# Constitutive Characteristics of Rock Damage under Freeze-Thaw Cycles
## 4 Constitutive Relationship and Damage Variables  
### Establishment of the Constitutive Model  
Damage theory is widely used in damage models, among which a large number of damage models are established based on the assumption of Lemaitre strain equivalence, as shown in Equation (3) below. This hypothesis shows that the attenuation of the shear strength of the rock is caused by the destruction of the internal mechanical structure of the rock mass. When the rock is subjected to shear failure, the material bearing the shear strength inside can be divided into two parts: the undamaged material and the cavity. The cavity is the part that does not bear any shear strength. As shown in Figure 5, the blank part is the undamaged part of the material, and the shaded part is the damaged part of the material. \[\tau_{1}=\tau_{1}^{\prime}(1-D) \tag{3}\]  
where \(\tau_{1}\) is the nominal shear stress, \(\tau_{1}^{\prime}\) is the effective shear stress, and D is the damage percentage. According to Equation (4), when the rock is destroyed, i.e., when D = 1, the  
Figure 4: Changes in static elastic modulus and freeze–thaw coefficient of rocks under different cyclic damage states of freeze–thaw: (**a**) Elastic modulus; (**b**) Freeze–thaw coefficient. nominal shear stress of the rock and soil must be 0, i.e., the residual strength must be 0, which is contrary to the actual situation, as shown in Equation (4) below. \[\tau_{1}=\tau_{1}^{\prime}(1-D)+\tau_{1}^{r}D \tag{4}\]  
where \(\tau_{1}^{\prime\prime}\) is the shear stress of the damaged part of the material, and the meaning of other physical quantities is the same as that of the model shown in Equation (4). According to Equation (4), when the rock and soil material is in a state of destruction, i.e., when D = 1, and \(\tau_{1}=\tau_{1}^{\prime}(1-D)=0\), so \(\tau_{1}=\tau_{1}^{\prime\prime}\). The stress state and rock strength parameters of some undamaged rocks meet the molar Coulomb failure criterion, as shown in Equations (5) and (6) and Figure 6.  
\[\tau^{\prime}=\frac{1}{2}(\sigma_{1}^{\prime}-\sigma_{3}^{\prime})\sin 2\theta_{r} \tag{5}\]  
\[\sigma^{\prime}=\frac{1}{2}(\sigma_{1}^{\prime}+\sigma_{3}^{\prime})+\frac{1} {2}(\sigma_{1}^{\prime}-\sigma_{3}^{\prime})\cos 2\theta_{r} \tag{6}\]  
\(\theta_{r}\) is the residual friction Angle of the fracture zone, \(C_{r}\) is the residual cohesion force, respectively, and \(\sigma_{1}^{r}\) is the first principal stress. Equations (7) and (8) can be obtained from Figure 5.  
\[\tau_{1}A=\tau_{1}^{\prime}A_{1}+\tau_{1}^{r}A_{2} \tag{7}\]  
\[A=A_{1}+A_{2} \tag{8}\]  
Equation (4) is a new damage model for rock materials. If \(\sigma^{\prime}=\sigma\), Equation (9) can be obtained by combining Equations (6) and (10). \[\sigma=\frac{1}{2}(\sigma_{1}^{\prime}+\sigma_{3}^{\prime})+\frac{1}{2}( \sigma_{1}^{\prime}-\sigma_{3}^{\prime})\cos 2\theta_{r} \tag{9}\]  
In addition, since it is the horizontal residual shear strength of the rock, it also conforms to the Mohr-Coulomb criterion, i.e., \(\tau_{1}^{r}\)  
Figure 4:0  
Figure 4:1  
where \(\sigma_{3}^{r}\) is the third principal stress determined according to the residual strength of the rock. Figure 4:2 Shear stress diagram of rock microelements. Therefore, by combining Equation (11) with Equation (10), Equation (12) can be obtained:  
Figure 4:3  
The shear stress of the undamaged part of the rock also conforms to the Mohr-Coulomb criterion under different normal stress conditions, as shown in Equation (13) below. Figure 4:4  
Thus, we can obtain Equations (14) and (15) by substituting Equations (12) and (13) into Equation (7). Figure 4:5  
Figure 4:6  
Figure 4:7  
Figure 4:8  
\[D=\left\{ \begin{array}{ll}1-\exp[-(F/F_{0})^{m}]&F>0\\ 0&F\leq 0\end{array} \right. \tag{18}\]  
In Equation (18), m, and F\({}_{0}\) are Weibull distribution parameters of rock microunit strength F.