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


Geometry
---------


.. image:: images/100022E6000069BB000070C26B7E3F82A208D5A2.svg
    :width: 364
    :height: 388


.. _RefImage_100022E6000069BB000070C26B7E3F82A208D5A2.svg:

**Shell features:**


* thickness :math:`h=0.05\mathrm{mm}`,


* radius of curvature :math:`R=1\mathrm{mm}`,


* width :math:`L=\mathrm{AB}=\mathrm{CD}=0.1\mathrm{mm}`,


* position of the first center of curvature:

    .. image: images/Object_1.svg
        :width: 364
        :height: 388


    .. _RefImage_Object_1.svg:

,


*the angle :math:`\alpha` is chosen so that the**upper** surface of the shell at point :math:`X` is at :math:`(y=0)`, that is to say aligned with :math:`A` and :math:`C`,


.. image:: images/Object_3.svg
    :width: 364
    :height: 388


.. _RefImage_Object_3.svg:


* position of the second center of curvature:

    .. image: images/Object_4.svg
        :width: 364
        :height: 388


    .. _RefImage_Object_4.svg:

.


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

:math:`E=\mathrm{2 000}\mathrm{MPa}`

:math:`\nu =0.3`

An elasto-plastic behavior law with the Von Misès criterion for linear isotropic work hardening is used: :math:`{\sigma }_{y}=100\mathrm{MPa}` :math:`{E}_{T}:200\mathrm{MPa}`.


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


* on :math:`\mathrm{AB}`: embedding: :math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0`,


* all over the shell: plane deformation according to :math:`\mathrm{Oz}` or :math:`\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=0`,


* on :math:`\mathrm{CD}`: linear force (per unit length :math:`\mathrm{Oz}`) according to :math:`\mathrm{Ox}` given by: :math:`{f}_{x}=50N/\mathrm{mm}` It is equivalent to a pressure of :math:`{p}_{X}={f}_{x}/h=100\mathrm{MPa}` exerted on the side :math:`\mathrm{CD}`,


* the load is gradually applied to the structure. The loading path is divided into 10 equal increments.


.. image:: images/Object_6.svg
    :width: 364
    :height: 388


.. _RefImage_Object_6.svg: