## 19 Theoretical Considerations

1.9.1 Fatigue Strength

In predicting fatigue life of structural components made of, for example, composites, there are at least two alternative methodologies that could be used depending on the damage tolerant or safe-life design concept adopted for the specific structural part. In the former case, it is assumed to consider a damage metric such as crack length, delaminated area, residual stiffness, or residual strength, and by means of a criterion correlate this metric to fatigue life. In safe-life design situations, cyclic stress or strain amplitude are directly associated to operational life through S-N or e-N curves. Under complex stress states, multiaxial limit state functions are introduced that are usually generalizations of static failure theories to take into account factors relevant to the fatigue life of the structure, that is, number of cycles, stress ratio, and loading frequency. Due to the fact that damage-tolerant fatigue design of composite structures is still in its infancy, and much more research is needed to establish reliable methodologies of general applicability, most of the industrial applications with this type of materials are safe-life parts.

One of the first attempts for generalizing a multiaxial static failure theory to account for fatigue, was made by Hashin and Rotem [23]. They presented a fatigue strength criterion based on the different damage modes developing upon failure. For unidirectional materials there are two such modes, mode I, or fiber failure mode, and mode II, or else matrix failure mode. The discrimination between these two modes is based on the off-axis angle of the reinforcement with respect to the loading direction. The critical angle, as shown in [23], is given by:

where Ts and osA stand for the static shear and longitudinal (axial) strength, respectively, while functions fx(R, N, v), f '(R, N, v) are the fatigue functions of the material along the same directions. The S-N curves of the material are given as the product of the static strength along any direction and the corresponding fatigue function. In the above equation R = omm /omax, N is the number of cycles and v the loading frequency.

If the reinforcement forms an angle less than 0C, with respect to the loading direction, then mode I is the prevailing mode of failure, else mode II is the one that leads to fatigue failure. Thus, the failure criterion has two forms:

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