## 3 Theoretical Formulations of Composite Plates
### MRRM Formulation  
Owing to the constant state matrix \(\overline{\mathbf{\Sigma}}\), we can describe the general solution of Equation (14) as  
\[\overline{\mathbf{\Theta}}(\xi)=\mathbf{\
abla}\mathrm{exp}(\Lambda_{\xi}^{\mathrm{T} })\mathbf{\gamma} \tag{16}\]  
AS stated in the eigenproblem in alignment of molecules [28], \(\mathbf{\Lambda}=\mathrm{diag}(\lambda_{1},\lambda_{2},\cdots,\lambda_{10})\) is a diagonal matrix with \(\lambda_{i}\) (\(i\) =1,2,...,10) being the eigenvalues of the coefficient matrix \(\overline{\mathbf{\Sigma}}\), \(\mathbf{\
abla}=[\mathbf{\
abla}_{1},\mathbf{\
abla}_{2},\mathbf{\
abla}_{3},\cdots,\mathbf{ \
abla}_{10}]\) are the eigenvectors, and \(\mathbf{\gamma}\) are the wave amplitude vectors. Due to the physical meaning of wave propagation, the coefficient matrix \(\overline{\mathbf{\Sigma}}\) is a Hamiltonian matrix, so the eigenvalue matrix can always be separated as follows,  
\[\mathbf{\Lambda}=\left[ \begin{array}{cc}\mathbf{\Lambda}_{+}&\mathbf{0}\\ \mathbf{0}&\mathbf{\Lambda}_{-}\end{array} \right] \tag{17}\]  
where \(\mathbf{\Lambda}_{+}\) comprises five eigenvalues with positive real part or positive imaginary part for pure imaginary, while \(\mathbf{\Lambda}_{-}\) contains the remaining five eigenvalues, and \(\mathbf{\Lambda}_{-}=-\mathbf{\Lambda}_{+}\). Similarly, we can divide the wave amplitudes \(\mathbf{\gamma}\) to \(\mathbf{\gamma}=\left[\left(\mathbf{\gamma}_{+}\right)^{\mathrm{T}},\left(\mathbf{\gamma}_ {-}\right)^{\mathrm{T}}\right]^{\mathrm{T}}\). So, the solution can be rewritten as  
\[\overline{\mathbf{\Theta}}(\xi)=\left[\mathbf{\
abla}_{+}\mathbf{\
abla}_{-}\right]\left[ \begin{array}{cc}\exp(\mathbf{\Lambda}_{+}\xi)&\mathbf{0}\\ \mathbf{0}&\exp(\mathbf{\Lambda}_{-}\xi)\end{array} \right]\left\{ \begin{array}{c}\bm {\gamma}_{+}\\ \mathbf{\gamma}_{-}\end{array} \right\} \tag{18}\]  
Before using MRRM, we should introduce the dual local coordinates first. For one element problem shown in Figure 3, the two local axes \(\xi_{1}\), \(\xi_{2}\) are in opposite directions. In the two local coordinates \(\xi_{1}\) and \(\xi_{2}\), the solution to Equation (18) can be rewritten as  
\[\overline{\mathbf{\Theta}}^{1}(\xi_{1})=\left[\mathbf{\
abla}_{+}^{1}\mathbf{\
abla}_{-} ^{1}\right]\left[ \begin{array}{cc}\exp(\mathbf{\Lambda}_{+}^{1}\xi_{1})&\mathbf{0} \\ \mathbf{0}&\exp(\mathbf{\Lambda}_{-}^{1}\xi_{1})\end{array} \right]\left\{ \begin{array}[] {c}\mathbf{\gamma}_{+}^{1}\\ \mathbf{\gamma}_{-}^{1}\end{array} \right\} \tag{19a}\] \[\overline{\mathbf{\Theta}}^{2}(\xi_{2})=\left[\mathbf{\
abla}_{+}^{2}\mathbf{\
abla}_{-} ^{2}\right]\left[ \begin{array}{cc}\exp(\mathbf{\Lambda}_{+}^{2}\xi_{2})&\mathbf{0} \\ \mathbf{0}&\exp(\mathbf{\Lambda}_{-}^{2}\xi_{2})\end{array} \right]\left\{ \begin{array}[] {c}\mathbf{\gamma}_{+}^{2}\\ \mathbf{\gamma}_{-}^{2}\end{array} \right\} \tag{19b}\]  
Figure 3: Dual-coordinate systems for a typical layer or element. According to \(1-\xi_{1}=\xi_{2}\) in the dual local coordinate system in Figure 3, the relationship between \(\overline{\mathbf{\Theta}}^{1}\) and \(\overline{\mathbf{\Theta}}^{2}\) can be derived as  
\[\overline{\mathbf{\Theta}}^{1}(1-\xi_{1})=\mathbf{T}_{f}\overline{\mathbf{\Theta}}^{2}( \xi_{1}) \tag{20}\]  
where \(\mathbf{T}_{f}\) is the transformation matrix. Plugging Equation (20) into Equation (14) yields the following relationships [29]  
\[\overline{\mathbf{\Sigma}}^{2}=-\mathbf{T}_{f}\overline{\mathbf{\Sigma}}^{1}\mathbf{T} _{f}^{-1},\ \mathbf{\Lambda}^{2}=-\mathbf{\Lambda}^{1},\ \mathbf{\Psi}^{2}=\mathbf{T}_{f}\mathbf{\Psi}^{1} \tag{21}\]  
Combining Equations (19)-(21) yields  
\[\left\{ \begin{array}{l}\gamma_{+}^{1}\\ \gamma_{+}^{2}\end{array} \right\}=\left[ \begin{array}{cc}\mathbf{0}&\exp(-\mathbf{ \Lambda}_{+}^{1})\\ \exp(\mathbf{\Lambda}_{-}^{1})&\mathbf{0}\end{array} \right]\left\{ \begin{array}{l} \gamma_{-}^{1}\\ \gamma_{-}^{2}\end{array} \right\}=\mathbf{P}\left\{ \begin{array}{l}\gamma_{ -}^{1}\\ \gamma_{-}^{2}\end{array}