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


Geometry
---------

According to modeling :math:`\mathrm{2D}` or :math:`\mathrm{3D}`, we consider respectively a square or a cube with side 2 :math:`\text{mm}`.


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

The material obeys the law of brittle elastic behavior ENDO_FISS_EXP with a damage gradient (D_ PLAN_GRAD_VARI and 3D_ GRAD_VARI). The macroscopic data correspond to:


.. csv-table::

    ":math:`E=30000\text{MPa}` ", "Young's module"
    ":math:`\nu =0.2` ", "Poisson's ratio"
    ":math:`{G}_{f}=0.1{\text{N/mm}}^{2}` ", "Cracking energy"
    ":math:`p=5` ", "Shape parameter"
    ":math:`{f}_{t}=2.986\text{MPa}` ", "Traction limit"
    ":math:`{f}_{c}=29.86\text{MPa}` ", "Compression limit"
    "", ""
    ":math:`D=50\text{mm}` ", "Half-width of the damage band"


The choice of these values of*ft* and *fc* in connection with François' criterion in ENDO_FISS_EXP corresponds to a confined traction limit of s*c* = 3 MPa. Moreover, the internal parameters of the model that are involved in the analytical solution are equal to:

:math:`k=1.5\times {10}^{-3}\text{MPa}`; :math:`m=11.111`; :math:`\gamma =9.534\times {10}^{3}`

In the test, they are calculated by the macro command DEFI_MATER_GC.


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

The displacements are imposed at all the nodes of the structure, so as to correspond to the desired homogeneous deformation. More precisely, the displacement in a node with coordinates :math:`X` is equal to: :math:`u(x)=\epsilon \cdot x`. We start by imposing a uniaxial tensile deformation and then unloading until a compression deformation is imposed in order to verify the effect of the restoration of stiffness under compression.


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

None.