## 3 Governing Equations  
According to the von Karman nonlinear strain-displacement relations, the strain components at the middle surface of equivalent circular cylindrical shells are of the form [49]  
\[ \begin{array}{l}\epsilon_{x}^{0}=\frac{\partial u}{\partial x}+\frac{1}{2} \big{(}\frac{\partial w}{\partial x}\big{)}^{2},\\ \epsilon_{y}^{0}=\frac{\partial v}{\partial y}-\frac{w}{R}+\frac{1}{2}\big{(} \frac{\partial w}{\partial y}\big{)}^{2},\\ \gamma_{xy}^{0}=\frac{\partial u}{\partial y}+\frac{\partial c}{\partial x}+ \frac{\partial w}{\partial x}\frac{\partial w}{\partial y},\\ \chi_{x}=\frac{\partial^{2}w}{\partial x^{2}},\ \ \chi_{y}=\frac{\partial^{2}w}{ \partial y^{2}},\ \ \chi_{xy}=\frac{\partial^{2}w}{\partial x\partial y},\end{array}  \tag{10}\]  
where \(\epsilon_{x}^{0}\) and \(\epsilon_{y}^{0}\) are normal strains, \(\gamma_{xy}^{0}\) is the shear strain at the middle surface of the equivalent shell, the changes of curvatures and twist of non-corrugated shell are denoted by \(\chi_{x},\chi_{y},\chi_{xy}\), respectively, and \(u=u(x,y)\), \(v=v(x,y)\), \(w=w(x,y)\) are the displacement terms in longitudinal, circumferential and inward directions, respectively. The equivalent strains of equivalent shell thickness at a distance \(z\) from the mid-surface are expressed as  
\[\epsilon_{x}=\epsilon_{x}^{0}-z\chi_{x},\ \ \ \epsilon_{y}=\epsilon_{y}^{0}-z\chi_{y},\ \ \ \gamma_{xy}=\gamma_{xy}^{0}-2z\chi_{xy}. \tag{11}\]  
By using Equation (10), the deformation compatibility equation of FG-CNTRC cylindrical shell is determined as  
\[\epsilon_{x,yy}^{0}+\epsilon_{y,xx}^{0}-\gamma_{xy,xy}^{0}+\frac{1}{R}w_{,xx}- w_{,xy}^{2}+w_{,xx}w_{,xy}=0. \tag{12}\]  
The nonlinear equilibrium equation system of the equivalent FG-CNTRC cylindrical shell in framework of the Donnell shell theory considered the von Karman nonlinearities is presented by [49]  
\[ \begin{array}{l}N_{x,x}+N_{xy,y}=0,\\ N_{y,y}+N_{xy,x}=0,\\ M_{x,xx}+2M_{xy,xy}+M_{y,yy}+N_{x}w_{,xx}+2N_{xy}w_{,xy}+N_{y}w_{,yy}+\frac{N_ {y}}{R}+q_{0}=0.\end{array}  \tag{13}\]  
Considering the first two equations of system (13), a stress function \(\varphi\) may be introduced as  
\[N_{x}=\frac{\partial^{2}\varphi}{\partial y^{2}},\ \ \ N_{y}=\frac{\partial^{2} \varphi}{\partial x^{2}},\ \ \ N_{xy}=-\frac{\partial^{2}\varphi}{\partial x\partial y}. \tag{14}\]  
Substituting Equations (9) and (14) into the third equation of (13), the stability equation is rewritten as  
\[ \begin{array}{l}-w_{,xxx}D_{11}-w_{,xxx}(2D_{16}+2D_{61})-w_{,xxy}(D_{12}+4D_ {66}+D_{21})\\ -w_{,xyyy}(2D_{62}+2D_{26})-w_{,yyyy}D_{22}\\ +\varphi_{,yy}w_{,xx}+\varphi_{,xx}w_{,yy}-2\varphi_{,xy}w_{,xy}+\frac{\varphi_ {,xx}}{R}+q_{0}=0.\end{array}  \tag{15}\]  
By using the constitutive relations (9) with the stress function expressions (14), the inverse relations can be established:  
\[\left\{ \begin{array}{l}\epsilon_{x}^{0}\\ \epsilon_{y}^{0}\\ \gamma_{xy}^{0}\end{array} \right\}=\left\{ \begin{array}{l}F_{11}\ \ \ F_{12}\ \ \ F_{16}\\ F_{21}\ \ \ F_{22}\ \ \ F_{26}\\ F_{61}\ \ \ F_{62}\ \ \ F_{66}\end{array} \right\}\left\{ \begin{array}{l}\varphi_{,yy}\\ \varphi_{,xx}\\ -\varphi_{,xy}\end{array} \right\}, \tag{16}\]  
where  
\( \begin{array}{l}F_{11}=\frac{A_{22}A_{66}-A_{62}A_{26}}{\Delta_{2}},F_{12}= \frac{A_{61}A_{26}-A_{21}A_{66}}{\Delta},F_{16}=\frac{A_{21}A_{62}-A_{61}A_{22} }{\Delta},\\ F_{21}=\frac{A_{62}A_{16}-A_{61}A_{16}}{\Delta},F_{22}=\frac{A_{11}A_{62}-A_{61}A_{ 16}}{\Delta},F_{26}=\frac{A_{61}A_{12}-A_{11}A_{62}}{\Delta},\\ F_{61}=\frac{A_{12}A_{26}-A_{22}A_{16}}{\Delta},F_{62}=\frac{A_{21}A_{16}-A_{11}A_{ 22}}{\Delta},F_{66}=\frac{A_{11}A_{22}-A_{21}A_{12}}{\Delta},\\ \Delta=(A_{11}A_{22}A_{66}+A_{12}A_{26}A_{61}+A_{21}A_{62}A_{16})-(A_{16}A_{22}A _{61}+A_{12}A_{21}A_{66}+A_{11}A_{62}A_{26}).\end{array} \)Substituting Equation (16) into Equation (12), the rewritten deformation compatibility equation is as:  
\[