# 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  
It can be known that fatigue crack life includes fatigue cracking and fatigue propagation, which are the relatively accurate cracking life of asphalt pavement before fracture and instability. The crack propagation life can be obtained by the Paris formula. When calculating the propagation life of cracks via the Paris formula, two assumptions should be made: (1) the initial fracture length at the fracture propagation stage is assumed to be 5 mm; (2) in the process of fracture propagation, the fracture angle remains constant and develops vertically upward. From the beginning of fatigue crack formation, cracks gradually spread to the middle layer and the upper layer under the action of temperature and load, and eventually run through the entire pavement structure. Therefore, in order to facilitate the calculation for crack propagation life, different structural layers need to be transformed into the same structural layer through certain transformations. The equivalent surface thickness theory proposed by [PERSON] et al. was used to carry out this conversion, which considered the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pavement Structure**} & \multirow{2}{*}{**Pitch Depth**} & \multicolumn{3}{c}{**Cracking Life \(N_{i}\) (time)**} \\ \cline{3-5}  & & **25 \({}^{\circ}\)C** & 0 \({}^{\circ}\)C & \(-\)**10 \({}^{\circ}\)C** \\ \hline \multirow{4}{*}{A} & 0–40 & 2240 & 1697 & 1487 \\  & 40–90 & 21,784 & 21,783 & 15,475 \\  & 90–150 & 391,312 & 208,632 & 54,454 \\  & 0–40 & 12,490 & 4874 & 3804 \\ \multirow{2}{*}{B} & 40–100 & 314,175 & 172,269 & 101,396 \\  & 100–180 & 5,766,217 & 1,068,115 & 492,909 \\ \multirow{2}{*}{C} & 0–60 & 3805 & 3513 & 3249 \\  & 60–140 & 86,089 & 73,513 & 54,454 \\ \multirow{2}{*}{C} & 140–240 & 3,898,005 & 1,947,968 & 1,068,115 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Fatigue cracking life of each structural layer of the flexible base of asphalt pavement. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pavement Structure**} & \multirow{2}{*}{**Pitch Depth**} & \multicolumn{3}{c}{**Cracking Life \(N_{i}\) (time)**} \\ \cline{3-5}  & & **25 \({}^{\circ}\)C** & 0 \({}^{\circ}\)C & \(-\)**10 \({}^{\circ}\)C** \\ \hline \multirow{4}{*}{A} & 0–40 & 73,513 & 54,454 & 41,100 \\  & 40–90 & 254,833 & 120,164 & 86,088 \\ \multirow{4}{*}{B} & 90–150 & 1,947,968 & 628,683 & 492,909 \\  & 0–40 & 73,513 & 63,112 & 63,112 \\ \multirow{4}{*}{B} & 40–100 & 314,174 & 208,632 & 172,269 \\  & 100–180 & 5,766,216 & 2,719,023 & 1,428,062 \\ \multirow{4}{*}{C} & 0–60 & 101,396 & 101,396 & 63,112 \\  & 60–140 & 1,068,115 & 628,683 & 314,174 \\ \multirow{4}{*}{C} & 140–240 & 24,168,688 & 8,854,377 & 1,947,968 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Fatigue cracking life of each structural layer of the semirigid base in asphalt pavement. influence of the change in the SIF in the mixture. When cracks eventually penetrated the entire surface thickness, the method would transform the lower and middle surface layers into the equivalent thickness of the upper layer, and the modulus remained unchanged, as shown in Figure 12.  
It is worth noting that there are two basic assumptions before the thickness conversion between the different layers is carried out: (1) only the elastic state of the first part of the dynamic modulus is considered, and the asphalt mixture is considered as quasielastic; therefore, the relationship between stress and strain is linear; (2) the surface layer is completely connected with the other layer, without considering the influence of contact between asphalt layers on crack propagation.