INTRODUCTION
Fatigue life prediction of engineering components is the end target of
numerous research efforts 1. The experimental
questions have been tackled mainly via constant amplitude loading (CAL),
and to a lesser extent via variable loading (VL) despite its more
relevance to the in-service conditions 2. Yet the VL
are typically simplified to repeated blocks of loading cycles3. Those CAL studies result typically in a reliable
S/N endurance data. The dependence on specific material properties,
geometrical, surface and environmental parameters of the fatigue life is
evaluated through extensive testing 4. With the
objective of understanding the effect of VL, two-step loading of either
the same type or different types with different sequences were
experimentally and theoretically investigated.
Fatigue damage develops with cycle-by-cycle accumulation, even when
stress levels are below the material’s fatigue limit for most of the
loading history. After the accumulated damage reaches a critical value
fracture would ensue 5. Cracking is a form of
accumulated damage.
The commonly held view that fatigue life of a component is divided into
crack initiation phase and a subsequent crack propagation phase have
witnessed over the last two decades several relaxations6. Specifically with the initiation period being
subdivided into initiation plus micro-structurally short crack growth
phases. Fatigue cracks initiate from the most favorable surface or
subsurface sites, e.g. 7, geometrical imperfections,
favorably oriented grains, non-metallic inclusions or brittle
precipitates. Below the fatigue limits of steels cracks are frequently
observed propagating but subsequently arrested. Thus, small crack growth6 are gradually becoming more popular. In un-notched
components fatigue cracks can originate from the roughness of their
surface 8 with initial sizes comparable to local
micro-structural features rendering the linear elastic fracture
mechanics characterization of crack growth invalid 9.
Elastic-plastic fracture mechanics has been used to characterize the
behavior of such short cracks 10, 11. In addition of
cracking, other forms of fatigue damage have been recognized.
Early on, it was recognized that an energy-based damage parameter can
unify the damage caused by different types of loading such as thermal
cycling, creep, and fatigue. A unified theory based on the total strain
energy density was presented by Ellyin and his coworkers12. The total strain energy per cycle is decomposed
into plastic and elastic strain energies. The plastic strain is the one
responsible for the damage, while the elastic portion associated with
the tensile stress drives the crack growth.
Another approach attempting such a unification of damage accumulation,
is the continuum damage mechanics (CDM) 13. This
approach was originally proposed by Kachanov 14 and
Rabotnov 15 in treating creep damage problems.
Chaboche and Lesne 16 were the first to apply CDM to
fatigue life prediction. For the one-dimensional case, they postulated
that fatigue damage evolution per cycle can be generalized by a function
of the load condition and damage state. By correlating the changes in
tensile load-carrying capacity and the effective stress concept, they
idealized damage evolution through a nonlinear continuous damage (NLCD)
model. Based on the CDM concept, many other forms of fatigue damage
equation have been developed, as described by Fatemi and Yang17. Basically, all these CDM-based approaches are very
similar to the NLCD model in both form and nature. The main differences
lie in the number and characteristics of the parameters used in the
model, the requirements for additional experiments, and their
applicability. CDM models were mainly developed for uniaxial fatigue
loading.
In the case of variable amplitude loading (VAL), fatigue life prediction
uses the full \(S/N\) curve established through CAL testing. The
Palmgren‑Miner linear damage rule was the first in that direction2, 18. That rule has the shortcomings of not
considering interaction and sequence effects 17 as
well as its inability to recognize damages taking place in the low-cycle
fatigue (LCF), where the dominant failure mechanism is identified as the
macroscopic strain. As a result, many different fatigue damage models
have been proposed to remedy those deficiencies. Some models, e.g. due
to Makkonen 19, require the evaluation of some
material-dependent parameters through extensive testing which may not be
available to design engineers, causing some difficulties in fatigue life
estimation. Other models do not need such extensive testing. An example
is the damage curve approach due to Gao et al. 20which is based on a similar work by Manson et al. 21.
Shang and Yao 22 proposed another simple continuum
fatigue damage model which considered the effect of mean stress.
Mesmacque et al. 23 developed a damage rule which
utilized only the full \(S/N\) curve of the material and the von Mises
stress as a damage indicator. Siriwardane et al. 24applied that rule to estimate the remaining fatigue life of railway
bridges and a significant deviation was reported between predicted and
in-service lives and, thus, they developed 25 a
similar rule using a plastic meso-strain as the damage indicator
instead. Other models applied the concept of iso-damage curves in a
nonlinear damage model utilizing the material full \(S/N\) curve to
analyze fatigue lifetime under VL 26. A bi-linear
cumulative damage model 27 is an example of such an
approach. This model was used to predict residual lifetimes for cases of
two-step loading (TSL) with high–low (H–L) and low–high (L–H)
sequences. Other nonlinear cumulative fatigue damage models28-33 were developed and used to show their
capabilities to predict the fatigue lifetime of notched and un-notched
specimens due to VL.
Some researchers considered the accumulation of fatigue damage by
deformation processes, e.g. 34. Other cumulative
fatigue damage models were based on ductility exhaust and crack
propagation, e.g. 35. The concept that fatigue
cracking damage accumulates with applied cycles was invoked in the
literature both experimentally and theoretically. Miller and Zachariah36 and Miller and Ibrahim 37 used an
exponential cumulative fatigue damage law to demarcate the fatigue crack
initiation and short crack propagation in un-notched round specimens in
torsion and push-pull. They (1) used two different laws for the fatigue
growth during the two phases of initiation and short crack propagation
and (2) assumed the roughness of the specimen surface resembling surface
micro cracks.
Some published theoretical and experimental researchers analyzed the
shape of the front of a surface crack 38, 39.The front
is curved rather than straight. Some woks in the literature assumed a
surface crack front with the geometry of a circular arc,40 - 46. In some other works, a crack depth equal to
half of the surface crack length is manipulated 42,
44.
Existing cumulative fatigue damage models based on cracking consider a
single or isolated dominant crack. This inevitably over-estimates the
fatigue lifetime of the specimen as the coalescence of interacting
in-plane and out-of-plane cracks shortens the time to failure47 - 51. Further, the individual short crack behavior
imposes great difficulty in regard to modeling the very early stages of
crack growth as individual cracks are susceptible to large changes in
growth rate depending on their exact lengths and position relative to
micro structural barriers, the features of the random distribution of
the grains along with other mechanical and metallurgical effects. Great
efforts are still needed to obtain an appropriate generalized prediction
model for cumulative fatigue damage.
Farag et al. 52 developed a numerical 2D
elastic-plastic fracture mechanics model, called herein after FRH
cracking model, to simulate the collective growth behavior of short
cracks originating from the surface roughness of un-notched round
specimens made of a two-phase alloy in constant amplitude axial fatigue
loading. The model assumed the pre-existence of tiny cracks of different
sizes and locations on the specimen surface due to its roughness. The
features of the random distribution of the material grains were
considered in terms of their sizes and micro-structural phases along
with their mechanical properties. For convenience, a succinct
description of the original model follows.