Reference solutions
======================


Isotropic plastic behavior
-------------------------------

The reference solution is calculated analytically.

We note

.. image:: images/Object_8.svg
    :width: 110
    :height: 29


.. _RefImage_Object_8.svg:

the temperature, the plastic deformation, the cumulative plastic deformation and the total deformation at the moment of calculation, and

.. image:: images/Object_9.svg
    :width: 110
    :height: 29


.. _RefImage_Object_9.svg:

the same quantities at the previous moment. :math:`{T}_{\mathrm{réf}}` refers to the reference temperature.

The solution is calculated as follows:


.. image:: images/Object_10.svg
    :width: 110
    :height: 29


.. _RefImage_Object_10.svg:

This calculation is done in each direction. For the case being treated,

.. image:: images/Object_11.svg
    :width: 110
    :height: 29


.. _RefImage_Object_11.svg:

at any moment.

The constraint stored in *Aster* is the real constraint existing in each grid in this direction. The internal variables (

.. image:: images/Object_12.svg
    :width: 110
    :height: 29


.. _RefImage_Object_12.svg:

) are calculated from the equations above.


Kinematic plastic behavior
-----------------------------------

The reference solution is calculated analytically.

We note

.. image:: images/Object_13.svg
    :width: 110
    :height: 29


.. _RefImage_Object_13.svg:

the temperature, the plastic deformation and the kinematic work hardening variable at the time of calculation, and

.. image:: images/Object_14.svg
    :width: 110
    :height: 29


.. _RefImage_Object_14.svg:

the same quantities at the previous moment.

The solution is calculated as follows:


.. image:: images/Object_15.svg
    :width: 110
    :height: 29


.. _RefImage_Object_15.svg:

This calculation is done in each direction. For the case being treated,

.. image:: images/Object_16.svg
    :width: 110
    :height: 29


.. _RefImage_Object_16.svg:

at any moment.

The constraint stored in *Aster* is the real constraint existing in each grid in this direction.


Pinto Ménégotto behavior
----------------------------

The reference solution is that obtained by an *Aster* calculation with the same mesh on which charge/discharge cycles are applied, in imposed displacement, making it possible to recreate the deformations resulting from the thermo-mechanical calculations presented below. The corresponding test is therefore only a non-regression test, by comparing the constraints obtained by these two types of modeling: on the one hand mechanical, and on the other hand thermo-mechanical.

The constraint stored in *Aster* is the real constraint existing in each grid in this direction.