## 5 Fatigue and Rutting Damage Analysis  
Rutting damage in the pavement occurs as a result of increased shear deformation and densification of the pavement structure. As the number of loading passes increases, longitudinal cracks start to develop under the wheel paths. Further interlocking of the cracks develop lead to formation of alligator tracking until the pavement is no longer functional, resulting in fatigue damage. The Asphalt Institute model can be used to calculate the number of passes to fatigue damage using horizontal tensile strains at the bottom of the asphalt layer, as shown in Equation (6) [39]. \[N_{f}=0.0796\times\varepsilon_{t}^{-3.291}\times E^{-0.854} \tag{6}\]  
where \(N_{f}\) is the number of load repetitions to fatigue damage, \(\varepsilon_{t}\) is the horizontal tensile strain at the bottom of the asphalt layer and \(E\) is the elastic modulus of asphalt mixture. The Asphalt Institute provides the relationship between compressive strain and number of passes to rutting damage, as shown in Equation (7). \[N_{f}=1.365\times 10^{-9}\times\varepsilon_{c}^{-4.477} \tag{7}\]  
where \(N_{f}\) is the number of load repetitions to rutting damage and \(\varepsilon_{c}\) is the vertical compressive strain on top of subgrade. Horizontal tensile strains and vertical compressive strains for each mixture type are shown in Table 11. As observed, the highest magnitude of \(\varepsilon_{c}\) and \(\varepsilon_{t}\) exists for SBS-7% mixture at 39.4 and 562.4, respectively. With the addition of fiber to the mixtures, better cohesion and interlocking of bitumen and aggregates occur, and \(\varepsilon_{c}\) and \(\varepsilon_{t}\) values further decrease, with the least magnitude exhibited by BF mixture at 410.6 and 20.1, respectively. The strain values in terms of et ad ec decrease by 35% and 31%, respectively compared to BF mixture. Both LF and PF mixtures have closer \(\varepsilon_{t}\) and \(\varepsilon_{c}\) values, with a difference of 8% in \(\varepsilon_{t}\) values. BF mixture outperforms both LF and PF mixtures by 15% and 8% in terms of reduced et values. The number of passes to fatigue and rutting damage also increases in case of fiber-modified asphalt mixtures. The comparison of the number of passes to fatigue damage for each mixture type is shown in Figure 9. As observed, there is a significant reduction in deceased number of passes for SBS-7% mixture, with only \(1.18\times 10^{10}\) passes. A slight improvement in increased number of passes can be observed for LF, with \(1.98\times 10^{10}\) passes. BF shows better overall performance in terms of increased number of passes to fatigue damage, with the magnitude of passes at 4.63 \(\times\) 10\({}^{10}\). Better cohesion and elastic response with the use of BF in asphalt mixture further increases the rutting resistance. \begin{table}
\begin{tabular}{c c c c c} \hline
**Depth** & **SBS-7\%** & **LF** & **BF** & **PF** \\ \hline
50 mm & 2.95 & 2.49 & 2.23 & 2.36 \\
450 mm & 0.36 & 0.29 & 0.21 & 0.24 \\
800 mm & 0.074 & 0.045 & 0.028 & 0.034 \\ \hline \end{tabular}
\end{table} 
Table 10: Von Mises stress values for different scenarios. The comparison of number passes to rutting damage is shown in Figure 10. All asphalt mixture types exhibit better performance against fatigue when compared to permanent deformation due to the presence of SBS modifier. However, the performance can be further improved with the use of fibers. As observed, the highest number of passes to rutting damage is exhibited by BF at \(6.57\times 10^{6}\) passes. PF follows the performance of BF modified SBS mixtures with a magnitude of \(5.63\times 10^{6}\) passes. The SBS-7% mixture, however, exhibits the lowest performance with only \(1.52\times 10^{6}\) passes to reach the rutting damage.