## 3 Results and Discussion
### Effect of Fine Fraction Content on Cracking Resistance
#### 3.1.1 Tensile Stiffness Index (TSI)  
The tensile stiffness index assesses the stiffness of the mixture. Figure 5 provides the average TSI values against temperature, obtained from the four replicates for each mix design.  
As observed in Figure 5, two main groups can be differentiated: mixtures manufactured with the polymer modified binder, PMB 45/80-65, and those manufactured with the conventional binder, B35/50. These results are consistent with those obtained in previous studies that stated the importance of selecting the optimal asphalt binder based on the environmental conditions to increase the cracking resistance of asphalt concrete [24].  
Figure 4: Stress-displacement curve at a test temperature of 5 \({}^{\circ}\)C.  
Figure 5: Tensile stiffness index versus temperature.  
Results show that the values of the tensile stiffness index increase with the decrease of temperature in all the evaluated mixtures. At a temperature of 20 \({}^{\circ}\)C all the mixtures present a similar behavior but as the temperature decreases clear differences can be observed. Mixtures manufactured with B35/50 increase stiffness more sharply than those manufactured with PMB 45/80-65, showing the greatest disparity at a temperature of 5 \({}^{\circ}\)C. Below this temperature, the trend is reversed and PMB 45/80-65 mixtures present the fastest growth in stiffness. However, B35/50 mixtures present the highest values over the whole temperature range, which is consistent with the properties of this binder, which has a greater stiffness compared with the polymer modified binder.  
Regarding the fine fraction content, the results show slight differences depending on the fine content leading to a higher tensile stiffness index as the content of fine aggregates increases.  
Statistical techniques have also been used to strengthen the conclusions drawn from the graphical interpretation. A multi-linear regression analysis has been used to fit the data and establish correlations between the tensile stiffness index and the predictors. R software (The R foundation(r)) was used for model estimation.  
\[y=\beta_{0}+\beta_{1}\times x_{1}+\beta_{2}\times x_{2}+\ldots+\beta_{i}\times x _{i}+\varepsilon \tag{1}\]  
where \(\beta_{i}\) = partial regression coefficients or estimates of the regression parameters and \(\varepsilon\) = random error term.  
It should be noted that only two types of asphalt binder were studied in this experimental work. So, in order to generalize these correlations other asphalt binders must be evaluated. The aim of these multi-linear relationships was to establish the weight of each studied factor and compare relevance between relevant properties.  
In this case, a multi-linear relationship between the tensile stiffness index and the temperature (T), the bitumen penetration at 25 \({}^{\circ}\)C (BG) and the fine fraction content (F) has been obtained.  
\[TSI=19.269-0.531\times T-0.137\times BG+0.049\times F\]  
\[R^{2}=0.897 \tag{2}\]  
The negative estimated regression coefficient for temperature implies that a temperature decrease would lead to an increase of the tensile stiffness index. Mixtures with higher fine fraction content presented higher values of stiffness, indicated by the positive coefficient.