## 2 Governing Equations of the Problem  
Let us consider a BD-FG porous cylindrical shell embedded in an elastic foundation with thickness \(h\), radius \(R\), and length \(L\), where two different porosity distributions of the constituent phases are accounted for the analysis, namely an even and an uneven distribution, see Figures 1 and 2. We assume a BD-FG material made of a metal (labeled as \(m\)) and a ceramic (labeled as \(c\)) in the inner and outer shell surfaces, respectively. While the material properties for a conventional FG model vary continuously along the thickness direction from a ceramic or metal to another one, the basic material properties selected herein, vary also along the shell length from the metal to the ceramic phase. To this end, the volume fractions of the ceramic and metal phases are defined as follows  
\[V_{c}(x,z)=\left(\frac{1}{2}+\frac{z}{h}\right)^{n_{z}}\left(\frac{x}{L} \right)^{n_{x}},V_{m}(x,z)=1-V_{c}(x,z), \tag{1}\]  
where \(n_{z}\) and \(n_{x}\) refer to the non-negative volume fraction exponents defining the profile variation of the material properties along the shell thickness and length directions, respectively. In addition, \(z\) and \(x\) stand for the radial distance from the mid-plane and longitudinal distance from the origin of the BD-FG cylindrical shell, respectively. The effective material properties (i.e., Yong's modulus, density and Poisson's ratio) of the BD-FG porous cylindrical shell are assumed to change according to a modified power law model with a linear algebraic combination of volume fractions of two basic materials. Two types of BD-FG material models include both even and/or uneven porosity distributions, i.e.,  
\[E(x,z) =E_{m}+(E_{c}-E_{m})\left(\frac{1}{2}+\frac{z}{h}\right)^{n_{z}} \left(\frac{x}{L}\right)^{n_{x}}-\frac{\zeta}{2}(  
evenly distributed across the radial direction, the porosities in an uneven model is mostly concentrated in the shell mid-plane. It is worth noting that the uneven porosity distribution is linearly reduced from a larger value at mid-plane to a smaller value at the top and bottom sides of the structure. In what follows, we apply the TSDT, such that the displacement field of an arbitrary point of the shell along the \(x\), \(y\), and \(z\) axes, is defined as  
\[ \begin{split} u(x,y,z,t)&=u_{0}(x,y,t)+z\varphi_{x }(x,y,t)-\overline{c}z^{3}\bigg{(}\varphi_{x}(x,y,t)+\frac{\partial u_{0}(x,y, t)}{\partial x}\bigg{)},\\ v(x,y,z,t)&=v_{0}(x,y,t)+z\varphi_{y}(x,y,t)- \overline{c}z^{3}\bigg{(}\varphi_{y}(x,y,t)+\frac{\partial u_{0}(x,y,t)}{ \partial y}\bigg{)},\\ w(x,y,z,t)&=w_{0}(x,y,t),\end{split}  \tag{4}\]  
where \(\overline{c}=\frac{4}{3h^{2}}\), and \(u\), \(v\), \(w\) stand for the longitudinal, circumferential, and transverse (radial) displacement components, respectively. These are determined, in turn, by means of the kinematic quantities \(u_{0}\), \(v_{0}\), and \(w_{0}\) at the middle surface, and the rotations \(\varphi_{x}\) and \(\varphi_{y}\) of a transverse normal section about the \(x\) and \(y\) axis, respectively. According to the TSDT, the strain components of the cylindrical shell can be written as [48]  
\[\varepsilon_{xx}=\frac{\partial u_{0}(x,y,t)}{\partial x}+z\frac{\partial \varphi_{x}(x,y,t)}{\partial x}-\overline{c}z^{3}\bigg{(}\frac{\partial \varphi_{x}(x,y,t)}{\partial x}+\frac{\partial^{2}w_{0}(x,y,t)}{\partial x^{ 2}}\bigg{)}, \tag{5a}\]  
Figure 1: Geometrical scheme of a bi-directional (BD)-functionally graded (FG) porous cylindrical shell embedded in an elastic foundation. Figure 2: Distribution of porosity: (**a**) even porosity, (**b**) uneven porosity.