# Analysis of Crack Formation and Growth in Tunnel Linings Using Double-K Fracture Criterion
## 4 Numerical Experiments of the Fracture Behavior of Tunnel Liniags
### Analysis of the Fracture Process of Tunnel Linings  
Figure 9 shows the propagation paths of the cracks at the arch crown and arch waist under the additional load. It is seen that the crack at crown is the I mode crack while the crack at waist is the I-II mixed mode crack.  
The relationship between the crack depth and eccentricity at the cross-section of the lining crack is shown in Figure 10. It is illustrated that the arch crown is subjected to an internal force of eccentric tension while the arch waist is subjected to an internal force of eccentric compression. The two groups of the resulting eccentricity increase with the additional load at first, and then decrease with the additional load when the additional load reaches a certain value, which is due to the decrease of the bending stiffness caused by crack propagation. In the initial stage of crack propagation, the decrease of the bending stiffness is negligible, so the eccentricity increases with the additional load. With the increasing crack depth, the bending stiffness decreases sharply, so the eccentricity begins to decrease with the additional load.  
Figure 8: Schematic diagrams of the calculation model: (**a**) crack at the arch crown, (**b**) crack at the arch waist.  
Figure 10: Relationship between the crack depth and eccentricity: (**a**) arch crown, (**b**) arch waist.  
Figure 9: Propagation paths of the cracks: (**a**) arch crown, (**b**) arch waist.  
Since the lining is a statically indeterminate structure, the external force acting on it is not equal to that acting on the crack. For convenience, let us introduce an equivalent load to characterize the force acting on the lining crack:  
\[\sigma_{e}=\frac{6M}{bh^{2}}+\frac{N}{bh}\frac{f_{N}(s)}{f_{M}(s)} \tag{20}\]  
This equivalent load is the weighted sum of the maximum stress caused by the bending moment and the axial force. The curves of the equivalent load-crack width and equivalent load-crack depth are respectively shown in Figures 11 and 12. The vertical dashed lines in Figure 11 represent two crack width indices calculated by the proposed method, and the horizontal dashed lines in Figure 12 represent the corresponding equivalent loads. The solid-line curves in Figures 11 and 12 represent the results obtained by the finite element method with the XFEM-FCM algorithm. It is seen that the proposed width indices accurately divide the fracture process of the lining into three stages. In the crack initiation stage, the crack depth is almost not developed, and the crack width increases linearly with the equivalent load. The crack initiation equivalent loads of the arch crown crack and arch waist crack are 0.68 and 0.51 Mpa, respectively. In the stable propagation stage, the fracture process zones of the arch crown crack and arch waist crack extend 59 and 54.6 mm, respectively, which are on the same order of magnitude as the free crack depth. Meanwhile, both load-width curves present the intense nonlinear feature. In the unstable fracture stage, the equivalent loads acting on the lining cracks begin to decrease, but the crack width and depth increase dramatically. It implies that the lining reaches the damage stage. The unstable fracture equivalent loads of the arch crown crack and arch waist crack are 1.30 and 1.17 Mpa, respectively.  
Overall, the proposed theory of the modified double-K fracture criterion can accurately identify two demarcation points of the three stages of the lining fracture process, including  
Figure 11: The curve of the equivalent load–crack width: (**a**) arch crown, (**b**) arch waist.  
Figure 12: The curve of the equivalent load–crack depth: (**a**) arch crown, (**b**) arch waist.  
the nonlinear starting point and the unstable fracture point. Therefore, use of the proposed criterion is feasible when analyzing the safety of tunnel cracks.