## 4 Numerical Results
In addition, the heat wave speed increases by increasing the metal content, or equivalently with a decreasing value of \(n\). Figure 5: (**a–c**): Effect of the material gradient index \(n\) on the distribution of the thermoelastic field variables through the radial direction for the FG cylindrical panels with piezoelectric layers \(\left(h_{c}/h_{p}=5\), \(\theta_{0}=75^{\circ}\), \(\psi=0.03\), \(\theta=0.5\theta_{0}\), \(F_{0}=0.8,\tau=0.2\right)\). Figures 7 and 8 also plot the effect of the dimensionless relaxation time \(\tau\) on the thermoelastic response of FG cylindrical panels with piezoelectric layers, revealing the through-the-thickness variations and time histories of the thermoelastic field variables of a material point (\(\xi=0.5\), \(\theta=0.5\theta_{0}\)), respectively. Based on Figures 7 and 8, it is clear that the use of either a classical thermoelasticity theory (i.e., \(\tau=0\)) or the L-S theory (i.e., the case with \(\tau\
eq 0\)) leads to significant differences in the numerical results, evaluated here in terms of temperature, radial displacement, and hoop stress. Moreover, the oscillating behavior of the temperature time histories for \(\tau\
eq 0\), (i.e., for a non-Fourier heat conduction law) plotted in Figure 8a, indicates the necessity of using the L-S theory for thermal shock loading problems. Figure 6: **(a–c)**: Effect of the material gradient index \(n\) on the time histories of the thermoelastic field variables for the FG cylindrical panels with piezoelectric layers \(\left(h_{c}/h_{p}=5\), \(\theta_{0}=75^{\circ}\), \(\psi=0.03\), \(\xi=0.5\), \(\theta=0.5\theta_{0}\), \(\tau=0.2\right)\). The through-the-thickness heat wave propagation and the resulting variations in the mechanical field variables of the FG cylindrical panels with piezoelectric layers are shown in Figure 9a-c for different non-dimensional time levels \(F_{o}\). The results clearly demonstrate that the model can simulate thermoelastic wave propagation in multilayer panels. It is worth observing that the temperature and the radial displacement increase with an increasing dimensionless time \(F_{o}\). Figure 7: (**a–c**): Effect of the relaxation time on the distribution of the thermoelastic field variables through the radial direction for the FG cylindrical panels with piezoelectric layers \(\left(p=1,\,h_{\rm c}/h_{p}=5,\,\theta_{0}=75^{\circ},\,\psi=0.03,\,\theta=0.5 \theta_{0},\,F_{0}=0.8\right)\). In Figure 10, the effect of the core thickness-to-piezoelectric thickness ratio \(h_{c}/h_{p}\) on the thermoelastic behavior of FG cylindrical panels is described, where an increased value of \(h_{c}/h_{p}\) yields a general increase of all the field variables. A further systematic study considers the effect of the panel opening angle \(\theta_{0}\) on the through-the-thickness variation of the dimensionless thermoelastic field variables, as depicted in Figure 11a-c. As expected, by increasing \(\theta_{0}\), the radial displacement increases, and the hoop stress component decreases, although the maximum values of the field variables involve the same locations of the panel independently of \(\theta_{0}\). Furthermore, in Figure 12a-c, we evaluate the effect of the opening angle \(\theta_{0}\) on the time histories for all thermoelastic field variables while considering the material points \(\xi=0.5\) and \(\theta=0.5\theta_{0}\). Also visible in Figures 11a and 12a, the distribution and time histories of the temperature seem to be unaffected by a variation in \(\theta_{0}\). Figure 10: Cont. Figure 9: (**a–c**): Effects of different time levels on the distribution of the thermoelastic field variables through the radial direction \(\left(n=1,\,h_{c}/h_{p}=5,\,\theta_{0}=75^{o},\,\theta=0.50_{0},\,\psi=0.03,\, \tau=0.2\right)\). Figure 11: (**a–c**): Effect of the opening angle \(\theta_{0}\) on the distribution of the thermoelastic field variables in the radial direction (\(n=1\), \(h_{c}/l_{p}=5\), \(\theta=0.5\theta_{0}\), \(\psi=0.03\), \(F_{0}=0.8\), \(\tau=0.2\)).