# Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation
## 3 Solution of Multilayer Pavement under Pavement-Unevenness Excitation
### Fundamental Governing Equation  
Since the maneuvering loads acting on the surface of the elastic half-space are multiple harmonic loads with angular frequency \(\omega_{l}\), then in the cylindrical coordinate system, all variables can be expressed as functions containing factors (\(\sum_{l=0}^{L}P_{l}\cdot e^{i(\omega_{l}t+\theta_{l})}\)) [34]; for example  
\[ \begin{array}{l}u_{x}(x,y,z,t)=u_{x}(x,y,z,\omega_{l})\sum_{l=0}^{L}P_{l} \cdot e^{i(\omega_{l}t+\theta_{l})}\\ u_{y}(x,y,z,t)=u_{y}(x,y,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i(\omega_{l} t+\theta_{l})}\\ u_{z}(x,y,z,t)=u_{z}(x,y,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i(\omega_{l} t+\theta_{l})}\end{array}  \tag{21}\]  
Therefore, we only need to ask for the displacement \(u(x,\,y,\,z,\,\omega_{l})\) and the stress-strain, which is independent of time and depends only on the angular frequency \(\omega_{l}\); unless otherwise stated, the following derived components are proportional to the factor \(\sum_{l=0}^{L}P_{l}\cdot e^{i(\omega_{l}t+\theta_{l})}\). The dynamic balance differential equation, without considering the physical space problem, is as follows:  
\[ \begin{array}{l}\frac{\partial\sigma_{rr}}{\partial t}+\frac{\partial\sigma _{\theta}}{\partial t\theta}+\frac{\partial\sigma_{zz}}{\partial z}+\frac{ \sigma_{rr}-\sigma_{\theta\theta}}{r}=\rho\frac{\partial^{2}u_{r}}{\partial t \theta^{2}}\\ \frac{\partial\sigma_{\theta}}{\partial t}+\frac{\partial\sigma_{\theta}}{ \partial t\theta}+\frac{\partial\sigma_{\theta}}{\partial t\theta}+\frac{ \partial\sigma_{\theta}}{\partial z}+\frac{2\sigma_{rr}}{r}=\rho\frac{ \partial^{2}u_{r}}{\partial t^{2}}\\ \frac{\partial\sigma_{rr}}{\partial r}+\frac{\partial\sigma_{\theta\theta}}{ \partial r\theta}+\frac{\partial\sigma_{\theta z}}{\partial z}+\frac{\sigma_{ r}}{r}=\rho\frac{\partial^{2}u_{r}^{2}}{\partial t^{2}}\end{array}  \tag{22}\]  
Due to  
\[ \begin{array}{l}\frac{\partial^{2}u_{r}(r,\theta,z,t)}{\partial t ^{2}}=\frac{\partial^{2}u_{r}(r,\theta,z,\omega_{l})\sum_{l=0}^{L}P_{l} \cdot e^{i(\omega_{l}t+\theta_{l})}}{\partial t^{2}}\\ =-\omega_{l}^{2}u_{r}(r,\theta,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i( \omega_{l}t+\theta_{l})}\\ \frac{\partial^{2}u_{\theta}(r,\theta,z,t)}{\partial t^{2}}=\frac{\partial^{2 }u_{\theta}(r,\theta,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i(\omega_{l}t+ \theta_{l})}}{\partial t^{2}}\\ =-\omega_{l}^{2}u_{\theta}(r,\theta,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i( \omega_{l}t+\theta_{l})}\\ \frac{\partial^{2}u_{z}(r,\theta,z,t)}{\partial t^{2}}=\frac{\partial^{2}u_{z} (r,\theta,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i(\omega_{l}t+\theta_{l})} }{\partial t^{2}}\\ =-\omega_{l}^{2}u_{z}(r,\theta,z,\omega_{l})\sum_{l=0}^{L}P_{l}\cdot e^{i( \omega_{l}t+\theta_{l})}\end{array}  \tag{23}\]  
The dynamic balance differential Equation (22) can be rewritten as  
\[ \begin{array}{l}\frac{\partial\sigma_{rr}}{\partial r}+\frac{\partial\sigma _{\theta}}{\partial t\theta}+\frac{\partial\sigma_{z}}{\partial z}+\frac{ \sigma_{rr}-\sigma_{\theta\theta}}{\partial z}+\rho\omega_{l}^{2}u_{r}=0\\ \frac{\partial\sigma_{\theta}}{\partial r}+\frac{\partial\sigma_{\theta}}{ \partial r\theta}+\frac{\sigma_{\theta z}}{\partial z}+\frac{2\sigma_{rr}}{r}+ \rho\omega_{l}^{2}u_{\theta}=0\\ \frac{\partial\sigma_{rr}}{\partial r}+\frac{\partial\sigma_{\theta z}}{ \partial r\theta}+\frac{\partial\sigma_{\theta z}}{\partial z}+\frac{\sigma_{ r}}{r}+\rho\omega_{l}^{2}u_{z}=0\end{array}  \tag{24}\]  
where \(\rho\) is the density, and \(\omega_{l}\) is the angular frequency. According to [PERSON]'s law, the equation of physics is as follows:  
\[ \begin{array}{l}\sigma_{rr}=c_{11}\varepsilon_{rr}+c_{12}\varepsilon_{\theta 0}+c_{13}\varepsilon_{zz}&\sigma_{\theta z}=2c_{44}\varepsilon_{\theta z}\\ \sigma_{yy}=c_{12}\varepsilon_{rr}+c_{11}\varepsilon_{\theta 0}+c_{13}\varepsilon_{zz}&\sigma_{rz}=2c_{44}\varepsilon_{rz}\\ \sigma_{zz}=c_{13}\varepsilon_{rr}+c_{13}\varepsilon_{\theta 0}+c_{33}\varepsilon_{zz}&\sigma_{r\theta}=2c_{66}\varepsilon_{r\theta} \end{array}