# Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams
## 4 Results and Discussion
### Thermal Buckling Analysis
Figure 3: Variation of the critical thermal loading vs. \(h/L\) for TD and TID properties, and different values of \(\varepsilon_{0}\) (\(n=1\), \(h_{c}/h=0.8\) and \(V_{SMA}=0.2\)). Such sensitivity of the response to the input value of \(n\) seems to reduce gradually, for higher geometrical values of \(L/h\), until the critical thermal buckling becomes totally insensitive to TID or TD material properties. The influence of linear and uniform thermal distributions on the critical thermal buckling is shown in Figure 6, while varying the geometrical ratio, \(L/h\), from 10 up to 100. As clearly visible from the plots in this figure, the critical thermal buckling obtained for a linear thermal distribution is always higher than the one associated with a uniform distribution, under the same value of \(V_{SMA}\). At the same time, for a fixed thermal distribution, an increased value of \(V_{SMA}\) enables increased values of the critical thermal buckling, especially for higher \(L/h\) ratios. The effect of \(\varepsilon_{0}\), \(V_{SMA}\), and \(n\) on the critical thermal buckling in linear thermal distribution is shown in Figures 7-9, for \(L/h\) ranging from 10 to 100. According to Figures 7 and 8, the initial strain and the volume fraction both provide a similar effect on the critical thermal buckling. Indeed, in both figures the critical thermal buckling increases by increasing \(\varepsilon_{0}\) and \(V_{SMA}\). The effect of \(\varepsilon_{0}\) and \(V_{SMA}\) on the critical thermal buckling seems to increase for higher values of \(L/h\). Another parameter affecting the critical thermal buckling response can be the SMA layers thickness (\(h_{a}\)), as depicted in Figure 9 for both linear and uniform thermal distributions, while assuming \(n=1,h/L=0.04\), \(V_{SMA}=0.2\), and \(\varepsilon_{0}=2\%\). More specifically, the critical thermal buckling increases by increasing \(h_{a}\), whereas the critical thermal buckling for a linear distribution is always higher than the one associated with a uniform distribution. Figure 8: Variation of the critical thermal buckling vs. \(L/h\) for a linear distribution and different values of \(V_{SMA}\) (\(n=1\), \(h_{c}/h=0.8\) and \(\varepsilon_{0}=2\%\)). Figure 9: Variation of the critical thermal buckling vs. \(h_{a}/h\) for a linear and uniform thermal distribution (\(n=1\), \(h/L=0.04\), \(V_{SMA}=0.2\) and \(\varepsilon_{0}=2\%\)). Figure 11: Variation of the first dimensionless frequency vs. temperature for different SMA volume fractions and a uniform temperature (\(\epsilon_{0}=2\%\), \(h_{c}/h=0.5\) and \(n=1\)). Figure 10: Variation of the first dimensionless frequency vs. temperature for different values of SMA initial pre-strain and a uniform temperature (\(V_{SMA}=0.2\), \(h_{c}/h=0.5\) and \(n=1\)). The sensitivity of the vibration response to the core thickness is plotted in Figure 13, under the following assumptions: \(V_{SMA}=0.4\), \(n=2\), \(\varepsilon_{0}=2\%\), and a uniform thermal distribution. As can be seen in this figure, the core thickness has a very pronounced effect on the first dimensionless frequency. An increased core thickness seems to provide  
Figure 12: Variation of the first dimensionless frequency vs. temperature for different FGM volume fractions and a uniform temperature (\(V_{SMA}=0.4\), \(h_{c}/h=0.5\) and \(\varepsilon_{0}=2\%\)). Figure 13: Variation of the first dimensionless frequency vs. temperature for different values of FGM core-to-total thickness ratio and a uniform temperature distribution (\(V_{SMA}=0.4\), \(n=2\) and \(\varepsilon_{0}=2\%\)). an increased frequency response for lower temperatures, with a contrary variation of the response for temperatures higher than 200 \({}^{\circ}\)C. In Figure 14, we finally plot the variation of the first dimensionless frequency versus temperature by considering TD and TID conditions, under the double assumption of \(V_{SMA}=0.4\) and \(V_{SMA}=0.6\), respectively.