## 2 Theoretical Formulation
### Material Properties
The influence of the material properties of open-cell metal foams [62; 63; 64] is defined mathematically by Equation (5), from which we can derive the Relation (6) between \(e_{0}\) and \(e_{m}\), as follows  
\[\frac{E(z)}{E^{*}}=\left(\frac{\rho(z)}{\rho^{*}}\right)^{2} \tag{5}\]  
\[\left\{ \begin{array}{c}1-e_{m}\cos\left(\frac{\pi}{2}-\frac{\pi x}{h}\right) =\sqrt{1-e_{0}\left(\frac{\pi}{2}-\frac{\pi x}{h}\right)}\\ 1-e_{m}^{*}\left(1-\cos\left(\frac{\pi}{2}-\frac{\pi x}{h}\right)\right)= \sqrt{1-e_{0}^{*}\left(1-\left(\frac{\pi}{2}-\frac{\pi x}{h}\right)\right)}\\ \Theta^{\prime}=\sqrt{\Theta}\end{array} \right. \tag{6}\]  
The mass of FG-GPL truncated cone is assumed to be identical, such that  
\[\left\{ \begin{array}{c}\int\limits_{0}^{h/2}\sqrt{1-e_{0}^{*}\left(1-\cos \left(\frac{\pi}{2}-\frac{\pi x}{h}\right)\right)}dx=\int\limits_{0}^{h/2} \sqrt{1-e_{0}\cos\left(\frac{\pi}{2}-\frac{\pi x}{h}\right)}dx\\ \int\limits_{0}^{h/2}\sqrt{\Theta}dx=\int\limits_{0}^{h/2}\sqrt{1-e_{0}\cos \left(\frac{\pi}{2}-\frac{\pi x}{h}\right)}dx\end{array} \right. \tag{7}\]  
here employed to determine \(e_{0}^{*}\) and \(\Theta\) for a known value of \(e_{0}\). It can be observed that \(e_{0}^{*}\) rises by enhancing \(e_{0}\). When \(e_{0}\) reaches \(0.6\), \(e_{0}^{*}\) (=0.9612) is near to the upper bound. This justifies the selection of \(e_{0}\in[0,\,0.6]\), as applied hereafter. Based on a Halpin-Tsai micromechanics model [65], the elasticity modulus for the nanocomposite without porosities \(E^{*}\) is defined as  
\[E^{*}=\frac{3}{8}\left(\frac{1+\epsilon_{L}^{GPL}\eta_{L}^{GPL}V_{GPL}}{1- \eta_{L}^{GPL}V_{GPL}}\right)E_{m}+\frac{5}{8}\left(\frac{1+\epsilon_{W}^{GPL} \eta_{W}^{GPL}V_{GPL}}{1-\eta_{W}^{GPL}V_{GPL}}\right) \tag{8}\]\[\epsilon_{L}^{GPL}=\frac{2l_{GPL}}{t_{GPL}} \tag{9}\] \[\epsilon_{w}^{GPL}=\frac{2w_{GPL}}{t_{GPL}}\] (10) \[\eta_{L}^{GPL}=\frac{E_{GPL}-E_{m}}{E_{GPL}+\epsilon_{L}^{GPL}E_{m}}\] (11) \[\eta_{w}^{GPL}=\frac{E_{GPL}-E_{m}}{E_{GPL}+\epsilon_{W}^{GPL}E_{m}} \tag{12}\]  
where \(E_{GPL}\) and \(E_{m}\) refer to the elasticity modulus of GPLs and metallic matrix, respectively, \(l_{GPL}\), \(w_{GPL}\), and \(t_{GPL}\) refer to the length, width, and thickness of nanofiller platelets, and \(V_{GPL}\) is the volume content of GPLs. The rule of mixture [66] is used to compute the mass density, Poisson's ratio, heat capacitance, heat conductivity coefficient, and heat expansion coefficient of the nanocomposite, namely,  
\[\rho^{*}=\rho_{GPL}V_{GPL}+\rho_{m}(1-V_{GPL}) \tag{13}\] \[v^{*}=v_{GPL}V_{GPL}+v_{m}(1-V_{GPL})\] (14) \[c^{*}=c_{GPL}V_{GPL}+c_{m}(1-V_{GPL})\] (15) \[k^{*}=k_{GPL}V_{GPL}+k_{m}(1-V_{GPL})\] (16) \[\alpha^{*}=\alpha_{GPL}V_{GPL}+\alpha_{m}(1-V_{GPL}) \tag{17}\]  
where \(\rho_{GPL},\rho_{m},v_{GPL},v_{m},c_{GPL},c_{m},k_{GPL},k_{m}\), \(\alpha_{GPL},\alpha_{m}\) stand for the mass density of GPLs, mass density of metal matrix, Poisson's ratio of GPLs, Poisson's ratio of metal matrix, specific heat capacity of GPLs, specific heat capacity of metal matrix, heat conduction coefficient of GPLs, heat conduction coefficient of metal matrix, thermal expansion coefficient of GPLs, thermal expansion coefficient of metal matrix, respectively. Moreover, the Poisson's ratio is kept constant for open-cell metal foams [67]. The shear modulus \(G^{*}\) of the nanocomposite reads as follows  
\[\epsilon_{L}^{GPL}=\frac{2l_{GPL}}{t_{GPL}} \tag{9}\]0  
The volume content of GPLs, \(V_{GPL}\), changes across the cone thickness for various dispersion patterns, as follows  
\[V_{GPL}(z)=\left\{ \begin{array}{ll}t_{i1}[1-\cos\left(\frac{\pi}{2}-\frac{ \pi x}{h}\right)]&\quad GPL\;X\\ t_{i2}[1-\cos\left(\frac{\pi x}{2h}\right)]&\quad GPL\;A\\ t_{i3}&\quad GPL\;Ud\\ t_{i4}[\cos\left(\frac{\pi x}{2h}\right)]&\quad GPL\;V\\ t_{i5}[\sin\left(\frac{\pi x}{h}\right)]&\quad GPL\;O\\ \end{array} \right\} \tag{19}\]  
where \(t_{i1}\), \(t_{i2}\), and \(t_{i3}\) denote the top limit of the \(V_{GPL}\), and \(i=1\), \(2\), \(3\) corresponding to porosity distributions 1, 2, and 3, respectively.