## 3 Theoretical Formulations of Composite Plates  
### Governing Equations in the State Space  
Based on the first-order shear deformation plate theory (FSDT), the displacements can be described as  
\[u_{1}(x,y,z,t)=u(x,y,t)+z\varphi_{x}(x,y,t) \tag{7a}\] \[u_{2}(x,y,z,t)=v(x,y,t)+z\varphi_{y}(x,y,t)\] (7b) \[u_{3}(x,y,z,t)=w(x,y,t) \tag{7c}\]  
where \(t\) is the time, \(u_{1}\), \(u_{2}\), \(u_{3}\) are the displacements at any point \((x,y,z)\), \(u\), \(v\), \(w\) are the displacements of the point \((x,y,0)\) on the mid-plane, \(\varphi_{x}\) and \(\varphi_{y}\) are the rotations of the normal about the \(y\)- and \(x\)-axis, respectively. The stress-strain relationship is given by the following expressions,  
\[\left[ \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \tau_{xy}\end{array} \right]=\left[ \begin{array}{ccc}Q_{11}&Q_{12}&0\\ Q_{21}&Q_{22}&0\\ 0&0&Q_{66}\end{array} \right]\left[ \begin{array}{c}u_{x}+z\varphi_{x,x}\\ v_{y}+z\varphi_{y,y}\\ u_{y}+v_{x}+z(\varphi_{x,y}+\varphi_{y,x})\end{array} \right] \tag{8a}\] \[\left[ \begin{array}{c}\tau_{xz}\\ \tau_{yz}\end{array} \right]=\kappa\left[ \begin{array}{cc}Q_{44}&0\\ 0&Q_{55}\end{array} \right]\left[ \begin{array}{c}w_{x}+\varphi_{x}\\ w_{y}+\varphi_{y}\end{array} \right] \tag{8b}\]  
Figure 2: Different GPL distribution patterns. in which the shear correction coefficient \(\kappa\) is assumed to be 5/6 for FSDT by [PERSON] [21]. The position-dependent elastic constants \(Q_{ij}\) are described as  
\[Q_{11}=\frac{E_{11}(z)}{1-\
u_{12}\
u_{21}},\ \ Q_{22}=\frac{E_{22}(z)}{1-\
u_{1 2}\
u_{21}},\ \ Q_{12}=\frac{\
u_{12}E_{22}(z)}{1-\
u_{12}\
u_{21}} \tag{9a}\] \[Q_{44}=G_{13}(z),\ \ Q_{55}=G_{23}(z),\ \ Q_{66}=G_{12}(z) \tag{9b}\]  
The governing equations of motion for the FG-GPLs reinforced plate [21] can be achieved  
\[N_{x,x}+N_{xy,y}=I_{1}\ddot{u}+I_{2}\tilde{\varphi}_{x} \tag{10a}\] \[N_{xy,x}+N_{y,y}=I_{1}\ddot{v}+I_{2}\tilde{\varphi}_{y}\] (10b) \[Q_{x,x}+Q_{y,y}=I_{1}\ddot{w}\] (10c) \[M_{x,x}+M_{xy,y}-Q_{x}=I_{2}\ddot{u}+I_{3}\ddot{\varphi}_{x}\] (10d) \[M_{xy,x}+M_{y,y}-Q_{y}=I_{2}\ddot{v}+I_{3}\ddot{\varphi}_{y} \tag{10e}\]  
where the over dot represents partial derivative of time \(t\). The stress or moment resultants and the inertia-related terms are defined by  
\[(N_{x},N_{y},N_{xy},Q_{x},Q_{y})=\int_{-h/2}^{h/2}\left(\sigma_{xx},\sigma_{yy },\tau_{xy},\tau_{xz},\tau_{yz}\right)\mathrm{d}z \tag{11a}\] \[(M_{x},M_{y},M_{xy})=\int_{-h/2}^{h/2}\left(\sigma_{xx},\sigma_{yy},\tau_{xy} \right)z\mathrm{d}z\] (11b) \[Q_{11}=\frac{E_{11}(z)}{1-\
u_{12}\
u_{21}},\ \ Q_{22}=\frac{E_{22}(z)}{1-\
u_{1 2}\
u_{21}},\ \ Q_{12}=\frac{\
u_{12}E_{22}(z)}{1-\
u_{12}\
u_{21}} \tag{9a}\]0  
Using Equations (7)-(9) and (11), we can reconstruct Equation (10) in the state space form as  
\[\frac{\partial}{\partial x}\mathbf{\Theta}=\mathbf{\Sigma}\mathbf{\Theta}= \left[ \begin{array}{cc}\mathbf{\Sigma}_{1}&\mathbf{\Sigma}_{2}\\ \mathbf{\Sigma}_{3}&\mathbf{\Sigma}_{4}\end{array} \right]\mathbf{\Theta} \tag{12}\]  
where \(\mathbf{\Theta}=[u,v,w,\varphi_{x},\,\varphi_{y},M_{x},M_{xy},N_{x},N_{xy},Q_{x} ]^{\mathrm{T}}\) is the vector, and the coefficient matrices \(\mathbf{\Sigma}_{i}\) (\(i=1,\,\ldots,\,4\)) are given in Appendix A. For the wave propagating along the \(y\) direction, the vector in the governing equations can be denoted in the following dimensionless form  
\[\mathbf{\Theta}=\left[ \begin{array}{c}u\\ v\\ w\\ \varphi_{x}\\ \varphi_{y}\\ M_{x}\\ M_{xy}\\ N_{x}\\ N_{x}\\ N_{xy}\\ Q_{x}\end{array} \right]=\left\{ \begin{array}{c}h\Pi(\xi)\\ h\vec{v}(\xi)\\ h\vec{w}(\xi)\\ q_{\xi}(\xi)\\ \varphi_{\eta}(\xi)\\ E_{\mathrm{M}}h^{2}\overline{M}_{\xi}(\xi)\\ E_{\mathrm{M}}h^{2}\overline{M}_{\xi\eta}(\xi)\\ E_{\mathrm{M}}h\overline{N}_{\xi}(\xi)\\ E_{\mathrm{M}}h\overline{N}_{\xi\eta}(\xi)\\ E_{\mathrm{M}}h\overline{N}_{\xi\eta}(\xi)\\ E_{\mathrm{M}}h\overline{Q}_{\xi}(\xi)\end{array}