Integrating behavioral relationships
=========================================

To achieve numerical integration of the law of behavior, time discretization is performed and an implicit Euler scheme is adopted, considered appropriate for elastoplastic behavior relationships. From now on, the following notations will be used: :math:`{A}^{\text{-}}`, :math:`A` and :math:`\Delta A` represent respectively the values of a quantity at the beginning and at the end of the time step in question as well as its increment during the step. The problem is then as follows: knowing the state at time :math:`{t}^{-}` as well as the deformation increments :math:`\Delta \varepsilon` (from the prediction phase (cf. reference documentation from STAT_NON_LINE [:ref:`R5.03.01 <R5.03.01>`])) and temperature :math:`\Delta T`, determine the state of the internal variables at time :math:`t` as well as the constraints :math:`\sigma`.

Variations in characteristics with respect to temperature are taken into account, noting that:


.. _Ref457380216:



.. _RefEquation 2.2-1:

:math:`{\sigma }^{H}\mathrm{=}\frac{K}{{K}^{\mathrm{-}}}{\sigma }^{{H}^{\mathrm{-}}}+K\text{tr}(\Delta \varepsilon \mathrm{-}\Delta {\varepsilon }^{\text{th}})` eq 2.2-1


.. _RefEquation 2.2-2:

:math:`\tilde{\sigma }\mathrm{=}\frac{\mu }{{\mu }^{\mathrm{-}}}{\tilde{\sigma }}^{\mathrm{-}}+2\mu (\Delta \tilde{\varepsilon }\mathrm{-}\Delta {\varepsilon }^{p})\mathrm{=}{\tilde{\sigma }}^{\varepsilon }\mathrm{-}2\mu \Delta {\varepsilon }^{p}` eq 2.2-2

with

:math:`{\tilde{\sigma }}^{\varepsilon }\mathrm{=}\frac{\mu }{{\mu }^{\mathrm{-}}}{\tilde{\sigma }}^{\mathrm{-}}+2\mu \Delta \tilde{\varepsilon }`

Looking at equation [:ref:`éq 2.2-1 <éq 2.2-1>`], it can be seen that the hydrostatic behavior is purely elastic if :math:`K` is constant. Only the treatment of the deviatoric component is delicate.

In the absence of a viscous term, the discretized coherence relationship is:

Elastic diet: :math:`F\mathrm{\le }0` and :math:`\Delta p\mathrm{=}0`

Plastic diet: :math:`F\mathrm{=}0` and :math:`\Delta p\mathrm{\ge }0`

On the other hand, in the presence of viscosity, the coherence condition is replaced by the equation [:ref:`éq2.1‑10 <éq2.1‑10>`] which, discretized, is written:

:math:`\frac{\Delta p}{\Delta t}\mathrm{=}{(\frac{\mathrm{\langle }F\mathrm{\rangle }}{K})}^{N}\mathrm{\iff }\mathrm{\langle }F\mathrm{\rangle }\mathrm{=}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}`

In other words, by asking:

:math:`\tilde{F}\mathrm{=}F\mathrm{-}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}`

the cumulative viscoplastic deformation increment is determined by:


.. _RefEquation 2.2-3:

:math:`\begin{array}{cc}\text{Régime élastique :}& \tilde{F}\mathrm{\le }0\text{et}\Delta p\mathrm{=}0\\ \text{Régime viscoplastique :}& \tilde{F}\mathrm{=}0\text{et}\Delta p\mathrm{\ge }0\end{array}` eq 2.2-3

Finally, by adopting an implicit discretization, the only difference between the laws of plastic and viscoplastic behavior lies in the form of the load function :math:`F`: we observe a complementary term in case of viscosity. In fact, incremental plasticity appears to be the borderline case for incremental viscoplasticity when :math:`K` tends to zero. This convergence has already been described by J.L.Chaboche and G. Cailletaud in [:ref:`bib3 <bib3>`].

In the rest of this paragraph, we will therefore detail the integration of the viscoplastic law. To find the case of plastic behavior, simply take :math:`K\mathrm{=}0` from the equations below (remember that the user must remove the LEMAITRE or LEMAITRE_FO keyword from the DEFI_MATERIAU command to enter this case).


.. _Ref457380797:

:math:`\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2}\mathrm{=}{\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}({C}_{1}\Delta {\alpha }_{1}+{C}_{2}\Delta {\alpha }_{2})`

The flow equations [:ref:`éq 2.1-6 <éq 2.1-6>`] and [:ref:`éq 2.1-7 <éq 2.1-7>`], once discretized, and the coherence condition [:ref:`éq 2.2-3 <éq 2.2-3>`] can be written (noting that :math:`p\mathrm{=}\lambda`):


.. _RefEquation 2.2-4:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}\frac{3}{2}\Delta p\frac{{\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2}}{{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2})}_{\text{eq}}}` eq 2.2-4


.. _RefEquation 2.2-5:

:math:`\tilde{F}\mathrm{\le }0\Delta p\mathrm{\ge }0\tilde{F}\Delta p\mathrm{=}0` eq 2.2-5

The treatment of the coherence condition (previous equation) is conventional. We start with an elastic test (:math:`\Delta p\mathrm{=}0`) which is indeed the solution if the plasticity criterion is not exceeded, that is to say if:


.. _RefEquation 2.2-6:

:math:`{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{C}_{1}({p}^{\mathrm{-}}){\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}({p}^{\mathrm{-}}){\alpha }_{2}^{\mathrm{-}})}_{\text{eq}}\mathrm{-}R({p}^{\mathrm{-}})<0` eq 2.2-6

