## 2 Theoretical Formula
### Vibration Equation
\begin{split}\delta\varphi_{y}:\ \Big{(}D_{12}-\frac{8}{3\mu^{2}}F_{12}+ \frac{16}{9\mu^{4}}H_{12}+D_{66}-\frac{8}{3\mu^{2}}F_{66}+\frac{16}{9\mu^{4}}H_ {66}\Big{)}\frac{\partial^{2}\varphi_{x}}{\partial x\partial y}\\ +\Big{(}D_{22}-\frac{8}{3\mu^{2}}F_{22}+\frac{16}{9\mu^{4}}H_{11} \Big{)}\frac{\partial^{2}\varphi_{y}}{\partial y^{2}}\\ +\Big{(}D_{66}-\frac{4}{3\mu^{2}}F_{66}+\frac{16}{9\mu^{4}}H_{66 }\Big{)}\frac{\partial^{2}\varphi_{y}}{\partial x^{2}}-\Big{(}A_{55}-\frac{8}{ \mu^{2}}D_{55}+\frac{16}{\mu^{4}}F_{55}\Big{)}\varphi_{y}\\ +\Big{(}-\frac{4}{3\mu^{2}}F_{11}+\frac{16}{9\mu^{4}}H_{12}- \frac{8}{3\mu^{2}}F_{66}+\frac{32}{9\mu^{4}}H_{66}\Big{)}\frac{\partial^{2}w} {\partial x^{2}\partial y}\\ +\Big{(}-\frac{4}{3\mu^{2}}F_{22}+\frac{16}{9\mu^{4}}H_{22}\Big{)} \frac{\partial^{2}w}{\partial y^{3}}-\Big{(}A_{55}-\frac{8}{\mu^{2}}D_{55}+ \frac{16}{\mu^{4}}F_{55}\Big{)}\frac{\partial w}{\partial y}\\ =\Big{(}I_{3}-\frac{8}{3\mu^{2}}I_{5}+\frac{16}{9\mu^{4}}I_{7} \Big{)}\frac{\partial^{2}\varphi_{y}}{\partial x^{2}}-\Big{(}\frac{4}{3\mu^{2 }}I_{5}-\frac{16}{9\mu^{4}}I_{7}\Big{)}\frac{\partial^{2}w}{\partial y \partial x^{2}}\\ \delta w:-\Big{(}-\frac{4}{3\mu^{2}}F_{11}+\frac{16}{9\mu^{4}}H_{ 11}\Big{)}\frac{\partial^{3}\varphi_{x}}{\partial x^{3}}\\ -\Big{(}-\frac{4}{3\mu^{2}}F_{12}+\frac{16}{9\mu^{4}}H_{12}- \frac{8}{3\mu^{2}}F_{66}+\frac{32}{9\mu^{4}}H_{66}\Big{)}\frac{\partial^{2} \varphi_{y}}{\partial x\partial y^{2}}\\ +\Big{(}A_{44}-\frac{8}{\mu^{2}}D_{44}+\frac{16}{\mu^{4}}F_{44} \Big{)}\frac{\partial^{2}\varphi_{x}}{\partial x}\\ -\Big{(}-\frac{4}{3\mu^{2}}F_{12}+\frac{16}{9\mu^{4}}H_{12}- \frac{8}{3\mu^{2}}F_{66}+\frac{32}{9\mu^{4}}H_{66}\Big{)}\frac{\partial^{2} \varphi_{y}}{\partial x^{2}\partial y}\\ -\Big{(}-\frac{4}{3\mu^{2}}F_{22}+\frac{16}{9\mu^{4}}H_{22}\Big{)} \frac{\partial^{3}\varphi_{y}}{\partial y^{3}}+\Big{(}A_{55}-\frac{8}{\mu^{2} }D_{55}+\frac{16}{\mu^{4}}F_{55}\Big{)}\frac{\partial\varphi_{y}}{\partial y} \\ -\frac{16}{9\mu^{4}}H_{11}\frac{\partial^{4}w}{\partial