Page 352 - Biaxial Multiaxial Fatigue and Fracture

336 S. POMMIER

shear stress than for the Tresca equivalent stress. Nevertheless this increase is not negligible,
since in tension, the standard deviation for 2,- is close to 12 % of < > while it approaches
15%of<&->inshear.
Although the scatter is higher in shear than in uniaxial extension, the maximum bound of
the distribution of the Tresca stress is found to be very similar in these two cases (Table 3). As
a matter of fact, the boundary conditions, applied on the model of the thin sheet, were not
adjusted to obtain similar mean values of the mean Tresca equivalent stress or of <Zmax> in the
two cases, but the same equivalent strain. Consequently, with such boundary conditions, the
higher level of the scatter in shear, as compared with a uniaxial extension, results in a lower
value of the mean Tresca equivalent stress, and not in an increase of the probabilities for high
values of the Tresca equivalent stress.

Table 3. Values calculated at the centre of the grain located at the centre of the model with a
crystalline orientation as follows: (VI, yf, cpz) =(O,O,O). The standard deviations are calculated
with 150 random configurations of its neighbours. The model is tested either in uniaxial
extension W=O.l% or in shear <y >=0.1%. The standard deviations (6) are given as a
percentage of the mean value of the distribution.

150 <OW> s( OW) <a,>( 1+3& ~q)) <%ax> s( %ax) < Zmax’(1+W Zmax))
analyses MPa (%) (MPa) (MPa) (%) (MPa)
Uniaxial 101.7 11.2 135.8 44.2 12.0 60.2
extension
Shear 87.6 17.2 132.6 33.5 14.8 48.3

It can be concluded fiom these calculations, that with a similar value of <T,,& in the
polycrystal, two different loading conditions are not equivalent in terms of the nucleation of
micro-cracks. As a matter of fact, these calculations show that, with the same value of <Tmax>,
the maximum resolved shear stress is significantly higher, within a few grains, in torsion as
compared with the case of tension.
However, the heavily loaded grains are sparse since they are located at the intersections
between the heavily loaded links associated with each principal direction. Therefore, on the one
hand, this effect should lead to a reduction of the threshold for crack nucleation in torsion as
compared with tension, if the threshold is given as a critical value for <T,,,~~>. However, on the
other hand, it may also modi@ the conditions for crack coalescence during subsequent crack
growth. Therefore, it is not easy at this stage, to provide defmite conclusions on the effect of
the increase of the scatter in an individual grain, on the fatigue life of the entire sample

Rotating loah
The last point that is worth to be discussed is the possible effect of the load percolation network
in the case of rotating loads.
In the first place, the previous results show that for a similar value of <T,,,~~>, there is a
higher probability for high values of the maximum resolved shear stress under multiaxial than
under uniaxial loading conditions (Fig. 12). Consequently, even if the mean values of T,,, are