## 2 Materials and Methods
### Governing Equations for Segmented Liniings
The strain-deformation relations of the segment are then derived using basic geometric assumptions:  
\[\varepsilon_{r}=\frac{\partial u_{r}}{\partial r} \tag{3}\]  
\[\varepsilon_{\theta}=\frac{u_{r}}{r}+\frac{1}{r}\frac{\partial u_{\theta}}{ \partial\theta} \tag{4}\]  
\[\gamma_{r\theta}=\frac{1}{r}\frac{\partial u_{r}}{\partial\theta}+\frac{ \partial u_{\theta}}{\partial r}-\frac{u_{\theta}}{r} \tag{5}\]  
Strains \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\) are linked to displacements \(u_{r}\) and \(u_{\theta}\) in the radial and circumferential directions, respectively. The lining behaves elastically before yielding, and the stress-strain relationships are governed by standard constitutive laws. \[\varepsilon_{r}=\frac{1-\mu^{2}}{E}\bigg{(}\sigma_{r}-\frac{1-\mu}{\mu} \sigma_{\theta}\bigg{)} \tag{6}\]  
\[\varepsilon_{\theta}=\frac{1-\mu^{2}}{E}\bigg{(}\sigma_{\theta}-\frac{1-\mu} {\mu}\sigma_{r}\bigg{)} \tag{7}\]  
Figure 1:0 Modeling shear interaction in segmented lining systems: (a) force analysis of segmented lining, (b) extracted unit segment, and (c) mechanical behavior of sliding layers. Figure 1:1  
A stress boundary is applied at \(r=R_{0}\), the liner's inner surface, with radial stress equal to the internal air pressure \(p_{\mathrm{a}}\). Figure 1:2  
Given the high internal pressure, both tension and compression damage in concrete are considered. The elastic response is governed by the modulus of elasticity and Poisson's ratio. In the plastic range, the uniaxial compressive stress-strain relationship is defined as follows. Figure 1:3  
\[d_{\mathrm{c}}= \begin{cases}1-\frac{\rho_{\mathrm{c}}r}{n-1+\frac{\rho_{ \mathrm{c}}r}{R_{0}}}&x_{\mathrm{c}}\leq 1\\ 1-\frac{\rho_{\mathrm{c}}r}{\kappa_{\mathrm{c}}(x_{\mathrm{c}}-1)^{2}+x_{ \mathrm{c}}}&x_{\mathrm{c}}>1\end{cases}  \tag{11}\]  
\[\rho_{\mathrm{c}}=\frac{f_{\mathrm{c},r}}{E_{\mathrm{c}}\varepsilon_{\mathrm{ c},r}} \tag{12}\]  
\[n=\frac{E_{\mathrm{c}}\varepsilon_{\mathrm{c},r}}{E_{\mathrm{c}}\varepsilon_{ \mathrm{c},r}-f_{\mathrm{c},r}} \tag{13}\]  
\[x=\frac{\varepsilon}{\varepsilon_{\mathrm{c},r}} \tag{14}\]  
In the compressive stress-strain curve of concrete, the descending branch is characterized by the parameter \(\alpha_{\mathrm{c}}\). The representative uniaxial compressive strength, denoted by \(f_{\mathrm{c},r}\), is typically taken as \(f_{\mathrm{c},r}=f_{\mathrm{c}}\). The corresponding peak strain is \(\varepsilon_{\mathrm{c},r}\), which marks the strain at peak compressive strength. The evolution of compressive damage is described using the parameter \(d_{\mathrm{c}}\). The uniaxial tensile stress-strain behavior of concrete, accounting for compressive damage, is defined by the following equations. \[\sigma=(1-d_{\mathrm{t}})E_{\mathrm{c}}\varepsilon \tag{15}\]  
\[d_{\mathrm{t}}= \begin{cases}1-\rho_{\mathrm{t}}(1.2-0.2x_{\mathrm{t}}{}^{5})& x_{\mathrm{t}}\leq 1\\ 1-\frac{\rho_{\mathrm{t}}}{a_{\mathrm{t}}(n-1)^{1/2}+x_{\mathrm{t}}}&x_{ \mathrm{t}}>1\end{cases}  \tag{16}\]  
\[\rho_{\mathrm{t}}=\frac{f_{\mathrm{t},r}}{E_{\mathrm{c}}\varepsilon_{\mathrm{ t},r}} \tag{17}\]  
\[x=\frac{\varepsilon}{\varepsilon_{\mathrm{t},r}} \tag{18}\]  
The descending branch of the tensile stress-strain curve in concrete is governed by the parameter \(\alpha_{\mathrm{t}}\). The representative tensile strength \(f_{\mathrm{t},r}\) is typically assumed equal to \(f_{\mathrm{t}}\). The strain at peak tensile strength is denoted as \(\varepsilon_{\mathrm{t},r}\). The evolution of damage under uniaxial tension is described using the parameter \(d_{\mathrm{t}}\). The mechanical properties of concrete used in this study are listed in Table 1.