## 2 Effective Material Properties  
Consider an FG-GRC plate with length \(a\), width \(b\), and thickness \(h\) (Figure 1) using the cartesian coordinate system \(x\), \(y\), \(z\), with plane \(z\) = 0, coinciding with the midplane of the plate. \(N_{L}\) is the total number of layers of FG-GRC plates, and it is worth noting that this paper is only for FG-GRC plates with an even number of layers. GPLs are used as fillers and four distribution patterns proposed by [PERSON] et al. [12; 43] are used in this paper. In Figure 2, taking the plate with \(N_{L}=6\) as an example, a schematic diagram of the four distribution modes of GPLs is given [12; 43]. In the UD mode, the GPLs content is uniformly distributed. In the FG-O mode, the GPLs content decrease from the middle surface to the surface of the plate. In the FG-X mode, the distribution is opposite to in the FG-O mode, and the GPLs content decreases from the surface to the middle surface of the plate. In the FG-A mode, the GPLs content increases from the upper surface to the lower surface. The effective Young's modulus of the \(k\)th layer of the FG-GRC plates, \(E_{c}^{(k)}\), can be calculated based on the Halpin-Tsai micromechanics model [44] and Voigt-Reuss model [45], as follows:  
\[E_{c}^{(k)}=\frac{3}{8}\times\frac{1+\xi_{\mathrm{L}}\eta_{\mathrm{L}}V_{ \mathrm{G}}^{(k)}}{1-\eta_{\mathrm{L}}V_{\mathrm{G}}^{(k)}}\times E_{\mathrm{M }}+\frac{5}{8}\times\frac{1+\xi_{\mathrm{W}}\eta_{\mathrm{W}}V_{\mathrm{G}}^{ (k)}}{1-\eta_{\mathrm{W}}V_{\mathrm{G}}^{(k)}}\times E_{\mathrm{M}} \tag{1}\]  
where  
\[\eta_{\mathrm{L}}=\frac{\left(E_{\mathrm{G}}/E_{\mathrm{M}}\right)-1}{\left( E_{\mathrm{G}}/E_{\mathrm{M}}\right)+\xi_{\mathrm{L}}},\eta_{\mathrm{W}}=\frac{ \left(E_{\mathrm{G}}/E_{\mathrm{M}}\right)-1}{\left(E_{\mathrm{G}}/E_{ \mathrm{M}}\right)+\xi_{\mathrm{W}}} \tag{2}\]  
\(k=1\), \(2\),..., \(N_{L}\), and \(E_{\mathrm{G}}\), \(E_{\mathrm{M}}\) represent [PERSON]'s modulus of GPLs and polymer matrix, respectively. \(\xi_{\mathrm{L}}\) and \(\xi_{\mathrm{W}}\) are the parameters associated with the geometry and size of the GPLs, defined as  
\[\xi_{\mathrm{W}}=2\left(\frac{w_{\mathrm{G}}}{h_{\mathrm{G}}}\right),\xi_{ \mathrm{L}}=2\left(\frac{l_{\mathrm{G}}}{h_{\mathrm{G}}}\right) \tag{3}\]  
with \(l_{\mathrm{G}}\), \(w_{\mathrm{G}}\), and \(h_{\mathrm{G}}\) being the average length, width, and thickness of the GPLs. In addition, the volume fraction \(V_{\mathrm{G}}^{(k)}\) of GPLs in the \(k\)th layer is defined as follows:  
\[V_{\mathrm{G}}^{(k)}=\frac{f_{\mathrm{G}}^{(k)}}{f_{\mathrm{G}}^{(k)}+\left( \rho_{\mathrm{G}}/\rho_{\mathrm{M}}\right)\left(1-f_{\mathrm{G}}^{(k)}\right)} \tag{4}\]  
where \(f_{\mathrm{G}}^{(k)}\) is GPLs weight fraction of the \(k\)th layer, \(\rho_{\mathrm{M}}\) and \(\rho_{\mathrm{G}}\) denote the mass densities of the polymer matrix and GPLs, respectively. According to the rule of mixture, the effective density and Poisson's ratio of the \(k\)th layer of the FG-GRC plates are defined as follows:  
\[\rho_{c}^{(k)}=\rho_{\mathrm{G}}V_{\mathrm{G}}^{(k)}+\rho_{\mathrm{M}}\Big{(} 1-V_{\mathrm{G}}^{(k)}\Big{)},v_{c}^{(k)}=v_{\mathrm{G}}V_{\mathrm{G}}^{(k)}+v _{\mathrm{M}}\Big{(}1-V_{\mathrm{G}}^{(k)}\Big{)} \tag{5}\]  
Figure 1: Mode of a functionally graded graphene reinforced composite (FG-GRC) plate. Figure 2: Different distribution modes of graphene nanoplatelets (GPLs). \(v_{\text{G}}\) and \(v_{\text{M}}\) are [PERSON]'s ratios of GPLs and polymer matrix, respectively. Assuming \(N_{L}\) as an even number, the GPLs weight fraction of the \(k\)th layer for the four distribution modes is as follows [43]:  
\[f_{\text{G}}^{(k)}=\left\{ \begin{array}{ll}f_{\text{G}}&\text{UD}\\ 4f_{\text{G}}\Big{(}\frac{N_{L}+1}{2}-\Big{|}k-\frac{N_{L}+1}{2}\Big{|}\Big{)} \big{/}(2+N_{L})&\text{FG}-\text{O}\\ 4f_{\text{G}}\Big{(}\frac{1}{2}+\Big{|}k-\frac{N_{L}+1}{2}\Big{|}\Big{)}\big{/} (2+N_{L})&\text{FG}-\text{X}\\ 2 kf_{\text{G}}/(N_{L}+1)&\text{FG}-\text{A}\end{array} \right.