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Arinan Dourado Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-8030 e-mail: [email protected] Firat Irmak Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-8030 e-mail: fi[email protected] Felipe A. C. Viana 1 Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-8030 e-mail: [email protected] Ali P. Gordon Professor Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-8030 e-mail: [email protected] A Nonstationary Uncertainty Model and Bayesian Calibration of Strain-Life Models The Coffin–Manson–Basquin–Haford (CMBH) model is a well-accepted strain-life rela- tionship to model fatigue life as a function of applied strain. In this paper, we propose a nonstationary uncertainty model for the CMBH model, alongside a Bayesian framework for model calibration and estimation of confidence and prediction intervals. Using Inconel 617 coupon test data, we compared our approach to traditional stationary var- iance models. The proposed uncertainty model successfully captures the fact that the var- iance of strain amplitude decreases with increasing fatigue life. Additionally, a discussion on how to use the proposed Bayesian framework to compensate for the lack of data by using prior information coming from similar alloys is also presented considering Hastealloy-X and Inconel 617 coupon data. [DOI: 10.1115/1.4049324] 1 Introduction Material uncertainty plays a significant role in modeling and predicting fatigue life. Often, coupon tests are used to calibrate proposed life models. Test data take the forms of deformation (e.g., stress–strain curves [1]) and crack initiation responses (e.g., strain-life [2]). Constitutive models can be tuned to match hystere- sis and used in corroboration with lifing models to regress fatigue life. Although several approaches have been developed, the prac- tice of testing, modeling, and applying calibrated curves has become established over the last decades as evidenced in recent reviews [35]. The calibration of lifing models using coupon data can be done through procedures as simple as minimizing squared errors between the predicted and observed values. This way, the cali- brated model becomes a predictor for expected behavior. Although this approach explains the dependence of fatigue life on the strain or stress levels, it does not account for the inherent coupon-to-coupon variation in fatigue life (due to factors such as variations in chemistry, heat treatment, and to some extent, mea- surement error). Pascual and Meeker [6] presented a statistical treatment for fatigue-limit characterization and used the approach to explain the variation of observed stress–life (S–N) curves. They used maxi- mum likelihood methods with normal and standardized smallest extreme value distributions to fit a model to stress versus life data. They detailed how to use suspension data and obtain likelihood- ratio-based confidence regions. Meggiolaro and Castro [7] pre- sented an extensive statistical evaluation of existing estimates of Coffin–Manson parameters of 845 different metals. From the col- lected data, the authors concluded that existing correlations between the fatigue ductility coefficient and the monotonic tensile properties were poor, and it would be more accurate to estimate the fatigue ductility coefficient based on the medians values from each of the 845 materials. Koelzow et al. [8] presented a probabil- istic workflow for lifetime predictions considering material and loading profile scatter. The analysis focused on creep rupture behavior considering experimental data of a 2%-Cr forged steel. Maximum-likelihood method is used for parameter estimation and a temperature-modified version of the Manson–Coffin–Basquin equation is used to represent the experimental data. Taking both the material and the load scatter into account, a reliability analysis is carried out to compute the probability of crack initiation. Using such probabilistic approaches yields curves describing the lower bounds (e.g., the 2.5 percentile) for fatigue life. Engi- neers often take such conservative models and embed them in general-purpose finite-element modeling packages for the design of components. Unfortunately, the stationary variance often adopted in such models makes them overly conservative, espe- cially in the low strain ranges. Most of the available fatigue test data in literature have a trend toward a nonstationary variance behavior in log life of metals. Based on this behavior, Nelson [9] modeled the mean value and standard deviation of log life as functions of the stress level. The author used maximum likelihood with log-normal distribution to fit the proposed mean value and standard deviation functions to a stress–life dataset. Fatigue life confidence intervals were derived through extrapolations on the considered mean and standard devi- ation functions. Despite being a promising methodology, the pre- sented results in terms of confidence intervals do not corroborate with expected physical behavior. Pascual and Meeker [10] applied an approach similar to Nelson [9], in fatigue-limit characteriza- tion. As in the previously mentioned works, maximum likelihood methods are used to fit the models and derive likelihood-ratio- based confidence regions. In this paper, we propose a nonstationary uncertainty model for the Coffin–Manson–Basquin–Haford (CMBH) model [1114], a double power-law equation correlating total strain amplitude with reversals to fatigue failure that is based on an exponential decay. More specifically, we propose the error between the CMBH model 1 Corresponding author. Manuscript received August 29, 2020; final manuscript received December 9, 2020; published online January 7, 2021. Assoc. Editor: Glenn B. Sinclair. Journal of Verification, Validation and Uncertainty Quantification MARCH 2021, Vol. 6 / 011002-1 Copyright V C 2021 by ASME Downloaded from http://asmedigitalcollection.asme.org/verification/article-pdf/6/1/011002/6612480/vvuq_006_01_011002.pdf by University Of Central Florida, Felipe Viana on 07 January 2021

