## 3 Analysis of Failure Characteristics of a Buried Waterstop
### Finite Element Analysis of Force and Deformation of Waterstop  
#### 3.2.1 Finite Element Model  
To analyze the impact of waterstop deformation on its waterproofing ability, ABAQUS/Standard is used to establish the plane strain finite element model of the buried waterstop in the concrete lining. The buried rubber waterstop is simulated by the Mooney-Rivlin model. The size is shown in Figure 8, and to distinguish it from the following, it is named waterstop I. The concrete damage-plastic model is used to simulate the tunnel lining, with a width of 300 mm and a height of 400 mm. The bonding of the waterstop and concrete is simulated by cohesive force contact. The construction quality of the waterstop is assumed to be good, and the finite element model is shown in Figure 9.  
The strength grade of concrete lining is considered as C30, the elastic modulus E\({}_{\text{c}}\) is 30 GPa, and [PERSON]'s ratio \(\
u_{\text{c}}\) is 0.2. Lining failure is simulated in ABAQUS using the concrete damage plastic model [31]. Among the damage plasticity parameters, the eccentricity e\({}_{\text{f}}\), the ratio of biaxial to uniaxial compressive strength f\({}_{\text{b}0}\)/f\({}_{\text{c}0}\), and the coefficient k use default values of 0.1, 1.16, and 0.667, respectively; and the viscosity parameter \(\mu_{\text{c}}\) is set to 0.005 based on ABAQUS implicit solution; the expansion angle \(\psi\) is set to 30\({}^{\circ}\); and the stress-strain curves of concrete uniaxial compression and tension are selected for the concrete design code [32], as shown in Table 2, where \(\sigma_{\text{c}}\) and \(\sigma_{\text{t}}\) are compressive stress and tensile stress and \(\varepsilon_{\text{c}}^{\text{in}}\), and \(\varepsilon_{\text{t}}^{\text{in}}\) are inelastic strain and cracking strain, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \(\sigma_{\text{c}}\) & \(\varepsilon_{\text{c}}^{\text{in}}\) & \(\sigma_{\text{t}}\) & \(\varepsilon_{\text{t}}^{\text{in}}\) \\ \hline
14.070 & 0 & 2.030 & 0 \\
20.100 & 0.000802 & 2.010 & 0.000028 \\
14.637 & 0.002456 & 1.232 & 0.000149 \\
10.073 & 0.004080 & 0.849 & 0.000257 \\
7.501 & 0.005638 & 0.661 & 0.000359 \\
5.931 & 0.007162 & 0.548 & 0.000458 \\
4.890 & 0.008668 & 0.473 & 0.000565 \\
4.153 & 0.010165 & 0.419 & 0.000653 \\
3.607 & 0.011655 & 0.378 & 0.000749 \\
3.186 & 0.013141 & 0.346 & 0.000846 \\ \hline \end{tabular}
\end{table} 
Table 2: Uniaxial stress–strain relationship of concrete. Figure 8: Waterstop I (unit: mm). Figure 9: Finite element analysis model. In the study of rubber material, strain energy function can be used to simulate its constitutive model, such as the Mooney-Rivlin model, the Ogden model, the Yeoh model, and the Arruda-Boyce model. Among them, the Mooney-Rivlin model is widely used. The rubber waterstop is simulated by the dual coefficient Mooney-Rivlin in ABAQUS, and the strain energy function relationship is:  
\[\text{W}=\text{C}_{10}(\text{l}_{1}-3)+\text{C}_{01}(\text{l}_{2}-3) \tag{3}\]  
where W is strain energy density, l\({}_{1}\) and l\({}_{2}\) are invariants of the strain tensor, and C\({}_{10}\) and C\({}_{01}\) are mechanical properties constants of materials. The relationship between elastic modulus E\({}_{0}\) and material constant can be obtained from the incompressibility of rubber:  
\[\text{E}_{0}=6(\text{C}_{10}+\text{C}_{01}) \tag{4}\]  
Rubber elastic modulus E\({}_{0}\) can be determined by rubber hardness H\({}_{\text{A}}\), and its relationship is as follows:  
\[\text{E}_{0}=(15.75+2.15\text{H}_{\text{A}})/(100-\text{H}_{\text{A}}) \tag{5}\]  
It can be seen from the specification that the hardness of the rubber waterstop of the tunnel joint is 60 \(\pm\) 5. In this simulation, the median hardness value of 60 is used for calculation. From experience, it is known that C\({}_{10}\)/C\({}_{01}\) generally takes 0.25 for better fitting [33]. It can be calculated that C\({}_{10}\) = 0.484 MPa and C\({}_{01}\) = 0.121 MPa.