## 2 Theoretical Formulation
### 3D-GrF Circular Cylindrical Microshell
\begin{array}{l}\delta\left(\frac{\partial w}{\partial x}\right)\bigg{|}_{ x=0,L}=0\quad\mbox{or}\quad\int_{\theta}\Bigl{[}(D_{1,2}+D_{5,0}l^{2}) \frac{\partial^{2}w}{\partial x^{2}}+\Bigl{(}\frac{D_{3,2}}{R^{2}}-\frac{D_{5,0 }l^{2}}{R^{2}}\Bigr{)}\frac{\partial^{2}w}{\partial\theta^{2}}\\ +\Bigl{(}\frac{D_{5,0}l^{2}}{R^{2}}-\frac{1}{R}D_{3,1}\Bigr{)}\frac{\partial v}{ \partial\theta}-D_{1,1}\frac{\partial u}{\partial x}-\frac{1}{R}D_{3,1}w \Bigr{]}\mathrm{d}\theta|_{x=0,L}=0\end{array}  \tag{49}\]  
Boundary conditions at the edges with \(\theta=\) constant are:  
\[ \begin{array}{l}\delta\left.u\right|_{\theta=0,\theta_{0}}=0\quad\mbox{or} \quad\int_{x}\Bigl{[}\frac{D_{5,0}l^{2}}{R^{2}}\Bigl{(}1+\frac{l^{2}}{R^{2}} \Bigr{)}\frac{\partial u}{\partial\theta}-\frac{D_{5,0}l^{2}}{4R^{2}}\frac{ \partial^{3}u}{\partial x^{2}\partial\theta}+\frac{D_{5,0}l^{2}}{4R^{3}}\frac{ \partial^{3}v}{\partial x\partial\theta^{2}}\\ +\frac{D_{5,0}l^{2}}{R^{2}}\frac{\partial v}{\partial x}+\Bigl{(}\frac{3D_{5,0}l^ {2}}{2R^{3}}-\frac{2D_{5,1}}{R^{2}}-\frac{l^{2}D_{5,1}}{R^{4}}\Bigr{)}\frac{ \partial^{2}w}{\partial x\partial\theta}\Bigr{]}\mathrm{d}x|_{\theta=0,\theta_{ 0}}=0\end{array}  \tag{50}\]  
\[ \begin{array}{l}\delta\left.v\right|_{\theta=0,\theta_{0}}=0\quad\mbox{or} \quad\int_{x}\Bigl{[}\frac{1}{R^{2}}D_{1,0}+\frac{D_{5,0}l^{2}}{R^{4}}\Bigr{)} \frac{\partial v}{\partial\theta}+\frac{1}{R}D_{3,0}\frac{\partial u}{ \partial x}\\ +\frac{1}{R^{2}}D_{1,0}w+\Bigl{(}\frac{D_{5,0}l^{2}}{2R^{2}}-\frac{1}{R}D_{3,1} \Bigr{)}\frac{\partial^{2}w}{\partial x^{2}}-\frac{D_{5,0}l^{2}}{4R^{2}}\frac{ \partial^{3}v}{\partial x^{2}\partial\theta}+\frac{D_{5,0}l^{2}}{4R^{3}}\frac{ \partial^{3}u}{\partial x\partial\theta^{2}}\\ -\Bigl{(}\frac{D_{5,0}l^{2}}{R^{4}}+\frac{1}{R^{3}}D_{1,1}\Bigr{)}\frac{ \partial^{2}w}{\partial\theta^{2}}\Bigr{]}\mathrm{d}x|_{\theta=0,\theta_{0}}=0 \end{array}  \tag{51}\]  
\[ \begin{array}{l}\delta\left(\frac{\partial u}{\partial\theta}\right)\bigg{|}_{ \theta=0,\theta_{0}}=0\quad\mbox{or}\quad\int_{x}\Bigl{(}\frac{D_{5,0}l^{2}}{4R^{4}} \frac{\partial^{2}u}{\partial\theta^{2}}-\frac{D_{5,0}l^{2}}{4R^{3}}\frac{ \partial^{2}v}{\partial x\partial\theta}-\frac{D_{5,0}l^{2}}{2R^{3}}\frac{ \partial w}{\partial x}\right)\mathrm{d}x|_{\theta=0,\theta_{0}}=0\end{array}  \tag{52}\]\[ \begin{array}{l}\delta w\big{|}_{\theta=0,\theta_{0}}=0\quad\text{or}\quad\int_{x} \left[-\left(\frac{1}{R^{4}}D_{1,2}+\frac{D_{5}\theta l^{2}}{R^{4}}\right) \frac{\partial^{3}w}{\partial\theta^{3}}+\frac{3D_{5}\theta l^{2}}{R^{2}}\frac{ \partial^{2}v}{\partial x^{2}}\right.\\ \left.-\left(\frac{1}{R^{2}}D_{3,2}+\frac{4D_{5,2}}{R^{2}}+\frac{3D_{5,\theta l ^{2}}}{R^{2}}+\frac{2D_{5,2}l^{2}}{R^{4}}\right)\frac{\partial^{3}w}{ \partial x^{2}\partial\theta}\right.\\ \left.+\left(\frac{D_{5,0}l^{2}}{R^{4}}+\frac{1}{R^{3}}D_{1,1}\right)\frac{ \partial^{2}v}{\partial\theta^{2}}+\left(\frac{1}{R^{2}}D_{3,1}-\frac{D_{5,0}l ^{2}}{R^{3}}\right)\frac{\partial^{2}u}{\partial x\partial\theta}\\ \left.+\frac{1}{R^{3}}D_{1,1}\frac{\partial w}{\partial\theta}\right]\mathrm{ d}x\big{|}_{\theta=0,\theta_{0}}=0\end{array}  \tag{53}\]  
\[ \begin{array}{l}\delta\left(\frac{\partial w}{\partial\theta}\right)\Big{|}_ {\theta=0,\theta_{0}}=0\quad\text{or}\quad\int_{x}\left[\left(\frac{1}{R^{4}} D_{1,2}+\frac{1}{R^{4}}D_{5,0}l^{2}\right)\frac{\partial^{2}v}{\partial\theta^{2}}+ \left(\frac{D_{3,2}}{R^{2}}-\frac{D_{5,0}l^{2}}{R^{2}}\right)\frac{\partial^{ 2}w}{\partial x^{2}}\right.\\ \left.-\left(\frac{D_{5,0}l^{2}}{R^{4}}+\frac{1}{R^{3}}D_{1,1}\right)\frac{ \partial v}{\partial\theta}-\frac{1}{R^{2}}D_{3,1}\frac{\partial u}{\partial x }-\frac{1}{R^{3}}D_{1,1}w\right]\mathrm{d}x\big{|}_{\theta=0,\theta_{0}}=0 \end{array}  \tag{54}\]