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


.. _Ref482175678:


Geometry
---------

Consider a :math:`\mathrm{2D}` square mesh:

Y

L32

P1

P2

L21

L43

P3

P4

X

L14

The sides :math:`\mathrm{L21}`, :math:`\mathrm{L32}`, :math:`\mathrm{L43}`, :math:`\mathrm{L14}` each measure :math:`10\mathrm{mm}`.


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

We take: :math:`E=200\mathrm{GPa}`, and :math:`\nu =\mathrm{0,3}`.

The traction curve used is given in the following table:


.. csv-table::

    ":math:`\varepsilon` ", "0.0001", "0.00338", "0.00338", "0.03", "0.03", "0.04", "0.07", "0.10", "0.15", "0.2", "0.2", "0.2", "0.2", "0.2", "0.3", "0.4"
    ":math:`\sigma` ", "27.30", "222.72", "222.72", "519.58", "519.58", "580.94", "633.48", "721.82", "828.96", "970.19", "970.19", "970.19", "1084.75", "1084.75", "1269.57", "1419.48"



.. csv-table::

    ":math:`\varepsilon` ", "0.5", "0.7", "1.0", "1.0", "1.5", "2.0", "", "", "", "", "", ""
    ":math:`\sigma` ", "1547.86", "1763.72", "1763.72", "2025.50", "2025.50", "", "", "", "", "", "", ""


The Rousselier model is used in three configurations with the following parameters:


.. csv-table::

    "**Elasto-plastic base model (** **ROUSS_PR** ****) **", "**Elasto-plastic model (**** ROUSS_PR **** **) with germination**", "**Visco-plastic model (** **** **** **)**) **", "**Visco-plastic model (**** **VISCOROUSS** **) and theta-method**"
    "1. :math:`D=2.`
    2. :math:`{\sigma }_{1}=600\mathrm{MPa}`
    3. :math:`\lambda =1.`
    4. :math:`\mathrm{f0}=1.e-4` (initial porosity)
    5. :math:`\mathrm{fc}=1.` (critical porosity)
    6. :math:`A=1.` ", "1. :math:`D=2.`
    2. :math:`{\sigma }_{1}=600\mathrm{MPa}`
    3. :math:`\lambda =1.`
    4. :math:`\mathrm{f0}=1.e-4`
    5. :math:`\mathrm{fc}=1.`
    6. :math:`A=1.`
    7. :math:`\mathrm{An}=0.6` ", "1. :math:`D=2.`
    2. :math:`{\sigma }_{1}=600\mathrm{MPa}`
    3. :math:`\lambda =1.`
    4. :math:`\mathrm{f0}=1.e-4`
    5. :math:`\mathrm{fc}=1.`
    6. :math:`A=1.`
    7. :math:`{\sigma }_{0}=27\mathrm{MPa}`
    8. :math:`{\varepsilon }_{0}=1.e-2`
    9. :math:`\theta =0.57`
    10. :math:`m=2`"



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

With reference to the figure in [:ref:`§1.1 <§1.1>`] the boundary conditions are as follows:


1. on the edge :math:`L32` move

    .. image:: images/Object_1.svg
        :width: 9
        :height: 17


    .. _RefImage_Object_1.svg:

imposed in direction :math:`\mathrm{OY}` (monotonic traction),


2. :math:`\mathrm{L21}` movements blocked following :math:`X`,


3. :math:`\mathrm{L14}` movements blocked following :math:`Y`.

The temporal evolution of elongation

.. image:: images/Object_2.svg
    :width: 9
    :height: 17


.. _RefImage_Object_2.svg:

are shown in the following table:


+-------------------------------------------------------------------------------------------------------------------------------------------------+++
|Time :math:`[s]`                                                                                                                                 |||
+-------------------------------------------------------------------------------------------------------------------------------------------------+++
|Displacement                                                                                                                                     |||
+                                                                                                                                                 +++
|                                                                                                                                                 |||
+ .. image:: images/Object_3.svg                                                                                                                  +++
|    :width: 9                                                                                                                                    |||
+    :height: 17                                                                                                                                  +++
|                                                                                                                                                 |||
+                                                                                                                                                 +++
|                                                                                                                                                 |||
+                                                                                                                                                 +++
|                                                                                                                                                 |||
+ :math:`[\mathrm{mm}]`                                                                                                                           +++
|                                                                                                                                                 |||
+-------------------------------------------------------------------------------------------------------------------------------------------------+++


The evolution is linear between the two moments.


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

Zero stresses and deformations.