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


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

It is a column with height :math:`\mathit{LZ}=5m`, length :math:`\mathit{LX}=1m`, and width :math:`\mathit{LY}=1m`. In :math:`Z=\frac{\mathit{LZ}}{2}`, this column has an interface-type discontinuity. The column is thus entirely crossed by the discontinuity.

The geometry of the column is shown in the figure.


.. image:: images/10000000000001A40000019F01972182FF094FCE.jpg
    :width: 3.7681in
    :height: 3.9571in


.. _RefImage_10000000000001A40000019F01972182FF094FCE.jpg:

**Figure** 1.1-a **: Problem geometry**


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

The parameters given in the Table correspond to the parameters used for modeling in the hydromechanical coupled case. The coupling law used is' LIQU_SATU '. The parameters specific to this coupling law are given but have no influence on the solution (because we chose to take a uniformly zero pore pressure throughout the domain). Only the elastic parameters have an influence on the solution of the pseudo-coupled problem. The contact used is of the 'MORTAR' type. The associated cohesive law is' CZM_LIN_MIX '.


.. csv-table::

    "Liquid (water)", "Viscosity :math:`{\mu }_{w}(\mathit{en}\mathit{Pa.s})`
    Compressibility module :math:`\frac{1}{{K}_{w}}(\mathit{en}{\mathit{Pa}}^{\text{-1}})`
    Liquid density :math:`{\rho }_{w}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})` "," :math:`{10}^{\text{-3}}`
    :math:`{5.10}^{\text{-10}}`
    :math:`1`"
    "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`"
    "Coupling parameters", "Biot coefficient :math:`b`
    Initial homogenized density :math:`{r}_{0}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})`
    Intrinsic permeability :math:`{K}^{\text{int}}(\mathit{en}{m}^{2})` "," :math:`1`
    :math:`\mathrm{2,5}`
    :math:`{\mathrm{1,01937}}^{\text{-19}}`"
    "Parameters of the cohesive law", "Critical constraint :math:`{\mathrm{\sigma }}_{c}(\mathit{en}\mathit{MPa})`
    Cohesive energy :math:`{G}_{c}(\mathit{en}\mathit{Pa}\mathrm{.}m)`
    Increase coefficient :math:`r` "," :math:`1.1`
    :math:`900`
    :math:`10`"


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

On the other hand, the forces associated with gravity (in the equation for the conservation of momentum) are neglected. The reference pore pressure is taken to be zero :math:`{p}_{1}^{\text{ref}}=0\mathit{MPa}` and the porosity of the material is :math:`\varphi =\mathrm{0,15}`.


Boundary conditions
----------------------

The following Dirichlet conditions apply:


* on the [ABCD] face, movements are blocked in all directions (:math:`{u}_{\text{x}}=0`, :math:`{u}_{\text{y}}=0` and :math:`{u}_{\text{z}}=0`),


* on the [EFGH] side, the movements following :math:`y` are blocked :math:`{u}_{\text{y}}=0` and the movements following :math:`x` and :math:`z` are imposed for each moment of calculation: :math:`{u}_{\text{x}}=f(t)`, :math:`{u}_{\text{z}}=g(t)`.

8 different loads are carried out. The values of the movements imposed on the upper face for each of the 8 calculation moments are summarized in the table below:


.. csv-table::

    "**Moment**", ":math:`{u}_{\text{x}}=0` "," :math:`{u}_{\text{z}}=0`"
    "**1**", ":math:`0` "," :math:`-0.0001`"
    "**2**", ":math:`0` "," :math:`0.0001`"
    "**3**", ":math:`0` "," :math:`0.001`"
    "**4**", ":math:`0` "," :math:`0.0005`"
    "**5**", ":math:`0` "," :math:`0.0012`"
    "**6**", ":math:`0` "," :math:`0.0017`"
    "**7**", ":math:`0.001` "," :math:`0.0017`"
    "**8**", ":math:`0` "," :math:`-0.0001`"


Table 1.3-1: Displacements imposed on the upper side