# Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core
## 4 Numerical Results
### Verification
Table 3 presents the comparison for the (2-2-1) FG sandwich with a softcore (i.e., \(\mathcal{LA}\) layer). The relative error uniformly decreased in proportion to the width-to-thickness ratio, \(b/h\), except for the volume fraction, \(n=0\), which demonstrated the reverse trend. Here, the (2-2-1) FG sandwich resembled a pure homogeneous plate of \(\mathcal{LA}\) when \(n\) is 0. Regarding the volume fraction, \(n\), the largest errors occurred at \(n=1\) for \(b/h=5\), \(n=5\) for \(b/h=10\), and \(n=10\) for \(b/h=100\). Thus, the present method using the (1,1,0) hierarchical model was in good agreement with the FSDT, such that the overall relative error ranged from 0.895% to 5.843%, with a relatively coarse NEM grid. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Constituents** & **Young’s Modulus \(E\) (GPa)** & **Poisson’s Ratio \(\
u\)** & **Density \(\rho\) (kg/m\({}^{3}\))** \\ \hline Metal & \(\mathcal{LA}\) & 70 & 0.3 & 2707 \\ Ceramic & \(\mathcal{LA}\)\(\mathcal{LA}\)\(\mathcal{LA}\)\(\mathcal{LA}\)\(\mathcal{LA}\) & 380 & 0.3 & 3800 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of metallic and ceramic constituents. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Hierarchical Models**} & \multicolumn{6}{c}{**Width-to-Thickness Ratio (b/h)**} \\ \cline{2-7}  & **3** & **5** & **10** & **30** & **100** & **500** \\ \hline (1,1,0) & 1.18979 & 1.44776 & 1.61397 & 1.69192 & 1.70425 & 1.70550 \\  & (28.350\%) & (13.687\%) & (6.604\%) & (3.167\%) & (2.808\%) & (2.773\%) \\ (1,1,2) & 1.15453 & 1.43293 & 1.60610 & 1.68401 & 1.69614 & 1.69736 \\  & (24.546\%) & (12.514\%) & (6.084\%) & (2.685\%) & (2.319\%) & (2.283\%) \\ (3,3,2) & 1.03051 & 1.33049 & 1.54583 & 1.64808 & 1.66357 & 1.66513 \\  & (11.167\%) & (4.470\%) & (2.103\%) & (0.494\%) & (0.354\%) & (0.340\%) \\ (3,3,4) & 1.00808 & 1.32349 & 1.54501 & 1.64804 & 1.66355 & 1.66510 \\  & (8.748\%) & (3.921\%) & (2.049\%) & (0.491\%) & (0.353\%) & (0.339\%) \\ (9,9,8) & 0.96431 & 1.29846 & 1.53366 & 1.64174 & 1.65786 & 1.65948 \\  & (4.026\%) & (1.955\%) & (1.299\%) & (0.107\%) & (0.001\%) & (0.0\%) \\
3-D elasticity & 0.92699 & 1.27356 & 1.51399 & 1.63998 & 1.65770 & 1.65948 \\ \hline \hline \end{tabular}  
* The values in parentheses are the relative percentage differences with respect to 3-D elasticity.  
\end{table} 
Table 2: _The model limits_ of the calibrated fundamental frequencies, \(\overline{\omega}\), of hierarchical models for simply supported, square FG sandwich plates with homogeneous softcores (2-2-1, \(n=1\) ). Figure 5: Variation of the calibrated fundamental frequency, \(\overline{\omega}\), with respect to the width-to-thickness ratio \(b/h\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**b/h**} & \multirow{2}{*}{**Method**} & \multicolumn{6}{c}{**\(n\)**} \\ \cline{3-6}  & & **0** & **0.5** & **1** & **5** & **10** \\ \hline \multirow{2}{*}{5} & Bennoun [10] & 0.85405 & 1.25633 & 1.36784 & 1.52684 & 1.55459 \\  & Present & 0.86375 & 1.31196 & 1.44776 & 1.61419 & 1.63609 \\  & (Relative error) & (1.136\%) & (4.428\%) & (5.843\%) & (5.721\%) & (5.243\%) \\ \hline \multirow{2}{*}{10} & Bennoun [10] & 0.92984 & 1.44124 & 1.58470 & 1.75986 & 1.78200 \\  & Present & 0.94416 & 1.45987 & 1.61397 & 1.79491 & 1.81655 \\  & (Relative error) & (1.540\%) & (1.293\%) & (1.847\%) & (1.993\%) & (1.939\%) \\ \hline \multirow{2}{*}{100} & Bennoun [10] & 0.96099 & 1.52702 & 1.68761 & 1.86908 & 1.88725 \\  & Present & 0.99343 & 1.54069 & 1.70425 & 1.89418 & 1.91633 \\ (Relative error) & (3.376\%) & (0.895\%) & (0.986\%) & (1.343\%) & (1.541\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of calibrated fundamental frequencies, \(\overline{\omega}\), for simply supported square FG sandwich plates with homogeneous softcores (2-2-1). case.