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


Geometry
---------


.. image:: images/100000000000029C00000258BC7ED10621CE11D6.png
    :width: 3.2681in
    :height: 2.7563in


.. _RefImage_100000000000029C00000258BC7ED10621CE11D6.png:

**Figure 1.1-a**

The geometry refers to a single finite isoparametric element of square shape, the length of each edge is equal to 1.


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

The mass consists of an elasto-plastic material modelled by a Drucker Prager law of behavior, associated or not.

The material elasticity parameters are as follows:


* Isotropic elasticity Young's modulus: :math:`E={10}^{9}\mathrm{Pa}`


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


* Real constant density: :math:`\mathrm{\rho }=2764\mathit{kg}\mathrm{.}{m}^{-3}`


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

The characteristics of work hardening are then given by:


* Pressure dependence coefficient: :math:`\alpha =\mathrm{0,328}`


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


* Ultimate constraint: :math:`{\mathrm{\sigma }}_{\mathit{ULTM}}=1.0\times {10}^{6}`

**For modeling A:**

Ultimate cumulative plastic deformation: :math:`{P}_{\mathrm{ULT}}=2`

**For B modeling:**

Ultimate cumulative plastic deformation: :math:`{P}_{\mathit{ULT}}\mathrm{=}1.225\mathrm{\times }{10}^{\mathrm{-}2}`


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

A unit displacement is imposed along the :math:`y` axis on the :math:`\mathrm{CD}` segment, zero along the :math:`y` axis on the :math:`\mathrm{AB}` segment and zero along the :math:`x` axis on the :math:`\mathrm{AD}` segment.

In this test case, concerning modeling A, we simulate a case of perfect plasticity with the two Drucker-Prager behavior laws in associated and non-associated conditions to verify the computational consistency of the results. To do this, simply take large values for :math:`{P}_{\mathit{ULT}}`.