## 2 Methodology
### Crack Simulation  
There is a section of circumferential surface crack in the wall of the shield tunnel lining. Its depth \(a_{1}\) is uniform along the circumferential direction, and the crack is always open during vibration. Figure 2 illustrates the crack morphology in a longitudinal element of unit width from the lining and the crack's location in the annular cross-section of the tunnel lining. The outer diameter of the tube is \(D_{o}\), and the inner diameter is \(D_{i}\). Meanwhile, the cracked lining unit is subjected to axial force \(N_{x}\), bending moment \(M_{x}\), effective radial shear \(S_{x}\), and effective circumferential shear \(S_{x\beta}\). The crack is symmetric about the lining tube axis A-A, and the crack distribution angle is \(45^{\circ}\) on both sides of the axis. The crack simulation uses LSM. It weakens the stiffness over a localized region. In addition, the local flexibility due to the crack results in discontinuities in the generalized displacements on both sides of the crack. The deformation on both sides of the crack attributed to the local flexibility is shown below [28]. \[\delta^{R}-\delta^{L}=M_{i\times i}P \tag{17}\]  
Figure 2: Microelements of crack and partly circumferential crack cross-section geometric dimensions. The generalized displacements on the left and right sides of the crack are \(\delta^{R}\) and \(\delta^{L}\), respectively. The local flexibility is \(M_{i\times i}\), and the generalized force is \(P\). The coupling among the bending moment, the axial shear force, the transverse shear force, and the corresponding bending and axial deformations provide information on the individual components in the local flexibility matrix of the cracked area. The local flexibility matrix and the relevant equations are presented below [32]. \[ \begin{split} F_{1}(a)=\sqrt{\tan\frac{\pi a}{2h}/\frac{\pi a}{2h} }\bigg{(}0.751+2.02\frac{a}{h}+0.37\Big{(}1-\sin\frac{\pi a}{2h}\Big{)}^{3} \bigg{)}/\cos\frac{\pi a}{2h}\\ F_{2}(a)=\sqrt{\tan\frac{\pi a}{2h}/\frac{\pi a}{2h}}\bigg{(}0.923 +0.199\Big{(}1-\sin\frac{\pi a}{2h}\Big{)}^{4}\bigg{)}/\cos\frac{\pi a}{2h}\\ F_{II}(a)=\bigg{(}1.122-0.561\frac{a}{h}+0.085\Big{(}\frac{a}{h} \Big{)}^{2}+0.18\Big{(}\frac{a}{h}\Big{)}^{3}\bigg{)}/\sqrt{1-\frac{a}{h}}\end{split}  \tag{18}\]  
Symbols \(F_{1}\), \(F_{2}\) and \(F_{II}\) are the correction factors of stress intensity. Symbol \(a\) is the variable for depth, and \(\theta_{1}\) is the variable for crack angle. For the plane stress, \(E^{\prime}=E\), and for the plane strain, \(E^{\prime}=E/\left(1-\mu^{2}\right)\). The elements of the local flexibility matrix are  
\[ \begin{split} M_{11}=\frac{16R}{E^{\prime}D_{o}^{4}(1-\eta^{2})^ {2}}\int_{0}^{a_{1}}aF_{1}^{2}(a)da,\\ M_{12}=\frac{128 RL}{\pi E^{\prime}D_{o}^{5}(\eta^{6}-\eta^{4}- \eta^{2}+1)}\int_{0}^{\pi/4}\cos^{2}\theta_{1}d\theta_{1}\int_{0}^{a_{1}}aF_{ 1}(a)F_{2}(a)da,\\ M_{13}=\frac{256R}{\pi E^{\prime}D_{o}^{5}(\eta^{6}-\eta^{4}- \eta^{2}+1)}\int_{0}^{\pi/4}\cos^{2}\theta_{1}d\theta_{1}\int_{0}^{a_{1}}aF_{ 1}(a)F_{2}(a)da,\\ M_{22}=\frac{64R}{\pi E^{\prime}D_{o}^{4}(1-\eta^{2})^{2}}\int_{ 0}^{\pi/4}\cos^{2}\theta_{1}d\theta_{1}\int_{0}^{a_{1}}aF_{II}^{2}(a)da\\ +\frac{257 RL^{2}}{\pi E^{\prime}D_{o}^{6}(1-\eta^{4})^{2}}\int_{ 0}^{\pi/4}\cos^{4}\theta_{1}d\theta_{1}\int_{0}^{a_{1}}aF_{2}^{2}(a)da,\\ M_{23}=\frac{514 RL}{\pi E^{\prime}D_{o}^{6}(1-\eta^{4})^{2}}\int_{ 0}^{\pi/4}\cos^{4}\theta_{1}d\theta_{1}\int_{0}^{a_{1}}aF_{2}^{2}(a)da,\\ M_{33}=\frac{1028R}{\pi E^{\prime}D_{o}^{6}(1-\eta^{4})^{2}}\int_{ 0}^{\pi/4}\cos^{4}\theta_{1}d\theta_{1}\int_{0}^{a_{1}}aF_{2}^{2}(a)da,\\ M_{44}=\frac{4(1+\mu)}{E^{\prime}h}\int_{0}^{a_{1}}\tan\frac{\pi a }{2h}da,\\ M_{12}=M_{21},M_{13}=M_{31},M_{23}=M_{32},M_{14}=M_{41},M_{24}=M_{42}, \\ M_{34}=M_{43},M_{14}=M_{24}=M_{34}=0,\eta=D_{i}/D_{o}\end{split}  \tag{19}\]  
The additional displacement in the region around the crack can be decomposed in axial \(u\), angular \(\partial w/\partial x\), radial \(w\), and circumferential \(v\) directions.