Otherwise, the solution is plastic (:math:`\Delta p>0`) and the consistency condition is reduced to :math:`\tilde{F}\mathrm{=}0`. To solve it, we show that we can reduce ourselves to a scalar problem by expressing :math:`\Delta {\varepsilon }^{p}` and :math:`\Delta {\alpha }_{1},\Delta {\alpha }_{2}` in terms of :math:`\Delta p`. By grouping together the equations of the problem resulting from the implicit discretization, we obtain the system of equations:


.. _RefEquation 2.2-7:

:math:`{({\tilde{\sigma }}^{{e}_{}}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2})}_{\text{eq}}\mathrm{=}R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}` eq 2.2-7


.. _RefEquation 2.2-8:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}\frac{3}{2}\Delta p\frac{{\tilde{\sigma }}^{{e}_{}}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2}}{R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}}` eq 2.2-8


.. _RefEquation 2.2-9:

:math:`\begin{array}{c}\Delta {\alpha }_{1}\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{1}{\alpha }_{1}\Delta p\\ \Delta {\alpha }_{2}\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{2}{\alpha }_{2}\Delta p\end{array}` eq 2.2-9

In this writing, it should be noted that :math:`p\mathrm{=}{p}^{\mathrm{-}}+\Delta p` and :math:`{\alpha }_{i}\mathrm{=}{\alpha }_{{i}^{\mathrm{-}}}+\Delta {\alpha }_{i}` and that :math:`{C}_{i},{\gamma }_{i}` are functions of :math:`p`. By considering the last three equations, this linear system in :math:`\Delta {\varepsilon }^{p}` and :math:`\Delta {\alpha }_{i}` can be solved to express these quantities in terms of :math:`\Delta p`. In fact, it is equivalent to:


.. _Ref457380206:



.. _RefEquation 2.2-10:

:math:`\Delta {\varepsilon }^{p}(R(p)+3\mu \Delta p+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N})\mathrm{=}\Delta p(\frac{3}{2}{\tilde{\sigma }}^{e}\mathrm{-}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}{C}_{2}\Delta {\alpha }_{2})` eq 2.2-10


.. _RefEquation 2.2-11:

:math:`\begin{array}{c}\Delta {\alpha }_{1}(1+{\gamma }_{1}\Delta p)\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{1}{\alpha }_{1}^{\mathrm{-}}\Delta p\\ \Delta {\alpha }_{2}(1+{\gamma }_{2}\Delta p)\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{2}{\alpha }_{2}^{\mathrm{-}}\mathit{Dp}\end{array}` eq 2.2-11

By calculating :math:`{C}_{1}\Delta {\alpha }_{1}` and :math:`{C}_{2}\Delta {\alpha }_{2}` and substituting them in the expression for :math:`\Delta {\varepsilon }^{p}`, we get an expression for :math:`\Delta {\varepsilon }^{p}` in terms of :math:`\Delta p` only:


.. _RefEquation 2.2-12:

:math:`\begin{array}{c}{C}_{1}\Delta {\alpha }_{1}\mathrm{=}(\frac{{C}_{1}}{1+{\gamma }_{1}\Delta p})\Delta {\varepsilon }^{p}\mathrm{-}(\frac{{C}_{1}{\gamma }_{1}{\alpha }_{1}^{\mathrm{-}}\Delta p}{1+{\gamma }_{1}\Delta p})\mathrm{=}{M}_{1}(p)\Delta {\varepsilon }^{p}\mathrm{-}{M}_{1}(p){\gamma }_{1}\Delta p{\alpha }_{1}^{\mathrm{-}}\\ {C}_{2}\Delta {\alpha }_{2}\mathrm{=}(\frac{{C}_{2}}{1+{\gamma }_{2}\Delta p})\Delta {\varepsilon }^{p}\mathrm{-}(\frac{{C}_{2}{\gamma }_{2}{\alpha }_{2}^{\mathrm{-}}\Delta p}{1+{\gamma }_{2}\Delta p})\mathrm{=}{M}_{2}(p)\Delta {\varepsilon }^{p}\mathrm{-}{M}_{2}(p){\gamma }_{2}\Delta p{\alpha }_{2}^{\mathrm{-}}\\ \text{avec}{M}_{i}(p)\mathrm{=}\frac{{C}_{i}(p)}{1+{\gamma }_{i}(p)\Delta p}\end{array}` eq 2.2-12

By putting this expression into the expression for :math:`\Delta {\varepsilon }^{p}` we find:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}\frac{1}{(R(p)+(3\mu +{M}_{1}+{M}_{2})\Delta p+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N})}(\frac{3}{2}\Delta p{\tilde{\sigma }}^{e}\mathrm{-}\Delta p(({C}_{1}\mathrm{-}{M}_{1}{\gamma }_{1}\Delta p){\alpha }_{1}^{\mathrm{-}}+({C}_{2}\mathrm{-}{M}_{2}{\gamma }_{2}\Delta p){\alpha }_{2}^{\mathrm{-}}))`

which is simplified to:


.. _RefEquation 2.2-13:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}\frac{1}{D(p)}(\frac{3}{2}\Delta p{\tilde{\sigma }}^{e}\mathrm{-}\Delta p({M}_{1}{\alpha }_{1}^{\mathrm{-}}+{M}_{2}{\alpha }_{2}^{\mathrm{-}}))` eq 2.2-13

with:

:math:`D(p)\mathrm{=}R(p)+(3\mu +{M}_{1}(p)+{M}_{2}(p))\Delta p+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}`

Now all that remains is to replace :math:`\Delta {\varepsilon }^{p}` in the expressions in :math:`{C}_{1}\Delta {\alpha }_{1}` and :math:`{C}_{2}\Delta {\alpha }_{2}` to express this term in terms of :math:`\Delta p` by:

