A probabilistic model for low-cycle fatigue crack initiation under variable load cycles
Introduction
Low-cycle fatigue (LCF) of materials subjected to variable amplitude loads always attracts interest of researchers, as in real world it is more common that variable rather than constant cyclic loads are applied to materials or structural components. The most classical and widely applied cumulative damage rule considering load variation should be the linear damage accumulation method, proposed by Palmgren [1] and mathematically formulated by Miner [2], known as Miner’s rule. Nevertheless, the main deficiency of Miner’s rule is that the simple linear damage accumulation cannot account effects such as the load level dependency [3], load interaction (load sequence dependency) [4], and contribution of small cycles [5]. Since then, other nonlinear cumulative damage models have been developed and reviews of existing damage models are provided by researchers such as Fatemi and Yang [6], Bandara et al. [7], Santecchia et al. [8], and Hectors and De Waele [9]. However, one of the common drawbacks of most nonlinear models is their complexity for engineering applications. Unlike Miner’s rule which directly sums up the LCF damage values which can be easily calculated from the LCF life values provided through the constant amplitude experiments, these models often require additional parameters only obtainable by conducting costly and sophisticated variable amplitude experiments. Kwofie and Rahbar [10], [11] introduced the concept of fatigue driving stress based on the Basquin equation and established a nonlinear damage accumulation model. Aeran et al. [12] also proposed a model solely based on S–N curves without need for additional parameters. The load interaction effects between two load levels are accounted via a factor determined by their stress amplitude ratio. Recently, Pavlou [13] proposed the theory of the S–N fatigue damage envelope which derives damage maps on the stress-life diagram using data from constant amplitude tests. The isodamage curves in the damage envelope can then be numerically solved from steady state heat transfer analysis and be used to assess fatigue life under given variable amplitude load sequences. This theory is a generalization of linear, double-linear, and non-linear damage models and successfully describes load sequence effects.
The LCF life of a smooth and defect-free specimen can be divided into two phases: the crack initiation phase in which micro-cracks are created on the specimen surface but retarded in their growth and none of them is dominant, and the crack growth phase in which the damage is localized and leads to the propagation of a dominant crack until the failure. It is worth noting that some authors like Miller et al. [14], [15], [16] highlighted that Miner’s rule is indeed a satisfactory cumulative damage model to depict the crack growth, and its deficiency is due to the nonlinear damage accumulation during the crack initiation phase. They showed that Miner’s rule can be used to depict the damage accumulation for the fatigue life spent in the short crack propagation phase associated with an exponential crack growth law. Ciavarella et al. [17] further demonstrated that Miner’s rule can be extended to the long crack propagation phase ruled by Paris’ law.
Conventional LCF assessment uses material stress-life (S–N) / Wöhler or strain-life (–N) curves fitted from experimental data of a material subject to several different levels of constant load cycles. The failure life data in those experiments are usually determined when a crack initiates and grows up to a predefined size, for example for a 0.4% C steel in the work of Ibrahim and Miller [15], [16]. However, it lacks universally agreed criteria to determine the crack initiation life alone, as mentioned by Kujawski and Ellyin [18] and Bhattacharya and Ellingwood [19]. Ibrahim and Miller [15], [16] proposed to determine the crack initiation life and size via multiple two-step fatigue tests, by modeling the short crack growth immediately after the crack initiation phase with an exponential law. Besides the basic fatigue models in terms of S–N or –N curves, advanced models are also proposed to describe the complex fatigue processes. For example, Wei and Liu [20] addressed the multiaxial fatigue under constant loadings and proposed an energy-based model which considers the damage contribution from hydrostatic stress/strain components that was ignored by most critical-plane based models. Cui et al. [21] investigated the corrosion-fatigue life of high-strength steel wires and introduced an improved continuum damage mechanics model quantifying the coupled effects.
