# Permanent Deformation Mechanism of Steel Bridge Deck
## 4 Results and Discussion
### Movement Characteristics of Coarse Aggregate  
Rutting is fundamentally the result of accumulated permanent deformation arising from the compaction and lateral flow of asphalt mixtures. Under repeated loading, coarse aggregates undergo both translational and rotational motions, which are regarded as primary contributors to the development of permanent deformation. As temperature increases, the viscosity of the asphalt mortar decreases, thereby facilitating the movement of coarse aggregates. Based on the data obtained from the PFC simulations--including the coordinates and angular velocities of coarse aggregate particle elements, as illustrated  
Figure 20: The results of average contact force in asphalt mixtures with loading time: (**a**) average contact pressure; (**b**) average contact tension. in Figure 21--the translational displacement of coarse aggregates was calculated using Equation (3), while the translational and rotational angles were determined according to Equations (4) and (5), respectively [48]. \[L=\sqrt{(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}+(z_{1}-z_{0})^{2}} \tag{3}\]  
\[\alpha=\arctan\left|\frac{y_{1}-y_{0}}{x_{1}-x_{0}}\right|\text{, \ \ \ }\beta= \arctan\left|\frac{z_{1}-z_{0}}{x_{1}-x_{0}}\right|\text{, \ \ \ }\gamma=\arctan\left|\frac{z_{1}-z_{0}}{y_{1}-y_{0}}\right| \tag{4}\]  
\[\theta_{j}=\sum w_{i}\Delta t_{i} \tag{5}\]  
where \(L\) is the translational distance of the coarse aggregate; \((x_{0},y_{0},z_{0})\) and \((x_{1},y_{1},z_{1})\) represent the centroid coordinates before and after aggregate translation, respectively; \(\alpha\), \(\beta\), and \(\gamma\) are the translation angles of the coarse aggregate in the XY, XZ, and YZ planes, respectively; \(\theta_{j}\) is the rotation angle of coarse aggregate around the \(j\)-axis (\(j=x\), \(y\), \(z\)) direction; \(w_{i}\) denotes the angular velocity at the \(i\)th time step; and \(\Delta t_{i}\) represents the time step at the \(i\)th iteration. To quantitatively assess the directional characteristics of particle translational displacement in asphalt mixtures under varying temperatures, the translational angles \(\alpha\), \(\beta\), and \(\gamma\) were employed as indicators to characterize the orientation of particle movement. The angular range from 0\({}^{\circ}\) to 200\({}^{\circ}\) was divided into 50 intervals, with the midpoint of each interval designated as the representative angle. For each interval, the average translational displacement \(L\) was computed to analyze the directional distribution patterns of coarse aggregate movements within the mixture. The results are illustrated in Figure 22, which presents the angular distribution of translational displacements at different temperatures. A comparison of the data at 30 \({}^{\circ}\)C and 45 \({}^{\circ}\)C reveals that displacement variations at 30 \({}^{\circ}\)C are relatively moderate, with a more uniform overall distribution. The average translational displacement \(L\) generally ranges between 0 and 0.25 mm, with translational angles primarily distributed between 2\({}^{\circ}\) and 158\({}^{\circ}\), indicating the presence of both vertical and horizontal aggregate motions. Notably, within the angular range of 45\({}^{\circ}\) to 135\({}^{\circ}\), the maximum displacement \(L_{\beta}\) in the z-direction increases from 0.27 mm at 30 \({}^{\circ}\)C to 0.64 mm at 45 \({}^{\circ}\)C. This increase indicates that higher temperatures intensify vertical particle motion, which, under vertical stress, promotes secondary compaction of the aggregate skeleton and contributes to irreversible plastic deformation--an essential mechanism driving rutting development. Figure 21: Illustration for the movement characteristics of coarse aggregates: (**a**) translation; (**b**) rotation. As temperature rises from 30 \({}^{\circ}\)C to 45 \({}^{\circ}\)C, the directional distribution of particle displacements becomes increasingly non-uniform, particularly in the XY plane, where displacement fluctuations become more pronounced.