# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 3 Implicit Gradient-Enhancement of the Damage-Plasticity Model for Intact Rock and Rock Mass  
Softening material behavior leads to an ill-posed initial boundary value problem and consequently to pathological mesh-sensitivity in finite element simulations. As a remedy, in [24], the crack band approach was employed for the RDP model. While the crack band approach is a rather simple regularization technique, it is also characterized by a number of shortcomings, which are addressed in [31]. Motivated by those deficiencies, in the following, a more sophisticated regularization technique based on the implicit gradient-enhanced formulation [32] is presented. To this end, the approach by [PERSON] and [PERSON] [17] proposed for a damage-plasticity model for concrete and based on the gradient of the internal softening variable, is adopted. By incorporating the gradient of an internal variable into the constitutive relations, nonlocality is introduced. Thus, the mechanical response of a material point does not exclusively depend on its local state, but is also influenced by the state in its neighborhood. Nonlocality is incorporated by replacing the local softening variable \(\alpha_{\mathrm{d}}\) by a weighted softening variable \(\hat{\alpha}_{\mathrm{d}}\) in the exponential damage law (11), which is expressed as  
\[\omega(\hat{\alpha}_{\mathrm{d}})=1-\exp(-\hat{\alpha}_{\mathrm{d}}/\varepsilon _{\mathrm{f}}) \tag{14}\]  
Therein, the softening modulus \(\varepsilon_{\mathrm{f}}\) is a material parameter. The weighted softening variable \(\hat{\alpha}_{\mathrm{d}}\) is calculated from a combination of the local softening variable \(\alpha_{\mathrm{d}}\) and its nonlocal counterpart \(\hat{\alpha}_{\mathrm{d}}\) as  
\[\hat{\alpha}_{\mathrm{d}}=m\,\bar{\alpha}_{\mathrm{d}}+(1-m)\,\alpha_{\mathrm{ d}} \tag{15}\]  
in which \(m\) denotes a weighting parameter. Choosing \(m\) larger than \(1\) yields the over-nonlocal formulation [33] to achieve full regularization of the problem, as proven in [34]. Furthermore, by ensuring that \(\hat{\alpha}_{\mathrm{d}}\) can only increase, damage is considered as an irreversible process [35]. According to the implicit approach, the field of the nonlocal softening variable \(\bar{\alpha}_{\mathrm{d}}\), henceforth simply denoted as the nonlocal field, is defined implicitly as the solution of a higher-order partial differential equation. Adopting the formulation by [PERSON] and [PERSON] [17], for the description of the nonlocal field a second order partial differential equation is employed as  
\[\bar{\alpha}_{\mathrm{d}}-l^{2}\,\Delta\bar{\alpha}_{\mathrm{d}}=\alpha_{ \mathrm{d}}\qquad\qquad\mathrm{in}\,\,\Omega \tag{16}\]  
in which \(l\) denotes a length scale parameter defining the radius of nonlocal interaction, \(\Delta\) is the Laplace operator and \(\alpha_{\mathrm{d}}\) represents the local softening variable of (12), and \(\Omega\) is the spatial domain occupied by the body under consideration. Equation (16) is a screened-Poisson equation, commonly denoted as the Helmholtz equation in the context of implicit gradient-enhanced formulations [12; 36]. It is apparent that nonlocality affects only the damage part of the model and the plasticity part of the RDP model remains local. As suggested in [32; 37], homogeneous Neumann boundary conditions are assumed as \(\
abla\bar{\alpha}_{\mathrm{d}}\cdot\mathbf{n}=0\) on the entire boundary \(\Gamma\) of the domain with the normal vector to the boundary \(\mathbf{n}\). This boundary condition was interpreted in [38] in the context of phase-field models enforcing cracks to occur perpendicular to the boundary. The set of governing equations is completed by the equilibrium equation  
\[\
abla\cdot\mathbf{\sigma}+\mathbf{\hat{f}}=\mathbf{0}\qquad\qquad\mathrm{in}\,\,\Omega \tag{17}\]in which \(\bar{\mathbf{f}}\) is the vector of body forces.