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Page 1: A Nonstationary Uncertainty Model and Bayesian Calibration

Arinan DouradoDepartment of Mechanical and

Aerospace Engineering,

University of Central Florida,

Orlando, FL 32816-8030

e-mail: [email protected]

Firat IrmakDepartment of Mechanical and

Aerospace Engineering,

University of Central Florida,

Orlando, FL 32816-8030

e-mail: [email protected]

Felipe A. C. Viana1

Department of Mechanical and

Aerospace Engineering,

University of Central Florida,

Orlando, FL 32816-8030

e-mail: [email protected]

Ali P. GordonProfessor

Department of Mechanical and

Aerospace Engineering,

University of Central Florida,

Orlando, FL 32816-8030

e-mail: [email protected]

A Nonstationary UncertaintyModel and Bayesian Calibrationof Strain-Life ModelsThe Coffin–Manson–Basquin–Haford (CMBH) model is a well-accepted strain-life rela-tionship to model fatigue life as a function of applied strain. In this paper, we propose anonstationary uncertainty model for the CMBH model, alongside a Bayesian frameworkfor model calibration and estimation of confidence and prediction intervals. UsingInconel 617 coupon test data, we compared our approach to traditional stationary var-iance models. The proposed uncertainty model successfully captures the fact that the var-iance of strain amplitude decreases with increasing fatigue life. Additionally, adiscussion on how to use the proposed Bayesian framework to compensate for the lack ofdata by using prior information coming from similar alloys is also presented consideringHastealloy-X and Inconel 617 coupon data. [DOI: 10.1115/1.4049324]

1 Introduction

Material uncertainty plays a significant role in modeling andpredicting fatigue life. Often, coupon tests are used to calibrateproposed life models. Test data take the forms of deformation(e.g., stress–strain curves [1]) and crack initiation responses (e.g.,strain-life [2]). Constitutive models can be tuned to match hystere-sis and used in corroboration with lifing models to regress fatiguelife. Although several approaches have been developed, the prac-tice of testing, modeling, and applying calibrated curves hasbecome established over the last decades as evidenced in recentreviews [3–5].

The calibration of lifing models using coupon data can be donethrough procedures as simple as minimizing squared errorsbetween the predicted and observed values. This way, the cali-brated model becomes a predictor for expected behavior.Although this approach explains the dependence of fatigue life onthe strain or stress levels, it does not account for the inherentcoupon-to-coupon variation in fatigue life (due to factors such asvariations in chemistry, heat treatment, and to some extent, mea-surement error).

Pascual and Meeker [6] presented a statistical treatment forfatigue-limit characterization and used the approach to explain thevariation of observed stress–life (S–N) curves. They used maxi-mum likelihood methods with normal and standardized smallestextreme value distributions to fit a model to stress versus life data.They detailed how to use suspension data and obtain likelihood-ratio-based confidence regions. Meggiolaro and Castro [7] pre-sented an extensive statistical evaluation of existing estimates ofCoffin–Manson parameters of 845 different metals. From the col-lected data, the authors concluded that existing correlationsbetween the fatigue ductility coefficient and the monotonic tensileproperties were poor, and it would be more accurate to estimate

the fatigue ductility coefficient based on the medians values fromeach of the 845 materials. Koelzow et al. [8] presented a probabil-istic workflow for lifetime predictions considering material andloading profile scatter. The analysis focused on creep rupturebehavior considering experimental data of a 2%-Cr forged steel.Maximum-likelihood method is used for parameter estimation anda temperature-modified version of the Manson–Coffin–Basquinequation is used to represent the experimental data. Taking boththe material and the load scatter into account, a reliability analysisis carried out to compute the probability of crack initiation.

Using such probabilistic approaches yields curves describingthe lower bounds (e.g., the 2.5 percentile) for fatigue life. Engi-neers often take such conservative models and embed them ingeneral-purpose finite-element modeling packages for the designof components. Unfortunately, the stationary variance oftenadopted in such models makes them overly conservative, espe-cially in the low strain ranges.

Most of the available fatigue test data in literature have a trendtoward a nonstationary variance behavior in log life of metals.Based on this behavior, Nelson [9] modeled the mean value andstandard deviation of log life as functions of the stress level. Theauthor used maximum likelihood with log-normal distribution tofit the proposed mean value and standard deviation functions to astress–life dataset. Fatigue life confidence intervals were derivedthrough extrapolations on the considered mean and standard devi-ation functions. Despite being a promising methodology, the pre-sented results in terms of confidence intervals do not corroboratewith expected physical behavior. Pascual and Meeker [10] appliedan approach similar to Nelson [9], in fatigue-limit characteriza-tion. As in the previously mentioned works, maximum likelihoodmethods are used to fit the models and derive likelihood-ratio-based confidence regions.