:math:`\begin{array}{c}{C}_{1}\Delta {\alpha }_{1}\mathrm{=}\frac{{M}_{1}}{D}(\frac{3}{2}\Delta p{\tilde{\sigma }}^{e}\mathrm{-}\Delta p({M}_{1}{\alpha }_{1}^{\mathrm{-}}+{M}_{2}{\alpha }_{2}^{\mathrm{-}}))\mathrm{-}{M}_{1}{\gamma }_{1}\Delta p{\alpha }_{1}^{\mathrm{-}}\\ {C}_{2}\Delta {\alpha }_{2}\mathrm{=}\frac{{M}_{2}}{D}(\frac{3}{2}\Delta p{\tilde{\sigma }}^{e}\mathrm{-}\Delta p({M}_{1}{\alpha }_{1}^{\mathrm{-}}+{M}_{2}{\alpha }_{2}^{\mathrm{-}}))\mathrm{-}{M}_{2}{\gamma }_{2}\Delta p{\alpha }_{2}^{\mathrm{-}}\end{array}`

then to substitute the expression obtained as well as :math:`\Delta {\varepsilon }^{p}` as a function of :math:`\Delta p` in the equation :math:`\tilde{F}\mathrm{=}0`, and we obtain a scalar equation in :math:`\Delta p` to solve, namely:

:math:`\tilde{F}(p)\mathrm{=}{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2})}_{\text{eq}}\mathrm{-}R(p)\mathrm{-}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\mathrm{=}0`

which is simplified to:


.. _RefEquation 2.2-14:

:math:`\tilde{F}(p)\mathrm{=}\frac{R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}}{D(p)}{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\mathrm{-}})}_{\text{eq}}\mathrm{-}R(p)\mathrm{-}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\mathrm{=}0` eq 2.2-14

This scalar equation in :math:`\Delta p` is solved numerically, by a method of finding function zeros (secant method that is briefly described in Appendix 2).

It is standardized as follows:


.. _RefEquation 2.2-15:

:math:`\tilde{\stackrel{ˆ}{F}}(p)\mathrm{=}1\mathrm{-}\frac{D(p)}{{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\mathrm{-}})}_{\text{eq}}}\mathrm{=}0` eq 2.2-15

Once :math:`\Delta p` is determined, :math:`\Delta {\varepsilon }^{p}` can be calculated using equation [:ref:`éq 2.2-13 <éq 2.2-13>`] and then :math:`\Delta {\alpha }_{1}` and :math:`\Delta {\alpha }_{2}` using equations [:ref:`éq 2.2-11 <éq 2.2-11>`]. All that remains is to calculate the stress tensor, using the equations [:ref:`éq 2.2-1 <éq 2.2-1>`] and [:ref:`éq2.2-2 <éq2.2-2>`], and to update the internal variables :math:`{\alpha }_{1}` and :math:`{\alpha }_{2}`.

**Notes:**


* *an interesting edge case (for the validation of this model) is presented by asking* :math:`{\gamma }_{i}\mathrm{=}0`\ *. We then find ourselves exactly in the situation of linear kinematic work hardening (si* :math:`R(p)\mathrm{=}{\sigma }_{y}`, [:ref:`R5.03.02 <R5.03.02>`]) or mixed work hardening for any :math:`R(p)` (cf. [:ref:`R5.03.16 <R5.03.16>`]),


* these models are also available in plane constraints, by a global method (static condensation due to R. de Borst) [:ref:`R5.03.03 <R5.03.03>`].




Integration of terms taking into account non-radiality
----------------------------------------------------------

Discretization leads to: :math:`{\Delta \alpha }_{i}\mathrm{=}{\Delta \varepsilon }^{p}\mathrm{-}{\gamma }_{i}\Delta p\left[{\delta }_{i}({\alpha }_{i}^{\text{-}}+\Delta {\alpha }_{i})+(1\mathrm{-}{\delta }_{i})(({\alpha }_{i}^{\text{-}}+\Delta {\alpha }_{i})\mathrm{:}n)n\right]`

Let's calculate :math:`\Delta {\alpha }_{i}\mathrm{:}n\mathrm{=}\sqrt{\frac{3}{2}}\Delta p\mathrm{-}{\gamma }_{i}\Delta p({\delta }_{i}\sqrt{\frac{3}{2}}{\beta }_{i}+{\delta }_{i}\Delta {\alpha }_{i}\mathrm{:}n+(1\mathrm{-}{\delta }_{i})\sqrt{\frac{3}{2}}{\beta }_{i}+(1\mathrm{-}{\delta }_{i})(\Delta {\alpha }_{i}\mathrm{:}n))`

by asking :math:`{\alpha }_{i}^{\text{-}}\mathrm{:}n\mathrm{=}\sqrt{\frac{3}{2}}{\beta }_{i}`. So we can express :math:`\Delta {\alpha }_{i}\mathrm{:}n` in terms of :math:`\Delta p` and :math:`{\beta }_{i}`

:math:`\Delta {\alpha }_{i}\mathrm{:}n(1+{\gamma }_{i}\Delta p)\mathrm{=}\sqrt{\frac{3}{2}}\Delta p(1\mathrm{-}{\gamma }_{i}{\beta }_{i})` or :math:`\Delta {\alpha }_{i}\mathrm{:}n\mathrm{=}\frac{\sqrt{\frac{3}{2}}\Delta p(1\mathrm{-}{\gamma }_{i}{\beta }_{i})}{(1+{\gamma }_{i}\Delta p)}`

We can therefore express :math:`\Delta {\alpha }_{i}` only as a function of :math:`\Delta p` and :math:`{\beta }_{i}\mathrm{=}\sqrt{\frac{2}{3}}{\alpha }_{i}^{\text{-}}\mathrm{:}n` and propagate these modifications in the resolution method used previously:

