Page 264 - Biaxial Multiaxial Fatigue and Fracture

248 M. END0

following superposition.

where FIA and FIB are the correction factors for the cases A and B in Fig. 3, respectively, and c
is the representative crack length. It is hypothesized that the threshold SIF range under biaxial
stress, A&h,bi, is equal to that under uniaxial stress, L&,mi, or:

AKt,,bi = AK Ih,um . (6)

This criterion has previously been used by Endo and Murakami [ 161 in the correlation of the
pure torsional fatigue limit, 7,; the biaxial fatigue limit, with the rotating bending fatigue limit,
aw; the uniaxial fatigue limit, for specimens having a small hole at the surface. Based upon this
criterion, Beretta and Murakami [2 1,221 predicted that 4, the ratio of the fatigue limit in torsion
to that in tension, ie., r, /a,, for a mode I crack emanating from a three-dimensional surface
defect under cyclic biaxial stressing should have a value between 0.83 and 0.87. They found
that the predicted value of 4 agreed well with previously reported experimental results for
various steel and cast iron specimens which contained small artificial defects. For fully reversed
loading; Le., R = -1, b&h,bj and AKtbuni were expressed using Eq.(5) as

where 01 and q are the maximum and minimum principal stress amplitudes resulting from the
combined stress at fatigue limit, respectively, and ow is the threshold stress amplitude for a
mode I crack under tension-compression cyclic loading; that is, the uniaxial fatigue limit of a
specimen containing the same sized defect under R = -1 loading.
When the crack length, c, under uniaxial loading is equal to that under biaxial loading,
Eq.(6) is reduced to

where k = Fle/F*, and represents the effect of stress biaxiality. If the torsional fatigue limit is
designated by rw, since UI = -m = r,, then 4 = rw/aw = 1/(1 - k). Equation (9) as well as Eq.(6)
provides a criterion for fatigue failure of specimens containing small defects when subjected to
multi-axial loading.
For round-bar specimens subjected to combined axial and torsional loading, Eq.(9) can be
expressed as

where and r, are the normal and shear stress amplitudes, respectively, at the fatigue limit
under combined loading. Equation (10) is identical in form to Gough and Pollard's "eIlipse arc"
relationship [29], which has been used to fit the experimental data for brittle cast irons and
specimens with a large notch [29,30]. The ellipse arc is empirical, and as such it requires
fatigue tests for the determination of a, and 7,. In contrast, in the case of small defects, a, can