# Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation
## 3 Solution of Multilayer Pavement under Pavement-Unevenness Excitation
### Interlayer Transfer Relationship Based on the DVP Method  
In this section, the DVP method [30; 31; 32] is used to deduce the transfer relationship of a multilayer elastic system. Take any layer \(j\) in the layered elastic system as an example; its upper-interface coordinate is \(z=z_{j-1}\), and the lower-interface coordinate is \(z=z_{j}\). For layer \(j\), the general solution of the LM equation in any layer can be expressed as follows:  
\[ \begin{bmatrix}\mathbf{U}(z)\\ \mathbf{T}(z)\end{bmatrix} = \begin{bmatrix}\mathbf{E}_{11}&\mathbf{E}_{12}\\ \mathbf{E}_{21}&\mathbf{E}_{22}\end{bmatrix}\begin{bmatrix}<e^{z_{12}(z-z_{j})}>&\mathbf{ 0}\\ \mathbf{0}&<e^{z_{34}(z-z_{j})}>\end{bmatrix}\begin{bmatrix}\mathbf{C}_{+}\\ \mathbf{C}_{-}\end{bmatrix}  \tag{38}\]  
In the above formula, \(s_{i}\) is the eigenvalue of the coefficient matrix in the system of ordinary differential equations of LM-type, Real (\(s_{1}\)) \(\geq\) Real (\(s_{2}\)) \(\geq\) Real (\(s_{3}\)) \(\geq\) Real (\(s_{4}\)) among the eigenvalues, and [\(E_{ij}\)] is the constant matrix composed of the eigenvectors corresponding to the eigenvalues. [C\({}_{+}\)] and [C\({}_{-}\)] represent matrices of undetermined coefficients determined by the interface and boundary conditions, and the diagonal matrix is defined as follows:  
\[ \begin{array}{l}<e^{s_{12}(z-z_{j})}>=diag[e^{s_{12}},e^{s_{22}}]\\ <e^{s_{34}(z-z_{j})}>=diag[e^{s_{32}},e^{s_{4}z}]\end{array}  \tag{39}\]  
First, the coefficient of the given \(j\) layer is eliminated, and the following layer relationship is obtained:  
\[ \begin{bmatrix}\mathbf{U}(z_{j-1})\\ \mathbf{T}(z_{j})\end{bmatrix} = \begin{bmatrix}\mathbf{S}_{11}^{j}&\mathbf{S}_{12}^{j}\\ \mathbf{S}_{21}^{j}&\mathbf{S}_{22}^{j}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j})\\ \mathbf{T}(z_{j-1})\end{bmatrix}  \tag{40}\]  
where the matrix [\(S_{ij}^{j}\)] is expressed as  
\[ \begin{bmatrix}\mathbf{S}_{11}^{j}&\mathbf{S}_{12}^{j}\\ \mathbf{S}_{21}^{j}&\mathbf{S}_{22}^{j}\end{bmatrix} = \begin{bmatrix}\mathbf{E}_{11}<e^{-s_ {12}h_{j}}>&\mathbf{E}_{12}\\ \mathbf{E}_{21}&\mathbf{E}_{22}<e^{s_{34}h_{j}}>\end{bmatrix}\begin{bmatrix}\mathbf{E}_{11} &\mathbf{E}_{12}<e^{s_{34}h_{j}}>\\ \mathbf{E}_{21}<e^{-s_{12}h_{j}}>&\mathbf{E}_{22}\end{bmatrix} ^{-1} \tag{41}\]  
For layer \(j+1\), the following relationship can be obtained:  
\[ \begin{bmatrix}\mathbf{U}(z_{j})\\ \mathbf{T}(z_{j+1})\end{bmatrix} = \begin{bmatrix}\mathbf{S}_{11}^{j+1}&\mathbf{S}_{12}^{j+1 }\\ \mathbf{S}_{21}^{j+1}&\mathbf{S}_{22}^{j+1}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j+1}) \\ \mathbf{T}(z_{j})\end{bmatrix}  \tag{42}\]  
When the interface \(z_{j}\) between layer \(j\) and layer \(j+1\) is a completely continuous interface, the transfer relationship between the interface \(\mathrm{z}_{j-1}\) and the interface \(\mathrm{z}_{j}\) is as follows:  
\[ \begin{bmatrix}\mathbf{U}(z_{j-1})\\ \mathbf{T}(z_{j+1})\end{bmatrix} = \begin{bmatrix}\mathbf{S}_{11}^{j+1}&\mathbf{S}_{12}^{j+1 }\\ \mathbf{S}_{21}^{j+1}&\mathbf{S}_{22}^{j+1}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j+1}) \\ \mathbf{T}(z_{j-1})\end{bmatrix}  \tag{43}\]  
where the transfer matrix [\(\mathbf{S}_{ij}^{j\pm 1}\)] is  
\[\begin{array}{l}\mathbf{S}_{11}^{j+1}=\mathbf{S}_{11}^{j}\left[\mathbf{I}-\mathbf{S}_{12}^{j +1}\mathbf{S}_{21}^{j}\right]^{-1}\mathbf{S}_{12}^{j+1}\\ \mathbf{S}_{12}^{j+1}=\mathbf{S}_{11}^{j}\left[\mathbf{I}-\mathbf{S}_{12}^{j+1}\mathbf{S}_{21}^{j} \right]^{-1}\mathbf{S}_{12}^{j}+\mathbf{S}_{12}^{j}\\ \mathbf{S}_{21}^{j+1}=\mathbf{S}_{21}^{j+1}+\mathbf{S}_{22}^{j+1}\left[\mathbf{I}-\mathbf{S}_{12}^{j +1}\mathbf{S}_{21}^{j}\right]^{-1}\mathbf{S}_{21}^{j}\mathbf{S}_{11}^{jWhen the interface is not completely continuous, the relationship between the expansion coefficients on both sides of the interface \(z_{j}\) (\(z_{j-}\) and \(z_{j+}\)) is as follows:  
\[ \begin{bmatrix}\mathbf{U}(z_{j-})\\ \mathbf{T}(z_{j+})\end{bmatrix} = \begin{bmatrix}\mathbf{I}&\mathbf{\alpha}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j+})\\ \mathbf{T}(z_{j-})\end{bmatrix} =