In this paper, we propose a nonstationary uncertainty model forthe Coffin–Manson–Basquin–Haford (CMBH) model [11–14], adouble power-law equation correlating total strain amplitude withreversals to fatigue failure that is based on an exponential decay.More specifically, we propose the error between the CMBH model

1Corresponding author.Manuscript received August 29, 2020; final manuscript received December 9,

2020; published online January 7, 2021. Assoc. Editor: Glenn B. Sinclair.

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and observed data to be normally distributed with zero-mean andvariance that is a function of fatigue life itself (i.e., the variance isnonstationary). This feature corroborates with experimental obser-vations. Although all parameters for our uncertainty model canalso be adjusted with observed data, we also present a way toreduce the number of calibration parameters, based on engineer-ing judgment.

As new alloys emerge as potential candidates for high tempera-ture applications, their adaptation in designs can be decelerateddue to the expensive test programs needed to achieve the valida-tion requirements. In such a scenario, reducing the number offatigue tests required for newer alloys characterization and valida-tion becomes almost economically imperative. Tests to fully char-acterize the material behavior especially in high-cycle fatigue canpotentially last for weeks and months making them extremelyexpensive. The potential savings in time and costs have motivatedthe development of approaches, such as the “fatigue to fracture”method [15]. Gong et al. [15] proposed the use of fracture data inthe low-cycle regime to derive life curves capable of potentiallypredicting high-cycle fatigue through extrapolation. Given ourimproved uncertainty estimates for fatigue life, our proposedmodel can potentially become a powerful tool for planning moreefficient test campaigns in support characterization and develop-ment of new materials. In this work, we advocate toward using aBayesian framework [16,17] for calibration of CMBH modelparameters as well as estimation of confidence and predictionintervals. Assuming that coupon data are being acquired in smallbatches at a time, we propose using prior information from similaralloys to compensate for the lack of initial available data. Thisstep is highly dependent on the alloy in case and the selection ofprior information is performed using proper knowledge of thealloys. Several studies can be found in literature addressing theuse of Bayesian model calibration frameworks to compensate forlack of data [18–21] and prior selection in model calibration[16,22,23] that can assist in the crucial step of our proposedframework. Our goal is to develop an approach that allows forstarting the collection of coupon data with only a few points athigh strain ranges and then adding coupon data at low strain ratesguided by the model convergence prediction uncertainties.

To demonstrate the performance of the proposed framework,predictions for low-cycle fatigue (LCF) of Hastelloy-X (Hast-X)with prior information coming from Inconel 617 (IN-617) are pre-sented. Both are Ni-base alloys, whereas the first (Hast-X) is anickel–chromium–iron–molybdenum alloy and the latter (IN-617)is a nickel–chromium–cobalt–molybdenum alloy. Since bothalloys belong to the same family and present similar properties,we opt to extract prior information readily available of IN-617 inthe Hast-X Bayesian calibration procedure. Although, in this con-tribution, we only show results for low-cycle fatigue data, webelieve the methodology could be applied to other failure modessuch as creep-fatigue [24–27], thermo-mechanical fatigue[28–31], or could be used for model verification [32].

The remaining of the paper is organized as follows: Section 2gives the background (strain-life and model estimation throughleast squares) and presents the Bayesian formulation (model form,likelihood, and posterior formulation) used in this work. Section 3describes the materials used in this study (typical service condi-tions, sources of uncertainty, and synopsis of available data). Sec-tion 4 presents and discusses the results. Finally, concludingremarks and future work are presented in Sec. 5.

2 Strain-Life Fatigue and Proposed Uncertainty

Model

2.1 Coffin–Manson–Basquin–Haford Model and ErrorWith Stationary Variance. In this paper, fatigue is used to definethe primary mechanism of crack initiation and early propagationof a material subjected to cyclic loading. Fatigue damage pro-gresses in metals by way of dislocation generation on

preferentially oriented grains leading to persistent slip bands facil-itating intrusion and extrusion development and ultimately a stageI crack. While both stress- and energy-life methods have beendeveloped extensively to predict the onset of fatigue cracks, theformer approach is most appropriate for high-cycle fatigue condi-tions, and energy methods have yet to be progressed for multi-axial states of the strain/strain [33]. The total strain approach,which merges contributions from Coffin, Manson, Basquin, andHaford, excels at life prediction for both low-cycle fatigue andhigh-cycle fatigue conditions. The CMBH model is given by

ea ¼rf

E2Nfð Þb þ e0f 2Nfð Þc (1)

where ea is the strain amplitude, 2Nf is the reversals to failure, rf

is the fatigue strength coefficient, E is the Young’s modulus, b isthe fatigue strength exponent, e0f is the fatigue ductility coefficient,and c is the fatigue ductility exponent.

