# Failure Mechanism of Rock Specimens with a Notched Hole under Compression--A Numerical Study
## 2 Stress State around Circular Holes and Cracks under Mode-I Loading Condition  
Prior to any excavation, rock mass is subjected to lithostatic stress field. Upon excavation, the stresses around the excavation are redistributed to reach an equilibrium state. The stress state after excavation depends on the geometry of the opening, in addition to the rock material properties and the boundary and loading conditions. Generally, the geometry of openings is designed in a way to minimize the stress concentration at the boundaries of the excavation to prevent consequent instabilities. Among the basic geometrical shapes, a hole with a circular cross-section has the minimum stress concentration factor at its boundary in a homogeneous, continuous, and isotropic medium [26]. In the classical mechanics, the stress state around a circular hole in an infinite plate can be calculated using the closed form solution. Figure 2a shows an infinite plate with a hole subjected to a two-directional stress field in which \(\sigma_{x}=K\sigma_{\infty}\) and \(\sigma_{y}=\sigma_{\infty}\). The stress state around a circular hole with a radius of \(a\) under a two-directional stress field in an infinite plate is [27]  
\[\sigma_{r}=\frac{\sigma_{\infty}}{2}\bigg{[}(1+K)\bigg{(}1-\frac{a^{2}}{r^{2} }\bigg{)}-(1-K)\bigg{(}1-4\frac{a^{2}}{r^{2}}+3\frac{a^{4}}{r^{4}}\bigg{)} cos2\theta\bigg{]} \tag{1}\]  
\[\sigma_{\theta}=\frac{\sigma_{\infty}}{2}\bigg{[}(1+K)\bigg{(}1+\frac{a^{2}}{ r^{2}}\bigg{)}+(1-K)\bigg{(}1+3\frac{a^{4}}{r^{4}}\bigg{)}cos2\theta\bigg{]} \tag{2}\]  
\[\tau_{r\theta}=\frac{\sigma_{\infty}}{2}\bigg{[}(1-K)\bigg{(}1+2\frac{a^{2}}{r ^{2}}-3\frac{a^{4}}{r^{4}}\bigg{)}sin2\theta\bigg{]} \tag{3}\]  
where \(\sigma_{r}\), \(\sigma_{\theta}\), and \(\tau_{r\theta}\) are radial, tangential, and shear stresses, respectively, and \(r\) is the distance to the hole center. It should be noted that the stress state around an opening is not dependent on the mechanical properties of the material. By placing \(r=a\) in Equations (1) through (3), the stress state at a circular hole boundary in an infinite plate is  
\[\sigma_{r}=0 \tag{4}\]  
\[\sigma_{\theta}=\sigma_{\infty}[(1+K)+2(1-K)cos2\theta] \tag{5}\]  
\[\tau_{r\theta}=0 \tag{6}\]  
Figure 2: The used coordinate system to express the stress state in a two directional stress field: (**a**) Around a circular hole and (**b**) near a crack tip. According to Equation (5), in a stress field in which K \(<\) 1, the maximum principal stress is concentrated at the side walls (\(\theta=0,\ \pi\)) and the minimum principal stress is located at the top (\(\theta=\frac{\pi}{2}\)) and bottom (\(\theta=\frac{3\pi}{2}\)) of the hole boundary. The fracture mechanics provides solutions for calculation of the stress and displacement in the solids in which cracks exist. Generally, the fracture mechanics provides accurate solutions for brittle elastic materials. For a given crack, its extension may be accomplished in one or more modes (Figure 1). The near crack tip (\(r<<\) 1) stresses under opening mode loads (Mode-I), and can be calculated using [PERSON]'s concept of the stress intensity factors, which characterize the strength of the singularity at a crack tip (Figure 2b) [28], as follows:  
\[\sigma_{xx}=\frac{\sigma_{\infty}\sqrt{\pi b}}{\sqrt{2\pi r}}cos\frac{\theta} {2}\bigg{[}1-sin\frac{\theta}{2}sin\frac{3\theta}{2}\bigg{]} \tag{7}\]  
\[\sigma_{yy}=\frac{\sigma_{\infty}\sqrt{\pi b}}{\sqrt{2\pi r}}cos\frac{\theta} {2}\bigg{[}1+sin\frac{\theta}{2}sin\frac{3\theta}{2}\bigg{]} \tag{8}\]  
\[\tau_{xy}=\frac{\sigma_{\infty}\sqrt{\pi b}}{\sqrt{2\pi r}}sin\frac{\theta} {2}cos\frac{3\theta}{2} \tag{9}\]  
where \(b\) is the half-length of the crack. Equations (7) through (9) suggest that the maximum stress concentration happens at the tips of the crack. Usually, the analytical methods give solutions with a good accuracy for calculation of the stress state around the openings for simple geometry.