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


Calculation method used for the reference solution [:ref:`bib1 <bib1>`]
-----------------------------------------------------------------------------


Isotropic work hardening: analytical
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`t=\mathrm{1s}` **(** :math:`T=0°C` **)**:

.. image:: images/Object_5.svg
    :width: 154
    :height: 50


.. _RefImage_Object_5.svg:

with

.. image:: images/Object_6.svg
    :width: 154
    :height: 50


.. _RefImage_Object_6.svg:

either

.. image:: images/Object_7.svg
    :width: 154
    :height: 50


.. _RefImage_Object_7.svg:

**Heating:** Maximum plastic deformation at :math:`t=\mathrm{1.45222s}` (:math:`T=45.222°C`):


.. image:: images/Object_8.svg
    :width: 154
    :height: 50


.. _RefImage_Object_8.svg:

either

.. image:: images/Object_9.svg
    :width: 154
    :height: 50


.. _RefImage_Object_9.svg:

Then: the plastic deformation no longer evolves.


Linear kinematic work hardening: analytics
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`t=\mathrm{1s}` **(** :math:`T=0°C` **)**:

.. image:: images/Object_10.svg
    :width: 154
    :height: 50


.. _RefImage_Object_10.svg:

**Heating:** Constant plastic deformation up to :math:`t=356/316=\mathrm{1.12658s}` (:math:`T=12.658°C`):

Then the plastic deformation decreases to reach :math:`t=\mathrm{2s}`:

.. image:: images/Object_11.svg
    :width: 154
    :height: 50


.. _RefImage_Object_11.svg:


Nonlinear kinematic hardening I: analytical
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`t=\mathrm{1s}` **(** :math:`T=0°C` **)**:

.. image:: images/Object_12.svg
    :width: 154
    :height: 50


.. _RefImage_Object_12.svg:

with :math:`A=\frac{C}{D}=100` either

.. image:: images/Object_14.svg
    :width: 154
    :height: 50


.. _RefImage_Object_14.svg:

**Heating:** Maximum plastic deformation at :math:`t=\mathrm{1.26011s}` (:math:`T=26.011°C`):


.. image:: images/Object_15.svg
    :width: 154
    :height: 50


.. _RefImage_Object_15.svg:

either

.. image:: images/Object_16.svg
    :width: 154
    :height: 50


.. _RefImage_Object_16.svg:

Plastic deformation no longer evolves until :math:`{t}_{1}=\mathrm{1.98332s}` (:math:`{T}_{1}=98.332°C`) where the other end of the elasticity domain is encountered.

Then: the plastic deformation decreases to reach :math:`t=\mathrm{2s}`:


.. image:: images/Object_17.svg
    :width: 154
    :height: 50


.. _RefImage_Object_17.svg:

with

.. image:: images/Object_18.svg
    :width: 154
    :height: 50


.. _RefImage_Object_18.svg:

either

.. image:: images/Object_19.svg
    :width: 154
    :height: 50


.. _RefImage_Object_19.svg:


Nonlinear kinematic hardening II
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Comparison to the reference solution proposed on day :math:`{\Phi }^{2}\mathrm{As}`. (numerical result obtained with 10 time steps), and comparison to the results obtained with *Code_Aster* with very fine time discretization


.. csv-table::

    "Plastic deformation :math:`\mathrm{YY}` ", "Fine Aster calculation: 100 steps up to :math:`\mathrm{1.26s}`,
    100 steps between 1.98 and 2s", "Reference :math:`{\Phi }^{2}\mathrm{As}`:
    Result for "10 steps"
    ":math:`t=\mathrm{1s}` ", "2.1072 10—03", "2.1072 10—03"
    ":math:`\mathrm{1.26s}<t<\mathrm{1.98s}` ", "4.18947 10—03", "4.38 10—03"
    ":math:`t=\mathrm{2s}` ", "4.12131 10—03", "4.32 10—03"



Precision on the reference results
----------------------------------------

We have an analytical solution for the first three behaviors, so the uncertainty is zero. It is estimated at :math:`\text{4\%}` for the fourth (difference between the result for 10 steps and the result for 200 steps, the solution strongly dependent on time discretization).


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


1. IPSI: Phi2as study day on 30 March 2000 on the nonlinear behaviors of materials.