A set of coupon tests provides observations for ea and 2Nf ,which are used to estimate the set of CMBH model parameters(i.e., rf, E, b, e0f , and c). For the sake of fitting the model parame-ters (and likely only then), fatigue life is taken as input and strainamplitude taken as output. Although this is not how Eq. (1) wouldbe commonly used (i.e., the strain is the input and fatigue life isthe output), this is a needed artifact for model parameter estima-tion. A common approach to estimate the parameters is by mini-mization of the sum of squared residuals (also known as, ordinaryleast square, or OLS for short). There is an agreement in the scien-tific community that for large enough sets of coupon data, theOLS offers a robust estimation method for the CMBH modelparameters.

Unfortunately, point-estimate approaches like OLS do notexplicitly incorporate uncertainty in their estimates (other thanthrough approximations such as first-order Taylor expansions orGaussian approximations using derivative information) [34].Nevertheless, the probabilistic treatment proposed by Pascual andMeeker [6] to treat uncertainty in S–N curves could be applied tostrain-life (after proper manipulation of the CMBH model andderivation of the likelihood function and likelihood-based inter-vals). For example, the likelihood can be formulated by assumingthat

eðobsÞa ¼ eðmodelÞ

a þ d; and

eðmodelÞa ¼ rf

E2Nfð Þb þ e0f 2Nfð Þc

(2)

where d is the observation error.Equation (2) shows that strain amplitude is output when it

comes to estimating the CMBH model parameters. However, it isworth mentioning that in the experimental settings for componentsstrain-life testing, the strain amplitude ea is the controlled vari-able, set to a given value, while the fatigue life Nf is actually theobserved output. Estimation error d is assumed to be normally dis-tributed with zero mean and constant variance k

d � Nðld ¼ 0; r2dÞ and r2

d ¼ k (3)

The likelihood for the data given the model parameters is

L eðobsÞa ;N

ðobsÞf jh; k

� �¼ 1

2pkð Þp=2exp � eTe

2k

� �(4)

where h ¼ ½rf ; b; e0f ; c; k�, e ¼ eðobsÞa � eðmodelÞ

a , eðobsÞa , and N

ðobsÞf is

the set of observed coupon data, eðmodelÞa is computed with the

observed NðobsÞf and the set of the CMBH model parameters h, and

p is the number of degrees-of-freedom in the model (in this case,p¼ 5, as we assume that E is known).

The main contribution of this work is the extension of theuncertainty model allowing for a nonstationary variance. This

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Page 3: A Nonstationary Uncertainty Model and Bayesian Calibration

would capture the fact that the variance of strain amplitudedecreases with increasing fatigue life we also propose using aBayesian framework to provide quantification of model and obser-vation uncertainties. We believe that the Bayesian approach canbe especially advantageous when it is possible to (a) specify priordistributions for the model parameters, and (b) update the modelwith the smart acquisition of new coupon data (for example, whenfatigue life uncertainty is used to assess sequential sampling).

2.2 Error With Nonstationary Variance. Although usefuland straightforward, the assumption of observation error normallydistributed with zero mean and stationary variance does not agreewith observed data. Taking fatigue life (Nf) as input and strainamplitude (ea) as output (i.e., setting a vertical line at a given 2Nf

value in the strain-life curve); coupon data show that observationerror decreases with increasing fatigue life (see Fig. 2; where inFig. 2(a) when 2Nf is approximately 300, ea varies from roughly0.005 to 0.010, while for 2Nf of 1000, ea hardly varies; and inFig. 2(b) in which the exponential decay behavior of the observa-tion error can be clearly observed). To overcome the stationaryvariance model limitations, in this paper, we proposed a variancemodel that emulates the behavior observed in the coupon data. Itconsists of an exponential decay model for the variance as a func-tion of fatigue life

r2d ¼ k1 þ k2 expð�k3ðlogð2Nf ÞÞ2Þ (5)

Although there are other possible model forms for this nonstation-ary variance model, we opted to select the form that better resem-bles the behavior expressed by the coupon data.

According to Eq. (5), k1 þ k2 is the upper limit for the variancewhen 2Nf approaches 1. In practice, this is an upper bound for thevariance in the plastic domain. On the other hand, k1 is the lowerlimit for the variance when 2Nf approaches1. In practice, this isthe lower bound for the variance in the elastic domain. Finally, k3

controls how the variance transitions between the plastic and elas-tic domains.

With the variance for the observation error d modeled throughEq. (5), the likelihood for the data given the model parameters isgiven by

L eðobsÞa ;N

ðobsÞf jh; k

� �¼ 1

2pjRjð Þp=2exp � 1

2eTR�1e

� �(6)

where R is a diagonal covariance matrix for the observed data,which entries are given by Eq. (5) and implies that observationsare independent. If priors for the CMBH model parameters andthe observation error are available; then, from Bayes’ rule, weobtain the posterior distribution of the parameters given observeddata

p h; kjeðobsÞa ;N

ðobsÞf

� �¼

L eðobsÞa ;N

ðobsÞf jh; k

� �p0 hð Þp0 kð Þð

L eðobsÞa ;N

ðobsÞf jh; k

� �p0 hð Þp0 kð Þdhdk

(7)

where p0ðhÞ and p0ðkÞ are the priors for the CMBH model andobservation error parameters, respectively.

