Tangent operator in speed
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The condition


.. _RefEquation 5-1:

:math:`\dot{f}=0` eq 5-1

is written:

:math:`\dot{f}=\frac{\partial f}{\partial {\sigma }_{\text{ij}}}\dot{{\sigma }_{\text{ij}}}+\frac{\partial f}{\partial {\gamma }^{p}}\dot{{\gamma }^{p}}=0`

From the expression of the cumulative plastic deviatoric deformation and :math:`{\gamma }^{p}=\sqrt{\frac{2}{3}{e}_{\text{ij}}^{p}{e}_{\text{ij}}^{p}}` from the :math:`\dot{{e}^{p}}=\dot{\lambda }\tilde{G}` relationship, we then find the condition:

:math:`\dot{f}=\frac{\partial f}{\partial {\sigma }_{\text{ij}}}\dot{{\sigma }_{\text{ij}}}+\frac{\partial f}{\partial {\gamma }^{p}}\dot{\sqrt{\frac{2}{3}}}\dot{\lambda }\tilde{{G}_{\text{II}}}=0`

This gives us for the plastic multiplier:

:math:`\dot{\lambda }=\frac{-\frac{\partial f}{\partial {\sigma }_{\text{ij}}}\dot{{\sigma }_{\text{ij}}}}{\sqrt{\frac{2}{3}}\frac{\partial f}{\partial {\gamma }^{p}}\tilde{{G}_{\text{II}}}}`

Then considering the constraints/deformations relationship:

:math:`\frac{\partial f}{\partial {\sigma }_{\text{ij}}}\dot{{\sigma }_{\text{ij}}}=\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}\dot{{\varepsilon }_{\text{kl}}}=\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}=\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}\dot{{\varepsilon }_{\text{kl}}}-\dot{\lambda }\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}{G}_{\text{kl}}`

and by including it in the expression for :math:`\dot{\lambda }` we can write:

:math:`\dot{\lambda }=-\frac{\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}\dot{{\varepsilon }_{\text{kl}}}-\dot{\lambda }\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}{G}_{\text{kl}}}{\sqrt{\frac{2}{3}}\frac{\partial f}{\partial {\gamma }^{p}}\tilde{{G}_{\text{II}}}}`

Either:


.. _RefEquation 5-2:

:math:`\dot{\lambda }=-\frac{\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}\dot{{\varepsilon }_{\text{kl}}}}{\sqrt{\frac{2}{3}}\frac{\partial f}{\partial {\gamma }^{p}}\tilde{{G}_{\text{II}}}-\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}{G}_{\text{kl}}}` eq 5-2

Reporting this result into the expression for :math:`\dot{{\sigma }_{\text{ij}}}` we find:


.. _RefEquation 5-3:

:math:`\dot{{\sigma }_{\text{ab}}}={D}_{\text{abcd}}(\dot{{\varepsilon }_{\text{cd}}}+\frac{\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}\dot{{\varepsilon }_{\text{kl}}}}{\sqrt{\frac{2}{3}}\frac{\partial f}{\partial {\gamma }^{p}}\tilde{{G}_{\text{II}}}-\frac{\partial f}{\partial {\sigma }_{\text{ij}}}{D}_{\text{ijkl}}{G}_{\text{kl}}}{G}_{\text{cd}})` eq 5-3