# Permanent Deformation Mechanism of Steel Bridge Deck
## 4 Results and Discussion
### Coupled Evolution of Rutting and Cracking  
Figure 13 presents the variation in rutting depth over time obtained from virtual rutting tests of the SACSS model under 30 \({}^{\circ}\)C and 45 \({}^{\circ}\)C conditions, comparing two scenarios: with and without surface cracks. As shown in Figure 13a, the SACSS model with a surface crack exhibited a faster rutting development rate from the early stages of loading. At 60 min, a rut depth of 3.598 mm was reached in the cracked model, while the rut depth in the model without cracks was 2.311 mm, indicating that the presence of surface cracks increased the rut depth by 35.77%. When the temperature was elevated to 45 \({}^{\circ}\)C, rut depths of 5.460 mm and 4.991 mm were observed in the cracked and crack-free models, respectively, with the presence of surface cracks contributing to a 9.4% increase in rut depth. These results indicate that the presence of a surface crack significantly accelerated rutting development. However, the influence of the crack was reduced under elevated temperature, at which point the rutting behavior was predominantly governed by the overall flow characteristics of the asphalt mixture.  
Figure 12: Rutting depth-loading time curve of laboratory and virtual tests at (**a**) 30 \({}^{\circ}\)C and (**b**) 45 \({}^{\circ}\)C.  
Figure 14 illustrates the relationship between particle contact failure and loading duration at a temperature of 30 \({}^{\circ}\)C. The evolution of meso-cracks under continuous loading can be categorized into two distinct stages: a gradual development phase and a rapid propagation phase.  
The initial gradual development phase corresponds approximately to the first 0-20 min of loading. During this period, 85 tensile contact failures and 38 shear contact failures were recorded. These results indicate that meso-cracks began to form within the specimen due to the applied external load; however, limited crack propagation was observed at the surface of the SBDP. At the mesoscopic scale, this can be attributed to the fact that only a small number of inter-particle contacts were fractured in the early stage, and the structural integrity of the asphalt mixture remained largely preserved.  
As the loading continued to intensify, the asphalt mixture entered the rapid propagation phase, spanning approximately 20 to 60 min. This stage exhibited a characteristic cyclic trend described as 'initial acceleration followed by deceleration'. From 20 to 46.9 min, a pronounced increase in contact failures was observed: tensile failures rose from 85 to 608, while shear failures increased from 38 to 225. In the subsequent interval (46.9 to 60 min), the growth rate of contact failures slowed, culminating in 918 tensile and 284 shear failures. Compared with the initial phase, a significant degradation of the mixture's internal structure occurred due to widespread inter-particle bond breakage. As a result, the formation  
Figure 14: Relationship between particle contact failure and the loading time at 30 \({}^{\circ}\)C.  
Figure 13: The effect of surface cracks on SBDP rutting at different temperatures: (**a**) 30 \({}^{\circ}\)C; (**b**) 45 \({}^{\circ}\)C.  
of meso-cracks surpassed a critical threshold and re-accelerated after a brief deceleration, indicating a recurrent escalation trend.  
Throughout the loading process, the number of tensile contact failures consistently exceeded that of shear failures, with the disparity becoming more pronounced during the rapid propagation phase. This observation suggests that loading duration had a more substantial impact on normal tensile failure than on shear failure in the asphalt mixture.