## 2 Numerical Model for Earth Pressure Balance (EPB) Shield Tunnelling Simulation
### Calibration of Contact Parameters  
As a series of calibration analyses have to be completed before generating particles for the designated numerical model, large-scale triaxial tests were conducted to determine the contact parameters [46; 47]. Table 3 lists details of the established triaxial test model. To apply the calibration results to the actual-size EPB shield tunnelling model, a relatively large specimen (diameter, 2 m; height, 4 m) and particle size are considered. Therefore, the soil specimen is modeled with ball elements with a single diameter of 20 cm. When DEM modeling with a single size of particle, their packing is a significant factor, because it determines the behavior of particle assemblies. Accordingly, this study generates a specimen with hexagonal-close-packed balls. After that, shell elements of FLAC3D are generated around the ball elements to act as membranes, which transmit the confining pressure to the specimen before and during the test.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value** \\ \hline Type of cutter head & Spoke-type \\ Diameter of cutter head [m] & 6.14 \\ Number of spokes & 6 \\ Number of cutting tools & 91 \\ Opening ratio [\%] & 75.5 \\ Type of screw conveyor & Shaft-type \\ Diameter of screw [m] & 0.9 \\ Length of screw conveyor [m] & 12 \\ Pitch of screw [m] & 0.5 \\ Inclination of screw conveyor [\({}^{\circ}\)] & 28 \\ Length of shield [m] & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dimensions of EPB shield model for numerical simulation.  
Figure 3: Configuration of EPB shield model consisting of wall elements for numerical simulation: (**a**) front view; (**b**) side and sectional views.  
Figure 4 shows the configuration of the simulated triaxial test model with the generated ball and shell elements. Analyses use different confining pressures of 125, 250, 375, and 500 kPa and a constant strain rate of 0.03%/min. With the fixed bottom plate, the load is applied by the top plate (consisting of wall elements) until 80% of the maximum deviatoric stress is observed. The analysis is repeated by trial and error with different combinations of contact parameters until the designated Mohr-Coulomb strength parameters, internal friction angle (\(\varphi\)), and cohesion (\(c\)) are obtained from a Mohr-Coulomb failure envelope plot.  
Figure 5 presents the results of the triaxial tests. Figure 5a shows four stress-strain curves at different confining pressures: the maximum deviatoric stress increases with increasing confining pressure. Low confining pressure induces plastic behavior with softening, while brittle behavior occurs at high confining pressure. The plotted Mohr-Coulomb failure envelope (Figure 5b) from the stress-strain relationship allows \(\varphi\) and \(c\) to be attained by trial and error. The absolute errors between the target and calibrated Mohr-Coulomb parameters are under 0.5%. Table 4 summarizes the determined contact parameters of the adhesive rolling resistance linear contact model and Mohr-Coulomb failure criterion parameters. The obtained calibrated values are consequently applied to the DEM domain of ground formation.  
\begin{table}
\begin{tabular}{c c} \hline
**Property** & **Value** \\ \hline Dimension of specimen, \(D\times H\) [m] & 2 \(\times\) 4 \\ Packing of ball element & Hexagonal close packing \\ Diameter of ball element [cm] & 20 \\ Number of ball element & 2101 \\ Density of ball element [kg/m\({}^{3}\)] & 2700 \\ Contact model & Adhesive rolling resistance linear \\ Number of shell element & 576 \\ Thickness of shell element [mm] & 6 \\ Density of shell element [kg/m\({}^{3}\)] & 90 \\ Elastic modulus of shell element [kPa] & 1400 \\ Confining pressure [kPa] & \(125,250,375,500\) \\ Strain rate [\%/min] & 0.03 \\ \hline \end{tabular}
\end{table}
Table 3: Details of large-scale triaxial test model for calibration of contact parameters.  
Figure 4: Large-scale triaxial test model for calibration of contact parameters.