LCF failure is rather a random than deterministic event as shown by life scatter exhibited in fatigue experiments. It is hence reasonable and necessary to introduce statistical methods to treat the randomness. For example, Schijve [22] compared three different probability distributions, log-normal, three-parameter Weibull and a shifted log-normal distribution, in data fitting with particular attention paid to very low failure probabilities. Babuška et al. [23] used Bayesian approach to fit the S–N curve as well as the fatigue limit parameter when considering fatigue run-outs. Weibull [24] introduced the weakest link principle which considers a component as a collection of elements and describes the survival probability of the entire component as the product of the survival probabilities of all elements. Weibull also suggested a two-parameter distribution to describe the failure probability. Since then, the weakest link approach and the Weibull distribution together have been widely used in the fatigue life study, so that these two terms become interchangeable, which is a misconception as pointed out by Zok [25]. The scatter of the LCF life under constant load cycles can be explained by local crack nucleation randomness. By citing various references, Miller and Ibrahim [16] stated that “the statistical nature of the early damage processes” is mainly “in initiation and not the propagation of cracks”. It is widely practiced that the local crack initiation randomness is modeled by the Poisson point processes (PPP), c.f. [26], [27] for example.
Schmitz et al. [28] adapted the weakest link principle and the Weibull distribution approaches to propose a probabilistic LCF model (named ProbLCF-CA hereafter, with “CA” standing for “constant amplitude”). One of the constitutional assumptions of the ProbLCF-CA model indicates that it considers the crack initiation life and its scatter. The ProbLCF-CA model calculates crack initiation failure probabilities over the entire component surface under the inhomogeneous strain field by summing the spatio-temporal local crack nucleation hazards over the surface and in time. It also enables a quantitative explanation of the size effect in LCF observed in fatigue tests [29], [30] via the scaling of the –N curve parameters over the component surface area. Mäde et al. [31] further considered the notch effect in fatigue tests by integrating the stress gradient factor into the ProbLCF-CA model.
Other stochastic and statistic theories have also been introduced to develop cumulative fatigue damage models. Shimokawa and Tanaka [32] proposed a statistical Miner’s rule based on the transfer law of reliability curve which assumes the unchanged order of damage and no load interaction. Ni [33], [34] developed the two-dimensional probabilistic Miner’s rule using the family of probabilistic strain-life (p––N) curves and reliability theory. Castillo and Fernandez-Catelli [35] applied the Buckingham theorem to build probabilistic fatigue models with dimensionless variables and set up compatibility conditions from reliability theory to ensure statistical and physical validity of the models. Jimenez-Martinez [36] published a review on statistical fatigue damage assessment under stochastic loadings for offshore structures. Böhm and Benasciutti [37] introduced a frequency-domain damage model for the strain energy density parameter under random loadings.
It is an important concern for engineering applications that a model could describe both the complexity yielded by variable amplitude loads and the randomness of fatigue crack initiation while still maintaining its simplicity. The models proposed in [10], [11], [12], [13] ignore the random nature of the fatigue process while the ProbLCF-CA model [28] only treats constant amplitude loadings. The objective of this work is to address both these two challenges. Focusing on the LCF crack initiation life, this paper aims to tackle the nonlinearity and randomness of damage accumulation under variable or random load sequences.
The main contribution of this work is a new theoretical model, named ProbLCF-VA (“VA” standing for “variable amplitude”), which describes the scatter in the LCF crack initiation life for homogeneous materials under variable load cycles. The ProbLCF-VA model assumes only surface LCF crack initiation and inherits the similar modeling approach of ProbLCF-CA developed in [28]. The ProbLCF-VA model applied to given load histories with variable load cycles yields a probabilistic cumulative damage rule. Furthermore, it provides a fatigue life prediction to account for uncertainties in load cycles, if distributional descriptions of random load spectra are provided. This latter application is useful for reliability analysis during the development phase of a product, since the exact load history of the product’s life-time is unknown yet but its application scenarios and its load spectrum can be statistically estimated with field experiences and/or engineering judgment. In practice, besides the LCF crack initiation lives from constant amplitude experiments, the ProbLCF-VA model only requires additional parameters describing the local crack nucleation randomness, which can be easily estimated from results of multiple constant amplitude fatigue tests. Therefore, the ProbLCF-VA model is applicable for engineering problems, either for product residual life evaluation in the operation phase, or for life prediction in the design phase of new products.
Since conventional fatigue experiments record failure life including both the crack initiation and the propagation phase, the proposed ProbLCF-VA model should be combined with crack propagation models for total fatigue life assessment. This can be easily realized by exploiting existing crack propagation models whether they are deterministic such as Paris’ law [38] and the exponential crack growth regime as described in [15], or probabilistic like the model proposed in [39]. Due to the volume limitation and focusing on the crack initiation phase, this paper will only introduce the theoretical part of the proposed model and try to validate the model using existing test data in the literature.
