## 3 Results and Analysis
### Freeze-Thaw Strain
At the end of this stage, the freezing contraction strain is markedly less than that observed under unidirectional freeze-thaw. 3. Melting phase transition stage. The higher heating rate results in more pore ice melting compared to unidirectional freeze-thaw. However, the melting process absorbs latent heat, causing a concomitant decrease in pore temperature. This temperature drop can lead to the re-freezing of pore water in some pores [22], resulting in violent fluctuations in the strain curve and even the appearance of two distinct troughs. 4. Thermal expansion stage. When the temperature rises above the freezing point of water, the pore ice in the red sandstone melts completely. Subsequently, only the thermal deformation of the rock skeleton particles occurs. As the difference between the sample temperature and the ambient temperature decreases, the thermal deformation activity of the rock particles weakens and the slopes of both the strain curve and the temperature curve gradually asymptote to zero. The maximum value of frost-heave strain influenced by pore water during freeze-thaw cycles was described above. With an increase in the number of freeze-thaw cycles, the maximum frost-heave strain exhibits minimal change. However, during the 5 th to 10 th unidirectional freeze-thaw cycles, the minimum strain value in the freezing contraction stage increases. The minimum strain in the freezing contraction stage under all-directional freeze-thaw does not decrease monotonically but fluctuates. Building on this, the governing  
Figure 4: Frost-heave strain in F-T cycle. (The abscissa is the time of a single freeze–thaw process, and the ordinate is the strain value). differential equation for heat conduction during a freeze-thaw cycle, derived from [PERSON]'s law of heat conduction and the principle of energy conservation, is given as follows [23]:  
\[\rho C(\theta)\frac{\partial T}{\partial t}=\lambda(\theta)\
abla^{2}T+L\cdot \rho_{i}\frac{\partial\theta_{i}}{\partial t}\]  
where \(C(\theta)\) is equivalent volumetric heat capacity, \(\lambda\) is thermal conductivity, \(L\) is the latent heat of phase change between ice water, \(\theta_{i}\) is volumetric ice content, \(\rho\) is bulk density of soil, and \(\rho_{i}\) is the bulk density of ice. Based on the Richard equation, the water migration control equation is as follows [23]:  
\[\frac{\partial\theta_{u}}{\partial t}=\frac{\partial}{\partial x}\bigg{[}D( \theta_{u})\frac{\partial\theta_{u}}{\partial x}\bigg{]}+\frac{\partial k_{y} (\theta_{u})}{\partial y}-\frac{\rho_{i}}{\rho_{w}}\frac{\partial\theta_{i}}{ \partial t}\]  
where \(D(\theta_{u})\) is the hydraulic diffusion coefficient of rock and soil and \(k_{y}(\theta_{u})\) is the permeability coefficient of rock and soil in the direction of gravity acceleration. At the same time, the thin film theory proposes that under the temperature gradient, the water film on the surface of the soil particles becomes asymmetric, leading to a thinner film at the cooler end and a thicker film at the warmer end, thereby creating an imbalance in the unfrozen water, ice, and soil particle system. Consequently, water molecules migrate within the film from the thicker to the thinner part to reach a new thermodynamic equilibrium driven by [24]. The relationship between the film thickness and temperature is as follows:  
\[d=\bigg{(}\frac{-A}{6\pi\rho_{s}L}\bigg{)}\bigg{(}\frac{T_{0}-T}{T_{0}}\bigg{)} ^{-1/3}\]  
where \(d\) is the thickness of water film, \(T\) is temperature, \(A\) is Hammock constant, \(\rho_{s}\) is the density of rock and soil, \(L\) is the latent heat of ice, and \(T_{0}\) is the freezing temperature of soil moisture. The film theory can describe the phenomenon of pore water migration to the freezing front. After each freeze-thaw cycle, micro-pores expand and micro-cracks develop in the rock, resulting in changes to pore water distribution.