## 4 Damage Constitutive Model of Deep Expansive Clay under Freeze-Thaw Cycles
### Establishment of Damage Variables  
The effective stress is the main factor affecting the strain of the soil body. When the material damage occurs, its constitutive relationship only needs to change the stress to the effective stress, which is the famous [PERSON]'s strain equivalence hypothesis, so as to establish the basic equation of the damage constitutive model of clay expansive soil:  
\[\sigma_{ij}=\sigma^{*}(1-D)=E_{ijkl}\epsilon_{kl}(1-D) \tag{1}\]  
where \(E_{ijkl}\) is the tensor component of material elastic stiffness, \(\sigma_{ij}\) is the stress tensor component, \(\sigma^{*}\) is the effective stress tensor component, \(\epsilon_{ij}\) is the strain tensor component, and \(D\) is the damage variable.  
In soil mechanics, soil is often considered a continuous medium from a macroscopic perspective. Due to the dispersion of soil particles, soil is also considered to be a porous medium. However, there are many kinds of soil particles and cement in it, and the existence of interlocking between soil particles makes the soil have good integrity and structure.  
Figure 6: The change diagram of mechanical index with the number of freeze–thaw cycles. (**a**) The cohesion, (**b**) The internal friction angle.  
Therefore, it is considered that the soil is continuous during a loading process of little structural damage, so the following assumptions are put forward:  
1. When the sample is subjected to the loading, macroscopically, it can be regarded as an isotropic microelement, while at the microscopic level it contains some of the basic information that causes damage, and at this point, it can be regarded as a nonhomogeneous microscopic material.
2. The clay microelement can be regarded as linearly elastic before damage is produced under loading. At this time, its strength follows the Hooke law, and the nonlinearity of stress-strain results from the generation of material damage. The initial tangential modulus can also be replaced by the elastic modulus of the undamaged material.
3. The damage caused by microelements is considered to be strength damage in clay materials.  
At the microscopic perspective, the ratio of the area in a damaged unit to the total cross-sectional area is called the damage variable. Macroscopically, it can be defined as the ratio of the number of self-damaged units, \(N_{f}\), to the total number of cells, \(N\). The damage equation is expressed as follows:  
\[D=\frac{N_{f}}{N} \tag{2}\]  
The former scholars demonstrated that the Weibull distribution is more appropriate to describe the pattern of strength distribution in clay under freeze-thaw cycles [33; 34]. By introducing elemental intensity as a random variable and combining it with the Weibull distribution function, the following can be obtained:  
\[\left\{P(F^{*})=\frac{\kappa}{F_{0}}\left(\frac{F^{*}}{F_{0}}\right)^{m-1} \exp\left[-\left(\frac{F^{*}}{F_{0}}\right)^{m}\right],\right.\right. \tag{3}\]  
The damage variable \(D\) can be expressed as follows:  
\[D=\frac{N_{f}}{N}=\int_{-\infty}^{F}P(x)dx=1-\exp\left[-\left(\frac{F^{*}}{F_ {0}}\right)^{m}\right] \tag{4}\]  
where \(m\) and \(F_{0}\) are parameters of the Weibull distribution. They are determined by the mechanical properties of the clay.