## 2 Methodology
### Tension-Compression Coupled Damage Equation  
The statistical damage variable may be defined as the ratio of the number of the destroyed micro-bodies to the total number of micro-bodies, denoted by Equation (12). \[D=\frac{N_{t}}{N} \tag{12}\]  
where \(N\) is the total number of micro-bodies; \(N_{t}\) is the number of destroyed micro-bodies. In this study, we assumed as follows [16]. 1. Rock micro-bodies and rock damage all are isotropic in the macroscopic. 2. Rock micro-bodies without damage obey [PERSON]'s law before micro-bodies are destroyed. 3. The strength of each micro-body obeys the Weibull probability distribution. Because the Weibull probability distribution meets the statistical characteristics of rock compression failure, the probability density function of each micro-body strength obeys the Weibull distribution of two parameters. The probability density function of the Weibull distribution \(p(F)\) is denoted by Equation (13). \[p(F)=\frac{m}{F_{0}}(\frac{F}{F_{0}})^{m-1}\exp(-(\frac{F}{F_{0}})^{m}) \tag{13}\]  
where \(F\) is the distribution variable of random distribution of rock micro-body strength; \(F_{0}\), \(m\) are [PERSON]'s distribution parameters related to rock mechanical properties. We assumed the failure criterion of rock denoted by Equation (14). \[f(\sigma)-k=0 \tag{14}\]  
where \(f(\sigma)\) is rock failure criteria; \(k\) is a constant. \(F=f(\sigma)\) is selected as the random distribution variable of rock micro-body strength. Assuming the distribution density of the probability of rock micro-body damage with \(F=f(\sigma)\) as \(p(F)\), the damage variable may be denoted by the failure probability as Equation (15). \[D=\int_{-\infty}^{F}p(x)dx \tag{15}\]  
where \(p(x)\) is the probability density distribution function. The number of destroyed micro-bodies under a certain level of load \(N_{t}(F)\) may be denoted by Equation (16). \[N_{t}(F)=\int_{0}^{F}Np(x)dx \tag{16}\]The damage variable may be denoted by Equation (17) by substituting Equations (13) and (16) into Equation (12) [5]. Equation (17) is the microscopic statistical damage evolution equation of rock under blasting load. \[D=\frac{N_{t}(F)}{N}=1-\exp(-(\frac{F}{F_{0}})^{m}) \tag{17}\]  
In the multi-axial stress state, the total damage to rock micro-body is the coupling action result of tensile stress damage and compressive stress damage, but they follow the different development laws separately. Rock tensile damage and compressive damage are alternately generated under blasting load. Therefore, we proposed a calculation method of the weighted damage degree to measure the coupling effect of tension stress damage and compression stress damage. An elastic-plastic damage coupling model under blasting load was established using the above-proposed scalar damage to investigate the nonlinear mechanical properties of strength degradation and stiffness softening of the rock mass. Rock micro-body compression damage may be denoted by Equation (18) [9; 11; 12]. \[D_{\text{cd}}=1-\exp[-(\frac{\sigma_{1}-\sigma_{3}}{A_{\text{c}}\sigma_{\text {cd}}})^{m_{\text{cd}}})] \tag{18}\]  
where the subscript c represents the compression; \(D_{\text{cd}}\) is compression damage degree rock micro-body (\(0<D_{\text{cd}}<1\)); \(A_{\text{c}}\) is a material constant related to crack damage strength; \(\sigma_{\text{cd}}\) is the uniaxial compressive strength of rock mass; \(m_{\text{cd}}\) is [PERSON]'s distribution parameter related to rock mechanical properties. Rock micro-body tension damage degree \(D_{\text{td}}\) may be denoted by Equation (19) [9; 11; 12]. \[D_{\text{td}}=1-\exp[-(\frac{\sigma_{1}}{\sigma_{\text{td}}})^{m_{\text{td}}} )],\;\varepsilon_{\text{V}}\leq 0 \tag{19}\]  
where the subscript t represents the tension; \(m_{\text{td}}\) is [PERSON]'s distribution parameter related to rock mechanical properties; \(\sigma_{1}\) is the maximum principal stress; \(\sigma_{\text{td}}\) is the uniaxial tensile strength of rock mass; \(\varepsilon_{\text{V}}\) is volumetric strain.