# 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} \hline \hline
**Forms** & **Author** & **Expression** \\ \hline \multirow{8}{*}{\(\Delta\varepsilon\sim N_{f}(R_{e}=-1)\)} & [PERSON] & \multirow{2}{*}{\(\varepsilon_{a}=\frac{\sigma^{\prime}{}_{f}}{E}\Big{(}2N_{f}\Big{)}^{2b}+ \varepsilon^{\prime}{}_{f}\Big{(}2N_{f}\Big{)}^{2c}\)} \\  & [PERSON] & & \\  & [PERSON] & & \\  & [PERSON] & \(\frac{1}{N_{f}}=2\Bigg{(}\frac{\sigma^{\prime}{}_{f}}{\varepsilon^{\prime}{}_{ f}}\Bigg{)}\frac{(\varepsilon_{pa})}{\varepsilon_{pa}}\Bigg{)}\frac{\sigma^{ \prime}{}_{f}}{\sigma^{\prime}{}_{f}-\sigma_{m}}\) \\  & [PERSON] & \(\frac{1}{N_{f}}=2\Bigg{(}\frac{\sigma_{\max}\varepsilon_{pa}}{\sigma^{\prime}{} _{f}\varepsilon^{\prime}{}_{f}E}\Bigg{)}^{1/(a+b)}\) \\  [PERSON] & \(\sigma_{\max}\varepsilon_{a}=\frac{\sigma^{\prime}{}^{2}}{E}\Big{(}2N_{f} \Big{)}^{2b}+\sigma^{\prime}{}_{f}\varepsilon^{\prime}{}_{f}\Big{(}2N_{f} \Big{)}^{b+c}\) \\  [PERSON] & \((\sigma_{a}+k\sigma_{m})\varepsilon_{a}=\frac{\sigma^{\prime}{}^{2}}{E}\Big{(}2 N_{f}\Big{)}^{2b}+\sigma^{\prime}{}_{f}\varepsilon^{\prime}{}_{f}\Big{(}2N_{f} \Big{)}^{b+c}\) \\  & [PERSON] & \(\sigma^{\prime}{}_{a}\sigma_{\max}^{(1-r)}-\varepsilon_{a}=\frac{\sigma^{ \prime}{}^{2}}{E}\Big{(}2N_{f}\Big{)}^{2b}+\sigma^{\prime}{}_{f}\varepsilon^{ \prime}{}_{f}\Big{(}2N_{f}\Big{)}^{b+c}\) \\  & [PERSON] & \(\frac{1}{N_{f}}=2\Big{(}4\sigma^{\prime}{}_{f}\varepsilon^{\prime}{}_{f}/A \Big{)}^{b+c}\) \\  & & \(A=\frac{\beta}{a}I,a=f(\sigma_{\max},\sigma_{\min})\) \\  & & \(I=\sum\limits_{k=1}^{k_{1}}(j+1-2k)\Big{(}\Delta\sigma_{k}-\Delta\sigma_{j+1}-k \Big{)}\Delta\varepsilon\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Strain fatigue curve. Fatigue test regulation for Manson-Coffin equation requires that \(R_{e}=-1\), otherwise, the equation should be modified, (see Table 7). In accordance with the first method, an equal strain fatigue test with strain levels ranging from 0.3 to 0.6 and a monotone direct tensile test were conducted on HMAC using the dynamic tester MTS810 to calculate the \(\varepsilon_{a}-N_{f}\) curve of asphalt concretes [33]. The \(\varepsilon_{a}-N_{f}\) curve was characterized by the Mason-Coffin equation, and the power function was adopted to describe the strain life curve:  
\[\varepsilon_{a}=\frac{\Delta\varepsilon}{2}=\frac{\Delta\varepsilon_{e}}{2}+ \frac{\Delta\varepsilon_{p}}{2}=\left(\frac{\sigma^{\prime}_{f}}{E}\right) \left(2N_{f}\right)^{b}+\varepsilon^{\prime}_{f}\left(2N_{f}\right)^{c} \tag{22}\]  
In the above equations, \(\sigma^{\prime}_{f}\) and \(\varepsilon^{\prime}_{f}\) are approximately equal to the fracture stress \(\sigma_{f}\) and fracture strain \(\varepsilon_{f}\)[34]. Fracture stress and fracture strain for HMAC were listed in the monotone direct tensile test (see Table 8). As seen in Table 8, \(\sigma^{\prime}_{f}\) of HMAC is 0.13295 MPa and \(\varepsilon^{\prime}_{f}\) is 0.00288. Constant strain fatigue tests at different stress-strain levels were conducted on HMAC, and the cyclic \(\sigma-\varepsilon\) curve at the strain level of 0.6 is shown in Figure 8. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Modified Manson-Coffin Equation** \\ \hline Morrow elastic stress & \(\varepsilon_{a}=\frac{\sigma^{\prime}_{f}-\sigma_{m}}{E}\left(2N_{f}\right)^{b }+\varepsilon^{\prime}_{f}\left(2N_{f}\right)^{c}\) \\ Cherber elastic stress curvature correction & \(\varepsilon_{a}=\frac{\sigma^{\prime}_{f}-\sigma_{m}^{2}}{E\sigma^{\prime}_{f}} \left(2N_{f}\right)^{b}+\varepsilon^{\prime}_{f}\left(2N_{f}\right)^{c}\) \\ Marrow total strain correction & \(\varepsilon_{a}=\frac{\sigma^{\prime}_{f}-\sigma_{m}}{\sigma^{\prime}_{f}} \frac{\sigma^{\prime}_{f}}{\varepsilon^{\prime}_{f}}\left(2N_{f}\right)^{b}+ \varepsilon^{\prime}_{f}\left(2N_{f}\right)^{c}\) \\ Sachs plastic correction & \(\Delta\varepsilon_{a}=\frac{\sigma^{\prime}_{f}}{E}\left(1-\frac{\varepsilon_ {a}}{\alpha_{a}}\right)^{b}\left(2N_{f}\right)^{c}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Modified Manson-Coffin equation when \(R_{e}\
eq-1\).