Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

It is a flat problem. We can look for the solution in the form of a rigid rotation and an elongation by a factor

.. image:: images/Object_2.svg
    :width: 14
    :height: 18


.. _RefImage_Object_2.svg:

In the direction

.. image:: images/Object_3.svg
    :width: 14
    :height: 18


.. _RefImage_Object_3.svg:

.


.. image:: images/Object_4.svg
    :width: 14
    :height: 18


.. _RefImage_Object_4.svg:

The transformation gradient and the Green-Lagrange deformation are then:


.. image:: images/Object_5.svg
    :width: 14
    :height: 18


.. _RefImage_Object_5.svg:

The behavioral relationship then leads to a diagonal Lagrangian stress tensor:


.. image:: images/Object_6.svg
    :width: 14
    :height: 18


.. _RefImage_Object_6.svg:

The boundary condition of the equilibrium equation then allows us to determine the value of the elongation

.. image:: images/Object_7.svg
    :width: 14
    :height: 18


.. _RefImage_Object_7.svg:

:


.. image:: images/Object_8.svg
    :width: 14
    :height: 18


.. _RefImage_Object_8.svg:

The Cauchy constraint is given by:


.. image:: images/Object_9.svg
    :width: 14
    :height: 18


.. _RefImage_Object_9.svg:

Finally, the force exerted on the faces:


* [:ref:`2,4 <2,4>`]:

    .. image:: images/Object_10.svg
        :width: 14
        :height: 18


    .. _RefImage_Object_10.svg:


* [:ref:`4,3 <4,3>`]:

    .. image:: images/Object_11.svg
        :width: 14
        :height: 18


    .. _RefImage_Object_11.svg:


* [:ref:`1,2,3,4 <1,2,3,4>`]:

    .. image:: images/Object_12.svg
        :width: 14
        :height: 18


    .. _RefImage_Object_12.svg:

where

.. image:: images/Object_13.svg
    :width: 14
    :height: 18


.. _RefImage_Object_13.svg:

represent the initial surfaces of the faces.


Benchmark results
----------------------

Displacements, the Cauchy constraint and the force exerted on the [:ref:`2,4 <2,4>`] and [:ref:`4,3 <4,3>`] faces are adopted as reference results.

**At time** :math:`t=\mathrm{2 }s` **:**

We're looking for

.. image:: images/Object_14.svg
    :width: 14
    :height: 18


.. _RefImage_Object_14.svg:

such as lengthening

.. image:: images/Object_15.svg
    :width: 14
    :height: 18


.. _RefImage_Object_15.svg:

be :math:`T=\mathrm{31 096}.154\mathrm{MPa}`.

The Cauchy constraint is then:


.. image:: images/Object_17.svg
    :width: 14
    :height: 18


.. _RefImage_Object_17.svg:

The forces exerted are:


.. image:: images/Object_18.svg
    :width: 14
    :height: 18


.. _RefImage_Object_18.svg:

**At time** :math:`t=\mathrm{3 }s` **:**

The bar is back in its original state:


.. image:: images/Object_19.svg
    :width: 14
    :height: 18


.. _RefImage_Object_19.svg:


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


1. Eric LORENTZ "A non-linear hyperelastic behavior relationship" Internal Note EDF/DER HI‑74/95/011/0