1 INTRODUCTION
Operating under variable fatigue loadings, the accumulation of fatigue
damage is one of the main reasons for the failure of structural
components. The loading sequence has been found to affect fatigue damage
accumulation significantly1–4. The influence of
loading sequence on fatigue damage has been widely investigated, and
many nonlinear cumulative damage models have been
developed5–10.
Depending on the relationship between fatigue damage and cycle life
fraction, existing cumulative damage models can be divided into linear
and nonlinear models. The Palmgren-Miner rule11 is the
earliest linear cumulative damage model, widely used due to its
simplicity. However, it has been found that Miner’s rule ignores the
effect of loading sequence on fatigue damage1; its
predictions tend to be non-conservative under high-low
(σ1>σ2) loading sequence
and conservative under low-high
(σ1<σ2) loading sequence.
Numerous nonlinear cumulative damage models have been developed to
overcome this shortcoming. Fatemi and Yang8comprehensively reviewed cumulative damage models. In addition, a review
of cumulative damage models for high-cycle fatigue was published by
Hectors and De Waele12. Nonlinear cumulative damage
models are generally categorized into five categories: (a) models based
on the S-N curve13–15, (b) models based on crack
growth concepts16,17, (c) models based on continuum
damage mechanics (CDM)18–20, (d) energy-based
theories21–23, (e) material degradation-based
models24,25. In the existing models, the nonlinear
continuous damage model developed by Chaboche18(the
Chaboche model) describes the progressive deterioration process before
macroscopic crack initiation. The Chaboche model presents several
advantages: First, the material parameters can be easily determined from
S-N curves. Second, the mean stress is included in the damage rate
equation. Finally, the model can be extended to account for additional
loading conditions and material properties, such as multi-axial
loading26 and strain hardening.
In most cases, cumulative damage models assume residual fatigue strength
decreases with loading cycles. However, it has been found that fatigue
strength could be improved by low-amplitude loads below the fatigue
limit. The strengthening effect of pre-cyclic stress on the residual
fatigue strength was earliest investigated by Gough27et al. and was called the ”coaxing effect.”
Investigations28–32 on the strengthening effect have
indicated that the fatigue strength improvement is attributed to
beneficial variations in the material microstructure instead of a purely
statistical phenomenon. Nakagawa33 pointed out that
the strengthening effect is induced by work hardening and strain aging.
It has been indicated that the strengthening effect is related to the
stress level, the number of cycles, and load
sequence34–36. In addition, the significance of the
strengthening effect varies with different
materials37. Lu and Zheng 38,39studied the strengthening effect of carbon steel. They concluded that
75% to 95% of the fatigue limit has strengthening effects, and the
cycle range of significant strengthening effects is from
2×105 to 4×105.
Some cumulative damage models have been developed to account for the
strengthening effect of low-amplitude loads below the fatigue limit. Zhu
et al.40 developed a modified Miner’s rule using the
fuzzy set method. Zhang et al.41 introduced a
strengthening function into the Chaboche model to account for
low-amplitude loads. Zheng et al.42 established the
three-dimensional surface equations for fatigue life, strengthening
loads, and strengthening cycles. However, existing models only consider
loads below the fatigue limit. Investigations have revealed that cyclic
loads over the fatigue limit also exhibit strengthening
effects43,44. Ishihara and McEvily45found that fatigue life can be improved by cyclic loading. The tests
show that the strengthening effect depends on the preload stress level,
i.e., the strengthening effect is most significant when the preload
stress level is slightly over the fatigue limit. It is reported that the
strengthening effect is attributed to the internal stress induced by
preloading44. In fact, due to the complexity of
material properties and loading conditions, it seems arbitrary to use
the fatigue limit as a threshold to distinguish between strengthening
and damaging effects in all cases.
Aeroengine turbine discs are usually subjected to variable amplitude
cycles under elevated temperatures. These cycles can be divided into
primary and sub-cycles, and the amplitude of most sub-cycles is over the
fatigue limit. Ignoring the strengthening effects of these sub-cycles
may underestimate the turbine disc fatigue life. Therefore, for
nickel-based superalloys, which are turbine disc materials, it is
necessary to investigate the influence of pre-applied low-amplitude
loads on the residual fatigue life under elevated temperatures. The
experimental procedure of powder metallurgy nickel-based superalloy
FGH96 is described in Section 2. Section 3 presents the test results,
followed by a detailed discussion of the test results and fracture
surface observations. In Section 4, the existing fatigue cumulative
damage models are first briefly reviewed. Then a novel cumulative damage
model was developed by introducing a strengthening factor into the
Chaboche model. In Section 5, the predictive capabilities of the
proposed model and the typical models were compared using experimental
data for various materials. The main contributions and limitations of
the study are discussed in Section 6. Finally, the conclusions are
presented in Section 7.