# Study on Damage Constitutive Model of Rock under Freeze-Thaw-Confining Pressure-Acid Erosion
### Determination of Rock Microelement Strength under Confining Pressure  
The failure of rock samples under load is basically a shear failure. The criteria for assuming the failure of rock samples are:  
\[f(\sigma^{*})=k_{0} \tag{15}\]  
where: \(\sigma^{*}\) is the effective stress of the rock and \(k_{0}\) is a constant. Because the Drucker-Prager (D-P) criterion considers the influence of the intermediate principal stress and the hydrostatic pressure, which overcomes the defect that the Mohr-Coulomb criterion does not consider the influence of the intermediate principal stress on the rock strength. Therefore, the D-P criterion is more suitable for geotechnical materials and has a wider application range. According to the D-P failure criterion, the rock element strength \(F\) is  
\[F=f(\sigma^{*})=\alpha I_{1}+\sqrt{J_{2}} \tag{16}\]  
where: \(\alpha\) is the strength parameter of the rock element, \(I_{1}\) and \(J_{2}\) are the first invariant of stress and the second invariant of deviator stress, and \(\alpha\), \(I_{1}\), and \(J_{2}\) are  
\[\alpha=\frac{\sin\varphi}{\sqrt{9+3\sin^{2}\varphi}} \tag{17}\]  
\[I_{1}=\sigma_{1}^{*}+\sigma_{2}^{*}+\sigma_{3}^{*} \tag{18}\]  
\[J_{2}=\frac{\left(\sigma_{1}^{*}-\sigma_{2}^{*}\right)^{2}+\left(\sigma_{2}^{* }-\sigma_{3}^{*}\right)^{2}+\left(\sigma_{3}^{*}-\sigma_{1}^{*}\right)^{2}}{6}, \tag{19}\]  
respectively. In formula (17), \(\varphi\) is the friction angle of the rock. According to the generalized Hooke's law and the principle of strain equivalence, and \(\sigma_{2}=\sigma_{3}\), there are  
\[\left. \begin{array}{c}\varepsilon_{1}=\left(\sigma_{1}^{*}-2\
u\sigma_{3}^{ *}\right)/E\\ \sigma_{1}^{*}=\frac{\sigma_{1}}{1-D}\\ \sigma_{2}^{*}=\frac{\sigma_{2}}{1-D}\\ \sigma_{3}^{*}=\frac{\sigma_{3}}{1-D}\\ 1-D=\frac{\left(\sigma_{1}^{*}-2\
u\sigma_{3}\right)}{E\varepsilon_{1}}\end{array}  \right\} \tag{20}\]  
Express the above \(I_{1}\) and \(J_{2}\) with the nominal stress \(\sigma_{i}\), namely  
\[I_{1}=E\varepsilon_{1}(\sigma_{1}+2\sigma_{3})/(\sigma_{1}-2\
u\sigma_{3}) \tag{21}\]  
\[\sqrt{J_{2}}=E\varepsilon_{1}(\sigma_{1}-\sigma_{3})/\sqrt{3}(\sigma_{1}-2\
u \sigma_{3}) \tag{22}\]Substituting Formulas (17), (21) and (22) into Formula (16) can obtain the rock element strength \(F\) under confining pressure, namely  
\[F=f(\sigma^{*})=\alpha I_{1}+\sqrt{I_{2}}=\frac{E\varepsilon_{1}\bigg{[}\sin\varphi (\sigma_{1}+2\sigma_{3})+\sqrt{3+\sin^{2}\varphi}(\sigma_{1}-\sigma_{3})\bigg{]} }{\sqrt{9+3\sin^{2}\varphi}(\sigma_{1}-2v\sigma_{3})} \tag{23}\]  
Damage Evolution Equation of Acid-Etched Rock under the Coupling Effect of Freeze-Thaw and Confining Pressure  
Substituting Equation (23) into Equation (14), the damage evolution equation of acid-etched rock under the coupling of freezing-thawing and confining pressure can be obtained, namely  
\[D=1-\frac{\rho_{\rm r}(2000+H_{1}+H_{2})}{2\rho_{0}(1000+H_{\rm r})}\frac{E_{ \rm n}}{E_{0}}\exp\left\{-\left\{\frac{E_{\rm n}\varepsilon_{1}\bigg{[}\sin \varphi(\sigma_{1}+2\sigma_{3})+\sqrt{3+\sin^{2}\varphi}(\sigma_{1}-\sigma_{3} )\bigg{]}}{F_{0}\sqrt{9+3\sin^{2}\varphi}(\sigma_{1}-2v\sigma_{3})}\right\}^{m}\right\} \tag{24}\]  
Damage Constitutive Model of Acid-Etched Rock under the Coupling Effect of Freezing-Thawing and Confining Pressure  
According to the Lemaitre strain equivalence principle and the concept of effective stress, the strain produced by the damaged rock under the nominal stress is equivalent to the effective strain produced by the undamaged rock under the effective stress (the nominal stress is the stress measured by the test). Then the rock damage constitutive relationship can be established as follows:  
\[\sigma_{i}^{*}=\frac{\sigma_{i}}{1-D};i=1,2,3 \tag{25}\]  
where: \(\sigma_{i}^{*}\) is the effective stress of the rock, \(\sigma_{i}\) is the nominal stress of the rock, and \(D\) is the damage variable of the rock.