:math:`\Delta {\alpha }_{i}(1+{\gamma }_{i}{\delta }_{i}\Delta p)\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{i}\Delta p{\delta }_{i}{\alpha }_{i}^{\text{-}}\mathrm{-}{\gamma }_{i}\Delta p(1\mathrm{-}{\delta }_{i})({\alpha }_{i}^{\text{-}}\mathrm{:}n)n\mathrm{-}{\gamma }_{i}\Delta p(1\mathrm{-}{\delta }_{i})(\Delta {\alpha }_{i}\mathrm{:}n)n`

Using the expression for :math:`\Delta {\alpha }_{i}\mathrm{:}n` in terms of :math:`\Delta p` and :math:`{\beta }_{i}`,

:math:`\Delta {\alpha }_{i}(1+{\gamma }_{i}{\delta }_{i}\Delta p)\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{i}\Delta p{\delta }_{i}{\alpha }_{i}^{\text{-}}\mathrm{-}{\gamma }_{i}\Delta p(1\mathrm{-}{\delta }_{i})\sqrt{\frac{3}{2}}{\beta }_{i}n\mathrm{-}{\gamma }_{i}\Delta p(1\mathrm{-}{\delta }_{i})\frac{\sqrt{\frac{3}{2}}\Delta p(1\mathrm{-}{\gamma }_{i}{\beta }_{i})}{(1+{\gamma }_{i}\Delta p)}n`

:math:`\Delta {\alpha }_{i}(1+{\gamma }_{i}{\delta }_{i}\Delta p)\mathrm{=}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{i}\Delta p{\delta }_{i}{\alpha }_{i}^{\text{-}}\mathrm{-}{\gamma }_{i}(1\mathrm{-}{\delta }_{i})\frac{{\beta }_{i}+\Delta p}{1+{\gamma }_{i}\Delta p}\Delta {\varepsilon }^{p}`

:math:`\Delta {\alpha }_{i}(1+{\gamma }_{i}{\delta }_{i}\Delta p)\mathrm{=}\Delta {\varepsilon }^{p}{N}_{i}(\Delta p,{\beta }_{i})\mathrm{-}{\gamma }_{i}\Delta p{\delta }_{i}{\alpha }_{i}^{\text{-}}` with

:math:`{N}_{i}(\Delta p,{\beta }_{i})\mathrm{=}\frac{1+{\gamma }_{i}\Delta p{\delta }_{i}\mathrm{-}{\gamma }_{i}(1\mathrm{-}{\delta }_{i}){\beta }_{i}}{1+{\gamma }_{i}\Delta p}`

Again, we can verify that if :math:`{\delta }_{i}\mathrm{=}1`, we find the equations without a non-radiality effect.

To continue solving, we need to calculate:

:math:`{C}_{i}\Delta {\alpha }_{i}\mathrm{=}{M}_{i}{N}_{i}\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{i}\Delta p{\delta }_{i}{M}_{i}{\alpha }_{i}^{\text{-}}` with :math:`{M}_{i}\mathrm{=}\frac{{C}_{i}}{(1+{\gamma }_{i}{\delta }_{i}\Delta p)}`

So much so that the calculation of the increase in plastic deformation is similar to the classical case:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}\sqrt{\frac{3}{2}}\Delta pn` with :math:`n\mathrm{=}\sqrt{\frac{3}{2}}\frac{\tilde{\sigma }\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}}{{(\tilde{\sigma }\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2})}_{\mathit{eq}}}`.

Using the expressions calculated previously as well as the criterion expression:

:math:`{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}2\mu \Delta {\varepsilon }^{p}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2})}_{\text{eq}}\mathrm{=}R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}` he comes:

:math:`\Delta {\varepsilon }^{p}(R(p)+3K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}+\Delta p(3\mu +{M}_{1}{N}_{1}+{M}_{2}{N}_{2}))\mathrm{=}\frac{3}{2}\Delta p({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\text{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\text{-}})`

So :math:`n\mathrm{=}\sqrt{\frac{3}{2}}\frac{{\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\text{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\text{-}}}{D}` with

:math:`D(\Delta p;{\beta }_{1};{\beta }_{2})\mathrm{=}R(p)+K{(\frac{\Delta p}{\Delta t})}^{\text{1/N}}+\Delta p(3\mu +{M}_{1}{N}_{1}+{M}_{2}{N}_{2})`

**Note**: again, we can check only if we don't take into account the non-radial effect, :math:`{\delta }_{i}\mathrm{=}1`, which results in :math:`N\mathrm{=}1`. We find the classic expression for normal :math:`n`.

In this case; there are 3 scalar unknowns: :math:`\Delta p`, :math:`{\beta }_{1}`, :math:`{\beta }_{2}`. In fact, it's possible to express :math:`{\beta }_{1}` and :math:`{\beta }_{2}` in terms of :math:`\Delta p` by noting that:

:math:`n\mathrm{=}\sqrt{(\frac{3}{2})}\frac{\tilde{{\sigma }^{e}}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\text{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\text{-}}}{{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\text{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\text{-}})}_{\mathit{eq}}}`. We can therefore determine :math:`n` based on :math:`\Delta p` only, then directly calculate :math:`{\beta }_{i}\mathrm{=}\sqrt{\frac{2}{3}}{\alpha }_{i}^{\text{-}}\mathrm{:}n`, which then become explicit functions of :math:`\Delta p`. To solve, simply replace the above expressions in the criterion (which is the same as writing :math:`n\mathrm{:}n\mathrm{=}1`):

