# 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
\[m(\sigma)=0.086+0.00002e^{2.74521\sigma} \tag{9}\]  
\[n(\sigma)=-23.514+23.549e^{0.00075\sigma} \tag{10}\]  
As there are four stress level (0.25, 0.5, 1, 1.5) of compression moduli, as measured by the indirect tensile test. Taking \(\sigma=1\) MPa into Equations (9) and (10), respectively, then the parameters \(m\) and \(n\) of uniaxial compression moduli under 1 MPa can be obtained, as shown in Table 13. \begin{table}
\begin{tabular}{c c c c} \hline
**Test Type** & \(m\) & \(n\) \\ \hline Compression moduli & 0.086 & 0.053 \\ Compression moduli measured by indirect tensile test & 1.472 & 0.215 \\ \hline \end{tabular}
\end{table} 
Table 13: Fitting parameters of fatigue moduli decay curve in different tests method under 1 MPa stress level. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Stress Level (MPa)** & **Parameters** & **0.25** & **0.5** & **1** & **1.5** & **2** & **2.5** & **3** & **3.5** \\ \hline \multirow{2}{*}{Compression moduli} & \(m\) & / & / & / & / & 0.087 & 0.106 & 0.144 & 0.319 \\  & \(n\) & / & / & / & / & 0.068 & 0.084 & 0.087 & 0.097 \\ \hline \multirow{2}{*}{Compression moduli measured by indirect tensile test} & \(m\) & 1.108 & 1.246 & 1.472 & 1.938 & / & / & / & / \\  & \(n\) & 0.191 & 0.203 & 0.215 & 0.223 & / & / & / & / \\ \hline \end{tabular}
\end{table} 
Table 12: Fitting parameters of fatigue moduli decay curves under different test conditions. Figure 9: Moduli decay patterns of compression moduli measured by the indirect tensile tests. The compression moduli measured by indirect tensile test and uniaxial compression moduli under 1 MPa stress level were compared, as shown in Figure 10. The decay patterns of the two compression moduli are experienced in three stages: migration stage, steady stage, and destructive stage, and the whole process is nonlinear. In order to compare the decay rate, substituting the parameters in Table 13 into Equation (8), the inflection point of the decay curve can be calculated. Then, the tangent line at the inflection point position can be implemented to compare the tangent slope. The result is as follows:  
From Figure 10 and Table 14, it can be notified that the compression moduli decay pattern of uniaxial compression and indirect tensile similarly exist three stages of migration stage, steady stage, and destructive stage, and the patterns are nonlinear, too. The inflection point of uniaxial compression moduli is 0.370. The decay rate decreases before the inflection point, while it increases after the inflection point. There is no inflection point of indirect tensile moduli. In order to compare the decay rate on same position, the tangency point of indirect test is also taken as 0.37. It can be observed that the tangent slope of indirect test is larger than the uniaxial compression test on tangency point position. During the migration stage, the decay rate of uniaxial compression is faster than that of the indirect tensile. During the steady stage, the decay rate of uniaxial compression is slower than that of the indirect tensile. The decay rate of uniaxial compression is also faster than that of the indirect tensile during the destructive stage. As the stress level of tensile moduli and compression moduli all is 1 MPa, the decay pattern of tensile moduli and compression moduli were compared simultaneously. The values of \(m\) and \(n\) in Tables 10 and 13 were substituted into Equation (6). Then, the fatigue moduli decay patterns were shown in Figure 11. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Type** & **Inflection Point** & **Tangency Point** & **Tangent Slope** \\ \hline Uniaxial compression & 0.370 & (0.370, 0.877) & \(-\)0.120 \\ Compression moduli measured by indirect tensile test & - & (0.370, 0.945) & \(-\)0.243 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Inflection points and slopes of decay curves of compression moduli under different test conditions. Figure 10: Decay pattern of compression moduli.