# Microscopic Numerical Analysis of Sand Liquefaction Under Subway Train Load
## 4 Microscopic Mechanism of Sand Liquefaction
### Microscopic Energy  
During cyclic loading, four interconnected energy components govern the phase transition from solid-like to fluid-like behavior: Strain Energy (E\({}_{k}\)) accumulates as elastic potential at particle contacts, reflecting skeleton integrity. Its abrupt collapse (>80% reduction) signals contact network failure when liquefaction initiates. Sliding Energy (E\({}_{\mu}\)) dissipates irreversibly through particle slippage, exhibiting a linear increase proportional to cumulative shear displacement. This constitutes the primary energy conversion pathway during fabric reorganization. Damping Energy (E\({}_{\beta}\)) arises from viscous particle-fluid inter  
Figure 13: Relationship curve between anisotropy and vibration frequency under cyclic loading: (**a**) anisotropy frequency relationship curve under different dynamic stress amplitudes; (**b**) anisotropy frequency relationship curve under different loading frequencies.  
actions, directly correlating with pore pressure generation. It demonstrates step-function surges at liquefaction onset as fluid pressure dominates stress transfer. Boundary energy represents external work input through wall motion, driving system transformation. Its continuous increase during loading phases provides the energy source sustaining the liquefaction process.  
Critically, the synchronous peak in pore pressure and collapse of E\({}_{\text{k}}\) marks the failure threshold where stored elastic energy is catastrophically converted to fluid kinetic energy. This energy cascade--from boundary input to frictional/viscous dissipation--mechanistically explains the sudden loss of shear resistance characteristic of liquefaction.  
Figure 14 shows the relationship curve between the internal microscopic energy of numerical specimens under cyclic loading and the number of vibrations, which are affected by different influencing factors. At the initial stage of loading, there is already a certain value of strain energy and boundary energy, with the boundary energy being about one-third of the strain energy. This is because after isostatic consolidation is completed, a certain amount of strain energy is stored inside the specimen. Meanwhile, due to the effect of static deviatoric stress, strain energy, sliding energy, damping energy, and boundary energy have all increased or decreased to a certain extent. As the number of vibrations increases, sliding energy, damping energy, and boundary energy increase at an approximately constant rate, while strain energy decreases at an approximately constant rate. After reaching the critical number of vibrations, these energies rapidly change--the sliding energy, damping energy, and boundary energy increase sharply, and the strain energy rapidly decreases until the sample reaches a state of liquefaction failure, at which point the strain energy is almost zero. During this process, the boundary energy is always higher than the damping energy and sliding energy. The variation in dynamic stress amplitude has a certain impact on damping energy and boundary energy. As the amplitude of dynamic stress increases, the damping energy and boundary energy generated decrease, which may be related to the anti-liquefaction ability of the sample. When the dynamic stress increases, the anti-liquefaction ability of the sample is weakened resulting in a reduction in the number of vibrations required for liquefaction. The resulting damping energy and boundary energy are also relatively small. However, under the action of seismic loads, although liquefaction damage is more likely to occur, it generates more microscopic energy. It is worth noting that in this case, the boundary energy is always lower than the damping energy. The effect of loading frequency on microscopic energy is similar to that of dynamic stress amplitude. The higher the loading frequency, the more vibrations are required for liquefaction, resulting in an increase in damping energy and boundary energy.  
Figure 14: _Cont._