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


Geometry
---------

We consider a rectangular bar oriented along the :math:`\mathrm{Oy}` axis.

.. code-block:: text

    .. image: images/10000000000001140000017190 FFA685A56FA0D0 .gif
        :width: 2.8752 in
        :height: 3.8437 in


    .. _refImage_10000000000001140000017190 FFA685A56FA0D0 .gif:


    The coordinates of the points are given in the following table:


.. csv-table::

    "**Point**", ":math:`\mathrm{N4}` "," :math:`\mathrm{N23}` "," ", "", ":math:`\mathrm{N27}` "," :math:`\mathrm{N31}` "," :math:`\mathrm{N1}`"
    "**Abscissa** (:math:`m`)", "0.5", "0.5", "0.5", "0.5", "0.5", "0.5"
    "**Ordered** (:math:`m`)", "5", "5", "2.5", "2.5", "2.5", "-5", "-5"


The problem is modelled on the time interval :math:`[0;\mathrm{10s}]`.


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

One gives here the parameters of the bar


.. csv-table::

    "Liquid water", ":math:`\rho`: density (:math:`{\mathrm{kg.m}}^{-3}`)
    :math:`1/{K}_{\mathrm{lq}}`: inverse of compressibility (:math:`{\mathrm{Pa}}^{-1}`)", "1000
    0.5E-9"
    "Material coefficients", ":math:`r`: homogenized density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`)
    :math:`E`: Young's modulus (:math:`\mathit{Pa}`)
    :math:`\nu`: Poisson's ratio (--)
    :math:`b`: Biot coefficient (—)
    :math:`{K}_{i}`: intrinsic permeability (:math:`{m}^{2}`)", "2800
    5.8E9
    0.
    1
    1.E-8"
    "Additional material coefficients for DRUCK_PRAGER "," :math:`\mathit{ECROUISSAGE}`: form of work hardening
    :math:`\alpha`: pressure dependence coefficient (--)
    :math:`{P}_{\mathit{ultm}}`: ultimate cumulative plastic deformation (--)
    :math:`{\sigma }_{Y}`: plasticity constraint (:math:`\mathit{Pa}`)
    :math:`H`: work hardening module (:math:`\mathit{Pa}`)", ":math:`\mathit{LINEAIRE}`
    0.33
    1.0
    1.E8
    0.0"



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

On :math:`\mathrm{HAUT}`, conditions :math:`\sigma \cdot n=0` and :math:`p=3.E6` Pa are imposed.

On :math:`\mathrm{GAUCHE}` and :math:`\mathrm{DROITE}`, the conditions :math:`{u}_{x}=0` and zero liquid flow :math:`\mathrm{M.n}=0` are imposed.

On :math:`\mathrm{BAS}`, the conditions :math:`{u}_{x}={u}_{y}=0` and zero liquid flow :math:`M\cdot n=0` are imposed.


Initial conditions
--------------------

The initial fluid pressure is taken to be equal to 2 :math:`\mathrm{MPa}`. The initial porosity is taken to be equal to 0.5.