Calculation of the tangent operator
=============================

In the case where :math:`{\sigma }_{\mathrm{eq}}^{e}=0` and :math:`x=0`, the elasticity tensor is taken as a tangent operator.

Otherwise, we get this operator by deriving the equation [:ref:`éq 4.3-1 <éq 4.3-1>`] with respect to :math:`\mathrm{\Delta }\mathrm{\epsilon }`:

:math:`\frac{\partial \stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial \mathrm{\Delta }\stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial b\left(x,{\sigma }_{\text{eq}}^{e}\right)}{\partial \mathrm{\Delta }\mathrm{\epsilon }}{\stackrel{~}{\mathrm{\sigma }}}^{e}+b\left(x,{\sigma }_{\text{eq}}^{e}\right)\frac{\partial {\stackrel{~}{\mathrm{\sigma }}}^{e}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}`

then also deriving [:ref:`éq 4.3-2 <éq 4.3-2>`] with respect to :math:`\mathrm{\Delta }\mathrm{\epsilon }`:

:math:`\frac{\partial \mathrm{\Delta }\mathrm{\sigma }}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial \mathrm{\Delta }\stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}+K{\mathrm{I}}_{3}\frac{\partial \text{Tr}\left(\mathrm{\Delta }\mathrm{\epsilon }\right)}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial \mathrm{\Delta }\stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}+K\mathrm{.}{\mathrm{I}}_{3}^{\mathrm{t}}{\mathrm{I}}_{3}`

It will be noted that, in these equations, the tensors of order 2 and of order 4 are respectively assimilated to vectors and to matrices. :math:`{\mathrm{I}}_{3}` is here a tensor of order 2, similar to a vector:

:math:`{}^{t}\text{}{\mathrm{I}}_{3}=\left(\mathrm{1,1,1,0,0,0}\right)`

In addition, we have:

:math:`\frac{\partial b(x,{\sigma }_{\text{eq}}^{e})}{\partial \Delta \varepsilon }=\frac{\partial b}{\partial x}(x,{\sigma }_{\text{eq}}^{e})\frac{\partial x}{\partial \Delta \varepsilon }+\frac{\partial b}{\partial {\sigma }_{\text{eq}}^{e}}(x,{\sigma }_{\text{eq}}^{e})\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }`

So you have to calculate :math:`\frac{\partial x}{\partial \Delta \varepsilon }`. To do this, we implicitly derive the scalar equation with respect to :math:`\Delta \varepsilon`.

To simplify, the parameter :math:`T` will be omitted later in the writing of :math:`g` and its derivatives.

We then have:

:math:`\left[3\mu \Delta t{G}^{\text{'}}(x)+1\right]\frac{\partial x}{\partial \Delta \varepsilon }+\Delta t\frac{\partial g}{\partial y}(x,y)\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }=\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }`

From where:

:math:`\frac{\partial x}{\partial \Delta \varepsilon }=\frac{1-\Delta t\frac{\partial g}{\partial y}(x,y)}{1+3\mu \Delta t{G}^{\text{'}}(x)}\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }`

:math:`\frac{\partial x}{\partial \Delta \varepsilon }=\frac{1-\Delta t\frac{\partial g}{\partial y}(x,y)}{1+3\mu \Delta t{G}^{\text{'}}(x)}\frac{3\mu }{{\sigma }_{\mathrm{eq}}^{e}}{\tilde{\sigma }}^{e}`

with the expression for :math:`\text{G'}(x)` obtained in [:ref:`§4.3 <§4.3>`].

The following expression for the tangent operator is finally obtained:

:math:`\frac{\partial \Delta \sigma }{\partial \Delta \varepsilon }=K{I}_{3}^{t}{I}_{3}+2\mu \left[\gamma {\tilde{\sigma }}^{e}{}^{t}\text{}{\tilde{\sigma }}^{e}+b(x,{\sigma }_{\text{eq}}^{e})A\right]`

with

:math:`A=\frac{\partial \Delta \tilde{\varepsilon }}{\partial \Delta \varepsilon }={J}_{6}-\frac{1}{3}{I}_{3}{}^{t}\text{}{I}_{3}` where :math:`{J}_{6}` is the rank 6 identity matrix.

:math:`\gamma =\frac{3}{2{({\sigma }_{\mathrm{eq}}^{e})}^{3}}\left[{\sigma }_{\mathrm{eq}}^{e}\frac{1-\Delta t\frac{\partial g}{\partial y}(x,y)}{1+3\mu \Delta t{G}^{\text{'}}(x)}-x\right]`

**Note:**

*In the case of the law* *VISC_IRRA_LOG, it is easy to verify that:*

:math:`\begin{array}{}{G}^{\text{'}}(x)=\frac{1}{{f}_{1}^{\text{'}}{g}_{1}+{f}_{2}^{\text{'}}{g}_{2}}[{g}_{1}{g}_{1}^{\text{'}}({f}_{1}^{{\text{'}}^{2}}-{f}_{1}{f}_{1}^{\text{''}})+{g}_{2}{g}_{2}^{\text{'}}({f}_{2}^{{\text{'}}^{2}}-{f}_{2}{f}_{2}^{\text{''}})+{g}_{1}{g}_{2}^{\text{'}}({f}_{1}^{\text{'}}{f}_{2}^{\text{'}}-{f}_{1}^{\text{''}}{f}_{2})\\ +{g}_{2}{g}_{1}^{\text{'}}({f}_{1}^{\text{'}}{f}_{2}^{\text{'}}-{f}_{1}{f}_{2}^{\text{''}})-\frac{1}{\mathrm{3m}}({f}_{1}^{\text{''}}{g}_{1}+{f}_{2}^{\text{''}}{g}_{2})]\\ \frac{\partial g}{\partial \lambda }(x,y,T)=\frac{{f}_{1}^{\text{''}}{g}_{1}+{f}_{2}^{\text{''}}{g}_{2}}{{f}_{1}^{\text{'}}{g}_{1}+{f}_{2}^{\text{'}}{g}_{2}}\end{array}`

*where* :math:`{f}_{\mathrm{1,}}f{\text{'}}_{\mathrm{1,}}f\text{'}{\text{'}}_{\mathrm{1,}}{f}_{\mathrm{2,}}f{\text{'}}_{\mathrm{2,}}f\text{'}{\text{'}}_{2}` *denote the values of* :math:`{f}_{1}` *and* :math:`{f}_{2}` *and their derivatives at the point* :math:`t(x,y,T)` *and where* :math:`{g}_{\mathrm{1,}}g{\text{'}}_{\mathrm{1,}}{g}_{\mathrm{2,}}g{\text{'}}_{2}` *refer to the values of* :math:`{g}_{1}` *and* :math:`{g}_{2}` *and their derivative with respect to* :math:`s` * *at the point* :math:`(x,T)` (see [:ref:`bib1 <bib1>`]).