## 5 Secondary lining Support Characteristic Curve
### Establishment of Numerical Model of Secondary lining
#### 5.1.2 Support Structure Parameters  
(1) Concrete Material Parameters  
Concrete parameters such as elastic modulus, peak compressive strength, and tensile strength were selected based on indoor test results. In addition, other parameters are required to describe the yield surface, plastic flow potential, and tensile and compressive stress-strain curves under composite stress in the CDP (Continuum Damage Plasticity) model. These include the ratio of biaxial compressive strength to uniaxial compressive ultimate strength  
Figure 8: Model of the secondary lining and stress loading diagram of the surrounding rock. (**a**) Secondary lining model. (**b**) Surrounding rock stress loading.  
(\(f_{\rm bo}/f_{\rm co}\)), characteristic length (\(I_{\rm c}\) or \(L_{\rm t}\)) under uniaxial compression (or tension), fracture energy (\(G_{\rm c}\) or \(G_{\rm t}\)) under uniaxial compression (or tension), and the compressive (tensile) damage parameters (\(D_{\rm c}\) or \(D_{\rm t}\)). The compressive yield strength (\(f_{\rm co}\)) and tensile yield strength (\(f_{\rm t0}\)) are selected from the CDP model calculation results. The ratio \(f_{\rm bo}/f_{\rm co}\), the material expansion coefficient (\(\alpha_{\rm p}\)), and the stiffness recovery coefficient (\(s_{0}\)) are chosen with default values \(f_{\rm bo}/f_{\rm co}=1.16\), \(\alpha_{\rm p}=0.2\), and \(s_{0}=0.2\), based on the literature [48]. Compression and tension damage parameters (\(D_{\rm c}\), \(D_{\rm t}\)), and characteristic lengths (\(L_{\rm c}\), \(L_{\rm t}\)) are selected from the recommended parameters of FLAC3D 9.0, and the fracture energies are adjusted to obtain the concrete stress-strain curve, which is validated by comparing it with laboratory test results. Based on the tensile and compressive stress-strain curves of concrete obtained from laboratory tests, and with reference to relevant literature [49; 50], the compressive stress (\(\sigma_{\rm s}\)) and inelastic compressive strain (\(\epsilon_{\rm cis}\)) curve, as well as the tensile stress (\(\sigma_{\rm t}\)) and inelastic tensile strain curve (\(\epsilon_{\rm its}\)), were derived, which were shown in Figure 9a,b, respectively. On this basis, the relationship between the compressive damage factor (\(d_{\rm c}\)) and \(\epsilon_{\rm its}\), as well as the relationship between the tensile damage factor (\(d_{\rm t}\)) and \(\epsilon_{\rm its}\), was established, and these relationships are shown in Figure 9c,d, respectively. The calculation parameters of concrete in the CDP model of FLAC3D 9.0 are summarized in Table 4.  
To accurately describe the shear properties of the interface between reinforcement and concrete, the interface stiffness (\(k\)), bond strength (\(c\)), internal friction angle (\(\varphi\)), and outer perimeter (\(p\)) need to be determined. These interface parameters can be derived from concrete material parameters. Interface parameters \(k\) and \(p\) are determined by the interface thickness (\(t\)), based on the following formula [47]:  
\[\varphi=\sin^{-1}(\frac{f_{c}-f_{t}}{f_{c}+f_{t}}) \tag{2}\]  
\[c=\frac{f_{c}}{2}\tan(\frac{\pi}{4}-\frac{\varphi}{2}) \tag{3}\]  
\[k=\frac{2\pi G}{10\ln(1+2t/D)} \tag{4}\]  
\[t=0.55r_{b} \tag{5}\]  
where \(G\) is the shear modulus, \(D\) is the diameter of the reinforcement, and \(r_{\text{b}}\) is the radius of the reinforcement.