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


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

The reference solution is generated from option POST_FATIGUE. The methodology adopted consists in defining a history of stress loading and recovering the evolution of the associated cumulative plastic deformation from a :math:`\mathrm{3D}` tensile test in thermo-viscoplasticity.

The loading history :math:`\sigma (t)` and :math:`p(t)` is then used in a POST_FATIGUE calculation with the material parameters presented in paragraph [:ref:`§1.1 <§1.1>`] to define a reference solution.


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

The Lemaître damage reference result is obtained for a tensile test with imposed deformation and at a constant temperature. The stress state and the cumulative plastic deformation resulting from this test are as follows:


.. csv-table::

    "**Time** :math:`[s]` ", "**Constraint**
    :math:`\mathrm{Sxx}(t)[\mathrm{Pa}]` ", "**Cumulative Plastic Deformation**
    :math:`P(t)`"
    "50", "7.15030E+06", "0.000000E+00"
    "100", "1.43006E+07", "0.000000E+00"
    "150", "2.14509E+07", "0.000000E+00"
    "200", "2.86012E+07", "0.000000E+00"
    "250", "3.57515E+07", "0.000000E+00"
    "300", "4.29018E+07", "0.000000E+00"
    "350", "5.00521E+07", "0.000000E+00"
    "400", "5.72024E+07", "0.000000E+00"
    "450", "6.43527E+07", "0.000000E+00"
    "500", "7.15030E+07", "0.000000E+00"
    "550", "7.86533E+07", "0.000000E+00"
    "600", "8.58036E+07", "0.000000E+00"
    "650", "9.29539E+07", "0.000000E+00"
    "700", "1.00091E+08", "9.547120E-08"
    "750", "1.06433E+08", "5.747160E-06"
    "800", "1.10614E+08", "2.650910E-05"
    "850", "1.12888E+08", "6.060610E-05"
    "900", "1.14130E+08", "1.019250E-04"
    "950", "1.14913E+08", "1.464460E-04"
    "1000", "1.15508E+08", "1.922890E-04"


This loading history is then used with the operator POST_FATIGUE option LEMAIT_S to estimate the damage over time with the material properties defined in paragraph [:ref:`§1.1 <§1.1>`]. The temperature is assumed to be constant and equal to :math:`20°C`. Depending on the value of the parameter :math:`s` used, the following damages are found:


+--------------------+----------------------+--------------------+
|                    |**Damage (reference)**                     |
+--------------------+----------------------+--------------------+
|**Time** :math:`[s]`|Case :math:`s=0.8`    |Case :math:`s=1.003`|
+--------------------+----------------------+--------------------+
|50                  |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|100                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|150                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|200                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|250                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|300                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|350                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|400                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|450                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|500                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|550                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|600                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|650                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|700                 |0.00000E+00           |0.00000E+00         |
+--------------------+----------------------+--------------------+
|750                 |5.43732E-03           |3.19264E-02         |
+--------------------+----------------------+--------------------+
|800                 |2.75450E-02           |1.90334E-01         |
+--------------------+----------------------+--------------------+
|850                 |6.75939E-02           |1.00000E+00         |
+--------------------+----------------------+--------------------+
|900                 |1.21543E-01           |1.00000E+00         |
+--------------------+----------------------+--------------------+
|950                 |1.87318E-01           |1.00000E+00         |
+--------------------+----------------------+--------------------+
|1000                |2.66202E-01           |1.00000E+00         |
+--------------------+----------------------+--------------------+


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

Digitally generated 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.