# Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams
## 2 Materials Properties and Basic Equations of SMA  
Let us consider a composite beam made of two layers and an FGM core. In more detail, the external layers are made of a graphite-epoxy composite with the addition of a SMA reinforcement. The selected structural member is here studied based on the [PERSON]'s model [30], due to its capability in considering the thermo-mechanical behavior at any temperature. According to this model, we introduce the following relation between the stress field, \(\sigma\), strain field, \(\varepsilon\), thermal field, \(T\), martensite fraction, \(\xi\), [PERSON]'s modulus of the SMA, \(E(\varepsilon,\xi,T)\), transformation tensor, \(\Omega(\xi)\), and thermal tensor, \(\Theta\), i.e.,  
\[d\sigma=E(\varepsilon,\xi,T)\;d\varepsilon+\Omega(\varepsilon,\xi,T)d\xi_{s} +\theta(\varepsilon,\xi,T)\;dT. \tag{1}\]  
The [PERSON]'s modulus of the SMA is expressed as [31]:  
\[E(\varepsilon,\xi,T)=E(\xi)=\frac{E_{A}}{1+\xi\Big{(}\frac{E_{A}}{E_{M}}-1 \Big{)}} \tag{2}\]  
where \(E_{A}\) and \(E_{M}\) are the [PERSON]'s moduli of a perfect austenite and martensite state, respectively. The martensite fraction is the summation of two parameters, i.e., the stress rise, \(\xi_{s}\), and temperature rise, \(\xi_{T}\). In addition, the transformation tensor is defined as:  
\[\Omega(\xi)=-\varepsilon_{L}E(\xi) \tag{3}\]  
where \(\varepsilon_{L}\) is the maximum residual strain of the SMA wires obtained during a uniaxial tension test. There are two phases to measure the martensite fractions, depending on temperature and stress: (I) the phase transformation to M (martensite), (II) the phase transformation from M (martensite) to A (austenite). Phase (I) is described by the following equations:  
If \(T>M_{s}\) and \(\sigma_{s}^{cr}+C_{M}(T-M_{s})<\sigma<\sigma_{f}^{cr}+C_{M}\Big{(}T-M_{f}\Big{)}\)  
\[ \begin{split}\xi_{s}=\frac{1-\xi_{s0}}{2}\cos\Big{(}\frac{\pi}{ \sigma_{s}^{cr}-\sigma_{f}^{cr}}(\sigma-\sigma_{f}^{cr}-C_{M}(T-M_{s}) )\Big{)}+\frac{1+\xi_{s0}}{2}\\ \xi_{T}=\xi_{T0}-\frac{\xi_{T0}}{1-\xi_{s0}}(\xi_{s}-\xi_{s0}) \end{split}  \tag{4}\]  
If \(T<M_{s}\) and \(\sigma_{s}^{cr}<\sigma<\sigma_{f}^{cr}\)  
\[ \begin{split}\xi_{s}=\frac{1-\xi_{s0}}{2}\cos\Big{(}\frac{\pi}{ \sigma_{s}^{cr}-\sigma_{f}^{cr}}(\sigma-\sigma_{f}^{cr})+\frac{1+\xi_{s0 }}{2}\\ \xi_{T}=\xi_{T0}-\frac{\xi_{T0}}{1-\xi_{s0}}(\xi_{s}-\xi_{s0})+ \Delta_{T\xi}\end{split}  \tag{5}\]  
If \(M_{f}<T<M_{s}\) and \(T<T_{0}\)  
\[\Delta_{T\xi}=\frac{1-\xi_{T0}}{2}\Big{(}\cos(a_{M}(T-M_{f}))+1\Big{)} \tag{6}\]  
Otherwise, \(\Delta_{T\xi}=0\). Phase (II), instead, accounts for the following relations:If \(T>A_{s}\) and \(C_{A}\Big{(}T-A_{f}\Big{)}<\sigma<C_{A}(T-A_{S})\)  
\[ \begin{split}\xi=&\frac{\xi_{0}}{2}\Big{(}\cos\Big{[} \frac{\pi}{A_{f}-A_{s}}(T-A_{s}-\frac{\sigma}{C_{A}})\Big{]}+1\Big{)}\\ &\xi_{s}=\xi_{s0}-\frac{\xi_{0}}{\xi_{0}}(\xi_{0}-\xi)\\ &\xi_{T}=\xi_{T0}-\frac{\xi_{0}}{\xi_{0}}(\xi_{0}-\xi)\\ & C_{A}=\Big{(}\frac{dA_{s}}{d\sigma}\Big{)}^{-1},\,C_{M}=\Big{(} \frac{dM_{s}}{d\sigma}\Big{)}^{-1}\end{split}  \tag{7}\]  
where the subscript \(0\) refers to the initial state and \(M_{f}\), \(M_{s}\), \(A_{s}\), and \(A_{f}\) are the martensite finish temperature, martensite start temperature, austenite start temperature, and austenite finish temperature, respectively. \(C_{A}\) and \(C_{M}\) refer to the transformation constants for the austenite and martensite phases, respectively. More details regarding the mechanical properties of SMAs can be found in Table 1, accordingly to [30]. The relations defining the properties of a graphite-epoxy reinforced with SMAs (NiTi wires) with TD conditions are mentioned in Appendix A, whose properties are reported in Table 2, in line with [31].