# Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core
## 2 FG Sandwich Plates with Homogeneous Cores  
Figure 1 represents a general rectangular FG sandwich plate composed of three layers, in which three layers are macroscopically homogeneous or inhomogeneous. The lengths of the two sides are \(a\) and \(b\), and the uniform thickness of plate is denoted by \(h\). A Cartesian coordinate system \((x,y,z)\) is positioned at the center, on the mid-surface of the plate such that the top and bottom surfaces of the plate are located at \(z=\pm h/2\) in the \(z-\)direction. The plate could be non-symmetric with respect to the material property and/or the geometric dimension. The three layers are numbered as _Layer 1_, _Layer 2_, and _Layer 3_ from the bottom to the top of the plate, and the thickness-wise intervals of each layer are set by \((l=1,2,3)\).  
\[\textit{Layer 1, }z\in[h_{I},h_{I+1}] \tag{1}\]  
Meanwhile, for the convenience of notation, the _pseudo-thickness ratio_ of each layer from the bottom to the top is denoted by \(H_{1}-H_{2}-H_{3}\)\((H_{I}=0,1,2,\cdots)\) in terms of three numbers \(H_{I}\). Then, the actual thickness ratios \(R_{I}\in[0,1]\) of each layer were calculated as following:  
\[R_{I}=\frac{H_{I}}{H_{1}+H_{2}+H_{3}} \tag{2}\]  
FG sandwich plates are classified into two types, one with FGM face sheets and a homogeneous core (_Type A_) and the other with homogeneous face sheets and an FGM core (_Type B_). In the current study, Type A is shown in Figure 2, where the white and black layers are homogeneous, while the layers with gray spectrum are FGM. In the figure, the plot indicates the volume fraction distributions, where power-law functions are adopted for FGM face sheets in Type A. Let us denote \(V(z)\in[0,1]\) and \(\widetilde{V}(z)\in[0,1]\) as two volume fractions which obey the following physical constraint, given by:  
\[V(z)+\widetilde{V}(z)=1.0,\quad z\in[-h/2,h/2] \tag{3}\]  
In the current study, the core material was calculated through \(V(z)\), as represented in Figure 2.  
Due to the physical constraints, either one of the two volume fractions was sufficient to identify the volume fraction distributions of the FG sandwich plate. The volume fraction, \(V(z)\), was calculated for the current study, then expressed by a linear combination of layer-wise continuous volume fractions, \(V^{(I)}(z)\), such that  
\[V(z)=V^{(1)}(z)+V^{(2)}(z)+V^{(3)}(z) \tag{4}\]  
Figure 1: Geometry and dimensions of a non-symmetric, rectangular FG sandwich plate with uniform thickness.  
where \(V^{(l)}(z)\) are defined by  
\[V^{(1)}=\left(\frac{z-h_{1}}{h_{2}-h_{1}}\right)^{n},\ z\in[h_{1},h_{2}] \tag{5}\]  
\[V^{(2)}=1,\ z\in[h_{2},h_{3}] \tag{6}\]  
\[V^{(3)}=\left(\frac{z-h_{3}}{h_{4}-h_{3}}\right)^{n},\ z\in[h_{3},h_{4}] \tag{7}\]  
Here, the core material could be metal or ceramic, where the former case is described as _softcore_ while the latter is described as _hardcore_. In other words, \(V(z)\) indicates the metal volume fraction for softcore cases and the ceramic volume fraction for hardcore cases.  
Next, the effective material properties \(\overline{M}_{eff}(z)\) at any point within of FG sandwich plate were evaluated as  
\[\overline{M}_{eff}(z)=V(z)M+[1-V(z)]\widetilde{M} \tag{8}\]  
according to the linear rule of mixtures [31]. \(M\) and \(\widetilde{M}\) indicate the properties of the metal and the ceramic cases, respectively. The variation of \(\overline{M}_{eff}(z)\) through the thickness was reflected in the process of numerical integration of mass and stiff matrices in natural element approximation, as explained in the next section.