Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Analytical solution, obtained with the hypothesis of uniaxiality of the constraints:


.. image:: images/Object_1.svg
    :width: 449
    :height: 25


.. _RefImage_Object_1.svg:

or in frame :math:`(A,L,T)`:


.. image:: images/Object_2.svg
    :width: 449
    :height: 25


.. _RefImage_Object_2.svg:

By the law of orthotropic elastic behavior, using the conventions of *Code_Aster* with respect to :math:`{\mathrm{NU}}_{LT}`, (cf. document for using the DEFI_MATERIAU [:ref:`§3.5.2 <§3.5.2>`] command), we obtain directly (see for example [:ref:`bib1 <bib1>`]):


.. image:: images/Object_3.svg
    :width: 449
    :height: 25


.. _RefImage_Object_3.svg:

,

.. image:: images/Object_4.svg
    :width: 449
    :height: 25


.. _RefImage_Object_4.svg:


.. image:: images/Object_5.svg
    :width: 449
    :height: 25


.. _RefImage_Object_5.svg:

with:


.. image:: images/Object_6.svg
    :width: 449
    :height: 25


.. _RefImage_Object_6.svg:


.. image:: images/Object_7.svg
    :width: 449
    :height: 25


.. _RefImage_Object_7.svg:


.. image:: images/Object_8.svg
    :width: 449
    :height: 25


.. _RefImage_Object_8.svg:

As the deformations are uniform in the plate, by integration, we obtain, by integration, the displacements in the reference frame :math:`(A,x,y)`:


.. image:: images/Object_9.svg
    :width: 449
    :height: 25


.. _RefImage_Object_9.svg:


.. image:: images/Object_10.svg
    :width: 449
    :height: 25


.. _RefImage_Object_10.svg:


Benchmark results
----------------------

Travel in frame :math:`(A,x,y)` (in :math:`m`):


+-----------------------------------------------------------------------------------+-----------+-----------+-----------+
|Point                                                                              |:math:`B`  |:math:`C`  |:math:`D`  |
+-----------------------------------------------------------------------------------+-----------+-----------+-----------+
|                                                                                   |           |5.917 10—7 |5.917 10—7 |
+ .. image:: images/Object_11.svg                                                   +           +           +           +
|    :width: 449                                                                    |           |           |           |
+    :height: 25                                                                    +           +           +           +
|                                                                                   |           |           |           |
+                                                                                   +           +           +           +
|                                                                                   |           |           |           |
+-----------------------------------------------------------------------------------+-----------+-----------+-----------+
|                                                                                   |—2.292 10—7|—5.028 10—7|—7.319 10—7|
+ .. image:: images/Object_12.svg                                                   +           +           +           +
|    :width: 449                                                                    |           |           |           |
+    :height: 25                                                                    +           +           +           +
|                                                                                   |           |           |           |
+                                                                                   +           +           +           +
|                                                                                   |           |           |           |
+-----------------------------------------------------------------------------------+-----------+-----------+-----------+


Constraints in the coordinate system linked to orthotropy:

:math:`{\sigma }_{LL}(x,y)=7500\mathrm{Pa}`, :math:`{\sigma }_{\mathrm{TT}}(x,y)=2500\mathrm{Pa}`, :math:`{\sigma }_{LT}(x,y)=4330.127\mathrm{Pa}`


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


1. GAY D: "Composite materials"; 3rd edition, Hermès