## 2 Problem Formulation  
Figure 1 shows a functionally graded graphene composites plate attached with a piezoelectric patch. This plate and piezoelectric patch system consist of the base plate and the piezoelectric patch, which was used for vibration control. Simply supported at four edges, the rectangular plate was subjected to the supersonic flow where the incoming flow speed is expressed as \(U_{\infty}\). Here, \(a\), \(b\) and \(h\) represent the length, width, and thickness of the base plate. Figure 1: The model of the functionally graded GPLs reinforced plate with a piezoelectric patch. Assuming the volume fraction of the GPLs linearly changes from layer to layer along with the thickness direction of the plate, two kinds of GPLs distribution patterns are considered in the present study-- O-GPLs and X-GPLs distribution pattern--as shown in Figure 2. The O-GPLs distribution pattern means the GPL content reduces from the highest on plate' midplane of the to the lowest on plate's top and bottom surfaces. In comparison, the GPL content in the X-GPLs distribution pattern augments from the midplane to the top and bottom surfaces. The modified Halpin-Tsai model was utilized to calculate the effective Young's modulus of the GPL composite. Assume that the GPLs are effective rectangular solid fillers and uniformly dispersed in a polymer matrix. Consequently, the effective Young's modulus \(E_{c}\) of the GPLs can be approximately obtained by the Voigt Reuss model [37]  
\[E_{c}=\frac{3}{8}E_{\Delta}+\frac{5}{8}E_{\perp} \tag{1}\]  
where \(E_{\Delta}\) is the longitudinal modulus, and \(E_{\perp}\) is the transverse modulus. They can be determined by Halpin--Tsai model  
\[E_{\Delta}=\frac{1+\xi_{L}\eta_{L}V_{GPL}}{1-\eta_{L}V_{GPL}}\times E_{M}, \hskip 14.226378 ptE_{\perp}=\frac{1+\xi_{W}\eta_{W}V_{GPL}}{1-\eta_{W}V_{GPL}} \times E_{M} \tag{2}\]  
Substituting Equation (2) into Equation (1) yields  
\[E_{c}=\frac{3}{8}\frac{1+\xi_{L}\eta_{L}V_{GPL}}{1-\eta_{L}V_{GPL}}\times E_{ M}+\frac{5}{8}\frac{1+\xi_{W}\eta_{W}V_{GPL}}{1-\eta_{W}V_{GPL}}\times E_{M} \tag{3}\]  
where  
\[\eta_{L}=\frac{(E_{GPL}/E_{M})-1}{(E_{GPL}/E_{M})+\xi_{L}},\hskip 14.226378 pt \eta_{W}=\frac{(E_{GPL}/E_{M})-1}{(E_{GPL}/E_{M})+\xi_{W}} \tag{4}\]  
where \(E_{GPL}\) and \(E_{M}\) denote the [PERSON]'s moduli of the GPLs and polymer matrix, respectively. \(V_{GPL}\) means the volume fraction of GPLs. \(\xi_{L}\) and \(\xi_{W}\) are the geometry and size parameters of GPL nanofillers, which can be expressed as  
\[\xi_{L}=2(\frac{l_{GPL}}{h_{GPL}}),\hskip 14.226378 pt\xi_{W}=2(\frac{w_{GPL}}{h _{GPL}}) \tag{5}\]  
in which \(l_{GPL}\), \(w_{GPL}\) and \(h_{GPL}\) represent the average length, width, and thickness of the GPLs, respectively. Mass density \(\rho_{c}\) and [PERSON]'s ratio \(\mu_{c}\) of the GPL composite can be calculated by rule of mixture  
\[\rho_{c}=\rho_{GPL}V_{GPL}+\rho_{M}V_{M},\hskip 14.226378 pt\mu_{c}=\mu_{GPL}V_{ GPL}+\mu_{M}V_{M} \tag{6}\]  
Figure 2: Distribution pattern of GPLs: (**a**) X-GPLs, (**b**) O-GPLs. in which \(V_{M}\) represents the volume fraction of polymer matrix, subscripts \"\(GPL\)\", \"\(M\)\" and \"\(c\)\" respectively stand for GPLs, polymer matrix, and GPL composite. The volume fraction of GPLs \(V_{GPL}\) is given as  
\[V_{GPL}=\frac{g_{GPL}}{g_{GPL}+(\rho_{GPL}/\rho_{M})(1-g_{GPL})} \tag{7}\]  
where \(g_{GPL}\) is the weight fraction of GPLs in the nanocomposite. The strain-displacement relationships of the base plate are presented as  
Figure 2.0  
where \(\epsilon_{11}\), \(\epsilon_{22}\) and \(\epsilon_{12}\) are strains in \(x\) and \(y\) direction and the shear strain, respectively. \(w\) is the transverse displacement of the middle surface of the plate. On the basis of Generalized Hooke's Law, the relationship between strains and stresses is obtained  
Figure 2.1  
where \(\sigma_{11}\), \(\sigma_{22}\) and \(\sigma_{12}\), respectively, represent the normal stress components in \(x\)- and \(y\)-direction and shear stress component.