## 3 Results and Discussion
### Investigating the Influence of SMA Volume Fraction  
In this section, the influence of the volume fraction of SMA in the surface layer on the natural frequency of composite sandwich plate is examined. We selected 20% and 40% as representative volume fractions of SMA. In the case of negative, uniform, and positive gradient distribution of graphite in the core layer, Figure 8 analyzes the variation of the natural frequency of 20% and 40% SMA volume fraction in the surface layer with temperature.  
Figure 7: Dimensionless natural frequencies vs. the graphite volume fraction subjected to different temperature.  
As shown in Figure 8, the effect of graphite distribution on natural frequency was maintained. Moreover, the volume fraction of SMA had no significant effect on the natural frequency in the temperature range of 15 \({}^{\circ}\)C to 40 \({}^{\circ}\)C. The modulus of SMA in the low temperature region was within 30 GPa, so the volume fraction of SMA had no observable effect on the stiffness. Meanwhile, the recovery stress generated by the pre-strain in this process had a very weak effect on the stiffness. As the temperature was higher than 40 \({}^{\circ}\)C, the conversion of SMA from martensitic phase to austenitic phase lead to the rapid increase of the modulus. The stiffness of the sandwich plate with 40% SMA volume fraction was increased markedly relative to 20%. Recovery stress generated by SMA pre-strain also increased with the increase of volume fraction, so that the natural frequency of the sandwich plate increased. In addition, as the temperature rises, the natural frequency of \(V\)s = 40% first decreased, then increased, and then decreased to near zero. From Equation (31), it can be seen that the thermal stress decreased the stiffness, while the recovery stress increased the stiffness. The variation of natural frequency with temperature was due to the dominant role of thermal stress at 15 \({}^{\circ}\)C to 40 \({}^{\circ}\)C, while the recovery stress at 40 \({}^{\circ}\)C to 55 \({}^{\circ}\)C suddenly increased, and the thermal stress resumed its dominant role again after the temperature was higher than 55 \({}^{\circ}\)C.  
The variations of dimensionless natural frequencies with SMA volume fraction are depicted in the Figure 9. The plot reveals that at 5% to 25% SMA volume fraction, natural frequencies decreased greatly with the increase of temperature. As the SMA volume fraction increased to 50%, the natural frequency decreased slowly with the temperature rise, that is, the dimensionless natural frequency decreased by only 0.42 during the temperature rise from 25 \({}^{\circ}\)C to 100 \({}^{\circ}\)C. This is due to the increase of the recovery stress with the increase of the volume fraction of SMA, which increased the stiffness of the sandwich plate and the natural frequency, as well as the critical buckling temperature. Therefore, the increase of SMA volume fraction reduced the difference of frequency variation of the natural frequency in the same heating process. In addition, the volume fraction of SMA had a weak effect on the natural frequency at 25 \({}^{\circ}\)C to 55 \({}^{\circ}\)C, while the natural frequency of the gradient sandwich composite plate increases with the increase of volume fraction of SMA at 70 \({}^{\circ}\)C to 100 \({}^{\circ}\)C.  
Figure 8: A comparison on dimensionless natural frequencies between the SMA volume fraction of 20% and 40% subjected to negative/non/positive gradient.