## 4 Finite Element Modelling
### Fatigue and Putting Damage Analysis  
Fatigue and rutting damage in terms of number of passes have been evaluated for each mixture type. Due to increased loads and high-temperature conditions, shear deformation and densification of the asphalt mixture occur, leading to the occurrence of permanent deformation. Furthermore, in terms of fatigue, the Asphalt Institute model provides a relationship between the number of passes to fatigue damage and horizontal tensile strain at the bottom of asphalt layer using Equation (4),  
\[N_{f}=0.0796\times e_{t}^{-3.291}\times E^{-0.854} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Depth** & **SBS-7\%** & **BF** & **CF** & **GF** \\ \hline
50 mm & 2.98 & 2.30 & 2.34 & 2.57 \\
450 mm & 0.33 & 0.25 & 0.23 & 0.25 \\
800 mm & 0.070 & 0.023 & 0.022 & 0.024 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Von Mises stress values for each scenario. Figure 3: Simulation results for the SBS 7% mixture. where, \(N_{f}\) is the number of load repititon to fatigue damage, \(\varepsilon_{t}\) is the horizontal tensile strain at the bottom of the asphalt layer, and \(E\) is the elastic modulus of the asphalt mixture in MPa. The Asphalt Institute model also provides a relationship between the number of passes to rutting damage and vertical compressive strain on top of the subgrade as shown in Equation (5),  
\[N_{f}=1.365\times 10^{-9}\times\varepsilon_{c}^{-4.477} \tag{5}\]  
where, \(N_{f}\) is the number of load repititon to rutting damage and \(\varepsilon_{c}\) is the vertical compressive strain on top of subgrade. The number of passes to fatigue damage along with the corresponding et and ec values are shown Table 8. As observed, the highest accumulation of both et and ec exist for SBS-7% at 34.1 MPa and 485.2 MPa, respectively, leading to further decreases in the number of passes to fatigue damage. With the use of fiber-reinforced mixtures, the magnitude of et and ec drops down to 18.2 MPa and 433.5 MPa in the case of the BF mixture, with the maximum number of passes to fatigue damage at 7.86 \(\times\) 10\({}^{10}\) and 8.96 \(\times\) 10\({}^{6}\), respectively. CF shows almost identical performance for BF in terms of the number of passes to fatigue damage. GF, however, shows the worst performance among the fiber-reinforced mixtures, with the number of passes to fatigue damage at 4.36 \(\times\) 10\({}^{10}\) and 5.84 \(\times\) 10\({}^{6}\)  
Visual representation of the comparison of number of passes to fatigue damage is shown in Figure 4. As observed, the highest number of passes to fatigue damage exists for BF mixture at 7.86 \(\times\) 10\({}^{10}\) passes. CF shows an almost identical performance at 6.79 \(\times\) 10\({}^{10}\). Therefore, BF outperforms CF by 13% and GF by 25%. However, the highest difference exists between the BF and SBS-7%, with an increase in the number of passes to fatigue damage for the BF-modified mixture by 61%. Furthermore, in terms of fatigue damage, both CF and BF mixtures show almost identical performance, where the addition of CF further improves the carbon chain links and reinforces the bitumen mixtures, thereby improving the elasticity of the asphalt layer. GF on the other hand, exhibits slight improvement to the number of passes to fatigue damage when compared to SBS-7% mixture. The visual representation of the comparison of compressive strain values is shown in Figure 5. As observed, SBS-7% shows the highest microstrains at 485.2 microns. Furthermore, BF outperforms CF to show reduced microstrains at 433.5 microns and 441.5 microns. Although both BF and CF show almost identical performance in terms of accumulated microstrains, BF performs significantly better by 13% than the CF mixture. This is due to the improvement of the viscous portion during the shear testing of bitumen where BF provides better performance against permanent deformation than the CF mixture.