# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 2 Damage-Plasticity Model for Intact Rock and Rock Mass
In the expression for \(m_{\mathrm{g1,m}}=\left(m_{\mathrm{b}}/m_{0}\right)m_{\mathrm{g1}}\), \(m_{\mathrm{g1}}\) is calibrated from experimental results (uniaxial tension, uniaxial compression, and triaxial compression tests) of intact rock specimens and \(m_{\mathrm{g2,m}}\) is determined from  
\[m_{\mathrm{g2,m}}=2\,m_{\mathrm{g1,m}}-6f_{\mathrm{tu}}/f_{\mathrm{cu}} \tag{6}\]  
such that the lateral plastic strain rate in uniaxial tension is zero. Uniaxial tensile strength \(f_{\mathrm{tu}}\) is calculated from (3) as  
\[f_{\mathrm{tu}}=-\frac{m_{\mathrm{b}}/m_{0}\,m_{0}f_{\mathrm{cu}}\left(e+1 \right)}{6e}+\sqrt{\left(\frac{m_{\mathrm{b}}/m_{0}\,m_{0}f_{\mathrm{cu}}\left( e+1\right)}{6e}\right)^{2}+s\,f_{\mathrm{cu}}^{2}} \tag{7}\]  
Hardening material behavior is described by means of the stress-like internal variable  
\[q_{\mathrm{h}}(\alpha_{\mathrm{p}})= \begin{cases}f_{\mathrm{\sigma}}/f_{ \mathrm{cu}}+\left(1-f_{\mathrm{\sigma}}/f_{\mathrm{cu}}\right)\alpha_{ \mathrm{p}}\left(\alpha_{\mathrm{p}}^{2}-3\,\alpha_{\mathrm{p}}+3\right)& \text{ if }\alpha_{\mathrm{p}}<1\\ 1&\text{ if }\alpha_{\mathrm{p}}\geq 1\end{cases} , \tag{8}\]  
which is conjugate to the strain-like hardening variable \(\alpha_{\mathrm{p}}\) and contains the yield stress in uniaxial compression \(f_{\mathrm{\sigma}}\). The evolution law for the strain-like hardening variable \(\alpha_{\mathrm{p}}\) is given as  
\[\dot{\alpha}_{\mathrm{p}}=\dot{\lambda}\,\frac{E_{\mathrm{m}}}{E_{\mathrm{i}}} \,\frac{1}{x_{\mathrm{h}}(\sigma_{\mathrm{m}})}\,\left(1+3\,\frac{\bar{\rho}^ {2}}{\bar{\rho}^{2}+f_{\mathrm{cu}}^{2}\cdot 10^{-8}}\,\cos^{2}(3\,\theta/2) \right)\,\left\|\frac{\partial g_{\mathrm{p}}(\bar{\sigma},q_{\mathrm{h}}( \alpha_{\mathrm{p}}))}{\partial\bar{\sigma}}\right\| \tag{9}\]  
in which \(\left\|\partial g_{\mathrm{p}}(\bar{\sigma},q_{\mathrm{h}}(\alpha_{\mathrm{p}} ))/\partial\bar{\sigma}\right\|\) is the norm of the gradient of the plastic potential function with respect to effective stress, \(E_{\mathrm{m}}/E_{\mathrm{i}}\) denotes the reduction of the Young's modulus of rock mass compared to intact rock [26] ranging from 0 (completely disintegrated) to 1 (intact rock), and \(x_{\mathrm{h}}(\sigma_{\mathrm{m}})\) is a measure for describing hardening ductility, defined as  
\[x_{\mathrm{h}}(\sigma_{\mathrm{m}})= \begin{cases}\left(B_{\mathrm{h}}-D_{ \mathrm{h}}\right)\,\exp\left(\frac{R_{\mathrm{h}}\left(A_{\mathrm{h}}-B_{ \mathrm{h}}\right)}{C_{\mathrm{h}}\left(B_{\mathrm{h}}-D_{\mathrm{h}}\right)} \right)+D_{\mathrm{h}}&\text{ if }R_{\mathrm{h}}<0\\ A_{\mathrm{h}}-\left(A_{\mathrm{h}}-B_{\mathrm{h}}\right)\,\exp\left(-R_{ \mathrm{h}}/C_{\mathrm{h}}\right)&\text{ if }R_{\mathrm{h}}\geq 0\end{cases}  \tag{10}\]  
with \(R_{\mathrm{h}}=-\sigma_{\mathrm{m}}/f_{\mathrm{cu}}-G_{\mathrm{h}}\). In (10), model parameters \(A_{\mathrm{h}}\), \(B_{\mathrm{h}}\), \(C_{\mathrm{h}}\), \(D_{\mathrm{h}}\), and \(G_{\mathrm{h}}\) control the hardening behavior. They are calibrated by experimental data from uniaxial tension, uniaxial compression, and triaxial compression tests. In [24; 25], default values of \(B_{\mathrm{h}}=10^{-5},D_{\mathrm{h}}=10^{-6}\) and \(G_{\mathrm{h}}=0\) are suggested in absence of respective experimental data. Damage is provoked when the hardening variable \(q_{\mathrm{h}}(\alpha_{\mathrm{p}})\) attains its maximum value of 1. At this stage, the scalar isotropic damage parameter \(\omega\) starts evolving dependent on the strain-like internal softening variable \(\alpha_{\mathrm{d}}\). This relation is described by means of an exponential softening law as  
\[\omega(\alpha_{\mathrm{d}})=1-\exp(-\alpha_{\mathrm{d}}/\varepsilon_{\mathrm{ f}}) \tag{11}\]  
with the softening modulus \(\varepsilon_{\mathrm{f}}\) controlling the slope of the softening curve. The rate of the strain-like internal softening variable \(\alpha_{\mathrm{d}}\) is computed from the volumetric part of the plastic strain rate \(\dot{\varepsilon}^{\mathrm{p,vol}}=\mathrm{tr}(\dot{\varepsilon}^{\mathrm{p}})\) as  
\[\dot{\alpha}_{\mathrm{d}}=