# Analysis of Crack Formation and Growth in Tunnel Linings Using Double-K Fracture Criterion
## 3 Modified Double-K Fracture Criterion for Cracks in Tunnel Linings  
If the whole lining is taken as the analysis object, it is difficult to evaluate the mechanical state of cracks by the analytical method. To overcome this, a small segment of lining containing a crack is separated from the whole model, as shown in Figure 4. This lining segment is subjected to the bending moment \(M\), axial force \(N\), shear force \(S\), and external  
Figure 3: Three-point bending beam specimen. load \(F\). Ignoring the effect of the shear force and external load on the crack, the crack width \(\delta\) and the stress intensity factor \(K\) can be given by the following equations [29]:  
\[\delta=\frac{24 Ma_{e}}{bh^{2}E}v_{M}(s)+\frac{4 Na_{e}}{bhE}v_{N}(s) \tag{5}\]  
\[K=\frac{6M}{bh^{2}}\sqrt{\pi a_{e}}f_{M}(s)+\frac{N}{bh}\sqrt{\pi a_{e}}f_{N}(s) \tag{6}\]  
The shape functions \(f_{M}(\cdot)\),\(f_{N}(\cdot)\), \(v_{M}(\cdot)\), and \(v_{N}(\cdot)\) are given as follows:  
\[f_{M}(s)=1.122-1.40s+7.33s^{2}-13.08s^{3}+14.0s^{4} \tag{7}\]  
\[f_{N}(s)=1.122-0.231s+10.550s^{2}-21.710s^{3}+30.382s^{4} \tag{8}\]  
\[v_{M}(s)=0.8-1.7s+2.4s^{2}+0.66/(1-s)^{2} \tag{9}\]  
\[v_{N}(s)=\left\{1.46+3.42[1-\cos(\pi s/2)]\right\}/\cos(\pi s/2)^{2} \tag{10}\]  
where \(b\) is the longitudinal length of the lining (\(b=1\)), \(h\) is the lining thickness, \(E\) is the elastic modulus, \(a_{e}\) is the equivalent crack depth, and \(\mathrm{s}=a_{e}/h\). The double-K fracture criterion adopts the stress intensity factor calculated by the elastic equivalent method to evaluate the mechanical state of cracks. Although use of the above formulas to calculate the stress intensity factor is convenient, the axial force \(N\) and bending moment \(M\) still cannot be measured accurately. On the other hand, as a statically indeterminate structure, the extension of cracks will change the stiffness of the lining, which results in redistribution of the internal force. Therefore, the stress intensity factor is not suitable for the mechanical analysis of lining cracks. As field monitoring of lining cracks usually only includes the width and depth, it is more appropriate to take the crack width as a mechanical state parameter to determine the fracture stage of the lining. To this end, the initiation toughness \(K_{ini}\) and unstable fracture toughness \(K_{un}\) presented in the double-K fracture criterion should be transformed into two crack width indices, i.e., initiation width \(\delta_{ini}\) and unstable width \(\delta_{un}\). The transformation method is described in detail below. Plug Equation (6) into Equation (5), the relationship between crack width and stress intensity factor can be expressed as:  
\[\delta=\frac{4a_{e}K}{E\sqrt{\pi a_{e}}}\frac{6 ev_{M}(s)+v_{N}(s)}{6 ef_{M}(s)+ f_{N}(s)} \tag{11}\]  
where \(e\) is the eccentricity, \(e=M/Nh\). The initiation and unstable widths can be obtained respectively by substituting the initiation toughness and unstable fracture toughness with the corresponding equivalent crack depths into the above formula. Figure 4: A lining segment containing a crack. For the initiation width, the size of the fracture process zone can be neglected, so the equivalent crack depth is equal to the free crack depth \(a_{0}\). It takes the form:  
Figure 3:0  
For the unstable width, the equivalent crack depth can be obtained by solving the following equation:  
Figure 3:1  
where the shape functions \(F(\cdot)\) and stress distribution \(\sigma(\cdot)\) in the fracture process zone are given as follows, respectively:  
Figure 3:2  
Figure 3:3  
Figure 3:4  
Figure 3:5  
where \(u=x/a_{e}\), \(s=a_{e}/h\), \(t=a_{0}/a_{e},f_{t}\) is the tensile strength of concrete; \(\sigma_{s}\) is the cohesive force at the physical crack tip; and \(w\) is the crack width at the physical crack tip. \(K_{c}\) is the additional fracture toughness caused by the cohesive force in the fracture process zone, namely, cohesion toughness.