## 4 Constitutive Relationship
### Constitutive Relationship after the Yield Point  
On the basis of the Lemaitre strain equivalence hypothesis [29; 30; 31; 32; 33; 34], the rock damage constitutive equation can be established as follows:  
\[[\sigma]=[E][1-D][\epsilon]=[1-D][\sigma^{*}] \tag{8}\]  
where \([\sigma]\) and \([\sigma^{*}]\) are the stress matrix and the effective stress matrix, respectively. \([E]\) is the elastic matrix, \([\epsilon]\) is the axial strain matrix, \((1-D)\) is the relative area of the rock that can effectively bear the internal force during compression, and \(D\) is the rock damage parameter. Assuming that the number of damaged micro-elements during rock compression is \(N_{h}\), the damage parameter \(D\) is defined as the ratio of \(N_{h}\) to the total number of micro-elements \(N\)[35; 36; 37], i.e.,  
\[D=\frac{N_{h}}{N} \tag{9}\]  
According to Figure 1, the red sandstone contains many internal pores and fissures, and there are weak layers with different micro-element strengths in the rock. Considering the continuous damage of rock during triaxial compression, it is assumed that the micro-element strength and micro-element force are isotropic and obey Hooke's Law. Before rock failure, the distribution variable \(F\) of the micro-element intensity follows Weibull distribution, and the probability density function is defined as follows:  
\[P(F)=\frac{m}{F_{0}}(\frac{F}{F_{0}})^{m-1}\exp[-(\frac{F}{F_{0}})^{m}] \tag{10}\]  
where \(m\) and \(F_{0}\) represent the _Weibull_ distribution parameters. Hence, the number of destroyed micro-elements in the interval \([F,F+dF]\) can be expressed as \(NP(y)dy\). When the rock is loaded to some degree \(F\), the number of damaged micro-elements can be obtained as follows:  
\[N_{h}(F)=\int_{0}^{F}NP(y)dy=N\bigg{\{}1-\exp[-(\frac{F}{F_{0}})]\bigg{\}} \tag{11}\]  
Substituting the above equation into Equation (9), the rock damage evolution equation can be derived as  
\[D=\frac{N_{h}}{N}=1-\exp[-(\frac{F}{F_{0}})^{m}] \tag{12}\]  
According to Equation (12), the micro-element strength \(F\) must be calculated to determine the damage parameter \(D\). The traditional method is to use uniaxial strain as the distribution variable, which cannot describe the influence of rock stress state on the micro  
Figure 12: Comparison between experimental data and model data. element strength. Therefore, the distribution variable of the micro-element intensity is used to establish the failure criterion of rock. On the basis of the simple parametric form of Drucker-Prager (D-P) criterion, the infinitesimal strength can be obtained. \[F=f(\sigma)=\alpha I_{1}+\sqrt{J_{2}} \tag{13}\]  
where \(\alpha\) is the cohesion, and \(\varphi\) is the internal friction angle of rock. Further, the first and second invariants of the stress tensor are represented by \(I_{1}\) and \(J_{2}\), respectively. \[\left. \begin{array}{l}\alpha=\frac{\sqrt{3}\sin\varphi}{3\sqrt{3+\sin^{2} \varphi}}\\ I_{1}=\sigma^{*}x+\sigma^{*}y+\sigma^{*}z=\sigma^{*}1+\sigma^{*}2+\sigma^{*}3 \\ J_{2}=\frac{1}{6}[{(\sigma^{*}}_{1}-{\sigma^{*}}_{2})^{2}+{(\sigma^{*}}_{2}- \sigma^{*}3)^{2}+{(\sigma^{*}}_{3}-{\sigma^{*}}_{1})^{2}]\end{array} \right\} \tag{14}\]  
where \(\sigma^{*}_{1}\), \(\sigma^{*}_{2}\), and \(\sigma_{3}^{*}\) are the effective stress values of rock under compression, and the corresponding nominal stress values are \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\). The nominal stress of rock can be measured by triaxial test, and \({\sigma_{1}}^{*}={\sigma_{2}}^{*}\), \(\sigma_{1}=\sigma_{2}\). According to the Hooke's Law and the concept of stress tensor,  
\[\left. \begin{array}{l}I_{1}=\frac{Er_{1}(\sigma_{1}+2\sigma_{3})}{\sigma_{1} -2\sigma_{3}}\\ \sqrt{J_{2}}=\frac{Er_{1}(\sigma_{1}-\sigma_{3})}{\sqrt{3}(\sigma_{1}-2\sigma _{3})}\end{array}