Reference problem
=====================

Geometry
---------

The stresses and deformations are homogeneous in the volume element. This can be represented by a plane or a solid element, for example:


.. image:: images/1000058200000FB600000CD1EFC9375AA8655098.svg
    :width: 202
    :height: 165


.. _RefImage_1000058200000FB600000CD1EFC9375AA8655098.svg:


Material properties
-----------------------

Law of elastoplastic behavior with linear kinematic work hardening.


.. image:: images/1000000000000120000000C1F5BD7A98ED2AF70B.png
    :width: 183
    :height: 116


.. _RefImage_1000000000000120000000C1F5BD7A98ED2AF70B.png:

:math:`\begin{array}{c}E\mathrm{=}195000\mathit{MPa}\\ \nu \mathrm{=}0.3\\ {\sigma }^{Y}\mathrm{=}181\mathit{MPa}\\ {E}_{T}\mathrm{=}1930\mathit{MPa}\end{array}`


Boundary conditions and loads
-------------------------------------

The volume element is locked along :math:`\mathrm{Ox}` along the :math:`[\mathrm{2,4}]` side while being subjected to a pull :math:`{\sigma }^{D}` and a shear force :math:`{\tau }^{D}`.


.. image:: images/10000C960000110E00000B93C8C5FAB7DA58B9F8.svg
    :width: 202
    :height: 165


.. _RefImage_10000C960000110E00000B93C8C5FAB7DA58B9F8.svg:

The loading path is as follows:

+---------+----------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------+
|         |:math:`{\sigma }^{D}` |:math:`{\tau }^{D}`                                                                                                                                        |
+         +                      +                                                                                                                                                           +
|         |                      |                                                                                                                                                           |
+         +                      + .. image:: images/1000000000000125000000B7625D352B37DA2858.png                                                                                            +
|         |                      |    :width: 183                                                                                                                                            |
+         +                      +    :height: 116                                                                                                                                           +
|         |                      |                                                                                                                                                           |
+         +                      +                                                                                                                                                           +
|         |                      |                                                                                                                                                           |
+         +                      +                                                                                                                                                           +
|         |                      |                                                                                                                                                           |
+---------+----------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------+
|         |(:math:`\mathit{MPa}`)|(:math:`\mathit{MPa}`)                                                                                                                                     |
+---------+----------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`A`|151.2                 |93.1                                                                                                                                                       |
+---------+----------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`B`|257.3                 |33.1                                                                                                                                                       |
+---------+----------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`C`|259.3                 |0                                                                                                                                                          |
+---------+----------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------+