The denominator in Eq. (7) is not available in close form and,in practice, one has to solve it numerically. Algorithms for numer-ical integration of the posterior distribution include the Markovchain Monte Carlo, particle filter, and others [34]. Numerical inte-gration of Eq. (7) generates a large set of samples for h and k(which characterizes the posterior distribution). Propagating thesesamples through the model defined in Eq. (1) and sweepingthrough 2Nf characterizes the distribution in the ea. Propagatingonly h ¼ ½rf ; b; e0f ; c� quantifies the uncertainty in the predicted ea

without considering unit-to-unit variation and observation error.

Intervals on this distribution are referred to as confidence inter-vals. If r2

d is also considered, then realizations of ea are drawnfrom

eðpredÞa � N rf

E2Nfð Þb þ e0f 2Nfð Þc;r2

d

� �(8)

and the intervals on this distribution are referred to as predictionintervals. Rigorously, while drawing from the posterior distribu-tion of eðpredÞ

a , one has to reinforce that eðpredÞa � 0.

2.3 Handling Calibration Parameters. As mentioned previ-ously, solving Eq. (7), with the likelihood given by either Eq. (4)or Eq. (6), is done numerically through methods such as Markovchain Monte Carlo and particle filter. These sampling-based algo-rithms need lower and upper bounds for calibration parameters. Inother words, we need to define the bounds for the CMBH modelparameters (rf, b, e0f , and c) as well as the variance parameters (k1

for the stationary model, and k1, k2, and k3 for the nonstationarymodel). The bounds for these parameters can be specified usingengineering judgment, or as illustrated in this contribution, onecan use estimates from similar alloys to help with this task.

Here, this task is carried out twice, once for the IN-617 calibra-tion and again for the Hast-X analysis. The Hast-X bounds aredefined based on the estimates for IN-617. For the IN-617 calibra-tion, we decide to use the following approach. For the CMBHmodel parameters (rf, b, e0f , and c), we consider that Young’smodulus (E) is previously known, and with the available data, wefit the CMBH model through ordinary least squares. Then, we usethis solution to build a reference for our lower and upper bounds(e.g., we make the interval to arbitrarily be 650% of the ordinaryleast square solution).

With regard to the variance parameters, we adopt differentstrategies depending on whether we are using the stationary or thenonstationary model. For the stationary model, we compute theresiduals at the observed data using the previously found ordinaryleast-square CMBH model parameters. We then compute the var-iance of the residuals at the observed data and use it as a referencefor k1 (e.g., we make the interval to be roughly 675% of thisvalue). For the nonstationary model, we impose the value of k3

based on a physical constrain derived from the percentage of theexponential decay related to each type of deformation (plastic orelastic). This constraint is derived from the normalized form ofEq. (5)

r�2 ¼ exp �k3uð Þ; where

r�2 ¼ r2

d � k1

k2

and u ¼ log 2Nf

log 2N�T

� �(9)

as illustrated by Fig. 1(a).The point 2N�T represents the CMBH model estimate of the

transition between plastic and elastic deformations (see Fig. 1(b)),and is obtained through

2N�T ¼rf

Ee0f

� � 1c�b

(10)

As demonstrated in Fig. 1(a), by imposing the derivative of r�2

to intercept u-axis at 1, i.e., the derivative intercepts u at 2N�T , wecan obtain k3 value. As derived, the constraint imposes k3 value tobe directly related to the deformation type that dominates theexponential decay. Depending on k3 value, ð1� r�ðu ¼ 1ÞÞ% ofthe exponential decay is related to plastic deformation whiler�ðu ¼ 1Þ% of the decay is related to elastic deformation. In thiscontribution, we set

r�ðu ¼ 1Þ ¼ 1=e (11)

Hence, k3 assumes a constant value of 1 (see Fig. 1) and roughly60% of the variance exponential decay is dominated by plastic

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deformations. Finally, to determine the remaining variance param-eters k1 and k2, we sampled k1 considering the same prior as forthe stationary variance model, and k2 value is obtained by solvingthe following mathematical constraint

ðr2

d ¼ Var½e� (12)

where e are the residuals at the observed data. In summary, k3 isset as a constant value based on a physical constraint, k1 issampled within a given range, and k2 value is obtained such that amathematical constraint is obeyed. Hence, when using the station-ary variance model, there is no reduction in the number of calibra-tion parameters, and engineering judgment is only used to definethe ranges of the parameters prior distributions. In this case, theresulting Bayesian calibration is carried considering then a total offive parameters. Now, when considering the proposed nonstation-ary variance model based on the aforementioned constraints, thetotal number of parameters for the Bayesian calibration is reducedfrom 7 (four for the CMBH model and three for the variancemodel) to 5. Once again, engineering judgment is used to definethe ranges of the related prior distributions.

