1. Introduction

Industrial experience shows that breakages of machine parts or structures during normal operation are most often due to fatigue. Its hidden progressive nature very often leads to a sudden breakup.

Fatigue refers to the modification of the properties of materials following the application of force cycles, cycles whose repetition can lead to the breakage of parts made of these materials [bib1].

Various methods are available for damage assessment. The second part of this document is devoted to the presentation of the oldest methods which are methods based on uniaxial tests: Wöhler method, Manson-Coffin method and more recently methods proposed by S. Taheri (EDF -R&D/ AMA).

What these methods have in common is that they determine a damage value from the evolution over time of a scalar component characterizing, for the calculation of the damage, the state of stresses or deformations of the structure.

The damage assessment is based on the use of material fatigue curves (Wöhler or Manson-Coffin), combining a variation in stress of a given amplitude to a number of admissible cycles.

To use these curves based on a real uniaxial load, it is necessary to treat the history of stresses or deformations by identifying elementary cycles (cf. [§2.2]).

The difficulty of defining a cycle for a complex signal explains the profusion of counting methods that have appeared in the literature [bib2].

Two of the most commonly used methods have been introduced in the*Code_Aster*:

• counting the ranges in cascade or method RAINFLOW,

• rule RCC_M.

A third method is added, which we will call a « natural » counting method, and which respects the order in which the loading cycles are applied.

For each elementary cycle, elementary damage is evaluated using methods based on Wöhler curves, Manson-Coffin curves, or both simultaneously.

For the Wöhler method (cf. [§2.3]) the user can correct the constraint to be integrated into the Wöhler curve by:

• a stress concentration factor \({K}_{T}\), to take into account the geometry of the part,

• an elastoplastic concentration coefficient \({K}_{e}\),

• a Goodman or Gerber correction in the Haigh diagram to take into account the mean stress of the cycle.

On the other hand, it is proposed to define the Wöhler curve in three different forms, one discretized form point by point and two analytical forms.

The Manson-Coffin method (cf. [§2.4]) applies to deformation loads. The Manson-Coffin curve is defined in a unique form, discretized form point by point.

Taheri methods (cf. [§2.5]) also apply to deformation loads and require data from the Manson-Coffin curve and possibly from the Wöhler curve. Their particularity is to take into account the order in which the elementary loading cycles are applied to the structure, unlike the other two methods.

Note:

Three methods for extracting elementary cycles are available: Rainflow method, RCC_M rule and « natural » counting .

The first two methods do not take into account the order in which the cycles are applied, which is irrelevant for the calculation of damage using the Wöhler or Manson-Coffin methods.

To calculate the damage by Taheri methods, the method of extracting cycles by « natural » counting [§2.2.3] must be used, which respects the order in which the cycles are applied.

For all of these methods, the calculation of the total damage suffered by the structure is determined by an accumulation method, the Miner rule.

The third part of this document presents the Lemaître and Lemaître-Sermage methods, which are « analytical » methods for calculating damage \(D\) (at any time \(t\)) from the data of the stress tensor \(\mathrm{\sigma }(t)\) and the cumulative plastic deformation \(p(t)\). These two methods apply to loads under any stress (uniaxial or multiaxial).

A linear accumulation rule can be used to determine the total damage to the structure.

Finally, the Crossland and Dang Van Papadopoulos criteria are presented in the fourth and last part of this document. They apply to any load (uniaxial or multiaxial) under stress and periodic loads. They provide a criterion value that indicates whether or not there is fatigue.

From the value of the criterion, it is possible to specify a scalar component characterizing the state of the structure for the calculation of the damage and to determine a damage value using the Wöhler curve of the material.