# Numerical Study on the Mechanical Behavior of Sand-Rubber Mixtures under True Triaxial Tests
## 3 Results
### Macroscopic Behavior
#### 3.1.2 Comparison of Internal Friction Angle Variation and Common Failure Criteria  
Figure 6 shows the relationship between the peak internal friction angles of samples with different rubber contents and \(b\) values. The angle of internal friction is an index of soil stress intensity and is often used to measure material strength. The peak internal friction angle is defined as follows [30]:  
\[\varphi_{b}=\arcsin\left(\frac{\sigma_{1}-\sigma_{3}}{\sigma_{1}+\sigma_{3}} \right)_{f} \tag{11}\]  
When the influence of intermediate principal stress is not considered, it is the same as the Mohr-Coulomb strength criterion. In order to compare and verify whether the true triaxial numerical model meets the requirements under equal stress path loading, the numerical simulation results when RC = 0% are compared with the indoor experimental results for pure sand of [PERSON] et al. [31] and [PERSON] et al. [32]. It can be seen from Figure 6 that the peak internal friction angle for the sample increases first and then decreases as the value of \(b\) increases. The peak point of the sample appears at \(b=0.6\). The overall trend is consistent with the results of the indoor experiment, thus verifying the feasibility of the discrete element model of RSM materials under different stress paths.  
Figure 5: Stress–strain behavior of rubber–sand mixtures with rubber contents of (a) 10%, (b) 10% and (c) 30%.  
Figure 6: Relationship between the peak internal friction angles and \(b\) values.  
When RC = 10% and RC = 30%, the peak points of the samples all appear at \(b\) = 0.8, which is different from the results of pure sand samples.  
In recent years, many scholars have proposed three-dimensional failure criteria that consider the influence of intermediate principal stress, such as the modified Lade-Duncan criterion [33], SMP criterion [34], FKZ criterion [35] and DP criterion [36]. In order to verify whether the rubber-sand mixture meets the existing failure criteria, these four common failure criteria are used for comparative verification.  
Figure 7 compares the numerical simulation results with these four commonly used failure criteria. As shown in Figure 7a, when RC = 0%, the results of the numerical simulation of the internal friction angle are basically consistent with the predicted results of the modified Lade-Duncan criterion and the SMP criterion, showing a trend of first increasing and then decreasing; at the same time, the FKZ criterion and the DP criterion are clearly not applicable. As shown in Figure 7b,\(c\), for the medium-rubber-content RSM (RC = 10-30%), when \(0\leq b\leq 0.4\), the simulation result is close to the SMP criterion result, and when \(0.8\leq b\leq 1.0\), the simulation result is close to the modified Lade-Duncan criterion. When \(0.4<b<0.8\), the peak internal friction angle is larger than the that of the SMP criterion and smaller than that of the modified Lade-Duncan criterion, placing it between the two criteria. In summary, when considering the strength of large-particle rubber and sand mixtures, there is no existing failure criterion that can be fully expressed, and further research is needed.  
Figure 7: \(Cont.\)