Implementation in Code_Aster
============================




.. _Ref228616190:


Internal variables
------------------

A number of model "internal variables" in the sense of *Code_Aster* are created and stored. The plastic deformation tensor :math:`{\varepsilon }^{p}` is not stored because it is obtained by post-treatment calculation from the total stresses and deformations. The values of the internal variables at the Gauss points (VARI_ELGA) are for all models:


.. csv-table::

    "VARI_1 "," :math:`{r}_{1}^{m}` (factor for mobilizing the monotonic deviatory mechanism, :math:`\text{k=}1`)"
    "VARI_2 "," :math:`{r}_{2}^{m}` (factor for mobilizing the monotonic deviatory mechanism, :math:`\text{k=}2`)"
    "VARI_3 "," :math:`{r}_{3}^{m}` (factor for mobilizing the monotonic deviatory mechanism, :math:`\text{k=}3`)"
    "VARI_4 "," :math:`{r}_{4}^{m}` (factor for mobilizing the monotonic consolidation mechanism)"
    "VARI_5 "," :math:`{r}_{1}^{c}` (factor of mobilization of the cyclical deviatory mechanism, :math:`\text{k=}1`)"
    "VARI_6 "," :math:`{r}_{2}^{c}` (factor of mobilization of the cyclical deviatory mechanism, :math:`\text{k=}2`)"
    "VARI_7 "," :math:`{r}_{3}^{c}` (factor of mobilization of the cyclical deviatory mechanism, :math:`\text{k=}3`)"
    "VARI_8 "," :math:`{r}_{4}^{c}` (factor in mobilizing the cyclical consolidation mechanism)"
    "VARI_9à VARI_12 ", "memory variables :math:`{({X}_{(1)}^{H})}_{\text{1111}}`, :math:`\sqrt{2}{({X}_{(1)}^{H})}_{\text{1212}}`, :math:`-{({S}_{(1)H}^{c})}_{\text{1111}}/{\parallel {S}_{(1)H}^{c}\parallel }_{\text{VM}}^{\text{2D}}`,, :math:`-\sqrt{2}{({S}_{(1)H}^{c})}_{\text{1212}}/{\parallel {S}_{(1)H}^{c}\parallel }_{\text{VM}}^{\text{2D}}` (incoming normal), for the cyclical deviatory mechanism of the :math:`\text{k=}1` plan"
    "VARI_13à VARI_16 ", "same for the cyclical deviatory mechanism of the :math:`k=2` plan"
    "VARI_17à VARI_20 ", "same for the cyclical deviatory mechanism of the :math:`k=3` plan"
    "VARI_21 ", "discontinuous memory variable :math:`{p}_{H}/(d\text{.}{P}_{\text{c0}}\text{.}{e}^{-\beta {\varepsilon }_{\text{vH}}^{p}})` from the consolidation mechanism"
    "VARI_22 ", "discontinuous memory variable :math:`{p}_{H}^{c}/\mid {p}_{H}^{c}\mid` of normal to the load surface of the consolidation mechanism"
    "VARI_23 ", "volume plastic deformation :math:`{\varepsilon }_{v}^{p}`"
    "VARI_24à VARI_27 ", "indicators of activation (1) or not (0) of monotonic or cyclical mechanisms (-1)"
    "VARI_28à VARI_31 ", "indicators of activation (1) or not (0) of cyclic mechanisms"
    "VARI_32 ", "normalized density for second order work :math:`\dot{\sigma }\text{.}\dot{\varepsilon }` (obtained by discretization: :math:`\Delta \sigma \text{.}\Delta \varepsilon \mathrm{/}\mathrm{\parallel }\Delta \sigma \mathrm{\parallel }\text{.}\mathrm{\parallel }\Delta \varepsilon \mathrm{\parallel }`)"
    "VARI_33 ", "Determinant of the tangent matrix :math:`\frac{d\sigma }{d\varepsilon }`}"
    "VARI_34 ", "Solution error indicator :math:`\sum _{i=1}^{\mathit{ITER}\text{\_}\mathit{INTE}\text{\_}\mathit{PAS}}\sum _{k\in \mathit{Actif}}{f}_{k}^{m,c}\left({\mathrm{\sigma }}^{\text{+}},{\mathrm{ϵ}}_{v}^{p},{r}_{k}^{\text{+}}\right)`"
    "VARI_35 ", "Explicit solution error indicator :math:`\frac{{N}_{\text{échec}}}{\mathit{ITER}\text{\_}\mathit{INTE}\text{\_}\mathit{PAS}}` (in%)"
    "VARI_36à VARI_41 ", "vector coordinates of the point of tangency :math:`{({X}_{(1)}^{H})}_{\text{1111}}`, :math:`\sqrt{2}{({X}_{(1)}^{H})}_{\text{1212}}` of the "father" mechanism to the charge surface of the deviatory mechanism from the plane :math:`\text{k=}1`; vector coordinates of the incoming normal to the load surface of the "father" mechanism of the deviatory mechanism; radius of the deviatory threshold reached by the charge surface before the discharge of this deviatory mechanism"
    "VARI_42à VARI_47 ", "same for the cyclical deviatory mechanism of the :math:`k=2` plan"
    "VARI_48à VARI_50 ", "same for the cyclical deviatory mechanism of the :math:`k=3` plan"


The VARI_34 status indicator for active mechanisms after convergence is a byte (8 digits) obtained by binary system. Each digit is associated with a plastic mechanism, by assigning the value 1 for an active mechanism and the value 0 for an unsolicited mechanism.

This makes it easy to visualize the active mechanisms with a simple scalar. For example, if VARI_34 = 10000010, then the active criteria are: deviatory :math:`\mathrm{XZ}` monotonic (:math:`{10}^{1}`) and isotropic cyclic (:math:`{10}^{7}`).