# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 2 Damage-Plasticity Model for Intact Rock and Rock Mass  
The damage-plasticity model for intact rock and rock mass, denoted as RDP model in the following, was proposed originally for intact rock in [24] and was further extended to rock mass in [25] by incorporating empirical down-scaling factors to account for the influence of discontinuities according to [26; 27; 28]. The RDP model is based on the theory of plasticity formulated in the effective stress space combined with continuum damage mechanics. The stress-strain relation is expressed as  
\[\sigma=\left(1-\omega\right)\mathbb{C}:\left(\epsilon-\epsilon^{\mathrm{p}}\right) \tag{1}\]  
in which \(\sigma\) describes the nominal Cauchy stress tensor (force per total area), \(\omega\) the scalar isotropic damage parameter ranging from 0 (undamaged material) to 1 (fully damaged material), \(\mathbb{C}\) the fourth order elastic stiffness tensor, \(\epsilon\) the total strain tensor, and \(\epsilon^{\mathrm{p}}\) the plastic strain tensor. The effective stress tensor \(\sigma\) (force per undamaged area) is linked to the nominal stress tensor by  
\[\sigma=\left(1-\omega\right)\bar{\sigma} \tag{2}\]  
The elastic domain is bounded by the smooth Hoek-Brown yield criterion [29; 30] formulated in the Haigh-Westergaard coordinates of the effective stress tensor, i.e., the mean stress \(\bar{\sigma}_{\mathrm{m}}\), the deviatoric radius \(\bar{\rho}\), and the Lode angle in the deviatoric plane \(\theta\). In addition, a stress-like hardening variable \(q_{\mathrm{h}}(\alpha_{\mathrm{p}})\) is incorporated leading to the definition of the yield function \(f_{\mathrm{p}}\) as  
\[f_{\mathrm{p}}(\bar{\sigma},q_{\mathrm{h}}(\alpha_{\mathrm{p}}))=\left(\frac {1-q_{\mathrm{h}}(\alpha_{\mathrm{p}})}{f_{\mathrm{cu}}^{2}}\left(\sigma_{ \mathrm{m}}+\frac{\bar{\rho}}{\sqrt{6}}\right)^{2}+\sqrt{\frac{3}{2}}\frac{ \bar{\rho}}{f_{\mathrm{cu}}}\right)^{2}+\frac{q_{\mathrm{h}}^{2}(\alpha_{ \mathrm{p}})}{f_{\mathrm{cu}}}\frac{m_{\mathrm{b}}}{m_{0}}\,m_{0}\,\left( \sigma_{\mathrm{m}}+r(\theta,e)\,\frac{\bar{\rho}}{\sqrt{6}}\right)-s\,q_{ \mathrm{h}}^{2}(\alpha_{\mathrm{p}}) \tag{3}\]  
Therein, \(f_{\mathrm{cu}}\) is the uniaxial compressive strength, \(m_{0}\) is the friction parameter, \(r(\theta,e)\) is the Willam-Warnke function to describe the shape of the yield surface in deviatoric planes, and parameters \(m_{\mathrm{b}}/m_{0}\) and \(s\) are empirical down-scaling factors to account for discontinuities in rock mass, the latter depending on the geological strength index \(GSI\) and the disturbance factor \(D\) according to [26]. A default value for the eccentricity \(e\) of 0.51 is proposed in [24]. For representing material behavior of intact rock, \(m_{\mathrm{b}}/m_{0}\) and \(s\) are equal to 1. The flow rule for describing the evolution of plastic strains is defined in the effective stress space as  
\[\dot{\epsilon}^{\mathrm{p}}=\dot{\lambda}\,\frac{\partial g_{\mathrm{p}}(\bar{ \sigma},q_{\mathrm{h}}(\alpha_{\mathrm{p}}))}{\partial\bar{\sigma}} \tag{4}\]  
with \(\dot{\lambda}\) denoting the consistency parameter and \(g_{\mathrm{p}}(\bar{\sigma},q_{\mathrm{h}}(\alpha_{\mathrm{p}}))\) the non-associated plastic potential function expressed as  
\[g_{\mathrm{p}}(\bar{\sigma},q_{\mathrm{h}}(\alpha_{\mathrm{p}}))=\left(\frac {1-q_{\mathrm{h}}(\alpha_{\mathrm{p}})}{f_{\mathrm{cu}}^{2}}\,\left(\bar{\sigma }_{\mathrm{m}}+\frac{\bar{\rho}}{\sqrt{6}}\right)^{2}+\sqrt{\frac{3}{2}}\, \frac{\bar{\rho}}{f_{\mathrm{cu}}}\right)^{2}+\frac{q_{\mathrm{h}}^{2}(\alpha_ {\mathrm{p}})}{f_{\mathrm{cu}}}\,\left(m_{\mathrm{gl,m}}\,\bar{\sigma}_{ \mathrm{m}}+m_{\mathrm{gl,m}}\,\frac{\bar{\rho}}{\sqrt{6}}\right) \tag{5}\]Therein, volumetric plastic flow is controlled by dilatancy parameters \(m_{\mathrm{g1,m}}\) and \(m_{\mathrm{g2,m}}\).