## 4 Numerical Results and Discussion  
For numerical illustration, a simply supported viscoelastic FGM plate attached to piezoelectric layers is considered. The thermo-mechanical properties of the piezoelectric layers are presented in Table 1[21; 38]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Elasticity Constant [\(10^{9}\)** & \(Q_{11}\) & \(Q_{12}\) & \(Q_{13}\) & \(Q_{22}\) & \(Q_{23}\) & \(Q_{33}\) & \(Q_{44}\) & \(Q_{55}\) & \(Q_{66}\) \\ \hline Sensor (PZT-4) & \(139\) & \(78\) & \(74\) & \(139\) & \(74\) & \(115\) & \(25.6\) & \(25.6\) & \(30.5\) \\ Actuator (Ba\({}_{2}\) NaNb\({}_{5}\) O\({}_{15}\)) & \(239\) & \(104\) & \(50\) & \(274\) & \(52\) & \(135\) & \(65\) & \(66\) & \(76\) \\ \hline Piezoelectric coefficients & \(\varepsilon_{31}\) & \(\varepsilon_{32}\) & \(\varepsilon_{33}\) & \(\varepsilon_{24}\) & \(\varepsilon_{15}\) & & & \\ \hline Sensor & \(-5.2\) & \(-5.2\) & \(15.1\) & \(12.7\) & \(12.7\) & & & & \\ Actuator & \(-0.4\) & \(-0.3\) & \(4.3\) & \(3.4\) & \(2.8\) & & & \\ \hline Dielectric constants & \(\eta_{1}\) & \(\eta_{2}\) & \(\eta_{3}\) & & & & & \\ \hline Sensor & \(6.5\) & \(6.5\) & \(5.6\) & & & & & \\ \hline Actuator & \(1.96\) & \(2.01\) & \(0.28\) & & & & & \\ \hline Thermal conductivity & \(\lambda_{x}=\lambda_{y}\) & \(\lambda_{z}\) & & & & & \\ \hline Sensor & \(2.1\) & \(3.15\) & & & & & \\ Actuator & \(8.6\) & \(12.9\) & & & & & \\ \hline Thermal expansion & \(\alpha_{x}=\alpha_{y}\) & \(\alpha_{z}\) & & & & & \\ \hline Sensor & \(1.97\) & \(2.62\) & & & & & \\ \hline Actuator & \(4.39\) & \(2.45\) & & & & & \\ \hline Piezoelectric modulus & \(d_{1}\) & \(p_{3}\) & & & & & \\ pyroelectric constant: & \(d_{1}\) & \(p_{3}\) & & & & & \\ \hline Sensor & \(-3.92\) & \(5.4\) & & & & & \\ \hline Actuator & \(-3.92\) & \(5.4\) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the piezoelectric sensor and actuator layers [21; 38]. The relaxation time constant, Poisson's ratio, and the parameters for the Prony series are [40]:  
\[E_{1m}=70\,GPa,\;E_{2m}=105\,GPa,\;E_{1c}=380\,GPa,\;E_{2c}=570\,GPa,\; \alpha_{c}=8.4\times 10^{-6}\frac{1}{k},\] \[\alpha_{m}=22.2\times 10^{-6}\frac{1}{\tau_{K}},\lambda_{m}=205 \frac{W}{m^{\prime}\,K},\;\lambda_{c}=35\frac{W}{m^{\prime}\,K},\tau=5s,\;v= 0.3\;.\]  
It is worth noting that all the presented results are the maximum values within the domain of the plate that are computed at \(t=2\) s. The geometrical dimensions of the plate and the bottom and top surface temperature of the plate are introduced as:  
\[a/b=1.5,\;a/h=10,\;h/h_{p}=25,\;V_{0}=0,\;T_{i}=400\,K,\;\text{and}\;T_{o}=600\,K.\]  
At first, a convergence study for the proposed analytical solution is conducted for normal transverse stress, deflection, and latitudinal displacement along the thickness direction for different values of half-wave numbers (\(m,\;n\)) and plotted in Figure 2a-c. As the figures show, transverse quantities are more affected by the half-wave numbers. Additionally, the figures indicate that by increasing the half-wave numbers up to \(m=n=11\), transverse normal stress, deflection, and latitudinal displacement perfectly converge to their corresponding constant values. This matter clearly shows the high convergence rate of the Fourier series technique. In other words, this shows the optimum pace of convergence for the applied solution, and it means that this method requires comparatively lower computational effort, which is highly essential for time-variant analysis of the thermoelastic response of the sandwich structure. Figure 2: _Cont._  
Figure 2: Through-thickness distribution of transverse normal stress, latitudinal displacement, and deflection of plate for different half wave numbers (\(m\), \(n\)).