# Modeling of Fatigue-Strength Development
## 3 Results and Discussions
### Prediction of Fatigue-Strength Development in Emulsion Asphalt.
#### 3.2.7 Validation of the Actual Fatigue-Maturity Models  
The models presented as Equations (15) and (16) were developed using fatigue data of specimens of cold-emulsion reclaimed asphalt mix cured at 5 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, and 40 \({}^{\circ}\)C. To validate these models, an independent set of fatigue data from specimens cured at 50 \({}^{\circ}\)C for up to 28 days was used. The coefficients in Table 10 were used in Equations (15) and (16) to predict fatigue strength values of the cold-emulsion reclaimed asphalt pavement mix at a cure temperature of 50 \({}^{\circ}\)C. Results of load cycles to failure for the specimens cured at 50 \({}^{\circ}\)C were determined using procedures set out in Section 2.3 and plotted alongside the predicted values as shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Strain Level, \(\epsilon\)** & **Linear Hyperbolic Model** & **Parabolic Hyperbolic Model** \\ \cline{2-5}
**(\(\mu\)m/m)** & **k (1/day)** & **Nf\({}_{\mathbf{u}}\) (\(\times\)10\({}^{-3}\))** & **k (1/day)** & **Nf\({}_{\mathbf{u}}\) (\(\times\)10\({}^{-3}\))** \\ \hline
125 & 0.35 & 79,050.35 & 0.91 & 40,  
Figure 8: Evolution of fatigue strength of cold-emulsion reclaimed asphalt mix cured at 50 \({}^{\circ}\)C.  
It can be observed in Figure 8 that both the linear hyperbolic and parabolic hyperbolic maturity functions give good prediction early-life and the limiting fatigue values but fail to accurately capture fatigue strengths in the intermediate cure durations. When visually inspected, both the linear hyperbolic and parabolic hyperbolic models in Figure 8 appear to almost coincide at test strain levels between 125 \(\upmu\)m/m and 200 \(\upmu\)m/m. To ascertain the accuracy of the prediction offered by each of these functions, the predicted fatigue strength values were plotted against the experimental fatigue values for all test strain levels. Figure 9 depicts such a plot, for specimens tested at 125 \(\upmu\)m/m.  
It can be observed in Figure 9 that, there was a rapid drop in fatigue strength of the cold-emulsion asphalt between the first and third day of cure. This is attributed to the rapid evaporation in this period, as reflected in the rate constants presented in Section 3.2.2. Coefficients of determination of the fit between predicted and experimental fatigue strengths were determined at the 6 test strain levels and recorded in Table 11.  
The coefficients of determination, R\({}^{2}\), were used to assess the accuracy with which the linear hyperbolic and the parabolic hyperbolic models predicted fatigue strength development in the cold-emulsion reclaimed asphalt pavement mixture. It can be deduced from Table 11 that the proposed models offered reasonable predictions for cold-mix asphalt fatigue strengths up to a strain level of 175 \(\upmu\)m/m for the linear hyperbolic model and up to a strain level of 200 \(\upmu\)m/m for the parabolic hyperbolic model. Generally, the parabolic hyperbolic model offered a better correlation to the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Strain Level, \(\epsilon\) (\(\upmu\)m/m)** & **Linear Hyperbolic Model** & **Parabolic Hyperbolic Model** \\ \hline
125 & 0.9755 & 0.9854 \\
150 & 0.9116 & 0.9823 \\
175 & 0.8078 & 0.9235 \\
200 & 0.6799 & 0.8202 \\
225 & 0.4495 & 0.4515 \\
250 & 0.3817 & 0.3932 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Model coefficients of determination at various test strain levels.  
Figure 9: Validation of fatigue strength prediction models for emulsion reclaimed asphalts.  
experimental data than the linear hyperbolic model at all strain levels. The reduction in the predictive accuracy of the models at load strain levels beyond 200 \(\upmu\)m/m can be explained by the manner in which asphalt mixtures behave under loads. At low strain levels, asphalt behaves in an almost linear elastic manner but tends to visco-elastic behavior as loads increase. This inevitably leads to high variability in results at higher load strain levels than in lower load strain levels.