## 3 Model Validation  
The model validation process involves three steps. First, validation of the dynamic model for free vibration of the undamaged shield tunnel lining is conducted based on the literature [11; 34]. The properties and dimensions of the shell are \(R=1\) m, \(L/R=20\), \(h=0.01\) m, \(\mu=0.3\), \(\rho=7850\) kg/m\({}^{3}\), and \(E=210\) GPa. The boundary condition is clamped\(-\)clamped. The natural frequencies of the analytical results, as presented in Table 1, are compared with corresponding reference values. The maximum difference between the analytical and reference results is less than 1%, affirming the accuracy of both the model and the natural frequency solution method. Second, no comparable literature is available to provide data for validating the effectiveness and precision of the proposed method. Consequently, this paper establishes the FEM of the shield tunnel lining with a partly circumferential crack. The FEM is utilized to ensure the accuracy of the results. Literature [34] serves as the basis for validating the FEM of the undamaged shield tunnel lining. Moreover, the FEM incorporates a cohesive element \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Analytical** & **Reference [11]** & **Reference [34]** & **Mode (n,m)** \\ \hline
12.14 Hz & 12.13 Hz & 12.17 Hz & (2,1) \\
28.06 Hz & 28.06 Hz & 28.06 Hz & (2,2) \\
19.61 Hz & 19.61 Hz & 19.61 Hz & (3,1) \\
23.28 Hz & 23.28 Hz & 23.28 Hz & (3,2) \\
31.97 Hz & 31.97 Hz & 31.98 Hz & (3,3) \\
36.47 Hz & 36.48 Hz & 36.47 Hz & (4,1) \\
37.37 Hz & 37.38 Hz & 37.37 Hz & (4,2) \\
39.77 Hz & 39.77 Hz & 39.78 Hz & (4,3) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the natural frequencies of the analytical and corresponding reference results. Figure 3: The location of the load, crack, and monitors is used to explore the vibration response under a harmonic point load in the shield tunnel lining with a partly circumferential crack. for modeling the partly circumferential crack. As shown in Figure 4, the shield tunnel lining is simulated using ABAQUS. The length-to-radius ratio \(L/R\) of the model is 20. The tunnel lining material is concrete, which has a density \(\rho\) of 2500 kg/m\({}^{3}\), [PERSON]'s modulus \(E\) of 34.5 GPa, and [PERSON]'s ratio \(\mu\) of 0.2. The lining is considered circular, with radius \(R\) of 2.85 m and lining thickness \(h\) of 0.3 m. The elements are 8-node linear brick elements (\(C3D8\)). As the mesh number increases, the first-order natural frequency can gradually converge, enabling the determination of the mesh configuration [11]. For mesh configurations exceeding 276,552 meshes, the natural frequencies remain virtually unchanged and are consistent with the analytical solution. Consequently, the 276,552 meshes configuration addresses the vibration response of the shield tunnel lining featuring a partly circumferential crack. Table 2 shows the natural frequencies of the analytical and corresponding FEM results. Extracting the first three orders of the natural frequency and setting two relative depths, with the crack positioned in the middle of the shield tunnel lining, the maximum difference between the analytical and FEM results is less than 1.5%. This verification confirms the appropriateness of the adopted lining model with a circumferential crack. Lastly, to enhance the applicability of the proposed approach, it is essential to validate the results of the forced vibration response of the shield tunnel lining with a partly circumferential crack. The properties and dimensions of the lining remain unchanged. Figure 5 displays one example of the validation procedure. The point load's frequency and amplitude are set at 100 Hz and 100 N. The load, crack, and monitor relative positions are 0.25, 0.5, and 0.55, respectively. Additionally, seven monitors are spaced at a circumferential angle of 30\({}^{\circ}\) between neighboring points. The figure indicates the amplitude difference in radial displacements obtained through both FEM and the proposed method across two relative depths.