# 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
The updated variables of the constitutive relations at time \(t^{(n)}\), i.e., \(\sigma^{(n)},\alpha_{\mathrm{p}}^{(n)},\alpha_{\mathrm{d}}^{(n)}\), are evaluated by means of a stress-update algorithm, employing an implicit integration scheme following the return-mapping approach [39] for the plastic regime and a subsequent explicit evaluation of the damage part. Finally, the incremental discretized weak form is obtained as  
\[\hat{\alpha}_{\mathrm{d}}=m\,\bar{\alpha}_{\mathrm{d}}+(1-m)\,\alpha_{\mathrm{ d}} \tag{15}\]0 \[\hat{\alpha}_{\mathrm{d}}=m\,\bar{\alpha}_{\mathrm{d}}+(1-m)\,\alpha_{\mathrm{ d}} \tag{15}\]1  
with \(\mathbf{f}_{\text{int}}^{u}\) and \(\mathbf{f}_{\text{ext}}^{u,(n)}\) denoting internal and external force vectors related to the displacement field, and \(\mathbf{R}_{\bar{u}_{d}}\) is the residual vector for the nonlocal field. Due to the nonlinear dependence of the system of Equations (26) and (27) on the unknown nodal solution vector  
\[\hat{\alpha}_{\mathrm{d}}=m\,\bar{\alpha}_{\mathrm{d}}+(1-m)\,\alpha_{\mathrm{ d}} \tag{15}\]2  
an iterative Newton-Raphson solution procedure is employed. The nodal unknowns at time \(t^{(n)}\) in the i-th iteration step are composed of \(\mathbf{q}^{(n)}=\mathbf{q}^{(n-1)}+\Delta\mathbf{q}^{(n,0)}\), where \(\Delta\mathbf{q}^{(n,0)}\) has to be determined. Linearization of Equations (26) and (27) at the state \(\mathbf{q}^{(n)}\) with the initial guess of the nodal unknowns \(\mathbf{q}^{(n,0)}=\mathbf{q}^{(n-1)}\) yields the iterative procedure for the correction of the nodal unknowns \(\Delta\Delta\mathbf{q}^{(n,0)}\) for time \(t^{(n)}\) after the i-th iteration step  
\[ \begin{bmatrix}\mathbf{K}_{uu}^{(n-1)}&\mathbf{K}_{uu_{d}}^{(n-1)}\\ \mathbf{K}_{\bar{u}_{d}u}^{(n-1)}&\mathbf{K}_{\bar{u}_{d}u_{d}}^{(n-1)}\end{bmatrix} \begin{bmatrix}\Delta\Delta\mathbf{q}_{u}^{(n)}\\ \Delta\Delta\mathbf{q}_{u}^{(n)}\end{bmatrix} = \begin{bmatrix}-\mathbf{R}_{u} \left(\mathbf{q}_{u}^{(n-1)},\mathbf{q}_{\mathbf{n}_{d}}^{(n-1)}\right)\\ -\mathbf{R}_{\bar{u}_{d}}\left(\mathbf{q}_{u}^{(n-1)},\mathbf{q}_{\bar{u}_{d}} ^{(n-1)}\right)\end{bmatrix} \,, \tag{29}\]  
with the submatrices of the system matrix given as  
\[\mathbf{K}_{uu}^{(n-1)} =\left.\frac{\partial\mathbf{R}_{u}}{\partial\mathbf{q}_{u}^{(n) }}\right|_{\mathbf{q}_{u}^{(n-1)}}=\int_{\Omega}\mathbf{B}_{u}^{\intercal} \left.\frac{\partial\sigma^{(n)}}{\partial\varepsilon^{(n)}}\right|_{\varepsilon ^{(n)}\mathbf{q}_{u}}d\Omega \tag{30}\] \[\mathbf{K}_{uu_{d}}^{(n-1)} =\left.\frac{\partial\mathbf{R}_{u}}{\partial\mathbf{q}_{u}^{(n) }}\right|_{\mathbf{q}_{u}^{(n-1)}}=\int_{\Omega}\mathbf{B}_{u}^{\intercal} \left.\frac{\partial\sigma^{(n)}}{\partial\bar{\alpha}_{d}^{(n)}}\right|_{ \mathbf{\bar{N}}_{\bar{u}_{d}}^{(n)}}\mathbf{\bar{d}}\Omega\] (31) \[\mathbf{K}_{\bar{u}_{d}u}^{(n-1)} =\left.\frac{\partial\mathbf{R}_{\bar{u}_{d}}}{\partial\mathbf{q }_{u}^{(n)}}\right|_{\mathbf{q}_{u}^{(n-1)}}=-\int_{\Omega}\mathbf{N}_{\bar{u} _{d}}^{\intercal}\left.\frac{\partial\alpha_{d}^{(n)}}{\partial\varepsilon^{(n )}}\right|_{\varepsilon^{(n)}\mathbf{q}_{u}}d\Omega\] (32) \[\mathbf{K}_{\bar{u}_{d}u_{d}}^{(n-1)} =\left.\frac{\partial\mathbf{R}_{\bar{u}_{d}}}{\partial\mathbf{q }_{\bar{u}_{d}}^{(n)}}\right|_{\mathbf{q}_{\bar{u}_{d}}^{(n-1)}}=\int_{\Omega} \mathbf{I}^{2}\mathbf{B}_{\bar{u}_{d}}^{\intercal}\mathbf{B}_{\bar{u}_{d}} \,d\Omega+\int_{\Omega}\mathbf{N}_{\bar{u}_{d}}^{\intercal}\mathbf{N}_{\bar{u} _{d}}\,d\Omega \tag{33}\]  
in which \(\partial\sigma^{(n)}/\partial\varepsilon^{(n)}\), \(\partial\sigma^{(n)}/\partial\bar{\alpha}_{d}^{(n)}\), and \(\partial\alpha_{d}^{(n)}/\partial\varepsilon^{(n)}\) are the consistent tangent stiffness submatrices. They represent the derivatives of the constitutive relations consistent with the numerical algorithm for integrating the path-dependent rate constitutive equations, which is essential for the full Newton-Raphson scheme in order to preserve a quadratic rate of convergence.