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


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

The reference solution is obtained by an analytical calculation.

The imposed displacement :math:`{u}_{y}^{D}` immediately provides the deformation :math:`{\mathrm{\epsilon }}_{\mathit{yy}}=\frac{{u}_{y}^{D}}{1}`. Only this component (corresponding to the non-zero component of the stress tensor) is of interest to us here.

On the other hand, to calculate the behavior, it is necessary to extract the isotropic work hardening function :math:`R(p)` from the data:


.. image:: images/Object_7.svg
    :width: 279
    :height: 249


.. _RefImage_Object_7.svg:



.. math::
 :label: eq-1

    {\ mathrm {\ sigma}} ^ {t} =F (\ mathrm {\ epsilon}) = {\ mathrm {\ sigma}} _ {y} +\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}} p

    R (p) = {\ mathrm {\ sigma}}} _ {y} +\ left (\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}} -\ frac {3} {2} C\ right) p



Elastic behavior
~~~~~~~~~~~~~~~~~~~~~~~~~~

The behavior is elastic up to :math:`t=1`. For :math:`t=1`, the :math:`{\mathrm{\sigma }}_{\mathit{yy}}=E{\mathrm{\epsilon }}_{\mathit{yy}}=400\mathit{MPa}` constraint, it just reaches the plasticity threshold.


Elastoplastic filler
~~~~~~~~~~~~~~~~~~~~~~~~

At :math:`t=2`, we reach the plasticity criterion (in charge or in discharge), we have:


.. math::
 :label: eq-2

    |\ mathrm {\ sigma} -\ frac {3} {2} C {\ mathrm {\ epsilon}} ^ {p} |=R (p)

    \ textrm {and}

    \ mathrm {\ sigma} =E (\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}}} ^ {p})


For the load, it is therefore necessary to solve the following equation:


.. math::
 :label: eq-3

    {\ mathrm {\ sigma}} ^ {t} =F (\ mathrm {\ epsilon}) = {\ mathrm {\ sigma}} _ {y} +\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}} p= {\ mathrm {\ sigma}} _ {y} +\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}}}\ left (\ mathrm {\ epsilon} -\ frac {{\ mathrm {\ sigma}}} ^ {t}} {T}} {E}\ right)


Which brings back :math:`{\mathrm{\sigma }}^{t}={\mathrm{\sigma }}_{y}+{E}_{T}\left(\mathrm{\epsilon }-\frac{{\mathrm{\sigma }}_{y}}{E}\right)`. We move on the traction curve to point :math:`A` such that:


.. math::
: label: eq-4

    {\ mathrm {\ sigma}} _ {A} ^ {t} =E\ left ({\ mathrm {\ epsilon}} _ {A} - {\ mathrm {\ epsilon}}} _ {A} ^ {p}\ right)



Elastic discharge
~~~~~~~~~~~~~~~~~~~~

At :math:`t=3`, the discharge is elastic up to point :math:`A’`:


.. math::
: label: eq-5

    - {\ mathrm {\ sigma}} ^ {A\ text {'}}} -\ frac {3} {2} C {\ mathrm {\ epsilon}}} _ {A} ^ {p} =R ({p} _ {A})

    {\ mathrm {\ sigma}} ^ {A\ text {'}} =E\ left ({\ mathrm {\ epsilon}}} _ {A\ text {'}}} - {\ mathrm {\ epsilon}}} - {\ mathrm {\ epsilon}}} _ {A} ^ {p}\ right)



Elastoplastic load under compression
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

At :math:`t=4`:


.. math::
 :label: eq-6

    {\ mathrm {\ sigma}} ^ {c} =\ frac {3} {2} C {\ mathrm {\ epsilon}} ^ {p} -R (p)



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


+---------+------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------+
|:math:`t`|                                                                                    |                                                                                                         |
+         + .. image:: images/Object_16.svg                                                    + .. image:: images/Object_17.svg                                                                         +
|         |    :width: 279                                                                     |    :width: 26                                                                                           |
+         +    :height: 249                                                                    +    :height: 26                                                                                          +
|         |                                                                                    |                                                                                                         |
+         +                                                                                    + :math:`(\mathrm{MPa})`                                                                                  +
|         |                                                                                    |                                                                                                         |
+---------+------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------+
|1        | #. 10—3                                                                            |400                                                                                                      |
+         +                                                                                    +                                                                                                         +
|         |                                                                                    |                                                                                                         |
+---------+------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------+
|2        |4.5d—3                                                                              |500                                                                                                      |
+---------+------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------+
|3        |0.1d—3                                                                              |—380                                                                                                     |
+---------+------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------+
|4        |—2. 10—3                                                                            |—464                                                                                                     |
+---------+------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------+


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

Analytical solution.


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


* Behavioral relationship to linear and isotropic nonlinear kinematic work hardening. Note [:external:ref:`R5.03.16 <R5.03.16>`].