# 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
For the boundary conditions, the surface traction vector \(\bar{\mathbf{t}}=\boldsymbol{\sigma}\cdot\mathbf{n}\) on \(\Gamma_{t}\) and prescribed displacements \(\mathbf{u}=\bar{\mathbf{u}}\) on \(\Gamma_{u}\) are assumed. Partial differential Equations (16) and (17) form a fully coupled system with the unknown displacement vector \(\mathbf{u}\) and the nonlocal field \(\bar{\alpha}_{\mathrm{d}}\), which is solved by means of the finite element method. To this end, the weak form is formulated, which is subsequently discretized in space and in time. To obtain the weak form, the set of partial differential equations is multiplied by test functions \(\delta\mathbf{u}\) for the displacement field and \(\delta\bar{\alpha}_{\mathrm{d}}\) for the nonlocal field. Integration over the domain, application of the divergence theorem, incorporation of the infinitesimal strain \(\epsilon\), and consideration of \(\delta\mathbf{u}=\mathbf{0}\) on \(\Gamma_{u}\) yields the weak form of the IBVP expressed in Voigt notation as  
\[\int_{\Omega}\delta\epsilon^{\intercal}\boldsymbol{\sigma}\,d \Omega-\int_{\Gamma_{t}}\delta\mathbf{u}^{\intercal}\,\bar{\mathbf{t}}\,d \Gamma-\int_{\Omega}\delta\mathbf{u}^{\intercal}\,\bar{\mathbf{t}}\,d\Omega =0\,, \tag{18}\] \[\int_{\Omega}\delta\bar{\alpha}_{\mathrm{d}}\,\bar{\alpha}_{ \mathrm{d}}\,d\Omega+\int_{\Omega}\,l^{2}\,\left(\
abla\delta\bar{\alpha}_{ \mathrm{d}}\right)^{\intercal}\,\
abla\bar{\alpha}_{\mathrm{d}}\,d\Omega- \int_{\Omega}\delta\bar{\alpha}_{\mathrm{d}}\,\alpha_{\mathrm{d}}\,d\Omega =0\,. \tag{19}\]  
The displacement field \(\mathbf{u}\) and the nonlocal field \(\bar{\alpha}_{\mathrm{d}}\) are approximated over the domain using a Bubnov-Galerkin approach as  
\[\mathbf{u} =\mathbf{N}_{u}\,\mathbf{q}_{u} \tag{20}\] \[\bar{\alpha}_{\mathrm{d}} =\mathbf{N}_{\bar{\alpha}_{\mathrm{d}}}\,\mathbf{q}_{\bar{\alpha} _{\mathrm{d}}} \tag{21}\]  
in which \(\mathbf{N}_{(\bullet)}\) contains the shape functions and \(\mathbf{q}_{(\bullet)}\) are column vectors of the nodal unknown parameters, both expressed in the global form employing the standard assembly procedure, with \((\bullet)\) standing for the displacement field \(\mathbf{u}\) and the nonlocal field \(\bar{\alpha}_{\mathrm{d}}\), respectively. The infinitesimal strain \(\epsilon\) and the gradient of the nonlocal field \(\
abla\bar{\alpha}_{\mathrm{d}}\) are discretized as  
\[\epsilon =\mathbf{B}_{u}\,\mathbf{q}_{u} \tag{22}\] \[\
abla\bar{\alpha}_{\mathrm{d}} =\mathbf{B}_{\bar{\alpha}_{\mathrm{d}}}\,\mathbf{q}_{\bar{\alpha} _{\mathrm{d}}} \tag{23}\]  
with the strain-displacement matrix \(\mathbf{B}_{u}\) and the row vector of the spatial derivatives of the shape functions for the field of the nonlocal softening variable \(\mathbf{B}_{\bar{\alpha}_{\mathrm{d}}}\), again both expressed in the global form employing the standard assembly procedure. It follows that the shape functions for both fields must meet the requirement of \(C^{0}\)-continuity. Due to the nonlinear and path-dependent character of the IBVP, an incremental solution procedure is employed. For the incremental solution procedure, a discrete (pseudo-)time interval \([t^{(n-1)},t^{(n)}]\) is considered such that the body under consideration is in equilibrium at time \(t^{(n-1)}\). At this time, the nodal values \(\mathbf{q}_{u}^{(n-1)}\) and \(\mathbf{q}_{\bar{\alpha}_{\mathrm{d}}}^{(n-1)}\), the stress \(\sigma^{(n-1)}\) and the internal variables \(\alpha_{\mathrm{p}}^{(n-1)}\) and \(\alpha_{\mathrm{d}}^{(n-1)}\) are known. An incremental load is applied such that the traction vector and the vector of body forces at time \(t^{(n)}\) are prescribed as \(\bar{\mathbf{t}}^{(n)}=\bar{\mathbf{t}}^{(n-1)}+\Delta\bar{\mathbf{t}}^{(n)}\) and \(\bar{\mathbf{t}}^{(n)}=\bar{\mathbf{t}}^{(n-1)}+\Delta\bar{\mathbf{t}}^{(n)}\).