## 2 Theoretical Formulation
### 3D-GrF Circular Cylindrical Microshell
 \tag{43}\]  
\[\begin{array}{ll}\bigl{(}D_{1,2}+l^{2}D_{5,0}\bigr{)}\frac{\partial^{4}v}{ \partial x^{4}}-\Bigl{(}\frac{2}{R}D_{3,1}+D_{5,0}\frac{l^{2}}{R^{2}}\Bigr{)} \frac{\partial^{2}w}{\partial x^{2}}+\frac{1}{R}D_{3,0}\frac{\partial u}{ \partial x}+\frac{1}{R^{2}}D_{1,0}w\\ +\frac{1}{R^{2}}\Bigl{(}2D_{3,2}+4D_{5,2}+\frac{l^{2}}{R^{2}}D_{5,2}+2l^{2}D_{5, 0}\Bigr{)}\frac{\partial^{4}w}{\partial x^{2}\partial\theta^{2}}+\frac{l^{2}}{ R^{2}}D_{1,0}\frac{\partial w}{\partial\theta}\\ -\frac{1}{R^{2}}\Bigl{(}D_{3,1}+2D_{5,1}+\frac{l^{2}}{R^{2}}D_{5,1}-D_{5,0} \frac{3l^{2}}{R^{2}}\Bigr{)}\frac{\partial^{3}w}{\partial x^{3}\partial \theta^{2}}-\Bigl{(}\frac{1}{R^{3}}D_{1,1}+\frac{l^{2}}{R}D_{5,0}\Bigr{)} \frac{\partial^{3}v}{\partial\theta^{3}}\\ -\frac{1}{R}\Bigl{(}D_{3,1}+2D_{5,1}+\frac{l^{2}}{R^{2}}D_{5,1}+D_{5,0}\frac{ 5l^{2}}{R^{2}}\Bigr{)}\frac{\partial^{3}v}{\partial x^{3}\partial\theta}+ \frac{1}{R}\bigl{(}D_{1,2}+l^{2}D_{5,0}\bigr{)}\frac{\partial^{4}w}{\partial \theta^{4}}\\ -D_{1,1}\frac{\partial^{3}u}{\partial x^{3}}-\frac{2}{R}D_{1,1}\frac{\partial^{2}w}{ \partial\theta^{2}}-I_{1,2}\frac{\partial w}{\partial\theta^{2}}-\frac{1}{R^{3}}I_{ 1,2}\frac{\partial^{4}w}{\partial\theta^{2}  
where  
\[ \begin{array}{l}D_{1,i}=\int_{-\frac{1}{2}}^{\frac{b}{2}}\frac{E(z)}{1-\
u^{2}} \;z^{i}\mathrm{d}z\qquad(i=0,1,2)\\ D_{3,i}=\int_{-\frac{1}{2}}^{\frac{b}{2}}\frac{E(z)\
u}{1-\
u^{2}}\;z^{i} \mathrm{d}z\qquad(i=0,1,2)\\ D_{5,i}=\int_{-\frac{1}{2}}^{\frac{b}{2}}\mu(z)\;z^{i} \mathrm{d}z\qquad(i=0,1,2)\\ I_{1,i}=\int_{-\frac{1}{2}}^{\frac{b}{2}}\rho(z)\;z^{i}\mathrm{d}z\qquad(i=0,1,2).\end{array} \]  
Boundary conditions at the edges with \(x=\) constant are  
\[ \begin{array}{l}\delta u_{x=0,L}=0\quad\mbox{or}\quad\int_{\theta}\Bigl{[}D _{1,0}\frac{\partial u}{\partial x}-\frac{D_{5,0}l^{2}}{4R^{2}}\frac{\partial^ {3}u}{\partial x\partial\theta^{2}}+\frac{D_{5,0}l^{2}}{4R}\frac{\partial^{ 3}v}{\partial x^{2}\partial\theta}+\\ \frac{1}{R}D_{3,0}\Bigl{(}\frac{\partial v}{\partial\theta}+w\Bigr{)}-D_{1,1} \frac{\partial^{2}w}{\partial x}-\frac{1}{R^{2}}D_{3,1}\frac{\partial^{2}w}{ \partial\theta^{2}}\Bigr{]}\mathrm{d}\theta|_{x=0,L}=0\end{array}  \tag{45}\]  
