# Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams
## 3 Governing Equations Derivation
### Equilibrium Equations  
The analysis of the SMAHC sandwich beam is based on a HSDT, such that the displacement field, \(\overline{u}(x,z,t)\), \(\overline{v}(x,z,t),\overline{w}(x,z,t)\), is defined as [34]:  
\[ \begin{array}{l}\overline{u}(x,z,t)=u_{0}(x,t)+\frac{5}{4}\Big{(}z-\frac{4}{3 \hbar}z^{3}\Big{)}\phi_{x}(x,t)+\Big{(}\frac{1}{4}z-\frac{5}{3\hbar^{2}}z^{3} \Big{)}\frac{\partial w_{0}(x,t)}{\partial x}\\ \overline{v}(x,z,t)=0\\ \overline{w}(x,z,t)=w_{0}(x,t)\end{array}  \tag{10}\]  
where \(u_{0}\) and \(w_{0}\) are the displacement components along the \(x\) and \(z\) coordinate directions, respectively, of a point on the mid-surface. In addition, \(\phi_{x}\) refers to the rotation about the \(x\) axis and \(h\) is the total beam thickness. The kinematic relations at Cartesian coordinates (\(x\)-\(y\)-\(z\)) are defined as [34]:  
\[ \begin{array}{l}\varepsilon_{x}=\frac{\partial\overline{u}}{\partial x}+ \frac{1}{2}\Big{(}\frac{\partial\overline{w}}{\partial x}\Big{)}^{2},\; \varepsilon_{y}=\frac{\partial\overline{w}}{\partial y}\;,\;\varepsilon_{z}= \frac{\partial\overline{w}}{\partial z}\;,\;\gamma_{xy}=\frac{\partial \overline{w}}{\partial y}+\frac{\partial\overline{w}}{\partial x},\\ \gamma_{xz}=\frac{\partial\overline{u}}{\partial z}+\frac{\partial\overline{ w}}{\partial x}+\frac{\partial\overline{w}}{\partial x}\frac{\partial\overline{w}}{ \partial x}\;,\;\gamma_{yz}=\frac{\partial\overline{w}}{\partial z}+\frac{ \partial\overline{w}}{\partial y}\end{array}  \tag{11}\]  
By substituting Equation (10) into Equation (11), we obtain the following equation:  
\[ \begin{array}{l}\varepsilon_{x}=\varepsilon_{x}^{(0)}+z\varepsilon_{x}^{(1 )}+z^{3}\varepsilon_{x}^{(3)}\\ \gamma_{xz}=\gamma_{xz}^{(0)}+z^{2}\gamma_{xz}^{(2)}\end{array}  \tag{12}\]  
where \(\varepsilon_{x}^{(0)}\), \(\varepsilon_{x}^{(1)}\), \(\varepsilon_{x}^{(3)}\), \(\gamma_{xz}^{(0)}\), and \(\gamma_{xz}^{(2)}\) are defined as:  
\[\left\{ \begin{array}{l}\varepsilon_{x}^{(0)}\\ \varepsilon_{x}^{(1)}\\ \varepsilon_{x}^{(3)}\end{array} \right\}=\left\{ \begin{array}{l}\frac{du}{ dx}+\frac{1}{2}\Big{(}\frac{dw_{0}}{dx}\Big{)}^{2}\\ \frac{5}{4}\frac{d\phi_{x}}{dx}+\frac{1}{4}\frac{d^{2}w_{0}}{dx^{2}}\\ -\Big{(}\frac{5}{3\hbar^{2}}\frac{d\phi_{x}}{dx}+\frac{5}{3\hbar^{2}}\frac{dw_{ 0}}{dx^{2}}\Big{)}\end{array} \right\},\left\{ \begin{array}{l}\gamma_{xz}^{( 0)}\\ \gamma_{xz}^{(2)}\end{array} \right\}=\left\{ \begin{array}{l}\frac{5}{4}\phi_{ x}+\frac{5}{4}\frac{dw_{0}}{dx}\\ -\Big{(}\frac{5}{\hbar^{2}}\phi_{x}+\frac{5}{\hbar^{2}}\frac{dw_{0}}{dx}\Big{)} \end{array} \right\} \tag{13}\]  
The equilibrium equations are obtained according to the HSDT and the Principal of Minimum Potential Energy (PMPE), expressed as [34]:  
\[\Pi=U-W \tag{14}\]  
where \(U\) and \(W\) refer to the strain energy and the external work, respectively, defined in variational form \(\delta\)[35] as follows:  
\[\delta U=\int\limits_{\Omega}{(\sigma_{x}\delta\varepsilon_{x}+\sigma_{xz} \delta\gamma_{xz})dxdz} \tag{15}\]  
or equivalently:  
\[ \begin{array}{l}\delta U=\int\limits_{0}^{L}{\int\limits_{0}^{L}{\int\limits_ {-(\frac{1}{2}\delta\varepsilon_{x}+h_{0})}^{L}\sigma_{x}\left(\frac{5}{3\hbar^ {2}}\frac{d\delta\phi_{x}}{dx}+\frac{5}{3\hbar^{2}}\frac{d^{2}\delta w_{0}}{dx ^{2}}\right)}}z-\\ \sigma_{xz}\Big{(}\frac{5}{3\hbar^{2}}\frac{d\delta\phi_{x}}{dx}-\Big{(}\frac {5}{\hbar^{2}}\delta\phi_{x}+\frac{5}{\hbar^{2}}\frac{d\delta w_{0}}{dx} \Big{)}}z^{2}\Big{)}dzdx\end{array} \right\}+\\ \sigma_{xz}\Big{(}\frac{5}{3\hbar^{2}}\delta\phi_{x}+\frac{5}{4}\frac{dw_{0} }{dx}-\Big{(}\frac{5}{\hbar^{2}}\delta\phi_{x}+\frac{5}{\hbar^{2}}\frac{d \delta w_{0}}{dx}\Big{)}}z^{2}\Big{)}dzdx\end{array} \tag{16}\]At the same time, Equation (16) can be redefined in terms of resultants (\(N_{x}\), \(M_{x}\), \(N_{xz}\), \(P_{x}\), \(R_{xz}\)), i.e.,  
\[