## 2 A Transversely Isotropic Damage-Plasticity Model for Intact Rock and Rock Mass
### Transversely Isotropic Plasticity Formulation
 \tag{9}\]  
According to (5), the mapped effective stress depends on the mapping tensor \(\mathbb{P}\) and thus incorporates the orientation of the principal material directions and the anisotropy parameters. The parameters of the yield function comprise the uniaxial compressive strength for loading perpendicular to the planes of isotropy \(f_{\rm cu}^{(1)}\), the friction parameter for intact rock \(m_{0}^{*}\) and two empirical reduction factors, \(r_{\rm m}\) and \(r_{\rm c}\), for considering the influence of discontinuities on the friction and cohesion in rock mass by analogy to the original isotropic RDP model. [PERSON] et al. [7] proposed the latter as  
\[r_{\rm m}=\exp\biggl{(}\frac{\text{GSI}-100}{28-14\,\text{D}}\biggr{)} \tag{10}\]  
and  
\[r_{\rm c}=\exp\biggl{(}\frac{\text{GSI}-100}{9-3\,\text{D}}\biggr{)}\,, \tag{11}\]  
in terms of the geological strength index GSI and the disturbance factor D. In Figure 3, the influence of GSI and D on the empirical reduction factors, defined by (4), (10) and (11), is depicted. The shape of the yield function in deviatoric planes is controlled by the Willam-Warnke [49] function \(r(\hat{\theta}^{*},e)\), defined by the eccentricity parameter \(e\) ranging from 0.5 (triangular shape) to 1 (circular shape). The evolution of the yield function (9) is controlled by the stress-like hardening variable \(0\leq q_{\rm h}(\alpha_{\rm p})\leq 1\), depending on the strain-like hardening variable \(\alpha_{\rm p}\). For \(e=0.5\), the ultimate strength surface \(f^{*}(\hat{\mathbf{\sigma}}^{*},q_{\rm h}=1)=0\) corresponds to the smooth version of the Hoek-Brown failure criterion, proposed by [PERSON] and [PERSON] [11]. For considering the influence of transverse isotropy on the plastic strain tensor \(\epsilon^{\rm(p)}\), the non-associated plastic potential function \(g^{*}\big{(}\hat{\mathbf{\sigma}}^{*},q_{\rm h}(\alpha_{\rm p})\big{)}\) is also formulated in terms of the mapped effective stress tensor, reading as  
\[ \begin{split} g^{*}\big{(}\hat{\mathbf{\sigma}}^{*},q_{\rm h}(\alpha_ {\rm p})\big{)}=\left(\frac{1-q_{\rm h}(\alpha_{\rm p})}{f_{\rm cu}^{(1)}} \left(\sigma_{\rm m}^{*}+\frac{\hat{p}^{*}}{\sqrt{6}}\right)^{2}& +\sqrt{\frac{3}{2}}\,\frac{\hat{p}^{*}}{f_{\rm cu}^{(1)}}\right)^{ 2}+\\ &+\frac{q_{\rm h}^{2}(\alpha_{\rm p})}{f_{\rm cu}^{(1)}}\left(m_{ \rm g1,rm}^{*}\cdot\sigma_{\rm m}^{*}+m_{\rm g2,rm}^{*}\,\frac{\hat{p}^{*}}{ \sqrt{6}}\right).\end{split}  \tag{12}\]  
\(m_{\rm g1,rm}^{*}=r_{\rm m}\cdot m_{\rm g1}^{*}\) represents a reduced dilatancy parameter, calculated by the dilatancy parameter for intact rock \(m_{\rm g1}^{*}\), which can be determined from uniaxial compression tests, loaded perpendicular to the planes of isotropy, and triaxial compression tests, loaded in the principal material directions of an intact rock specimen. As described in detail by [PERSON] et al. [50], the parameter \(m_{\rm g2,rm}^{*}\) denotes a dependent dilatancy parameter, which can be calculated from \(m_{\rm g1,rm}^{*}\) such that the radial plastic strain rate vanishes in uniaxial tension perpendicular to the planes of isotropy. For calculating the rate of the plastic strain tensor, the plastic potential function (12) is employed in the non-associated flow rule  
\[\hat{\epsilon}^{\rm(p)}=\hat{\lambda}\,\mathbb{P}\colon\frac{\partial g^{*} \big{(}\hat{\mathbf{\sigma}}^{*},q_{\rm h}(\alpha_{\rm p})\big{)}}{\partial\, \hat{\mathbf{\sigma}}^{*}}, \tag{13}\]  
with \(\hat{\lambda}\) denoting the consistency parameter. Figure 3: Illustration of the range of the empirical reduction factors \(r_{\rm s}\), \(r_{\rm m}\) and \(r_{\rm c}\) for the elastic stiffness, the friction parameter and the cohesion parameter, respectively, for \(0\leq D\leq 1\) in terms of the geological strength index GSI. The stress-like hardening variable  
\[q_{\rm h}(\alpha_{\rm P})= \begin{cases}\dfrac{f_{\rm cy}^{(1)}}{f_{\rm cu}^{(1)}}+ \left(1-\dfrac{f_{\rm cy}^{(1)}}{f_{\rm cu}^{(1)}}\right)\alpha_{p}\left(\alpha_{p }^{2}-3\,\alpha_{p}+3\right)&\alpha_{p}<1\\ 1&\alpha_{p}\geq 1\end{cases}