\[ \begin{array}{l}\delta v_{x=0,L}=0\quad\mbox{or}\quad\int_{\theta}\Bigl{[}- \frac{D_{5,0}l^{2}}{4}\frac{\partial^{3}v}{\partial x^{3}}+\Bigl{(}1+\frac{3l^ {2}}{R^{2}}\Bigr{)}D_{5,0}\frac{\partial v}{\partial x}-\frac{D_{5,0}l^{2}}{4 R^{2}}\frac{\partial^{3}v}{\partial x\partial\theta^{2}}\\ +\frac{D_{5,0}l^{2}}{4R^{2}}\frac{\partial^{3}u}{\partial x^{2}\partial\theta }+\frac{D_{5,0}l^{2}}{4R^{3}}\frac{\partial^{3}u}{\partial x^{3}}+\frac{D_{5,0 }}{4R^{6}}\frac{\partial u}{\partial\theta}\\ -\Bigl{(}\frac{7l^{2}D_{5,0}}{2R^{2}}+\frac{D_{5,1}}{R}+\frac{D_{5,1}}{R^{3}} \Bigr{)}\frac{\partial^{2}v}{\partial x\partial\theta}\Bigr{]}\mathrm{d} \theta|_{x=0,L}=0\end{array}  \tag{46}\]  
\[ \begin{array}{l}\delta\left(\frac{\partial v}{\partial x}\right)\bigg{|}_{ x=0,L}=0\quad\mbox{or}\quad\int_{\theta}\Bigl{(}\frac{D_{5,0}l^{2}}{4} \frac{\partial^{2}v}{\partial x^{2}}-\frac{D_{5,0}l^{2}}{4R}\frac{\partial^{2 }u}{\partial x\partial\theta}\Bigr{)}\mathrm{d}\theta|_{x=0,L}=0\end{array}  \tag{47}\]  
\[ \begin{array}{l}\delta\left(\frac{\partial w}{\partial x}\right)\bigg{|}_{ x=0,L}=0\quad\mbox{or}\quad\int_{\theta}\Bigl{[}(D_{1,2}+D_{5,0}l^{2}) \frac{\partial^{2}w}{\partial x^{3}}+\Bigl{(}\frac{D_{5,0}l^{2}}{R^{2}} \Bigr{)}\frac{\partial w}{\partial x}\\ -\Bigl{(}\frac{1}{R^{2}}D_{3,2}+\frac{4D_{5,2}}{R^{2}}+\frac{3D_{5,0}l^{2}}{R^{ 2}}+\frac{2D_{5,2}l^{2}}{R^{4}}\Bigr{)}\frac{\partial^{3}w}{\partial x \partial\theta^{2}}+D_{1,1}\frac{\partial^{2}u}{\partial x^{2}}\\ +\Bigl{(}\frac{5D_{5,0}l^{2}}{2R^{2}}+\frac{1}{R}D_{3,1}+\frac{2}{R}D_{5,1}+ \frac{l^{2}}{R}D_{5,1})\frac{\partial^{2}v}{\partial x\partial\theta}\\ +\Bigl{(}\frac{2}{R^{2}}D_{5,1}+\frac{l^{2}}{R^{4}}D_{5,1}-\frac{3D_{5,0}l^{2} }{2R^{3}}\Bigr{)}\frac{\partial^{2}u}{\partial\theta^{2}}\Bigr{]}\mathrm{d} \theta|_{x=0,L}=0\end{array}  \tag{48}\]  
\[