## 4 Results and Discussion
### Transient and Steady Responses  
The study continuous with the analysis of the transient thermal response of a FG-GPL porous rotating truncated cone, with material properties in Table 1 for both GPLs and metal matrix. As plotted in Figure 3, a systematic study checks for the influence of the GPL pattern on the thermal time history (Figure 3a), radial displacement (Figure 3b), and stress state (Figure 3c-e) for the centered point of the selected truncated cone. Hereafter, the stress state \([\sigma]^{T}=[\sigma_{rr}~{}~{}\sigma_{\theta\theta}~{}\sigma_{zz}~{}~{}\tau_{ z}]^{T}\) will be labeled as \([\sigma]^{T}=[\sigma,~{}~{}\sigma_{1}~{}\sigma_{z}~{}~{}\tau]^{T}\) for simplicity reasons. Based on the plots in Figure 3, the effect of the GPL pattern is much more pronounced on the transient thermal and kinematic response compared to the stress one. More specifically, it is worth observing that the temperature response maintains almost the same for a GPLUD and GPLV distribution. At the same time, GPLUD, GPLV, and GPLO patterns yield the same effect on the radial stress response. Besides, the effect of GPLUD and GPLV patterns on the other stress components seems almost the same. The results also denote that the maximum and minimum compressive axial and radial stress is reached for a GPLA and GPLO, respectively. In addition, the maximum and minimum steady state of radial, axial, and shear stresses belongs to a GPLO and GPLA distribution, respectively, while the maximum and minimum steady state of tangential stress and radial displacement belongs to a GPLO and GPLX distribution. Figure 4 indicates the influence of porosity distribution on the time history of temperature (Figure 4a), radial displacement (Figure 4b),  
Figure 2: Time history of temperature (**a**), radial displacement (**b**), and radial stress (**c**) for a center point of cone compared with ANSYS Workbench. and stresses (Figure 4c-e) for the central point of the truncated cone. The results show that the temperature and displacement field will be steady sooner for D1, and the maximum and minimum magnitude of steady state temperature and displacement belongs to D1 and D3, respectively. It is interesting to note that the distribution of porosity has a meaningful effect on the radial and tangential stress time histories, which, in turn, show a relaxation for a D1 porosity, since they change from a compressive to a traction state. Figure 3: Influence of GPL pattern on the transient temperature (**a**), radial displacement (**b**), and stress state (**c**-**f**) for a center point of truncated cone (D1; \(e_{0}=0.4\); \(\phi=30^{\circ}\); \(\gamma_{CPL}=0.01\); \(\omega=100\) rad/s). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Material Property** & **E (GPa)** & \(\rho\) [\(\frac{\text{Kg}}{m^{2}}\)] & \(v\) & \(k\) [\(\frac{W}{m^{2}}\)] & \(\alpha\) [\(k^{-1}\)] & \(C\) [\(\frac{j}{Kg}\)] \\ \hline GPL & 1.01 TPa & 1062.5 & 0.186 & 3000 & \(2.35\times 10^{-5}\) & 644 \\ Metal & 130 GPa & 8960 & 0.34 & 250 & \(8.2\times 10^{-5}\) & 896 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material property of metal matrix and GPLs. Figure 4: Influence of porosity distribution on the transient temperature (**a**), radial displacement (**b**), and stress state (**c**–**f**) for center point of truncated cone (GPL V; \(\varepsilon_{0}=0.4\); \(\phi=30^{\circ}\); \(\gamma_{GPL}=0.01\); \(\omega=100\) rad/s). In Figure 5 we also plot the geometrical effect of a semi-vertex angle on the time history of temperature (Figure 5a), radial displacement (Figure 5b), and stresses (Figure 5c-e) at the central point of the truncated cone. It can be seen that, by increasing a semi-vertex angle, the cone behavior will be steady sooner. Figure 6 also shows the effect of the porosity coefficient on the same time histories for the same central point of the selected structure. It is worth observing that an increased porosity coefficient yields an increased steady state radial displacement and stress field.