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

Geometry
---------

Volume element materialized by a square plate :math:`\mathrm{ABCD}` with a unit side :math:`1\mathrm{mm}`:


.. image:: images/Object_1.svg
    :width: 151
    :height: 115


.. _RefImage_Object_1.svg:


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

:math:`E={2.10}^{5}\mathrm{MPa}`, :math:`\nu =0.3`, :math:`\alpha ={2.10}^{5}°{C}^{-1}`

The material is elastoplastic with linear kinematic work hardening:

:math:`\sigma =\pm {\sigma }_{y}(T)+C(T){\varepsilon }^{p}`

:math:`\mathrm{SIGY}=200.-\mathrm{1.7.T}` (in :math:`\mathrm{Mpa}`)

Work hardening module D_ SIGM_EPSI = :math:`C(T)=1000+\mathrm{2990.T}` (in :math:`\mathrm{MPa}`)


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

Such that the state of stress and deformation are uniform in the volume element:

Point :math:`A` stuck in :math:`x` and :math:`y`.

:math:`\mathrm{DY}=0` out of :math:`\mathrm{AB}`

Force distributed over :math:`\mathrm{CD}`: :math:`\mathrm{Fy}`

Uniform temperature :math:`T` out of :math:`\mathrm{ABCD}`. The reference temperature is :math:`0°C`.

:math:`\mathrm{Fy}` and :math:`T` vary over time as follows:

.. csv-table::

    "instant :math:`t` ", "0"," :math:`\mathrm{1s}` "," :math:`\mathrm{2s}`"
    ":math:`\mathrm{Fy}(t)` ", "0"," :math:`210\mathrm{MPa}` "," :math:`210\mathrm{MPa}`"
    ":math:`T(t)` ", "0"," :math:`0°C` "," :math:`100°C`"