Reference problem
=====================


Modeling A
--------------

The analysis consists in determining the damage suffered by a structure subjected to a history of deformation loading. The Rainflow method is used to determine the number of elementary cycles and the half-amplitude of each cycle.

For the loads in question, the Rainflow method determines 5 half-amplitude cycles:

:math:`\frac{\Delta {\varepsilon }_{1}}{2}=0.25`, :math:`\frac{\Delta {\varepsilon }_{2}}{2}=0.25`, :math:`\frac{\Delta {\varepsilon }_{3}}{2}=0.75`, :math:`\frac{\Delta {\varepsilon }_{4}}{2}\mathrm{=}0.25`, and :math:`\frac{\Delta {\varepsilon }_{5}}{2}=1.75`.

Then the damage suffered by the structure is calculated using the TAHERI_MANSON method and the TAHERI_MIXTE method.

As long as the amplitude of deformations of the various cycles applied to the structure remains increasing :math:`\frac{\Delta {\varepsilon }_{1}}{2}\mathrm{\le }\frac{\Delta {\varepsilon }_{2}}{2}\mathrm{\le }\mathrm{...}\mathrm{\le }\frac{\Delta {\varepsilon }_{n}}{2}`, the methods of TAHERI_MANSON and TAHERI_MIXTE are identical to the method of MANSON_COFFIN (calculation of the number of cycles at break, :math:`{N}_{\mathrm{rupt}}`, by interpolation on the Manson-Coffin curve and calculation of the damage by :math:`1/{N}_{\mathrm{rupt}}`).

On the other hand, if a :math:`i` cycle has a :math:`\frac{\Delta {\varepsilon }_{i}}{2}` half-amplitude less than :math:`\frac{\Delta {\varepsilon }_{i-1}}{2}`, the Taheri methods differ from the Manson-Coffin method.


* The TAHERI_MANSON method consists in determining an amplitude of stress :math:`\frac{\Delta {\sigma }_{i}}{2}` from :math:`\frac{\Delta {\varepsilon }_{i}}{2}` and :math:`{\varepsilon }_{\mathrm{max}}` (maximum value of the half amplitude of deformation encountered before the cycle :math:`i`).

To do this, the user must provide a :math:`\frac{\Delta \sigma }{2}(\frac{\Delta \varepsilon }{\mathrm{2,}}{\varepsilon }_{\mathrm{max}})` tablecloth under the TAHERI_NAPPE operand.

Starting with :math:`\frac{\Delta {\sigma }_{i}}{2}`, a half-amplitude of :math:`\frac{\Delta {\varepsilon }_{i}^{\text{*}}}{2}` deformations is determined using a function introduced under operand TAHERI_FONC. The value of the elementary damage for cycle :math:`i` is determined by interpolation of :math:`\frac{\Delta {\varepsilon }_{i}^{\text{*}}}{2}` on the Manson_Coffin curve.


* For the method 'TAHERI_MIXTE', the same procedure is carried out for the determination of the half-amplitude of stress :math:`\frac{\Delta {\sigma }_{i}}{2}`, then the value of the elementary damage of the cycle :math:`i` is determined by interpolation of :math:`\frac{\Delta {\sigma }_{i}}{2}` on the Wöhler curve.

This method therefore requires the data from the Wöhler and Manson_Coffin curves.

The total damage is determined by the linear accumulation of elementary damage.


Material properties
~~~~~~~~~~~~~~~~~~~~~~~~~~

To calculate the damage of TAHERI_MANSON, we need the Manson_Coffin curve, a table to calculate :math:`\frac{\Delta \sigma }{2}` from (:math:`\frac{\Delta \varepsilon }{2}` and :math:`{\varepsilon }_{\mathrm{max}}`) and a function to calculate :math:`\frac{\Delta {\varepsilon }^{\text{*}}}{2}` from :math:`\frac{\Delta \sigma }{2}`. The sheet (cyclic work hardening curve with pre-work hardening) is introduced under operand TAHERI_NAPPE and the function (cyclic work hardening curve) under operand TAHERI_FONC. For its part, the Manson_Coffin curve is introduced in DEFI_MATERIAU.

To calculate the damage of TAHERI_MIXTE, we need the Manson_Coffin curve, the Wöhler curve and a sheet (cyclic work hardening curve with pre-work hardening) allowing :math:`\frac{\Delta \sigma }{2}` to be calculated from (:math:`\frac{\Delta \varepsilon }{2}` and :math:`{\varepsilon }_{\mathrm{max}}`). The tablecloth is introduced under operand TAHERI_NAPPE. The Manson_Coffin curve and the Wöhler curve are introduced in DEFI_MATERIAU.


Loading history
~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    ":math:`t` ", "0. ", "1. ", "2. ", "3. ", "4. ", "5. ", "6."
    ":math:`\varepsilon (t)` ", "0. ", "3.5", "3. ", "3.5", "3. ", "3.5", "1."



.. csv-table::

    "7. ", "8. ", "9."
    "2.5", "0. ", "0.5"



B modeling
--------------

 The analysis consists of a special case where the load history is constant (for example, average load applied). Code_Aster will count the entire load history as a zero amplitude cycle for counting methods RAIN_FLOW, NATUREL, and RCCM.


Material properties
~~~~~~~~~~~~~~~~~~~~~~~~~~

 Same as those in modeling A.


Loading history
~~~~~~~~~~~~~~~~~~~~~~~~~





.. csv-table::

    ":math:`t` ", "0. ", "1. ", "2. ", "3."
    ":math:`\varepsilon (t)` ", "1", "1", "1", "1"