# A Study of Constitutive Model of Rock Damage under the Joint Effect of Load and Moisture
## 2 Damage Constitutive Model of Waterstone  
### The Mechanical Damage Characteristics of Rock  
Considering that the evolution of rock micro-defects is often random, the evolution process of the micro-defect system can be regarded as a non-equilibrium statistical process [2; 31], with the following assumptions, including 1 the rock materials are macroscopically isotropic; 2 the rock micro-units follow [PERSON]'s law; 3 the micro-unit strength follows Weibull distribution, with the probability density function as:\[\varphi(F)=\frac{m}{F_{0}}(\frac{F}{F_{0}})^{m-1}exp[-(\frac{F}{F_{0}})^{m}] \tag{1}\]  
In Formula (1) listed above, \(F\) is the random distribution variable of Weibull distribution of microelements; m refers to the parameter reflecting rock brittleness; \(F_{0}\) represents the macroscopic average strength of rock; and \(\varphi(F)\) refers to a measurement of elemental damage rate during the loading. The damage to the macro elements leads to the macroscopic deterioration of rock specimens. The damage parameter \(D\) is a measure of the damage degree of the material, following the relationship with the probability density of micro-element damage listed below:  
\[\varphi(F)=\frac{dD}{dF} \tag{2}\]  
\[D=\int_{0}^{F}\varphi(F)dF=1-exp[-(\frac{F}{F_{0}})^{m}] \tag{3}\]  
According to the equivalent strain assumption, the damage can only affect the strain behavior of the rock through the effective stress, which serves as the foundation of the damage constitutive relationship establishment. \[\sigma=E_{0}\epsilon(1-D) \tag{4}\]  
In Formula (4), \(E_{0}\) is the elastic module of the rock in the dry (no damage) status. Substituting the damage variable \(D\) into Formula (4), the constitutive formula of the sandrock damage can be obtained as follows. \[\sigma=E_{0}\epsilon exp[-(\frac{F}{F_{0}})^{m}] \tag{5}\]  
In Formula (5), the damage principle of the rock follows the Drucker-Prager yield rule, which takes the main middle stress, and the hydrostatic pressure into consideration, resulting in the selection of the micro-strength of the rock. \[F=f(\sigma)=\alpha I_{1}+J_{2}\frac{1}{2} \tag{6}\]  
whereas, \(I_{1}=\sigma_{ii}=\sigma_{1}+\sigma_{2}+\sigma_{3}\), is the first invariant of the stress; \(J_{2}=\frac{1}{6}[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+( \sigma_{3}-\sigma_{1})^{2}]\), is the second invariant of the stress; \(\alpha=\frac{\text{tan}\phi}{(9+12\text{tan}^{2}\phi)^{2}}\), is constant related to the internal friction angle of rock (\(\varphi=37.27^{\circ}\)). 1. Uniaxial compression status Under the uniaxial compression, the stress status follows \(\sigma=\sigma_{1}\), \(\sigma_{2}=\sigma_{3}=0\). \[F=f\big{(}\sigma^{\prime}\big{)}=\bigg{(}\alpha+\frac{1}{\sqrt{3}}\bigg{)}\frac {\sigma}{1-D(\sigma)} \tag{7}\]  
Taking the effective stress into consideration, which allows the substitution of Formula (4) into Formula (7), resulting in:  
\[F=f(\sigma)=\bigg{(}\alpha+\frac{1}{\sqrt{3}}\bigg{)}E_{0}\epsilon \tag{8}\]  
Substituting Formula (8) into Formula (5), the following formula can be developed. \[\sigma=E_{0}\epsilon e^{-(\frac{\alpha+\frac{1}{\sqrt{3}}\epsilon_{0}}{F_{0} })^{m_{1}}} \tag{9}\]\[\frac{d\sigma}{d\varepsilon}=E_{0}e^{-[\frac{(\alpha+1,\frac{1}{\sqrt{3}})E_{0} \varepsilon}{V_{0}}]^{m_{1}}}\{1-m_{1}[\frac{\left(\alpha+\frac{1}{\sqrt{3}} \right)E_{0}\varepsilon}{F_{0}}]^{m_{1}}\} \tag{10}\]  
Considering that in the conventional compression test, in the curve, when \(\varepsilon=\varepsilon_{max}\), \(\sigma=\sigma_{max}\), \(\left.\frac{d\sigma}{d\varepsilon}\right|_{\varepsilon=\varepsilon_{max}}=0\), the following formula can be obtained. \[\varphi(F)=\frac{dD}{dF} \tag{2}\]0  
\[\varphi(F)=\frac{dD}{dF} \tag{2}\]1  
The constitutive formula of rock under the uniaxial compression is obtained:  
\[\varphi(F)=\frac{dD}{dF} \tag{2}\]2  
Combining Formulas (12) and (13), the mechanical damage of rock due to load can be obtained.