.. _R3.06.08:

**r3.06.08** Finite elements dealing with almost incompressibility
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**Summary:**

In certain situations, the mechanical behavior of the material requires that the volume expansion remains zero,
In other words, the deformation takes place at constant volume: isotropic elasticity with Poisson's ratio
equal to :math:`0.5`, perfect plastic flows in limit analysis...

It is proposed here to treat this condition of "incompressibility" or "almost incompressibility" using a
formulation valid both in the compressible case and in the almost incompressible case. To do this, we use
a variational formulation with 3 fields where the unknowns are displacement, volume deformation and
the associated Lagrange multiplier (which would correspond to the pressure in the incompressible case).
Two versions of this formulation are proposed: one for small deformations, the other valid in the presence of
large deformations. In the situation of a one-to-one relationship between pressure and swelling,
In the case of Von Mises plasticity, it is possible to eliminate the unknown of swelling.
We then have a formulation with two displacement/pressure fields.

After a few reminders of the difficulties of solving incompressible problems, we describe
implanted mixed finite elements (in 3D and 2D, plane and axisymmetric in small and large deformations),
and we also present the main lines of integration in *code_aster* (models "INCO_UP", "INCO_UPG", "INCO_UPO").

This modeling is necessary to practice boundary analyses and to model elastic behaviors.
for Poisson coefficients close to :math:`0.5`. It can also be useful in the case of models
causing strong plastic deformations and for which traditional models may be insufficient
and generate stress oscillations.


.. toctree::
    :hidden:

    self

.. toctree::
    :maxdepth: 2
    :numbered:

    Difficult_s_li_es_au_traitement_de_l_incompressibilit_
    Formulation_variationnelle_mixte_du_probl_me
    Discr_tisation_par__l_ments_finis_mixtes
    Int_gration_dans_Code_Aster_des__l_ments_finis_incompressibles
    Validation_
    Bibliographie
    Description_des_versions