# 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
Equation (6) is simplify to \(y=\left(1-\chi^{m}\right)^{n}\), then,  
\[y^{\prime}=-mn\chi^{m-1}(1-\chi^{m})^{n-1} \tag{7}\]  
\[y^{\prime\prime}=m^{2}n\chi^{2m-2}(n-1)(1-\chi^{m})^{n-2}-mn\chi^{m-2}(m-1)(1- x^{m})^{n-1} \tag{8}\]  
where, \(y\) is the ratio of the modulus of the material to the initial modulus of the undamaged state after loading the specimen to N, \(\chi\) is the ratio of the loading cycles to the fatigue life, \(m\) and \(n\) are the material parameters related to the stress, \(y^{\prime}\) is the first derivative of \(y^{\prime}\), and \(y^{\prime\prime}\) is the two derivative of \(y^{\prime}\). While substituting parameters under the stress level of 1 MPa into Equation (8), the inflection point can be calculated, then the tangent line at the inflection point position can be adopted to compare the tangent slope. The result is as follows:  
It can be noticed from Figure 7 and Table 11 that the decay patterns of tensile moduli, as measured by the indirect tensile test and the direct tensile test are nonlinear. The moduli parameters are different. The inflection point of direct tensile test is 0.381. The decay rate of direct tensile moduli decreases gradually before the inflection, while it increases after the inflection point. The inflection point of tensile moduli, as measured by the indirect tensile test is 0.114. In addition, the decay rate shows an increasing trend. There are three stages of migration stage, steady stage and destructive stage in the decay curves of the direct tensile tests and the indirect tensile tests. The decay rate of the direct tensile moduli is more quickly than that of the indirect tensile moduli during the migration state, but it is contrary during the steady stage. In addition, the decay rate of the direct tensile moduli is quicker than that of the indirect tensile moduli at the destructive stage. Similarly, the decay patterns of the compression moduli were fitted as shown in Figures 8 and 9, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Type** & **Inflection Point** & **Tangency Point** & **Tangent Slope** \\ \hline Direct tensile & 0.381 & (0.381, 0.727) & \(-\)0.416 \\ Tensile moduli measured by indirect tensile test & 0.114 & (0.114, 0.908) & \(-\)0.791 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Inflection points and slopes of decay curves of tensile moduli under different test conditions. Figure 7: The decay pattern of tensile moduli measured by the indirect tensile test and direct tensile test. Figure 8: Moduli decay patterns of uniaxial compression tests. Figure 9: _Cont._  
From Figures 8 and 9, it can be observed that the decay patterns of compression moduli also exist three stages of migration stage, steady stage, and destructive stage, and the patterns are nonlinear. But, the decay rates are different under different stress state. For uniaxial compression test, the specimen is in a compression stress state. The skeleton structure, composition form, and friction coefficient of aggregates determine the inter-particle frictional resistance. The main factors determining its fatigue properties are the cohesion of the aggregate particles [31]. In the indirect tensile test, fatigue properties mainly depend on the interlocking effect of aggregate when the specimens are under the compression state [32]. In summary, when the materials are in a different stress state, the factors that determine the fatigue properties are different. In order to compare the decay pattern more obviously, this paper compared the decay pattern of compression moduli under same stress level. The parameters of the fitted curves for different stress levels are shown in Table 12:  
It is difficult to compare the moduli decay patterns. In this paper, the parameters \(m\) and \(n\) of compression moduli were fitted with stress levels. The fitting results were shown in Equations (9) and (10).