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


.. image:: images/10000200000004530000015FCDEB098627D618FE.png
    :width: 6.4638in
    :height: 2.0484in


.. _RefImage_10000200000004530000015FCDEB098627D618FE.png:


.. _DébuBut:


Geometry
---------

 Figure1: Reference problem (for a :math:`90°` rotation)

We consider a cubic element of matter with a side of :math:`1000\mathrm{mm}` subjected alternately to a tensile force and then to an overall rotation of :math:`45°`. It undergoes a total of 4 traction/rotation cycles.


Material data
----------------

Here we consider 6 elasto-plastic behavior laws with kinematic or combined kinematic/isotropic work-hardening of the von Mises type:

VMIS_CINE_LINE, VMIS_ECMI_LINE, VMIS_ECMI_TRAC,

VMIS_CIN1_CHAB and VMIS_CIN2_CHAB VMIS_CIN2_MEMO.

The table below lists the parameters used; in order to reinforce the comparison, the parameters used result in laws of behavior that are identical in all 5 cases (linear work hardening).


+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|**Keyword**|**Setting** |**Value**                                                                                                                                |
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|ELAS       |E           |:math:`200000\mathrm{MPa}`                                                                                                               |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |NUDE        |:math:`\mathrm{0,3}`                                                                                                                     |
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|TRACTION   |SIGM        |:math:`(0.001\mathrm{,200});(0.002\mathrm{,202})`                                                                                        |
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|ECRO_LINE  |D_ SIGM_EPSI|:math:`2000\mathrm{MPa}`                                                                                                                 |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |SY          |:math:`200\mathrm{MPa}`                                                                                                                  |
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|PRAGER     |C           |:math:`\frac{2}{3}\frac{E\mathrm{\ast }\text{D\_SIGM\_EPSI}}{E\mathrm{-}\text{D\_SIGM\_EPSI}}\mathrm{\simeq }\mathrm{1346,8}\mathit{MPa}`|
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|CIN1_CHAB  |C_I         |:math:`\frac{E\mathrm{\ast }\text{D\_SIGM\_EPSI}}{E\mathrm{-}\text{D\_SIGM\_EPSI}}\mathrm{\simeq }\mathrm{2020,2}\mathit{MPa}`           |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |R_0         |:math:`200\mathit{MPa}`                                                                                                                  |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |R_I         |:math:`200\mathit{MPa}`                                                                                                                  |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |G_0         |:math:`0`                                                                                                                                |
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|CIN2_CHAB  |C1_I        |:math:`\frac{1}{2}\frac{E\ast \text{D\_SIGM\_EPSI}}{E-\text{D\_SIGM\_EPSI}}\simeq \mathrm{1010,1}\mathrm{MPa}`                           |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |C2_I        |:math:`\frac{1}{2}\frac{E\ast \text{D\_SIGM\_EPSI}}{E-\text{D\_SIGM\_EPSI}}\simeq \mathrm{1010,1}\mathrm{MPa}`                           |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |R_0         |:math:`200\mathrm{MPa}`                                                                                                                  |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |R_I         |:math:`200\mathrm{MPa}`                                                                                                                  |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |G1_0        |:math:`0`                                                                                                                                |
+           +------------+-----------------------------------------------------------------------------------------------------------------------------------------+
|           |G2_0        |:math:`0`                                                                                                                                |
+-----------+------------+-----------------------------------------------------------------------------------------------------------------------------------------+


+---------+---+---------+
|MEMO_ECRO|MU |0        |
+         +---+---------+
|         |Q_M|0        |
+         +---+---------+
|         |Q_0|0        |
+         +---+---------+
|         |ETA|:math:`0`|
+---------+---+---------+


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

Two types of phases must be distinguished: traction phases and rotation phases. During the traction phases, the normal movements of the front and rear faces are blocked.

Traction phases:

First traction phase


.. csv-table::

    "**Entity**", "**Load Type**", "**Value**"
    "Underside", "FACE_IMPO "," :math:`\mathrm{DNOR}=0`"
    "Top side", "FACE_IMPO "," :math:`\mathrm{DNOR}=\mathrm{500mm}`"
    "Rotation axis", "DDL_IMPO "," :math:`\mathrm{DX}=0`"
    "Front panel", "FACE_IMPO "," :math:`\mathrm{DNOR}=0`"
    "Back side", "FACE_IMPO "," :math:`\mathrm{DNOR}=0`"


Next pull-ups:


.. csv-table::

    "**Entity**", "**Load Type**", "**Value**"
    "Underside", "LIAISON_OBLIQUE "," :math:`\mathrm{DZ}=0`"
    "Top side", "LIAISON_OBLIQUE "," :math:`\mathit{DZ}=300\mathit{mm}`"
    "Side :math:`X\mathrm{=}0`; :math:`Z\mathrm{=}\mathrm{1mm}` "," LIAISON_OBLIQUE "," :math:`\mathrm{DX}=0`"
    "Rotation axis", "DDL_IMPO "," :math:`\mathrm{DX}=\mathrm{0,}\mathrm{DZ}=0`"
    "Front panel", "DDL_IMPO "," :math:`\mathrm{DY}=0`"
    "Back side", "DDL_IMPO "," :math:`\mathrm{DY}=0`"


Each traction phase is composed of 5 identical increments.

Rotation phase:

Boundary conditions


.. csv-table::

    "**Entity**", "**Load Type**", "**Value**"
    "Rotation axis", "DDL_IMPO "," :math:`\mathrm{DX}=\mathrm{0,}\mathrm{DZ}=0`"
    "Front panel", "DDL_IMPO "," :math:`\mathrm{DY}=0`"
    "Back side", "DDL_IMPO "," :math:`\mathrm{DY}=0`"


The rotation load is imposed via a macro named CHAR_ROTA; an overall rotation of :math:`45°` per phase is imposed, divided into 5 increments of :math:`9°`.

At the end of the loading, a deformation of 2,145 is obtained.