## 2 A Transversely Isotropic Damage-Plasticity Model for Intact Rock and Rock Mass
### Transversely Isotropic Gradient-Enhanced Damage Behavior  
Damage material behavior is initiated when the yield function (9) attains the ultimate strength surface, i.e., at \(q_{\rm h}=1\). The evolution of the damage variable \(\omega\) is described by the exponential relation  
\[\omega(\alpha_{\rm d})=1-\exp\!\left(\dfrac{-\alpha_{\rm d}}{\varepsilon_{\rm f }^{(1)}}\right) \tag{18}\]  
as a function of the local damage driving variable \(\alpha_{\rm d}\). It is controlled by the softening modulus \(\varepsilon_{\rm f}^{(1)}\) for the loading direction perpendicular to the planes of isotropy. The latter can be calibrated by the specific mode I fracture energy \(G_{\rm fl}^{(1)}\) for loading perpendicular to the planes of isotropy. The rate equation for the local damage driving variable is given as  
\[\dot{\alpha}_{\rm d}=\begin{cases}0&\alpha_{\rm p}<1\\ \dfrac{\dot{\varepsilon}_{\rm vol}^{(p)}}{x_{\rm s}\!\left(\dot{\varepsilon}^{( p)}\right)}&\alpha_{\rm p}\geq 1\end{cases}, \tag{19}\]  
with \(\dot{\varepsilon}_{\rm vol}^{(p)}=\text{tr}\!\left(\dot{\varepsilon}^{(p)}\right)\) and \(x_{\rm s}\!\left(\dot{\varepsilon}^{(p)}\right)=1+A_{\rm s}\left(\dot{ \varepsilon}_{\odot,\rm vol}^{(p)}/\dot{\varepsilon}_{\rm vol}^{(p)}\right)^{B_ {\rm s}}\) denoting the volumetric plastic strain rate and the softening ductility measure, respectively. \(\dot{\varepsilon}_{\odot,\rm vol}^{(p)}=\text{tr}\!\left(\left\langle-\dot{ \varepsilon}^{(p)}\right\rangle\right)\) represents the compressive part of the volumetric plastic strain rate and \(A_{\rm s}\) and \(B_{\rm s}\) are softening parameters, which can be calibrated from uniaxial compression tests, loaded perpendicular to the planes of isotropy, and triaxial compression tests, loaded in the principal material directions.  
For obtaining mesh-independent results in finite element simulations considering strain softening, the TI-RDP model is regularized by an over-nonlocal implicit gradient enhancement, following [PERSON] et al. [12]. For this purpose, in (18) \(\alpha_{\mathrm{d}}\) is replaced by the over-nonlocal softening variable  
\[\hat{\alpha}_{\mathrm{d}}=m\,\tilde{\alpha}_{\mathrm{d}}+\left(1-m\right) \alpha_{\mathrm{d}}. \tag{20}\]  
\(\tilde{\alpha}_{\mathrm{d}}\) represents the nonlocal counterpart of \(\alpha_{\mathrm{d}}\), and \(m\) is the weighting parameter. The nonlocal softening variable \(\tilde{\alpha}_{\mathrm{d}}\) is implicitly determined by the screened Poisson equation  
\[\alpha_{\mathrm{d}}=\tilde{\alpha}_{\mathrm{d}}-l^{2}\,\
abla^{2}\,\tilde{ \alpha}_{\mathrm{d}}\, \tag{21}\]  
with \(l\) denoting an internal length scale parameter. The weak form of (21) together with the weak form of the equilibrium equations yields a coupled system of differential equations with the displacements \(\mathbf{u}(\mathbf{x})\) and the nonlocal softening variable \(\tilde{\alpha}_{\mathrm{d}}(\mathbf{x})\) as the primary unknowns. A detailed description of the over-nonlocal implicit gradient enhancement can be found in [PERSON] et al. [12].  
Note that according to (13), the rate of the plastic strain tensor depends on the mapping tensor \(\mathbb{P}\). Hence, the evolution of \(\hat{\alpha}_{\mathrm{d}}\) and consequently of the damage variable \(\omega\) are influenced by the direction-dependent material behavior.