# Numerical Study on the Mechanical Behavior of Sand-Rubber Mixtures under True Triaxial Tests
## 3 Results
### Micro-Mechanical Response  
#### 3.2.1 Coordination Number  
At the microscopic level of discrete element numerical simulations, the coordination number is often used to quantify the average contact number of each particle in the sample, which is defined as  
\[Z=2N_{c}/N_{p} \tag{12}\]  
where \(N_{c}\) is the number of contacts and \(N_{p}\) is the number of particles. According to the research of [PERSON] et al. [17; 18; 19], there are three different contact types of RSM (sand-sand contact, rubber-rubber contact and sand-rubber contact), and the coordination numbers can be defined according to these three kinds of contacts:  
\[Z^{s-s}=2N_{c}^{s-s}/N_{p}^{s} \tag{13}\]  
\[Z^{r-s}=2N_{c}^{r-s}/N_{p} \tag{14}\]  
\[Z^{r-r}=2N_{c}^{r-r}/N_{p}^{r} \tag{15}\]  
where \(N_{p}^{s}\) is the number of sand particles and is \(N_{p}^{r}\) the number of rubber particles. Since the different preparation methods of numerical simulation samples will have a certain impact on the coordination number, the same sample preparation method is used in this study, and the consistency of the samples may reduce the impact on the coordination number.  
Figure 8 shows the change curve of the overall coordination number in the shearing process of samples with different RCs. Figure 8a shows that the coordination number of pure sand is 5.4 after isotropic consolidation. As the shear progresses, the coordination number \(Z\) gradually decreases to a stable value. For different intermediate principal stresses, as the value of \(b\) increases, the sample changes little at the initial stage of loading, and then increases, and its corresponding residual value is similar to the peak deviatoric stress. In Figure 8b,c, due to the addition of large rubber particles, the coordination number of the sample decreases significantly after isotropic consolidation. During the shearing process, it first increases and then decreases. The peak number of digits increases as the middle principal stress coefficient increases. Analyzing the reasons for the above phenomenon, when the RCs in the RSM are different, different pore states will appear in the sample, and with particle compression, climbing and rotation occur in the particle assembly during the shearing process, resulting in a change in pore volume and a coordination number change; the influence mechanism of intermediate principal stress in RSMs involves the large pores inside the sample occupying an absolute advantage. As the loading progresses, the large  
Figure 7: Comparisons of the peak internal friction angles with the four common strength criteria of samples with different rubber contents: (**a**) 0%; (**b**) 10%; and (**c**) 30%.  
pores are first filled with smaller particles of smaller pores to make the sample denser. The homogenization process is not greatly affected by the ratio of intermediate principal stress.  
Figure 8: The evolution of coordination number versus major principal strain for samples with various RCs: (**a**) 0%; (**b**) 10%; and (**c**) 30%.  
Figure 9 shows the variation curve of peak coordination number with \(b\) value for different contact types. As shown in the figure, the peak coordination number is almost unaffected by the intermediate principal stress ratio. As RC increases, \(Z^{s-s}\) decreases, while \(Z^{r-r}\) and \(Z^{r-s}\) gradually increase. When RC = 10%, \(Z^{r-r}\) and \(Z^{r-s}\) are both less than 0.5, and \(Z^{s-s}\) is less than 3.5, indicating that the system is highly unstable.