Validation of a predictor of fatigue failure in the high-cycle range

This paper is dedicated to Professor Ernst Rank on the occasion of his 65th birthday
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Abstract

A predictor of failure initiation in high-cycle fatigue was calibrated on the basis of uniaxial experimental data and its predictive performance tested against biaxial data. The procedures of solution verification, calibration, ranking, validation and uncertainty quantification were employed. A condition that defines the domain of the predictor is inferred from the results of validation experiments.

Introduction

Mathematical models formulated for the prediction of fatigue life of mechanical components in the high cycle range comprise three sub-models: (a) A predictor of damage accumulation for the generalization of fatigue data obtained from notch-free coupons to notched specimens. (b) A deterministic model of linear elasticity, the solution of which characterizes the predictor. (c) A statistical model for the characterization of the material-specific randomness of mechanical fatigue. Various predictors and statistical models can be formulated. The models are ranked on the basis of an objective criterion that measures their predictive performance by the value of the likelihood function.

There is a rich technical literature on metal fatigue. For a comprehensive survey we refer to [1] and to [2] for a review of models based on the theory of critical distances. The primary aim of this paper is to outline and illustrate by examples how the procedures of verification, validation and uncertainty quantification (VVUQ) provide an objective basis for ranking alternative phenomenological models formulated for the prediction of fatigue life of metallic objects subjected to high-cycle loading.

Traditionally, the predictor of fatigue life was the maximum normal stress, shear stress or octahedral stress, scaled by a fatigue stress concentration factor, calibrated so as to account for the markedly different values of the expected number of cycles to failure given the maximum stress for notched and notch-free specimens, Neuber [3] and Peterson [4] defined fatigue stress concentration factors in terms of fatigue notch factors that depend on a local geometrical feature characterized by a radius and a material-dependent length. They gave different definitions for the functional form of the fatigue notch factor.

It was reported in [5] that the assumption that a material-dependent length characterizes fatigue life is not supported by experimental evidence. This should be understood in the following way: In the calibration process the predictor is trained to match the calibration data. This means that the statistical model and the functional form of the predictor are assumed to be correct. The parameters that characterize the predictor are determined such that the predicted and experimental outcomes maximize the likelihood function. Since the predicted quantities are smooth functions of the model parameters, the calibrated predictor interpolates or approximates in some sense the calibration data within the domain of calibration. Therefore any predictor can be expected to pass validation tests within its domain of calibration. In other words, a calibrated predictor is a validated predictor within its domain of calibration and, furthermore, any reasonable model can be calibrated on a sufficiently small domain.

Multiplication of the maximum stress by the fatigue stress concentration factor can be understood as averaging stress over an area characterized by a material-dependent critical distance. Therefore the classical methods used for accounting for notch sensitivity in high-cycle fatigue can be understood as applications of the theory of critical distances (TCD). Details are available in the Appendix.

The fatigue stress concentration factor Kf, as defined by the notch sensitivity indices proposed by Neuber and Peterson, had not been tested outside of a small domain of calibration until recently. The intended use of those predictors was for the interval of notch radii commonly used in machine elements. Based on the results of calibration data for a larger interval of notch radii, it was concluded in [5] that Kf cannot be validated. However, using an alternative definition for the notch sensitivity index, it can be and has been validated for notch radii in the range of 0.003 to 1.5 inches (0.08 to 38 mm).

For notches that have much smaller radii than the notch radii for which the traditional predictors were calibrated, and notches that cannot be characterized by a single geometric parameter, a new kind of predictor, based on integral averages over a domain that depends on the material and the local stress distribution, was proposed in [6] and calibrated for 75S-T6 (7075-T6) aluminum alloy under uniaxial conditions.

In this paper the development of calibration data for the new predictor, using data for 24S-T3 (2024-T3) aluminum alloy under substantially uniaxial conditions, is described. The performance of the new predictor is tested against uniaxial and biaxial data published in [7]. It is shown that, subject to a restriction on the size of the plastic zone, the predictor passes validation tests and, furthermore, it performs better than Kf based on the revised notch sensitivity index defined in [6].

This paper is organized as follows: (a) A predictor of failure initiation, characterized by three parameters, is described and measures that define the domain of the predictor are defined. (b) A statistical model characterized by five parameters is defined. (c) The calibration of the predictors, corresponding to a sequence of model form parameters, is described. (d) The predictors are ranked on the basis of the likelihood function. (e) The highest ranking predictor is tested against published records of experiments. Restrictions on the domain of the predictor are inferred from the results of validation experiments.

