}\frac{ds}{\left(a_{i}^{2} +s\right)\left(a_{j}^{2}+s\right)\sqrt{\left(a_{1}^{2}+s\right)\left(a_{2}^{2} +s\right)\left(a_{3}^{2}+s\right)}} \tag{17}\]  
where \(\eta\) is the largest positive root of the following virtual ellipsoid equation:  
\[\frac{x^{2}}{a_{1}^{2}+\eta}+\frac{y^{2}}{a_{2}^{2}+\eta}+\frac{z^{2}}{a_{3}^ {2}+\eta}=1 \tag{18}\]  
Simplifying the elliptical inclusions into cylindrical inclusions, if we let \(a_{1}=a_{2}=a\) and \(a_{3}\rightarrow\infty\), we can obtain:  
\[\eta=r^{2}-a^{2} \tag{19}\]  
Integrating the ellipsoid integral Equations (16) and (17), and substituting them into Equation (15), the Eshelby tensor \(S_{ijkl}\) can be obtained:  
\[S_{1111}(r_{n})=S_{2222}(r_{n})=\frac{1}{8(1-\
u)}\bigg{[}\frac{3a^{4}}{r_{n}^ {4}}+(1-2\
u)\frac{2a^{2}}{r_{n}^{2}}\bigg{]} \tag{20}\]  
\[S_{1122}(r_{n})=S_{2211}(r_{n})=\frac{1}{8(1-\
u)}\bigg{[}\frac{a^{4}}{r_{n}^ {4}}-(1-2\
u)\frac{2a^{2}}{r_{n}^{2}}\bigg{]} \tag{21}\]  
\[S_{1212}(r_{n})=\frac{1}{2(1-\
u)}\bigg{[}\frac{a^{4}}{r_{n}^{4}}+(1-2\
u) \frac{2a^{2}}{r_{n}^{2}}\bigg{]} \tag{22}\]  
When \(r=a\), by substituting into Equations (11)-(13), the eigenstrain \(\epsilon_{ij}^{*}\) of the rigid element can be obtained. \(D_{ijkl}\) is the Eshelby tensor in the membrane structures, which can be obtained from the Somigliana equation. The Eshelby tensor of the membrane structures with cylindrical inclusions can be expressed as [17]:  
\[ \begin{array}{rl}D_{ijkl}(r_{n})&=S_{ijkl}(r)+\frac{1}{8\pi(1-v)}\{2v\delta_{kl }x_{i}I_{i,j}(\eta)+(1-v)[\delta_{il}x_{k}I_{kj}(\eta)\] \[+\delta_{ik}x_{i}I_{i,l}(\eta)+\delta_{jk}x_{l}I_{i,l}(\eta)]-\delta_{ il}x_{k}[I_{k}(\eta)-a^{2}I_{kl}(\eta)]_{,l}\\ &-(\delta_{k}x_{j}+\delta_{jk}x_{i})[I_{j}(\eta)-a^{2}I_{ij}(\eta)]_{,l}-( \delta_{il}x_{j}+\delta_{jl}x_{i})[I_{j}(\eta)\\ &\qquad\qquad\qquad\qquad-a^{2}I_{ij}(\eta)]_{,k}-x_{i}x_{j}[I_{j}(\eta)-a^{2 }I_{ij}(\eta)]_{,l}\end{array}