## 2 Theoretical Basis
### Equivalent Strain-Softening Model for Jointed Rock Mass
#### 2.1.2 Strain-Softening Model of Equivalent Jointed Rock Mass  
A circular roadway is shown in Figure 3a with a radius \(R\) and original stress \(\sigma_{0}\). The resistance of the virtual support after excavation is \(P_{i}\), which represents the spatial binding force of the working face before and after excavation. The surrounding rock of the roadway is composed of three parts, namely an elastic zone, a softening zone, and a residual zone. During rock excavation unloading, the jointed plutons change from the peak \(GSI^{p}\) in the elastic stage (point B) to the residual \(GSI^{r}\) in the plastic residual stage (point D) after joint fissure extension and softening (point C). This process can be equivalently formulated using continuum theory. [PERSON] et al. [23] proposed a modified numerical model (a damage strain-softening model) based on continuum fast Lagrangian analysis secondary development to model the fracture development of jointed rock mass from a microscopic damage viewpoint of the damage evolution and strain-softening process. The rock mass is assumed to be a homogeneous, isotropic continuum described by Mohr-Coulomb and Hoek-Brown model parameters. Figure 2: Two-dimensional visualization of the GSI system. Among the plastic strain-softening models used for rock materials, the yield criterion and plastic potential energy are jointly determined by the stress tensor \(\sigma_{ij}\) and softening parameter \(\eta\), which are expressed as:  
\[f(\sigma_{\theta},\sigma_{r},\eta)=0 \tag{5}\]  
The unloading process involving rock excavation is shown in Figure 3b. When \(\eta=0\), the plutons are in the elastic deformation stage, with \(0<\eta<\eta^{*}\) for the strain-softening stage and \(\eta>\eta^{*}\) for the residual phase, where \(\eta^{*}\) is the critical softening parameter value for the transformation of the plutons from strain softening to the residual stage. The softening process of plutons is determined by the slope \(M\). The simplified piecewise linear strain-softening model (Figure 3b) states that ideal brittle failure occurs when \(M\)\(\rightarrow\)\(\infty\) and ideal plastic behavior occurs if \(M=0\). The Hoek-Brown (H-B) criterion is one of the most widely used yield criteria for plutons and has been extensively developed and refined [24; 25]. The 2018 version of the H-B yield criterion is adopted in this paper [25], namely:  
\[\sigma_{1}=\sigma_{3}+\sigma_{ci}(m_{b}\frac{\sigma_{3}}{\sigma_{ci}}+s)^{a} \tag{6}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses during pluton destruction, respectively; \(\sigma_{ci}\) is the uniaxial compressive strength of the rock mass, which remains unchanged during the pre- and post-peak periods; \(m_{b}\) is the H-B constant, which takes the value of \(m_{i}\) that reflects the degree of vulnerability of the pluton and should also remain unchanged during the pre- and post-peak periods; \(s\) reflects the degree of pluton destruction, which ranges from 0 to 1. The criterion introduces the perturbation coefficient \(D\) into the pluton intensity calculation, and the parameters can be expressed by:  
\[\left. \begin{array}{l}m_{b}=m_{i}\exp(\frac{GSI-100}{28-140})\\ s=\exp(\frac{GSI-100}{9-3D})\\ a=\frac{1}{2}+\frac{1}{6}\left(e^{-CSI/15}-e^{-20/3}\right)\end{array} \right\} \tag{7}\]  
Figure 3: Strain-softening behavior of plutons: (**a**) Roadway surrounding rock zoning; (**b**) The simplified piecewise linear strain-softening model. The H-B strain-softening model is obtained by assuming that \(m_{b}\) and \(s\) in the H-B constant decay linearly with \(\eta\), and is given as:  
\[\omega(\eta)=\left\{ \begin{array}{ll}\omega^{p}-\frac{\omega^{p}-\omega^{r}}{ \eta^{*}}\eta,&0<\eta<\eta^{*}\\ \omega^{r},&\eta\geq\eta^{*}\end{array} \right. \tag{8}\]  
where \(\omega^{p}\) is the peak parameter, \(\omega^{r}\) is the residual parameter, and\(\cdot\omega\) can be substituted for \(m_{b}\).