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

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

Volume element materialized by a cube with a unit side (:math:`m`):


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


.. _RefImage_10000FE4000069BB0000437DD411376829BB2C70.svg:

**Figure 1.1-a: Geometry**

The load is such that a state of uniform stress and deformation is obtained in the volume.

The bottlenecks are as follows:


* side :math:`\mathit{ABCD}`: :math:`\mathit{DZ}\mathrm{=}0`


* side :math:`\mathit{BCGF}`: :math:`\mathit{DX}\mathrm{=}0`


* side :math:`\mathit{ABFE}`: :math:`\mathit{DY}\mathrm{=}0`


* face :math:`\mathit{EFGH}`: movement :math:`\mathit{Uz}(t)`

Temperature :math:`T(t)` is assumed to be uniform across the cube; the reference temperature is :math:`0°C`.

:math:`\mathit{Uz}` and :math:`T` vary over time as follows:


.. csv-table::

    "**instant** :math:`t` ", "0", "0", "100", "200", "300"
    ":math:`\mathit{Uz}(t)` "," :math:`0\mathit{m.}` "," :math:`–{10}^{\mathrm{-}3}\mathit{m.}` "," "," :math:`–{10}^{\mathrm{-}3}\mathit{m.}` "," :math:`–{10}^{\mathrm{-}3}\mathit{m.}`"
    ":math:`T(t)` "," :math:`0°C` "," :math:`0°C` "," "," :math:`200°C` "," :math:`0°C`"


A purely mechanical loading is therefore carried out, then heating is carried out by blocking the direction :math:`\mathit{Uz}`, before cooling. This makes it possible to verify the separation of thermal and mechanical deformations as well as the non-recovery of mechanical properties after heating.


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

For the Mazars model, the following parameters were used (value at :math:`0°C`):

Elastic behavior:


.. image:: images/Object_1.svg
    :width: 256
    :height: 163


.. _RefImage_Object_1.svg:

Thermal characteristics:


.. image:: images/Object_2.svg
    :width: 256
    :height: 163


.. _RefImage_Object_2.svg:

Damaging behavior:

:math:`{\epsilon }_{d0}=1.0{\text{10}}^{-4};\phantom{\rule{2em}{0ex}}{A}_{c}=1.15;\phantom{\rule{2em}{0ex}}{A}_{t}=1.0;\phantom{\rule{2em}{0ex}}{B}_{c}=2000.0;\phantom{\rule{2em}{0ex}}{B}_{t}=10000.0;\phantom{\rule{2em}{0ex}}k=0.7`

It is also considered that :math:`E` and :math:`{B}_{c}` vary with temperature. Their evolution is given in figures [:ref:`Figure 1.2-a <Figure 1.2-a>`] and [:ref:`Figure 1.2-b <Figure 1.2-b>`].


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


.. _RefImage_Object_4.svg:

**Figure 1.2-a:** Evolution **of Young's modulus with temperature**


.. image:: images/Object_5.svg
    :width: 256
    :height: 163


.. _RefImage_Object_5.svg:

**Figure 1.2-b:** Evolution **of** :math:`{B}_{C}` **with temperature**