7. Calculation of the tangent operator in speed

The speed tangent operator is given for information purposes only. In programming, the operator corresponding to RIGI_MECA is calculated using the same formulas as those providing FULL_MECA in which we set: \(\mathrm{\Delta \lambda }=0\), \(H\) and \({\gamma }^{\text{+}}\text{=}{\gamma }^{\text{-}}\).

If the stress tensor represents the effective stress:

Condition \(\dot{F}\text{=}0\) is written as: \(\dot{F}\text{=}\frac{\partial F}{\partial \sigma }\dot{\sigma }\text{+}\frac{\partial F}{\partial \gamma }\dot{\gamma }=0\). So this gives us:

\(\dot{\gamma }=-\frac{\sum _{\text{ij}}\frac{\partial F}{\partial {\sigma }_{\text{ij}}}{\dot{\sigma }}_{\text{ij}}}{\frac{\partial F}{\partial \gamma }}\)

Now, we have \(\dot{\sigma }\text{=}H(\dot{\varepsilon }\text{-}{\dot{\varepsilon }}^{p})\) where \(H\) is the Hooke matrix. Also, \({\dot{\varepsilon }}^{p}=\dot{\lambda }\frac{\partial G}{\partial \sigma }=\frac{\dot{\gamma }}{(\eta (\gamma )+1)}\frac{\partial G}{\partial \sigma }\), from where

\(\dot{\gamma }=-\frac{\sum _{\text{ijkl}}\frac{\partial F}{\partial {\sigma }_{\text{ij}}}{H}_{\text{ijkl}}{\dot{\varepsilon }}_{\text{kl}}}{\frac{\partial F}{\partial \gamma }-\frac{1}{\eta (\gamma )+1}\sum _{\text{ijkl}}\frac{\partial F}{\partial {\sigma }_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\sigma }_{\text{kl}}}}\text{avec}\frac{\partial G}{\partial {\sigma }_{\text{ij}}}=\eta (\gamma ){\delta }_{\text{ij}}+\sqrt{\frac{3}{2}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}},\)

And finally

\(\begin{array}{}{\dot{\mathrm{\sigma }}}_{\text{ab}}={H}_{\text{abcd}}({\dot{\mathrm{\varepsilon }}}_{\text{cd}}-\dot{\mathrm{\lambda }}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{cd}}})=\sum _{\text{cd}}{D}_{\text{abcd}}{\dot{\mathrm{\varepsilon }}}_{\text{cd}}\\ \text{où}{D}_{\text{abcd}}\text{}={H}_{\text{abcd}}+\frac{(\sum _{\text{kl}}{H}_{\text{abkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}})(\sum _{\text{ij}}\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijcd}})}{(\mathrm{\eta }(\mathrm{\gamma })+1)\frac{\partial F}{\partial \mathrm{\gamma }}-\sum _{\text{pqmn}}\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{pq}}}{H}_{\text{pqmn}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{mn}}}}\end{array}\).

If the stress tensor represents the total stress:

We have \(\dot{\sigma }=H(\dot{\varepsilon }\text{-}{\dot{\varepsilon }}^{p})\text{+}{\dot{\sigma }}_{p}I\) where \(H\) is the Hooke matrix. In subscribed notation, we can write:

\({\dot{\mathrm{\sigma }}}_{\text{ij}}={H}_{\text{ijkl}}({\dot{\mathrm{\varepsilon }}}_{\text{kl}}-{\dot{\mathrm{\varepsilon }}}_{\text{kl}}^{p})+{\dot{\mathrm{\sigma }}}_{p}{\mathrm{\delta }}_{\text{ij}}\)

In addition, \({\dot{\varepsilon }}^{p}=\dot{\lambda }\frac{\partial G}{\partial \sigma }=\frac{\dot{\gamma }}{(\eta (\gamma )+1)}\frac{\partial G}{\partial \sigma }\),

Condition \(\dot{F}\text{=}0\) is written as: \(\dot{F}=\frac{\partial F}{\partial \sigma }\dot{\sigma }+\frac{\partial F}{\partial \gamma }\dot{\gamma }=0\). So this gives us:

