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


Geometry
---------


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The simple tensile test is carried out on a single isoparametric finite element of cubic shape :math:`\mathit{CUB}4`. The length of each edge is 1. The different facets of this cube are mesh groups named :math:`\mathrm{HAUT}`, :math:`\mathrm{BAS}`,, :math:`\mathrm{DEVANT}`, :math:`\mathrm{DERRIERE}`, :math:`\mathrm{DROIT}`, and :math:`\mathrm{GAUCHE}`. The mesh group *SYM* also contains the mesh groups :math:`\mathrm{BAS}`, :math:`\mathrm{DEVANT}` and :math:`\mathrm{GAUCHE}`; the group of elements :math:`\mathrm{COTE}` the mesh groups :math:`\mathrm{DERRIERE}` and :math:`\mathrm{DROIT}`.


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

The elastic properties are:


*
    * Young's modulus: :math:`E=1\mathit{MPa}`


    * Poisson's ratio: :math:`\mathrm{\nu }=\mathrm{0,25}`

The traction limit is equal to :math:`{\mathrm{\sigma }}_{t}=1\mathit{kPa}`


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

The simple tensile test consists in imposing a vertical elongation on the specimen while maintaining the lateral pressure constant and equal to the initial isotropic stress :math:`{P}_{0}=10\mathit{kPa}`

In the model under consideration, the cubic element represents one eighth of the sample. The boundary conditions are therefore as follows:


*
    *
        * Symmetry conditions:


            * :math:`{u}_{z}=0` on mesh group :math:`\mathrm{BAS}`


            * :math:`{u}_{x}=0` on mesh group :math:`\mathrm{GAUCHE}`


            * :math:`{u}_{y}=0` on mesh group :math:`\mathrm{DEVANT}`


        * Lateral pressure conditions:


            * :math:`{P}_{n}={P}_{0}=10\mathit{kPa}` on mesh groups :math:`\mathit{DROIT}` and :math:`\mathit{ARRIERE}`


        * Loading conditions:


            * :math:`{u}_{z}=+1` on mesh group :math:`\mathrm{HAUT}`

The loading takes place in 30 steps of time between :math:`t=0` and :math:`t=30` during which the displacement imposed on the group of elements :math:`\mathrm{HAUT}` varies from :math:`{u}_{z}=0` to :math:`{u}_{z}=0.3` (total vertical deformation of :math:`30\text{\%}`).


Results
---------

The solutions are post-treated at point :math:`C`, in terms of:


* vertical constraint :math:`{\mathrm{\sigma }}_{\mathit{zz}}`;


* horizontal deformation :math:`{\mathrm{ϵ}}_{\mathit{xx}}`;


* :math:`{e}^{P}=\Vert {e}^{P}\Vert` deviatoric plastic deformation norm

They are compared to an analytical solution (described in the next paragraph) in terms of *maximum difference between* :math:`t=0` *and* :math:`t=20`.