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


Geometry
---------

Volume element materialized by a cube with a unit side:


.. image:: images/10000FE4000069BB0000437DD411376829BB2C70.svg
    :width: 256
    :height: 163


.. _RefImage_10000FE4000069BB0000437DD411376829BB2C70.svg:


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

:math:`E={2.10}^{5}\mathrm{MPa}`, :math:`\nu =0.3`, :math:`\alpha ={10}^{-5}°{C}^{-1}`

The material is elastoplastic with different types of behavior:

:math:`\mathrm{C1}`: Isotropic hardening: the traction curve is of the form:

:math:`\sigma ({\varepsilon }^{P},T)={\sigma }_{y}(T)+Q(T)(1-{e}^{-b(T){\varepsilon }^{P}})`

.. csv-table::

    ":math:`\mathrm{SIGY}=200.-\mathrm{1.7.T}` ", "(in :math:`\mathrm{MPa}`)"
    ":math:`Q(T)=100.+\mathrm{1.7.T}` ", "(in :math:`\mathrm{MPa}`)"
    ":math:`b(T)=50.+\mathrm{2.T}` ", ""


:math:`\mathrm{C2}`: Linear kinematic hardening:

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

.. csv-table::

    ":math:`\mathrm{SIGY}=200.-\mathrm{1.7.T}` ", "(in :math:`\mathrm{MPa}`)"
    ":math:`C(T)=1000+\mathrm{2990.T}` ", "(in :math:`\mathrm{MPa}`)"


:math:`\mathrm{C3}`: Nonlinear kinematic hardening (:math:`I`):


.. image:: images/Object_3.svg
    :width: 256
    :height: 163


.. _RefImage_Object_3.svg:


.. csv-table::

    ":math:`\mathrm{SIGY}=200.-\mathrm{1.7.T}` ", "(in :math:`\mathrm{MPa}`)"
    ":math:`C(T)=(100+\mathrm{1.7.T})(50+\mathrm{2.T})` ", "(in :math:`\mathrm{MPa}`)"
    ":math:`D(T)=50` ", ""


:math:`\mathrm{C4}`: Nonlinear kinematic hardening (:math:`\mathrm{II}`):


.. image:: images/Object_4.svg
    :width: 256
    :height: 163


.. _RefImage_Object_4.svg:

same characteristics as for behavior :math:`\mathrm{C3}`, except :math:`D(T)=50+\mathrm{2T}`


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

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

Point :math:`B` stuck in :math:`x`, :math:`y` and :math:`z`. Point :math:`A` stuck in :math:`z`, :math:`\mathrm{DY}=0` on the side :math:`\mathrm{ABFE}`

Force distributed on the face :math:`\mathrm{CDHG}`: :math:`\mathrm{Fy}`

Uniform temperature :math:`T(t)` on the cube. The reference temperature is :math:`0°C`.

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


.. csv-table::

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