## 2 A Transversely Isotropic Damage-Plasticity Model for Intact Rock and Rock Mass  
In the following, the transversely isotropic damage-plasticity (TI-RDP) model, proposed by [PERSON] et al. [32], is described briefly and extended for considering the mechanical behavior of rock mass. A description of the used symbols is provided in Table 1 in Appendix A. The TI-RDP model is based on a combination of the isotropic damage-plasticity model, proposed by [PERSON] et al. [9], extended by an implicit gradient-enhanced regularization scheme in [PERSON] et al. [12], and the approach by [PERSON] [25] for capturing transversely isotropic material behavior. Based on the theories of plasticity and damage mechanics, the stress-strain relation of the TI-RDP model, defined in the global coordinate system, reads as  
\[\mathbf{\sigma}=(1-\omega)\,\mathbb{C}^{\rm(ti)}\colon\left(\mathbf{\varepsilon}- \mathbf{\varepsilon}^{\rm(p)}\right). \tag{1}\]  
where \(\mathbf{\sigma}\) denotes the nominal Cauchy stress tensor (force per total area), \(\omega\) is the scalar isotropic damage variable ranging from 0 (undamaged material) to 1 (completely damaged material), \(\mathbf{\varepsilon}\) and \(\mathbf{\varepsilon}^{\rm(p)}\) represent the total and the plastic strain tensor, respectively. The fourth-order transversely isotropic elastic stiffness tensor with respect to the global coordinate system  
\[\mathbb{C}^{\rm(ti)}_{ijkl}=a_{mi}\,a_{nj}\,a_{pk}\,a_{ql}\,\mathbb{C}^{\prime \rm(ti)}_{mmpq} \tag{2}\]  
is obtained by transforming the transversely isotropic elastic stiffness tensor \(\mathbb{C}^{\prime\rm(ti)}\), aligned with the principal directions of the material, into the global coordinate system by means of the orthogonal transformation matrix \(\mathbf{a}\). In Voigt notation, \(\mathbb{C}^{\prime\rm(ti)}\) reads as  
\[\mathbb{C}^{\prime\rm(ti)}=r_{\rm s} \begin{bmatrix}\frac{1}{E_{1}}&-\frac{v_{ 22}}{E_{2}}&-\frac{v_{22}}{E_{2}}&0&0&0\\ -\frac{v_{22}}{E_{2}}&\frac{1}{E_{2}}&-\frac{v_{22}}{E_{2}}&0&0&0\\ -\frac{v_{22}}{E_{2}}&-\frac{v_{23}}{E_{2}}&\frac{1}{E_{2}}&0&0&0\\ 0&0&0&\frac{1}{C_{12}}&0&0\\ 0&0&0&0&\frac{2(1+v_{23})}{E_{2}}\end{bmatrix} ^{-1}, \tag{3}\]  
characterized by five independent material parameters, i.e., the Young's modulus \(E_{1}\) in the direction of the normal vector \(\mathbf{n}\) to the planes of isotropy (planes with the same material parameters in all directions within those planes), i.e, \(\mathbf{n}\) represents the axis of material symmetry, as illustrated in Figure 2, the Poisson's ratio \(v_{12}\) and the shear modulus \(G_{12}\) in the planes containing the axis of material symmetry, and the Young's modulus \(E_{2}\) as well as the Poisson's ratio \(v_{23}\), both in the plane of isotropic material behavior. As shown in Figure 2, in contrast to intact rock, which is characterized by a stratification or a pervasive foliation, the macro structure of rock mass is governed by mechanical discontinuities, such as joints and faults, resulting in a strong dependence of the material properties of rock mass on the surface conditions of those discontinuities. Thus, by analogy to the isotropic version of the RDP model, the elastic stiffness properties of intact rock, i.e., the [PERSON]'s moduli \(E_{1}\) and \(E_{2}\) as well as the shear modulus \(G_{12}\) are multiplied by the empirical stiffness reduction factor  
\[r_{\rm s}=0.02+\frac{1-\mathrm{D}/2}{1+\exp\biggl{(}\frac{60+15\cdot\mathrm{D}- \mathrm{GSI}}{11}\biggr{)}}+\biggl{(}0.98-\frac{1}{1+\exp(-40/11)}\biggr{)} \biggl{(}\frac{\mathrm{GSI}}{100}\biggr{)}^{10} \tag{4}\]  
according to [PERSON] et al. [6]. The latter is expressed in terms of the geological strength index (GSI) for considering the geological conditions of rock mass, ranging from 1 (intact rock) to 0 (completely disintegrated rock), and the disturbance factor \(\mathrm{D}\) for taking into account the disturbance of the in situ stress in the rock mass due to the excavation process with \(0\,(\mathrm{undisturbed})\leq\mathrm{D}\leq 1\,(\mathrm{significantly\ disturbed})\).