## 5 Anti-Rutting Analysis of HMAC Pavement under Continuous Temperature Variation
### The Establishment of Rutting Calculation Model  
The deformation of asphalt pavement includes creep deformation and elastic deformation. Rutting is a severe road distress that is mainly caused by the creep deformation [45]. In this study, the time-hardening creep model was used to simulate the rutting. The creep deformation of pavement (\(\varepsilon_{cr}\)) is related to the temperature (\(T\)), stress (\(q\)), and time (\(t\)), and was obtained from Equation (1):  
\[\varepsilon_{cr}=f(T,q,t) \tag{1}\]  
In this study, the Norton Bailey creep law was applied to simulate the creep characteristics of asphalt mixture under loading conditions. In the analysis process, the stress is assumed to be constant; as a result, the above equation can be expressed as:  
\[\varepsilon_{cr}=C_{1}q^{C_{2}}t^{C_{3}} \tag{2}\]  
where \(q\) and \(t\) refer to the stress and time, respectively; meanwhile, \(C1\), \(C2\), and \(C3\) are the temperature-dependent model parameters, and were determined based on the lab tests. Furthermore, Equation (2) can be expressed as:  
\[\varepsilon_{cr}=Aq^{n}t^{m} \tag{3}\]  
Equation (3) is the time-stress-creep model of ABAQUS, in which \(A\), \(m\), and \(n\) are the model parameters:  
\[A=C_{1}C_{3},\ n=C_{2},\ m=C_{3}-1 \tag{4}\]  
In order to simulate road rutting, the creep parameters of the asphalt surface layers were used in the model. The selected parameter values are shown in Table 11, based on references [45; 46; 47; 48; 49]. In the analysis step module, 24 analysis steps, step-1 h through step-24 h, were set. These step represented 24 hours in a day. In order to simulate the formation of rutting of asphalt pavement during a continuous temperature change in one day, the cumulative loading time of the pavement structure needed to be input into each analysis step. The cumulative loading time of the wheel load was calculated as follows:  
\[t=\frac{0.36 NP}{n_{w}pBv} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Structural Layer**} & \multirow{2}{*}{**Temperature (\({}^{\circ}\)C)**} & \multicolumn{3}{c}{**Creep Parameter**} \\ \cline{3-5}  & & \(A\) & \(n\) & \(m\) \\ \hline \multirow{6}{*}{AC-13} & 20 & 6.54E-11 & 0.937 & \(-\)0.592 \\  & 30 & 3.33E-09 & 0.862 & \(-\)0.587 \\  & 40 & 1.45E-08 & 0.792 & \(-\)0.577 \\  & 50 & 1.39E-06 & 0.414 & \(-\)0.525 \\  & 60 & 1.46E-05 & 0.336 & \(-\)0.502 \\ \hline \multirow{6}{*}{HMAC-20} & 20 & 1.46E-12 & 0.962 & \(-\)0.601 \\  & 30 & 5.80E-10 & 0.896 & \(-\)0.599 \\ \cline{1-1}  & 40 & 5.85E-09 & 0.823 & \(-\)0.598 \\ \cline{1-1}  & 50 & 6.10E-07 & 0.415 & \(-\)0.532 \\ \cline{1-1}  & 60 & 6.50E-06 & 0.350 & \(-\)0.522 \\ \hline \multirow{6}{*}{ATB-25} & 20 & 4.59E-11 & 0.922 & \(-\)0.581 \\  & 30 & 3.46E-09 & 0.859 & \(-\)0.576 \\ \cline{1-1}  & 40 & 1.96E-08 & 0.83 & \(-\)0.562 \\ \cline{1-1}  & 50 & 1.20E-06 & 0.322 & \(-\)0.522 \\ \cline{1-1}  & 60 & 3.76E-05 & 0.21 & \(-\)0.418 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Creep parameters of asphalt mixture at different temperatures. where \(t\) (s) is the cumulative loading time of the wheel load; \(N\) is the number of wheel-loading cycles; \(P\) (kN) is the axle weight of the vehicle; \(n_{w}\) is number of tires on the axle; \(p\) (MPa) is the tire contact pressure; \(B\) (cm) is the width of tire-ground contact; and \(v\) (km/h) is driving speed. According to the above formula, the required calculation parameters were determined, as shown in Table 12.  
Substituting the above parameters into Formula (5), it can be calculated that the time during which a one-wheel load acts on the road surface is 0.01006 s, and the cumulative loading time for 500,000-axle loads is 5030 s. A traffic volume distribution of a certain section of road over 24 hours was selected, and an approximated value of the segmented cumulative loading time of each hour in one day was obtained, as shown in Table 13.  
Two load belts were installed in the middle of the road surface to apply the moving load.