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

Geometry and boundary conditions
-----------------------------------


.. image:: images/10028CBC00000A8B00000B791E48614B1DA079E2.svg
    :width: 170
    :height: 187


.. _RefImage_10028CBC00000A8B00000B791E48614B1DA079E2.svg:

**Figure 1.1-a: Geometry and boundary conditions**


.. csv-table::

    "Edge length:", ":math:`a=0.56m`"
    "Thickness:", ":math:`0.1m`"


The loading is such that a state of homogeneous stress and of the plane stress type is obtained even if the modeling in Code_Aster was carried out in 3D. Loading is imposed in the form of trips imposed in two stages:

• direction :math:`(\Delta {\varepsilon }_{\mathit{xx}},\Delta {\varepsilon }_{\mathit{yy}},\Delta {\varepsilon }_{\mathit{xy}})\mathrm{=}(1,\mathrm{-}\nu ,0)`, up to maximum stress (initiation of the softening phase)

• direction :math:`(\Delta {\varepsilon }_{\mathrm{xx}},\Delta {\varepsilon }_{\mathrm{yy}},\Delta {\varepsilon }_{\mathrm{xy}})=(\mathrm{1 ,}1.5,1)`, up to :math:`{\varepsilon }_{\mathrm{xx}}=0.0015`

Where

• :math:`{\varepsilon }_{\mathrm{xx}}`: movement of the side delimited by the side

.. image:: images/Object_1.svg
    :width: 170
    :height: 187


.. _RefImage_Object_1.svg:

in the direction :math:`\mathrm{OX}`

• :math:`{\varepsilon }_{\mathrm{yy}}`: movement of the side delimited by the side

.. image:: images/Object_2.svg
    :width: 170
    :height: 187


.. _RefImage_Object_2.svg:

in the direction :math:`\mathrm{OY}`

• :math:`{\varepsilon }_{\mathrm{xy}}`: movement of the side delimited by the side

.. image:: images/Object_3.svg
    :width: 170
    :height: 187


.. _RefImage_Object_3.svg:

in the direction :math:`\mathrm{OY}`

Let :math:`\mathit{P5P6P7P8}` be the plane of the cube in :math:`z=0.1`.

In practice, the following conditions are imposed:

.. csv-table::

    "* while loading:", ":math:`\mathrm{P1P4P8P5}`: :math:`\mathrm{dx}=0`
    :math:`\mathrm{P1}`: :math:`\mathrm{dy}=\mathrm{dz}=0`
    :math:`\mathrm{P5}`: :math:`\mathrm{dy}=0`"
    "* during phase 1:", ":math:`\mathrm{P2}`, :math:`\mathrm{P6}`: :math:`\mathrm{dy}=0`
    :math:`\mathit{P2P3P7P6}`: :math:`\mathrm{dx}=1`
    :math:`\mathit{P3P4P8P7}`: :math:`\mathrm{dy}=-0.2`"
    "* during phase 2:", ":math:`\mathit{P2P3P7P6}`: :math:`\mathrm{dx}=1`
    :math:`\mathrm{P4}`, :math:`\mathrm{P8}`: :math:`\mathrm{dy}=1.5`
    :math:`\mathrm{P3}`, :math:`\mathrm{P7}`: :math:`\mathrm{dy}=3.5`
    :math:`\mathrm{P2}`, :math:`\mathrm{P6}`: :math:`\mathrm{dy}=2.`"



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

For the Mazars model, the following parameters were used:

Elastic behavior:

:math:`E\mathrm{=}32000\mathit{MPa},\nu \mathrm{=}0.2`

Damaging behavior:

:math:`{\varepsilon }_{\mathit{d0}}\mathrm{=}9\text{.}\text{375}\text{.}{\text{10}}^{\mathrm{-}4};\text{Ac}\mathrm{=}1\text{.}\text{15};\text{At}\mathrm{=}0\text{.}8;\text{Bc}\mathrm{=}\text{1391}\text{.}3;\text{Bt}\mathrm{=}\text{10}\text{000};k\mathrm{=}0\text{.}\text{7}`