## 4 Finite Element Modeling  
### Data for Modeling  
The behavior of viscoelastic materials can be modeled by the [PERSON]'s logit model for understanding the stress-strain response [22]. Furthermore, the data from dynamic modulus testing is used to evaluate the parameters for the [PERSON]'s logit model, as shown in Equation (5). [37]. [PERSON]'s model consists of Maxwell and Kelvin models, where it can reflect both elastic deformation and stress relaxation, thereby efficiently modeling the mechanical behavior of asphalt mixture [38]. The modified [PERSON]'s logit model is used to incorporate the effect of decreasing viscous deformation with increasing time, thereby stabilizing the viscous deformation [37]. A generalized dashpot is used in case of modified [PERSON]'s logit model, where the strain further decreases and stabilizes.  
\[\varepsilon(t)=\left[\frac{1}{E_{0}(1-\omega)}+\frac{1}{E_{2}}\bigg{(}1-e^{ \frac{E_{2}}{n_{2}}}\bigg{)}+\frac{1}{A^{\varepsilon_{B}}}\right] \tag{5}\]  
where \(\varepsilon(t)\) is the creep strain of modified [PERSON]'s model, \(E_{0}\) is the dynamic modulus, \(E_{2}\) is the internal damper parameter, \(n_{2}\), \(A\) and \(B\) are the model viscoelastic fitted parameters and \(\omega\) is the damage value. Creep strain values are evaluated at temperature of 30 \({}^{\circ}\)C in order to reduce bias regarding rutting progression at high temperatures. This model utilizes the material decay function with the use of external damper and external spring for accurate evaluation of deformation in viscoelastic materials.  
The [PERSON]'s logit model parameters are shown in Table 8. As observed, a gradual increase in n\({}_{2}\) values occurs for the BF mixture. SBS-7% yields the least n\({}_{2}\) value of 6482.36 MPa, while the addition of fiber increases both A and n\({}_{2}\) parameters for all mixture types, with the maximum value of A exhibited by BF at 167.89 MPa. The performance of both LF and PF modified mixtures stay close with A parameter values at 158.36 MPa and 161.25 MPa, respectively. LF yields the least magnitude of n\({}_{2}\) parameter among the fiber-modified mixtures at 6986.45 MPa.  
Loading simulations were conducted using a cumulative axle load of 100 kN at 50,000 passes with a traffic speed of 90 km/h. A dual axle wheel was considered for exerting the load on the developed finite element model. Further loading details are shown in Table 9. Loading details.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mixture Types** & **n\({}^{2}\) [MPa]** & **A [MPa]** & **B** & **R\({}^{2}\)** \\ \hline SBS-7\% & 6482.36 & 154.20 & 0.001124 & 0.9907 \\ LF & 6986.45 & 158.36 & 0.001075 & 0.9986 \\ PF & 7031.25 & 161.25 & 0.001067 & 0.9971 \\ BF & 7585.63 & 167.89 & 0.001043 & 0.9989 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Modified Burger’s logit model parameters.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Tire Type** & **Dual Tire** \\ \hline Ground pressure & 0.6 MPa \\ \hline Axle load & 100 kN \\ \hline Speed & 90 km/h \\ \hline Number of axle passes & 50,000 \\ \hline Single loading time & 0.21457 \\ \hline Cumulative loading time & 11,524.11 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Loading details.