## 3 Results and Discussion
### Definition of a Mixture Index  
In an attempt to obtain a single index that would represent the mixture composition and that could explain the strength values well, a commonly used parameter for cemented soils [15; 16] was investigated: the volumetric cement content, \(C_{iv}\), given by Equation (8):  
\[C_{iv}=\frac{V_{c}}{V} \tag{8}\]  
where \(V_{c}\) is the cement volume given by Equation (9), and \(V\) is the specimen volume. \[V_{c}=\frac{w_{FA}+w_{SH}}{\left(\frac{w_{FA}\times\gamma_{sFA}+w_{SH}\times \gamma_{sSH}}{w_{FA}+w_{SH}}\right)} \tag{9}\]  
where \(w_{FA}\) is the fly ash weight; \(w_{SHs}\) is the solid sodium hydroxide weight; and \(\gamma_{sFA}\) and \(\gamma_{sSHs}\) are the particles unit weight of fly ash and solid sodium hydroxide, respectively. After calculating the \(C_{iv}\) for the mixtures indicated in Tables 3 and 6, another statistical model, denoted Model 3, was therefore obtained by Equation (10):  
\[q_{u}=10^{1.389}10^{3.831\cdot 10^{-6}\cdot c_{ip}} \tag{10}\]  
The volumetric cement content is currently used together with the mixture porosity, in the so-called porosity/cement ratio, as the effectiveness of cementation also depends on the number of contact points between particles [15]. Applying this other index to the data presented in Tables 3 and 6, following the same type of mathematical equation proposed by [34], led to the model proposed in Equation (11) (Model 4). \[q_{u}=10^{13.795}\Big{(}\frac{\eta}{c_{ip}0.0830}\Big{)}^{-7.065} \tag{11}\]  
Figure 17: Strength evolution depending on the individual variation in each input variable. If another type of appropriate mathematical function is used (\"S\" curve), a slightly better fit is found, as given by Equation (12) (Model 5). \[q_{u}=0.0561\cdot 10^{\frac{(-15),234.}{i_{w}(10,830)}} \tag{12}\]  
Table 7 summarizes the values of the coefficient of determination (\(R^{2}\)) and \(R^{2}\) adjusted associated with Models 3, 4, and 5, from which the following comments follow:  
* Models 3 to 5 provide similar coefficients of determination, all of the three slightly higher than those of the previous models (Models 1 and 2);
* Even in an experimental plan where all the mixtures have approximately the same porosity (see Figure 11), the porosity/cement ratio (Model 4) seems to perform slightly better than \(C_{iv}\) (Model 3);
* Model 5 has slightly higher values of \(R^{2}\) and \(R^{2}_{adj}\), and the estimates of the constants in the model are not very large values, as is the case on the other models (e.g., \(10^{13.795}\) in Model 4), which is in favor of Model 5 for practical use. In obtaining all the three expressions above (Models 3, 4, and 5), it was checked that assumptions of the regression modelling regarding residuals, outliers, and leverage points hold. Considering that, in this experimental plan, the water content and liquid content of the mixtures were not constant at the optimum points, as in previous studies [18; 34], several attempts were made to include this effect on the mixture index. In a preliminary approach, the experimental points were plotted together with Models 4 and 5 (see Figure 18), using a different marker for mixtures associated with each liquid content to evaluate if the mixtures with certain liquid contents would be farther away from the model curve. From Figure 18, it seems that the scatter is similar for all liquid contents. Subsequently, other indexes were evaluated. One of the indexes analyzed (\(w/b_{c}{}^{n}\)) is based on the two mass ratios defined in Equations (13) and (14), the water content (\(w\)) and the binder content (\(b_{c}\)), adjusted by an exponent, \(n\). \[w=\frac{w_{w}}{w_{s}} \tag{13}\]  
\[b_{c}=\frac{w_{FA}+w_{SH_{u}}}{w_{s}} \tag{14}\]  
where \(w_{w}\) is the water weight, \(w_{s}\) is the soil weight, \(w_{FA}\) is the fly ash weight, and \(w_{SHs}\) is the sodium hydroxide solid weight. For this index, Model 6 was fitted as expressed in Equation (15).