3 Considered Materials: Inconel 617 and Hastealloy-X

For this study, we considered two Ni-base alloys Inconel 617(also known as IN-617) and Hastealloy-X (also known as Hast-X). Both materials are imparted with considerable strength andoxidation resistance at elevated temperatures making them idealmaterials for applications in reactor/combustion devices, gas tur-bines, and aircraft components. IN-617 is a solid-solution

strengthened, nickel–chromium–cobalt–molybdenum alloy, andwith its high chromium and aluminum content, it displays a signif-icant oxidation resistance at high temperatures, and creep proper-ties at elevated temperatures due to its high nickel and chromiumcontent.

Many fatigue studies were performed considering IN-617 toinvestigate the influences of temperature [35–38], strain rate [39],and dwell period [25,40]. Several others investigated creep defor-mation and rupture [41–45]. Nonisothermal, low-cycle fatiguealso termed thermo-mechanical fatigue has been studied as well[46]. Across studies, the grain morphologies of IN-617 specimensare generally identical. In addition, this material has a relativelylow density when compared to others in its class. This makes IN-617 suitable for applications where components are subjected toconstant or cyclic mechanical loads at temperatures up to 980 �C,and where weight is a factor. Given its longer history in this studyIN-617 is used as the “base” alloy that provides the information tocompensate for the lack of data for a “newer” similar alloy. Wederived a database for IN-617 at 850 �C using Refs. [36], [38],[40], and [47].

In this paper, we also studied how CMBH model parameters forHast-X can be calibrated using prior information obtained fromIN-617. Hast-X is a solid-solution strengthened, nickel–chromium–iron–molybdenum alloy, with its strength increased bythe addition of molybdenum and iron, while corrosion resistanceis increased by chromium addition. Components made of Hast-Xare primarily exposed to LCF conditions, making LCF an essen-tial concern for safe design and operation. Despite its importance,relatively limited data regarding Hast-X LCF are available in theliterature. Thus, in this contribution, we evaluate the use of anovel Bayesian framework for the calibration of Hast-X CMBHparameters. In this Bayesian calibration procedure, the requiredprior distributions for the Hast-X CMBH parameters are derivedconsidering the calibration results for IN-617 (i.e., derived uponIN-617 CMBH parameters posterior distributions). The Hast-Xcoupon dataset required for the calibration procedure was createdbased on Refs. [48] and [49], with temperatures at 870 and900 �C.

4 Results and Discussion

4.1 Model Calibration: Inconel 617. Figure 2(a) shows theset of 61 IN-617 coupons at 850 � C that was used in this study aswell as the CMBH model with the ordinary least square parame-ters. The ordinary least squares estimates for the CMBH modelparameters are presented in Table 1. Young’s modulus is assumedto be E¼ 152,000 MPa. Figure 2(b) shows the squared residuals atthe observed data versus 2Nf . This clearly illustrates the limita-tions of the stationary variance model as the variation in thesquare residuals decreases dramatically with 2Nf . With the resultsfrom Fig. 2, we proceed to the Bayesian calibration of the CMBHmodel. In other words, these results serve as the initial point forthe Bayesian analysis as it helps to define the bounds for the cali-bration parameters. Here, we assume the priors for all calibrationparameters to be uniform distributions defined between the lowerand upper bounds illustrated in Table 1 (using with the approachdiscussed in Sec. 2.3).

Table 2 shows the 95% intervals for the posterior distributionof the CMBH model parameters obtained after we performed theBayesian calibration (keep in mind this is only a partial view ofthe results, as all calibration parameters are jointly distributed).While the distributions for parameters rf, b, and c seem to beheavily tailed (as 2.5 and 97.5 percentiles are close to the lowerand upper bounds, respectively), the distributions for e0f and k1 arenarrow. Overall, the medians of the calibration parameters are rel-atively close to the point estimator returned by ordinary leastsquares (presented only for comparison and do not represent anaccuracy metric for the proposed framework). As previously men-tioned, one main advantage of the Bayesian approach over pointestimator procedures is the inherent ability to return uncertainty

Fig. 1 Coffin–Manson–Basquin–Haford model and normalizednonstationary variance model. (a) Normalized nonstationaryvariance model. (b) Graphical interpretation of the model.

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estimates (not always available in point estimator procedures,unless approximations are used).

