## 4 Stress-Strain Curves after Freeze-Thaw Cycles of Expansive Clay  
The shape of the stress-strain curves shown in Figure 3 is very similar to that of the nonlinear elastic model proposed by [PERSON] and [PERSON] [57]. The relationship between the deviatoric stress and axial strain is expressed as Equation (2):  
\[(\sigma_{1}-\sigma_{3})=\frac{\varepsilon_{1}}{a+be_{1}} \tag{2}\]  
where \(\sigma_{1}-\sigma_{3}\) is the deviatoric stress, \(\varepsilon_{1}\) is the axial strain, and \(a\) and \(b\) are the two model parameters. The \(a\) and \(b\) in Equation (2) correspond to the initial elastic modulus and ultimate strength in the deviatoric stress-axial strain curve, respectively, and this correspondence is expressed in Equation (3):  
\[\left. \begin{array}{c}E_{i}=\frac{1}{a}\\ \left(\sigma_{1}-\sigma_{3}\right)_{\text{ult}}=\frac{1}{b}\end{array} \right\} \tag{3}\]  
where \(E_{\text{i}}\) is the initial elastic modulus, and \(\left(\sigma_{1}-\sigma_{3}\right)_{\text{ult}}\) is the ultimate strength. Equation (2) converted to Equation (4) is expressed as follows:  
\[\frac{\varepsilon_{1}}{(\sigma_{1}-\sigma_{3})}=a+b\varepsilon_{1} \tag{4}\]  
Equation (4) shows that there is a linear relationship between \(\varepsilon_{1}/(\sigma_{1}-\sigma_{3})\) and \(\varepsilon_{1}\). The \(\varepsilon_{1}/(\sigma_{1}-\sigma_{3})\)-\(\varepsilon_{1}\) curves are drawn according to the test results of rubber fiber-reinforced expansive clay and the results of unreinforced expansive clay conducted by [PERSON] et al. [49] and [PERSON] et al. [50]. The partial results are displayed in Figure 8. The intercept a and slope b were obtained by linear fitting of the \(\varepsilon_{1}/(\sigma_{1}-\sigma_{3})\)-\(\varepsilon_{1}\) curves, and the initial elastic modulus and ultimate strength after different numbers of freeze-thaw cycles at different confining pressures were obtained by substituting the correspondence expressed in Equation (3). The results are summarized in Table 3.  
Analysis of the data shows that there is a proportional relationship between the initial elastic modulus and ultimate strength, and the correspondence is expressed as Equation (5):  
\[\frac{(\sigma_{1}-\sigma_{3})_{\text{ult}}}{E_{\text{i}}}=M \tag{5}\]  
where \(M\) is the coefficient between the initial elastic modulus and ultimate strength. Analysis of the data reveals a certain regularity in the variation in the ultimate strength. The normalized ultimate strength is defined as the ratio of the ultimate strength after freeze-thaw cycles over the ultimate strength without freeze-thaw cycles, expressed as Equation (6):  
\[R=\frac{(\sigma_{1}-\sigma_{3})_{\text{ult},\;N}}{(\sigma_{1}-\sigma_{3})_{ \text{ult},\;0}} \tag{6}\]  
where \(R\) is the normalized ultimate strength, \((\sigma_{1}-\sigma_{3})_{\text{ult},\;N}\) is the ultimate strength after \(N\) freeze-thaw cycles, and \((\sigma_{1}-\sigma_{3})_{\text{ult},\;0}\) is the ultimate strength without freeze-thaw cycles. The normalized ultimate strengths were averaged (average values for different confining pressures) and are shown in Figure 10. The relationship between the average value of the normalized ultimate strength and the number of freeze-thaw cycles was established and shown as Equation (7), where the parameter \(\lambda\) is a deterioration parameter. \[R_{\text{ave}}=\lambda\Big{(}\text{e}^{-\lambda N}-1\Big{)}+1 \tag{7}\]  
where \(R_{\text{ave}}\) is the average normalized ultimate strength, \(\lambda\) is the deterioration parameter of the ultimate strength under freeze-thaw cycles, and \(N\) is the number of freeze-thaw cycles. In Equation (7), \(\lambda\) reflects the soil ultimate strength as affected by freeze-thaw cycles. The value of \(\lambda\) is between 0 and 1. The ultimate strength is less affected by freeze-thaw cycles when the value of \(\lambda\) is close to 0. The ultimate strength is more affected by freeze-thaw cycles when the value of L is close to 1.