FRH CRACKING MODEL
Material grains of different sizes, phases and mechanical properties are
assumed randomly arranged along the specimen minimum circumference
(SMC). The specimen surface roughness resembles multi coplanar micro
cracks with different sizes existing at the center of each grain along
the SMC. The front of a crack is assumed having the geometry of a
circular arc 40 - 46. The crack depth is assumed equal
to half of the surface crack length as manipulated in some relevant
works found in the literature 42, 44. The crack length
determines the corresponding crack area. A specimen with an initial
configuration as described is cyclically loaded.
Consequently, some grains experience plastic deformation. The tip of a
crack in an elastically deformed grain has a plastically deformed zone
given by the crack size, the applied stresses and the grain yield
stresses. Thus, the stress-strain regime existing at the two opposing
tips of two adjacent cracks in two neighboring grains enables the
estimation of the extents of the plastic zones at both tips. The rate of
advance of a crack tip is estimated by making use of the surface crack
length, the extents of the monotonic and cyclic plastic zones ahead of
the tip and the average shear strain within the plastically deformed
zones.
After N load cycles, (1) some surface cracks become longer, (2) the
total cracked area increases (3) the area of the supporting ligament
decreases, (4) the tip of a crack has the possibility of either
advancing or stop propagating and (5) the two opposing tips of some
adjacent cracks approach and possibly stop at a distance or merge to
form one crack. A linear coordinate system is devised along the SMC to
continuously trace the location of the tips of the existing cracks.
As a result of that cracking activity the following two scenarios are
possible, i.e. (1) when the applied stress is greater than the
material’s fatigue limit \(\sigma_{F}\), the supporting ligament becomes
unable to withstand the applied load after the application of a finite N
load cycles and, hence, the specimen breaks, (2) In case of the applied
stress be less than \(\sigma_{F}\), all existing cracks become
non-propagating at some load cycles and, thus, the specimen remains
unbroken after the application of a relatively large N load cycles. The
details of the original model can be found elsewhere52.
The model was applied to predict the fatigue lifetime of un-notched
round specimens made of a ferrite-pearlite 0.4C-70/30 carbon steel in
push-pull axial loading. This cracking simulation enabled an assessment
of the specimen’s fatigue lifetime and the endurance \(S/N\) curve of
the material with its fatigue limit could, thus, be assessed. Figure 1
presents the prediction of that simulation in the HCF regime compared
with published experimental results 3. Further, the
model recognized the effect of surface roughness, specimen size and mean
stress on fatigue lifetimes 52.