## 2 Governing Equations of the Problem
\[\varepsilon_{yy}=\frac{\partial v_{0}(x,y,t)}{\partial y}+z\frac{ \partial\varphi_{y}(x,y,t)}{\partial y}-\overline{c}z^{3}\bigg{(}\frac{\partial \varphi_{y}(x,y,t)}{\partial y}+\frac{\partial^{2}w_{0}(x,y,t)}{\partial y^{2} }\bigg{)}+\frac{w}{R}, \tag{5b}\] \[\gamma_{xy}=\frac{\partial u_{0}(x,y,t)}{\partial y}+\frac{ \partial v_{0}(x,y,t)}{\partial x}+2\bigg{(}\frac{\partial\varphi_{z}(x,y,t)} {\partial y}+\frac{\partial\varphi_{y}(x,y,t)}{\partial x}\bigg{)}\] (5c) \[-\overline{c}z^{3}\bigg{(}\frac{\partial\varphi_{x}(x,y,t)}{ \partial y}+\frac{\partial\varphi_{y}(x,y,t)}{\partial x}+2\frac{\partial^{2 }w_{0}(x,y,t)}{\partial y\partial x}\bigg{)}\] \[\gamma_{xz}=\varphi_{x}(x,y,t)-2\overline{c}z^{2}\bigg{(}\varphi_{x}( x,y,t)+\frac{\partial w_{0}(x,y,t)}{\partial x}\bigg{)}+\frac{\partial w_{0}(x,y,t)}{ \partial x},\] (5d) \[\gamma_{yz}=\varphi_{y}(x,y,t)-2\overline{c}z^{2}\bigg{(}\varphi_{y}(x,y,t)+ \frac{\partial w_{0}(x,y,t)}{\partial y}\bigg{)}+\frac{\partial w_{0}(x,y,t) }{\partial y}, \tag{5e}\]  
which are related to the stress components as follows  
\[\left\{ \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xz}\\ \sigma_{yz}\\ \sigma_{xy}\end{array} \right\}=\left[ \begin{array}{cccccc}Q_{11}&Q_{12}&0 &0&0&0\\ Q_{21}&Q_{22}&0&0&0\\ 0&0&Q_{44}&0&0\\ 0&0&0&Q_{55}&0\\ 0&0&0&0&Q_{66}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \varepsilon_{xz}\\ \varepsilon_{yz}\\ \varepsilon_{xy}\end{array} \right\}, \tag{6}\]  
where the stiffness \(Q_{ij}\) is defined as  
\[ \begin{split} Q_{11}(x,z)=Q_{22}(x,z)=\frac{E(x \varepsilon)}{1-(\
u(x,z))^{2}},Q_{12}(x,z)=Q_{21}(x,z)=\
u(x,z)Q_{11}(x,z),\\ Q_{44}(x,z)=Q_{55}(x,z)=Q_{66}(x,z)=\frac{E(x,z)}{2(1+\
u(x,z))}.\end{split}  \tag{7}\]  
The strain energy of the BD-FG porous cylindrical shell is expressed as follow  
\[\Pi_{S}=\frac{1}{2}\int\limits_{0}^{L}\int\limits_{0}^{2\pi R}\int\limits_{- \frac{h}{2}}^{\frac{h}{2}}(\sigma_{xx}\varepsilon_{xx}+\sigma_{yy}\varepsilon _{yy}+\tau_{xy}\gamma_{xy}+\tau_{xz}\gamma_{xz}+\tau_{yz}\gamma_{yz})dzdydx, \tag{8}\]  
and the kinetic energy of the cylindrical shell reads  
\[\Pi_{T}=\frac{1}{2}\int\limits_{0}^{L}\int\limits_{0}^{2\pi R}\int\limits_{- \frac{h}{2}}^{\frac{h}{2}}\rho(x,z)\bigg{(}\bigg{(}\frac{\partial u(x,y,z,t)}{ \partial t}\bigg{)}^{2}+\bigg{(}\frac{\partial v(x,y,z,t)}{\partial t}\bigg{)} ^{2}+\bigg{(}\frac{\partial w(x,y,z,t)}{\partial t}\bigg{)}^{2}\bigg{)}dzdydx. \tag{9}\]  
The total potential energy corresponding to the axial compressive load \(F_{a}(t)\) together with the [PERSON] and/or [PERSON] elastic foundation, can be written as follow  
\[\Pi_{E}=\frac{1}{2}\int\limits_{0}^{L}\int\limits_{0}^{2\pi R}\int\limits_{- \frac{h}{2}}^{\frac{h}{2}}\Big{(}k_{w}{w_{0}}^{2}+k_{S}\bigg{(}\bigg{(}\frac{ \partial w_{0}}{\partial x}\bigg{)}^{2}+\bigg{(}\frac{\partial w_{0}}{ \partial y}\bigg{)}^{2}\bigg{)}+F_{a}(t)\bigg{(}\frac{\partial w_{0}}{\partial x }\bigg{)}^{2}\bigg{)}dzdydx, \tag{10}\]  
where \(k_{w}\) and \(k_{g}\) refer to the Winkler foundation stiffness and shear layer stiffness of the elastic foundation, respectively. Consistently with the [PERSON]'s principle, the following governing equations of motion for the BD-FG cylindrical shell are determined  
\[\int\limits_{t_{1}}^{t_{2}}(\delta\Pi_{T}-\delta\Pi_{S}-\delta\Pi_{E})dt=0, \tag{11}\]  
where the symbol \(\delta\) denotes the variation of the energy quantities. By combination of Equations (8)-(10) and Equation (11), after integration by parts, we get the following governing equations of motion, under the assumption of a null value for \(u_{0}\), \(v_{0}\), \(w_{0}\), \(\varphi_{x}\), and \(\varphi_{y}\).