\(\dot{\mathrm{\gamma }}=-\frac{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{\dot{\mathrm{\sigma }}}_{\text{ij}}}{\frac{\partial F}{\partial \mathrm{\gamma }}}\)

by replacing \({\dot{\mathrm{\sigma }}}_{\text{ij}}\) and \({\dot{\mathrm{\varepsilon }}}_{\text{kl}}^{p}\) with their expressions we can deduce:

\(\dot{\mathrm{\gamma }}\frac{\partial F}{\partial \mathrm{\gamma }}=-\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}({\dot{\mathrm{\varepsilon }}}_{\text{kl}}-\frac{\dot{\mathrm{\gamma }}}{(\mathrm{\eta }(\mathrm{\gamma })+1)}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}})-\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{\dot{\mathrm{\sigma }}}_{p}{\mathrm{\delta }}_{\text{ij}}\)

and then the expression for \(\dot{\mathrm{\gamma }}\):

\(\begin{array}{}\dot{\mathrm{\gamma }}=-\frac{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}{\dot{\mathrm{\varepsilon }}}_{\text{kl}}}{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}-\frac{1}{(\mathrm{\eta }(\mathrm{\gamma })+1)}\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}}\text{-}\frac{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{\dot{\mathrm{\sigma }}}_{p}{\mathrm{\delta }}_{\text{ij}}}{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}-\frac{1}{(\mathrm{\eta }(\mathrm{\gamma })+1)}\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}}\text{}\\ \\ \text{avec}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{ij}}}=\mathrm{\eta }(\mathrm{\gamma }){\mathrm{\delta }}_{\text{ij}}+\sqrt{\frac{3}{2}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}},\end{array}\)

And finally

\(\begin{array}{}{\dot{\mathrm{\sigma }}}_{\text{ab}}={H}_{\text{abcd}}({\dot{\mathrm{\varepsilon }}}_{\text{cd}}-\dot{\mathrm{\lambda }}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{cd}}})+{\dot{\mathrm{\sigma }}}_{p}{\mathrm{\delta }}_{\text{ab}}={D}_{\text{abcd}}{\dot{\mathrm{\varepsilon }}}_{\text{cd}}+{E}_{\text{ab}}{\dot{\mathrm{\sigma }}}_{p}\\ {\dot{\mathrm{\sigma }}}_{\text{ab}}={H}_{\text{abcd}}({\dot{\mathrm{\varepsilon }}}_{\text{cd}}+\frac{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}{\dot{\mathrm{\varepsilon }}}_{\text{kl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{cd}}}}{(\mathrm{\eta }(\mathrm{\gamma })+1)\frac{\partial F}{\partial \mathrm{\gamma }}-\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}}+\frac{\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{\mathrm{\delta }}_{\text{ij}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{cd}}}{\dot{\mathrm{\sigma }}}_{p}}{(\mathrm{\eta }(\mathrm{\gamma })+1)\frac{\partial F}{\partial \mathrm{\gamma }}-\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}})+{\dot{\mathrm{\sigma }}}_{p}{\mathrm{\delta }}_{\text{ab}}\\ \text{où}\\ {D}_{\text{abcd}}\text{}={H}_{\text{abcd}}+\frac{{H}_{\text{abkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijcd}}}{(\mathrm{\eta }(\mathrm{\gamma })+1)\frac{\partial F}{\partial \mathrm{\gamma }}-\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}}\\ \text{et}\\ {E}_{\text{ab}}\text{}=\frac{{H}_{\text{abcd}}\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{\mathrm{\delta }}_{\text{ij}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{cd}}}}{(\mathrm{\eta }(\mathrm{\gamma })+1)\frac{\partial F}{\partial \mathrm{\gamma }}-\frac{\partial F}{\partial {\mathrm{\sigma }}_{\text{ij}}}{H}_{\text{ijkl}}\frac{\partial G}{\partial {\mathrm{\sigma }}_{\text{kl}}}}+{\mathrm{\delta }}_{\text{ab}}\end{array}\)