# Microscopic Numerical Analysis of Sand Liquefaction Under Subway Train Load
## 4 Microscopic Mechanism of Sand Liquefaction
### Coordination Number and Mechanical Coordination Number  
The coordination number, also known as the average number of contacts, reflects the average number of other particles that each particle contacts. This value can to some extent reflect the compaction degree of the sample. By observing the changes in coordination number, it is possible to understand whether the sample becomes increasingly compact or gradually breaks down. The calculation formula is shown in Formula (8). During the cyclic loading process, there are some particles that only come into contact with another particle or do not come into contact with other particles at all, and these particles cannot contribute to the load-bearing capacity of the sample. In order to better reflect the compaction degree of the soil skeleton, [PERSON] [47] proposed the mechanical coordination number based on the coordination number. The mechanical coordination number excludes particles with only one contact or no contact, therefore it has better persuasiveness compared to the coordination number. The calculation formula is shown in Equation (9). \[Z=\frac{2C}{N} \tag{8}\]  
\[Z_{m}=\frac{2C-N_{1}}{N-N_{1}-N_{0}} \tag{9}\]  
In the formula, \(Z\) represents the coordination number; \(Z_{m}\) represents the mechanical coordination number; \(C\) represents the number of contacts; \(N\) represents the number of particles; \(N_{1}\) represents the number of particles with only one contact; and \(N_{0}\) represents the number of particles without contact. Figure 9 shows the relationship between the internal coordination number and mechanical coordination number of numerical specimens affected by different influencing factors under cyclic loading with the number of vibrations. It can be clearly seen from the figure that the mechanical coordination number is always higher than the coordination number, which is due to the presence of a large number of suspended particles in the numerical model. As the number of vibrations increases, the coordination number and mechanical coordination number show a gradually decreasing trend. The cyclic loading adopts a bias sine wave, which keeps the specimen under compression, further causing the specimen to remain in a state of shear dilation. In the loading stage where the dynamic stress is greater than the static deviatoric stress, the shear dilation state intensifies, and the particles increase in contact due to compression, thereby improving the coordination \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working** & **Experimental** & **Simulated** & **Peak Pore** & **Peak Pore** \\
**Condition** & **Liquefaction** & **Liquefaction** & **Pressure Ratio** & **Pressure Ratio** \\
**Parameters** & **Cycle Number** & **Cycle Number** & **(Experimental)** & **(Simulated)** \\ \hline \hline \(\sigma_{d}=10\) kPa, & 59 & 65 & 0.99 & 0.95 \\ \(\text{f}=0.5\) Hz & 103 & 92 & 1.01 & 0.97 \\ \(\text{f}=0.5\) Hz & & & & \\ \(\sigma_{d}=20\) kPa, & 170 & 185 & 0.98 & 0.94 \\ \(\text{f}=0.5\) Hz & & & & \\ Seismic load & 13 & 11 & 1.02 & 0.99 \\ (\(\sigma_{d}=60\) kPa) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of liquefaction cycles and peak pore pressure ratio between experiment and simulation. number and mechanical coordination number of the numerical sample. On the contrary, during the unloading stage where the dynamic stress is less than the static deviatoric stress, the shear dilation state weakens, and the contact between particles decreases, resulting in a corresponding decrease in coordination number and mechanical coordination number. Overall, cyclic loading causes fluctuations in the coordination number and mechanical coordination number inside the specimen, but still shows a decreasing trend with increasing vibration frequency.