Section snippets

Predictors of fatigue life

Predictors of fatigue failure are formulated with the objective to generalize results of fatigue tests performed on coupons subjected to constant or smoothly varying uniaxial or, less frequently, biaxial stress conditions to arbitrary triaxial stress conditions.

We assume that the propensity of a material to fail in fatigue is characterized by records of fatigue experiments performed on notch-free coupons under constant-cycle loading. These records contain the details of test conditions, the

The source of calibration data

Our source of the calibration data is a series of reports issued by NACA and later NASA in the period 1951–1959 see Refs. [10], [11], [12], [13], [14]. These reports provide detailed documentation of experiments performed on notch-free and notched specimens cut from 0.09 inch (2.3 mm) thick aluminum sheets. In this paper we use the data for 24S-T3 (2024-T3) aluminum alloy. The source data are in US customary units. For that reason we also use US customary units.

The available data are summarized

Statistical model

The random nature of the outcome of fatigue experiments is taken into account by means of a statistical model. Many plausible choices are possible however those choices can be objectively ranked on the basis of the maximum likelihood function, given a set of experimental data. Based on the results of investigation reported in [15], we selected a model from the family of random fatigue limit models proposed by Pascual and Meeker [16]. Additional details are available in the appendix of Ref. [5].

Verification

The relative error in all numerically computed quantities of interest (QoI) have been verified to be less than one percent. This is necessary in order to ensure that numerical errors are negligible in comparison with the uncertainties associated with the physical experiments, such as uncertainties in loading conditions and variations in the dimensions of the test articles.

Verification was based on p-extension: A sequence of solutions corresponding to increasing polynomial degree on a fixed

Calibration of β(V,α)

Each test record contains information on the maximum stress σmax(i), the cycle ratio Ri, the number of cycles ni and an indication of whether the specimen failed δi. The goal of calibration is to find βi by iteration such that for a fixed value of α Gα(i)=μ1(log10ni)=A3+10A1ni1A2see Eq. (6). This condition is equivalent to selecting βi such that the probability of the specimen surviving ni cycles is 50%. Notched specimens that did not fail are not considered in the calibration of β.

The

Ranking

We evaluate for each specimen type the log likelihood function for a sequence of the model form parameter α=αj. Specifically, the sequence αj=0.2(j1),j=1,2,,6will be used. We compute LLk(j)(αjθ)=i=1mk[(1δi)lnϕM(wi,Gj(i))+δiln1ΦM(wi,Gj(i))],k=2,3,,10 where θ is the vector of the five statistical parameters that characterize the random fatigue limit model (see Table 2), mk is the number of qualified records available for specimen type k, wi=log10ni and Gj(i) is the function defined by Eq. 

Validation

In this section we address the question of whether the predictor Gα (α=0) calibrated under uniaxial loading will pass validation tests under biaxial loading. We will use the results of uniaxial and biaxial fatigue tests performed on 2024-T3 aluminum alloy specimens in the intermediate to high cycle fatigue regime published in [7].

There are several sources of uncertainty and bias in the mathematical model coming from the data itself, the choice of statistical model, the choice of the predictor

The domain of the predictor

In Ref. [5] fatigue test records for notched specimens that failed at fewer than 8500 cycles, the lower limit of the number of cycles for which S–N data are available, were disregarded in the calibration process. The rationale was that the predictor Gα was conceived for the purpose of generalization of the S–N data to notched specimens in the high cycle regime and therefore the range of the predictor must be in the high cycle regime, the lower limit of which is usually and somewhat arbitrarily

Summary and conclusions

We have demonstrated a process by which mathematical models for the prediction of fatigue life of notched mechanical components in the high cycle regime are formulated, calibrated, ranked and tested in validation experiments.

Of necessity, several assumptions have to be made in the formulation of predictors and selection of statistical models. These assumptions are based on prior experience, insight and intuition. It is possible to objectively rank models based on their predictive performance,

Acknowledgment and disclaimer

The work reported in this paper was supported in part under Navy SBIR 2008.2 Phase II.5 - Topic N08-131 Flaw-Tolerant Design & Certification of Airframe Components , Contract N68335-19-C-0239.

The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring agency or the U.S. Government.

CRediT authorship contribution statement

Barna Szabó: Conceptualization, Methodology, Writing - original draft. Ricardo Actis: Methodology, Data curation, Investigation analysis, Reviewing and editing. David Rusk: Methodology, Data curation, Investigation, Writing - review & editing.

References (18)

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