Page 345 - Biaxial Multiaxial Fatigue and Fracture

Variability in Fatigue Liues: An Effect of the Elastic Anisotropy of Grains? 329

relative difference between the maximum principal stress in overstressed and understressed
links is lower than 15% in aluminium, it exceeds 70% in copper. It is worth noticing that the
length of an overstressed links is approaching 15 grains for example in copper, but there is no
proof that it could not be larger if the number of grains in the model was increased. The
distribution of the maximum principal stress in zirconium, titanium and zinc is also self-
organized (Fig. 7 d,e,f). As for the cubic crystals, the distributions are similar in the three cases
expect for the intensity of the load links, which is maximum in the case of zinc.
The mechanism leading to the formation of a load percolation network in a polycrystal can
be explained as follows. Let consider a particle with a high rigidity embedded within a
homogeneous material (Fig. 8.). The material is loaded homogeneously far from the particle.
Away from the particle the isostatic lines are aligned with the maximum principal stress
direction b, but they turn in the vicinity of the particle, which is a stress concentrator. The
curvature of the isostatic lines corresponds to a load transfer by shear within the homogeneous
material. If, in the planes that contain the principal stress direction b, the shear modulus of the
material is low, the load transfer by shear is inefficient. Consequently, the stress state is
disturbed by the particle over a distance I, which is large. In a polycrystal, the formation of load
links is due, on the one hand, to the existence of grains (or defects) with a higher or a lower
rigidity related to the principal stress direction than the mean one, and on the other hand, to the
inefficiency of the material to homogenize the stress state around that grain. This inefficiency is
directly connected to the shear modulus of the material in the planes that contain the maximum
principal stress direction. If the texture of the material is random, the length of the links is
dependent on the probability to find consecutive grains with a low shear modulus related to the
principal stress direction [ 151, i.e. to the width of the distribution of the shear modulus.

Material with a low shear modulus
and a particle with a high rigidity

Fig. 8. Illustration of the mechanism leading to the formation of a load percolation network in
a polycrystal.

It can be concluded from these first results that the anisotropy of the grains in their elastic
domain lead to the formation of a load percolation network, analogous to that observed in a
granular material. This network possesses an intrinsic scale larger than the grain size.