# Peridynamics for Fracture Analysis of Reflective Cracks in Semi-Rigid Base Asphalt Pavement
## 4 Pavement Simulation Model
### Influence of Friction  
Since the friction between the vehicle and road surface is much smaller than that of the vehicle load, most scholars ignore the friction between the wheel and road surface when studying the influence of vehicle load on reflective crack propagation. Without considering the friction between wheels and pavement, due to the symmetry of the structure, when studying the most unfavorable position of load, the two cases of vehicles approaching cracks and vehicles far from cracks can be simplified to analyze the asymmetrical load simultaneously. Once friction is taken into account, the two cases are no longer equivalent (Figure 13). In this section, the friction between the tire and pavement is considered to analyze the crack propagation under the action of two different asymmetrical loads: approaching the crack and far from the crack. The numerical simulation model was selected as the case of an asphalt overlay with a thickness of 15 cm under asymmetrical loading in Section 4.4, and the material parameter settings of the model are the same as those in Section 4.4. The friction coefficient between the tire and pavement was selected as 0.1 [15], and the direction of friction is opposite to the direction of vehicle operation. To accurately explore the effect of friction between the tire and pavement on reflective crack propagation, a group of frictionless examples were added as the control group. Three groups of different working conditions are set in this section. In case (a), the friction between the wheels and road surface is ignored, and only the vertical vehicle load is considered. In case (b) and case (c), the friction between the tire  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Thickness of Asphalt Overlay** & **Expansion Behavior of Reflective Cracks** \\ \hline T = 6 cm & At t = 0.8 ms, the initial cracks in the semi-rigid base begin to expand and spread along the upward sloping direction, and run through the asphalt overlay at t = 1.6 ms. \\ \hline T = 9 cm & At t = 1.0 ms, the initial cracks begin to extend horizontally along the interface between semi-rigid base and asphalt overlay. At t = 1.6 ms, the fracture propagation direction changes and begins to extend upward at a certain angle to the base. At t = 6.0 ms, cracks run through the asphalt overlay. \\ \hline T = 12 cm & At t = 1.4 ms, the initial cracks begin to extend horizontally. It changes and begins to expand along the upward sloping direction at t = 2.  
and pavement is considered, and the direction of friction in both working conditions is always left. However, the position of the vehicle load changes: one is on the left side of the crack, and the other is on the right. In case (b), the vehicle approaches the crack, and the vertical asymmetric load is to the left of the crack. According to Section 4.4, the crack has a tendency to expand to the right. At this point, the direction of friction is opposite to the trend of crack propagation. In case (c), the vehicle is away from the crack, the vertical asymmetric load is on the right side of the crack, the crack has a tendency to expand to the left, and the direction of friction is the same as the trend of crack propagation. Numerical simulation results of the crack propagation trajectory of the three groups of calculation models at the same time are shown in Figure 14.  
In the numerical simulation case (a), the initial crack begins to extend horizontally at t = 1.8 ms. It begins to expand along the upward sloping direction at t = 2.6 ms. At t = 16 ms, the structure is stable, and the crack does not run through the asphalt overlay. In case (b), the crack does not expand, indicating that the friction force in this case has a restraining effect on the crack growth. In case (c), the crack begins to expand at t = 1.4 ms and then begins to tilt upward at t = 2.2 ms until the crack runs through the asphalt overlay at t = 7.6 ms.