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


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

We are looking for the :math:`U` displacement field in the form:


.. image:: images/Object_6.svg
    :width: 132
    :height: 76


.. _RefImage_Object_6.svg:

The transformation gradient, the deformation and its mechanical part are then:


.. image:: images/Object_7.svg
    :width: 132
    :height: 76


.. _RefImage_Object_7.svg:

with:


.. image:: images/Object_8.svg
    :width: 132
    :height: 76


.. _RefImage_Object_8.svg:

**Note:**

:math:`{({E}^{m})}_{\text{eq}}\mathrm{=}\mathrm{\mid }a\mathrm{-}b\mathrm{\mid }\mathrm{=}a\mathrm{-}b` (it is assumed that :math:`a>b`)

The behavioral relationship is written as:


.. image:: images/Object_10.svg
    :width: 132
    :height: 76


.. _RefImage_Object_10.svg:

with:


.. image:: images/Object_11.svg
    :width: 132
    :height: 76


.. _RefImage_Object_11.svg:

To determine

.. image:: images/Object_12.svg
    :width: 132
    :height: 76


.. _RefImage_Object_12.svg:

taking into account linear work hardening, the following are introduced:


* the shear modulus:

    .. image:: images/Object_13.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_13.svg:


* the work hardening module:

    .. image:: images/Object_14.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_14.svg:

,

The "internal pseudo-variable"

.. image:: images/Object_15.svg
    :width: 132
    :height: 76


.. _RefImage_Object_15.svg:

It is then worth:


.. image:: images/Object_16.svg
    :width: 132
    :height: 76


.. _RefImage_Object_16.svg:

Finally,

.. image:: images/Object_17.svg
    :width: 132
    :height: 76


.. _RefImage_Object_17.svg:

is written:


.. image:: images/Object_18.svg
    :width: 132
    :height: 76


.. _RefImage_Object_18.svg:

Taking into account boundary conditions:


.. image:: images/Object_19.svg
    :width: 132
    :height: 76


.. _RefImage_Object_19.svg:

The system to be solved is written as:


.. image:: images/Object_20.svg
    :width: 132
    :height: 76


.. _RefImage_Object_20.svg:

It is also written:


.. image:: images/Object_21.svg
    :width: 132
    :height: 76


.. _RefImage_Object_21.svg:

A

.. image:: images/Object_22.svg
    :width: 132
    :height: 76


.. _RefImage_Object_22.svg:

fixed, so it is a non-linear system in

.. image:: images/Object_23.svg
    :width: 132
    :height: 76


.. _RefImage_Object_23.svg:

and

.. image:: images/Object_24.svg
    :width: 132
    :height: 76


.. _RefImage_Object_24.svg:

, since

.. image:: images/Object_25.svg
    :width: 132
    :height: 76


.. _RefImage_Object_25.svg:

Is quadratic in

.. image:: images/Object_26.svg
    :width: 132
    :height: 76


.. _RefImage_Object_26.svg:

and

.. image:: images/Object_27.svg
    :width: 132
    :height: 76


.. _RefImage_Object_27.svg:

quadratic in

.. image:: images/Object_28.svg
    :width: 132
    :height: 76


.. _RefImage_Object_28.svg:

.

However, one can choose to fix

.. image:: images/Object_29.svg
    :width: 132
    :height: 76


.. _RefImage_Object_29.svg:

(so

.. image:: images/Object_30.svg
    :width: 132
    :height: 76


.. _RefImage_Object_30.svg:

) and solve a linear system by

.. image:: images/Object_31.svg
    :width: 132
    :height: 76


.. _RefImage_Object_31.svg:

and

.. image:: images/Object_32.svg
    :width: 132
    :height: 76


.. _RefImage_Object_32.svg:

(from which we deduct

.. image:: images/Object_33.svg
    :width: 132
    :height: 76


.. _RefImage_Object_33.svg:

and

.. image:: images/Object_34.svg
    :width: 132
    :height: 76


.. _RefImage_Object_34.svg:

):


*

    .. image:: images/Object_35.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_35.svg:


*

    .. image:: images/Object_36.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_36.svg:


*

    .. image:: images/Object_37.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_37.svg:


*

    .. image:: images/Object_38.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_38.svg:

It then remains to express Cauchy's constraint:


.. image:: images/Object_39.svg
    :width: 132
    :height: 76


.. _RefImage_Object_39.svg:

Or here:


.. image:: images/Object_40.svg
    :width: 132
    :height: 76


.. _RefImage_Object_40.svg:

As for the force exerted on the face [:ref:`3,4 <3,4>`], due to the hypothesis of dead charges, it is simply written:


.. image:: images/Object_41.svg
    :width: 132
    :height: 76


.. _RefImage_Object_41.svg:


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

As reference results, we will adopt displacements, the Cauchy constraint and the force exerted on face :math:`\mathrm{[}\mathrm{3,4}\mathrm{]}` (in :math:`\mathrm{3D}` only):

**At time** :math:`t\mathrm{=}\mathrm{2 }s` (:math:`\Delta T\mathrm{=}100°C`, traction

.. image:: images/Object_43.svg
    :width: 132
    :height: 76


.. _RefImage_Object_43.svg:

)

In fact, we're looking for

.. image:: images/Object_44.svg
    :width: 132
    :height: 76


.. _RefImage_Object_44.svg:

such as elongation:


.. image:: images/Object_45.svg
    :width: 132
    :height: 76


.. _RefImage_Object_45.svg:


* :math:`K\mathrm{=}166\mathrm{666 }\mathit{MPa}` :math:`\mu \mathrm{=}\mathrm{76 923 }\mathit{MPa}` :math:`R\text{'}\mathrm{=}\mathrm{2 020 }\mathit{MPa}`


* :math:`a\mathrm{=}0.095`


*

    .. image:: images/Object_50.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_50.svg:


*

    .. image:: images/Object_51.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_51.svg:


*

    .. image:: images/Object_52.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_52.svg:


*

    .. image:: images/Object_53.svg
        :width: 132
        :height: 76


    .. _RefImage_Object_53.svg:

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

.. image:: images/Object_54.svg
    :width: 132
    :height: 76


.. _RefImage_Object_54.svg:

The bar is back in its original state:


.. image:: images/Object_55.svg
    :width: 132
    :height: 76


.. _RefImage_Object_55.svg:


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

The solution is analytical. Except for rounding errors, it can be considered accurate.


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

Reference may be made to:


1. E. LORENTZ: A nonlinear hyperelastic behavior relationship - Internal note EDF DER HI-74/95/011/0