## 4 Numerical Results
### Effect of the Power-Law Parameter  
Here, we study the effect of the power-law parameter on the nonlinear vibration behavior of the nanocomposite structure. Figure 5 depicts the variation of the nonlinear frequency as a function of the power-law parameter for different thickness ratios, \(h_{c}/h_{t}\), and vibration amplitudes \(a/h_{t}\). Note that, for a small amplitude of vibration, the nonlinear frequency of the system increases by increasing the power-law parameter. The effect of an increasing power-law parameter on the vibration response of the system is different depending on the amount of the thickness ratio. For an increased amplitude of the vibration up to a threshold value, the results become non-uniform for an increased power-law parameter. An increased amplitude of vibration will completely change the behavior of the system; namely, by increasing the power-law parameter, the nonlinear frequency of the GPL-reinforced sandwich beam decreases for a large amplitude of vibration.  
In Figure 6, we plot the variation of the nonlinear-to-linear frequency ratio of the nanocomposite structure vs. the power-law parameter. It is worth noting that an increased power-law parameter has a decreasing effect on the nonlinear-to-linear frequency ratio of the system. These effects become even more pronounced for larger vibration amplitudes, while leaving the overall behavior almost unaltered.  
Figure 5: Variation of the first-order nonlinear frequency of a GPL-reinforced sandwich beam due to increasing power-law parameter for different core-to-beam thickness ratios and amplitudes of vibration (\(N=10,\Gamma_{b}=1\%,L/h_{t}=10\)).  
In Figures 7 and 8, the nonlinear frequency and the nonlinear-to-linear frequency ratio of a GPL-reinforced sandwich beam are plotted vs. the power-law parameter, while assuming different thickness ratios and total weight fractions of the GPL phase in the beam.  
Figure 6: Variation of the first-order nonlinear-to-linear frequency ratio of a GPL-reinforced sandwich beam due to an increased power-law parameter for different core-to-beam thickness ratios and amplitudes of vibration (\(N=10,\Gamma_{b}=1\%,L/h_{t}=10\)).  
Figure 7: Variation of the first-order nonlinear frequency of a GPL-reinforced sandwich beam due to an increasing power-law parameter, for different core-to-beam thickness ratios and total amount of GPLs reinforcement in the beam (\(N=10,a/h_{t}=1,L/h_{t}=10\)).  
According to Figure 7, for a small amount of the GPLs weight fraction, the nonlinear frequency increases by increasing the power-law parameter. Moreover, the effect of an increasing power-law parameter becomes more pronounced for sandwich beams with thick face sheets. As the total weight fraction of GPLs increases, a different response is noticed, in terms of nonlinear frequency, by increasing the power-law parameter. More specifically, for a large GPL weight fraction, we notice a threshold value after which the nonlinear frequency decreases by increasing the power-law parameter. Of course, the value of the threshold point varies with the thickness of the face sheets. On the other hand, the results for the nonlinear-to-linear frequency ratio show some opposite effects for an increasing power-law parameter. An increased total amount of GPL reinforcement phase in the face sheets has a pronounced effect on the general behavior of the system, which has to be studied carefully.