# Wrinkling Patterns and Stress Analysis of Tensile Membrane with Rigid Elements
## 2 Non-Uniform Stress Field
 \tag{23}\]  
where \(I_{i,j}(\eta)\), \(I_{ij,k}(\eta)\), \(I_{i,kl}(\eta)\), and \(I_{ij,kl}(\eta)\) can be obtained by taking the partial derivatives of Equations (6) and (7) with respect to \(x_{n}\) and \(y_{n}\) respectively, and, substituting them into Equation (23), the Eshelby tensor \(D_{ijkl}\) can be obtained:  
\[D_{1111}(r_{n})=S_{1111}(r_{n})+\frac{1}{1-v}\bigg{[}(1+v)\frac{a^{2}}{r_{n}^{ 4}}x_{n}^{2}-\frac{3a^{2}}{r_{n}^{6}}x_{n}^{2}-\frac{2a^{2}}{r_{n}^{6}}x_{n}^{ 4}+\frac{3a^{4}}{r_{n}^{8}}x_{n}^{4}\bigg{]} \tag{24}\]  
\[D_{1122}(r_{n})=S_{1122}(r_{n})+\frac{1}{2(1-v)}\bigg{\{}\Big{[}(1-2v)x_{n}^{ 2}+y_{n}^{2}\Big{]}\frac{a^{2}}{r_{n}^{4}}-\bigg{(}y_{n}^{2}+\frac{1}{a^{2}}x_ {n}^{2}y_{n}^{2}\bigg{)}\frac{a^{4}}{r_{n}^{6}}+\frac{6a^{4}}{r_{n}^{8}}x_{n}^ {2}y_{n}^{2}\bigg{\}} \tag{25}\]  
\[D_{2211}(r_{n})=S_{2211}(r_{n})+\frac{1}{2(1-v)}\bigg{\{}\Big{[}(1-2v)y_{n}^{ 2}+x_{n}^{2}\Big{]}\frac{a^{2}}{r_{n}^{4}}-\bigg{(}x_{n}^{2}+\frac{1}{a^{2}}x_ {n}^{2}y_{n}^{2}\bigg{)}\frac{a^{4}}{r_{n}^{6}}+\frac{6a^{4}}{r_{n}^{8}}x_{n}^ {2}y_{n}^{2}\bigg{\}} \tag{26}\]  
\[D_{2222}(r_{n})=S_{2222}(r_{n})+\frac{1}{1-v}\bigg{[}(1+v)\frac{a^{2}}{r_{n}^{ 4}}y_{n}^{2}-\frac{3a^{2}}{r_{n}^{6}}y_{n}^{2}-\frac{2a^{2}}{r_{n}^{6}}y_{n}^{ 4}+\frac{3a^{4}}{r_{n}^{8}}y_{n}^{4}\bigg{]} \tag{27}\]  
\[D_{1212}(r_{n})=S_{1212}(r_{n})+\frac{1}{1-v}\bigg{[}v\Big{(}x_{n}^{2}+y_{n}^{ 2}\Big{)}\frac{a^{2}}{r_{n}^{4}}-\Big{(}x_{n}^{2}+y_{n}^{2}\Big{)}\frac{a^{4}} {r_{n}^{6}}\bigg{]} \tag{28}\]  
When \(r_{n}>a\), by substituting the eigenstrain \(\epsilon_{kl}^{*}\) and Equations (24)-(28) into the Equation (14), we can obtain the disturbance strain \(\epsilon_{ij}^{*}\) caused by the cylindrical inclusion. Due to the stress concentration in the corners of the square rigid element, we introduce concentrated stress after finite-element analysis to correct the disturbance strain \(\epsilon_{ij}^{\prime}\) caused by cylindrical inclusions, and then obtain the stress distribution of the membrane as:  
\[\sigma_{ij}=\sigma_{ij}^{0}+\sigma_{ij}^{\prime}+\sigma_{ij}^{c} \tag{29}\]  
where \(\sigma_{ij}^{c}\) is the concentrated stress, can be expressed as:  
\[\sigma_{ij}^{c}=\frac{a^{2}}{\pi\eta}\frac{||x|-|y||}{r}\sigma_{ij}^{\prime} \tag{30}\]  
In order to obtain the stress distribution of the membrane structures with rigid elements, we first deduced the stress function of the uniform membrane structure using the inverse method and solved the stress distribution \(\sigma_{ij}^{0}\) of the uniform thin film by the stress function and the stress superposition principle. Next, the Eshelby tensor \(S_{ijkl}\) of the membrane and the Eshelby tensor \(D_{ijkl}\) of the rigid element were calculated according to the Eshelby elastic inclusion theory. The uniform membrane-stress distribution \(\sigma_{ij}^{0}\) and the Eshelby tensor \(S_{ijkl}\) of the thin film were substituted for the strain equilibrium equation to obtain the eigenstrain of the rigid element \(\epsilon_{ij}^{*}\). According to the Eshelby tensor \(D_{ijkl}\) and the eigenstrain \(\epsilon_{ij}^{*}\) of the rigid element, the perturbation stress of the rigid element to the membrane structure was calculated. However, the elastic inclusion method of this paper is based on the cylindrical inclusion structure, and there was a large stress concentration at the apex of the square rigid element. Therefore, we calculated the stress distribution of the membrane structure containing the circular rigid element and the square rigid element, respectively, through finite-element simulation, and fitted the stress concentration factor at the apex of the rigid element by comparing the stress distribution of the two structuresat the apex of the rigid element. Finally, the stress distribution of the membrane structure with square rigid elements was obtained by stress superposition. The above theory was realized by MATLAB programming, and the calculation results were compared with the calculation results of finite-element analysis.