# Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams
## 4 Results and Discussion
### Thermal Buckling Analysis  
In this section, we assess the sensitivity of the buckling response to the length ratio (\(h/L\)), volume fraction power (\(n\)), initial strain (\(\epsilon_{0}\)), volume fraction (\(V_{SMA}\)), and thermal distribution. Figures 3 and 4 show the monotone variation of the critical thermal buckling versus \(h/L\) for different values of initial strain and volume fraction of SMA wires within the composite layers, respectively. In both figures, we account for both the TD- and TID- properties of the material, together with a uniform thermal distribution. According to the figures, by increasing \(\epsilon_{0}\) and \(V_{SMA}\), the critical thermal buckling increases. In addition, the value of the critical thermal buckling for TID-properties is higher than that one for TD properties. Based on the plots in both figures, the critical solutions based on TD and TID material properties assume the same value for low geometrical ratios (i.e., for \(h/L=0.01\)), while increasing their differences for higher values of \(h/L\) ratios. An increased value of \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(L/h\)} & \multicolumn{6}{c}{\(n\)} \\ \cline{2-9}  & \multicolumn{2}{c}{**0**} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**2**} & \multicolumn{2}{c}{**10**} \\ \cline{2-9}  & **Current Study** & [39] & **Current Study** & [39] & **Current Study** & [39] & **Current Study** & [39] \\ \hline
0.1 & 2365.8 & 2212.88 & 1030.1 & 959.01 & 854.7 & 797.03 & 901 & 851.48 \\
0.05 & 596.40 & 545.72 & 255.09 & 232.72 & 211.17 & 192.66 & 225.20 & 206.22 \\
0.025 & 142.40 & 128.93 & 57.03 & 51.14 & 46.45 & 41.56 & 50.08 & 44.91 \\
0.013 & 33.40 & 29.51 & 9.53 & 7.84 & 6.93 & 5.52 & 7.93 & 6.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Critical bucking load of FGM beam with S-S supports for different value of n and L\(h\). \(\varepsilon_{0}\) and \(V_{SMA}\) enables a shift of the curves upwards, with an overall increase in the critical thermal buckling for both TD and TID material properties. In addition, Figure 5 depicts the variation of the critical thermal buckling versus \(L/h\) for different values of \(n\), under the double assumption of TD or TID material, together with a uniform thermal distribution and fixed values of \(V_{SMA}=0.2\), \(\varepsilon_{0}=2\%\), and \(h_{c}/h=0.8\). As shown in the plots of Figure 5, an enhanced value of \(n\) yields an overall increase in the critical thermal buckling, for both TID and TD material properties. Additionally, for a fixed value of \(n\) and \(L/h\), the critical thermal buckling associated with TID material properties is always higher than the one related to TD material properties. The main differences among TD- and TID-based results rely on the different phenomenological models, where stress and strain distributions can be clearly affected by temperature together with any possible phase transformation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(L/h\)} & \multirow{2}{*}{**Result**} & \multicolumn{5}{c}{\(n\)} \\ \cline{3-8}  & & **0** & **0.1** & **1** & **2** & **10** \\ \hline \multirow{3}{*}{5} & [PERSON] et al. [41] & 6.8470 & 6.4990 & 4.8210 & 4.2510 & 3.7370 \\  & [PERSON] et al. [40] & 6.5632 & 6.2372 & 4.6533 & 4.1025 & 3.5610 \\  & Present work & 6.5134 & 6.1445 & 4.6234 & 4.0671 & 3.4966 \\  & [PERSON] et al. [41] & 6.9510 & 6.5990 & 4.9070 & 4.3340 & 3.8040 \\
20 & [PERSON] et al. [40] & 6.9313 & 6.5808 & 4.8950 & 4.3234 & 3.7914 \\  & Present work & 6.9301 & 5.5251 & 4.8942 & 4.2330 & 3.7913 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: First dimensionless frequency for FGM beam with an S-S boundary condition. Figure 2: Variation of the critical temperature versus \(h/L\) for two types of thermal distribution. Figure 4: Variation of the critical thermal loading vs. \(h/L\) for TD and TID properties and different values of \(V_{SMA}\) (\(n=1\), \(h_{c}/h=0.8\) and \(\varepsilon_{0}=2\%\)).