## 2 Theoretical Formulation
### Material Properties  
Three different porosity distributions are assumed throughout the cone thickness (Figure 1), i.e., two types of non-uniform symmetric distribution and a uniform one. In the first distribution (labeled as I-D1), the porosity is nonlinearly symmetric, with a higher density around the mid-plane rather than the inner and outer surfaces of the cone. In the second distribution (labeled as II-D2), a non-uniform symmetric porosity is assumed, with a higher density near the inner and outer surfaces rather than the mid-plane. For such two cases, the material properties with porosities are defined in Equations (1) and (2), respectively, whereas for a uniform distribution of porosity (labeled as D3) the material properties read as in Equation (3). At the same time, five different GPL patterns are assumed along the cone thickness, as represented in Figure 1 and defined in [43; 60]. More specifically:  
Porosity distribution 1 (Non-uniform symmetric I-D1)  
\[ \begin{split}\alpha(x)&=\alpha^{*}[1-e_{0}\cos( \pi/2-\pi x/h)]\\ c(x)&=c^{*}[1-e_{0}\cos(\pi/2-\pi x/h)]\\ k(x)&=k^{*}[1-e_{0}\cos(\pi/2-\pi x/h)]\\ E(x)&=E^{*}[1-e_{0}\cos(\pi/2-\pi x/h)]\\ G(x)&=G^{*}[1-e_{0}\cos(\pi/2-\pi x/h)]\\ \rho(x)&=\rho^{*}[1-e_{m}\cos(\pi/2-\pi x/h)]\\ \end{split}  \tag{1}\]  
Porosity distribution 2 (Non-uniform symmetric II-D2)  
\[ \begin{split}\alpha(x)&=\alpha^{*}[1-e_{0}^{*}(1- \cos(\pi/2-\pi x/h))]\\ c(x)&=c^{*}[1-e_{0}^{*}(1-\cos(\pi/2-\pi x/h))]\\ k(x)&=k^{*}[1-e_{0}^{*}(1-\cos(\pi/2-\pi x/h))]\\ E(x)&=E^{*}[1-e_{0}^{*}(1-\cos(\pi/2-\pi x/h))]\\ G(x)&=G^{*}[1-e_{0}^{*}(1-\cos(\pi/2-\pi x/h))]\\ \rho(x)&=\rho^{*}[1-e_{0}^{*}(1-\cos(\pi/2-\pi x/h))] \\ \rho(x)&=\rho^{*}[1-e_{0}^{*}(1-\cos(\pi/2-\pi x/h))] \\ \end{split}  \tag{2}\]  
Figure 1: Geometry of rotating truncated cone for various porosity and GPL patterns across the x axis (thickness direction). Uniform porosity distribution (D3)  
\[ \begin{array}{l}\alpha(z)=\alpha^{*}\Theta\\ c(z)=c^{*}\Theta\\ k(z)=k^{*}\Theta\\ E(z)=E^{*}\Theta\\ G(z)=G^{*}\Theta\\ \rho(z)=\rho^{*}\Theta^{\prime}\end{array}  \tag{3}\]  
The following equations are also applied for evaluating \(x\)[61]  
\[ \begin{array}{l}x=\sqrt{\left(r_{n}-r_{in}\right)^{2}+\left(z_{n}-z_{in} \right)^{2}}\\ r_{in}=\frac{\left(\tan(\pi-\varphi)r_{n}-z_{n}-a\tan\left(\frac{\pi}{2}- \varphi\right)\right)}{\tan(\pi-\varphi)-\tan\left(\frac{\pi}{2}-\varphi \right)}\\ z_{in}=\tan\left(\frac{\pi}{2}-\varphi\right)\left(r_{in}-a\right)\end{array}  \tag{4}\]  
when \(\varphi=0\) (cylinder), it is \(x=r_{n}-a\), \(h=b-a\). In the previous relations, \(x\) is the normal space of points through the thickness of the cone from the inner surface; \(r_{n}\) and \(z_{n}\) refer to the radius and height of an arbitrary point in the domain; \(r_{in}\) and \(z_{in}\) refer to the radius and height of points at the inner surface changing from \(a\) to \((a+L\tan\varphi)\) and from \(0\) to \(L\), respectively. Moreover, \(E(x)\), \(G(x)\), \(\rho(x)\), \(c(x)\), \(\alpha(x)\), \(k(x)\) stand for the modulus of elasticity, shear stiffness, mass density, specific heat capacity, thermal expansion coefficient, and heat conduction coefficient of the porous nanocomposite cone, respectively. At the same time, \(E^{*}\), \(G^{*}\), \(\rho^{*}\), \(c^{*}\), \(\alpha^{*}\), \(k^{*}\) are the similar properties of GPL cone without any porosity. Furthermore, \(e_{0}\) and \(e_{0}^{*}\) (\(0\leq e_{0}(e_{0}^{*})<1\)) are the coefficients of porosity belonging to D1 and D2, respectively; \(e_{m}\) and \(e_{m}^{*}\) stand for the coefficients of mass density related to D1 and D2, respectively; \(\Theta\) and \(\Theta^{\prime}\) stand for the variables of a uniform porosity distribution. As the size and density of the internal cavities increase, the porosity increases, leading to a reduction in the material properties.