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


Geometry
---------

It is an elastic cube (or square in 2D), with a side of 1,000 mm and crossed by an oblique 45° interface. The geometry is shown in figure, with the interface in yellow and the associated coordinate system at the bottom left. The cube is positioned so that :math:`0\le x\le 1000`, :math:`0\le y\le 1000`, and :math:`0\le z\le 1000`.


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+ .. image:: images/100000000000044D000002725A31EA5E5ADE5C47.png                                                              +
|    :width: 3.9366in                                                                                                         |
+    :height: 2.2165in                                                                                                        +
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Figure 1.1-a: geometry of the structure studied


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

The volume material obeys an elastic law and the interface to the CZM_ELAS_MIX law. The characteristics are shown below:


.. csv-table::

    ":math:`E=30000\text{MPa}` ", "Young's module"
    ":math:`\nu =0.2` ", "Poisson's ratio"
    ":math:`{k}_{n}=20\text{MPa/mm}` ", "Interface stiffness in the normal direction"
    ":math:`{k}_{t}=40\text{MPa/mm}` ", "Interface stiffness in the tangent direction"


Moreover, the Lagrangian coefficient of increase (PENA_LAGR_ABSO), which has no impact here because the solution is homogeneous, is fixed at :math:`r=600\text{MPa/mm}`.

Finally, the six possible combinations of adhesion are examined: normal elastic, unilateral or perfect adhesion and elastic or perfect tangential adhesion.


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

The cube is subject to a :math:`\sigma ={\sigma }_{0}\text{}{e}_{x}\otimes {e}_{x}` constraint. More specifically:


* Side :math:`x=0` is stuck in :math:`{u}_{x}=0`


* Side :math:`y=0` is stuck in :math:`{u}_{y}=0`


* Side :math:`z=0` is stuck in :math:`{u}_{z}=0`


* Face :math:`x=1000\text{mm}` is subject to surface force :math:`T={\sigma }_{0}{e}_{x}`

These conditions also make it possible to block rigid body movements. The intensity of the stress :math:`{\sigma }_{0}` takes the values +2 MPa and -2 MPa successively.


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

Since this is an elastic problem, there is no need for initial conditions.