## 4 Numerical Analysis and Failure Mechanism of a Single-Crack Specim
### Failure Mechanism of a Non-Closed Single-Crack Concrete Sample  
The mode-I stress intensity factor for the non-closed crack is negative when the crack is under compression. The dimensionless SIF at the crack tip achieved from maximum circumferential tensile stress will be less than that from the radial shearing stress when the pre-crack angle is small (Figure 9). Thus, mode-II fracture in a non-closed crack will occur only if \(f_{r\theta\max}/f_{\theta\max}>1\) and \(f_{r\theta\max}/f_{\theta\max}>K_{\rm{IIC}}/K_{\rm{IC}}\). Therefore, the mode domain can be divided into two regions--the mode-I and mode-II fracture regions--as shown in Figure 10.  
Figure 8: Distribution curve of dimensionless intensity factor for a closed crack (\(f\) denotes dimensionless SIF).  
Figure 9: Relationship between the maximum mode-I and mode-II dimensionless stress intensity factors.  
A mode-I fracture will occur in a sample with a non-closed crack when the crack angle is less than 45\({}^{\circ}\) (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the crack rupture angles of the specimens with small pre-crack angles do not agree with the initiation angles obtained by the maximum circumferential stress theory. However, as \(\beta\) approaches 45\({}^{\circ}\), the experimental fracture initiation angles are closer to those obtained by the maximum circumferential stress theory. For crack angles greater than 30\({}^{\circ}\) (specimens 9-11), there are considerable differences between the rupture angles of the non-closed and closed cracks because the stress patterns are different for the two conditions. The initiation angles of the non-closed cracks are close to the results obtained by the radial shear stress criterion.  
It can be seen in Table 5 that under the same crack width, as the crack angle increases, the maximum circumferential stress and the maximum radial shear stress at the crack tip first increase and then decrease. For closed cracks (crack width 0.5 mm), the maximum circumferential stress at the crack tip is greater than the radial shear stress at the same crack angle and the maximum of both stresses appears at 45\({}^{\circ}\) (case 3). In the case of non-closed cracks, the maximum radial shear stress is greater than the maximum circumferential tensile stress (cases 8, 9, and 10) after the crack angle is greater than 45\({}^{\circ}\). When the crack angle is 75\({}^{\circ}\), the maximum circumferential tensile stress is only 1.55 MPa. There is good agreement  
Figure 11: Comparison between theoretical and experimental initiation angles.  
Figure 10: Mode-I and mode-II fracture regions for hollow cracks.  
between the rupture angles derived from the numerical and experimental tests and those calculated by the maximum circumferential stress theory. However, for specimens 9-11, the maximum circumferential tensile stress is smaller than the maximum radial shear stress even though \(f_{r\theta\max}/f_{\theta\max}>1\) and \(f_{r\theta\max}/f_{\theta\max}>K_{\rm{IIC}}/K_{\rm{IC}}\). The rupture angles of the experimental results are close to those of the maximum radial stress for a mode-II fracture.  
The results presented above indicate that the maximum circumferential stress theory has some limitations in interpreting the fracture propagation of the non-closed cracks under compressional loading. The initiation angles approach those of the radial shear stress criterion if \(f_{r\theta\max}/f_{\theta\max}>1\) and \(f_{r\theta\max}/f_{\theta\max}>K_{\rm{IIC}}/K_{\rm{IC}}\).