# A Macro-Micro Damage Model for Rock under Compression Loading
## 2 Wing Crack Model Analysis  
The wing crack model has been widely applied in the damage and fracture analysis of rock-like materials under compressive load, as shown in Figure 1. When rock materials are subjected to the external load, its internal crack surface will slide over each other, resulting in the initiation and propagation of wing cracks. The stress intensity factor at the wing crack tip decreases with the propagation of wing crack. If the external load does not increase, the wing crack reaches a stable state. When the external load continues to increase, the stress intensity factor at the wing crack tip increases. When the stress intensity factor reaches or exceeds the fracture toughness \(K_{\text{IC}}\), the wing crack will continue to grow and gradually bend in the direction which is consistent with the maximum principal stress. However, because the real wing crack is not a simple straight line, complex numerical calculation has to be carried out in order to obtain the accurate solution of stress intensity factor at the wing crack tip. This is not convenient for practical application. Therefore, based on assumption that the real curved wing crack is replaced by a straight one (an approximated wing crack) by connecting the tip of curved wing crack with the tip of compressive-shear main crack, many scholars have proposed several common approximate solutions [16; 19; 20]. Due to the limited space of this paper, this paper only lists three calculation models that were widely used in compression-shear fracture and damage analysis of rock materials. [PERSON] and [PERSON] [1] analyzed propagation process of the wing crack by using the complex function analysis method. However, the calculation process is complex, which brings inconvenience to practical application. Therefore, [PERSON] and [PERSON] [2] proposed an approximate equation for calculating stress intensity factor \(K_{\text{I}}\text{wing}\) (HN model):  
\[K_{\text{I}}\text{wing}=\frac{2\pi\tau_{eff}\sin\theta}{\sqrt{\pi(l+0.27a)}}- \sigma_{\text{n}}^{\prime}\sqrt{\pi l} \tag{1}\]  
where \(\theta\) is the orientation of straight wing crack against main crack, \(\tau_{eff}\) is the effective shear stress acting on the main crack surface, \(\sigma_{\text{n}}^{\prime}\) is the normal stress acting on the single isolated crack of length \(2l\) (see Figure 2b). \[\tau_{eff}=\tau-\mu\sigma_{\text{n}}=(\sigma_{1}-\sigma_{3})\sin(2\beta)/2-\mu [(\sigma_{1}+\sigma_{3})+(\sigma_{1}-\sigma_{3})\cos(2\beta)]/2 \tag{2}\]  
\[\sigma_{\text{n}}^{\prime}=\{(\sigma_{1}+\sigma_{3})+(\sigma_{1}-\sigma_{3}) \cos[2(\theta+\beta)]\}/2 \tag{3}\]  
Figure 1: Wing crack model. where \(\mu\) is the friction coefficient of the main crack surface, \(\beta\) is the main crack inclination angle, \(\tau\) and \(\sigma_{\rm n}\) are the shear stress and normal stress acting on the main crack surface. The calculation results show that when the wing crack length \(l\) is rather short (\(l\)\(\rightarrow\)0), the cracking angle \(\theta_{\rm c}\) corresponding to the extreme value of the stress intensity factor \(K_{\rm I}^{\rm wing}\) is inconsistent with the theoretical cracking angle of wing crack. Nevertheless, Equation (1) generally has high accuracy in most cases, and has also been used by many researchers to analyze the mechanical and deformation properties of rock materials [21,22]. By simplifying the curved wing crack into a straight wing crack with a length of 2\(l\), [PERSON] [19] derived expression of stress intensity factor \(K_{\rm I}^{\rm wing}\) (S model) at the wing crack tip with the assumption that the middle of the straight wing crack is affected by the relative sliding displacement of the initial main crack.