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

Geometry
---------

The geometry of the test case studied consists of a rectangular plate with a hole in its center subjected to traction.

For obvious reasons of symmetry, only a quarter of the plate is studied.


.. image:: images/Object_2.png
    :width: 4.6319in
    :height: 3.0299in


.. _RefSchema_Object_2.png:

**Figure** 1.1-a **: Problem studied**


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

Since the objective is to test the different laws of behavior, the different models use different laws of behavior.

In the case of models :math:`A`, :math:`B` and :math:`E`, the material that constitutes the plate is modelled by an elastoplastic behavior law. The plasticity criterion is that of von Mises and linear isotropic work hardening is considered (VMIS_ISOT_LINE). The values of the various parameters are summarized in the following table.


.. csv-table::

    "**Settings**", "**Symbol**", "**Values**"
    "Young's module", ":math:`E` "," :math:`70\mathrm{MPa}`"
    "Fish Coeff.", ":math:`\nu` "," :math:`\mathrm{0,2}`"
    "Elastic limit", ":math:`{\sigma }_{y}` "," :math:`\mathrm{0,24}\mathrm{MPa}`"
    "Work hardening slope", ":math:`H` "," :math:`\mathrm{2,24}\mathrm{MPa}`"


**Table** 1.2-1 **: Material parameters for A, B, and E models**

The values of the various parameters are summarized in the following table.


.. csv-table::

    "**Settings**", "**Symbol**", "**Values**"
    "Young's module", ":math:`E` "," :math:`20000\mathrm{MPa}`"
    "Fish Coeff.", ":math:`\nu` "," :math:`0`"
    "Elastic limit", ":math:`{\sigma }_{y}` "," :math:`2\mathrm{MPa}`"
    "Work hardening slope", ":math:`H` "," :math:`-2000\mathrm{MPa}`"


**Table** 1.2-2 **: Material parameters for C models**

In the case of modeling :math:`D`, the plate is made of fragile concrete (ENDO_ISOT_BETON). The values of the various parameters are summarized in the following table.


.. csv-table::

    "**Settings**", "**Symbol**", "**Values**"
    "Young's module", ":math:`E` "," :math:`20000\mathrm{MPa}`"
    "Fish Coeff.", ":math:`\nu` "," :math:`0`"
    "Elastic limit", ":math:`{\sigma }_{y}` "," :math:`2\mathrm{MPa}`"
    "Work hardening slope", ":math:`H` "," :math:`-2000\mathrm{MPa}`"


**Table** 1.2-3 **: Material parameters for D models**


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

They are identical for the 5 models.

In order to recreate the symmetry conditions, the movements are blocked:


* following :math:`y` out of :math:`\mathrm{AB}`,


* following :math:`x` on :math:`\mathrm{DE}`.

Loading is defined by requiring a movement of :math:`\mathrm{0,3}\mathrm{mm}` along the :math:`y` axis to the :math:`\mathrm{DC}` border.


Initial conditions
--------------------

At moment :math:`0`, the system is in balance and is not subject to any prestress.