## 4 Damage Constitutive Model of Deep Expansive Clay under Freeze-Thaw Cycles
### Determination of the Clay Microelement Strength  
The Druck-Prager damage criterion can be used to calculate the microelement strength of clay, and the basic form of clay microelement damage is selected as follows:  
\[F^{*}=\alpha I_{1}^{*}+\sqrt{I_{2}^{*}} \tag{5}\]  
In this formula,  
\[I_{1}^{*}=\sigma_{1}^{*}+\sigma_{2}^{*}+\sigma_{3}^{*} \tag{6}\]  
\[J_{2}^{*}=\frac{1}{6}\left[(\sigma_{1}^{*}-\sigma_{2}^{*})^{2}+(\sigma_{2}^{* }-\sigma_{3}^{*})^{2}+(\sigma_{1}^{*}-\sigma_{3}^{*})^{2}  
Equation (9) is obtained based on the previous assumptions about the damage variables and then combined with the Lemaitre strain equivalence principle as:  
\[\frac{1}{1-D}=\exp\left[\left(\frac{\alpha I_{1}+\sqrt{I_{2}}}{F_{0}(1-D)}\right)^ {m}\right] \tag{9}\]  
where  
\[I_{1}=\sigma_{1}+\sigma_{2}+\sigma_{3} \tag{10}\]  
\[I_{2}=\frac{1}{6}\left[\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2} -\sigma_{3}\right)^{2}+\left(\sigma_{1}-\sigma_{3}\right)^{2}\right] \tag{11}\]  
By \(F=\alpha I_{1}+\sqrt{I_{2}}\), we get:  
\[\frac{1}{D}=\exp\left[\left(\frac{F}{F_{0}(1-D)}\right)^{m}\right] \tag{12}\]  
When \(D=0\), the soil is not damaged; \(F=0\) is a prerequisite that the equation mentioned above is valid. During triaxial compression, the clay was subjected to stresses in three dimensions. There is \(\alpha>0\) in the Druck-Prager damage criterion; a necessary condition is followed as:  
\[\sigma_{1}=\sigma_{2}=\sigma_{3}=0 \tag{13}\]  
By analyzing the stress-strain curve, it can be seen that the stress-strain can be regarded as a linear relationship at the beginning of axial stress loading [35]. As the stress increases, the slope starts to get smaller and smaller at a certain point, which is nonlinear. It is possible to identify the specific point in the stress-strain curve for clay at which damage occurs. The stress value that corresponds to this point can be considered as the damage stress. So, Equation (13) derived above is unreasonable. If the stress is below the stress threshold corresponding to a certain point, the stress-strain curve shows a linear relationship, and it can be inferred that no damage happens in the clay at this point. Once the stress point surpasses the damage threshold value, damage emerges within the clay body, and it intensifies throughout the loading process. Combining Equation (1) with Equation (9), the stress-strain relationship of the soil under triaxial stress can be expressed as:  
\[\sigma=E\epsilon_{1}\exp\left[-\left(\frac{F^{*}}{F_{0}}\right)^{m}\right]+2 \mu\sigma_{3} \tag{14}\]  
\[\sigma=E\epsilon_{1}\exp\left[-\left(\frac{\alpha I_{1}+\sqrt{I_{2}}}{F_{0}} \right)^{m}\right]+2\mu\sigma_{3} \tag{15}\]  
where \(E\) is the elastic modulus of soil, \(\epsilon_{1}\) is the axial strain of soil, \(\mu\) is the Poisson ratio of soil, and \(\sigma_{3}\) is the surrounding pressure of soil. When considering the damage caused by freeze-thaw action, the freeze-thaw damage variable can be depicted by using a macroscopic scale. The damage variable incurred after a certain number of freeze-thaw experiences can be expressed as:  
\[D_{n}=1-\frac{E_{n}}{E_{0}} \tag{16}\]  
where \(E_{0}\) is the elastic modulus of the soil without freeze-thaw cycles, and \(E_{n}\) is the elastic modulus of the soil with freeze-thaw cycles of n times. [PERSON] et al. [36] proposed that the concept of the benchmark damage state is that the soil is initially damaged under the external loading. As the damage intensifies, the constitutive relationship of the soil damage under the combined effects of freeze-thaw action and loading can be defined as:  
\[I_{1}^{*}=\sigma_{1}^{*}+\sigma_{2}^{*}+\sigma_{3}^{*} \tag{6}\]0  
\[I_{1}^{*}=\sigma_{1}^{*}+\sigma_{2}^{*}+\sigma_{3}^{*} \tag{6}\]1  
where \(D_{m}\) is the damage variable under the combined loading and freeze-thaw action; \(D\) is the damage variable under loading; \(D_{n}\) is the damage variable under freeze-thaw actions alone.