# Structural Stress Characteristics and Joint Deformation of Shield Tunnels Crossing Active Faults
## 2 Numerical Modeling
### Constitutive Model and Contact Settings  
We introduce the plastic damage constitutive of concrete to simulate the mechanical behavior of the tunnel segments. By introducing the uniaxial compressive damage factor \(d_{\rm c}\) and the uniaxial compressive damage factor \(d_{\rm t}\) (GB50010-2010, 2015) [33], the plastic damage constitutive of concrete is applied to characterize the stiffness degradation characteristics of concrete structures caused by damage in the plastic stage. In ABAQUS, the concrete tensile and compressive damage factor \(D_{\rm k}\) can be determined by the following formula:  
\[D_{\rm t}=\left\{ \begin{array}{ll}1-\sqrt{\frac{\rho_{\rm t}\pi}{n-1+x^{ \rm t}}}&(x\leq 1)\\ 1-\sqrt{\frac{\rho_{\rm t}}{\pi_{c}(x-1)^{2}+x}}&(x>1)\end{array} \right. \tag{1}\]  
\[D_{\rm c}=\left\{ \begin{array}{ll}1-\sqrt{\frac{\rho_{\rm c}\pi}{n-1+x^{\rm t }}}&(x\leq 1)\\ 1-\sqrt{\frac{\rho_{\rm c}}{\pi_{c}(x-1)^{2}+x}}&(x>1)\end{array} \right. \tag{2}\]  
\[\rho_{k}=\frac{f_{k,r}}{E_{\rm c}\epsilon_{k,r}} \tag{3}\]  
\[x_{k}=\frac{\varepsilon}{\varepsilon_{k,r}} \tag{4}\]  
\[n=\frac{E_{\rm c}\epsilon_{c,r}}{E_{\rm c}\epsilon_{c,r}-f_{c,r}} \tag{5}\]  
where \(D_{\rm t}\) and \(D_{\rm c}\) are the tensile and compressive damage factors of concrete, respectively; \(f_{k,r}\) is the representative value of the uniaxial tensile or compressive strength of concrete; \(\varepsilon_{k,r}\) is the representative value of the uniaxial tensile or compressive strength of the concrete corresponding to peak strain; \(E_{\rm c}\) is the elastic modulus of concrete; and \(\alpha_{\rm t}\) and \(\alpha_{\rm c}\) are the values of the descending section of the uniaxial tensile stress-strain curve and the uniaxial compressive stress-strain curve, respectively. [PERSON] et al. (2007) [34] proposed a damage-plastic constitutive model based on the equivalent material theory of reinforced concrete. In this constitutive model, the steel bars are replaced by an equivalent reinforced concrete model, and the strengthening characteristics are realized by defining the plastic stress-strain curve for tensile strengthening. The constitutive model can reasonably reflect the tensile failure characteristics of concrete. Figure 3 presents the equivalent plastic damage constitutive relation of concrete adopted in the study, in which the parameters of the tensile constitutive model of the equivalent materials can be determined by the following formula:  
\[\varepsilon_{\rm y1}=\frac{\sigma_{\rm y1}}{E}=\frac{f_{\rm ct}}{E_{\rm c}} \tag{6}\]  
\[\varepsilon_{\rm y2}=\frac{\sigma_{\rm y2}}{E}=\frac{f_{\rm s}}{E_{\rm c}} \tag{7}\]  
\[\sigma_{\rm y1}=[E_{\rm s}S+E_{\rm c}(1-S)]\frac{f_{\rm ct}}{E_{\rm c}} \tag{8}\]  
\[\sigma_{\rm y2}=Sf_{\rm s} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Materials** & **Gravity Density** & **Elastic Modulus** & **Poisson Ratio** & **Angle of Internal** & **Cohesive Force** & **Yield Stress** & **Ultimate Stress** \\  & **(QN=\(-3\))** & **(MPa)** & **Poisson Ratio** & **Friction (\(\tau\))** & **(RPa)** & **(MPa)** & **(MPa)** \\ \hline Soil & 2080 & 50 & 0.2 & 35 & 20 & - & - \\ Rock stratum & 2720 & \(8\times 10^{4}\) & 0.2 & 40 & 500 & - & - \\ Bolts & 7800 & \(170\times 10^{3}\) (after reduction) [32] & 0.2 & - & - & 480 & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical parameters. In ABAQUS, the material parameters required for plastic damage constitutive of concrete include a dilation angle \(\Psi=15^{\circ}\); flow potential offset \(\kappa=0.1\); biaxial and uniaxial ultimate compressive strength ratio \(\sigma_{\text{p0}}/\sigma_{\text{c0}}=1.16\); invariant stress ratio \(K_{\text{c}}=0.667\); viscosity coefficient \(\mu=0.0005\).