:math:`\tilde{\stackrel{ˆ}{F}}(p)\mathrm{=}{(\tilde{{\sigma }^{e}}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\text{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\text{-}})}_{\mathit{eq}}\mathrm{-}D(\Delta p;{\beta }_{1}(\Delta p);{\beta }_{2}(\Delta p))\mathrm{=}0`




Integration of the memory effect
---------------------------------

In the case of the memory effect, function :math:`R(p)` is no longer known explicitly, but through the system of equations:

1 eq: :math:`f({\varepsilon }^{p},\xi ,q)\mathrm{=}\frac{2}{3}{J}_{2}({\varepsilon }^{p}\mathrm{-}\xi )\mathrm{-}q\mathrm{=}\frac{2}{3}\sqrt{\frac{3}{2}({\varepsilon }^{p}\mathrm{-}\xi )\mathrm{:}({\varepsilon }^{p}\mathrm{-}\xi )}\mathrm{-}q\mathrm{\le }0`

6 eq: :math:`\Delta \xi \mathrm{=}(1\mathrm{-}\eta )H(F)\mathrm{\langle }{\text{n:n}}^{\text{*}}\mathrm{\rangle }\Delta p{n}^{\text{*}}\mathrm{=}(1\mathrm{-}\eta )\frac{\Delta q}{\eta }{n}^{\text{*}}`

With

:math:`\Delta R\mathrm{=}b(Q\mathrm{-}R)\Delta p` :math:`Q\mathrm{=}Q{}_{0}\text{}+({Q}_{m}\mathrm{-}{Q}_{0})(1\mathrm{-}{e}^{\mathrm{-}2\mu ({q}^{\mathrm{-}}+\Delta q)})` :math:`{n}^{\text{*}}\mathrm{=}\frac{3}{2}\frac{{\varepsilon }^{p}\mathrm{-}\xi }{{J}_{2}({\varepsilon }^{p}\mathrm{-}\xi )}`

Knowing :math:`\Delta p`, we start by calculating :math:`f({\varepsilon }^{p},{\xi }^{\mathrm{-}},{q}^{\mathrm{-}})`.

If this quantity is negative, then the system solution managing the memory effect is: :math:`\mathrm{\Delta q}=\mathrm{0,}\mathrm{\Delta \xi }=0`.

Otherwise, knowing :math:`\Delta p`, we must find :math:`\Delta q` and :math:`\Delta \xi` such as:

:math:`f({\varepsilon }^{p},\xi ,q)\mathrm{=}\frac{2}{3}{J}_{2}({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}}\mathrm{-}\Delta \xi )\mathrm{-}{q}^{\mathrm{-}}\mathrm{-}\Delta q\mathrm{=}0`

:math:`\Delta \xi \mathrm{=}\frac{(1\mathrm{-}\eta )}{\eta }\Delta q{n}^{\text{*}}\mathrm{=}\frac{(1\mathrm{-}\eta )}{\eta }\Delta q\frac{3}{2}\frac{{\varepsilon }^{p\mathrm{-}}+\Delta {\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}}\mathrm{-}\Delta \xi }{\frac{3}{2}({q}^{\mathrm{-}}+\Delta q)}`

*Because :math:`\Delta {\varepsilon }^{\mathrm{p}}\mathrm{=}\frac{1}{D(p)}(\frac{3}{2}\Delta p{\tilde{\sigma }}^{e}\mathrm{-}\Delta p({M}_{1}{\alpha }_{1}^{\mathrm{-}}+{M}_{2}{\alpha }_{2}^{\mathrm{-}}))` can be explicitly calculated from :math:`\Delta p`.

There is left:

:math:`\Delta \xi (1+\frac{(1\mathrm{-}\eta )\Delta q}{\eta ({q}^{\mathrm{-}}+\Delta q)})\mathrm{=}\frac{(1\mathrm{-}\eta )}{\eta }\Delta q\frac{{\varepsilon }^{p\mathrm{-}}+\Delta {\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}}}{({q}^{\mathrm{-}}+\Delta q)}\mathrm{\Rightarrow }` :math:`\Delta \xi (\eta {q}^{\mathrm{-}}+\Delta q)\mathrm{=}(1\mathrm{-}\eta )\Delta q({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}})\mathrm{\Rightarrow }\Delta \xi \mathrm{=}\frac{(1\mathrm{-}\eta )\Delta q({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}})}{\eta {q}^{\mathrm{-}}+\Delta q}`

by entering into the threshold area equation: :math:`f({\varepsilon }^{p},\xi ,q)\mathrm{=}0`

:math:`\begin{array}{c}\frac{2}{3}{J}_{2}({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}}\mathrm{-}\Delta \xi )\mathrm{-}{q}^{\mathrm{-}}\mathrm{-}\Delta q\mathrm{=}0\mathrm{=}\frac{2}{3}{J}_{2}({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}})\mathrm{\mid }(1\mathrm{-}\frac{(1\mathrm{-}\eta )\Delta q}{\eta {q}^{\mathrm{-}}+\Delta q})\mathrm{\mid }\mathrm{-}{q}^{\mathrm{-}}\mathrm{-}\Delta q\mathrm{=}0\\ \mathrm{\iff }\frac{2}{3}{J}_{2}({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}})\mathrm{\mid }\eta ({q}^{\mathrm{-}}+\Delta q)\mathrm{\mid }\mathrm{-}({q}^{\mathrm{-}}+\Delta q)({\mathit{\eta q}}^{\mathrm{-}}+\Delta q)\mathrm{=}0\text{si}\eta {q}^{\mathrm{-}}+\Delta q>0\end{array}`

which makes it possible to explicitly calculate :math:`\Delta q` from :math:`\Delta p`:

