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


Geometry
---------


.. image:: images/100000000000012D0000019FF96BCAF98D7BE59F.png
    :width: 2.1654in
    :height: 2.8819in


.. _RefImage_100000000000012D0000019FF96BCAF98D7BE59F.png:


+--------------------+----------+
|Cylinder dimensions:           |
+--------------------+----------+
|:math:`{R}_{0}`     |:math:`1m`|
+--------------------+----------+
|:math:`{R}_{1}`     |:math:`2m`|
+--------------------+----------+

**Figure 1.1-a: Hollow cylinder cutting and loading**


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

Young's module: :math:`E\mathrm{=}1\mathit{MPa}`

Poisson's ratio: :math:`\nu \mathrm{=}0.3`

Law of LEMAITRE:

:math:`g(\sigma ,\lambda ,T)\mathrm{=}{(\frac{1}{K}\frac{\sigma }{{\lambda }^{\frac{1}{m}}})}^{n}` with :math:`\frac{1}{K}\mathrm{=}1`, :math:`\frac{1}{m}\mathrm{=}0`, :math:`n\mathrm{=}1`

Law LEMA_SEUIL:

:math:`g(\sigma ,\lambda ,T)\mathrm{=}A(\frac{2}{\sqrt{3}}\sigma )\Phi` with :math:`A\mathrm{=}\frac{\sqrt{3}}{2}`, :math:`\Phi \mathrm{=}1` all over the mesh

:math:`S\mathrm{=}{10}^{\mathrm{-}10}`

given the value of the various material parameters, the two laws are absolutely identical and can therefore be compared to the same analytical solution.


Boundary conditions and loading
------------------------------------

**Boundary conditions:**

The cylinder is locked at :math:`\mathit{DY}` on the :math:`[\mathrm{AB}]` and :math:`[\mathrm{CD}]` sides.

**Charging:**

The cylinder is subjected to internal pressure on :math:`\mathrm{[}\mathit{DA}\mathrm{]}\mathit{P0}\mathrm{=}1.E\mathrm{-}3\mathit{MPa}`