## 3 Results and Discussion  
### Under the Condition of Similar Void Ratios  
The effects of the carbonate sand content on the stress-strain behaviour of carbonate sand-fine mixtures were very similar for the range of confining pressures studied. Thus, only the triaxial drained test results obtained under an effective confining pressure of 0.5 MPa are presented, as shown in Figure 3. The specimen identities of the three specimens tested are C5_V0, C5_V20 and C5_V40. In the figure, deviator stress (\(q\)), which is defined as the difference between the major and minor principal stresses, represents the strength of specimens. Specimens were prepared at a similar initial void ratio ranging between 0.573 and 0.583. As observed in Figure 3, the peak strength of the specimen decreased with increasing carbonate sand content, and both deviator stress and volumetric strain maintain a steady value around an axial strain of 30%. In addition, the specimen changed from dilative to contractive behaviour with an increase in carbonate sand content. As the carbonate sand content increased, the specimen seemed to become \"looser\" despite being under the condition of similar initial void ratios. The three specimens all exhibited strain-softening behaviour. However, the shear strength of the three specimens approached almost the same value at large deformation under the same effective confining pressure. Considering the significant convergence of the stress-strain and volumetric deformation curves, it is assumed that the critical state has been achieved at the end of shearing for all specimens. [53]. The critical state data of triaxial tests under the condition of similar void ratios are summarised in Table 2. The mean effective stress (\(p^{\prime}\)) is defined as \(p^{\prime}=(\sigma_{1}{}^{\prime}+2\sigma_{3}{}^{\prime})/3\), where \(\sigma_{1}{}^{\prime}\) and \(\sigma_{3}{}^{\prime}\) are the major and minor principal stresses, respectively.  
Figure 4a,b show the critical state data of the triaxial drained tests in the \(p^{\prime}\)-\(q\) and \(e\)-In \(p^{\prime}\) planes for specimens with similar initial void ratios, respectively. The critical state in the \(p^{\prime}\)-\(q\) plane is represented by a linear best-fit line passing through the origin. Clearly, the carbonate sand content had no significant effect on the critical state stress ratio for carbonate sand-fine mixtures. For the range of the carbonate sand contents tested, this finding is consistent with the aforementioned assumption that for sand-fine mixtures where fines played a dominant role in the soil skeleton, carbonate sand did not contribute to the force chains of the fine matrix, leading to a similar stress ratio at the critical state for the three tested carbonate sand-fine mixtures. Regarding the critical state data in the \(e\)-In \(p^{\prime}\) plane, a linear relationship can also be used to represent the critical state for each carbonate  
Figure 3: Triaxial test results of carbonate sand-fine mixtures with similar void ratios under a confining pressure of 0.5 MPa: (**a**) stress–strain curves; (**b**) volumetric deformation curves.  
sand-fine mixture. The critical state lines (CSLs) of carbonate sand-fine mixtures with 20 and 40% carbonate sand content were parallel to that of fines only (i.e., 0% carbonate sand content). In addition, the CSL shifted downwards with an increase in carbonate sand content. In other words, the carbonate sand-fine mixture with a higher carbonate sand content exhibited a lower void ratio at the critical state for a given mean effective stress. In the \(e\)-\(\ln p^{\prime}\) plane, the CSL of carbonate sand-fine mixtures was clearly influenced by the carbonate sand content, and no single CSL was observed with different carbonate sand content.