:math:`\Delta q\mathrm{=}\eta \frac{2}{3}{J}_{2}({\varepsilon }^{\mathrm{p}}\mathrm{-}{\xi }^{\mathrm{-}})\mathrm{-}\eta {q}^{\mathrm{-}}`

It then remains to modify the isotropic work hardening function by calculating:

:math:`Q\mathrm{=}Q{}_{0}\text{}+({Q}_{m}\mathrm{-}{Q}_{0})(1\mathrm{-}{e}^{\mathrm{-}2\mu ({q}^{\mathrm{-}}+\mathit{\Delta q})})` then :math:`\Delta R\mathrm{=}b(Q\mathrm{-}R)\Delta p`


.. _RefEquationuation scalaire en :math:`\Delta p` ( éq 2.2-14) en utilisant les expressions ci-dessus.:

We can therefore use the resolution of the scalar equation in :math:`\Delta p` (eq 2.2-14) using the expressions above.

**Notes:**


*In [:ref:`bib2 <bib2>`] we also find the expression*: *:* :math:`\text{dq}\mathrm{=}\eta H(f)\mathrm{\langle }{\text{n:n}}^{\text{*}}\mathrm{\rangle }\text{dp}`* . *

This last equation results from the expression for multiplier speed. In the implicit discretization carried out here, it is not used for the solution (since then the system would have more equations than unknowns). Moreover, the 3 equations given in [:ref:`bib2 <bib2>`] are redundant: in fact, knowing :math:`\Delta {\varepsilon }^{p}`, it is necessary to determine a tensor variable :math:`\Delta \xi` and a scalar variable. :math:`\Delta q` Now we have a tensor equation and two scalar equations.

This is due to the fact that equation :math:`\text{dq}\mathrm{=}\eta H(f)\mathrm{\langle }{\text{n:n}}^{\text{*}}\mathrm{\rangle }\text{dp}` comes from the coherence condition :math:`\mathit{df}\mathrm{=}0` (which is specified in [:ref:`bib2 <bib2>`]) but is not used for the implicit resolution of the problem. :math:`\text{df}({\varepsilon }^{p},\xi ,q)\mathrm{=}\frac{{\varepsilon }^{p}\mathrm{-}\xi }{{J}_{2}({\varepsilon }^{p}\mathrm{-}\xi )}d{\varepsilon }^{p}\mathrm{-}\frac{{\varepsilon }^{p}\mathrm{-}\xi }{{J}_{2}({\varepsilon }^{p}\mathrm{-}\xi )}d\xi \mathrm{-}\text{dq}\mathrm{=}n\mathrm{:}{n}^{\text{*}}\text{dp}\mathrm{-}{n}^{\text{*}}\mathrm{:}{n}^{\text{*}}\text{dq}\mathrm{-}\text{dq}\mathrm{=}n\mathrm{:}{n}^{\text{*}}\text{dp}\mathrm{-}2\text{dq}\mathrm{=}0`

It would be useful for an explicit resolution, by expressing the derivatives with respect to time of all the variables sought.


* an interesting criterion, given in [:ref:`bib2 <bib2>`], allows you to adjust the parameters of the memory effect. Indeed, considering a simple traction-compression loading, we must find :math:`q\mathrm{=}\frac{1}{2}\Delta {\varepsilon }^{{p}_{\text{max}}}` (by choosing :math:`\eta \mathrm{=}\frac{1}{2}`). For a material point under uniaxial load, the (uniform) fields have as components: :math:`\sigma \mathrm{=}\sigma (\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array})` :math:`{\varepsilon }^{p}\mathrm{=}p(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{-}\frac{1}{2}& 0\\ 0& 0& \mathrm{-}\frac{1}{2}\end{array})` In this case, during the first uniaxial load in the direction :math:`x`: :math:`\begin{array}{c}{\xi }^{\mathrm{-}}\mathrm{=}0\\ {q}^{\mathrm{-}}\mathrm{=}0\\ \Delta q\mathrm{=}\eta {\varepsilon }_{x}^{p}\end{array}` In this case, :math:`q\mathrm{=}\frac{1}{2}\Delta {\varepsilon }^{{p}_{\text{max}}}`, implies that :math:`\eta \mathrm{=}\frac{1}{2}` and :math:`\Delta \xi \mathrm{=}\frac{1}{2}({\varepsilon }^{p})` Moreover, in the case of a cycle of traction symmetric compression (in plastic deformation), we obtain, when of the first symmetric discharge (with :math:`\eta \mathrm{=}\frac{1}{2}`): :math:`\begin{array}{c}{\xi }^{\mathrm{-}}\mathrm{=}\frac{1}{2}{\varepsilon }^{{p}_{\text{max}}}\\ {q}^{\mathrm{-}}\mathrm{=}\frac{1}{2}{\varepsilon }_{\text{xx}}^{{p}_{\text{max}}}\\ \Delta q\mathrm{=}\eta (\frac{2}{3}{J}_{2}({\varepsilon }^{p})\mathrm{-}{q}^{\mathrm{-}})\mathrm{=}\eta (\mathrm{\mid }{\varepsilon }_{\text{xx}}^{{p}_{\text{min}}}\mathrm{-}{\xi }^{\mathrm{-}}\mathrm{\mid }\mathrm{-}\frac{1}{2}{\varepsilon }_{\text{xx}}^{{p}_{\text{max}}})\mathrm{=}\frac{1}{2}\mathrm{\mid }{\varepsilon }_{\text{xx}}^{{p}_{\text{min}}}\mathrm{\mid }\end{array}` :math:`q\mathrm{=}{q}^{\mathrm{-}}+\Delta q\mathrm{=}{\varepsilon }_{\text{xx}\text{max}}^{p}\mathrm{=}\frac{1}{2}\Delta {\varepsilon }_{\text{xx}}^{p}` :math:`\Delta \xi \mathrm{=}\frac{(1\mathrm{-}\eta )\Delta q({\varepsilon }^{p}\mathrm{-}{\xi }^{\mathrm{-}})}{\eta {q}^{\mathrm{-}}+\Delta q}\mathrm{=}\mathrm{-}\frac{1}{2}\Delta {\varepsilon }_{\text{xx}\text{max}}^{p}` :math:`\xi \mathrm{=}{\xi }^{0}+\Delta \xi \mathrm{=}0`: which corresponds well to the expected result (cf. [:ref:`bib2 <bib2>`]): domain :math:`F\mathrm{=}0` centered on the origin, and radius the half-amplitude of plastic deformation.




