Tangent operator
=================

If the option is: RIGI_MECA_TANG, option used during the prediction, the tangent operator calculated at each Gauss point is said to be in speed:

:math:`{\dot{\sigma }}_{\text{ij}}={D}_{\text{ijkl}}^{\text{elp}}{\dot{\epsilon }}_{\text{kl}}`

In this case, :math:`{D}_{\text{ijkl}}^{\text{elp}}` is calculated from the non-discretized equations.

If the option is: FULL_MECA, option used when updating the tangent matrix at each iteration by updating the constraints and the internal variables:

:math:`{\mathrm{d\sigma }}_{\text{ij}}={A}_{\text{ijkl}}{\mathrm{d\epsilon }}_{\text{kl}}`

In this case, :math:`{A}_{\text{ijkl}}` is calculated from the implicitly discretized equations.




Nonlinear elastic tangent operator
----------------------------------------

The elastic relationship in speed is written as:


.. _RefEquation 5.1-1:

:math:`{\dot{\sigma }}_{\text{ij}}=-\dot{P}{\delta }_{\text{ij}}+{\dot{s}}_{\text{ij}}=({k}_{0}P+{K}_{\text{cam}})\text{tr {}\dot{\epsilon }{\delta }_{\text{ij}}+\mathrm{2\mu }\tilde{\dot{\epsilon }}\text{}}` eq 5.1-1


.. _RefEquation 5.1-2:

:math:`{\dot{\sigma }}_{\text{ij}}=({k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu )\text{tr {}\dot{\epsilon }{\delta }_{\text{ij}}+\mathrm{2\mu }{\dot{\epsilon }}_{\text{ij}}\text{}}` eq 5.1-2

The elastic tangent operator of the Cam_Clay law, noted :math:`{D}^{e}`, is therefore deduced from the following matrix writing:

:math:`\left\{\begin{array}{}{\dot{\sigma }}_{\text{11}}\\ {\dot{\sigma }}_{\text{22}}\\ {\dot{\sigma }}_{\text{33}}\\ \sqrt{2}{\dot{\sigma }}_{\text{12}}\\ \sqrt{2}{\dot{\sigma }}_{\text{23}}\\ \sqrt{2}{\dot{\sigma }}_{\text{31}}\end{array}\right\}=\underset{{D}^{e}}{\underset{\underbrace{}}{\left[\begin{array}{cccccc}{k}_{0}P+{K}_{\text{cam}}+\frac{4}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & 0& 0& 0\\ {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}+\frac{4}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & 0& 0& 0\\ {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}-\frac{2}{3}\mu & {k}_{0}P+{K}_{\text{cam}}+\frac{4}{3}\mu & 0& 0& 0\\ 0& 0& 0& \mathrm{2\mu }& 0& 0\\ 0& 0& 0& 0& \mathrm{2\mu }& 0\\ 0& 0& 0& 0& 0& \mathrm{2\mu }\end{array}\right]}}\left\{\begin{array}{}{\dot{\epsilon }}_{\text{11}}\\ {\dot{\epsilon }}_{\text{22}}\\ {\dot{\epsilon }}_{\text{33}}\\ \sqrt{2}{\dot{\epsilon }}_{\text{12}}\\ \sqrt{2}{\dot{\epsilon }}_{\text{23}}\\ \sqrt{2}{\dot{\epsilon }}_{\text{31}}\end{array}\right\}` eq 5.1-3




Plastic tangent operator in speed. Option RIGI_MECA_TANG
-------------------------------------------------------------

In this case, the global tangent operator is :math:`{K}_{i-1}` (the option RIGI_MECA_TANG called at the first iteration of a new load increment) based on the results known at time :math:`{t}_{i-1}` [:ref:`bib3 <bib3>`].

If the stress tensor at :math:`{t}_{i-1}` is on the border of the elasticity domain, we write the condition: :math:`\dot{f}=0` which must be satisfied in conjunction with the condition :math:`f=0`. If the stress tensor at :math:`{t}_{i-1}` is inside the domain, :math:`f<0`, then the tangent operator is the elasticity operator.


