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


Geometry of the problem
---------------------

It is one-eighth of a sphere with inner radius :math:`{R}_{i}=1m`, outer radius :math:`{R}_{e}=2m`, and center :math:`O(\mathrm{0,0}\mathrm{,0})`. This portion of the sphere is crossed by an interface-type discontinuity (a non-meshed interface that is introduced into the model through level-sets using the DEFI_FISS_XFEM operator), concentric with radius :math:`R=\mathrm{1.5m}`.

The geometry of the column is shown in the figure.


.. image:: images/1000000000000235000001B8CB74839425719A67.png
    :width: 4.4374in
    :height: 3.5972in


.. _RefImage_1000000000000235000001B8CB74839425719A67.png:


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

The parameters given in the Table correspond to the parameters used for the 4 models. The behavior is elastic ('ELAS').


.. csv-table::

    "", "", ""
    "Elastic parameters", "Young's modulus :math:`E(\mathit{en}\mathit{MPa})`
    Poisson's ratio :math:`\nu`
    Thermal expansion coefficient :math:`\alpha (\mathit{en}{K}^{\text{-1}})` "," :math:`5800`
    :math:`0`
    :math:`0`"


**Table** 1.2-1 **: Material Properties**


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

The following Dirichlet conditions apply:


* on the lower face [ABDC], the movements following :math:`z` are blocked :math:`{u}_{\text{z}}=0`,


* on the [ABFE] side, the movements following :math:`y` are blocked :math:`{u}_{\text{y}}=0`,


* on the [CDFE] side, the movements following :math:`x` are blocked :math:`{u}_{\text{x}}=0`,


* on outer shell BDF and inner shell ACE, movements are blocked in all directions (:math:`{u}_{\text{x}}=0`, :math:`{u}_{\text{y}}=0` and :math:`{u}_{\text{z}}=0`).

The load is as follows:


* On each of the lips of the interface at :math:`r=R` a uniform distributed pressure is imposed :math:`p=10\mathit{MPa}` by means of AFFE_CHAR_MECA and the keyword FISSUREdePRES_REP.