Calculation of tangent stiffness
------------------------------

In order to allow a resolution of the global problem (equilibrium equations) by a Newton method [:ref:`R5.03.01 <R5.03.01>`], it is necessary to determine the coherent tangent matrix of the incremental problem.

This matrix is classically composed of an elastic contribution and a plastic contribution:


.. _RefEquation 2.3-1:

:math:`\frac{\delta \sigma }{\delta \varepsilon }\mathrm{=}\frac{\delta {\sigma }^{e}}{\delta \varepsilon }\mathrm{-}2\mu \frac{\delta \Delta {\varepsilon }^{p}}{\delta \varepsilon }` eq 2.3-1

with :math:`{\sigma }^{e}\mathrm{=}\sigma +2\mu \Delta {\varepsilon }^{p}`, which in particular returns :math:`{\tilde{\sigma }}^{e}\mathrm{=}\frac{\mu }{{\mu }^{\mathrm{-}}}{\tilde{\sigma }}^{\mathrm{-}}+2\mu \Delta \tilde{\varepsilon }`

It is immediately deduced that under elastic regime (classical or pseudo-discharge), the tangent matrix is reduced to the elastic matrix:


.. _RefEquation 2.3-2:

:math:`\frac{\delta \sigma }{\delta \varepsilon }\mathrm{=}\frac{\delta {\sigma }^{e}}{\delta \varepsilon }` eq 2.3-2

To do this, we once again adopt the convention for writing symmetric tensors of order 2 in the form of vectors with 6 components. So for a :math:`a` tensor:

:math:`a\mathrm{=}{}^{t}\text{}\left[\begin{array}{ccc}{a}_{\text{xx}}& {a}_{\text{yy}}& {a}_{\text{zz}}\end{array}\begin{array}{ccc}\sqrt{2}{a}_{\text{xy}}& \sqrt{2}{a}_{\text{xz}}& \sqrt{2}{a}_{\text{yz}}\end{array}\right]` eq 2.3-3

If we also introduce the hydrostatic vector :math:`1` and the deviatoric projection matrix :math:`P`:

:math:`1\mathrm{=}{}^{t}\text{}\left[\begin{array}{ccc}1& 1& 1\end{array}\begin{array}{ccc}0& 0& 0\end{array}\right]` eq 2.3-4


.. _RefEquation 2.3-5:

:math:`P\mathrm{=}\text{Id}\mathrm{-}\frac{1}{3}1\mathrm{\otimes }1` eq 2.3-5

where :math:`\mathrm{\otimes }` is the tensor product

So the coherent tangent stiffness matrix is written for elastic behavior:


.. _RefEquation 2.3-6:

:math:`\frac{\mathrm{\partial }{\sigma }^{e}}{\mathrm{\partial }\Delta \varepsilon }\mathrm{=}K1\mathrm{\otimes }1+2\mu P` eq 2.3-6

On the other hand, under the plastic regime, the variation in plastic deformation is no longer zero.

We are drifting away from :math:`\tilde{{\sigma }^{e}}`, knowing that we have:


.. _RefEquation 2.3-7:

:math:`\frac{\delta \Delta {\varepsilon }^{p}}{\delta \varepsilon }\mathrm{=}\frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}\text{.}\frac{\delta {\tilde{\sigma }}^{e}}{\delta \varepsilon }\mathrm{=}2\mu \frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}\text{.}P` eq 2.3-7


.. csv-table::

    ":math:`s` ", "symmetric tensor space"
    ":math:`P` ", "spotlight on diverters"


To calculate :math:`\frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}`, we use the expression for :math:`\Delta {\varepsilon }^{p}` in terms of :math:`{\tilde{\sigma }}_{e}` and :math:`p`:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}\frac{1}{D(p)}(\frac{3}{2}\Delta p{\tilde{\sigma }}_{e}\mathrm{-}\Delta p({M}_{1}{\alpha }_{1}^{\mathrm{-}}+{M}_{2}{\alpha }_{2}^{\mathrm{-}}))`

which is written in the form:

:math:`\Delta {\varepsilon }^{p}\mathrm{=}A(p){\tilde{\sigma }}_{e}+{B}_{1}(p){\alpha }_{1}^{\mathrm{-}}+{B}_{2}(p){\alpha }_{2}^{\mathrm{-}}`

So:

:math:`\frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}A(p)\text{Id}+{\tilde{\sigma }}^{e}\mathrm{\otimes }\frac{\delta A(p)}{\delta {\tilde{\sigma }}^{e}}+\frac{\delta {B}_{1}(p)}{\delta {\tilde{\sigma }}^{e}}\mathrm{\otimes }{\alpha }_{1}^{\mathrm{-}}+\frac{\delta {B}_{2}(p)}{\delta {\tilde{\sigma }}^{e}}\mathrm{\otimes }{\alpha }_{2}^{\mathrm{-}}`

