# Free Vibration Analysis of Functionally Graded Sandwich Plates with a Homogeneous Core
## 3 Natural Element Free Vibration Approximation  
Viewing the FG sandwich plate as a 3-D, linearly elastic body, its harmonic motion was denoted by \(\mathbf{u}(\mathbf{x};t)=\overset{-}{\mathbf{u}}(\mathbf{x})e^{i\omega t}\). Then, the free vibration of FG sandwich plate was determined to be governed by (\(\alpha\), \(\beta=x,y,z\))  
\[\sigma_{\alpha\beta}\binom{-}{\mathbf{u}}_{,\beta}=\omega^{2}\rho\overline{u}_{ \alpha}\ in\ \alpha\times[-h/2,h/2] \tag{9}\]  
with the displacement boundary condition:  
\[\overset{-}{\mathbf{u}}=\overset{\mathbf{u}}{\mathbf{u}}\ on\ \partial\omega_{D}\times[-h/2,h/2] \tag{10}\]  
where, \(\sigma_{\alpha\beta}\) and \(\rho\) are Cauchy stresses and the effective density, and \(\omega\) and \(\partial\omega_{D}\) indicate the plate mid-surface and the displacement boundary region, respectively. Meanwhile, in the \(\left(q_{x},q_{y},q_{z}\right)\) hierarchical model, the triple-vector amplitude function, \(\overset{-}{\mathbf{u}}(\mathbf{x})\), is expressed as [28]  
\[\overline{u}_{\alpha}(\mathbf{x})=\sum_{\ell=0}^{q_{\alpha}}\overline{U}_{\alpha }^{\ell}(x,y)\cdot\left(\frac{2z}{h}\right)^{\ell} \tag{11}\]  
Figure 2: Volume fraction distributions through the thickness of an FG sandwich plate with two FGM facesheets and a homogeneous core (Type A). where, \(\overline{U}_{a}^{\ell}(x,y)\) are in-plane amplitude functions defined on the plate's mid-surface, which were determined through the 2-D natural element method. For the natural approximation, the mid-surface of the sandwich plate was discretized by \(N\) grid points (i.e., nodes) and \(N_{D}\) Delaunay triangles. This construction of grid points and Delaunay triangles was generated by an NEM grid, as shown in Figure 2(a). To each grid point \(J\), Laplace interpolation functions, \(\phi_{J}(x)\), shown in Figure 2(b), were assigned. The definition and characteristics of these functions may be found at the following references: [32; 33]. Using these interpolation functions, both the trial and test functions, \(\mathbf{u}^{-q,h}\) and \(\mathbf{v}^{q,h}\), were approximated as  
\[\overline{u}_{a}^{q,h}(\mathbf{x})=\sum_{J=1}^{N}\overline{U}_{a,J}^{0}\phi_{J}(x, y)\cdot\left(\frac{2z}{d}\right)^{0}+\cdots=\sum_{m=0}^{q_{x}}\left(\sum_{J=1}^{N} \overline{U}_{a,J}^{m}\phi_{J}(x,y)\right)\cdot\left(\frac{2z}{d}\right)^{m} \tag{12}\]  
\[v_{\beta}^{q,h}(\mathbf{x})=\sum_{I=1}^{N}V_{\beta,I}^{0}\phi_{I}(x,y)\cdot\left( \frac{2z}{d}\right)^{0}+\cdots=\sum_{\ell=0}^{q_{\beta}}\left(\sum_{I=1}^{N}V _{\beta,I}^{\ell}\phi_{I}(x,y)\right)\cdot\left(\frac{2z}{d}\right)^{\ell} \tag{13}\]  
Assuming that \(q_{x}=q_{y}=q_{z}=q\) for the concise expression purpose, the virtual strain vector \(\mathbf{\varepsilon}\left( \begin{matrix}-q,h\\ \mathbf{v}\end{matrix} \right)\) and the actual stress vector \(\mathbf{\sigma}\left( \begin{matrix}-q,h\\ \mathbf{u}\end{matrix} \right)\) were expressed by  
\[\mathbf{\varepsilon}\left( \begin{matrix}-q,h\\ \mathbf{v}\end{matrix} \right)= \sum_{\ell=0}^{q}\sum_{I=1}^{N}\mathbf{L}\phi_{I}\overline{V}_{I}^ {\ell}\cdot\left(\frac{2z}{d}\right)^{\ell}=\sum_{\ell=0}^{q}\sum_{I=1}^{N} \mathbf{B}_{I}^{\ell}\overline{V}_{I}^{\ell}\cdot\left(\frac{2z}{d}\right)^{\ell} \tag{14}\] \[\mathbf{\sigma}\left( \begin{matrix}-q,h\\ \mathbf{u}\end{matrix} \right)=\mathbf{E}\mathbf{\varepsilon}\left(\mathbf{u}^{q,h}\right)= \sum_{m=0}^{q}\sum_{J=1}^{N}\mathbf{B}_{J}^{m}\mathbf{U}_{J}^{m}\cdot\left(\frac{2z}{ d}\right)^{m} \tag{15}\]  
in which the matrix \(\mathbf{B}_{J}^{m}\), containing partial differential operators, is defined by  
\[\mathbf{B}_{J}^{m}= \begin{bmatrix}\partial\phi_{J}/\partial x&0&0\partial\phi_{J }/\partial y&0&m\phi_{J}/z\\ 0&\partial\phi_{J}/\partial y&0&\partial\phi_{J}/\partial x&m\phi_{J}/z&0\\ 0&0&m\phi_{J}/z&0&\partial\phi_{J}/\partial y&\partial\phi_{J}/\partial x \end{bmatrix}