## 2 Statistical Thermal Damage Constitutive Modeling of Rocks
### Establishment of a Constitutive Model  
In order to better simulate the stress-strain relationship of rock under high confining pressures, a statistical thermal damage constitutive model of rock based on the above modified strength criterion and the statistical damage theory of rock is proposed. Equation (3) is transformed into a stress-invariant representation:  
\[ \begin{array}{ll}F(I_{1}^{*},J_{2}^{*},\theta_{\sigma})&=\frac{\varphi \psi_{T}}{k(\sqrt{1+\varphi^{2}-\varphi})}\sigma_{u}\frac{I_{1}^{*}}{3}+4J_{2} ^{*}\cos^{2}\theta_{\sigma}\\ &+\frac{\varphi\psi_{T}}{k(\sqrt{1+\varphi^{2}-\varphi})}\sigma_{u}\sqrt{J_{2 }^{*}}\Big{(}\cos\theta_{\sigma}+\frac{\sin\theta_{\sigma}}{\sqrt{3}}\Big{)} \\ &=\sigma_{u}^{2}\end{array}  \tag{7}\]  
Here,  
\[I_{1}^{*}=\sigma_{1}^{*}+\sigma_{2}^{*}+\sigma_{3}^{*}=\frac{(\sigma_{1}+ \sigma_{2}+\sigma_{3})E\epsilon_{1}}{\sigma_{1}-\mu(\sigma_{2}+\sigma_{3})} \tag{8}\]  
\[ \begin{array}{ll}J_{2}^{*}&=\frac{1}{6}\Big{[}(\sigma_{1}^{*}-\sigma_{2}^{* })^{2}+\left(\sigma_{1}^{*}-\sigma_{3}^{*}\right)^{2}+\left(\sigma_{2}^{*}- \sigma_{3}^{*}\right)^{2}\Big{]}\\ &=\frac{E_{1}^{2}(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}-\sigma_{1} \sigma_{2}-\sigma_{2}\sigma_{3}-\sigma_{1}\sigma_{3})}{3[\sigma_{1}-\mu( \sigma_{2}+\sigma_{3})]^{2}}\\ \end{array}  \tag{9}\]  
\[\tan\theta_{\sigma}=\frac{2\sigma_{2}-\sigma_{1}-\sigma_{3}}{\sqrt{3}(\sigma _{1}-\sigma_{3})} \tag{10}\]  
The nominal stress values \(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\) and the strain value \(\epsilon_{1}\) corresponding to the principal stress can be obtained from the conventional triaxial test of the rock, and the result of the corresponding effective stresses are \(\sigma_{1}^{*}\), \(\sigma_{2}^{*}\) and \(\sigma_{3}^{*}\). \(\theta_{\sigma}\) is known as the Lode Angle, and \(E\) is the elastic modulus of the rock. In conventional triaxial tests, \(\sigma_{2}=\sigma_{3}\) and \(\sigma_{2}^{*}=\sigma_{3}^{*}\), so Equations (8)-(10) can be transformed into  
\[I_{1}^{*}=\frac{(\sigma_{1}+2\sigma_{3})E\epsilon_{1}}{\sigma_{1}-2\mu\sigma_ {3}} \tag{11}\]  
\[J_{2}^{*}=\frac{E^{2}\epsilon_{1}^{2}(\sigma_{1}-\sigma_{3})^{2}}{3(\sigma_{1 }-2\mu\sigma_{3})^{2}} \tag{12}\]  
\[\theta_{\sigma}=30^{\circ} \tag{13}\]  
By substituting Equations (11)-(13) into Equation (7),  
\[f(\sigma)=\sigma_{u}^{2}=\frac{\varphi\psi_{T}}{k(\sqrt{1+\varphi^{2}}-\varphi )}\frac{\sigma_{u}\sigma_{1}E\epsilon_{1}}{\sigma_{1}-2\mu\sigma_{3}}+\left( \frac{(\sigma_{1}-\sigma_{3})^{2}E^{2}\epsilon_{1}^{2}}{\sigma_{1}-2\mu\sigma _{3}}\right)^{2} \tag{14}\]  
In this paper, the mechanical damage variable \(D_{M}\) is defined as the ratio of damaged elements (\(N_{dam}\)) to the total number of elements (\(N_{tol}\)) of rock material under a load:  
\[D_{M}=\frac{N_{dam}}{N_{tol}} \tag{15}\]  
Suppose that the strength of rock elements follows the Weibull distribution:  
\[N_{dam}=\int_{0}^{f}N_{tol}\frac{n}{F}\left(\frac{f}{F}\right)^{n-1}\exp\left[ -\left(\frac{f}{F}\right)^{n}\right]df \tag{16}\]where \(n\) and \(F\) are the distribution parameters subject to the Weibull distribution, and the mechanical damage value under a normal temperature can be obtained by connecting Equations (15) and (16):  
\[D_{M}=\int_{o}^{f(\sigma)}P(f)df=1-\exp\left[-\left(\frac{f(\sigma)}{F}\right)^{ n}\right] \tag{17}\]  
The thermal damage value of rocks at high temperatures \(D_{T}\) is defined as  
\[D_{T}=1-\frac{E_{T}}{E_{0}} \tag{18}\]  
\(E_{T}\) is the elastic modulus at high temperatures, and \(E_{0}\) is the rock elastic modulus at room temperature. Considering the coupling of the thermal damage and mechanical damage of the rocks under high temperature conditions, the total damage value of the rocks is defined as [33]  
\[D=D_{M}+D_{T}-D_{M}D_{T} \tag{19}\]  
Considering that the initial thermal damage value of rock at normal temperatures is 0, the damage value of rock at normal temperatures is therefore considered \(D=D_{M}\).