# Characteristics of Moduli Decay for the Asphalt Mixture under Different Loading Conditions
## 3 Test Result and Analysis
### Analysis of the Fitting Results of Fatigue Tests under Different Stress Levels  
In order to compare the moduli decay pattern under the same stress state, the tensile moduli and compression moduli were compared, respectively. Real-time ratio \(E(N)/E_{0}\) for tensile moduli, as measured by direct tensile test and indirect tensile tests, was fitted with the cycle ratio by Equation (6). The fitting results of tensile moduli from the direct tensile and the indirect tensile fatigue tests in different stress levels are shown in Figures 5 and 6, respectively. Figure 5: Moduli decay pattern of the direct tensile fatigue tests. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Stress Level (MPa)** & **0.25** & **0.5** & **1** & **1.5** & **2** & **2.5** & **3** & **3.5** \\ \hline \multirow{2}{*}{Tensile moduli} & \(E_{\text{min}}\) (MPa) & 609 & 716 & 869 & 961 & / & / & / & / \\  & \(CV\) (\%) & 0.049 & 0.057 & 0.052 & 0.051 & / & / & / & / \\ \hline \multirow{2}{*}{Compression moduli} & \(E_{\text{min}}\) (MPa) & / & / & / & / & 7480 & 8086 & 8528 & 8787 \\  & \(CV\) (\%) & / & / & / & / & 0.021 & 0.022 & 0.009 & 0.010 \\ \hline \multirow{2}{*}{Indirect tensile moduli} & \(E_{\text{min}}\) (MPa) & 895 & 958 & 1040 & 1106 & / & / & / & / \\  & \(CV\) (\%) & 0.021 & 0.019 & 0.053 & 0.042 & / & / & / & / \\ \hline \multirow{2}{*}{Compression moduli measured by indirect tensile test} & \(E_{\text{min}}\) (MPa) & 5073 & 5593 & 6184 & 6563 & / & / & / & / \\  & \(CV\) (\%) & 0.027 & 0.025 & 0.018 & 0.018 & / & / & / & / \\ \hline \multirow{2}{*}{Tensile moduli measured by indirect tensile test} & \(E_{\text{min}}\) (MPa) & 560 & 616 & 693 & 733 & / & / & / & / \\  & \(CV\) (\%) & 0.036 & 0.024 & 0.036 & 0.021 & / & / & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The critical moduli value of fatigue test. It can be noticed from Figures 5 and 6 that the decay patterns of tensile moduli that were obtained from these two test methods are similar. There are three stages: migration stage, steady stage, and destructive stage, and the whole process is nonlinear. However, the decay rates of different stress state are different, which are caused by different fatigue resistance of asphalt. In the direct tensile test, the specimen is within a uniform tensile condition. The main factors determining its fatigue properties are the cohesion of the asphalt mortar (the adhesion of the aggregate with the asphalt and its internal friction based on the aggregate gradation). The effect of intrusion between aggregates is relatively weak [30]. During the indirect tensile tests, the transverse tensile fatigue properties mainly depend on the adhesion (between asphalt mortar and aggregates) and the internal frictional resistance. Figures 5 and 6 reflect the decay pattern of different stress levels under different stress states. In order to compare the decay pattern more obviously, this paper compared the decay pattern of tensile moduli under the same stress level. The parameters of the fitted curves for tensile stress were shown in the Table 10:  
The tensile moduli measured by indirect tensile test and direct tensile under the same stress level \(1\,\mathrm{MPa}\) were compared. As shown in Figure 7: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Stress Level (MPa)** & **Parameters** & **0.25** & **0.5** & **1** & **1.5** \\ \hline \multirow{2}{*}{Tensile moduli} & \(m\) & 0.222 & 0.265 & 0.337 & 0.413 \\  & \(n\) & 0.229 & 0.235 & 0.248 & 0.2636 \\ \hline \multirow{2}{*}{Tensile moduli measured by indirect tensile test} & \(m\) & 0.638 & 0.743 & 0.964 & 1.255 \\  & indirect tensile test & \(n\) & 0.503 & 0.586 & 0.732 & 0.844 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Fitting parameters of the tensile moduli decay curves under different test conditions. Figure 6: Moduli decay pattern of tensile moduli measured by the indirect tensile fatigue tests. In order to compare the decay pattern more clearly, the decay Equation is simplified.