# Stress-Strain Model for Freezing Silty Clay under Frost Heave Based on Modified [PERSON]'s Equation
## 2 Freezing Experiments  
A set of one-dimensional freezing test experiments for silty clay columns were carried out in an open frost heave test system. The one-dimensional frost heave test system (Figure 1) consisted of an insulated environmental chamber, a freezing cell, two cryostats with a temperature accuracy of \(\pm 0.05\)\({}^{\circ}\)C, and various sensors necessary for measuring the soil temperature, frost heave, and water intake. The sample was placed in an organic cylindrical glass container with inner dimensions of \(10\times 20\) cm (diameter \(\times\) height). To enhance the cooling efficiency, the circumference of the container was insulated by a 50-mm-thick layer of Styrofoam, and the container with the prepared specimens was placed in a temperature-controlled chamber at a constant temperature of \(-2\)\({}^{\circ}\)C to avoid heat loss. Alcohol was used as the working fluid circulating through the plate's bottom and top (Figure 2), with temperatures ranging from \(-40\)\({}^{\circ}\)C to \(+40\)\({}^{\circ}\)C, with an accuracy of \(\pm 0.1\)\({}^{\circ}\)C. In addition, the water supply system was composed of a Mariotte bottle connected to the bottom plate through a plastic tube. In all experiments, an unpressurised water supply process was employed. Figure 1: Experiment setup in one-dimensional freezing heave test: (**a**) soil sample, (**b**) experimental apparatus, (**c**) water supply, (**d**) loading device, (**e**) data acquisition, and (**f**) cold bath. Figure 2: Schematic diagram of the freezing experiment apparatus. Frost-susceptible silt clay obtained from the city of Harbin in Northeast China was used to prepare the reconstituted samples. The grain size distribution of this clay is shown in Figure 3. The grain size characteristics of silt clay are as follows: curvature coefficient = 1.44; nonuniformity index = 11.85; plasticity index = 10.67%. All circular solid samples were 15 cm in height and 10 cm in diameter. These soil samples were compacted under slight vibration at an optimal moisture content of approximately 14% to achieve a density of approximately 1670 kg/m\({}^{3}\). The soil samples were then saturated through the vacuum method, bringing the water content of the sample close to 23.59%. Table 1 lists the five freezing tests and their experimental conditions (i.e., test FT1 to FT5) in this study. The saturated soil column was subjected to a range of overburden stress. The tops of the samples were kept cold by subjecting them to a negative temperature, while the bottoms were kept warm by subjecting the samples to a positive temperature. After 24 h of consolidation, a cooling temperature (\(T_{\mathrm{c}}\)) was applied to the top of the soil column, whereas the bottom of the sample was subjected to a warm temperature (\(T_{\mathrm{w}}\)) of 2 \({}^{\circ}\)C. To characterize the natural freezing environment and the sequence of freezing temperatures, the temperature of the top portion (i.e., \(T_{\mathrm{c}}\)) varied with time and underwent three freezing stages (i.e., stages 1, 2, and 3). The \(T_{\mathrm{c}}\) values in stages 1, 2, and 3 were \(-\)5, \(-\)10, and \(-\)15 \({}^{\circ}\)C, respectively, and each freezing stage lasted 72 h. These identical thermal conditions are illustrated in Figure 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test No.** & **Overburden Stress, \(\sigma_{\mathrm{v}}\) (MPa)** & **Freezing Time at Every Stage, \(t\) (h)** \\ \hline FT1 & 0 & 72 \\ FT2 & 0.015 & 72 \\ FT3 & 0.028 & 72 \\ FT4 & 0.041 & 72 \\ FT5 & 0.054 & 72 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental conditions. Figure 3: Grain size distribution of silt clay. For each sample, the temperature gradient was calculated from the temperature difference between the warm end and the cold end divided by the height of the sample. The initial temperatures were the temperatures of the samples before testing.