The posterior distribution of the CMBH model parameters canbe propagated through the model defined in Eqs. (2) and (8) tocharacterize uncertainty in the strain-based fatigue lifing. Figures3(a) and 3(b) show the posterior median and confidence bands forthe strain amplitude versus fatigue life, considering both station-ary and nonstationary variance models, respectively. The confi-dence interval (dashed lines) quantifies the uncertainty in themean predictions for ea. This is the uncertainty in the CMBHmodel parameters and does not consider the coupon-to-couponvariation or the observation error. This uncertainty depends on theavailable data as well as the model form (this uncertaintydecreases with increasing 2Nf ). The prediction interval (dash-dot-ted lines) quantifies the uncertainty in the predicted ea while alsoconsidering the observation error. In the statistical model

expressed in Eq. (2), the observation error d � Nðld ¼ 0;r2dÞ is

the term that accounts for everything that is not covered inEq. (1). In other words, any specimen-to-specimen variation (heattreatment, composition, etc.) as well as experimental measure-ment error. Figure 3(a) highlights the effect of the stationary var-iance model, which returns an interval resembling a “constant”band around the median prediction. While this does not seem tobe a problem for short fatigue lives, it is definitely not an adequaterepresentation for long fatigue lives (let us say for ea < 0:005, forthis particular dataset). As illustrated in Fig. 3(b), this problem isovercome when we use the proposed nonstationary variance

Table 1 Bounds for the prior distribution of IN-617Coffin–Manson–Basquin–Haford model parameters and obser-vation error (OLS estimates are rf 51;875; b520:21; �0f 51:342,and c520:897)

Parameter Lower bound Upper bound

CMBH modelrf 1000 3000b �0.35 �0.15�0f 0.5 2.5c �0.9 �0.8Stationary variance modelk1 5� 10�7 2� 10�6

Nonstationary variance modelk1 5� 10�7 2� 10�6

Table 2 95% intervals for the posterior distribution of IN-617Coffin–Manson–Basquin–Haford model parameters consider-ing the available data (61 coupons at 850 �C)

Parameter 2.5th percentile 50th percentile 97.5 percentile

Stationary variance modelrf 1061 1891 2930b �0.329 �0.252 �0.176�0f 0.803 1.092 1.428c �0.898 �0.853 �0.804k1 1:623� 10�6 1:791� 10�6 1:843� 10�6

Nonstationary variance modelrf 1043 1098 2568b �0.282 �0.184 �0.162�0f 0.914 1.337 1.421C �0.892 �0.875 �0.816K1 4:671� 10�7 4:886� 10�7 4:996� 10�7

Fig. 3 Posterior confidence and prediction intervals for strainamplitude versus fatigue life for IN-617 (61 coupons at 850 �C).(a) Stationary variance model. (b) Nonstationary variancemodel.

Fig. 2 Ordinary least square and residuals at observed data(IN-617). (a) IN-617 data points at 850 �C and ordinary leastsquare model. (b) Squared residuals at observed data.

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model. In this case, the prediction interval decreases with increas-ing 2Nf (converging to a small band in the elastic domain). Theseresults indicate that the proposed nonstationary variance model isbetter suited to quantify uncertainty in the CMBH model.

Confidence bands, lower ones, in particular, are useful forunderstanding reliability curves and implementing reliability-based design and optimization. Conservatism in this predictioncan yield significant financial losses since “healthy” assets can beunnecessarily subject to inspection, repair, or even decommis-sioned. Figure 4 shows a comparison between the lower band (2.5percentile) predictions of both uncertainty models. The proposednonstationary model significantly reduces the conservatism in lifeprediction especially in the elastic region (long fatigue liferegion). In some cases, the nonstationary model predicts fatiguelives 2.5 times greater than the stationary model.

Additionally, in terms of engineering applications, uncertaintybands (both confidence and prediction intervals) can be used forprobabilistic design. A designer can take the probabilistic CMBHmodel and use it to assess the probabilistic fatigue life distributionof a critical component. Another interesting use of confidence andprediction intervals is in the characterization of fatigue life itself.For example, when conducting an experiment at very small strainlevels, one can use the prediction interval to assess the probabilitythat coupon will lead to run-out after a given number of cycles.Finally, uncertainty bands could be used to guide sequential sam-pling (adding coupons to the dataset and updating the model).

4.2 Model Calibration: Hastelloy-X. To illustrate how ourproposed Bayesian framework can be used to compensate for thelack of available data, here we show the CMBH calibration for theHast-X alloy with prior information coming from the IN-617 cali-bration results. Figure 5(a) shows the considered dataset of 14Hast-X coupons at 870 �C and 900 �C (in addition to the previous61 IN-617 coupons at 850 �C) that was used in this study. A goodagreement between both data sets can be observed. Figure 5(b)illustrates the strain versus fatigue life results for Hast-X using thenonstationary variance model, when uniform prior distributionswith limits derived from the bounds shown in Table 1 are consid-ered. For comparison purposes, Fig. 5(b) is presented on the samescale as Fig. 3 and will be used here just as a reference. In otherwords, Fig. 5(b) represents the calibration results considering datafor 14 Hast-X coupons without any prior information on theuncertainty in CMBH parameters.

