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


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

It is a flat problem. We can look for the solution in the form of a rigid rotation followed by an expansion by a factor :math:`a` in one direction and :math:`b` in the other:


.. image:: images/Object_4.svg
    :width: 449
    :height: 76


.. _RefImage_Object_4.svg:

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


.. image:: images/Object_5.svg
    :width: 449
    :height: 76


.. _RefImage_Object_5.svg:

The behavioral relationship leads to a diagonal Lagrangian stress tensor (with :math:`\lambda` and :math:`\mu` the Lamé coefficients):


.. image:: images/Object_7.svg
    :width: 449
    :height: 76


.. _RefImage_Object_7.svg:

We deduce the Cauchy stress tensor, which is also diagonal:


.. image:: images/Object_8.svg
    :width: 449
    :height: 76


.. _RefImage_Object_8.svg:

Finally, the boundary conditions are written as:


.. image:: images/Object_9.svg
    :width: 449
    :height: 76


.. _RefImage_Object_9.svg:

It is also possible to calculate the forces exerted on the faces:


.. image:: images/Object_10.svg
    :width: 449
    :height: 76


.. _RefImage_Object_10.svg:

where

.. image:: images/Object_11.svg
    :width: 449
    :height: 76


.. _RefImage_Object_11.svg:

represent the initial surfaces of the faces.

Model B reference
------------------------------



The reference results for this modeling are provided by the same calculation to which the rotation step was not applied.


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

As reference results, we adopt the displacements, the Grenn-Lagrange deformations, the Cauchy stresses and the forces exerted on faces :math:`[\mathrm{1,}3]`, :math:`[\mathrm{3,}4]` and :math:`[\mathrm{1,}\mathrm{2,}\mathrm{3,}4]` at the end of loading (:math:`t=2s`).

We're looking for

.. image:: images/Object_12.svg
    :width: 449
    :height: 76


.. _RefImage_Object_12.svg:

such as dilation

.. image:: images/Object_13.svg
    :width: 449
    :height: 76


.. _RefImage_Object_13.svg:

be :math:`p=–26610.3\mathrm{MPa}`.

Dilation :math:`b` and displacements are then:


.. image:: images/Object_16.svg
    :width: 449
    :height: 76


.. _RefImage_Object_16.svg:

The Cauchy constraints are as follows:


.. image:: images/Object_17.svg
    :width: 449
    :height: 76


.. _RefImage_Object_17.svg:

Finally, the forces exerted are:


.. image:: images/Object_18.svg
    :width: 449
    :height: 76


.. _RefImage_Object_18.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