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

The problem comes from an article by J.C. Simo *and*F*. Armero :ref:`[1] <[1]>`.


Geometry
---------

The geometry can be visualized in the figure and will be used for all models.


.. image:: images/10000201000002100000022AE3CEFA7E1D6B4F0C.png
    :width: 3.0626in
    :height: 3.2134in


.. _RefImage_10000201000002100000022AE3CEFA7E1D6B4F0C.png:

Figure 1.1-1: Membrane geometry

Coordinate of Node A (48.60) and Node C (24.30).


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

The material is elastic:


* Modulus of elasticity: :math:`E=\mathrm{206,9}\mathit{GPa}`


* Poisson's ratio: :math:`\mathrm{\nu }=\mathrm{0,29}`

The plasticity model is a Von Mises model with linear isotropic nonlinear work hardening. The work-hardening curve is of the form:


.. math::
 :label: eq-1

    R (p) = {\ mathrm {\ sigma}}} _ {y\ mathrm {\ sigma}} _ {\ mathrm {\ sigma}} _ {y,\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ sigma}}} _ {y\ mathrm {\ sigma}} _ {y\ mathrm {\ sigma}}} _ {y\ mathrm {\ sigma}} _ {y\ mathrm {e}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ sigma}} mathrm {\ delta} p})



* Elastic limit: :math:`{\mathrm{\sigma }}_{y\mathrm{,0}}=450\mathit{MPa}`


* Work hardening limit: :math:`{\mathrm{\sigma }}_{y,\mathrm{\infty }}=715\mathit{MPa}`


* Saturation setting: :math:`\mathrm{\delta }=16.93`


* Work hardening parameter: :math:`H=129.2`

The work hardening curve has been finely tabulated in order to correctly describe the curve with the keyword TRACTION. The deformation model used is GDEF_LOG.


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

We embed the left side of the membrane (:math:`\mathit{DX}=\mathit{DY}=0`), and we apply a vertical surface force on the right side such as :math:`\mathit{FY}=\mathrm{0,3125}\mathit{kN}/\mathit{mm}\mathrm{²}`


Extension to 3D
---------------

The 3D extension is obtained by extruding in the :math:`z` direction (:math:`1\mathit{mm}` thickness). In addition, the movements along DZ are blocked in order to be reduced to the solution in plane deformations.


.. _refheading__26035_1128081541: