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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 596
    :height: 426


.. _RefImage_Object_1.svg:

Pinched cap:


.. image:: images/Object_2.svg
    :width: 596
    :height: 426


.. _RefImage_Object_2.svg:

With

.. image:: images/Object_3.svg
    :width: 596
    :height: 426


.. _RefImage_Object_3.svg:


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

Elastic behavior:


.. image:: images/Object_4.svg
    :width: 596
    :height: 426


.. _RefImage_Object_4.svg:


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

We are looking for the successive states of equilibrium under the load


.. image:: images/Object_5.svg
    :width: 596
    :height: 426


.. _RefImage_Object_5.svg:

applied in :math:`{P}_{1}{P}_{6}{P}_{2}{P}_{7}`

Because of the geometric and physical symmetry of the problem, only quarter :math:`{P}_{1}{P}_{2}{P}_{3}{P}_{4}` is modeled, taking into account symmetry conditions. These conditions eliminate 5 rigid body movements. The last rigid body movement is eliminated by blocking the next movement.

.. image:: images/Object_8.svg
    :width: 596
    :height: 426


.. _RefImage_Object_8.svg:

. at point :math:`{P}_{5}`,


+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+
| .. code::                                      |:math:`{P}_{5}`:      |                                                                 |:math:`\mathrm{DZ}=0`  |
+                                                +                      +                                                                 +                       +
|    Conditions aux limites :                    |                      |                                                                 |                       |
+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+
|                                                |                      |                                                                 |                       |
+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+
|Symmetry:                                       |:math:`\mathrm{P2P3}`:|                                                                 |:math:`\mathrm{DY}=0.` |
+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+
|                                                |                      |:math:`\mathrm{DRX}=0.`                                          |:math:`\mathrm{DRZ}=0.`|
+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+
|                                                |:math:`\mathrm{P4P1}`:|:math:`\mathrm{DX}=0.`                                           |                       |
+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+
|                                                |                      |:math:`\mathrm{DRY}=\mathrm{0}\mathrm{.}` :math:`\mathrm{DRZ}=0.`|                       |
+------------------------------------------------+----------------------+-----------------------------------------------------------------+-----------------------+


We are particularly interested in the movements of points :math:`{P}_{1}` and :math:`{P}_{3}` according to the loading directions.