## 3 Numerical Modeling
### Material Structure and Parameters  
The structural material parameters are shown in Table 1.  
To reflect the damage variation in concrete segments in the simulation, the constitutive model of Concrete Damaged Plasticity (CDP) is applied for the concrete segments [16]. When concrete is in the linear elastic stage, the material is in a non-damaged state, but when the material enters the yield stage, the damage factor is introduced to reduce the elastic modulus of concrete, and the stress-strain relation is as follows:  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Structure Name** & **Material** & **Modulus of Elasticity (GPa)** & **Density (t/m\({}^{3}\))** & **Poisson’s Ratio** \\ \hline Plain segment & C50 concrete & 34.5 & 2.42 & 0.2 \\ Steel blend special piece & C50 concrete, Q235 steel & 58 & 5.13 & 0.25 \\ Inner steel plate, bolts & Q235 steel & 206 & 7.85 & 0.25 \\ \hline \end{tabular}
\end{table}
Table 1: Structural material parameters1.  
Figure 5: Schematic diagram of cutting side inner steel plate.  
Figure 6: Flat-face cutterhead for opening of the cross passage portal.  
\[\sigma=(1-d)E_{0}:(\varepsilon-\varepsilon^{pl}) \tag{1}\]  
where \(\sigma\) denotes the stress tensor; d is the concrete damage factor (\(d\in[0,1]\)); \(\varepsilon\) is the total strain tensor; \(\varepsilon^{pl}\) is the plastic strain of concrete; \(E_{0}\) is the initial elastic modulus of concrete.  
The damage factor is calculated using the strain energy calculation method with reference to the classical damage theory of Najar, the calculation method is as follows:  
\[d=1-2\int f(\varepsilon)d\varepsilon/E_{0}\varepsilon^{2} \tag{2}\]  
where \(\int f(\varepsilon)d\varepsilon\) is the Gauss integral of strain corresponding to the stress from the original point to the peak stress on the stress-strain curve of concrete.  
The values of d range from 0 to 1. And 0 and 1 represent the initial no damage state and the theoretical complete damage state, respectively. In general, a damage factor is generally defined before approaching 1, which is set as the damage threshold of material progressive failure. The mesh that reaches the damage threshold in the software calculation will fail and automatically exit the calculation.  
The linear elastic constitutive model is used for the steel-concrete composite segments. According to the classic Halpin-Tsai model, the steel-concrete composite segments are simplified to short fiber composite materials, and their equivalent elastic modulus is obtained by the following formula [17]:  
\[E=\frac{1+\xi\eta V_{f}}{1-\eta V}E_{m} \tag{3}\]  
\[\eta=\frac{E_{f}/E_{m}-1}{E_{f}/E_{m}+\xi} \tag{4}\]  
where \(E\) is the equivalent elastic modulus of composite material; \(E_{m}\) is the elastic modulus of steel segment; \(E_{f}\) is elastic modulus of concrete; \(\xi\) is the reinforcement degree of steel segment; \(V_{f}\) is the volume specific gravity of steel fiber in the segment.  
The Drucker-Prager (D-P) constitutive model is applied for the concrete material of the cutting portal. The D-P model is widely used in brittle materials such as rocks, ceramics and concrete, which can reflect the change trend of the strength characteristics of brittle materials under multi-axial stress. The Shear Damage model is selected to define damage, and the combination of damage and fracture is used for cutting brittle materials. In addition, the process of cutting tool is realized by setting the advancing speed and rotation speed of the tool model. The advancing speed is 0.05 mm/s and the rotation speed is 1 RPM. The damage threshold is set for the concrete material of the machinable tunnel portal. Make the material at the hole door under the cutting action of the cutter head, when the material reaches the damage point, the unit is automatically deleted and exits the calculation.