## 4 Damage Constitutive Model and Damage Evolution Law
#### 4.1.1 Establishment of Constitutive Equation  
The recent damage constitutive model of single-fissured quasi-sandstone under freezing-thawing and static loading has difficulty in accurately reflecting the postpeak deformation characteristics. Therefore, based on the research by [PERSON] et al. [46; 47], it is assumed that there are damage elements and nondamage elements in the single-fissured quasi-sandstone after freezing-thawing and triaxial compression and that the damage elements can also bear part of the axial stress (that is, the constant quasi-linear section after the stress-strain curve descends). The damaged part includes the fissured damage, freeze-thaw damage, damage from static loading, single-fissure and freeze-thaw coupling damage, single-fissure and load coupling damage, freeze-thaw and static loading coupling damage, and the coupling damage of the three. Assuming that the damaged part can also bear residual stress, a constitutive model of single-fissured quasi-sandstone subjected to freezing-thawing and static load considering residual strength was established. When the single-fissured quasi-sandstone is frozen-thaawed and loaded, in the direction of the principal stress \(\sigma_{1}\), assuming that the undamaged part of the single-fissured quasi-sandstone bears effective stress \(\sigma_{1}^{*}\), the corresponding infinitesimal area is \(A_{1}\). The damaged part is subjected to residual stress \(\sigma_{r}\) and the corresponding area is \(A_{r}\). Then,  
\[\sigma_{1}A= \sigma_{1}^{*}A_{1}+\sigma_{r}A_{r} \tag{1}\] \[A= A_{1}+A_{r} \tag{2}\]  
In short, the ultimate damage of the quasi-sandstone is determined by the single fissure, freezing-thawing and static loading. Therefore, the general damage variable \(D_{g}\) is defined as  
\[D_{g}=\frac{A_{r}}{A} \tag{3}\]  
Based on Equations (1)-(3),  
\[\sigma_{1}=\sigma_{1}^{*}(1-D_{g})+\sigma_{r}D_{g} \tag{4}\]  
The constitutive relation of the undamaged part follows the generalised Hooke's law. \[\sigma_{1}^{*}=E_{0}\epsilon_{1}^{*}+\mu_{ft}(\sigma_{2}^{*}+\sigma_{3}^{*}) \tag{5}\]  
where \(E_{0}\) is the static elastic modulus of intact quasi-sandstone at a normal temperature, \(\epsilon_{1}^{*}\) is the effective axial strain of the undamaged elements of the single-fissured quasi-sandstone, and \(\mu_{ft}\) is the Poisson ratio of single-fissured quasi-sandstone after freezing-thawing. According to the coordination relationship of quasi-sandstone deformation,  
\[\epsilon_{1}^{*}=\epsilon_{1} \tag{6}\]  
Under the conventional state of three-dimensional stress, the quasi-sandstone is subjected to increased axial stress until failure under the confining pressure. Therefore, the lateral residual strength can be ignored, that is, the lateral load is resisted by the undamaged part. \[\sigma_{2}^{*}=\sigma_{2},\sigma_{3}^{*}=\sigma_{3} \tag{7}\]  
Substituting Equations (6) and (7) into Equation (5) results in  
\[\sigma_{1}^{*}=E_{0}\epsilon_{1}+\mu_{ft}(\sigma_{2}+\sigma_{3}) \tag{8}\]  
The intact quasi-sandstone is cut by the structural plane to form macroscopic defects, which are distributed in the quasi-sandstone in different forms, resulting in fissured damage of the quasi-sandstone. Furthermore, its strength and deformation characteristics have a strong response. Therefore, the static elastic modulus can be used to measure the fissure damage of single-fissured quasi-sandstone. The fissure damage variable \(D_{f}\) is defined as  
\[D_{f}=1-\frac{E_{f}}{E_{0}} \tag{9}\]  
where \(E_{f}\) is the static elastic modulus of quasi-sandstone with a single fissure at different angles. Single-fissured quasi-sandstone shows the heterogeneity of the mesostructure, which gives the strength of micro-element within the material a probability distribution. During the loading process, the internal microflaws continue to be generated, develop, and evolve into new macroflaws, and the failure of micro-units shows randomness.