Reference problem
---------------------

.. _Toc93483749:

Geometry and meshing
~~~~~~~~~~~~~~~~~~~~~~~

.. figure:: images/10000201000005AD0000022F6C851186B1143700.png
 :name: fig1-geom-mail
    :width: 80%

    **Geometry and mesh of the embankment dam**

.. _RefImage_10000201000005AD0000022F6C851186B1143700.png:


We consider an embankment dam with a clay core, composed of the foundation,
gravel between the foundation and the body of the dam, of a clay core characterized by a
relatively low permeability coefficient, and backfill refills upstream and downstream.

The mesh is shown in :numref:`fig1-geom-mail`, where the various
Components of the dam are shown
by different colors.


.. _Toc93483751:


Material properties
~~~~~~~~~~~~~~~~~~~~~~

.. _table1_v101135 :

**Table 1**: Property of dam materials

.. csv-table::
 :name: tab-1
    :header-rows: 1
    :widths: auto
    :align: center

    "", "**Foundation**", "**Gravel**", "**Core**", "**Recharge**"
    "Density :math:`\rho` (kg/m3)", "2250", "2450", "1900", "2450"
    "Young's Module :math:`E` (MPa)", "250", "250", "250", "250", "250"", "250"
    "Poisson's Ratio :math:`\nu` ", "0.3", "0.3", "0.3", "0.3", "0.3"
    "Permeability :math:`k` (cm/s)", "1e-5", "1e-5", "1e-3", "2e-6", "1e-3"
    "Cohesion force :math:`c` (kPa)", "40", "40", "44.5", "40"
    "Friction angle :math:`\varphi` (deg)", "35", "35", "20", "23.3", "20"


The mechanical and hydraulic properties of the materials in the model are shown in :ref:`Tableau 1<table1_v101135>`.

For the Mohr-Coulomb law, we hypothesize the associated plastic flow law.
The parameters of the Drucker-Prager law were obtained by the following formulas:

:math:`A=\frac{2\mathrm{sin}(\varphi )}{3-\mathrm{sin}(\varphi )},{\sigma }_{y}=\frac{6c\mathrm{cos}(\varphi )}{3-\mathrm{sin}(\varphi )}`

where the cohesive force and the friction angle take the values in :ref:`Tableau 1<table1_v101135>`.
The angle of expansion is equal to the angle of friction in the Drucker-Prager law.

In order to simplify the calculation, the ultimate cumulative plastic deformation :math:`{p}_{\mathit{ultm}}` is
considered null (= 0).

The porosity is uniform in the model and is equal to 0.672.


.. _Toc93483752:


Boundary conditions and loads
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The boundary conditions applied to the model are as follows:


* Bottom of the embedded foundation, and the vertical sides normally blocked.
* Hydraulic pressure from the upstream coast of 77.19m (the elevation of the peak = 94.2m).
* Hydraulic pressure defined from the level of the groundwater downstream, which is equal to 43m.

The following mechanical loads are applied to the model:

* Gravity with :math:`g=\mathrm{9,81}m/{s}^{2}`
* The hydrostatic pressure on the upper surface of the dam.