## 2 Establishment of a Method for Evaluating Rock Damage Effects
### Basis of Evaluation
#### 2.1.2 Evaluation Method Based on Damage Fracture Theory  
Based on damage fracture theory, various dynamic damage models have been developed to describe brittle materials. The damage variables are treated statistically by treating rock dynamic fracture as a damage accumulation process. The rock damage mechanism is attributed to the dynamic evolution of cracks within the rock. When the damage variable reaches a threshold value, it is implied that the cracks cross each other and the rock will fracture and destroy. [PERSON] et al. [23] and [PERSON] et al. [24] established the expression form of damage variables, on which a variety of dynamic damage models were established [25; 26; 27; 28]. The damage variables consider not only the strain generated by the internal stress of the rock but also other factors, such as the strain of the rock body. The accuracy of the damage variables has been improving, reflecting the distributional characteristics of rock damage in more detail, and damage variables are embedded in well-established commercial software, providing reliable support for the analysis of rock damage effects. The HJC model can better describe the dynamic mechanical behavior of rock materials, such as nonlinear deformation and fracture characteristics under large strains, high hydrostatic pressures, and high strain rates, and has been widely used in the study of dynamic damage behavior of rocks under blasting [29]. In the HJC material model, rock damage is caused by the cumulative formation of equivalent plastic strain and plastic volumetric strain, and Equation (3) represents the damage variables:  
\[D=\sum\Bigl{(}(\Delta\varepsilon_{p}+\Delta\mu_{p})/(\varepsilon_{p}^{f}+\mu _{p}^{f})\Bigr{)} \tag{3}\]  
\[\varepsilon_{p}^{f}+\mu_{p}^{f}=D_{1}(p^{*}+T^{*})^{D_{2}}\geq EF_{\text{min}} \tag{4}\]  
where \(\Delta\varepsilon_{p}\), \(\Delta\mu_{p}\) are the equivalent plastic strain increment and equivalent plastic volume strain increment; \(\varepsilon_{p}^{f}\), \(\mu_{p}^{f}\) are the equivalent plastic strain and equivalent plastic volume strain at the current integration step; \(D_{1}\), \(D_{2}\) are material damage parameters; \(p^{*}\) is the normalized pressure; \(T\)* is the maximum characteristic tensile stress; and \(EF_{\min}\) is the minimum plastic strain at which the material breaks. In plastic mechanics, the strain increment is linearly proportional to the corresponding stress bias when plastic deformation is present in a rock, as shown in Equation (5):  
\[\Delta\varepsilon_{ij}=\sigma_{ij}^{\prime}\Delta\lambda \tag{5}\]  
where \(\Delta\varepsilon_{ij}\) is the strain increment, \(\sigma_{ij}^{\prime}\) is the stress bias, and \(\Delta\lambda\) is the instantaneous constant. The equivalent plastic strain increment is obtained by substituting the above equation into the Mises yield equation with the expression in Equation (6):  
\[\Delta\varepsilon_{p}=(\sqrt{2}/3)\left[\left(\Delta\varepsilon_{1}-\Delta \varepsilon_{2}\right)^{2}+\left(\Delta\varepsilon_{2}-\Delta\varepsilon_{3} \right)^{2}+\left(\Delta\varepsilon_{3}-\Delta\varepsilon_{1}\right)^{2} \right]^{1/2} \tag{6}\]  
where \(\Delta\varepsilon_{1}\), \(\Delta\varepsilon_{2}\), \(\Delta\varepsilon_{3}\) is the plastic strain increment under the principal stress. Equations (2) and (6) show that equivalent plastic stresses lead to equivalent plastic strain increments. Equation (3) reveals that the damage variable \(D\) in the HJC model incorporates not only the strain resulting from the equivalent stress but also includes the volume strain generated during plastic deformation. Utilizing the damage variable to assess the destructive impact of the rock aligns more closely with the real-world scenario. The study shows that by combining compressive damage variable and tensile damage variable, the total damage variable \(D\) is defined as Equation (7) [30]:  
\[D=\max(D_{c},D_{t}) \tag{7}\]  
where \(D_{c}\) is the compressive damage variable. \(D_{t}\) is the tensile damage variable.