## 2 Theoretical Formula
### Vibration Equation  
According to the [PERSON] shear deformation theory in a composite sandwich plate, the following assumptions are considered:  
1. Transverse strain of sandwich plate \(\varepsilon_{\text{z}}=0\). 2. The straight line perpendicular to the neutral surface before deformation will no longer maintain a straight line after deformation. Figure 1: Schematic of a composite sandwich plate with functionally graded core and shape memory alloy (SMA)-composite face sheets. The displacement caused by bending is  
\[u(x,y,z,t)=z\varphi_{x}(x,y,t)-\frac{4z^{3}}{3h^{2}}\bigg{(}\varphi_{x}(x,y,t)+ \frac{\partial w(x,y,t)}{\partial x}\bigg{)}\]  
\[v(x,y,z,t)=z\varphi_{y}(x,y,t)-\frac{4z^{3}}{3h^{2}}\bigg{(}\varphi_{y}(x,y,t)+ \frac{\partial w(x,y,t)}{\partial y}\bigg{)} \tag{7}\]  
\[w(x,y,z,t)=w(x,y,t)\]  
where \(\varphi_{x}(x,y,t)\), \(\varphi_{y}(x,y,t)\) are the angular displacement components along the X, Y coordinates, respectively and '\(w\)' is the displacement components of along Z coordinate. According to the small deformation theory, the strain at any point on the outer surface of the sandwich plate can be expressed as  
\[\left\{ \begin{array}{c}\varepsilon_{x}=z\frac{\partial\varphi_{x}}{\partial x }-\frac{4z^{3}}{3h^{2}}\bigg{(}\frac{\partial\varphi_{x}}{\partial x}+\frac{ \partial^{2}w}{\partial x^{2}}\bigg{)}\\ \\ \varepsilon_{y}=z\frac{\partial\varphi_{y}}{\partial y}-\frac{4z^{3}}{3h^{2}} \bigg{(}\frac{\partial\varphi_{y}}{\partial y}+\frac{\partial^{2}w}{\partial y ^{2}}\bigg{)}\\ \\ \gamma_{xy}=z\bigg{(}\frac{\partial\varphi_{x}}{\partial y}+\frac{\partial \varphi_{y}}{\partial x}\bigg{)}-\frac{4z^{3}}{3h^{2}}\bigg{(}\frac{\partial \varphi_{x}}{\partial y}+\frac{\partial\varphi_{y}}{\partial x}+2\frac{ \partial^{2}w}{\partial x\partial y}\bigg{)}\\ \\ \gamma_{xz}=\bigg{(}1-\frac{4z^{2}}{h^{2}}\bigg{)}\bigg{(}\varphi_{x}+\frac{ \partial w}{\partial x}\bigg{)}\\ \\ \gamma_{yz}=\bigg{(}1-\frac{4z^{2}}{h^{2}}\bigg{)}\bigg{(}\varphi_{y}+\frac{ \partial w}{\partial y}\bigg{)}\end{array} \right. \tag{8}\]  
Assuming that the functionally gradient sandwich composite plate embedded SMA wires in surface layer shown in Figure 1 contain face sheets and core, the stress-strain relation of the kth layer can be given as [25]  
\[\sigma^{(k)}=\widetilde{Q}^{(k)}\varepsilon^{(k)}-\sigma^{(k)}_{T}+k_{SMA} \sigma^{(k)}_{r} \tag{9}\]  
where \(k\) takes \(t\), \(b\) and \(c\) represent the top layer, bottom layer and core layer, respectively. \[\widetilde{Q}^{(k)}=\left[ \begin{array}{ccccc}\widetilde{Q}^{(k)}_{11}& \widetilde{Q}^{(k)}_{12}&0&0&0\\ \widetilde{Q}^{(k)}_{12}&\widetilde{Q}^{(k)}_{22}&0&0&0\\ 0&0&\widetilde{Q}^{(k)}_{44}&0&0\\ 0&0&0&\widetilde{Q}^{(k)}_{55}&0\\ 0&0&0&0&\widetilde{Q}^{(k)}_{66}\end{array} \right] \tag{10}\]  
The term \(\widetilde{Q}^{(k)}\) denotes the modulus matrix, of which \(\text{Q}^{(k)}_{\text{ij}}\) are modulus elements. The top and bottom surfaces of the sandwich plate were orthogonal structures composed of epoxy resin embedded shape memory alloy wire, wherein the modulus element is quoted from [26]. For the gradient core,since the elastic modulus and Poisson's ratio are both functions of the Z coordinate, and the core layer is considered to be an isotropic, the modulus matrix elements of Equation (10) are obtained as  
\[\left\{ \begin{array}{c}\widetilde{Q}_{11}^{(c)}(z)=\widetilde{Q}_{22}^{(c)}(z)= \frac{E^{(c)}(z)}{1-\left[v^{(c)}(z)\right]^{2}}\\ \widetilde{Q}_{12}^{(c)}(z)=\frac{E^{(c)}(z)v^{(c)}(z)}{1-\left[v^{(c)}(z) \right]^{2}}\\ \widetilde{Q}_{66}^{(c)}(z)=\widetilde{G}_{12}^{(c)}(z)=\frac{E^{(c)}(z)}{2 \left(1-v^{(c)}(z)\right)}\\ \widetilde{Q}_{44}^{(c)}(z)=\widetilde{G}_{23}^{(c)}(z)\\ \widetilde{Q}_{55}^{(c)}(z)=\widetilde{G}_{13}^{(c)}(z)\end{array} \right. \tag{11}\]  
The recovery stress induced by pre-strain of SMA and SMA coefficient are represented as \(\sigma_{r}^{(k)}\) and \(k_{SMA}\), respectively, as \(k\) takes \(t\) or \(b\). When the shape memory alloy is contained in the layer, \(k_{SMA}=1\), and if not, \(k_{SMA}=0\).