RESULTS AND DISCUSSION
Some issues relevant to the present work have been addressed for
discussion in the previous publication due to Farag et al.52 on cracking simulation-based fatigue life
assessment due to CAL. Examples are (1) the inability of the stress
intensity factor to correlate the behavior of short cracks at high
stresses, (2) the utilization of a correlation between fatigue crack
growth (FCG) rates and a devised crack tip deformation parameter equal
to δ 0.46Δδ 0.5 for both long
and short cracks, where δ and Δδ are the maximum and
cyclic crack tip opening displacements, (3) the plastic zone interaction
between the two opposing tips of adjacent cracks when the two tips are
approaching each other, (4) The similar behavior of \(A_{C}\) and the
total length of cracks existing on the specimen surface and (5) the
extreme difficulty in the present model of considering the evidence of
microstructure evolution of crack initiation. In this publication52, the present authors reviewed some works in the
literature in relation to the fatigue behavior of long and short cracks
in un-notched specimens under CAL. Here, results of the present
simulation due to the application of some patterns of VAL are presented
and discussed. As demonstrated below, the discussion of the present
results is in need of some results due to CAL 52.
The present work utilized one numerical simulation considering one
surface finish to have all the present results. Different specimens were
randomly configured, virtually tested at different stresses and
consequently their \(S/N\) curve was constructed as shown in Fig. 1. A
single experimental endurance data point was utilized to adjust the
proposed modeling to the experimental data. The necessity of such an
adjustment had been expected in advance before the first run of the
present simulation. The expressions for the crack tip deformation
utilized in the present model 52 are the result of an
elastic-plastic finite element analyses FEA previously performed for a
trough-thickness crack in a plate 56. The present
crack system is different and, thus, the geometrical factor used in the
mathematical form of the stress intensity factors was employed.
Alternatively, similar 3D FEA have to be performed to analyze curved
surface crack from the surface of round bars.
The model considers growing fatigue micro cracks, \(\leq\ \)0.1 µm,
originating only from the roughness of a round specimen surface. Many
metallurgical and mechanical factors, manufacturing operations, surface
treatments, cleaning and handling have strong influences on surface
roughness. For example, fine turning with a tool peak radius of 0.5 mm,
and feed 0.13 mm can achieve an average peak-to-valley height of 4 µm.
Values as small as 0.05 µm can be achieved by honing. In between values
are obtainable by appropriate surface finish processes such as grinding
and lapping. In order to correlate these surface roughness parameters to
the present model, the metrologist and the production engineer can
easily measure the surface roughness in terms of maximum peak-to-valley
height and mean surface roughness. From those two values the standard
deviation of the surface roughness can be determined. In the present
model the average surface roughness is identified as an average initial
crack depth, and its standard deviation given by a typical statistical
analysis of metrological surface measurement. The maximum value of crack
length was 0.1 µm according to the relevant experimental data available
for the same specimens 3.
Some of the variables utilized to devise the present model are assumed
randomly distributed. These include grain size, starting crack size and
monotonic and cyclic yield stress of the grains of the two phases of the
specimens’ material. The parameters of the mean value and standard
deviation of the grain size and material’s straight can be controlled
with proper heat treatments whilst those parameters controlling the
starting crack size are given by the roughness of the specimen surface.
With this assumption, the present simulation is closer to reality.
Further, randomization is invoked (1) to distribute the grains of the
two phases of the material along the minimum circumference of the
specimen and (2) to select one of the virtually configured specimens to
run a specific virtual test. This process is uncontrolled.
The present simulation is based on the collective behavior of growing
short fatigue cracks originating from the specimen surface. The model
utilizes \(A_{C}\) as a measure of fatigue damage. The results of the
present work can easily show that the behavior of \(A_{C}\) with the
number of cycles is similar to that of the total length of the existing
cracks. On the other hand, the cracking-based cumulative fatigue damage
models existing in the literature consider the length of a single
isolated dominant crack 34, 36, 37, 47-51. This is
expected to over-estimate the fatigue lifetime for the interaction and
coalescence of existing small cracks shorten the time to failure47 – 51. Further, an individual crack shows large
changes in its growth rate. This depends on the features of that crack
in terms of its size and position relative to micro structural barriers
along with other mechanical and metallurgical effects.
Figures 3I-3IV present examples to demonstrate the anomalous growth
behavior of four short surface cracks, labeled as crack 4, crack 306,
crack 406 and crack 556, individually traced throughout one of the
virtual CAL tests with a stress amplitude equal 390 MPa. Fifteen,
nineteen, eleven and twenty-three neighboring short cracks are involved
in the cracking activities illustrated respectively in Figs. 3I-3IV.
As an example, a description of the cracking activities demonstrated in
Fig. 3I follows. The involved fifteen neighboring short cracks are
sequentially numbered from left to right. The growth of the crack
labelled 4 is assumed continuously monitored throughout the test. The
two tips of that crack commence their advance with the start of the test
till N = 35200 cycles when its length becomes 24 µm, see Fig. 3I(A). The
crack length, then, suddenly achieves a length of 110.9 µm, see Fig.
3I(B), due to its coalescence with a neighboring crack 86.8 µm long
located at its right and composed of the two already merged cracks 5 and
6. The resulting crack continues its growth, see Fig. 3I(B), from its
two tips to reach 136.4 µm at N = 54647 cycles when its left tip merges
crack 3 to result in a crack 136.512 µm long, see Fig. 3I(C), which
grows continuously from both tips to achieve 145 µm at N = 54650 cycles
before it coalesces with the growing coalesced cracks 2 and 1 to have a
crack of a total length equal to 241.88 µm. As N increases, the two tips
of the resulting crack advance to achieve a length equal to 246.78 µm at
N = 182364 cycles, see Fig. 3I(C), when its right tip merges the left
tip of a growing adjacent crack composing of nine coalesced small cracks
to suddenly reach a length of 561.1 µm. For a better illustration, Fig.
3I(A-C) are plotted with different scales.