Page 354 - Biaxial Multiaxial Fatigue and Fracture

338 S. POMMIER

sufficient to let a load percolation network appear. The second reason is that micro-crack
nucleation occurs mostly at the surfaces of the samples. However, in a massive sample, grains
located under the surface, may contribute to homogenize the stress state within the polycrystal.
It was shown before [ 151, with a rough model of a massive sample, that the scatter is lower in
the bulk than at the surfaces. Nevertheless, the scatter at the surfaces is very similar in the case
of a thin sheet and in the case of a grain located at the surface of a massive sample. This point
would need further investigation, since the difference between the stress and strain
heterogeneity at the surfaces and in the bulk of a sample could contribute, with environmental
effects, to explain the preferential nucleation of micro-cracks at the surfaces.

CONCLUSIONS

Probabilistic approaches are developed to describe scale effects and scatter in fatigue. When
fatigue cracks are nucleated on defects, the origin of the scatter is clear. When defects are non-
damaging, micro-cracks are usually nucleated by cyclic slip in “weak” grains. In this case, the
origins of the scatter in fatigue lives remain unclear.
Under bulk elastic conditions, a “weak” grain is a grain within which the maximum resolved
shear stress on the slip systems is the highest, which happens when two conditions are
satisfied: the Schmid factor of the grain is high and the stress applied on the “weak” grain is
high. The object of the paper was to discuss the second condition. Since the elastic behaviour
of the grains is anisotropic, the stress and strain distribution is heterogeneous in a polycrystal.
The spatial distribution of this heterogeneity was studied using experiments and finite element
analyses.
The spatial distribution of strain at the surface of a sample of TA6V titanium alloy was
observed using the photostress method. Though the material is fully elastic, fine inclined lines
appeared at the surface of the sample, where the strain is higher than the mean one in the
sample. However the direction of the principal strain remains mostly coincident with the load
axis. This experiment showed that there is a scale associated with the strain heterogeneity in
the TA6V titanium alloy, which is larger than the grain size, approaching 10 grains.
In order to reveal a scale for the spatial distribution of strain in a different material, thin
sheet of OFHC polycrystalline copper have been subjected to a cyclic creep test. After failure,
fine inclined lines forming a regular pattern are observed at the surface of the sample, revealing
a scale for the heterogeneity of strain larger than one millimetre.
Finite elements calculations were performed, in order to understand the above-mentioned
effects.
A polycrystalline thin sheet was modelled by FEM analysis. These computations showed
that a load percolation network, analogous to that observed in a granular material, is formed
through the polycrystal. The load is transferred through heavily loaded links whose direction is
coincident with the principal stress directions of the equivalent homogeneous problem. This
network possesses an intrinsic scale larger than the grain size.
The probability of a given value of the maximum principal stress within a grain was
calculated using the FEM. One grain located at the centre of the thin sheet was set to have a
fixed orientation, while the crystalline orientations of the other grains in the model were
selected randomly before each calculation. The variability of the maximum principal stress
under uniaxial loading conditions depends on the elastic anisotropy of the grain. For a given
crystal orientation, the maximum principal stress vary up to +I- 35 % for zinc and copper and