Page 338 - Biaxial Multiaxial Fatigue and Fracture

322 S. ’ POMMIER

loading, an equivalent shear stress invariant [I] is sufficient. Under non-proportional loading
conditions, a critical plane has usually to be determined and the damage is calculated in this
plane. The Dang Van criterion [2] belongs to this category. The damage is calculated using a
combination of the shear stress on the critical plane and of the hydrostatic pressure.
On the other hand, probabilistic approaches that are developed to predict scale effect and
scatter in fatigue rely on the assumption that cracks are nucleated on defects. Murakami and co-
authors [3] clearly showed that the fatigue limit is decreasing when the defect size is increased
and that under a critical dimension defects become non-damaging. Four main sources of scatter
are usually acknowledged. The first one is the probability to find a defect within a given
volume of material. The others are the variability of the defect’s size, of their distance to the
surfaces and of their mutual distance [4]. The Dang Van’s criterion can be successfully
employed even when cracks are initiated on defects [5], since cracks are nucleated by cyclic
slip within grains around the defects. The scatter can be introduced in the model through a
statistic distribution of the fatigue limit, which is evaluated, for example, from the distribution
of the defects sizes, through the widely used model by MurakamiLkEndo [6].
When the material is “defect free” or at least when defects are non-damaging, crack
nucleation occurs in “weak” grains. This is the case in particular in titanium alloys, in
aluminium alloys and in superalloys used by the aeronautic industry, possibly because, on the
one hand, the defects sizes are small in these alloys, and because, on the other hand, fatigue
crack nucleation in grains is assisted by environmental effects at the surfaces.
In any case, when crack nucleation occurs in “weak” grains, the scatter in fatigue lives does
not completely disappear and should be evaluated from life prediction methods, which is not
the case by now. For example, Susmel8zPetrone [7] have found in a 6082 T6 aluminium alloys
fatigued under various testing condition a scatter factor equal to +/-2 on the experimental
fatigue lives. In their experiments, the surface of material in the gauge length is close to 15700
mm‘. In this case, the authors mention clearly that crack nucleation does not occur on defects. It
would be useful to be able to predict the scatter in such cases, and to identify its origins.
Concerning crack nucleation in “weak” grains, if an analogy with defects is made the scatter
should be controlled by: the volume fraction of “weak” grains, the variability of the size of
“weak” grains, of their distance to the surfaces and of their mutual distance. The variability of
the grain sizes is easy to determine from micrographs. The effect of the surfaces is clear, since
in a “defect-free” material, cracks are almost always nucleated on the surfaces. In order to
estimate the volume fraction of “weak” grains and the distance between “weak” grains, we
should at first identify clearly which are the “weak” grains.
A “weak” grain is usually a grain within which cyclic slip occurs during fatigue cycling.
Therefore, under bulk elastic conditions, this is a grain within which the maximum resolved
shear stress over all the slip systems is higher than elsewhere. The resolved shear stress is high
when two conditions are satisfied: 1- the crystalline orientation of the grain towards the
principal stress directions is favourable for slip and 2- the stress applied on the “weak” grain is
high.
1- Concerning the crystalline orientation of the grain, if the texture of the material is known,
the statistic distribution of the Schmid factor can be calculated (Fig. 1. (a)). This distribution
allows determining the probability to find a grain with a high value of the Schmid factor within
a given volume of material. The mutual distance between grains with a high Schmid factor
could also be determined, for example, using crystal orientation maps as measured from
electron back scattering patterns. This information is important to evaluate the probability of
micro-cracks coalescence during the first stages of crack growth.