# Investigation of the Fatigue Life of Bottom-Up Cracking in Asphalt Concrete Pavements
## 4 Prediction of Fatigue Crack Life of Asphalt Pavements
### Crack Propagation Life
Therefore, according to the equivalent layer thickness transformation theory, three surface layer structures, A, B, and C, are respectively transformed into the equivalent thicknesses related to the upper layer [25]:  
For structure A: \(H_{1}^{{}^{\prime}}=H_{1}+H_{2}\sqrt[3]{\frac{E_{2}}{E_{1}}}+H_{3}\sqrt[3]{ \frac{E_{3}}{E_{1}}}=17.9\)cm  
For structure B: \(H_{1}^{{}^{\prime}}=H_{1}+H_{2}\sqrt[3]{\frac{E_{2}}{E_{1}}}+H_{3}\sqrt[3]{ \frac{E_{3}}{E_{1}}}=21.7\)cm  
For structure C: \(H_{1}^{{}^{\prime}}=H_{1}+H_{2}\sqrt[3]{\frac{E_{2}}{E_{1}}}+H_{3}\sqrt[3]{ \frac{E_{3}}{E_{1}}}=28.7\)cm  
The Paris formula is transformed into an integral formula and is combined with the equivalent thickness of structures A, B, and C; then, the crack propagation life of the asphalt pavement structure with different surface layers can be calculated. When calculating the crack propagation life, the initial crack length was assumed to be 5 mm and the crack ran through the whole pavement structure. Therefore, the crack propagation life of different asphalt pavement structures under 25 \({}^{\circ}\)C, 0 \({}^{\circ}\)C, and \(-\)10 \({}^{\circ}\)C are shown in Table 10, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Pavement**} & \multirow{2}{*}{\(\mathbf{C_{\text{o}}}\) **(mm)**} & \multirow{2}{*}{\(\mathbf{C_{\text{i}}}\) **(mm)**} & \multirow{2}{*}{\(\mathbf{E}\) **(MPa)**} & \multirow{2}{*}{\(\mathbf{MOD}\) **(mm)**} & \multirow{2}{*}{\(\mathbf{N_{p}}\)**(time)**} & \multirow{2}{*}{\(\mathbf{E}\) **(MPa)**} & \multirow{2}{*}{\(\mathbf{MOD}\) **(mm)**} & \multirow{2}{*}{\(\mathbf{N_{p}}\)**(time)**} & \multirow{2}{*}{\(\mathbf{E}\) **(MPa)**} & \multirow{2}{*}{\(\mathbf{MOD}\) **(mm)**} & \multirow{2}{*}{\(\mathbf{N_{p}}\)**(time)**} \\ \cline{4-4} \cline{6-11}  & & & & & & & & & & 25,149 & 0.625 & \\ A & 5 & 50 & 1497 & 0.625 & 14,628,099 & 13,009 & 0.625 & 47,998 & 21,916 & 0.625 & 78 \\  & 5 & 60 & 1850 & 0.625 & & 15,813 & 0.625 & & 25,843 & 0.625 & \\  & 5 & 40 & 832 & 0.625 & & 14,690 & 0.625 & & 25,149 & 0.625 & \\ B & 5 & 60 & 1497 & 0.625 & 18,600,394 & 13,009 & 0.625 & 58,390 & 21,916 & 0.625 & 97 \\  & 5 & 80 & 1850 & 0.625 & & 15,813 & 0.625 & & 25,843 & 0.625 & \\  & 5 & 60 & 832 & 0.625 & & 14,690 & 0.625 & & 25,149 & 0.625 & \\ C & 5 & 80 & 1497 & 0.625 & 26,937,407 & 13,009 & 0.625 & 80,363 & 21,916 & 0.625 & 134 \\  & 5 & 100 & 1850 & 0.625 & & 15,813 & 0.625 & & 25,843 & 0.625 & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Crack propagation life for each structure of asphalt pavements. Figure 12: Schematic diagram of pavement structure and equivalent layer thickness conversion. It can be seen from Table 10 that the crack propagation life of the asphalt pavement increased with the increase in surface thickness. For the sample with the same grade and the same oil content, the crack propagation life decreased sharply with the decrease in temperature. When the temperature dropped from 25 \({}^{\circ}\)C to 0 \({}^{\circ}\)C, the crack propagation life of the three layers decreased by more than 300 times. When the temperature dropped from 0 \({}^{\circ}\)C to \(-\)10 \({}^{\circ}\)C, the crack propagation life of the three layers decreased by more than 600 times. This phenomenon has the same variation trend with the number of load cycles of the OTs at different temperatures in the room.