# Characteristics of Moduli Decay for the Asphalt Mixture under Different Loading Conditions
## 3 Test Result and Analysis
### The Establishment of the Decay Model for Moduli  
The conventional \(S\)-\(N\) fatigue Equation is widely used to analyze the fatigue performance of asphalt mixtures [26; 27]. [PERSON] [28] defined the traditional \(S\)-\(N\) fatigue Equation as:  
\[N_{f}=k(\frac{1}{t})^{n} \tag{1}\]  
or  
\[N_{f}=k(\frac{1}{\sigma})^{n} \tag{2}\]  
where, \(N_{f}\) is fatigue life, \(t\) is stress ratio, \(\sigma\) is stress level. \(K\), and \(n\) are the material parameters of asphalt mixtures.  
According to Equations (1) and (2), the fatigue curves characterized by stress ratios and stress levels were shown in Figure 4a,b, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stress Level (MPa)** & **Maximum Load Value (KN)** & **Stress Ratio** & **Average Fatigue Life (Times)** & \(CV\)\%** \\ \hline
2 & 15.7 & 0.178 & 797,470 & 0.098 \\
2.5 & 19.625 & 0.223 & 198,977 & 0.074 \\
3 & 23.55 & 0.268 & 91,067 & 0.086 \\
3.5 & 27.475 & 0.312 & 26,734 & 0.105 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Test result of uniaxial compression.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stress Level (MPa)** & **Maximum Load Value (KN)** & **Stress Ratio** & **Average Fatigue Life (Times)** & \(CV\)\%** \\ \hline
0.25 & 2.386 & 0.11 & 1,283,789 & 0.021 \\
0.5 & 4.772 & 0.22 & 50,062 & 0.166 \\
1 & 9.544 & 0.439 & 2236 & 0.079 \\
1.5 & 14.315 & 0.659 & 435 & 0.167 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Test result of indirect tensile.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stress Level (MPa)** & **Maximum Load Value (KN)** & **Stress Ratio** & **Average Fatigue Life (Times)** & \(CV\)\%** \\ \hline
0.25 & 0.625 & 0.223 & 398,879 & 0.176 \\
0.5 & 1.25 & 0.446 & 41,129 & 0.203 \\
1 & 2.5 & 0.892 & 3686 & 0.188 \\
1.5 & 3.75 & 1.338 & 993 & 0.176 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Test result of direct tensile.  
From Figure 4, it can be observed that the \(S\)-\(N\) fatigue curves of asphalt mixtures show great difference under different stress conditions. It is difficult to evaluate the fatigue performance of asphalt mixtures in different test methods [7]. Given that, in this paper, the nonlinear fatigue damage model was implemented to simulate the moduli decay patterns. Defining the moduli as the damage variable, the damage model based on the moduli decay was established.  
The damage variable could be expressed, as shown in Equation (3).  
\[D(N)=1-\frac{E(N)}{E_{0}} \tag{3}\]  
where, \(D(N)\) is damage variable, \(E(N)\) is modulus in loading cycle \(N_{f}\) the specimen, and \(E_{0}\) is the initial value of modulus.  
[PERSON] [28] established another fatigue damage model. As shown in Equation (4).  
\[D(N)=1-\left[1-(\frac{N}{N_{f}})^{\frac{1}{1-\alpha}}\right]^{\frac{1}{1+ \gamma}} \tag{4}\]  
where, \(N_{f}\) is the fatigue life, \(N\) is the loading cycles, and \(\alpha\) and \(\gamma\) are the material parameters related to the stress.  
Figure 4: (**a**) Fatigue curves characterized by stress ratio (**b**) Fatigue curves characterized by stress level.  
Based on Equations (3) and (4), in this paper, Equation (5) is deduced as the decay Equation of the moduli for asphalt mixtures.  
\[\frac{E(N)}{E_{0}}=\left[1-\left(\frac{N}{N_{f}}\right)^{\frac{1}{1-\alpha}} \right]^{\frac{1}{1+\gamma}} \tag{5}\]  
The fitting parameters are replaced by \(m\), \(n\), where, \(m=\frac{1}{1-\alpha}\), \(n=\frac{1}{1+\gamma}\).  
Equation (5) can be simplified as:  
\[\frac{E(N)}{E_{0}}=\left[1-\left(\frac{N}{N_{f}}\right)^{m}\right]^{n} \tag{6}\]