# Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core
## 4 Numerical Results  
### Verification  
A simply supported, rectangular FG sandwich plate, shown in Figure 4a, was taken to verify the proposed numerical method, where the lengths of the two sides, \(a\) and \(b\), were \(1.0\)m and the thickness \(h\) was taken as variable. The two constituent materials were \(A\mathcal{L}\) and \(A\mathcal{\ell}_{2}O_{3}\), and their mechanical properties are given in Table 1. The metallic \(A\mathcal{L}\) was used for the softcore while the ceramic \(A\mathcal{\ell}_{2}O_{3}\) was used for the hardcore, respectively. To numerically implement the \((q_{x},q_{y},q_{z})\) hierarchical model, the mid-surface of the sandwich plate was uniformly discretized by an \(11\times 11\) set of grid points, as shown in Figure 4b. Figure 4: A simply supported, rectangular FG sandwich plate: (**a**) the geometric dimensions and loading/boundary conditions, (**b**) an \(11\times 11\) uniform NEM grid. The simply supported constraint was implemented as follows: \(\overline{U}_{z}^{0}=0\) for all four sides of the mid-surface, \(\overline{U}_{x}^{0}=\cdots=\overline{U}_{x}^{dx}=0\) for the two sides labelled 1 and 3, and \(\overline{U}_{y}^{0}=\cdots=\overline{U}_{y}^{dy}=0\) for the other two sides, 2 and 4. The stiffness and mass matrices in Equations (18) and (19) were computed by combining the 2-D, in-plane Gaussian integration rule (7 points) and the thickness-wise trapezoidal rule (30 uniform segments). The lowest nine natural frequencies and natural modes were solved through the Lanczos transformation method and the Jacobi iteration method. The fundamental frequencies, \(\omega_{(1,1)}\), of the (2-2-1) FG sandwich plates with softcores (i.e., \(Ni\) cores) were computed using the hierarchical models and calibrated according to  
\[\overline{\omega}=\frac{\omega b^{2}}{h}\sqrt{\frac{\rho_{0}}{E_{0}}} \tag{22}\]  
where \(\rho_{0}=1.0\) kg/m\({}^{3}\) and \(E_{0}=1.0\) GPa. The numerical experiments were parametrically carried out with respect to the width-to-thickness ratio \(b/h\) for the same volume fraction index, \(n\), of unity. The computed calibrated fundamental frequencies are given in Table 2 and represented in Figure 5, where the solutions of 3-D elasticity were obtained by ANSYS [35] using 10,000 3-D shell elements. All the hierarchical models showed the remarkable relative differences in \(\overline{\omega}\) for thick plates, such that they overestimate the fundamental frequencies with non-negligible differences. However, for a specific value of \(b/h\), the hierarchical models showed the spectral model accuracy such that their relative differences uniformly decreased, proportional to the model level, \(q\). Furthermore, the relative differences of all the hierarchical models monotonically decreased in proportion to the width-to-thickness ratio \(b/h\). In particular, it was observed that the relative difference became remarkably smaller when the model level equaled or exceeded (3,3,2), regardless of the width-to-thickness ratio. Meanwhile, both the (9,9,8) hierarchical model and the 3-D elasticity approach to the same limit equated to 1.6595, which was the solution provided by the classical laminated plate theory (CLPT) [36]. Meanwhile, it was observed that the limit values, \(\overline{\omega}\), of other hierarchical models became slightly larger as the model level, \(q\), decreased. However, the relative difference of the (1,1,0) model was 2.773% at \(b/h=500\); hence, the lower-order models were not expected to cause inaccuracy problems in the free vibration analysis for thin FG sandwich plates with homogeneous cores. The generic constant, \(C\), which was involved in the modified shear correction factor (MSCF), was evaluated as 3/2 through this limit experiment. Next, the (1,1,0) hierarchical model was compared with the five-variable, refined plate theory of [PERSON] et al. [10]. The former was equivalent to the first-order shear deformation theory (FSDT) and the latter was refined from the FSDT.