x^{4}}- \frac{32}{9\mu^{4}}(H_{12}+2H_{66})\frac{\partial^{4}w}{\partial x^{2} \partial y^{2}}-\frac{16}{9\mu^{4}}H_{22}\frac{\partial^{4}w}{\partial y^{4}} \\ +\Big{(}A_{44}-\frac{8}{\mu^{2}}D_{44}+\frac{16}{\mu^{4}}F_{44} \Big{)}\frac{\partial^{2}w}{\partial x^{2}}+\Big{(}A_{55}-\frac{8}{\mu^{2}}D_ {55}+\frac{16}{\mu^{4}}F_{55}\Big{)}\frac{\partial^{2}w}{\partial y^{2}}\\ +\Big{(}N_{x}^{r}-N_{x}^{r}-N_{x}^{r}\Big{)}\frac{\partial^{2}w}{ \partial x^{2}}+\Big{(}N_{y}^{r}-N_{y}^{r}-N_{y}^{r}\Big{)}\frac{\partial^{2} w}{\partial y^{2}}+q\\ =\Big{(}\frac{4}{3\mu^{2}}I_{5}-\frac{16}{9\mu^{4}}I_{7}\Big{)} \frac{\partial^{2}\varphi_{x}}{\partial x\partial t^{2}}+\Big{(}\frac{4}{3\mu^ {2}}I_{5}-\frac{16}{9\mu^{4}}I\Big{)}\frac{\partial^{3}\varphi_{y}}{\partial y \partial t^{2}}+I_{1}\frac{\partial^{2}w}{\partial t^{2}}\\ -\frac{16}{9\mu^{4}}I_{7}\Big{(}\frac{\partial^{4}w}{\partial x ^{2}\partial t^{2}}+\frac{\partial^{4}w}{\partial y^{2}\partial t^{2}}\Big{)} \end{split}  \tag{15}\]  
where, \(q\), \(A\), \(D\), \(F\), \(H\) and \(I\) are the transverse load, stiffness terms and inertia terms, respectively; \(N^{r}\), \(N^{T}\), and \(N^{rT}\) represent SMA recovery stress resultants, surface thermal stress resultants and core thermal stress resultants, respectively, which are all defined as  
\[ \begin{split}\Big{(}A_{ij,}D_{ij}F_{ij,}H_{ij}\Big{)}& =\int_{-h_{c}/2}^{h_{c}/2}\widetilde{Q}_{ij}^{(c)}(z)\Big{(}1,z^{2},z^{4},z^{6 }\Big{)}dz+\int_{h_{c}/2}^{h/2}\widetilde{Q}_{ij}^{(t)}\Big{(}1,z^{2},z^{4},z^{6 }\Big{)}dz\\ &+\int_{-h/2}^{-h_{c}/2}\widetilde{Q}_{ij}^{(b)}\Big{(}1,z^{2},z^ {4},z^{6}\Big{)}dz\ \ (i,j=1,2,4,5,6)\\ (I_{1}I_{3}I_{5}I_{7})&=\int_{-h_{c}/2}^{h_{c}/2} \overline{\rho}^{(c)}(z)\Big{(}1,z^{2},z^{4},z^{6}\Big{)}dz+\int_{h_{c}/2}^{h/2} \overline{\rho}^{(t)}\Big{(}1,z^{2},z^{4},z^{6}\Big{)}dz\\ &\qquad\qquad+\int_{-h_{c}/2}^{-h_{c}/2}\overline{\rho}^{(b)}\Big{(} 1,z^{2},z^{4},z^{6}\Big{)}dz\end{split}  \tag{16}\]\[\left\{ \begin{array}{l}N_{x}^{T}\\ N_{y}^{T}\end{array} \right\}=\sum\limits_{k=l,b}\left[