## 3 Elasto-Plastic Constitutive Model
### Plastic Strain Analyses of UGM  
#### 3.5.1 Horizontal Normal Plastic Strain  
It is generally believed that UCMs do not produce tensile strain, but in an actual situation, UGMs produce plastic tensile or compressive strain due to the particle rotation or movement under the load effect. Therefore, nominal plastic strains were used to represent the plastic strains of the UGM layer in this study. With the three elasto-plastic constitutive models, the UGM layer produced obvious plastic strains, including horizontal nominal tensile and compressive strain. Near the center of the loading area, the tensile strain in the middle and bottom part of UGM layer was obviously greater than that in other areas (shown in Figure 4). Comparing the nominal tensile strain from the three models, the distribution of plastic strain from the Mohr-Coulomb model was relatively concentrated. It had a sharp increase in the middle layer. This indicated that the Mohr-Coulomb model was prone to produce rapid plastic strain growth areas, which would lead to structural damage. For the Druck-Page model, there were only tensile strains and the strain of the UGM surface layer tended to 0. For the revised SMP model, the distribution of plastic strain was uniform with an obvious gradient. Moreover, it had a good relationship with the loading area, but it was not quite obvious for the Mohr-Coulomb and Druck-Page models. The nominal tensile strain occurred under the loading area, resulting in the lateral movement of particles, and then a nominal compressive strain appears on both sides of the loading area. This was closely consistant with the actual situation of vehicular loading on a pavement surface [11; 33; 34]. However, the distribution of plastic strain from the revised SMP model was different from the predictions using the nonlinear anisotropic elastic-plastic model from the literature [6]. The plastic strains of the latter model were mainly distributed in the upper part of UGM layer. This was due to the anisotropic elasticity modulus, which had different stiffness modulus in horizontal and vertical directions [6; 12; 13; 14]. The strain distribution of Druck-Page and revised SMP models was smooth and continuous due to the fact the criteria had no angle. The stiffness modulus near the loading area was greater than that in other areas. UGM layer was the stress-dependent material, and the modulus associated with stress state. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter Type**} & \multicolumn{2}{c}{**Elastic Part**} & \multicolumn{2}{c}{**Plastic Part**} \\ \cline{2-6}  & **Young’s** & **Possion’s** & **Friction** & **Cohesion** & **Dilation** \\  & **Modulus (MPa)** & **Ratio** & **Angle (\({}^{\circ}\))** & **(kPa)** & **Angle (\({}^{\circ}\))** \\ \hline Value & 400 & 0.35 & 49.06 & 19.6 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of Mohr-Coulomb model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter Type**} & \multicolumn{2}{c}{**Elastic Part**} & \multicolumn{2}{c}{**Plastic Part**} \\ \cline{2-6}  & **Young’s** & **Possion’s** & **Friction** & **Flower Stress** & **Dilation** \\  & **Modulus (MPa)** & **Ratio** & **Angle(\({}^{\circ}\))** & **Ratio** & **Angle (\({}^{\circ}\))** \\ \hline Value & 400 & 0.35 & 49.06 & 0.8 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of Druck-Prager model. Figure 4: Distribution of horizontal nominal plastic strain of UGM layer with the three models (**a**) Mohr-Coulomb model, (**b**) Druck-Prage model, (**c**) revised SMP model. Figure 5 suggests that little horizontal plastic strain appears on the top surface of UGM layer when Mohr-Coulomb and Druck-Prage models are adopted. The greatest horizontal nominal tensile strain of the Mohr-Coulomb model was up to 298.5 \(\upmu\epsilon\) at the bottom layer in the center of loading area. This was about two times greater than that obtained from the revised SMP model, which indicated that Mohr-Coulomb model would produce greater plastic strains.