## 4 Consistent Tangent Stiffness
\(\mathbf{n}\) does not depend on time and its variation with \(\delta\mathbf{\epsilon}\) can be obtained starting from its definition (8)\({}_{2}\):  
\[\delta\mathbf{n}=\frac{3}{2}\left(\frac{1}{q^{\text{trial}}}\delta\mathbf{s} ^{\text{trial}}-\frac{\mathbf{s}^{\text{trial}}}{q^{\text{trial}}2}\delta q^{ \text{trial}}\right) \tag{40}\]  
with:  
\[\delta\mathbf{s}^{\text{trial}}=2G\,\mathbf{Dev}:\delta\mathbf{\epsilon} \tag{41}\]  
where \(\mathbf{Dev}\) is the deviatoric operator defined as:  
\[\mathbf{Dev}=\mathbf{II}-\frac{1}{3}\,\mathbf{I}\otimes\mathbf{I} \tag{42}\]  
Replacing (41) in (40) and taking into account that \(\mathbf{s}^{\text{trial}}=2/3q^{\text{trial}}\mathbf{n}\), one has:  
\[\delta\mathbf{n}=\frac{3G}{q^{\text{trial}}}\Big{(}\mathbf{Dev}-\frac{2}{3} \mathbf{n}\otimes\mathbf{n}\Big{)}:\delta\mathbf{\epsilon} \tag{43}\]  
The variations \(\delta p\) and \(\delta q\) are obtained from (18) and (17):  
\[\delta p =\delta p^{\text{trial}}-K\Big{(}p_{,p\rho}^{\text{eq}}\delta p +p_{,p\delta}^{\text{eq}}\delta q\Big{)}\Delta\gamma-Kp_{,p}^{\text{eq}}\, \delta\Delta\gamma \tag{44}\] \[\delta q =\delta q^{\text{trial}}-3G\Big{(}p_{,p\delta}^{\text{eq}}\delta p +p_{,p\delta}^{\text{eq}}\delta q\Big{)}\Delta\gamma-3 Gp_{,q}^{\text{eq}}\, \delta\Delta\gamma\]  
with \(\delta q^{\text{trial}}\) given in (41)\({}_{2}\) and:  
\[\delta p^{\text{trial}}=3K\mathbf{m}:\delta\mathbf{\epsilon} \tag{45}\]  
After rearranging, Equation (44) can be rewritten as:  
\[\delta\mathbf{\sigma}=\tfrac{2}{3}\,\mathbf{n}\,\delta q+\tfrac{2}{3}\,q\,\delta \mathbf{n}+\mathbf{I}\,\delta p=\tfrac{2}{3}\,\mathbf{n}\left(\tfrac{\partial \mathbf{q}}{\partial\mathbf{\epsilon}}:\delta\mathbf{\epsilon}+\tfrac{\partial \mathbf{q}}{\partial\mathbf{\epsilon}}\,\delta t\right)+\tfrac{2}{3}\,q\,\tfrac{ \partial\mathbf{n}}{\partial\mathbf{\epsilon}}:\delta\mathbf{\epsilon}+\left(\mathbf{ I}\,\tfrac{\partial p}{\partial\mathbf{\epsilon}}:\delta\mathbf{\epsilon}+\mathbf{I}\, \tfrac{\partial p}{\partial t}\,\delta t\right) \tag{38}\]0  
with:  
\[\delta\mathbf{\sigma}=\tfrac{2}{3}\,\mathbf{n}\,\delta q+\tfrac{2}{3}\,q\,\delta \mathbf{n}+\mathbf{I}\,\delta p=\tfrac{2}{3}\,\mathbf{n}\left(\tfrac{\partial \mathbf{q}}{\partial\mathbf{\epsilon}}:\delta\mathbf{\epsilon}+\tfrac{\partial \mathbf{q}}{\partial\mathbf{\epsilon}}\,\delta t\right)+\tfrac{2}{3}\,q\,\tfrac{ \partial\mathbf{n}}{\partial\mathbf{\epsilon}}:\delta\mathbf{\epsilon}+\left(\mathbf{ I}\,\tfrac{\partial p}{\partial\mathbf{\epsilon}}:\delta\mathbf{\epsilon}+\mathbf{I}\, \tfrac{\partial p}{\partial t}\,\delta t\right) \tag{38}\]1 (48) \[\delta\left[\left(\frac{p^{\text{eq}}}{p_{p}^{\text{eq}}}\right)^ {\frac{\lambda^{*}-\kappa^{*}}{p^{*}}}\right] =\frac{\lambda^{*}-\kappa^{*}}{\mu^{*  
The present expression of \(\delta\Delta\gamma\) can be replaced in (46), obtaining (after some algebra) the following system of equations in the unknown variations \(\delta p\) and \(\delta q\):  
\[ \begin{split} F_{11}\,\delta p+F_{12}\,\delta q&=3\,K \,\mathbf{m}:\delta\mathbf{\epsilon}-\frac{K}{A}\,\frac{\mu^{*}}{\tau}\left[\frac{p ^{\text{eq}}}{p^{\text{eq}}_{p}}\right]^{\frac{A^{*}-\mu^{*}}{p^{\mu}}}\delta t \\ F_{21}\,\delta p+F_{22}\,\delta q&=2\,G\,\mathbf{n}: \delta\mathbf{\epsilon}-3\,G\,\frac{p^{\text{eq}}_{A}}{Ap^{\text{eq}}_{p,p}}\frac {\mu^{*}}{\tau}\left[\frac{p^{\text{eq}}}{p^{\text{eq}}_{p}}\right]^{\frac{A^ {*}-\mu^{*}}{p^{\mu}}}\delta t\end{split}  \tag{55}\]  
with:  
\[ \begin{split} F_{11}&=1+K\,p^{\text{eq}}_{pp}\, \frac{B}{A}\\ F_{12}&=K\,p^{\text{eq}}_{A}\,\frac{B}{A}\\ F_{21}&=3\,G\,p^{\text{eq}}_{ap}\,\Delta\gamma+3\,G \,p^{\text{eq}}_{A}\,\frac{B}{A}-3\,G\,p^{\text{eq}}_{A}\,\frac{p^{\text{eq }}_{A}}{p^{\text{eq}}_{p,p}}\,p^{\text{eq}}_{p}\,\Delta\gamma\\ F_{22}&=1+3\,G\,p^{\text{eq}}_{A\!I}\Delta\gamma+ 3\,G\,\frac{\left(p^{\text{eq}}_{A\!I}\right)^{2}}{p^{\text{eq}}_{p,p}}\, \frac{B}{A}-3\,G\,\frac{p^{\text{eq}}_{A}}{p^{\text{eq}}_{p,p}}\,p^{\text{eq }}_{A}\,\Delta\gamma\end{split}