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


Geometry
---------


.. image:: images/10000000000000E50000010ECCE334BD207B5CD4.jpg
    :width: 1.9098in
    :height: 2.252in


.. _RefImage_10000000000000E50000010ECCE334BD207B5CD4.jpg:


.. csv-table::

    "Point coordinates", ":math:`X` "," :math:`Y`"
    ":math:`A` ", "0", "0"
    ":math:`B` ", "0.5", "0"
    ":math:`C` ", "0.5", "1.0"
    ":math:`D` ", "0", "1,0"


The geometry is rectangular in shape.


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

For modeling A, the mass consists of an elasto-plastic material with linear negative work-hardening.

For modeling B, the mass consists of an elasto-plastic material with parabolic negative work hardening.

The elastic parameters of the material are as follows:


* Young's module: :math:`E=5800\mathrm{MPa}`


* Poisson's ratio: :math:`\nu =\mathrm{0,3}`


* Real constant density: :math:`\rho \mathrm{=}2764`


* Isotropic thermal expansion coefficient: :math:`\alpha =0`

The characteristics of work hardening are then given by:


* Pressure dependence coefficient: :math:`\alpha =\mathrm{0,33}`


* Ultimate cumulative plastic deformation: :math:`{P}_{\mathrm{ULT}}=\mathrm{0,01}`


* Elastic limit: :math:`{\sigma }_{y}\mathrm{=}2.11{10}^{6}`


* Elastic limit for the mesh concerned: :math:`{\sigma }_{y}\mathrm{=}2.\mathrm{\times }{10}^{6}`

For modeling A:


* Work hardening module for the entire mesh: :math:`H\mathrm{=}\mathrm{-}200.\mathrm{\times }{10}^{6}`

For modeling B:


* Ultimate constraint: :math:`{\sigma }_{\mathit{yULT}}\mathrm{=}0.47\mathrm{\times }{10}^{6}`


* Ultimate constraint for the mesh concerned: :math:`{\sigma }_{\mathit{yULT}}\mathrm{=}0.44\mathrm{\times }{10}^{6}`


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

A drained biaxial test (D_ PLAN) is modelled. The trips normal to the study plan are therefore zero. A vertical displacement is imposed on :math:`\mathrm{[}\mathit{DC}\mathrm{]}` while maintaining the constant lateral pressure (:math:`2\mathit{MPa}`) in the study design. The boundary conditions are therefore as follows:

:math:`{u}_{y}\mathrm{=}0` out of :math:`\mathrm{[}\mathit{AB}\mathrm{]}` (mesh group :math:`\mathit{BAS}`)

:math:`{u}_{x}\mathrm{=}0` out of :math:`\mathrm{[}\mathit{AD}\mathrm{]}` (mesh group :math:`\mathit{GAUCHE}`)

:math:`{\sigma }_{n}\mathrm{=}2.\mathrm{\times }{10}^{6}` out of :math:`\mathrm{[}\mathit{BC}\mathrm{]}` (mesh group :math:`\mathit{EXTREM}`)

A vertical displacement is then imposed on :math:`\mathrm{[}\mathit{DC}\mathrm{]}` (mesh group :math:`\mathit{HAUT}`) to apply vertical deformation up to :math:`\text{3\%}`.


Results
---------

The solutions are post-treated in order to control the evolution of the quantities of INDL_ELGA and PDIL_ELGA in non-regression.