# Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams
## 3 Governing Equations Derivation
### Equilibrium Equations
\begin{array}{ll}\delta U=&\int\limits_{0}^{L}\left(N_{x}\left(\frac{d\delta u }{dx}+\frac{dw_{0}}{dx}\frac{d\delta w_{0}}{dx}\right)+M_{x}\left(\frac{5}{4} \frac{d\delta\phi_{x}}{dx}+\frac{1}{4}\frac{d^{2}\delta w_{0}}{dx^{2}}\right)+ \\ &P_{x}\left(\frac{5}{3h^{2}}\frac{d\delta\phi_{x}}{dx}+\frac{5}{3h^{2}}\frac{d^ {2}\delta w_{0}}{dx^{2}}\right)+N_{xx}\left(\frac{5}{4}\delta\phi_{x}+\frac{5} {4}\frac{d\delta w_{0}}{dx}\right)+\\ &R_{xz}\left(-\left(\frac{5}{h^{2}}\delta\phi_{x}+\frac{5}{h^{2}}\frac{d\delta w _{0}}{dx}\right)\right)\right)dx\end{array}  \tag{17}\]  
where \(N_{x}\) is the in-plane force resultants; \(M_{x}\) is the moment resultants; \(N_{xz}\) is the final transverse force resultants; and \(P_{x}\) and \(R_{xz}\) are the stressed higher order resultants determined as a sum of the mechanical, thermal, and recovery resultants (denoted with superscript M, T, and r, respectively). Such parameters are defined as:  
\[ \begin{array}{ll}N_{x}=N_{x}^{M}+N_{x}^{T}+N_{x}^{T}=\int_{-(h_{c}/2+h_{a})}^ {h_{c}/2+h_{a}}\left(\sigma_{x}^{M}+\sigma_{x}^{T}\right)dz+\int_{-(h_{c}/2+h_ {a})}^{-h_{c}/2}\sigma_{x}^{r}dz+\int_{h_{c}/2+h_{a}}^{h_{c}/2+h_{a}}\sigma_{ x}^{r}dz\\ M_{x}=M_{x}^{M}+M_{x}^{T}+M_{x}^{r}=\int_{-(h_{c}/2+h_{a})}^{h_{c}/2+h_{a}}z \left(\sigma_{x}^{M}+\sigma_{x}^{T}\right)dz+\int_{-(h_{c}/2+h_{a})}^{-h_{c}/2 }z\sigma_{x}^{r}dz+\int_{h_{c}/2+h_{a}}^{h_{c}/2+h_{a}}z\sigma_{x}^{r}dz\\ N_{xz}=N_{xz}^{M}+N_{xz}^{T}+N_{xz}^{r}=\int_{-(h_{c}/2+h_{a})}^{h_{c}/2+h_{a}} \left(\sigma_{xz}^{M}+\sigma_{xz}^{T}\right)dz+\int_{-(h_{c}/2+h_{a})}^{-h_{c}/ 2}\sigma_{xz}^{r}dz+\int_{h_{c}/2+h_{a}}^{h_{c}/2+h_{a}}\sigma_{xz}^{r}dz\\ P_{x}=P_{x}^{M}+P_{x}^{T}+P_{x}^{r}=\int_{-(h_{c}/2+h_{a})}^{h_{c}/2+h_{a}}z^{ 3}\left(\sigma_{x}^{M}+\sigma_{x}^{T}\right)dz+\int_{-(h_{c}/2+h_{a})}^{-h_{c}/ 2}z^{3}\sigma_{x}^{r}dz+\int_{h_{c}/2+h_{a}}^{h_{c}/2+h_{a}}z^{3}\sigma_{x}^{r} dz\\ R_{xz}=R_{xz}^{M}+R_{xz}^{T}+R_{xz}^{r}=\int_{-(h_{c}/2+h_{a})}^{h_{c}/2+h_{a}}z^{ 2}\left(\sigma_{xz}^{M}+\sigma_{xz}^{T}\right)dz+\int_{-(h_{c}/2+h_{a})}^{-h_{c }/2}z^{2}\sigma_{xz}^{r}dz+\int_{h_{c}/2+h_{a}}^{h_{c}/2+h_{a}}z^{2}\sigma_{xz} ^{r}dz\end{array}  \tag{18}\]  
Finally, from Equation (17), the equilibrium equations can be obtained as:  
\[ \begin{array}{ll}\delta u_{0}=0\Rightarrow\frac{dN_{x}}{dx}=0\\ \delta\phi_{x}=0\Rightarrow-\frac{5}{4}\frac{dM_{x}}{dx}+\frac{5}{3h^{2}} \frac{dP_{x}}{dx}+\frac{5}{4}N_{xz}-\frac{5}{h^{2}}R_{xz}=0\\ \delta w_{0}=0\Rightarrow-\frac{d}{dx}\left(N_{x}\frac{dw_{0}}{dx}\right)+ \frac{1}{4}\frac{d^{2}M_{x}}{dx^{2}}-\frac{5}{3h^{2}}\frac{d^{2}P_{x}}{dx^{2}} -\frac{5}{4}\frac{dN_{xz}}{dx}+\frac{5}{h^{2}}\frac{dR_{xz}}{dx}=0\end{array}  \tag{19}\]  
In addition, the stress-strain (\(\sigma-\epsilon\)) relation involving the recovery stress and thermal strain at Cartesian coordinates are defined as:  
\[\left\{ \begin{array}{ll}\sigma_{x}\\ \sigma_{y}\\ \sigma_{z}\\ \sigma_{yz}\\ \sigma_{xz}\\ \sigma_{xy}\end{array} \right\}=\left[ \begin{array}{cccccc}C_{11}&C_{12}&C_{ 13}&0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{array} \right]\left(\left( \begin{array}{c}\varepsilon_{x }\\ \varepsilon_{y}\\ \varepsilon_{z}\\ \gamma_{yz}\\ \gamma_{xz}\\ \gamma_{xy}\end{array} \right)-\left\{ \begin{array}{ll}\alpha_{x}\\ \alpha_{y}\\ \alpha_{z}\\ 0\\ 0\\ 0\end{array} \right\}\Delta T\right)+V_{s}\left\{ \begin{array}{ll}\sigma_{1}^{r} \\ \sigma_{2}^{r}\\ 0\\ 0\\ 0\\ \sigma_{12}^{r}\end{array} \right\} \tag{20}\]  
where \(C_{ij}\) (_i,j_ = 1,2,..,6) are the stiffness matrix components. In addition, \(\alpha_{i}\)(_i_ = \(x\), \(y\), _z_), \(\Delta T\) stand for the thermal expansion and thermal variation, and \(V_{s}\) and \(\sigma_{ij}^{r}\) (_i,j_ = 1,2,3) refer to the volume fraction and recovery stress of the shape memory alloy, respectively. Thus, the mechanical and thermal stresses are defined as:  
\[\left\{ \begin{array}{ll}\sigma_{x}^{M}\\ \sigma_{xz}^{M}\end{array} \right\}=\left\{