## 3 Elasto-Plastic Constitutive Model
### Plastic Strain Analyses of UGM
#### 3.5.2 Vertical Normal Plastic Strain  
Like the horizontal plastic strain, Figure 6 shows that the vertical plastic strain of the Mohr-Coulomb model was distributed in a highly concentrated way, and that of the revised SMP and Druck-Prage models was relatively uniform. In addition, the curves of vertical plastic strain in the Mohr-Coulomb model were not smooth. The maximum nominal vertical compressive strain occurred at the bottom in the centerline of loading area. For the Mohr-Coulomb and Druck-Prage models, all the vertical plastic strains were compressive strains, while, there were smaller tensile strains on both sides of the loading area in the revised SMP model. These results are in line with the loading state of a pavement structure under actual conditions. Bulging appears on both sides of the loading area due to the greater shear stress, showing that the revised SMP model can describe the dilatancy effect caused by the loads [35; 36]. The variation of the vertical plastic strain along the depth is shown in Figure 7, where the strains increase gradually and the change rate of plastic strains is higher near the centerline of the loading area. When the Mohr-Coulomb and Druck-Prage models were used, there was almost no vertical plastic strain in the depth range of 0-7.5 cm from UGM top surface. After that, the plastic strain produced by the Mohr-Coulomb model increased sharply. The plastic compressive strain at the bottom of UGM layer reached 595 \(\upmu\epsilon\) in R0, and it was up to 509.8 \(\upmu\epsilon\) in R15. The plastic compressive strain for the revised SMP model was 303.4 \(\upmu\epsilon\) at R0, and 253.7 \(\upmu\epsilon\) at R15, respectively. The plastic strains were about half of that from Mohr-Coulomb model. However, the plastic strain of Druck-Prage model was much smaller than that of the other two models. Along with the increase of depth, the variation of plastic strain linearly increased. This was due to the Druck-Prage criterion considered the intermediate principle stress compared with the Mohr-Coulomb model. Under condition of three-direction stress, the strength and stiffness of the UGM layer was  
Figure 5: Horizontal plastic strain of an UGM layer using the three models. enhanced so that the plastic strain was much smaller. In addition, the Druck-Prage criterion had no angle resulting in a smooth distribution of the plastic strain. As for the revised SMP model, the stiffness of the revised SMP model increased gradually along with the loading cycles. In the initial state, the calculated modulus of the revised SMP model was relatively lower. However, the Druck-Prage model had a sharp increased stiffness due to the three-direction stress, and it hardly varied along with the loading cycles. Therefore, the plastic strain of Druck-Prage model was smaller than that of the revised SMP model. Figure 6: Distribution of vertical nominal plastic strain of UGM layer with the three models. (**a**) Mohr-Coulomb model, (**b**) Druck-Prager model, (**c**) Revised SMP model. Using the revised SMP model, the vertical plastic tensile strains were 55.5 \(\mu\varepsilon\) and 42.3 \(\mu\varepsilon\) at two sides of the loading area on the surface of UGM layer. This showed that the revised SMP model had the possibility of arching. It had strong relevance with the shear failure at the edge loading area under the actual load action, showing that the revised SMP model can describe the dilatancy effect caused by loads. However, there was no vertical plastic strain at the top of UGM layer using Mohr-Coulomb and Druck-Prage models, which was inconsistent with the actual conditions. The above results show that Mohr-Coulomb model would not produce horizontal and vertical plastic strains at the top surface of UGM layer, but very high ones at the bottom of the UGM layer. Moreover, it had a sharp change at the position of 5-10 cm depth of the UGM layer. The plastic strains in the Druck-Prage model were the smallest among the three models.