1. Reference problem#
1.1. 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:
\(\frac{\Delta {\varepsilon }_{1}}{2}=0.25\), \(\frac{\Delta {\varepsilon }_{2}}{2}=0.25\), \(\frac{\Delta {\varepsilon }_{3}}{2}=0.75\), \(\frac{\Delta {\varepsilon }_{4}}{2}\mathrm{=}0.25\), and \(\frac{\Delta {\varepsilon }_{5}}{2}=1.75\).
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 \(\frac{\Delta {\varepsilon }_{1}}{2}\mathrm{\le }\frac{\Delta {\varepsilon }_{2}}{2}\mathrm{\le }\mathrm{...}\mathrm{\le }\frac{\Delta {\varepsilon }_{n}}{2}\), the methods of TAHERI_MANSON and TAHERI_MIXTE are identical to the method of MANSON_COFFIN (calculation of the number of cycles at break, \({N}_{\mathrm{rupt}}\), by interpolation on the Manson-Coffin curve and calculation of the damage by \(1/{N}_{\mathrm{rupt}}\)).
On the other hand, if a \(i\) cycle has a \(\frac{\Delta {\varepsilon }_{i}}{2}\) half-amplitude less than \(\frac{\Delta {\varepsilon }_{i-1}}{2}\), the Taheri methods differ from the Manson-Coffin method.
The TAHERI_MANSON method consists in determining an amplitude of stress \(\frac{\Delta {\sigma }_{i}}{2}\) from \(\frac{\Delta {\varepsilon }_{i}}{2}\) and \({\varepsilon }_{\mathrm{max}}\) (maximum value of the half amplitude of deformation encountered before the cycle \(i\)).
To do this, the user must provide a \(\frac{\Delta \sigma }{2}(\frac{\Delta \varepsilon }{\mathrm{2,}}{\varepsilon }_{\mathrm{max}})\) tablecloth under the TAHERI_NAPPE operand.
Starting with \(\frac{\Delta {\sigma }_{i}}{2}\), a half-amplitude of \(\frac{\Delta {\varepsilon }_{i}^{\text{*}}}{2}\) deformations is determined using a function introduced under operand TAHERI_FONC. The value of the elementary damage for cycle \(i\) is determined by interpolation of \(\frac{\Delta {\varepsilon }_{i}^{\text{*}}}{2}\) 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 \(\frac{\Delta {\sigma }_{i}}{2}\), then the value of the elementary damage of the cycle \(i\) is determined by interpolation of \(\frac{\Delta {\sigma }_{i}}{2}\) 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.
1.1.1. Material properties#
To calculate the damage of TAHERI_MANSON, we need the Manson_Coffin curve, a table to calculate \(\frac{\Delta \sigma }{2}\) from (\(\frac{\Delta \varepsilon }{2}\) and \({\varepsilon }_{\mathrm{max}}\)) and a function to calculate \(\frac{\Delta {\varepsilon }^{\text{*}}}{2}\) from \(\frac{\Delta \sigma }{2}\). 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 \(\frac{\Delta \sigma }{2}\) to be calculated from (\(\frac{\Delta \varepsilon }{2}\) and \({\varepsilon }_{\mathrm{max}}\)). The tablecloth is introduced under operand TAHERI_NAPPE. The Manson_Coffin curve and the Wöhler curve are introduced in DEFI_MATERIAU.
1.1.2. Loading history#
\(t\) |
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\(\varepsilon (t)\) |
3.5 |
3.5 |
3.5 |
2.5 |
0.5 |
1.2. 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.
1.2.1. Material properties#
Same as those in modeling A.
1.2.2. Loading history#
\(t\) |
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\(\varepsilon (t)\) |
1 |
1 |
1 |
1 |