## 4 Consistent Tangent Stiffness
 \tag{56}\]  
The system (55) can be solved analytically for \(\delta p\) and \(\delta q\):  
\[\begin{split}\delta p&=c_{11}\left(3\,K\,\mathbf{m} :\delta\mathbf{\epsilon}-\frac{K}{A}\,\frac{\mu^{*}}{\tau}\left[\frac{p^{\text{ eq}}}{p^{\text{eq}}_{p}}\right]^{\frac{A^{*}-\mu^{*}}{p^{\mu}}}\delta t \right)+c_{12}\left(2\,G\,\mathbf{n}:\delta\mathbf{\epsilon}-3\,G\,\frac{p^{\text {eq}}_{A}}{Ap^{\text{eq}}_{p,p}}\,\frac{\mu^{*}}{\tau}\left[\frac{p^{\text{ eq}}}{p^{\text{eq}}_{p}}\right]^{\frac{A^{*}-\mu^{*}}{p^{\mu}}}\delta t \right)\\ \delta q&=c_{21}\left(3\,K\,\mathbf{m}:\delta\mathbf{ \epsilon}-\frac{K}{A}\,\frac{\mu^{*}}{\tau}\left[\frac{p^{\text{eq}}}{p^{\text {eq}}_{p}}\right]^{\frac{A^{*}-\mu^{*}}{p^{\mu}}}\delta t\right)+c_{22}\left( 2\,G\,\mathbf{n}:\delta\mathbf{\epsilon}-3\,G\,\frac{p^{\text{eq}}_{A}}{Ap^{\text {eq}}_{p,p}}\,\frac{\mu^{*}}{\tau}\left[\frac{p^{\text{eq}}}{p^{\text{eq}}_{ p}}\right]^{\frac{A^{*}-\mu^{*}}{p^{\mu}}}\delta t\right)\\ &\text{where the terms }c_{11},c_{12},c_{21},c_{22}\text{ are the coefficients of the matrix }\mathbf{C}=\mathbf{F}^{-1},\text{ namely:}\\ \mathbf{C}= \begin{bmatrix}c_{11}&c_{12}\\ c_{21}&c_{22}\end{bmatrix} =\mathbf{F}^{-1}=\frac{1}{\det\mathbf{F}} \begin{bmatrix}F_{ 22}&-F_{12}\\ -F_{21}&F_{11}\end{bmatrix}  \tag{58}\]  
Rearranging (57), one obtains:  
\[\begin{split}\delta p&=[c_{11}\,K\,\mathbf{I}+c_{12}\,2\,G\, \mathbf{n}]:\delta\mathbf{\epsilon}-\left[c_{11}\,\frac{K}{A}\,\frac{\mu^{*}}{ \tau}\left(\frac{p^{\text{eq}}}{p^{\text{eq}}_{p}}\right)^{\frac{A^{*}-\mu^{ *}}{p^{\mu}}}+c_{12}\,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}}}\right]:\delta t\\ \delta q&=[c_{21}\,K\,\mathbf{I}+c_{22}\,2\,G\,\mathbf{n}]: \delta\mathbf{\epsilon}-\[\delta\sigma =\underbrace{\left[\left(\frac{2}{3}c_{12}\,K\,\mathbf{n}\otimes I+c_{12 }\,2\,G\,\mathbf{I}\otimes\mathbf{n}\right)+\frac{4}{3}\,G\,\Big{(}c_{22}-\frac{ q}{q^{\text{trial}}}\Big{)}\,\mathbf{n}\otimes\mathbf{n}+2\,G\,\frac{q}{q^{\text{ trial}}}\,\mathbf{Dev}+c_{11}\,K\,\mathbf{I}\otimes\mathbf{I}\right]} :\delta\epsilon+ \tag{60}\] \[+\left\{-I\Big{[}c_{11}\,\frac{K}{A}\,\frac{\mu^{*}}{\tau}\left( \frac{p^{\text{eq}}}{p^{\text{eq}}_{p}}\right)^{\frac{1+p^{*}+p^{*}}{p^{*}}}+ c_{12}\,3\,G\,\frac{p^{\text{eq}}_{q}}{A\,p^{\text{eq}}_{p^{\text{eq}}}}\, \frac{\mu^{*}}{\tau}\left(\frac{p^{\text{eq}}}{p^{\text{eq}}_{p}}\right)^{ \frac{1+p^{*}+p^{*}}{p^{*}}}\Big{]}+\right.\] \[\left.-\frac{2}{3}\mathbf{n}\Big{[}c_{21}\,\frac{K}{A}\,\frac{\mu ^{*}}{\tau}\left(\frac{p^{\text{eq}}}{p^{\text{eq}}_{p}}\right)^{\frac{1+p^{* }+p^{*}}{p^{*}}}+c_{22}\,3\,G\,\frac{p^{\text{eq}}_{q}}{A\,p^{\text{eq}}_{p^{ \text{eq}}}}\,\frac{\mu^{*}}{\tau}\left(\frac{p^{\text{eq}}}{p^{\text{eq}}_{ p}}\right)^{\frac{1+p^{*}+p^{*}}{p^{*}}}\Big{]}\right\}\delta t\]  
The term inside the curly brackets in (60) is the partial derivative of the stress with respect to time, i.e. \(\partial\sigma/\partial t\). The stress variation in (60) consists of two contributions. The first, \(\partial\sigma/\partial\epsilon\), is the tangent stiffness operator to be used for the predictor step in the global Newton-Raphson iteration scheme. This is the term that is required by Abaqus(tm) to be computed at each Gauss point in a User Material subroutine and to be passed to the code for global iteration. The second term, \(\partial\sigma/\partial t\), accounts for the stress relaxation at constant strain and does not have to be passed to Abaqus(tm) by the User Material subroutine. However, it has to be properly taken into account for the validation of the tangent stiffness operator, as will be discussed in the next section. It should be noted that the tangent stiffness operator \(\partial\sigma/\partial\epsilon\) in (60) is not symmetric. However, it can be shown that the coefficients \(c_{12}\) and \(c_{21}\) of the non-symmetric part in (60) vanish for vanishing \(\Delta\gamma\), a result in agreement with the findings in [20].