## 4 Damage Constitutive Model and Damage Evolution Law
#### 4.1.2 Determining Model Parameters  
The model parameter is an important indicator of the damage evolution characteristics of materials, and it is essential for reflecting the nonlinear deformation of single-fissured quasi-sandstone. The undetermined parameters must be determined according to the basic mechanical variables of single-fissured quasi-sandstone. Using the extremum condition at the peak of the model curve, the theoretical expressions of the parameters \(m\) and \(F_{0}\) of the constitutive model of single-fissured quasi-sandstone subjected to static loading after freezing-thawing can be obtained. The two extremum conditions of the curve \(\sigma_{1}-\varepsilon_{1}\) at the peak point are  
\(\text{\textcfract}\)\(\epsilon_{1}=\epsilon_{c}\), \(\sigma_{1}=\sigma_{c}\); \(\textcfract}\)\(\epsilon_{1}=\epsilon_{c}\), \(\frac{\partial\sigma_{1}}{\partial\varepsilon_{1}}=0\)  
From the extremum condition 1 and Equation (14), one can obtained  
\[\left(\frac{F^{*}}{F_{0}}\right)^{m}=ln\left[\frac{E_{f}E_{\bar{\imath}}\left( E_{0}\varepsilon_{c}\ -\ \sigma_{r}+2\mu_{f\bar{\imath}}\sigma_{3}\right)}{E_{0}^{2}(\sigma_{c}\ -\ \sigma_{r})}\right] \tag{20}\]  
According to extremum conditions 2 and Equations (14) and (19),  
\[\left(\frac{F^{*}}{F_{0}}\right)^{m}=\frac{\left(\alpha\ +\ \frac{\sqrt{3}}{3} \right)E_{0}\varepsilon_{c}+\left(2\alpha\ +\ \frac{2\sqrt{3}}{3}\right)\mu_{f\bar{\imath}}\sigma_{3}\ +\ \left(2\alpha\ -\ \frac{\sqrt{3}}{3}\right)\sigma_{3}}{\left(\alpha\ +\ \frac{\sqrt{3}}{3}\right)m(E_{0} \varepsilon_{c}\ -\ \sigma_{r}+2\mu_{f\bar{\imath}}\sigma_{3}>)} \tag{21}\]By combining Equations (20) and (21), one can obtain the expressions of undetermined parameters \(m\) and \(F_{0}\)  
\[m= \frac{\left(\alpha+\frac{\sqrt{3}}{3}\right)E_{0}\varepsilon_{c}\ +\ \left(2\alpha+\frac{2\sqrt{3}}{3}\right)\mu_{ft} \sigma_{3}\ +\ \left(2\alpha-\frac{\sqrt{3}}{3}\right)\sigma_{3}}{\left(\alpha+\frac{\sqrt{3} }{3}\right)>\ \left(\ E_{0}\varepsilon_{c}-\sigma_{r}\ +\ 2\mu_{ft}\sigma_{3}\right)ln\left[ \frac{E_{f}E_{t}\left(E_{0}\varepsilon_{c}-\sigma_{r}\ +2\mu_{ft}\sigma_{3}\right)}{E_{0}^{2} \left(\varepsilon_{c}-\sigma_{r}\right)}\right]} \tag{22}\]  
\[F_{0}=\left\{\left(\alpha+\frac{\sqrt{3}}{3}\right)E_{0}\varepsilon_{c}\ +\ \left(2\alpha+\frac{2\sqrt{3}}{3}\right)\mu_{ft} \sigma_{3}\ +\ \left(2\alpha-\frac{\sqrt{3}}{3}\right)\sigma_{3}\right\}\times \left\{\frac{1}{ln\left[\frac{E_{f}E_{t}\left(E_{0}\varepsilon_{c}-\sigma_{r} +2\mu_{ft}\sigma_{3}\right)}{E_{0}^{2}\left(\varepsilon_{c}-\sigma_{r}\right) }\right]}\right\}^{\frac{1}{\alpha}} \tag{23}\]  
By substituting Equations (22) and (23) into Equations (13) and (14), a complete damage evolution model and constitutive equation of single-fissured quasi-sandstone subjected to static loading after freezing-thawing can be obtained. Equation (13) shows that, when the axial strain, fissure angle, and freeze-thaw temperature of quasi-sandstone change, the evolution of the general damage rate is  
\[\overset{\bullet}{D}_{g}=(1-D_{f})(1-D_{t})\frac{\partial D_{l}}{\partial \varepsilon_{1}}+(1-D_{l})(1-D_{t})\frac{\partial D_{f}}{\partial f}+(1-D_{f} )(1-D_{l})\frac{\partial D_{t}}{\partial t} \tag{24}\]  
Equation (24) indicates that the general damage of quasi-sandstone changes with the increase or decrease in axial strain, fissure angle, and frozen temperature, and the damage mechanical behaviours of single-fissured quasi-sandstone are coupled and influenced by the three changes.