## 4 Damage Constitutive Model of Deep Expansive Clay under Freeze-Thaw Cycles
### Determination of the Parameters of the Damage Constitutive Equation  
According to the identity \(\sqrt{J_{2}}=sI_{1}+t\), we can get the parameters of this identity in Table 3.  
According to Table 3 above, the relationship of \(s\) and \(t\) and the content of moisture \(\omega\) are obtained as follows:  
\[s=0.6745\exp(-0.0474\omega) \tag{21}\]  
\[t=589.47\omega^{-0.48} \tag{22}\]  
The damage conditions of deep expansive clay can be expressed as follows:  
\[G=\sqrt{J_{2}}-sI_{1}-t \tag{23}\]  
Combining Equations (21), (22), and (23), we get the damage conditions of deep expansive clay; that is:  
\[G=\sqrt{J_{2}}-0.6745\exp(-0.0474\omega)I_{1}-589.47\omega^{-0.48} \tag{24}\]  
According to the damage threshold value relationship, the damage evolution equation of the damage variable can be obtained as:  
\[D=\left\{ \begin{matrix}0&G<0\\ 1-\exp\left[-\left(\frac{\alpha I_{1}^{*}+\sqrt{J_{2}^{*}}-F_{0}^{*}}{F_{0}} \right)^{m}\right]&G\geq 0\end{matrix} \right. \tag{25}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Content of Moisture \(\omega\) (\%)** & \(s\) & \(t\) & \(\mathbf{R^{2}}\) \\ \hline
14 & 0.2254 & 132.145 & 0.9845 \\
17 & 0.1647 & 112.014 & 0.9756 \\
21 & 0.1225 & 97.014 & 0.9874 \\
24 & 0.1014 & 80.257 & 0.9565 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The linear relationship coefficients of \(\sqrt{J_{2}}\sim I_{1}\). where \(F_{G}^{*}\) is the strength value of deep expansive clay of the damage valve. By combining Equations (1), (5), (20), (24), and (25), the damage constitutive equation of deep expansive clay, considering the damage threshold value under freeze-thaw cycle conditions, can be expressed as:  
\[\varepsilon=\left\{ \begin{array}{rl}\frac{\sigma-2\mu\sigma_{3}}{E_{n}}&G<0\\ \frac{\sigma-2\mu\sigma_{3}}{E_{n}}\exp\left[\left(\frac{\left(\frac{\alpha I_ {1}+\sqrt{I_{2}-\left(\alpha+0.6745e^{-0.0474\omega}\right)}I_{1G}-589.47\omega ^{-0.48}P_{a}\right)E_{1}}}{F_{0}\left(\sigma-2\mu\sigma_{3}\right)}\right)^{m }\right]&G\geq 0\end{array} \right. \tag{26}\]  
where \(E_{n}\) and \(\mu\) are the elastic modulus and the Poisson ratio, respectively, under freeze-thaw cycles of n times. The above expression can be simplified as follows:  
\[\text{In}a+m\text{In}b=\text{In}\left[-\text{In}\left(\frac{\sigma-2\mu\sigma _{3}}{E_{n}\varepsilon}\right)\right] \tag{27}\]  
where  
\[a=\left(\frac{1}{F_{0}}\right)^{m} \tag{28}\]  
\[b=\frac{\left[\alpha I_{1}+\sqrt{I_{2}}-\left(\alpha+0.6745e^{-0.0474\omega} \right)I_{1G}-589.47\omega^{-0.48}P_{a}\right]E_{n}\varepsilon_{1}}{\left( \sigma-2\mu\sigma_{3}\right)} \tag{29}\]  
The above expression can be simplified further as follows:  
\[Y=mX+c \tag{30}\]  
where \(Y=\text{In}\left[-\text{In}\left(\frac{\sigma-2\mu\sigma_{3}}{E_{n} \varepsilon}\right)\right]\); \(X=\text{In}b\); \(c=\text{In}a\). The values of \(m\) and \(c\) can be obtained by linearly fitting the data measured in the experiment to the above formula. According to \(F_{0}=\exp(-\frac{c}{m})\), \(F_{0}\) can be obtained as follows in the table. The Poisson ratio of the deep expansive clay is taken to be 0.45, and then, based on the Druck-Prager expression of failure criterion, \(\alpha I_{1}+\sqrt{I_{2}}=k\), the value of \(\alpha\) and k can be obtained by linearly fitting. After the mechanical parameters of the damage model under the combined action of freeze-thaw cycles and loading were determined, they were listed in Table 4; the stress values under three confining pressures (100 kPa, 200 kPa, and 300 kPa) can be determined by using the final damage constitutive model. The measured data with a water content of 17% and six freeze-thaw cycles were compared with the theoretical data. As shown in Figure 7, when the confining pressure was 100 kPa, 200 kPa, and 300 kPa, the actual strength values were 118.4 kPa, 152.3 kPa, and 184.1 kPa, respectively, and the theoretical strength values are 120 kPa, 150 kPa, and 186 kPa. The fitting degree of the strength value was as high as 99%. The theoretical curve was basically consistent with the measured curve.