Quantities of type :math:`\frac{\delta A(p)}{\delta {\tilde{\sigma }}^{e}}` are calculated using: :math:`\frac{\delta A(p)}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\frac{\delta A(p)}{\delta p}\frac{\delta p}{\delta {\tilde{\sigma }}^{e}}`

Finally, all that's left is to calculate the variation of :math:`p`: :math:`\frac{\delta p}{\delta {\tilde{\sigma }}^{e}}`

To do this, we use: :math:`\tilde{F}(p,{\tilde{\sigma }}^{e})\mathrm{=}0`

:math:`\tilde{F}(p,{\tilde{\sigma }}_{e})\mathrm{=}\frac{R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}}{D(p)}{({\tilde{\sigma }}^{e}\mathrm{-}\frac{2}{3}{M}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{M}_{2}{\alpha }_{2}^{\mathrm{-}})}_{\text{eq}}\mathrm{-}R(p)\mathrm{-}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\mathrm{=}0`


.. _RefEquation 2.3-8:

:math:`{\tilde{F}}_{,p}(p,{\tilde{\sigma }}^{e})\delta p\mathrm{=}\mathrm{-}{\tilde{F}}_{,{\tilde{\sigma }}^{e}}(p,{\tilde{\sigma }}^{e})\delta {\tilde{\sigma }}^{e}\mathrm{\Rightarrow }\frac{\delta p}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\mathrm{-}\frac{{\tilde{F}}_{,{\tilde{\sigma }}^{e}}(p,{\tilde{\sigma }}^{e})}{{\tilde{F}}_{,p}(p,{\tilde{\sigma }}^{e})}` eq 2.3-8

Details of the calculations are given in Appendix 1.

The initial tangent matrix, used by option RIGI_MECA_TANG, is obtained by adopting the behavior of the previous step (elastic or plastic, signified by an internal variable :math:`\xi` equal to 0 or 1) and by making :math:`\Delta p` tend towards zero in the preceding equations.




Meaning of internal variables
------------------------------------

The internal variables of the two models at the Gauss points (VELGA) are:


* V1= :math:`p`: the cumulative plastic deformation (positive or zero)


* V2 = :math:`\xi`: equal to :math:`n` (number of internal iterations) if the Gauss point has plasticized during the increment or 0 otherwise.

For 3D modeling, the following internal variables are:


* For model VMIS/VISC_CIN1_CHAB


* V3 = :math:`{\alpha }_{1\text{xx}}`


* V4 = :math:`{\alpha }_{1\text{yy}}`


* V5 = :math:`{\alpha }_{1\text{zz}}`


* V6 = :math:`{\alpha }_{1\text{xy}}`


* V7 = :math:`{\alpha }_{1\text{xz}}`


* V8 = :math:`{\alpha }_{1\text{yz}}`


* For model VMIS/VISC_CIN2_CHAB


* V3 = :math:`{\alpha }_{1\text{xx}}`


* V4 = :math:`{\alpha }_{1\text{yy}}`


* V5 = :math:`{\alpha }_{1\text{zz}}`


* V6 = :math:`{\alpha }_{1\text{xy}}`


* V7 = :math:`{\alpha }_{1\text{xz}}`


* V8 = :math:`{\alpha }_{1\text{yz}}`


* V9 = :math:`{\alpha }_{2\text{xx}}`


* V10 = :math:`{\alpha }_{2\text{yy}}`


* V11 = :math:`{\alpha }_{2\text{zz}}`


* V12 = :math:`{\alpha }_{2\text{xy}}`


* V13 = :math:`{\alpha }_{2\text{xz}}`


* V14 = :math:`{\alpha }_{2\text{yz}}`


*

For C_ PLAN, D_ PLAN, and AXIS models:


* V7 = 0


* V8 = 0


* V13 = 0


* V14 = 0


* For model VMIS/VISC_CIN2_MEMO


* V3 = :math:`{\alpha }_{1\text{xx}}`


* V4 = :math:`{\alpha }_{1\text{yy}}`


* V5 = :math:`{\alpha }_{1\text{zz}}`


* V6 = :math:`{\alpha }_{1\text{xy}}`


* V7 = :math:`{\alpha }_{1\text{xz}}`


* V8 = :math:`{\alpha }_{1\text{yz}}`


* V9 = :math:`{\alpha }_{2\text{xx}}`


* V10 = :math:`{\alpha }_{2\text{yy}}`


* V11 = :math:`{\alpha }_{2\text{zz}}`


* V12 = :math:`{\alpha }_{2\text{xy}}`


* V13 = :math:`{\alpha }_{2\text{xz}}`


* V14 = :math:`{\alpha }_{2\text{yz}}`


* V15 = :math:`R(p)`


* V16 = :math:`q`


* V17 = :math:`{\xi }_{\text{xx}}`


* V18 = :math:`{\xi }_{\text{yy}}`


* V19 = :math:`{\xi }_{\text{zz}}`


* V20 = :math:`{\xi }_{\text{xy}}`


* V21 = :math:`{\xi }_{\text{xz}}`


* V22 = :math:`{\xi }_{\text{yz}}`


*


* V23 = :math:`{\epsilon }^{{p}_{\text{xx}}}`


* V24 = :math:`{\varepsilon }^{{p}_{\text{yy}}}`


* V25 = :math:`{\varepsilon }^{{p}_{\text{zz}}}`


* V26 = :math:`{\varepsilon }^{{p}_{\text{xy}}}`


* V27 = :math:`{\varepsilon }^{{p}_{\text{xz}}}`


* V28 = :math:`{\varepsilon }^{{p}_{\text{yz}}}`