# Laboratory Evaluation of Rejuvenating Agent on Reclaimed SBS Modified Asphalt Pavement
## 2 Materials and Methods
### Dynamic Modulus Test  
Dynamic modulus (\(E\)*) is a fundamental property of asphalt mixture that defines the viscoelastic stiffness as a function of loading rate and temperature. In this study, test samples were fabricated using SGC at 7 \(\pm\) 0.5% air void. SGC-compacted samples were 17 cm in height and 15 cm in diameter, and the samples were cored and trimmed to 15 cm in height and 10 cm in diameter. DM tests were conducted at three temperatures (4 \({}^{\circ}\)C, 20 \({}^{\circ}\)C, 40 \({}^{\circ}\)C) and six frequencies (25 Hz, 10 Hz, 5 Hz, 1 Hz,  
Figure 1: (**a**) Sample dimensions; (**b**) semi-circular bending (SCB) test configuration; (**c**) samples with a notch depth; (**d**) typical SCB load versus displacement (\(P\)-\(u\)) curve.  
0.5 Hz, 0.1 Hz) using asphalt mixture performance test (AMPT, IPC Global, Victoria, Australia) and following AASHTO T342-11, as shown in Figure 2.  
The stress level and the number of loading cycles in the test are shown in Tables 4 and 5 respectively.  
Using the dynamic modulus test results measured at three different temperatures and six different loading frequencies, the master curves of recycled mixtures based on the time-temperature superposition principle were constructed for a reference temperature of 20 \({}^{\circ}\)C [27]. The sigmoidal function is used to fit and develop the master curve, as provided in Equation (4).  
\[\log\lvert E^{*}\rvert=\delta+\frac{a}{1+\exp(\beta+\gamma(\log f_{r}))} \tag{4}\]  
where \(\lvert E^{*}\rvert=\) dynamic modulus (MPa);\(f_{r}=\) reduced frequency at reference temperature; \(\delta=\) minimum value of \(\lvert E^{*}\rvert\); \(\delta+\alpha=\) maximum value of \(\lvert E^{*}\rvert\); \(\beta\), \(\gamma=\) fitting parameters.  
Equation (5) shows the temperature shift factor.  
\[\log(f_{r})=\log(a(T))+\log(f) \tag{5}\]  
where \(a(T)=\) temperature shift factor; \(f=\) frequency at a particular temperature.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Loading Frequency/Hz**} & \multicolumn{3}{c}{**Test Temperature/\({}^{\circ}\)C**} \\ \cline{2-4}  & **4** & **20** & **40** \\ \hline
25 & 1400 & 700 & 250 \\
10 & 1260 & 630 & 228 \\
5 & 1120 & 560 & 206 \\
1 & 980 & 490 & 184 \\
0.5 & 840 & 420 & 162 \\
0.1 & 700 & 350 & 140 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The stress level (kPa) of dynamic modulus (DM) test.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Loading Frequency/Hz**} & \multicolumn{2}{c}{**Number of Loading Cycles**} \\ \hline
25 & 200 \\
10 & 200 \\
5 & 100 \\
1 & 20 \\
0.5 & 15 \\
0.1 & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The number of loading cycles of DM test.  
Figure 2: (a) Three parallel sample; (b) asphalt mixture performance test (AMPT) equipment.  
The temperature shift factor may be written in the form of Equation (6) by using the Arrhenius time-temperature superposition model [28].  
\[\log(a(T))=c\Bigg{(}\frac{1}{T_{k}}-\frac{1}{T_{ref}}\Bigg{)} \tag{6}\]  
where \(T_{k}\) = the test temperature in Kelvin temperature; \(T_{ref}\) = the reference temperature in Kelvin temperature; \(c\) = nonlinear fitting curve coefficient.  
A nonlinear optimization program in 1 stOpt software was used to determine the master curve coefficients.