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.