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


.. _Ref482175678:


Geometry
---------

The geometry is chosen deliberately simple, to translate a state of homogeneous stresses and deformations, as is the case in uniaxial traction. In the case :math:`1D` and :math:`\mathrm{3D}`, this is a bar with a diameter :math:`\varphi =6\mathit{mm}` and a length of :math:`L=0.1m`. In :math:`\mathrm{3D}`, we only mesh a quarter of the bar. The traction is carried out at an imposed displacement in :math:`\mathrm{1D}` and :math:`\mathrm{3D}` with the length of the test piece. In case COQUE_3D, the side of the plate is square, and :math:`\pi \frac{{\varphi }^{2}}{4}` thick (so that the forces are identical to case :math:`\mathrm{1D}`).


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:


.. image:: images/Shape3.gif


.. _RefSchema_Shape3.gif:

For the case of flexure with thermal expansion, 4 bars are used instead of one, offset symmetrically in Y and Z of :math:`{e}_{y}={e}_{z}=2.5\mathit{mm}`.


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

YOUNG module = :math:`2.E11\mathrm{Pa}`

Poisson's ratio :math:`\nu =0.3`

Thermal expansion coefficient :math:`\alpha =1.2E-5`

Keyword CORR_ACIER:

Damage coefficient D_ CORR = 0.2

Work hardening parameters ECRO_K = :math:`500\mathrm{MPa}`

.. csv-table::

    "Elastic limit", "ECRO_M = :math:`2.781`"
    "", "SY = :math:`500.\mathrm{MPa}`"



Pure traction modeling
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The degree of corrosion is determined using the CREA_CHAM command:

NOM_CMP = 'CORR' VALE = 0.0%, 2.5%, and 13%


Flexural modeling with thermal expansion
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The degree of corrosion is zero:

NOM_CMP = 'CORR' VALE = 0.0% The imposed temperature field varies linearly from :math:`{T}_{\mathit{ini}}=20°` to :math:`{T}_{\mathit{final}}=120°`.


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


1D modeling
~~~~~~~~~~~~~~~~

Moves :math:`\mathit{DX}` :math:`\mathit{DZ}` and :math:`\mathit{DY}` blocked at point :math:`A`

Displacement :math:`\mathit{DX}` and :math:`\mathit{DZ}` imposed at point :math:`B`


3D modeling
~~~~~~~~~~~~~~~~

Displacement :math:`\mathit{DX}` prevented on :math:`{\mathit{Surf}}_{\text{int}}`

Displacement :math:`\mathit{DY}` prevented on :math:`{\mathit{Surf}}_{1}`

Displacement :math:`\mathit{DZ}` prevented on :math:`{\mathit{Surf}}_{2}`

Displacement :math:`\mathit{DX}` imposed on :math:`{\mathit{Surf}}_{\mathit{ext}}`

Modeling COQUE_3D:

Displacement :math:`\mathit{DX}` prevented on :math:`\mathit{Li4}`

Displacement :math:`\mathit{DY}` prevented on :math:`\mathit{Li1}`

Displacement :math:`\mathit{DX}` imposed on :math:`\mathit{Li3}`


Flexural modeling with thermal expansion
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Embedding at point :math:`A`,

According to :math:`Y`, a force :math:`{F}_{y}` is imposed at the point :math:`B` in order to be placed at the elastic limit.

Knowing that at point :math:`A` we have :math:`{M}_{z}={F}_{y}L` on the one hand, and :math:`{M}_{z}=4{e}_{y}{\sigma }_{\mathit{xx}}\pi \frac{{\varphi }^{2}}{4}` on the other hand, and that at the elastic limit :math:`{\sigma }_{\mathit{xx}}=\mathit{SY}` we then impose :math:`{F}_{z}=4{e}_{y}\mathit{SY}\pi \frac{{\varphi }^{2}}{4}\frac{1}{L}\simeq 1413.7167N`.


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

Zero stresses and deformations.