# Experimental Study on Damage Constitutive Model of Rock under Thermo-Confining Pressure Coupling
## 4 Damage Constitutive Model of Rock
### Rock Damage Theory Based on the Drucker-Prager Criterion
#### 4.1.3 Total Damage of Rock under Thermo-Confining Pressure Coupling  
During triaxial compression tests under high temperature and high confining pressure, the primary causes of rock damage are the destructive effects of temperature and confining pressure on the internal structure of the rock. High temperatures cause thermal expansion of internal materials and cracks within the rock, usually leading to internal damage. Confining pressure compresses and closes the microcracks and pores within the rock, thereby enhancing its overall stiffness and density. Due to the uncertainty in the extent of rock damage caused by the coupling of temperature and stress, certain mathematical methods are needed to quantitatively analyze the total damage. Assuming that the effective volumes of undamaged rock, thermally damaged rock, and rock damaged are \(V_{0}\), \(V_{1}\), and \(V_{2}\) respectively, then we have:  
\[D_{T}=1-\frac{V_{1}}{V_{0}} \tag{16}\]  
\[D_{m}=1-\frac{V_{2}}{V_{1}} \tag{17}\]  
By combining the above assumptions, we can establish the following relationship:  
\[D_{s}=D_{T}+D-D_{T}D \tag{18}\]  
By substituting Equations (14) and (15) into the overall equation, the total damage variable for rock under the coupled effects of temperature and confining pressure can be obtained as:  
\[D_{s}=1-\frac{E_{T}}{E_{0}}\exp[-(\frac{F}{F_{0}})^{m}] \tag{19}\]  
According to [PERSON]'s strain equivalence hypothesis, the constitutive relationship of the rock damage can be expressed as follows:  
\[\sigma_{i}=\sigma_{i}^{\prime}(1-D_{s});\ \ i=1,2,3 \tag{20}\]  
According to the generalized Hooke's law, we have:  
\[\epsilon_{i}^{\prime}=\frac{\sigma_{i}^{\prime}-v(\sigma_{j}^{\prime}+\sigma_ {k}^{\prime})}{E_{T}} \tag{21}\]where \(\epsilon_{i}^{\prime}\) is the effective strain corresponding to the effective stress \(\sigma_{i}^{\prime}\). According to the deformation compatibility conditions, we have:  
\[\epsilon_{i}^{\prime}=\epsilon_{i} \tag{22}\]  
From Equations (20)-(22), the damage constitutive relation of rock can be derived as follows:  
\[\sigma_{1}=E_{T}\epsilon_{1}(1-D_{s})+v(\sigma_{2}+\sigma_{3}) \tag{23}\]  
The initial axial strain generated under the initial confining pressure is:  
\[\sigma_{s}=\sigma_{1}-\sigma_{3} \tag{24}\]  
\[\epsilon_{0}=\frac{1-2v}{E_{T}}\sigma_{3} \tag{25}\]  
\[D_{m}=1-\frac{V_{2}}{V_{1}} \tag{17}\]0  
By substituting Equations (10) and (24)-(26) into Equation (23), the damage constitutive model of rock under thermo-confining pressure coupling can be obtained as follows:  
\[ \begin{split}&\sigma_{1}=[E_{T}\epsilon_{1}+(1-2v)\sigma_{3}] \frac{E_{T}}{E_{0}}\times\\ &\exp[-\left\{\frac{[E_{T}\epsilon_{1}+(1-2v)\sigma_{3}][( \sigma_{1}+3\sigma_{3})\sin\varphi+\sigma_{1}\sqrt{3+\sin^{2}\varphi}]}{F_{0} \sqrt{9+3\sin^{2}\varphi}[\sigma_{1}+(1-2v)\sigma_{3}]}\right\}^{m}]+(2v-1) \sigma_{3}\end{split}  \tag{27}\]  
In summary, the parameters that need to be determined are \(E_{T}\), m and \(F_{0}\). The elastic modulus under the influence of temperature, \(E_{T}\), can be determined using the fitting model discussed in Section 3.3.1. The parameters \(m\) and \(F_{0}\) can be determined from the peak points (\(\epsilon_{p}\),\(\sigma_{p}\)) of the rock stress-strain curves.