## 4 Numerical Examples
### Rectangular E-FGM Plate  
Consider bending of simply supported exponentially functionally graded material (E-FGM) rectangular plates with Poisson's ratio \(\
u=0.3\) under a bi-sinusoidal transverse load. The nondimensional responses are defined as \([\overline{u},\overline{v},\overline{w}]=\frac{10\varrho_{0}h^{3}}{a^{4} \varrho_{0}}[u,v,w](z)\), \([\overline{\sigma}_{xx},\overline{\sigma}_{yy}]=\frac{h^{2}}{a^{2}\varrho_{ 0}}[\sigma_{xx},\sigma_{yy}](z)\), \(\overline{\tau}_{xy}=\frac{10h^{2}}{a^{2}\varrho_{0}}\tau_{xy}(z)\), \([\overline{\tau}_{yz},\overline{\tau}_{xz}]=\frac{h}{a\varrho_{0}}\big{[} \tau_{yz},\tau_{xz}](z)\), where \(a\) is the width, \(h\) is the thickness, \(\varrho_{0}\) is the amplitude of the load. \(E_{0}\) is the elasticity modulus on the bottom surface of the plate; however, the nondimensional responses are irrelevant to \(E_{0}\) in practice; see Zenkour [15] for more details. Using fourth-order NURBS basis functions and setting the SDF truncation numbers \(M=5\), \(N=6\) as described in Section 4.1. Assuming the gradient index \(k=0.5\), the width-to-thickness ratio \(a/h=4\), and the length-to-width ratio \(b/a\) ranging from 1 to 4, Figure 8 compares the displacement and stress results obtained using the SDF-based IGA  
Figure 5: Three-dimensional geometry and 2D NURBS midplane of a square plate. (**a**) Three-dimensional geometry; (**b**) 2D NURBS midplane. with the exact 3D elasticity solutions provided by Zenkour [16]. It can be seen that the predicted results agree perfectly with the exact solutions, demonstrating the ability of the proposed method to supply sufficiently accurate and reliable results. Figure 6: Convergence of nondimensional responses with respect to the SDF truncation numbers for Al/Al2O3 square plate. (**a**) Convergence of \(\overline{w}\); (**b**) convergence of \(\overline{\sigma}_{xx}\); (**c**) convergence of \(\overline{\tau}_{xz}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(g\)} & \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{\(\overline{w}(\frac{a}{2},\frac{a}{2},0)\)} & \multicolumn{3}{c}{\(\overline{\sigma}_{xx}^{-}\left(\frac{a}{2},\frac{a}{2},\frac{h}{3}\right)\)} \\ \cline{3-8}  & & _a/h = 4_ & _a/h = 10_ & _a/h = 100_ & _a/h = 4_ & _a/h = 10_ & _a/h = 100_ \\ \hline \multirow{4}{*}{1} & Quasi-3D [15] & 0.7171 & 0.5875 & 0.5625 & 0.6221 & 1.5064 & 14.9690 \\  & TSDT-IGA [42] & 0.7284 & 0.5889 & 0.5625 & 0.5783 & 1.4816 & 14.8890 \\  & FSDT [15] & 0.7291 & 0.5889 & 0.5625 & 0.8060 & 2.0150 & 20.1500 \\  & Present & 0.7172 & 0.5875 & 0.5625 & 0.6219 & 1.5064 & 14.9692 \\  & Quasi-3D [15] & 1.1585 & 0.8821 & 0.8286 & 0.4877 & 1.1971 & 11.9230 \\  & TSDT-IGA [42] & 1.1598 & 0.8815 & 0.8287 & 0.4406 & 1.1711 & 11.8436 \\  & FSDT [15] & 1.1125 & 0.8736 & 0.8286 & 0.6420 & 1.6049 & 16.0490 \\  & Present & 1.1589 & 0.8823 & 0.8287 & 0.4884 & 1.1974 & 11.9198 \\  & Quasi-3D [15] & 1.3745 & 1.0072 & 0.9361 & 0.3695 & 0.8965 & 8.9077 \\  & TSDT-IGA [42] & 1.3908 & 1.0087 & 0.9362 & 0.3230 & 0.8733 & 8.8582 \\  & FSDT [15] & 1.3178 & 0.9966 & 0.9360 & 0.4796 & 1.1990 & 11.9900 \\  & Present & 1.3757 & 1.0074 & 0.9362 & 0.3657 & 0.8947 & 8.9076 \\ \hline \end{tabular}
\end{table} 
Table 2: Nondimensional deflection and normal stress of Al/Al\({}_{2}\)O\({}_{3}\) square plates with various gradient indexes and aspect ratios. Figure 7: Converged 3D responses of Al/Al\({}_{2}\)O\({}_{3}\) square plate under a bi-sinusoidal load. (**a**) Three-dimensional responses of \(\overline{w}\); (**b**) 3D responses of \(\overline{\sigma}_{xx}\); (**c**) 3D responses of \(\overline{\tau}_{xz}\). Figure 8: \(Cont.\)  
For further verification, various values of the gradient index \(k\) are selected (\(a/h=4\), \(b/a=3\)). In Figure 9, the displacement and stress results obtained using the SDF-based IGA method are compared with the exact 3D solutions [16]. It can be observed that the SDF-based IGA results closely match the exact solutions.