## 2 A Transversely Isotropic Damage-Plasticity Model for Intact Rock and Rock Mass
### Transversely Isotropic Plasticity Formulation  
The plastic part of the TI-RDP model is formulated in terms of the effective stress tensor \(\widetilde{\mathbf{\sigma}}\) (force per undamaged area), which is linked to the nominal stress tensor \(\mathbf{\sigma}\) by \(\widetilde{\mathbf{\sigma}}=\mathbf{\sigma}/(1-\omega)\). For considering the transversely isotropic behavior, a fictitious effective stress tensor  
\[\mathbf{\sigma}^{*}=\mathbb{P}:\widetilde{\mathbf{\sigma}} \tag{5}\]  
is defined by mapping the effective stress tensor into a fictitious isotropic configuration, denoted by the superscript \((\bullet)^{*}\), as originally proposed by [PERSON] [25] and for instance applied by [PERSON] et al. [28], [PERSON] et al. [46] and [PERSON] [29; 47]. \(\mathbb{P}\) denotes the fourth-order linear mapping tensor with major and minor symmetry, reading in terms of the Walpole algebra [48] as  
\[\mathbb{P}=\mathbb{E}^{(1)}+\gamma\left(\mathbb{E}^{(2)}+\mathbb{F}+\delta \,\mathbb{G}\right) \tag{6}\]  
with the anisotropy parameters \(\gamma\) and \(\delta\). The latter are linked to the uniaxial compressive strength for intact rock perpendicular to the planes of isotropy \(f_{\rm cu}^{(1)}\), the uniaxial compressive  
Figure 2: Schematic sketch of (**i**) the discontinuities in rock mass, (**ii**) the stratification or foliation planes in intact rock and its consideration in the TI-RDP model by (**iii**) the transversely isotropic equivalent continuum for stratified or foliated intact rock. strength parallel to the latter \(f_{\rm cu}^{(2)}\), the shear strength in planes perpendicular to the planes of isotropy \(f_{\rm sh}^{(12)}\) and the shear strength in the plane of isotropy \(f_{\rm sh}^{(23)}\) by  
\[\gamma=\frac{f_{\rm cu}^{(1)}}{f_{\rm cu}^{(2)}}\quad\text{ and }\quad\delta= \frac{f_{\rm sh}^{(23)}}{f_{\rm sh}^{(12)}}\,. \tag{7}\]  
\(\gamma\) can be determined by means of uniaxial compression tests with loading directions perpendicular and parallel to the planes of isotropy of intact layered rock. \(\delta\) can be determined by simple shear tests with loading in planes perpendicular to the planes of isotropy and in the plane of isotropy of intact layered rock. According to (7), \(\gamma\) and \(\delta\) are limited to values greater than \(0\) for a meaningful physical interpretation. The sub-tensors  
\[ \begin{split}\mathbb{E}_{ijkl}^{(1)}&=\phi_{ij} \phi_{kl}\,\quad\mathbb{E}_{ijkl}^{(2)}=\frac{1}{2}\,\chi_{ij}\,\chi_{kl}\,\quad\mathbb{F}_{ijkl}=\frac{1}{2}\left(\chi_{ik}\,\chi_{jl}+\chi_{il}\, \chi_{jk}-\chi_{ij}\,\chi_{kl}\right)\,,\\ \mathbb{G}_{ijkl}&=\frac{1}{2}\left(\phi_{ik}\,\chi_ {jl}+\phi_{il}\,\chi_{jk}+\chi_{ik}\,\phi_{jl}+\chi_{il}\,\phi_{jk}\right) \end{split}  \tag{8}\]  
describe the influence of the stratification or foliation planes on the material behavior, based on the microstructure tensor \(\mathbf{\phi}=\mathbf{n}\otimes\mathbf{n}\) and its orthogonal tensor \(\mathbf{\chi}=\mathbf{1}-\mathbf{\phi}\), which are both defined by the normal vector \(\mathbf{n}\) of the planes of isotropy. For delimiting the elastic domain, a direction-dependent yield criterion is formulated by expressing the yield function of the original isotropic RDP model in terms of the Haigh-Westergaard coordinates \(\varphi_{\rm m}^{*}\), \(\tilde{\rho}^{*}\) and \(\tilde{\theta}^{*}\) of the mapped effective stress tensor \(\tilde{\sigma}^{*}\):  
\[ \begin{split} f^{*}\big{(}\tilde{\sigma}^{*},q_{\rm h}(\alpha_{ \rm p})\big{)}=\left(\frac{1-q_{\rm h}(\alpha_{\rm p})}{{f_{\rm cu}^{(1)}}^{2 }}\left(\sigma_{\rm m}^{*}+\frac{\tilde{\rho}^{*}}{\sqrt{6}}\right)^{2}+\sqrt {\frac{3}{2}}\,\frac{\tilde{\rho}^{*}}{f_{\rm cu}^{(1)}}\right)^{2}+\\ +\frac{q_{\rm h}^{2}(\alpha_{\rm p})}{f_{\rm cu}^{(1)}}\,r_{\rm m }\cdot m_{0}^{*}\left(\tilde{\sigma}_{\rm m}^{*}+r(\tilde{\theta}^{*},e)\, \frac{\tilde{\rho}^{*}}{\sqrt{6}}\right)-r_{\rm c}\cdot q_{\rm h}^{2}(\alpha_ {\rm p})\,.\end{split}