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


Geometry
---------

We consider a 1×1 mm square test piece comprising 4 elements.


.. image:: images/10000201000002FC0000024632A555F784A31609.png
    :width: 3.8028in
    :height: 2.7563in


.. _RefImage_10000201000002FC0000024632A555F784A31609.png:

Figure 1-1: Square of 4 elements in tension


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

We adopt a Von Mises elasto-plastic behavior law with isotropic work hardening TRACTION whose traction curve is given point by point:


.. csv-table::

    ":math:`\mathrm{\epsilon }` ", "0.0027", "0.005", "0.005", "0.01", "0.01", "0.015", "0.02", "0.03", "0.04", "0.05", "0.05", "0.05", "0.075", "0.1"
    ":math:`\mathrm{\sigma }` :math:`\mathit{MPa}` ", "555", "589", "589", "631", "631", "657", "676", "691", "704", "725", "741", "741", "741", "772", "794"



.. csv-table::

    "0.125", "0.15", "0.2", "0.2", "0.3", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.8", "0.9"
    "812", "827", "851", "887", "887", "97", "912", "933", "950", "965", "978", "990"


The deformations used in the behavior relationship are linearized deformations. Young's modulus :math:`E` is :math:`200\mathit{GPa}` while the Poisson's ratio :math:`\mathrm{\nu }` is equal to :math:`\mathrm{0,3}`.

The Weibull model coefficients used are as follows:

:math:`m=8`,

:math:`{V}_{0}=125\mathrm{\mu }m`,

:math:`{\mathrm{\sigma }}_{u}=2630\mathit{MPa}`.


.. _RefNumPara__11653_1489988600:


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

With reference to Figure 1-1, the boundary conditions are as follows:


* :math:`\mathit{BAS}`: movements blocked according to :math:`(X)\mathit{et}(Y)`,


* :math:`\mathit{GAUCHE},\mathit{DROITE}`: movements blocked following :math:`(X)`.

     The pressure is affected on segment :math:`\mathit{HAUT}` via PRES_REP following :math:`(Y)` and varies as a function of time as follows:


.. csv-table::

    "instant :math:`t` ", "0", "1"
    ":math:`P(t)` ", "0", "-0.5x104"


The pressure is imposed using 5 steps up to 0.2 s and 10 steps up to 10 s.


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

Zero stresses and deformations.