## 5 Fracture Analysis
### Fracture Propagation Analysis  
Based on the numerical simulation results from this paper, the evolutions of crack length, displacement, and loading during the SCB simulation were obtained. Figure 8 describes the relationship between the crack length and the displacement, as well as the relationship between the load and the displacement.  
As shown in Figure 8, the fracture process could be divided into three stages from a macro perspective. Stage I was regarded as the crack initiation stage. Stage II represented the crack propagation stage before failure. While stage III was the crack propagation stage after failure. In order to clarify the differences between these stages, the local damage and crack characteristics in these stages at 0 \({}^{\circ}\)C were compared in Figure 9 from a mesoscopic perspective.  
In Figure 9, the deformation was enlarged three times. The gray elements represent the cohesive damage in asphalt mastic, while the orange elements represent the adhesive damage at the aggregate-asphal interface. The local damage in Stage I is shown in Figure 9a. It was found that during Stage I, both the gray elements and orange elements were observed near the open end, which means cohesive and adhesive damage occurred due to stress concentration around the open end. However, since stress could still be transmitted through these elements with cohesive and adhesive damage, the load applied to the specimen could still increase.  
Figure 9b shows the local damage and cracking in Stage II. It can be seen that some orange elements, representing adhesive damage, turned into a white color, which represents discrete micro-cracks. This means the micro-cracks gradually formed at the aggregate-asphal interface as the displacement increased. Subsequently, some gray elements, representing cohesive damage, at the tip of the opening changed to a white color, implying the generation of continuous macroscopic cracks. The appearance of cracks further increased  
Figure 8: Evolutions of crack length, displacement, and loading during SCB simulation at (a) 0 \({}^{\circ}\)C; (b) \(-\)10 \({}^{\circ}\)C.  
the deformation of the adjacent orange and gray elements, indicating a higher degree of adhesive damage at the nearby aggregate-asphal interface and cohesive damage in the asphalt mortar. These damages would gradually turn into cracks with increased displacement. The whole specimen would finally reach the maximum load capacity with the growth of cracks. In this process, scalar stiffness degradation (SDEG) was analyzed to describe the damage distribution inside the asphalt concrete specimen. A larger value of SDEG means more serious element damage. In numerical simulations, element damage is judged to be complete when SDEG reaches a critical value [30,50,51]. In this paper, this value was taken as 0.99. Figure 10 shows the distribution of SDEG for cohesive- and adhesive-damaged elements (SDEG > 0) of the entire model at the peak load. The minimum SDEG value of the adhesive elements at the aggregate-asphal interface was 0.00325, while the minimum SDEG value of the cohesive elements inside the asphalt mortar was 0.00059. The average value of the adhesive elements was 0.441, while the average value of the cohesive elements was 0.424. The damage of the adhesive elements was greater compared to the cohesive elements, indicating that the interface adhesion at the peak load is the weaker part.  
Figure 10: SDEG range of damaged elements at peak loads.  
Figure 9: Crack propagation process at 0 \({}^{\circ}\)C: (**a**) Stage I: Crack initiation; (**b**) Stage II: Crack propagation stage before failure; (**c**) Stage III: Crack propagation.  
In Stage III, the cracks continued to increase, as shown in Figure 9c. Due to the increasing macroscopic cracks, the bearing capacity of the specimen continued to decrease.