# Value-Added Use of Waste PET in Rubberized Asphalt Materials for Sustainable Payement
## 2 Materials and Research Methodology
### Dynamic Modulus and Phase Angle Master Curves of Asphalt Mixtures  
For the target asphalt mixtures, the dynamic modulus ( \(|E^{*}|\) ) and phase angle (\(\phi\)) were obtained using a dynamic modulus testing system at temperatures of 7 \({}^{\circ}\)C, 20 \({}^{\circ}\)C, and 40 \({}^{\circ}\)C and frequencies of 0.1 Hz, 1 Hz, and 10 Hz without a confining pressure. The measured results are summarized in Table 4.  
According to the above test results, the dynamic modulus and phase angle master curves of the target asphalt mixtures were fitted using Equations (4) and (5):  
\[\log|E^{*}|=\delta+\frac{(\text{Max }-\ \delta)}{1+e^{\delta+\gamma\log f_{r}}} \tag{4}\]  
\[\Phi=-90\times\alpha\gamma\frac{e^{[\beta+\gamma(\log f_{r})]}}{\left[1+e^{[ \beta+\gamma(\log f_{r})]}\right]^{2}} \tag{5}\]  
where \(|E^{*}|\) = the dynamic modulus, psi; \(\delta,\beta,\) and \(\gamma\) = the fitting parameters; Max = the limiting maximum modulus, psi; \(\alpha\) = Max \(-\)\(\delta\); and \(f_{r}\) = the reduced frequency, Hz. To calculate the \(\log f_{r}\) and \(|E^{*}|_{\text{max}}\), Equations (6)-(8) below were adopted. \[\log f_{r}=\log f+\frac{\Delta E_{a}}{19.14714}\bigg{(}\frac{1}{T}-\frac{1}{T_ {r}}\bigg{)}=\log f+\log[a(T)] \tag{6}\]  
where \(f_{r}\) = the reduced frequency at the reference temperature, Hz; \(f\) = the loading frequency at the test temperature, Hz; \(\Delta E_{a}\) = the activation energy (treated as a fitting parame \begin{table}
\begin{tabular}{c c c c c} \hline
**Item** & **Temperature**, \({}^{\circ}\)**C** & **Frequency, Hz** & **Average \(|E^{*}|\)**, kPa** & **Average \(\phi_{r}\)** \({}^{\circ}\) \\ \hline \multirow{8}{*}{18 CRMA} & 7 & 10 & 13,272,900 & 9.92 \\  & 7 & 1 & 10,033,875 & 14.10 \\  & 7 & 0.1 & 6,847,996 & 18.87 \\  & 20 & 10 & 7,024,812 & 18.36 \\  & 20 & 1 & 4,160,425 & 24.82 \\  & 20 & 0.1 & 2,192,737 & 29.93 \\  & 40 & 10 & 1,830,607 & 30.48 \\  & 40 & 1 & 795,003 & 30.52 \\  & 40 & 0.1 & 447,988 & 23.87 \\ \hline \multirow{8}{*}{18 CRMA2 PET-TETA} & 7 & 10 & 11,200,800 & 10.85 \\  & 7 & 1 & 8,129,153 & 15.71 \\  & 7 & 0.1 & 5,298,624 & 20.43 \\  & 20 & 10 & 6,341,018 & 18.76 \\  & 20 & 1 & 3,688,934 & 25.92 \\  & 20 & 0.1 & 1,915,439 & 29.63 \\  & 40 & 10 & 1,699,514 & 29.85 \\  & 40 & 1 & 761,905 & 29.88 \\  & 40 & 0.1 & 446,045 & 23.07 \\ \hline \multirow{8}{*}{18 CRMA2 PET-EA} & 7 & 10 & 12,392,450 & 10.27 \\  & 7 & 1 & 9,193,799 & 14.48 \\ \cline{1-1}  & 7 & 0.1 & 6,217,086 & 19.18 \\ \cline{1-1}  & 20 & 10 & 6,858,531 & 18.34 \\ \cline{1-1}  & 20 & 1 & 4,059,905 & 25.26 \\ \cline{1-1}  & 20 & 0.1 & 2,129,238 & 30.72 \\ \cline{1-1}  & 40 & 10 & 1,846,554 & 31.39 \\ \cline{1-1}  & 40 & 1 & 768,575 & 31.91 \\ \cline{1-1}  & 40 & 0.1 & 325,905 & 27.02 \\ \hline \end{tabular}
\end{table} 
Table 4: Dynamic modulus and phase angle of the target asphalt mixtures. ter); \(T\) = the test temperature, \({}^{\circ}\)K; \(T_{r}\) = the reference temperature, \({}^{\circ}\)K; and \(a(T)\) = the shift factor at temperature \(T\). \[|E^{*}|_{\text{max}}=\text{P}_{\text{c}}\Bigg{[}4,200,000\Big{(}1-\frac{\text{VMA }}{100}\Big{)}+435,000\Big{(}\frac{\text{VFA}\times\text{VMA}}{10,000}\Big{)}+ \frac{1-\text{P}_{\text{c}}}{\frac{(1-\text{VMA})}{1000}+\frac{\text{VMA}}{15 000\text{VFA}}}\Bigg{]} \tag{7}\]  
where  
\[\text{P}_{\text{c}}=\frac{\Big{(}20+\frac{435,000\text{(VFA)}}{\text{VMA}} \Big{)}^{0.58}}{650+\Big{(}\frac{435,000\text{(VFA)}}{\text{VMA}}\Big{)}^{0.58}} \tag{8}\]  
\(|E^{*}|_{\text{max}}\) = the limiting maximum asphalt mixture dynamic modulus, psi; VMA = the voids in the mineral aggregate, %; and VFA = the voids filled with asphalt, %. To better fit the dynamic modulus master curve, the fitting parameters (\(\beta\), \(\gamma\), \(\delta\), and \(\Delta E_{\text{a}}\)), standard error (\(S_{\text{e}}\)), and explained variance (\(R^{2}\)) were optimized and checked following AASHTO P61-13.