# Fatigue Life Prediction of High Modulus Asphalt Concrete Based on the Local Stress-Strain Method
## 6 Strain-Life Fatigue Equation
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen** & **Fracture Strain** & **Fracture Stress/MPa** & **Secant Modulus/MPa** \\ \hline SJ-1 & 0.00308 & 0.1588 & 351.55 \\ SJ-2 & 0.00384 & 0.13672 & 385.60 \\ SJ-3 & 0.0023 & 0.13465 & 413.54 \\ SJ-4 & 0.0023 & 0.10162 & 345.18 \\ Average & 0.00288 & 0.13295 & 373.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: HMAC parameters of monotone direct tensile test. Figure 8: Cyclic \(\sigma-\varepsilon\) curve at the strain level of 0.6. The fatigue parameters strain and fatigue life of the specimen were then obtained, as shown in Table 9.  
As seen in Table 9, four common fitting algorithms were used to fit the data from Equation (1). The relationship between the fitting parameters and the equation was studied, as seen in Table 10.  
As shown in Table 10, the fitting parameters (\(b\), \(c\)) of the four fitting methods are nearly the same--\(b=-0.1023\) and \(c=-0.1026\). When \(\sigma^{\prime}{}_{f}\), \(\varepsilon^{\prime}{}_{f}\), \(b\), and \(c\) are put into Equation (22), then Equation (22) changes into the following form which predicts the fatigue life of HMAC in the strain-controlled mode:  
\[\varepsilon_{a}=\frac{0.13295}{E}\times\left(2N_{f}\right)^{-0.1023}+0.00288 \times\left(2N_{f}\right)^{-0.1026} \tag{23}\]  
**7. Fatigue Life Prediction Based on Local Stress-Strain Method**  
The fatigue life of HMAC was predicted based on the modified Neuber equation. First, the load-time process was transformed into a nominal-time history. Second, local stress and strain of the notched specimen were transformed to nominal stress and strain using the cyclic \(\sigma-\varepsilon\) curve and the modified Neuber equation. Fatigue damage was calculated according to the strain fatigue life curve, and the fatigue life was then obtained based on cumulative damage theory. Specific steps for fatigue life prediction based on the modified Neuber equation and local stress-strain method are as follows:  
(1) Local strain and local stress in the plane strain state were solved using the results of the load strain and temperature strain, cyclic \(\sigma-\varepsilon\) curve and Neuber hyperbolic equations. \[\varepsilon=\frac{\sigma}{E}+\left(\frac{\sigma}{26.847}\right)^{1.3793} \tag{24}\]  
\[\varepsilon\sigma=0.92\times 1.976^{2}se \tag{25}\]  
(2) The fatigue life \(N_{f}\) was determined on the basis of the \(\varepsilon_{a}-N_{f}\) curve equation. \[\varepsilon_{a}=\frac{0.13295}{E}\times\left(2N_{f}\right)^{-0.1023}+0.00288 \times\left(2N_{f}\right)^{-0.1026} \tag{26}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fitting Algorithms** & \(b\) & \(c\) & \(\mathbf{R^{2}}\) \\ \hline Marquardt method & \(-0.10234\) & \(-0.10263\) & \(0.981687\) \\ Quasi-newton method & \(-0.10233\) & \(-0.10264\) & \(0.981688\) \\ Evolution algorithm & \(-0.10231\) & \(-0.10267\) & \(0.981681\) \\ Max inherit optimization & \(-0.10237\) & \(-0.10261\) & \(0.981687\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parameters b and c using different fitting algorithms. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specimen** & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **S7** & **S8** \\ \hline HMAC & & & & & & & & \\ \(\varepsilon_{a}\) & 0.32 & 0.53 & 0.74 & 0.91 & 1.15 & 1.392 & 1.56 & 1.66 \\ \(N_{f}\) & 18242 & 15124 & 7692 & 4543 & 3692 & 2420 & 1502 & 1031 \\ \hline Matrix asphalt concretes & & & & & & & & \\ \(\varepsilon_{a}\) & 1.28 & 2.17 & 3.04 & 3.75 & 4.18 & 5.37 & 6.17 & 6.89 \\ \(N_{f}\) & 4862 & 3941 & 1920 & 1137 & 1972 & 1143 & 108 & 84 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Strain amplitude and fatigue life of HMAC in equal strain fatigue test.