## 6 Results and Discussion
### Parametric Study  
We perform, now, a parametric study focusing on the sensitivity of the frequency response \(\Delta\omega\%=\left(\omega_{c}-\omega_{epoxy}\right)/\omega_{epoxy}\times 100\) against the weight fraction of GPLs for each selected distribution pattern (Figure 7), with a clear beneficial effect on the structural stiffness and stability as \(\Lambda_{GPL}\) increases within the material. Moreover, due to the presence of high normal stresses in the upper and lower sides of multilayered plates, a GPL-X distribution with an increased quantity of GPLs causes a hardening effect on the system, together with higher natural frequencies. On the other hand, a higher concentration of the reinforcing phase in the middle surface of the plate increases the structural deformability monotonically, whose variation rate is plotted in Figure 7, with a perfect agreement with predictions by [PERSON] et al. [16]. A further parameter affecting the frequency response is the number of layers \(N_{L}\) in the structure, as plotted in Figure 8 for three different reinforcement distributions. Except for the uniform reinforcement case for which the response is unaffected by \(N_{L}\), a small increase of this parameter can cause some hardening or softening effects on the structural stiffness, with a sharp increase or decrease of the frequency, for a GPL-X or GPL-O distribution, respectively. Even for these two distributions, the solutions stabilize for a number of layers equal or higher than 10, as also predicted by [PERSON] et al. [16]. The sensitivity of the response to the number of layers in the thickness direction is plotted in Figure 9 in terms of dimensionless vibrational frequency, for a square plate with \(a/h=25\) and \(\Lambda_{GPL}=0.3\%\) under a completely-clamped (CCCC) and simply-supported (SSSS) boundary condition. Based on a comparative evaluation of the plots in these two figures, the best stability response seems to be reached for a CCCC multilayered structure with \(N_{L}=10\) and a GPL-X reinforcement distribution, in terms of vibration frequency (see Figure 9). At the same time, an increased number of layers more than 10 becomes deleterious for the overall stability of plates with a GPL-O type distribution, both for CCCC and SSSS boundary conditions. Figure 10 depicts the variation of the first vibration frequency against the thermal gradient \(\Delta T\) of moderately thick square plates for different GPL weight fractions. Due to the variation of the thermoelastic properties of the system, together with the presence of initial internal stresses and strains in the structure by thermal attacks, this complex environment causes a combined hardening-softening impact on the system. More specifically, an increased thermal variation decreases the fundamental frequency of the system up to a certain value of \(\Delta\Gamma\), for which the fundamental frequency becomes zero, and the structure undergoes a static instability phenomenon. This critical temperature moves towards higher values for increased weight fractions of GPLs (from 0.2% up to 0.8%). It is worth also noticing that in the pre-divergence zone, by approaching the static instability phenomenon, an enhanced temperature causes a very fast reduction of the vibrational frequency of the system. In order to survey the effect of the Kerr foundation on the vibrational behavior and static instability of GPL-reinforced plates, we plot the variation of dimensionless first fundamental frequency versus the thermal variation for different substrate coefficients while selecting an SSSS and CCCC boundary condition, respectively, in Figure 11a,b. As the presence of an elastic foundation can vary the bending stiffness of a structure, we note that an enhanced value of the foundation constants improves the vibrational behavior of the system. Meanwhile, the divergence instability can be delayed by increasing the substrate coefficients values. In other words, the presence of an elastic medium gets higher critical values at which the static instability phenomenon takes place.