# A Macro-Micro Damage Model for Rock under Compression Loading
## 3 Description of the Damage Model  
### Relationship between Stress and Crack Growth in Compression  
At present, many scholars have carried out relevant research on rock with multi-cracks. However, little consideration was given to interaction between the microcracks. Assuming that the propagation direction of wing crack is parallel to maximum principal stress, the stress intensity factor at the wing crack tip could be calculated by Equation (16) when the crack spacing is large. As the crack spacing decreases, the interaction between the cracks will lead to a damaged connection and result in a break of the rock bridge. The rock bridge interaction mechanical model of the multi-cracks rock as the wing crack expands is shown in Figure 6.  
The crack interaction is considered by drawing the internal stress \(\sigma^{\prime}{}_{3}\) in the rock bridge. It is noteworthy that the internal stress \(\sigma^{\prime}{}_{3}\) has a great significance for evaluating the rock failure from crack interaction. This internal stress \(\sigma^{\prime}{}_{3}\) is determined as [16]:  
\[\sigma^{\prime}{}_{3}=\frac{T_{\text{e}}\cos\beta}{N_{\text{A}}^{-1/2}-2(l+a \sin\beta)} \tag{16}\]  
where \(N_{\text{A}}\) is the number of initial compression-shear cracks per unit area; \(F=T_{\text{e}}\text{cos}\beta=2a\tau_{\text{eff}}\text{cos}\beta\) is a wedging force acting at the midpoint of each crack.  
Acting on the wing crack, the internal stress \(\sigma^{\prime}{}_{3}\) produces an additional intensity factor at the crack tip.  
\[K^{\prime}{}_{1}=\frac{2a\tau_{eff}\cos\beta}{N_{\text{A}}^{-1/2}-2(l+a\sin \beta)}\sqrt{\pi l} \tag{17}\]  
Considering the damage to the rock bridge by the wing crack interaction, the stress intensity factor at the wing crack tip is obtained by Equations (15) and (17) for the case where the wing cracks are aligned in the direction of the principal compressive stress.  
\[K_{\text{I}}=K_{\text{i}}{}^{\text{wing}}+K^{\prime}{}_{1}=2\tau_{eff}|\frac{ 3}{4}(\sin\frac{\theta}{2}+\sin\frac{3\theta}{2})\,e^{-\frac{l}{2}}+\sin\theta (1-e^{-\frac{l}{2}})|\sqrt{\frac{a+l}{\pi}}\text{arc}\sin(\frac{a}{a+l})- \sigma_{3}\sqrt{\pi l}+\frac{2a\tau_{eff}\cos\beta}{N_{\text{A}}^{-1/2}-2(l+a \sin\beta)}\sqrt{\pi l} \tag{18}\]  
Based on Equation (18), it can be seen that the interaction of multiple wing cracks resulting in rock bridge damage makes the stress intensity factor at the crack tip larger than that of a single wing crack.  
Based on the fracture mechanics criterion, it is well known that wing crack will propagate when the mode-I stress intensity factor \(K_{\text{I}}\) at the wing crack tip reaches rock fracture toughness \(K_{\text{IC}}\). According to the Equation (18), the linkage of stress state (\(\sigma_{1}\), \(\sigma_{3}\)) and crack growth is obtained as:  
\[\sigma_{1}=\frac{\Big{\{}2 BC+\sqrt{\pi l}(1+BE)\Big{\}}\sigma_{3}+K_{\text{IC}} }{2 AC+AE\sqrt{\pi l}} \tag{19}\]  
where  
\[A=\frac{\sin(2\beta)-\mu\{1+\cos(2\beta)\}}{2} \tag{20}\]  
\[B=\frac{\sin(2\beta)+\mu\{1-\cos(2\beta)\}}{2} \tag{21}\]  
Figure 6: Sketch of interaction of multiple-cracks.  
\[C=\left[\frac{3}{4}(\sin\frac{\theta}{2}+\sin\frac{3\theta}{2})\;e^{-\frac{l}{2}}+ \sin\theta(1-e^{-\frac{l}{2}})\right]\sqrt{\frac{a+l}{\pi}}\text{arc}\sin(\frac {a}{a+l}) \tag{22}\]  
\[E=\frac{2a\cos\beta}{N_{\text{A}}^{-1/2}-2(l+a\sin\beta)} \tag{23}\]