.. _RefEquation 5.2-1:

:math:`\dot{f}=(\frac{\partial f}{\partial \sigma })\dot{\sigma }+\frac{\partial f}{\partial {P}_{\text{cr}}}{\dot{P}}_{\text{cr}}=0` eq 5.2-1

like :math:`{\dot{P}}_{\text{cr}}=\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}{\dot{\epsilon }}_{v}^{p}`, so:


.. _RefEquation 5.2-2:

:math:`\dot{f}=(\frac{\partial f}{\partial \sigma })\dot{\sigma }+\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}{\dot{\epsilon }}_{v}^{p}=0` eq 5.2-2

**On the other hand** :math:`{\dot{\epsilon }}^{e}=\dot{\epsilon }-{\dot{\epsilon }}^{p}`

So:


.. _RefEquation 5.2-3:

:math:`{D}^{{e}^{-1}}\dot{\sigma }=\dot{\epsilon }-\dot{\Lambda }\frac{\partial f}{\partial \sigma }`, eq 5.2-3

**that is to say:**


.. _RefEquation 5.2-4:

:math:`{\dot{\sigma }}_{\text{ij}}={D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}-\dot{\Lambda }{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}` eq 5.2-4

The plastic work hardening module is written according to equation [:ref:`éq 3.5-7 <éq 3.5-7>`] and using the flow rule:


.. _RefEquation 5.2-5:

:math:`{H}_{p}=\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}\frac{\partial F}{\partial {P}_{\text{cr}}}=-\frac{1}{\dot{\Lambda }}\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{\partial {P}_{\text{cr}}}{\partial {\epsilon }_{v}^{p}}{\dot{\epsilon }}_{v}^{p}` eq 5.2-5

The equations [:ref:`éq 5.2-1 <éq 5.2-1>`] and [:ref:`éq 5.2-5 <éq 5.2-5>`] give:


.. _RefEquation 5.2-6:

:math:`{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{\dot{\sigma }}_{\text{ij}}-\dot{\Lambda }{H}_{p}=0` eq 5.2-6

Multiplying equation [:ref:`éq 5.2-4 <éq 5.2-4>`] by :math:`{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}` gives:


.. _RefEquation 5.2-7:

:math:`{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{\dot{\sigma }}_{\text{ij}}={(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}\dot{\epsilon }-\dot{\Lambda }{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}` eq 5.2-7

**The two previous equations allow us to find:**


.. _RefEquation 5.2-8:

:math:`{H}_{p}\dot{\Lambda }={(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}-\dot{\Lambda }{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}` eq 5.2-8

**Hence the expression for the plastic multiplier:**


.. _RefEquation 5.2-9:

:math:`\dot{\Lambda }=\frac{{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}}{{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}+{H}_{p}}` eq 5.2-9

**Let H be the elastoplastic module defined as:**


.. _RefEquation 5.2-10:

:math:`H={(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}+{H}_{p}` eq 5.2-10

The plastic multiplier is written as:


.. _RefEquation 5.2-11:

:math:`\dot{\Lambda }=\frac{{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}}{H}` eq 5.2-11

Replacing :math:`\dot{\Lambda }` with its expression in equation [:ref:`éq 5.2-4 <éq 5.2-4>`], we get:

:math:`{\dot{\sigma }}_{\text{ij}}={D}_{\text{ijkl}}^{e}{\dot{\epsilon }}_{\text{kl}}-\frac{1}{H}\left[{(\frac{\partial f}{\partial \sigma })}_{\text{mn}}{D}_{\text{mnop}}^{e}{\dot{\epsilon }}_{\text{op}}\right]\text{.}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}` eq 5.2-12

The elastoplastic operator :math:`{D}^{\text{elp}}={D}^{e}-{D}^{p}` is therefore deduced from this:

:math:`{\dot{\sigma }}_{\text{ij}}=\underset{{D}^{\text{elp}}}{\underset{\underbrace{}}{\left[{D}_{\text{ijkl}}^{e}-\frac{1}{H}{(\frac{\partial f}{\partial \sigma })}_{\text{op}}{D}_{\text{ijop}}^{e}{D}_{\text{mnkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{mn}}\right]}}{\dot{\epsilon }}_{\text{kl}}` eq 5.2-13

with,


.. _RefEquation 5.2-14:

:math:`{D}_{\text{ijkl}}^{p}=\frac{1}{H}{(\frac{\partial f}{\partial \sigma })}_{\text{op}}{D}_{\text{ijop}}^{e}{D}_{\text{mnkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{mn}}` eq 5.2-14

Calculation of :math:`H`:


.. _RefEquation 5.2-15:

:math:`{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}=-\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}}){\delta }_{\text{ij}}+{\mathrm{3s}}_{\text{ij}}`, eq 5.2-15

Which is written in vector notation:

