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


Geometry
---------

It is a cube with 8 nodes, where three faces have a zero normal displacement and the three opposite faces have an imposed and identical normal displacement.

The cube has a side length of :math:`1\mathit{mm}`. In modeling A, the cube is oriented according to the coordinate system :math:`\mathit{Oxyz}`.

**Modeling A**


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

:math:`{U}_{2}\mathrm{=}{U}_{1}\mathrm{=}{U}_{3}`


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

To test the irreversible evolution of mechanical characteristics 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\mathrm{=}32000\mathit{MPa}` ", "from", ":math:`0°C` to :math:`20°C`"
    "", ":math:`E\mathrm{=}15000\mathit{MPa}` ", "to", ":math:`400°C` (linear decay)"
    "", ":math:`E\mathrm{=}5000\mathit{MPa}` ", "to", ":math:`800°C` (linear decay)"
    "Poisson's ratio:", ":math:`\nu \mathrm{=}0.18` ", "", ""
    "Thermal expansion coefficient:", ":math:`a\mathrm{=}{10}^{\mathrm{-}5}\mathrm{/}°C` ", "", ""
    "Desiccation shrinkage coefficient:", ":math:`k\mathrm{=}{10}^{\mathrm{-}5}` ", "", ""


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


+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|Uniaxial compression strength:            |:math:`f\text{'}c\mathrm{=}40N\mathrm{/}{\mathit{mm}}^{2}`         |of  |:math:`0°C` to :math:`400°C`|
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|                                          |:math:`f\text{'}c\mathrm{=}15N\mathrm{/}{\mathit{mm}}^{2}`         |unto|:math:`800°C` (linear decay)|
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|Uniaxial tensile strength:                |:math:`f\text{'}t\mathrm{=}4N\mathrm{/}{\mathit{mm}}^{2}`          |of  |:math:`0°C` to :math:`400°C`|
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|                                          |:math:`f\text{'}t\mathrm{=}1.5N\mathrm{/}{\mathit{mm}}^{2}`        |unto|:math:`800°C` (linear decay)|
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|Compressive strength ratio                                                                                    |    |                            |
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|biaxial/uniaxial compression:             |:math:`b\mathrm{=}1.16`                                            |    |                            |
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|Breakdown energy in compression:          |:math:`{G}_{c}\mathrm{=}10\mathit{Nmm}\mathrm{/}{\mathit{mm}}^{2}` |    |                            |
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|Tensile break energy:                     |:math:`{G}_{t}\mathrm{=}0.1\mathit{Nmm}\mathrm{/}{\mathit{mm}}^{2}`|    |                            |
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|Ratio of the elastic limit to the strength                                                                    |    |                            |
+------------------------------------------+-------------------------------------------------------------------+----+----------------------------+
|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:`\mathit{facexy}`):              |blocked next :math:`\mathit{oz}`.                             |
+--------------------------------------------------------------+--------------------------------------------------------------+
|Top side of the cube (:math:`\mathit{face1xy}`):              |Displacement :math:`{U}_{z}\mathrm{=}\mathrm{0,15}\mathit{mm}`|
+--------------------------------------------------------------+--------------------------------------------------------------+
|Left side of the cube (:math:`\mathit{faceyz}`):              |blocked next :math:`\mathit{ox}`.                             |
+--------------------------------------------------------------+--------------------------------------------------------------+
|Right side of the cube (:math:`\mathit{face1yz}`):            |Displacement :math:`{U}_{x}\mathrm{=}\mathrm{0,15}\mathit{mm}`|
+--------------------------------------------------------------+--------------------------------------------------------------+
|Front side of the cube (:math:`\mathit{facexz}`):             |blocked next :math:`\mathit{oy}`.                             |
+--------------------------------------------------------------+--------------------------------------------------------------+
|Back side of the cube (:math:`\mathit{face1xz}`):             |Displacement :math:`{U}_{y}\mathrm{=}\mathrm{0,15}\mathit{mm}`|
+--------------------------------------------------------------+--------------------------------------------------------------+