## 3 Numerical Examples
### Discussion of Results  
Based on the above procedures, the results of the two numerical interconversion methods can be analyzed. Before comparing the results of relaxation modulus converted from dynamic modulus and creep compliance, it needs to determine which method is better to represent the creep to relaxation conversion results. The collocation method was adopted in the transferring dynamic modulus to relaxation modulus, whereas both collocation method and least squares method were adopted in the transferring creep compliance to relaxation conversion. Figure 4 compares the relaxation modulus obtained by the collocation method and squares method. The relaxation modulus from collocation method was compared with relaxation modulus from least squares method based on statistical \(t\)-test at a significance level of 0.05. Results indicate that there was no statistical difference between relaxation modulus based on the two methods. In other words, the relaxation modulus converted from creep compliance through the collocation method and least squares method is almost the same. Figure 4: Relaxation modulus master curve converted from creep compliance through collocation method and least squares method in log-log scale. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & \multicolumn{3}{c}{\(E_{i}\) (MPa)} \\ \cline{4-7}
**No.** & \(\tau_{j}\) **(s)** & \(D_{j}\) **(l/MPa)** & \(\rho_{i}\) **(s)** & \multicolumn{2}{c}{**Dynamic to**} & \multicolumn{2}{c}{**Creep to Relaxation**} \\  & & & & **Relaxation** & \multicolumn{2}{c}{**Collocation Method**} & **Least Squares** \\ \hline
1 & \(2\times 10^{-6}\) & \(2.14\times 10^{-6}\) & \(2\times 10^{-6}\) & 2888.60 & 1542.07 & 2339.76 \\
2 & \(2\times 10^{-5}\) & \(9.26\times 10^{-6}\) & \(2\times 10^{-5}\) & 3858.85 & 7967.98 & 7236.69 \\
3 & \(2\times 10^{-4}\) & \(2.07\times 10^{-5}\) & \(2\times 10^{-4}\) & 4791.64 & 6962.31 & 7381.14 \\
4 & \(2\times 10^{-3}\) & \(3.91\times 10^{-5}\) & \(2\times 10^{-3}\) & 4175.53 & 6426.10 & 6218.63 \\
5 & \(2\times 10^{-2}\) & \(1.00\times 10^{-4}\) & \(2\times 10^{-2}\) & 3510.67 & 4146.14 & 4243.43 \\
6 & \(2\times 10^{-1}\) & \(2.89\times 10^{-4}\) & \(2\times 10^{-1}\) & 1669.80 & 1712.93 & 1750.13 \\
7 & \(2\times 10^{0}\) & \(8.35\times 10^{-4}\) & \(2\times 10^{0}\) & 530.77 & 616.41 & 626.74 \\
8 & \(2\times 10^{1}\) & \(2.63\times 10^{-3}\) & \(2\times 10^{1}\) & 144.70 & 197.50 & 200.97 \\
9 & \(2\times 10^{2}\) & \(6.64\times 10^{-3}\) & \(2\times 10^{2}\) & 42.56 & 64.27 & 67.28 \\
10 & \(2\times 10^{3}\) & \(1.30\times 10^{-2}\) & \(2\times 10^{3}\) & 18.22 & 34.14 & 32.71 \\
11 & \(2\times 10^{4}\) & \(2.92\times 10^{-2}\) & \(2\times 10^{4}\) & 9.63 & 7.63 & 8.84 \\  & \(D_{g}\) (1/MPa) & \(3.33\times 10^{-5}\) & \(E_{e}\) (MPa) & 21.83 & 18.94 & 18.94 \\ \hline \hline \end{tabular} Note: \(r_{j}\) and \(\rho_{i}\) are the relaxation time for creep compliance and dynamic modulus, respectively; \(D_{j}\) is the Prony series for creep compliance.  
\end{table} 
Table 3: Prony Series Coefficients of Relaxation Modulus Obtained from Dynamic Modulus and Creep Compliance. Assuming the collocation method is more reliable, the relaxation modulus converted through it can be selected to represent the creep to relaxation conversion results. Combined with the dynamic to relaxation conversion results, these two curves can be compared in Figure 5 in semi-log scale. Figure 5a is plotted by using semi-log scale in which a clear pattern of relaxation can be observed: the relaxation modulus dramatically reduced within a short period and then tends to plateau. Figure 5b is plotted by using log-log scale in which the relationship between relaxation modulus and reduced time at a wide range is available. The relaxation modulus difference based on dynamic modulus and creep compliance was also calculated. As seen, such a difference was relatively small at the short- or long-term region, whereas the difference greatly increased in the middle part of reduced time. The difference curve also presents to be fluctuated.