The rest of the paper is organized as follows. In Section 2, the ProbLCF-CA model proposed in [28] is briefly reviewed and then a mathematical formulation of ProbLCF-VA is developed to enable the consideration of variable load cycles. Section 3 employs several case studies to qualitatively validate the proposed model with data in the literature and show the prediction results of ProbLCF-VA. Compared to other cumulative damage rules, the ProbLCF-VA model is capable of quantitatively depicting load interaction effects and small cycle contributions as well as crack nucleation randomness. Finally, the paper is summarized in Section 4.
Section snippets
Model description
This section presents the formulation and properties of the ProbLCF-VA model for homogeneous materials subjected to variable/random cyclic loads. The ProbLCF-CA model [28] is reviewed first and important modeling assumptions and properties which are useful in the derivation of ProbLCF-VA (Section 2.1) are highlighted. The review especially shows that the physical interpretation of the Weibull parameters is also mathematically grounded with the modeling assumptions, and proposes that the
Applications of ProbLCF-VA
In this section, several case studies will be presented to illustrate the behavior and properties of crack initiation predicted using the ProbLCF-VA model. Load histories with two-step sequences will be introduced first to demonstrate the load interaction effect on crack initiation. These results are compared to counterparts from Miner’s rule and qualitatively tested by experimental data from the literature. Then the proposed model is applied to load histories inserted with different number of
Conclusion
This paper presents the ProbLCF-VA model to describe the statistical nature of LCF crack initiation, either with given load histories of variable load cycles or under random load spectra. A review of the ProbLCF-CA model is provided to clarify its physical and mathematical foundations. Based on the review, the ProbLCF-VA model is built as a sequence of intra-cyclic ProbLCF-CA models concatenated via the inter-cyclic independence assumption. Several artificial examples are used to illustrate the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank Sebastian Schmitz, Lucas Mäde, Georg Rollmann and Yetao Zhou from the gas turbine R&D department of Siemens Energy for fruitful discussions and management supports. We also thank the anonymous reviewers for their time and effort to provide valuable feedback on the first submission.
Permission for Use: The content of this paper is copyrighted by Siemens Energy, Inc. and is licensed to Elsevier for publication and distribution only. Any inquiries regarding permission to use the content of
References (42)
Cumulative fatigue damage modeling—crack nucleation and early growth
Int J Fatigue
(1997)- et al.
Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials
Int J Fatigue
(1998) - et al.
Fatigue failure predictions for steels in the very high cycle region – A review and recommendations
Eng Fail Anal
(2014) - et al.
An equivalent driving force model for crack growth prediction under different stress ratios
Int J Fatigue
(2011) - et al.
A new nonlinear fatigue damage model based only on S–N curve parameters
Int J Fatigue
(2017) The theory of the S–N fatigue damage envelope: Generalization of linear, double-linear, and non-linear fatigue damage models
Int J Fatigue
(2018)- et al.
Continuum damage mechanics analysis of fatigue crack initiation
Int J Fatigue
(1998) Statistical distribution functions and fatigue of structures
Int J Fatigue
(2005)- et al.
BayesIan inference and model comparison for metallic fatigue data
Comput Methods Appl Mech Engrg
(2016) - et al.
Spatial Poisson processes for fatigue crack initiation
Comput Methods Appl Mech Engrg
(2019)
A probabilistic model for LCF
Comput Mater Sci
Combined notch and size effect modeling in a local probabilistic approach for LCF
Comput Mater Sci
A statistical consideration of Miner’s rule
Int J Fatigue
Strain-based probabilistic fatigue life prediction of spot-welded joints
Int J Fatigue
Die Lebensdauer von Kugellagern
Z Des Vereines Deutscher Ingen
Cumulative damage in fatigue
J Appl Mech
A concept of fatigue damage
Trans ASME
Fatigue of structures and materials
A review on fatigue life prediction methods for metals
Adv Mater Sci Eng
Cumulative damage and life prediction models for high-cycle fatigue of metals: A review
Metals
A fatigue driving stress approach to damage and life prediction under variable amplitude loading
Int J Damage Mech
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These authors contribute to the work equally and should be regarded as co-first authors.