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 (σ12) loading sequence and conservative under low-high (σ12) 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.