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


Geometry
---------


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.. _RefImage_100068F8000069D5000024E6F4E9873107F659D0.svg:


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

*Modeling A*

:math:`E=1.0\mathrm{MPa}`

:math:`\nu =0.3`

:math:`k=1.0W/(\mathrm{m.}°C)`

:math:`{C}_{p}=0J/(°{\mathrm{C.m}}^{3})`

*B Modeling*

:math:`{E}_{L}=1.0\mathrm{MPa}`

:math:`{E}_{T}=0.9\mathrm{MPa}`

:math:`{E}_{N}=0.8\mathrm{MPa}`

:math:`{\nu }_{\text{LT}}=0.1`

:math:`{\nu }_{\text{LN}}=0.25`

:math:`{\nu }_{\mathrm{TN}}=0.3333333`

:math:`k=1.0W/(\mathrm{m.}°C)`

:math:`{C}_{p}=0J/(°{\mathrm{C.m}}^{3})`


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


* 3D mechanics:


+------------------------------+---------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------+
|Plan :math:`z=0`:             |:math:`\mathrm{dz}=0`                                                            |for membrane loading;                                                                                          |
+                              +                                                                                 +                                                                                                               +
|                              | :math:`\mathrm{dx}=0`, :math:`\mathrm{dy}=0`                                    | for flexure loading                                                                                           |
+                              +                                                                                 +                                                                                                               +
|                              |                                                                                 |                                                                                                               |
+------------------------------+---------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------+
|Maps :math:`y=0`, :math:`y=1`:|:math:`\mathrm{dy}=0`                                                            |                                                                                                               |
+------------------------------+---------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------+
|Maps :math:`x=0`, :math:`x=1`:|:math:`\mathrm{dx}=0`                                                            |                                                                                                               |
+------------------------------+---------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------+
|Bow: :math:`O`                |:math:`\mathrm{dz}=0`                                                            |(for flexure loading only)                                                                                     |
+------------------------------+---------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------+
|Loading:                      |                                                                                 |                                                                                                               |
+                              +                                                                                 + .. image:: images/Object_1.svg                                                                                +
|                              |                                                                                 |    :width: 674                                                                                                |
+                              +                                                                                 +    :height: 235                                                                                               +
|                              |                                                                                 |                                                                                                               |
+                              +                                                                                 + :math:`{z}_{0}=\mathrm{1m}`                                                                                   +
|                              |                                                                                 |                                                                                                               |
+------------------------------+---------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------+



* 2D mechanics, plane constraints:


.. csv-table::

    "Axis: :math:`x=0`
    Node: :math:`O` "," :math:`\mathrm{dx}=0`
    :math:`\mathrm{dy}=0` ", "(these conditions do not correspond to the application of the homogenization method)."


Loading: deformation

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* 3D and 2D thermal:


.. csv-table::

    "Plan :math:`x=0` "," :math:`\mathrm{temp}=0` ", "(this condition does not correspond to the application of the homogenization method)."


Loading: gradient

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    :height: 235


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imposed uniform.