## 4 Numerical Results and Discussion
\[E_{1m}=70\,GPa,\;E_{2m}=105\,GPa,\;E_{1c}=380\,GPa,\;E_{2c}=570\,GPa,\; \alpha_{c}=8.4\times 10^{-6}\frac{1}{k},\]0 Effect of piezoelectric thickness variation on through-thickness distribution of dimensionless transverse normal stress, in-plane shear stress, and longitudinal displacement. \[E_{1m}=70\,GPa,\;E_{2m}=105\,GPa,\;E_{1c}=380\,GPa,\;E_{2c}=570\,GPa,\; \alpha_{c}=8.4\times 10^{-6}\frac{1}{k},\]1 \(Cont.\)  
The time history of the stress and displacement components for different relaxation time constants \(\tau\) is depicted in Figure 7a-c. The figures show that stress and displacement increase with time regardless of the value of \(\tau\) and finally converge to their steady-state constant values. Moreover, it is seen that for all quantities for which their time history is  
\[E_{1m}=70\,GPa,\;E_{2m}=105\,GPa,\;E_{1c}=380\,GPa,\;E_{2c}=570\,GPa,\; \alpha_{c}=8.4\times 10^{-6}\frac{1}{k},\]2 Effect of time constant variation, \(\tau\), on the transverse normal and shear stresses, as well as latitudinal displacement and transverse displacements. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(T_{o}\) & \multicolumn{5}{c}{\(z\)/\(h\)} \\ \cline{3-8}  & \(0\) & \(0\) & \(0.2\) & \(0.4\) & \(0.6\) & \(0.8\) & \(1\) \\ \hline \multirow{3}{*}{\(\overline{\pi}\left(0,\frac{b}{2}\right)\)} & \(600\) & \(-20.620\) & \(-20.850\) & \(-21.100\) & \(-21.392\) & \(-21.706\) & \(-22.040\) \\ \cline{2-8}  & \(800\) & \(-15.547\) & \(-16.507\) & \(-17.509\) & \(-18.574\) & \(-19.576\) & \(-20.704\) \\ \cline{2-8}  & \(900\) & \(-13.856\) & \(-15.088\) & \(-16.403\) & \(-17.635\) & \(-18.929\) & \(-20.265\) \\ \hline \multirow{3}{*}{\(\overline{\pi}\left(\frac{b}{2},\frac{b}{2}\right)\)} & \(600\) & \(5.6785\) & \(6.0438\) & \(6.1482\) & \(6.3048\) & \(6.4092\) & \(6.5658\) \\ \cline{2-8}  & \(800\) & \(22.015\) & \(22.276\) & \(22.380\) & \(22.484\) & \(22.589\) & \(22.797\) \\ \cline{2-8}  & \(900\) & \(27.443\) & \(27.704\) & \(27.756\) & \(27.860\) & \(28.017\) & \(28.173\) \\ \hline \multirow{3}{*}{\(\overline{\sigma}_{z}\left(\frac{b}{2},\frac{b}{2}\right)\)} & \(600\) & \(0\) & \(0.4945\) & \(0.7732\) & \(0.7438\) & \(0.4281\) & \(0\) \\ \cline{2-8}  & \(800\) & \(0\) & \(0.4155\) & \(0.7031\) & \(0.7167\) & \(0.4301\) & \(0\) \\ \cline{2-8}  & \(900\) & \(0\) & \(0.3892\) & \(0.6798\) & \(0.7076\) & \(0.4307\) & \(0\) \\ \hline \multirow{3}{*}{\(\tau_{xy}(0,0)\)} & \(600\) & \(-199.79\) & \(-181.63\) & \(-239.87\) & \(-257.41\) & \(-258.66\) & \(-202.92\) \\ \cline{2-8}  & \(800\) & \(-137.79\) & \(-124.63\) & \(-187.27\) & \(-225.47\) & \(-248.64\) & \(-199.16 depicted in this presentation, the rate of convergence to the ultimate values decreases when the time constant increases. From Figure 6(c), it is seen that the influence of the relaxation time constant,\(\tau\), on the deflection is more significant compared to other quantities. \[E_{1m}=70\,GPa,\;E_{2m}=105\,GPa,\;E_{1c}=380\,GPa,\;E_{2c}=570\,GPa,\; \alpha_{c}=8.4\times 10^{-6}\frac{1}{k},\]3 Cont. Figure 8a-c demonstrate the effect of the applied voltage, \(\overline{\
abla}_{0}\), on the variation of stresses and displacement along the thickness direction. According to the figures, increasing the voltage causes stresses and deflections to be increased. Moreover, it is seen that the effect of the applied voltage near the outer region of the FGM layer is more significant, which is due to the effect of the actuator layer. From Figure 8a it is observed that the normal longitudinal stress, \(\overline{\sigma}_{x}\), is discontinuous at two interfaces due to the different elastic constants at the interfaces. Additionally, it is concluded that by moving from the lower surface to the top surface of the sandwich plate, the sign of this stress changes from tension to compression. Moreover, one can say that the maximum value of the normal longitudinal stress \(\overline{\sigma}_{x}\) has a linear-like relation with the intensity of the applied voltage.