## 2 Material Properties and Geometry  
A nanocomposite sandwich beam with length \(L\), thickness \(h_{t}\) and width \(b\) is considered, as shown in Figure 1. The Cartesian coordinate system \((x,z)\) is here used to derive the equations of motion, where the structural mid-plane is parallel to the \(x\)-axis. The beam is made of a homogeneous core and two face sheets with a symmetric GPL-based reinforcement, whose weight fraction satisfies the following power-law:  
\[\Gamma(z)=\Gamma_{\text{max}}\!\left(\frac{2|z|-h_{c}}{h_{t}-h_{c}}\right)^{ \kappa}, \tag{1}\]  
where \(\Gamma_{\text{max}}\) is the maximum value of the distribution function and \(\kappa\) is a power-law parameter, which defines the GPLs dispersion throughout the thickness of the face sheets.  
For a proper analysis, the total amount of GPLs in the beam remains constant independently of the distribution pattern (see Figure 2). This means that, if we keep constant the total amount of GPLs in the beam, \(\Gamma_{b}\), the maximum value of the GPL weight fraction, \(\Gamma_{\text{max}}\), increases by increasing the power-law parameter. Note that the total amount of reinforced GPLs decreases by increasing the power-law parameter if the maximum value of the GPL weight fraction is kept constant.  
Due to possible difficulties during the manufacturing process of a functional reinforced lamina, each face sheet is considered to be made of N layers with equal thickness and the GPL reinforcement is assumed to be uniform within each layer (see Figure 1).  
Therefore, the GPL weight fraction in the \(k\)th lamina can be defined as  
\[\Gamma^{(k)}=\Gamma_{\text{max}}\!\left(\frac{2|h_{k}|-h_{c}}{h_{t}-h_{c}} \right)^{\kappa}, \tag{2}\]  
where \(h_{k}\) is the distance between the mid-plane of the beam and the mid-plane of the \(k\)th layer.  
The volume fraction of the reinforced GPLs can be related to their weight fraction as  
\[V^{(k)}_{\text{GPL}}=\frac{\Gamma^{(k)}}{\Gamma^{(k)}+(\rho_{\text{GPL}}/\rho _{M})\!\left(1-\Gamma^{(k)}\right)}, \tag{3}\]  
Figure 1: General configuration and graphene nanoplatelet (GPL) dispersion description in a laminated GPL-reinforced sandwich beam.  
Figure 2: Variation of the dimensionless GPL weight fraction (\(\overline{\Gamma}=\Gamma(z)/\Gamma_{b}\)) through the thickness of the top face sheet with respect to various power-law parameters (\(\overline{h}=(2z-h_{c})/2h_{f}\)).  
\(\rho_{M}\) and \(\rho_{\rm GPL}\) being the mass density of the matrix and GPLs, respectively. Here, the Halpin-Tsai micromechanical model is adopted to define the effective elastic modulus of the reinforced face sheets [54]. Moreover, the GPL reinforcement is assumed to be randomly oriented in each lamina [27].  
Therefore, the elastic modulus for the \(k\)th layer can be expressed as  
\[E_{C}^{(k)}=\frac{3}{8}\bigg{(}\frac{1+\xi_{L}\eta_{L}V_{\rm GPL}^{(k)}}{1- \eta_{L}V_{\rm GPL}^{(k)}}\bigg{)}E_{M}+\frac{5}{8}\bigg{(}\frac{1+\xi_{W}\eta _{W}V_{\rm GPL}^{(k)}}{1-\eta_{W}V_{\rm GPL}^{(k)}}\bigg{)}E_{M}, \tag{4}\]  
where  
\[\eta_{W}=\frac{(E_{\rm GPL}/E_{M})-1}{(E_{\rm GPL}/E_{M})+\xi_{W}},\ \ \eta_{L}= \frac{(E_{\rm GPL}/E_{M})-1}{(E_{\rm GPL}/E_{M})+\xi_{L}}, \tag{5}\]  
\[\xi_{W}=\frac{2w_{\rm GPL}}{h_{\rm GPL}},\ \ \xi_{L}=\frac{2l_{\rm GPL}}{h_{\rm GPL }}, \tag{6}\]  
and \(h_{\rm GPL}\), \(l_{\rm GPL}\), \(w_{\rm GPL}\) stand for the average thickness, length, and width of GPLs, respectively; \(E_{M}\) and \(E_{\rm GPL}\) denote the Young modulus of the matrix and GPLs, respectively.  
Using the rule of mixtures, the effective mass density and Poisson's ratio for the \(k\)th layer can be defined as  
\[\rho_{C}^{(k)}=\rho_{\rm GPL}V_{\rm GPL}^{(k)}+\rho_{M}\bigg{(}1-V_{\rm GPL}^ {(k)}\bigg{)}, \tag{7}\]  
\[\
u_{C}^{(k)}=\
u_{\rm GPL}V_{\rm GPL}^{(k)}+\
u_{M}\bigg{(}1-V_{\rm GPL}^{(k) }\bigg{)}, \tag{8}\]  
\(\
u_{M}\) and \(\
u_{\rm GPL}\) being the Poisson's ratio of the matrix and GPLs, respectively.