:math:`\left[\begin{array}{c}-\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{3s}}_{\text{11}}\\ -\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{3s}}_{\text{22}}\\ -\frac{2}{3}{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{3s}}_{\text{33}}\\ 3\sqrt{2}{s}_{\text{12}}\\ 3\sqrt{2}{s}_{\text{23}}\\ 3\sqrt{2}{s}_{\text{31}}\end{array}\right]` eq 5.2-16

Hence the expression of:

:math:`{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}:\left[\begin{array}{c}-{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{6\mu s}}_{\text{11}}\\ -{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{6\mu s}}_{\text{22}}\\ -{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\mathrm{6\mu s}}_{\text{33}}\\ \mathrm{6\mu }\sqrt{2}{s}_{\text{12}}\\ \mathrm{6\mu }\sqrt{2}{s}_{\text{23}}\\ \mathrm{6\mu }\sqrt{2}{s}_{\text{31}}\end{array}\right]` eq 5.2-17

and

:math:`{(\frac{\partial f}{\partial \sigma })}_{\text{ij}}{D}_{\text{ijkl}}^{e}{(\frac{\partial f}{\partial \sigma })}_{\text{kl}}={\mathrm{4k}}_{0}{M}^{4}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}}{)}^{2}+\text{12}{\mathrm{\mu Q}}^{2}` **where**

:math:`\text{12}{\mathrm{\mu Q}}^{2}=\text{18}\mu \text{tr}(s\text{.}s)` **eq 5.2-18**

From the equations [:ref:`éq 3.5-7 <éq 3.5-7>`] and [:ref:`éq 5.2-18 <éq 5.2-18>`], we can deduce the expression for :math:`H`:


.. _RefEquation 5.2-19:

:math:`H={\mathrm{4M}}^{4}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}})(k{}_{0}\text{}(P-{P}_{\text{trac}}-{P}_{\text{cr}})+{\text{kP}}_{\text{cr}})+\text{12}{\mathrm{\mu Q}}^{2}` eq 5.2-19

By posing:


.. _RefEquation 5.2-20:

:math:`{A}_{\text{ij}}=-{\mathrm{2k}}_{0}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{P}_{\text{cr}}){\delta }_{\text{ij}}+{\mathrm{6\mu s}}_{\text{ij}}`, eq 5.2-20

we can write the following symmetric plastic matrix:

:math:`{D}^{p}=\frac{1}{H}\left[\begin{array}{cccccc}{A}_{\text{11}}^{2}& {A}_{\text{11}}{A}_{\text{22}}& {A}_{\text{11}}{A}_{\text{33}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{11}}{s}_{\text{12}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{11}}{s}_{\text{23}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{11}}{s}_{\text{31}}\\ \text{.}& {A}_{\text{22}}^{2}& {A}_{\text{22}}{A}_{\text{33}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{22}}{s}_{\text{12}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{22}}{s}_{\text{23}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{22}}{s}_{\text{31}}\\ \text{.}& \text{.}& {A}_{\text{33}}^{2}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{33}}{s}_{\text{12}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{33}}{s}_{\text{23}}& 6\sqrt{2}{\mathrm{\mu A}}_{\text{33}}{s}_{\text{31}}\\ \text{.}& \text{.}& \text{.}& \text{36}{\mu }^{2}{s}_{\text{12}}^{2}& \text{36}{\mu }^{2}{s}_{\text{12}}{s}_{\text{23}}& \text{36}{\mu }^{2}{s}_{\text{12}}{s}_{\text{31}}\\ \text{.}& \text{.}& \text{.}& \text{.}& \text{36}{\mu }^{2}{s}_{\text{23}}^{2}& \text{36}{\mu }^{2}{s}_{\text{23}}{s}_{\text{31}}\\ \text{.}& \text{.}& \text{.}& \text{.}& \text{.}& \text{36}{\mu }^{2}{s}_{\text{31}}^{2}\end{array}\right]` eq 5.2-21




Implicit tangent operator. Option FULL_MECA
------------------------------------------------

The coherent tangent operator in option FULL_MECA is calculated as the speed tangent operator for the current stress state.

However, theoretical elements for calculating it are given in the appendix, in paragraph :ref:`8 <Ref191144469>`. Note that the equations in the appendix assume that the criterion passes through a zero stress state, :math:`{P}_{\text{trac}}` and :math:`{K}_{\text{cam}}` were not yet introduced. It is necessary to remember to take them into account and if necessary to reactivate these equations for the coherent tangent operator.