## 3 Numerical Results
### Time Responses of Results  
Figure 2 presents the time responses of w(\(a/2\),\(b/2\)) and \(\sigma_{x}\) for \(t=0.1\) s-3.0 s under the TSDT with \(c_{1}=0.925925/\)mm\({}^{2}\), advanced \(k_{\alpha}\), \(L/h^{*}=5\), \(h_{3}=0.1\) mm, \(k_{c}c(t)=0\), \(R_{n}=1\), \(T=600\) K, and \(\overline{T}_{1}=100\) K. The w(\(a/2\),\(b/2\)) vs. \(t\) of \(\theta_{\rm L}=30^{\circ}\) and \(60^{\circ}\) are presented. The greatest value at \(t=0.1\) s for w(\(a/2\),\(b/2\)) is \(-9.459201\) mm under \(\theta_{\rm L}=30^{\circ}\). The values of  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(l_{3}\) (mm)} & \multirow{2}{*}{\(R_{n}\): 0.1} & \multirow{2}{*}{0.2} & \multirow{2}{*}{0.5} & \multirow{2}{*}{1} & \multirow{2}{*}{2} & \multirow{2}{*}{5} & \multirow{2}{*}{10} \\ \cline{2-2} \cline{6-8}  & & & & & & & & \\ \hline
0.1 & 0.037104 & 0.039685 & 0.049168 & 0.069195 & 0.117301 & 0.245510 & 0.334065 \\ \hline
0.2 & 0.041771 & 0.045295 & 0.058288 & 0.086731 & 0.161186 & 0.384678 & 0.521903 \\ \hline
0.3 & 0.049420 & 0.054397 & 0.072819 & 0.114345 & 0.229530 & 0.566865 & 0.694518 \\ \hline
0.4 & 0.062453 & 0.069776 & 0.096835 & 0.158129 & 0.323746 & 0.677580 & 0.722486 \\ \hline
0.5 & 0.084918 & 0.095833 & 0.134965 & 0.216903 & 0.390003 & 0.571030 & 0.560417 \\ \hline
0.6 & 0.120294 & 0.134674 & 0.180651 & 0.254273 & 0.344935 & 0.381031 & 0.368428 \\ \hline
0.7 & 0.157532 & 0.169341 & 0.199188 & 0.230172 & 0.251058 & 0.251351 & 0.245202 \\ \hline
0.8 & 0.167441 & 0.171462 & 0.179415 & 0.184980 & 0.186852 & 0.184625 & 0.182127 \\ \hline
0.9 & 0.155595 & 0.155993 & 0.156575 & 0.156684 & 0.156272 & 0.155241 & 0.154367 \\ \hline
1.0 & 0.147206 & 0.147117 & 0.146916 & 0.146709 & 0.146496 & 0.146213 & 0.145987 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Advanced nonlinear \(k_{\alpha}\) vs. \(R_{n}\), \(h_{3}\) and \(T=100\) K.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(L/h^{*}\)} & \multirow{2}{*}{\(\theta_{\rm L}\)} & \multirow{2}{*}{\(\mathbf{Grids}\)} & \multirow{2}{*}{\(\mathbf{w(a/2\),\(b/2\))(mm)}} \\ \cline{3-5}  & & & \(N\times M\) & \(R_{n}=0.5\) & \(R_{n}=1\) & \(R_{n}=2\) \\ \hline \multirow{4}{*}{5} & \multirow{4}{*}{\(30^{\circ}\)} & \(7\times 7\) & 0.039876 & 0.048409 & 0.066685 \\  & & \(9\times 9\) & 0.065312 & 0.047408 & 0.066193 \\  & & \(11\times 11\) & 0.032934 & 0.042817 & 0.058517 \\  & & \(13\times 13\) & 0.031425 & 0.043521 & 0.055847 \\  & & \(7\times 7\) & \(-0.004903\) & 0.081978 & 0.122267 \\  & & \(9\times 9\) & 0.009086 & 0.019592 & 0.031389 \\  & & \(11\times 11\) & 0.015974 & 0.023372 & 0.032928 \\  & & \(13\times 13\) & 0.016464 & 0.022055 & 0.030752 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Convergence in magnetostrictive FGM plates/cylindrical shells for advanced \(k_{\alpha}\) at \(T=6\) s.  
w(\(a/2\),\(b/2\)) rapidly converge to 0.108490 mm for \(\theta_{\rm L}=30^{\circ}\) and converge to 0.066303 mm for \(\theta_{\rm L}=60^{\circ}\). Also, the \(\sigma_{x}\) versus \(t\) on \(z=0.5h^{*}\) are presented. The greatest value of \(\sigma_{x}\) is \(-1.3909\times 10^{-3}\) GPa, found at \(t=0.1\) s under \(\theta_{\rm L}=30^{\circ}\) Values of \(\sigma_{x}\) have small oscillation and converge to \(-7.9112\times 10^{-4}\) GPa for \(\theta_{\rm L}=30^{\circ}\) and to \(-8.1968\times 10^{-4}\) GPa for \(\theta_{\rm L}=60^{\circ}\).