## 3 Elasto-Plastic Constitutive Model
### Finite Element Model of Pavement Structure  
Using the general-purpose FE software ABAQUS, a 2-dimensional model was used to simulate the pavement structure, which was simplified into a three-layer system with 15 cm asphalt surface layer HMA + 25 cm unbound gravel layer UGM + 140 cm soil subgrade. The cross-section of the pavement structure was 6 m width and 1.8 m depth. Due to the symmetry of pavement structure and vehicular loading condition, the 1/2 2-Dimensional plane strain model was adopted, seen in Figure 3. The right and bottom boundary was designed as the encastre boundary condition (ENCASTRE), and XYMM boundary condition (\(U_{1}=0\)) was adopted for the symmetry axis in the left side. There were two contact surfaces in the FE model for the three layers. The types of the two contact surfaces were both the completely continuous in the FE model.  
With the standard axial load of 100 kN, the equivalent load pressure was 25 \(\times\) 10\({}^{3}\)/0.213 = 117.4 kPa\(\cdot\)m based on the static equivalence principle [31]. The total loading time was 20 s, and each analysis step time period was 0.2 s with periodic amplitude in a half sinusoid wave. The current analyses didn't consider the dynamic issue caused by the vehicular load.  
In the 'Step' module of the FE model, 'Static, General' was chosen as the procedure type. The element type CAX8 was adopted for the finite domains. The size of mesh was 2.5 cm \(\times\) 2.5 cm near the loading areas. The single mode biased seeding was used, and the seed became 2.5 cm to 10 cm along the loading area to the right boundary.  
As for the model parameters of the structure layers, the linear viscoelastic model was adopted for HMA layer referenced from [12,13]. Poisson's ratio and transient modulus of elasticity were 0.35 and 9840 MPa, respectively. The Prony series of generalized Maxwell solid model was simulated by shear and bulk relaxation modulus, listed as Table 1. An elastic model was adopted for soil subgrade layer, with Poisson's ratio of 0.4 and elastic modulus of 45 MPa. The parameters of the Mohr-Coulomb and Druck-Prager constitutive models were shown in Tables 2 and 3. The parameters of the revised SMP model can be found in [5].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(i\) & **1** & **2** & **3** & **4** & **5** \\ \hline \(G_{i}/K_{i}\) & \(6.31\times 10^{-1}\) & \(2.51\times 10^{-1}\) & \(8.47\times 10^{-2}\) & \(2.67\times 10^{-2}\) & \(6.66\times 10^{-3}\) \\ \(\tau_{i}\) & \(2.06\times 10^{-2}\) & \(1.73\times 10^{-1}\) & 1.29 & 5.35 & \(1.06\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Prony series parameter of linear viscoelastic model.  
Figure 3: 2-Dimensional model of pavement structure.  
In the pavement structure, the plastic strains of the UGM layer model were compared using the three elasto-plastic models during the first 100 loading cycles. Taking the symmetry axis as zero point, 0, 0.15, 0.30 and 0.60 m from the symmetry axis were selected as analyses paths which denoted as R0, R15, R30 and R60, respectively, as seen in Figure 3. The analysis points started from the surface of UGM layer, which were at 0.15, 0.20, 0.25, 0.30, 0.35 and 0.40 m along the axial direction, respectively.