The internal variables
======================

For the IT implementation, we retained the following 4 internal variables:




V1: the cumulative plastic deviatory deformation
------------------------------------------------

The work hardening variable :math:`{\gamma }^{p}` is proportional to the second invariant of the deviatory deformation tensor.

:math:`{\gamma }^{p}=\sqrt{\frac{2}{3}{e}_{\text{ij}}^{p}{e}_{\text{ij}}^{p}}`

with :math:`{e}_{\text{ij}}^{p}={\varepsilon }_{\text{ij}}^{p}-\frac{\text{tr}({\varepsilon }_{\text{ij}}^{p})}{3}{\delta }_{\text{ij}}`




V2: the cumulative plastic volume deformation
------------------------------------------------

Plastic volume deformation is defined by the relationship presented in paragraph [:ref:`§3.2.4 <§3.2.4>`] on the law of evolution of the plastic mechanism: :math:`\dot{{\varepsilon }_{v}^{p}}=\dot{\lambda }G`




V3: the domains of rock behavior
---------------------------------------------

Five behavior domains, numbered from 0 to 4 (cf. figure), are identified to allow a relatively simple representation of the state of damage of the rock, from intact rock to rock in residual state. These domains are a function of the cumulative plastic deviatory deformation :math:`{\gamma }^{p}` and the stress state. Each domain number increment defines the transition to a higher damage domain.


* If the deviator is less than 70% of the peak deviator, then the material is in the 0 range;


* If not:


* If :math:`{\gamma }^{p}=0` then the material is in domain 1;


1. If :math:`0<{\gamma }^{p}<{\gamma }^{e}` then the material is in domain 2;


* If :math:`{\gamma }_{e}<{\gamma }^{p}<{\gamma }_{\text{ult}}` then the material is in domain 3;


*

    .. image:: images/Object_260.svg
        :width: 215
        :height: 148


    .. _RefImage_Object_260.svg:

If :math:`{\gamma }^{p}>{\gamma }_{\text{ult}}` then the material is in domain 4.


.. image:: images/Object_262.svg
    :width: 215
    :height: 148


.. _RefImage_Object_262.svg:




V4: the state of plasticization
-----------------------------

It is an internal indicator of *Code_Aster*. It is 0 if the gauss point is in elastic charge or under discharge, and is equal to 1 if the gauss point is in plastic charge.