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


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

Since the loading history is very simple, the reference results can be obtained manually by applying the algorithms presented in the reference document [:ref:`R7.04.01 <R7.04.01>`].


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


Modeling A
~~~~~~~~~~~~~~~

.. csv-table::

    ":math:`t` "," :math:`D(t)` **** (Too bad) **"
    "43.11", "0."
    "100. ", "0.000848907"
    "1000. ", "0.014474925"
    "10000. ", "0.178374238"
    "20000. ", "0.524693005"
    "21000. ", "0.602827469"
    "2000. ", "0.73829052"
    "2200. ", "0.792149807"
    "22400. ", "0.967604351"



B modeling
~~~~~~~~~~~~~~~

The reference results for test case number 2 are obtained using a spreadsheet in which the Lemaître-Sermage damage expression was implemented according to a numerical integration diagram identical to that used in routine POST_FATIGUE of*Code_Aster*.

First, we check that the uncertainty in the results obtained for the value :math:`s=1.0` via the spreadsheet is acceptable:


.. csv-table::

    "**Damage**
    **(Excel calculation)**", "**Damage**
    **(reference solution)**", "**Difference (** :math:`\text{\%}` **)**"
    "0.0000000000", "0.0000000000"," 0.00000%"
    "0.0008489062", "0.0008489070"," -0.00010%"
    "0.0144749268", "0.0144749250"," 0.00001%"
    "0.1783742841", "0.1783742380"," 0.00003%"
    "0.5246932887", "0.5246930050"," 0.00005%"
    "0.6028278917", "0.6028274690"," 0.00007%"
    "0.7382915411", "0.7382905200"," 0.00014%"
    "0.7921514337", "0.7921498070"," 0.00021%"
    "0.9676720845", "0.9676043510"," 0.00700%"


In a second step, a reference solution is generated for a value of

.. image:: images/Object_21.svg
    :width: 12
    :height: 14


.. _RefImage_Object_21.svg:

= 1.003:


.. csv-table::

    ":math:`t` "," :math:`D(t)` **(Damage — Excel solution)**"
    "43.11", "0.0"
    "100. ", "0.004742198"
    "1000. ", "0.083020455"
    "10000. ", "1.809947268"
    "20000. ", "2.003578566"
    "21000. ", "0.083020455"
    "2000. ", "0.178700399"
    "2200. ", "0.199207053"
    "22400. ", "0.220252827"



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

Analytical solution.


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


1. A.M. DONORE: Estimation of lifespan in fatigue with large numbers of cycles and in oligocyclic fatigue. Note [:ref:`R7.04.01 <R7.04.01>`] Index B.