## 3 Numerical Modeling  
### IMASS Constitutive Model  
The brittle failure characteristics of rock mass, which exhibit material-softening and -bulking behaviors, are essentially important for modeling the rock damage around an underground excavation. A new constitutive model named the IMASS model (Itasca Model for Advanced Strain Softening model [14]) has been developed by [PERSON] et al. (2020) to consider the progressive failure of the rock mass. It is able to represent the damage around an excavation, slope, or caving process where we must account for the progressive failure and disintegration of the rock mass from an intact/jointed condition to a bulked material. The conceptual model has been introduced by [PERSON] and [PERSON] (1999) [15] based on four main behavior zones, as shown in Figure 4a. An elastic zone means undisturbed rock mass; a seismogenic zone may generate microseismic activity due to discontinuity damage; a yielded zone is subject to significant damage, is fractured, and has lost most of its cohesion and provides minimal support to the excavation; and a mobilized zone gives an estimate as to the portion of the collapsed rock mass.  
Regarding groundwater in this region, it consists of pore water within the loose collapsed mass layer and fissured water within the bedrock. The pore water primarily exists within the Quaternary loose collapsed mass layer, sourced from atmospheric precipitation and surface water runoff. The groundwater volume contained within the pores is relatively small and exhibits seasonal variation. The average permeability coefficient of the medium-weathered argillaceous siltstone is calculated to be 0.0512 m/day, classifying it as a slightly permeable layer.  
The peak behavior of the IMASS model is grounded in the Hoek-Brown [18; 19] constitutive model. The intricate process of rock strain softening induces strain-dependent material characteristics within the damaged rock, as illustrated in Figure 4b. The IMASS constitutive model has been developed to simulate the response of such rock properties to changes in stress. The model employs an indicator termed \"Sloss\" to quantify the extent of softening or weakening experienced by a rock mass, with its range being \([-1,1]\).  
Between peak and post-peak points in Figure 4b,  
\[\text{Sloss}=1-\frac{\text{plastic shear strain}}{\text{critical plastic shear strain}}\]  
Figure 4: IMASS schematic diagram: (**a**) Schematic representation of sloss for various degrees of damage in a rock mass (modified from [PERSON] et al., 2020 [17]); (**b**) material response in the IMASS constitutive model.  
Between post-peak and ultimate strength points,  
\[\text{Sloss}=-\frac{\text{volumetric strain}}{\text{maximum allowable volumetric strain}}\]  
Figure 4 schematically portrays the sloss parameter across various degrees of damage and disturbance within a rock mass. At point \"C\", the rock mass is assumed to have fractured, yet the resulting rock fragments remain fully interlocked (with cohesion fully lost but high friction). At this stage, the stress surpasses the spalling or fracturing strength of the rock. In contrast, at point \"D\", the interlocking of rock fragments is minimized, resulting in maximized porosity. Excavations that lack support within this stress state are deemed unstable.