## 5 Conclusions  
In this study, drained triaxial tests were first carried out on carbonate sand-fine mixtures with carbonate sand contents ranging from 0 to 40%. As a potential density parameter, the interfine void ratio was selected as a control variable in addition to the traditional void ratio. Therefore, two series of triaxial tests were conducted by controlling similar void ratios and interfine void ratios. Based on the test results, the effect of carbonate sand content on the mechanical behaviour of carbonate sand-fine mixtures was investigated. Using computer tomography, the internal porosity of carbonate sand grains was measured, and the microscopic structure of carbonate sand-fine mixtures was studied. Furthermore, the equation of the interfine void ratio was modified and proved effective in determining a state variable to describe the density of carbonate sand-fine mixtures. The conclusions are as follows:  
1. For carbonate sand-fine mixtures with a similar initial void ratio, all mixtures exhibited strain-softening behaviour, and the peak strength decreased with an increase in carbonate sand content. The shear strength value was almost the same at large deformation under the same effective confining pressure. In addition, the volume changed from dilative to contractive behaviour with an increase in carbonate sand content. As the carbonate sand content increased, the mixture seemed to become \"looser\". In the \(p^{\prime}\)-\(q\) plane, a unique CSL was observed despite different carbonate sand contents, while the CSL shifted downwards with an increase in carbonate sand content in the \(e\)-\(\ln\,p^{\prime}\) plane. 2. For carbonate sand-fine mixtures with similar initial interfine void ratios, the peak strength increased with an increase in carbonate sand content, and the volume changed from dilative to contractive behaviour with a decrease in carbonate sand content. The mixture became \"denser\" with an increase in carbonate sand content. In the \(e\)-\(\ln\,p^{\prime}\) plane, the CSL shifted upwards with an increase in carbonate sand content. 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 interfine void ratio. 3. Based on the two-dimensional cross-sectional CT images, it was found that most of the tested carbonate sand grains contained a substantial number of internal pores. For the tested carbonate sand, the internal porosities of one hundred carbonate sand grains were computed using an innovative integral calculation method. A normal  
Figure 11: Critical state lines of carbonate sand-fine mixtures in (a) \(p^{\prime}\)–\(q\) and (b) \(e_{\zeta}\)–\(\ln\,p^{\prime}\) planes. distribution of internal porosity was observed, with a mean value of 0.238. In addition, most of the carbonate sand grains were not in direct contact with each other, and the space around the contact angles between carbonate sand grains was tightly filled with fine particles. Carbonate sand grains likely did not contribute to the force chains of the fine matrix when the carbonate sand content was less than 40%. 4. According to the original definition of the interfine void ratio, except for the volume of carbonate sand grains, their internal pores should also be neglected. Considering the influence of internal pores, the equation of the modified interfine void ratio for carbonate sand-fine mixtures was established for the first time. The mixture with the smallest modified interfine void ratio exhibited the highest peak strength and strongest dilative volumetric behaviour. Using the modified interfine void ratio, unique CSLs were achieved in both the \(p^{\prime}\)-\(q\) and \(e_{f}\)-In \(p^{\prime}\) planes, representing the critical state. As a result, the modified interfine void ratio can be used as a state variable to coherently characterise the mechanical behaviour of carbonate sand-fine mixtures with a carbonate sand content of no more than 40%. 5.