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


Geometry
---------

It is an 8-node cube, where two faces have zero normal displacement, and the two opposite faces have an imposed normal displacement, different from each other by a factor of 2.

The cube has a side of :math:`1\mathrm{mm}`. The test cases are composed of a charge, followed by a discharge. In modeling A, the cube is oriented according to the coordinate system :math:`\mathit{Oxyz}`. In model B, it is rotated :math:`30°` by around the :math:`\mathrm{Oy}` axis.


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

**Modeling A**

**B Modeling**


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:

:math:`{U}_{2}\mathrm{=}{\mathrm{2.U}}_{1}`


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

To test the evolution of mechanical characteristics irreversibly with temperature, a decreasing temperature field is applied. Some variables depend on temperature, others on drying. Finally, a non-zero desiccation shrinkage coefficient, equal to the thermal expansion coefficient, is applied to test "computer" operation. The thermal deformations will thus be equal and opposite to the deformations due to desiccation shrinkage. These dependencies only occur for purely computer checks; the mechanical characteristics can be considered to be constant.

**For the usual linear mechanical characteristics:**


.. csv-table::

    "Young's module:", ":math:`E=32000\mathrm{MPa}` ", "*of*", ":math:`0°C` *to* :math:`20°C`"
    "", ":math:`E=15000\mathrm{MPa}` ", "*to*", ":math:`400°C` *(linear decay)*"
    "", ":math:`E=5000\mathrm{MPa}` ", "*to*", ":math:`800°C` *(linear decay)*"
    "*Poisson's ratio:*", ":math:`\nu =0.18` ", "", ""
    "*Coefficient of thermal expansion:*", ":math:`\alpha ={10}^{-5}` ", "", ""
    "*Desiccation shrinkage coefficient:*", ":math:`\kappa ={10}^{-5}` ", "", ""


**For the non-linear mechanical characteristics of the model** **** BETON_DOUBLE_DP :**


.. csv-table::

    "*Uniaxial compression strength:*", ":math:`f\text{'}c=40N/{\mathrm{mm}}^{2}` ", "*of*", ":math:`0°C` *to* :math:`400°C`"
    "", "*f'c = 15 N/mm^2*", "*to*", ":math:`800°C` *(linear decay)*"
    "*Uniaxial tensile strength:*", ":math:`f\text{'}t=4N/{\mathrm{mm}}^{2}` ", "*of*", ":math:`0°C` *to* :math:`400°C`"
    "", ":math:`f\text{'}t=1.5N/{\mathrm{mm}}^{2}` ", "*to*", ":math:`800°C` *(linear decay)*"
    "*Ratio of resistances in biaxial compression/uniaxial compression:*", ":math:`\beta \mathrm{=}1.16` ", "", ""
    "*Breakdown energy in compression:*", ":math:`\mathit{Gc}\mathrm{=}10\mathit{Nmm}\mathrm{/}{\mathit{mm}}^{2}` ", "", ""
    "*Tensile breaking energy:*", ":math:`\text{Gt}\mathrm{=}0.1\mathit{Nmm}\mathrm{/}{\mathit{mm}}^{2}` ", "", ""
    "*Ratio of the elastic limit to the resistance in uniaxial compression:*", "*30%*", ""



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


+--------------------------------------------------------------+--------------------------------------------------------------------------------------+
|Temperature field decreasing from :math:`20°C` to :math:`0°C`.                                                                                       |
+--------------------------------------------------------------+--------------------------------------------------------------------------------------+
|Underside of the cube (:math:`\mathrm{facexy}`):              |blocked next :math:`\mathrm{oz}`.                                                     |
+--------------------------------------------------------------+--------------------------------------------------------------------------------------+
|Top side of the cube (:math:`\mathrm{face1xy}`):              |forced displacement :math:`0.30\mathrm{mm}` followed by a :math:`0.1\mathrm{mm}` dump |
+--------------------------------------------------------------+--------------------------------------------------------------------------------------+
|Left side of the cube (:math:`\mathrm{faceyz}`):              |blocked next :math:`\mathrm{ox}`.                                                     |
+--------------------------------------------------------------+--------------------------------------------------------------------------------------+
|Right side of the cube (:math:`\mathrm{face1yz}`):            |forced displacement :math:`0.15\mathrm{mm}` followed by a :math:`0.05\mathrm{mm}` dump|
+--------------------------------------------------------------+--------------------------------------------------------------------------------------+
|Lower front nodes (:math:`{N}_{1}`, :math:`{N}_{2}`):         |stuck next :math:`\mathrm{oy}` (Suppression of solid body movements).                 |
+--------------------------------------------------------------+--------------------------------------------------------------------------------------+