# Stress-Strain Model for Freezing Silty Clay under Frost Heave Based on Modified [PERSON]'s Equation
## 4 Modified Takashi's Equation  
In these experiments, the freezing front in stage 1 gradually moves downward with time until the freezing rate \(U\) decreases to zero and the temperature \(T_{\text{soil}}\) becomes stable. In this stage, the soil samples can be divided into the freezing and unfrozen zones, as shown in Figure 7(a). The temperature (\(T_{\text{fre,c1}}\)) at the top of the first freezing soil layer equals \(-5\)\({}^{\circ}\)C (i.e., \(T_{\text{fre,c1}}=T_{\text{c}}=-5\)\({}^{\circ}\)C at that moment). Similarly, the freezing front continues to move downward with time. The soil samples in stage 2 can be divided into frozen, freezing, and unfrozen zones (Figure 7(b)). The frozen zone remains unchanged with time, the second freezing soil layer gradually moves downward, and the unfrozen zone gradually shrinks. Meanwhile, the temperature (\(T_{\text{fre,c2}}\)) at the top of the second freezing soil layer under \(T_{\text{c}}=-10\)\({}^{\circ}\)C is \(-1.6\)\({}^{\circ}\)C.  
The soil samples in stage 3 can also be divided into frozen, freezing, and unfrozen zones, as shown in Figure 7(c). Similarly, the temperature (\(T_{\text{fre,c3}}\)) at the top of the third freezing soil layer under  
Figure 8: Fundamental principle of the generated frost heave during the multistage freezing during (**a**) stage 1, (**b**) stage 2, and (**c**) stage 3. \(T_{\rm c}=-15\)\({}^{\circ}\)C is \(-0.6\)\({}^{\circ}\)C. Overall, three successive freezing soil layers (i.e., the first, second, and third freezing soil layers) appeared from stage 1 to stage 3 during the testing. When the temperature field changes from \(t\) to \(t+dt\), the increment (dS) of frost heave at the top of the soil sample is equal to the sum of the frost heave increments of the frozen soil layer (dS\({}_{\rm fro}\)) and the freezing soil layer (dS\({}_{\rm fre}\)). Therefore:  
\[\rm{dS}=\rm{dS}_{\rm fro}+\rm{dS}_{\rm fre} \tag{1}\]  
As a result of the existence of the frozen fringe, which has a scant amount of unfrozen water and low permeability, water cannot migrate from the unfrozen area to the frozen area. Therefore, we assumed that frozen soils do not permeate water migration and that the water migration occurs only in the freezing soil layers. Thus, dS\({}_{\rm fro}=0\) in Equation (1). Then:  
\[\rm{dS}=\rm{dS}_{\rm fre} \tag{2}\]  
This means the increment of frost heave is mainly derived from the degree of frost heave in the freezing soil layer. The increments of frost heave in the freezing soil layer (i.e., frost heave increments at the surfaces of soil samples) during each freezing stage are represented as dS\({}_{\rm fre,1}\), dS\({}_{\rm fre,2}\), and dS\({}_{\rm fre,3}\), respectively. These values are calculated with Equation (2) and the experimental results in Figure 8, and are plotted in Figure 9. From this calculation, the increments of frost heave generated in the freezing zone eventually stabilize with time. Figure 9: _Cont._  
The increments of \(h_{\text{f}}\) during stages 1, 2, and 3 (\(\text{d}h_{1}\), \(\text{d}h_{2}\), and \(\text{d}h_{3}\), respectively), which can be calculated using the results in Figure 6, are presented in Figure 10. The frost heave strain (\(\varepsilon\)) can be defined as the ratio of frost heave to the freezing soil layer thickness (China Construction Ministry, China, 1998). As such:  
\[\varepsilon_{i}=\ \frac{dS_{\text{f}re,i}}{dh_{i}} \tag{3}\]  
where \(\varepsilon_{i}\), \(\text{d}S_{\text{f}re,i}\), \(\text{d}h_{i}\) are the frost heave strain, frost heave increment, and thickness of the \(i\)th freezing soil layer during stage \(i\) (where \(i=1\), \(2\) and \(3\)). Thus, \(\varepsilon_{i}\) can be calculated with Equation (3), the calculation results for which are shown in Figure 11.  
Figure 10: _Cont._  
Figure 9: Frost heave increment generated in the freezing soil layers during (**a**) stage 1, (**b**) stage 2, and (**c**) stage 3.