# Microscopic Numerical Analysis of Sand Liquefaction Under Subway Train Load
## 4 Microscopic Mechanism of Sand Liquefaction
### Structural Anisotropy  
Texture anisotropy refers to the directional differences in certain physical or mechanical properties of particle aggregates, which are determined by factors such as particle arrangement, orientation distribution, and mutual contact. And this difference is used to describe the directional characteristics of the microstructure vector of granular materials. Anisotropy can be divided into primary anisotropy and induced anisotropy. Primary anisotropy mainly describes the anisotropy generated during the preparation process of the sample, while induced anisotropy is the anisotropy formed by the change in particle position of the sample when subjected to external forces.  
Regarding the anisotropy of particles, [PERSON] et al. [48] proposed the texture tensor method for quantifying texture anisotropy.  
\[A_{ij}=\frac{1}{C}\sum_{k=1}^{c}n_{i}^{(k)}n_{j}^{(k)} \tag{10}\]  
In the formula, \(C\) represents the number of contacts and \(n^{(k)}\) represents the kth texture vector. This second-order tensor corresponds to a 3 \(\times\) 3 matrix with three eigenvalues  
Figure 12: Contact force chain vibration frequency relationship curve under cyclic loading: (**a**) the number of strong/weak force chains under different dynamic stress amplitudes; (**b**) the total force chain number under different dynamic stress amplitudes; (**c**) the number of strong/weak force chains under different frequencies; (**d**) the total force chain number under different frequencies.  
(\(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\)) and three corresponding eigenvectors. o determines the magnitude of anisotropy, a partial eigenvalue \(\lambda_{d}\)[49] can be introduced.  
\[\lambda_{d}=\frac{1}{\sqrt{2}}\sqrt{\left(\lambda_{1}-\lambda_{2}\right)^{2}+ \left(\lambda_{1}-\lambda_{3}\right)^{2}+\left(\lambda_{2}-\lambda_{3}\right)^{ 2}} \tag{11}\]  
Figure 13 shows the variation in internal structural anisotropy of numerical specimens under cyclic loading with different influencing factors as a function of vibration frequency. The anisotropy is calculated using Equations (10) and (11). From the graph, it can be seen that as the number of vibrations increases, the anisotropy gradually increases. Before reaching a certain critical number of vibrations, the anisotropic change curve is relatively flat, indicating that the sample has a high load-bearing capacity at this stage and no significant deformation has occurred. However, when the number of vibrations exceeds this critical value, the anisotropy rapidly increases, indicating that the particle position has undergone significant changes under cyclic loading, and the bearing capacity of the sample has decreased, ultimately leading to the sample reaching a state of liquefaction failure. In addition, different dynamic stress amplitudes and loading frequencies have a significant impact on the anisotropy of the sample after liquefaction failure. Under the action of a lower dynamic stress amplitude and higher loading frequency, the anisotropy generated after liquefaction failure of the sample is greater, which means that the particle position has undergone significant changes, and therefore the deformation of the sample also increases accordingly. It is worth noting that the anisotropy under an earthquake load and the anisotropy under a subway load show different trends of change. The anisotropy under a seismic load shows the same trend as porosity with the increase in vibration frequency, that is, it first increases and then decreases.