# Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams
## 3 Governing Equations Derivation
### Stability Equations  
We now determine the stability equations of the beam under a thermal condition. The expansion of the total potential energy (\(V\)) based on the Taylor series reads as follows [36]:  
\[\Delta V=\delta V+\frac{1}{2}\delta^{2}V+\frac{1}{6}\delta^{3}V+\ldots\ldots. \tag{22}\]  
where \(\delta V\) is the first variation of the total potential energy, which is related to the equilibrium state, whereas the second variation, \(\delta^{2}V\), is representative of the stability in the neighborhood of the equilibrium state. For all virtual displacements, if \(\delta^{2}V>0\) the beam remains stable, otherwise (\(\delta^{2}V<0\)) it becomes unstable. The last condition, \(\delta^{2}V=0\), is associated with the critical buckling condition [36]. In fact, the critical thermal buckling load disrupts the stability in equilibrium state. The main reason for obtaining the stability equations is related to the nonlinearity of the equilibrium equations. Let us assume that the displacement components (\(u_{0},\ \phi_{x},\ w_{0}\)) undergo a small increment:  
\[ \begin{split} u_{0}&\to u_{0}^{0}+u_{0}^{1}\\ \phi_{x}&\to\phi_{x}^{0}+\phi_{x}^{1}\\ w_{0}&\to w_{0}^{0}+w_{0}^{1}\end{split}  \tag{23}\]  
where \(u_{0}^{0}\), \(\phi_{x}\), \(w_{0}^{0}\) refer to the displacement field in the equilibrium state, whereas \(u_{0}^{1}\), \(\phi_{x}^{1}\), \(w_{0}^{1}\) are the associated components of a neighboring stable state in respect of the equilibrium state. The force resultants of a neighboring state are related to the equilibrium state as:  
\[ \begin{split} N_{x}&\to N_{x}^{0}+N_{x}^{1}\\ M_{x}&\to M_{x}^{0}+M_{x}^{1}\\ P_{x}&\to P_{x}^{0}+P_{x}^{1}\\ N_{xz}&\to N_{xz}^{0}+N_{xz}^{1}\\ R_{xz}&\to R_{xz}^{0}+R_{xz}^{1}\end{split}  \tag{24}\]  
Superscript '1' refers to the stability state and superscript '0' refers to the equilibrium state. By substituting Equations (23) and (24) into the equilibrium equation, Equation (19), a deviation of the force resultants from the initial equilibrium state can be obtained, as detailed in Appendix A, (A8). Finally, the stability equations can be obtained as follows:  
\[ \begin{split}\delta u_{0}&=0\Rightarrow\frac{dN_{ x}^{1}}{dx}=0\\ \delta\phi_{x}&=0\Rightarrow-\frac{5}{4}\frac{dM_{ x}^{1}}{3h^{2}}+\frac{5}{3h^{2}}\frac{dP_{1}^{1}}{dx}+\frac{5}{4}N_{xz}^{1}- \frac{5}{h^{2}}R_{xz}^{1}=0\\ \delta w_{0}&=0\Rightarrow-N_{x0}\frac{d^{2}w_{0}^ {1}}{dx^{2}}+\frac{1}{4}\frac{dM_{x}^{1}}{dx^{2}}-\frac{5}{3h^{2}}\frac{dP_{ 1}^{1}}{dx^{2}}-\frac{5}{4}\frac{dN_{x}^{1}}{dx}+\frac{5}{h^{2}}\frac{dR_{1}^{ 1}}{dx}=0\end{split}  \tag{25}\]  
where \(N_{x0}\), for a uniform and linear temperature rise, is defined as:  
\[N_{x0}=\int_{-(k_{x}/2+h_{a})}^{(k_{x}/2+h_{a})}C_{ij}(z)\alpha_{i}(z)(T-T_{0}) dz,\ i,j=x,y,z \tag{26}\]  
and \(T_{0}\) is the reference temperature. When the beam buckles at \(T=T_{0}+\Delta T\), such thermal value will be updated in Equation (26). Let us now consider a sandwich beam whose temperature at the top and bottom layer surfaces is labeled as \(T_{t}\) and \(T_{b}\), respectively. The temperature distribution for fixed boundary conditions is obtained by solving the heat conduction equation along the beam thickness. If the beam thickness is thin enough, the temperature distribution is approximated as linear through the thickness direction z. For a simply supported beam, the displacement components are assumed as:  
\[PE_{p}=\int_{0}^{L}\int_{-(l_{c}/2+l_{h_{a}})}^{l_{c}/2+h_{a}}\left(\sigma_{xx} \varepsilon_{xx}^{L}+\sigma_{xz}\gamma_{xz}^{L}\right)dxdz \tag{33}\]  
\[PE_{T}=\int_{0}^{L}\int_{-(l_{c}/2+l_{h})}^{l_{c}/2+h_{a}}\left(\sigma_{0 xx} \varepsilon_{xx}^{NL}+\sigma_{0 xx}\gamma_{xz}^{NL}\right)dxdz \tag{34}\]  
where \(\sigma_{ij}\)oij and \(\sigma_{0 ij}\)\((i,j=x,z)\) refer to the stress tensor and pre-stress components due to the applied temperature field, respectively.