## 3 Numerical Examples
### Validation of the Methodology  
To validate the relaxation modulus conversion methods mentioned above, data points from another research project are adopted in this paper for validation purposes. The test specimen was gyratory compacted using loose mixture from field asphalt plant. Three types of additives were adopted to modify neat binders, namely Evotherm, rubber, and polymer. Both dynamic modulus and creep compliance of specimen was tested using the same test apparatus, test temperatures and frequencies as specified in Section 3.1. Relaxation modulus was converted by using both dynamic modulus and creep compliance. Table 4 presents the Prony series coefficients of relaxation modulus of the three mixture types.  
Figure 6a-c show the master curve and percentage difference caused by dynamic modulus and creep compliance, respectively. As can be seen, the two master curves of relaxation modulus obtained by dynamic modulus and creep compliance are close to each by visual observation. The difference ranges are 1% to 28% for the specimens using Evotherm modified binders, 2% to 35% for the specimens using rubber modified binders, and 7% to 32% for the specimens using polymer modified binders, respectively. The validation results confirm that the methods used in this paper are robust.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{\(E_{i}\) (MPa)} \\ \cline{3-8}
**No.** & \(\rho_{i}\) **(s)** & \multicolumn{2}{c}{**Evotherm**} & \multicolumn{2}{c}{**Rubber**} & \multicolumn{2}{c}{**Polymer**} \\ \cline{3-8}  & & \begin{tabular}{c} **Dynamic to** \\ **Relaxation** \\ \end{tabular} & \begin{tabular}{c} **Creep to** \\ **Relaxation** \\ \end{tabular} & \begin{tabular}{c} **Dynamic to** \\ **Relaxation** \\ \end{tabular} & \begin{tabular}{c} **Creep to** \\ **Relaxation** \\ \end{tabular} & \
\begin{tabular}{c} **Dynamic to** \\ **Relaxation** \\ \end{tabular} &
\begin{tabular}{c} **Creep to** \\ **Relaxation** \\ \end{tabular} \\ \hline
1 & \(2\times 10^{-6}\) & 2887.79 & 1628.27 & 2757.41 & 2889.77 & 2888.43 & 2677.87 \\
2 & \(2\times 10^{-5}\) & 4120.24 & 7975.11 & 3796.04 & 7164.79 & 3842.27 & 6895.38 \\
3 & \(2\times 10^{-4}\) & 4511.36 & 7458.14 & 4191.18 & 7465.54 & 4325.67 & 7378.90 \\
4 & \(2\times 10^{-3}\) & 3532.57 & 4518.68 & 3605.38 & 5829.68 & 3549.11 & 5891.67 \\
5 & \(2\times 10^{-2}\) & 2838.39 & 3347.47 & 3139.01 & 4460.67 & 3207.17 & 3560.67 \\
6 & \(2\times 10^{-1}\) & 1558.67 & 1965.74 & 1744.32 & 2274.53 & 1663.13 & 1633.28 \\
7 & \(2\times 10^{0}\) & 632.04 & 788.14 & 672.53 & 793.78 & 652.71 & 755.31 \\
8 & \(2\times 10^{1}\) & 202.66 & 234.71 & 198.79 & 227.54 & 213.06 & 261.15 \\
9 & \(2\times 10^{2}\) & 71.68 & 69.27 & 47.66 & 78.26 & 71.329 & 85.80 \\
10 & \(2\times 10^{3}\) & 25.76 & 22.50 & 39.76 & 42.71 & 30.29 & 39.23 \\
11 & \(2\times 10^{4}\) & 3.01 & 7.09 & 8.32 & 9.45 & 6.90 & 11.19 \\ \(E_{e}\) (MPa) & 22.38 & 22.90 & 23.15 & 23.66 & 23.14 & 25.46 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Prony Series Coefficients of Relaxation Modulus Obtained from Dynamic Modulus and Creep Compliance for new data points from previous publications.  
Figure 6: Relaxation modulus master curve converted from dynamic modulus and creep compliance in log-log scale for **(a)** Evotherm; **(b)** Rubber; **(c)** Polymer.