## 4 Modified Interfine Void Ratio  
Compared to the mixture in which the density was controlled by the void ratio, the change rule of mechanical behaviour influenced by the carbonate sand content was precisely the opposite in specimens controlled by the interfering void ratio. Therefore, by adopting different density parameters, the effect of carbonate sand content on the mechanical properties of the mixtures was also very different. Past studies have proved the rationality of the interfine void ratio as the density parameter for sand-fine mixtures. Therefore, an appropriate density parameter should be determined that can be used independently to describe the state (including initial and critical states) and characterise the mechanical properties of carbonate sand-fine mixtures. For carbonate sand-fine mixtures, the internal pores are inactive. Based on the concept of the interfine void ratio defined by [PERSON] et al. [7, 8], their volume should be neglected when the interfine void ratio is evaluated. Thus, for carbonate sand-fine mixtures, the interfine void ratio is expressed as follows:  
\[e_{f}=\frac{V_{int}}{V_{f}} \tag{4}\]  
where \(V_{int}\) is the volume of intergranular pores, and \(V_{f}\) is the volume of fines. The void ratio of carbonate sand-fine mixtures is calculated as follows:  
\[e=\frac{V_{int}+V_{int}}{V_{f}+V_{c}} \tag{5}\]  
where \(V_{in}\) and \(V_{c}\) are volumes of the internal pores and solid parts of carbonate sand grains, respectively. The relationship between \(V_{in}\) and \(V_{c}\) can be expressed using the following equation:  
\[V_{in}=\frac{n_{in}}{1-n_{in}}V_{c} \tag{6}\]  
Substituting Equations (5) and (6) into Equation (4) yields the following:  
\[e_{f}=\bigg{(}1+\frac{V_{c}}{V_{f}}\bigg{)}e-\frac{n_{in}}{1-n_{in}}\frac{V_ {c}}{V_{f}} \tag{7}\]  
If specific gravities of carbonate sand and fines are the same, the ratio of the volume of carbonate sand and fines (\(V_{c}/V_{f}\)) is equal to the ratio of their mass (\(CS/(1-CS)\)). As mentioned previously, a certain deviation was observed in the specific gravity of carbonate sand and fines tested in this study. Accordingly, considering the influence of the specific gravity, the interfine void ratio for carbonate sand-fine mixtures was modified as follows:  
\[e_{fc}=\bigg{(}\frac{G_{f}CS}{G_{\rm s}(1-CS)}+1\bigg{)}e-\frac{n_{in}}{1-n_{ in}}\frac{G_{f}CS}{G_{\rm s}(1-CS)} \tag{8}\]  
where \(e_{fc}\) is the modified interfine void ratio for carbonate sand-fine mixtures. \(G_{s}\) and \(G_{f}\) are specific gravities of carbonate sand and fines, respectively. When the carbonate sand content is zero, the modified interfine void ratio is still equal to the conventional void ratio. Based on Equation (8), the modified interfine void ratios of all fifteen tested specimens at the initial and critical states were calculated, and the results are summarised in Table 4. The drained triaxial test results of five carbonate sand-fine mixtures with different carbonate sand content shown in Figures 2 and 3 were analysed using the framework of the modified interfine void ratio. Among the tested specimens with similar void ratios (Figure 2), specimen C5_V0 had the smallest modified interfine void ratio \(e_{fc}\) after consolidation (the densest specimen), resulting in the highest peak strength and strongest dilative volumetric behaviour. In addition, all three specimens had similar \(e_{fc}\) (0.597-0.618) at the critical state, leading to a similar critical state shear strength (0.895-0.913 MPa). The same analysis results were obtained for those specimens with similar interfine void ratios (Figure 3). Thus, the modified interfine void ratio \(e_{fc}\) can be used as a state variable to coherently characterise the mechanical behaviour of carbonate sand-fine mixtures with carbonate sand contents no more than 40%. The critical state data shown in Figures 3 and 4 are presented in Figure 11 in terms of the modified interfine void ratio.