## 3 Macroscopic Behavior  
### Influence of Volume Fraction of Cubic Particles  
Figures 4a and 5a illustrate the evolution of deviatoric stress as a function of axial strain for binary mixtures under different confinement conditions of 1.0 MPa and 2.0 MPa, respectively. The data exhibit a strain-softening behavior with increasing axial strain, which aligns with previous investigations on dense granular materials [36; 37; 38]. The porosity values of the sample vary from 0.44 to 0.51 during the simulation. Notably, when the volume fraction of cubes (FC) decreases from 100% to 80%, the deviatoric stress at the peak rises from 3.8 MPa to 4.2 MPa. However, as FC continues to decrease, the deviatoric stress at the peak decreases from 4.2 MPa to 3.9 MPa. For FC values below or equal to 60%, the deviatoric stress at an axial strain of 30% remains relatively constant. In contrast, the deviatoric stress at the residual state generally exhibits an increasing trend as FC increases. Consequently, the volume fraction of cubes significantly influences both the peak and critical-state shear strength of the mixtures. Furthermore, as the confining pressure  
increases from 1.0 MPa to 2.0 MPa, the shear strength of granular mixtures nearly doubles. This finding underscores the substantial impact of confining pressure on the shear behavior of granular mixtures. Figures 4b and 5b present the evolution of volumetric strain as a function of axial strain for the binary mixtures subjected to confining pressures of 1.0 MPa and 2.0 MPa, respectively. Initially, the mixture samples undergo slight contraction followed by dilation. As the sample approaches the residual state, the slope of the volume strain curve gradually decreases, consistent with previous findings. This behavior can be attributed to the easier compaction of spherical particles, which necessitates more void space for shear, leading to enhanced dilation in granular systems with lower cubic particle content. Consequently, the volumetric strain of the sample increases as the volume fraction of spheres during shearing increases. Moreover, the confining pressure influences the volumetric strain response. Samples exhibit greater resistance to deformation and dilation under higher confining  
Figure 4: Comparisons of (**a**) deviatoric stress and (**b**) volumetric strain between different mixture samples under a confining pressure of 1.0 MPa. Figure 5: Comparisons of (**a**) deviatoric stress and (**b**) volumetric strain between different mixture samples under a confining pressure of 2.0 MPa. pressures, as shown by the consistently higher volumetric strain under 1.0 MPa compared to that under 2.0 MPa. The deviatoric stress and volumetric strain curves obtained from the triaxial shear test provide valuable insights into the intrinsic properties of granular materials, as demonstrated in Figure 6. These mechanical parameters, such as the friction angle (\(\varphi\)), dilatancy angle (\(\psi\)), Poisson's ratio (\(\
u\)), and stiffness modulus (\(E\)) offer further understanding of the shear behavior of granular media under specific test conditions. The friction angle \(\varphi\) can be obtained from the deviatoric stress curve as a function of the effective stress ratio \(M^{\prime}\), Equations (16) and (17):  
\[M^{\prime}=q/p^{\prime} \tag{16}\]  
\[\varphi=\arcsin\frac{3M^{\prime}}{M^{\prime}+6} \tag{17}\]  
where \(q\) denotes the deviatoric stress, \(p^{\prime}\) indicates the effective average stress and \(M^{\prime}\) denotes the effective stress ratio.