## 2 Functionally Graded Porous GPLRC Cylindrical Panel  
Figure 1a represents a cylindrical panel in which GPLs are distributed with a specific distribution pattern through the thickness. Cartesian coordinates \((x,y,z)\) are introduced on the corner of the neutral surface \(\varpi\) of the panel with the \(x\)-axis along the cylinder axis and the \(z\)-axis through the thickness. The radial distance between the neutral surface and the mid-surface is denoted by \(e\)[42]. The geometry of cylindrical panel is characterized by length \(L\), radius \(R\), sub-tended angle \(\theta_{0}\), and uniform thickness \(h\). Then, the material domain \(\Omega\in\Re^{3}\) can be expressed by \(\Omega=\varpi\times[-h/2,h/2]\). Four thickness-wise functional distribution patterns of GPLs are depicted in Figure 1b--FG-U, FG-O, FG-X, and FG-\(\Lambda\)--where the GPLs are uniform in FG-U, biased towards the neutral surface in FG-O, concentrated in the top and bottom regions in FG-X, and biased towards the bottom in FG-\(\Lambda\). In addition, pores and a central crack are included within the panel, as will be described below. The thickness-wise volume fractions of the GPLs and the underlying matrix are denoted by \(f_{GPL}(z)\) and \(f_{m}(z)\). Then, either one is enough to identify both volume fractions as two volume fractions satisfy the physical constraint given by  
\[f_{GPL}(z)+f_{m}(z)=1 \tag{1}\]The GPL volume fraction \(f_{GPL}(z)\) is chosen in this study and its mathematical expression becomes  
\[f_{GPL}(z)= \begin{cases}V_{GPL}^{*},&\text{FG}-\text{U}\\ 2(1-2|z|/h)V_{GPL}^{*},&\text{FG}-\text{O}\\ 2(2|z|/h)V_{GPL}^{*},&\text{FG}-\text{X}\\ (1-2z/h)V_{GPL}^{*},&\text{FG}-\Lambda\end{cases}  \tag{2}\]  
depending on the GPL distribution pattern, where the total GPL volume fraction \(V_{GPL}^{*}\) is calculated by  
\[V_{GPL}^{*}=\frac{g_{GPL}}{g_{GPL}+\rho_{GPL}(1-g_{GPL})/\rho_{m}} \tag{3}\]  
using the GPL mass fraction \(g_{GPL}^{*}\), and the two densities \(\rho_{GPL}\) and \(\rho_{m}\) of the GPL and matrix material. GPLs are modeled as a rectangular solid with length \(l_{GPL}\), width \(w_{GPL}\), and thickness \(t_{GPL}\), and their distribution is assumed to be uniform within the underlying matrix in _the local sense_. So, the graphene-reinforced composites are usually considered isotropic and their effective Young's modulus \(E_{\text{C}}\) is evaluated using the Halphin-Tsai approach [20]:  
\[E_{\text{C}}=\frac{3}{8}\cdot\frac{1+\xi_{L}\eta_{L}f_{GPL}}{1-\eta_{L}f_{GPL} }E_{m}+\frac{5}{8}\cdot\frac{1+\xi_{T}\eta_{T}f_{GPL}}{1-\eta_{T}f_{GPL}}E_{m} \tag{4}\]  
with  
\[\eta_{L}=\frac{E_{GPL}-E_{m}}{E_{GPL}+\xi_{L}E_{m}},\;\eta_{T}=\frac{E_{GPL}-E _{m}}{E_{GPL}+\xi_{T}E_{m}} \tag{5}\]  
in which \(E_{GPL}\) and \(E_{m}\) are the elastic moduli of GPLs and matrix material, and the relative geometry ratios \(\xi_{L}\) and \(\xi_{T}\) are defined by  
\[\xi_{L}=\frac{2l_{GPL}}{t_{GPL}},\;\xi_{T}=\frac{2w_{GPL}}{t_{GPL}} \tag{6}\]  
In a similar manner, the effective values of mass density \(\rho_{\text{C}}\) and Poisson's ratio \(\
u_{\text{C}}\) of GPLRC are determined by  
\[\rho_{\text{C}}=V_{GPL}\rho_{GPL}+V_{m}\rho_{m} \tag{7}\]  
\[\
u_{\text{C}}=V_{GPL}\
u_{GPL}+V_{m}\
u_{m} \tag{8}\]  
using the simplest linear rule of mixture. Figure 1: A cylindrical panel reinforced with graphene platelets: (**a**) geometry and dimensions, and (**b**) GPL distribution patterns. The porosity within GPLRC is generally characterized by size, shape, orientation, and dispersion structure of pores. In this study, the former three factors are assumed to be uniform in space, but only the relative volume of pore and the dispersion structure are considered for the parametric investigation of pores.