Now, we demonstrate how our proposed Bayesian frameworkcan help to guide the CMBH calibration for Hast-X by leveraginginformation coming from IN-617, by emulating a sequential sam-pling procedure. Consider that the data acquisition for Hast-Xbegins with coupon tests performed at ea � 0:004. There is noparticular reason for this arbitrary threshold other than conven-ience. It is high enough to lead to fast coupon data gathering(around 1000 cycles). In general, engineering knowledge about

the specific fatigue behavior of the alloy to be studied can help toset this threshold up. For the Hast-X dataset considered, ea �0:004 leads to five data points, which are used to calibrate theCMBH model and variance parameters. Figure 6(a) illustrates thestrain versus fatigue life results for Hast-X considering the nonsta-tionary variance model with only these five coupon data (remain-ing data showed only for illustration purposes) and uniform priordistributions with limits derived from the bounds shown in Table 1are considered. As compared with Fig. 5(b), the curves forFig. 6(a) exhibit significantly large uncertainties at low strainranges as well as nonconservative estimates for the mean life.Figure 6(b) illustrates the strain versus fatigue life results forHast-X considering the same five coupon data but with CMBHand variance parameters prior distributions built upon IN-617 pos-terior distributions (see summary shown in Table 2). This timearound, the mean and spread in fatigue life are very well character-ized due to similarities with IN-617. Additionally, the originalHast-X dataset (all 14 coupons) used in this study lacks coupondata with ea � 0:007; however, the fact that we used priors derivedfrom IN-617 reduced the spread in predictions even at that range.

Another interesting comparison is related to the lower boundfor the confidence interval. Figure 7 illustrates the lower boundsobtained with the full dataset (14 coupon tests) as well as the pre-viously mentioned five data points (coupon tests with ea � 0:004)when both uniform priors and priors derived from IN-617 posteri-ors are used. To better illustrate the results shown in Fig. 7 con-sider the comparison for fatigue life prediction when ea ¼ 0:004.The lower bound prediction with the full dataset (14 coupon tests)is around 1500 cycles. The prediction when the model is

Fig. 4 Comparison of lower bands (2.5 percentile) of predic-tion intervals for IN-617 between both variance models

Fig. 5 Available data sets for Hast-X CMBH Bayesian calibra-tion (IN-617 data obtained at 850 �C and 14 Hast-X couponsobtained at 870 �C and 900 �C); and posterior confidence andprediction intervals for strain amplitude versus fatigue life forHast-X considering the nonstationary variance model. (a) IN-617 and Hast-X data sets. (b) Nonstationary variance model forHast-X complete dataset.

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calibrated considering uniform priors is of approximately 2600cycles. In the scenario where priors derived from IN-617 posteri-ors are used, the prediction is roughly 1200 cycles. Despite appa-rently leading to a more conservative bound (it is worthmentioning that in this scenario only five coupon tests were con-sidered in the CMBH calibration), in this scenario, the predictionsare closer to the reference lower bound (calibration with all 14coupon tests) estimates. This sequential sampling Bayesian frame-work can potentially be very useful in assessing the fatigue life of

components when the alloy characterization of strain-life curvesis still in its preliminary stages.

The summation of all of the previously described features ofour proposed Bayesian calibration framework combined with anonstationary variance model allows engineers to design moreefficient experimental campaigns guided by probabilistic fatiguelife estimations. Starting from a small collection of coupon data athigh strain ranges, and guided by the model predictions and uncer-tainties convergence, coupon tests at low strain ranges can beselected at specific conditions that allows the “maximum” of“information” to be gained by these tests and reduce the overallnumber of coupon tests for the material characterization. As illus-trated by the Hast-X analysis using an extremely oversimplifiedprocedure that borrows some of the ideas of the above procedure,the material behavior can be sufficiently characterized with fewcoupon data. Additionally, using the upper band of the confidenceinterval, one can assess the probability that coupon will lead torun-out after a given number of cycles. For instance, using theconfidence bands illustrated in Fig. 6(b) for ea of 0.003, there is97.5% chance to a coupon to fail before 10,000 cycles and only2.5% chance to fail above this value. Hence, if one is running anexperimental test at this strain value and the coupon has not failedeven after 10,000 cycles, one can tag this test as a run-out with97.5% of confidence. Such feature can potentially help to savecosts concerning experimental campaigns even further.

5 Conclusions

Building strain-life relationships to model materials subjectedto low-cycle fatigue is a widely accepted practice. The CMBHmodel is a popular choice that employs a double power-law equa-tion. In this paper, we proposed a nonstationary uncertainty modelfor the CMBH model. We also proposed a Bayesian frameworkfor calibration of CMBH model parameters as well as estimationof confidence and prediction intervals in strain versus fatigue lifecurves.

We used a set of available Inconel 617 and Hastelloy-X coupontest data to study (a) uncertainty quantification (i.e., model cali-bration) and propagation (strain-life uncertainty intervals) undertwo distinct variance models, (b) transferability of uncertaintieswithin the same family of alloys to compensate for the lack ofavailable data, and (c) model update through the sequential acqui-sition of coupon data. We concluded that the proposed uncertaintymodel successfully captures the fact that the variance of strainamplitude decreases with increasing fatigue life. We also con-cluded that the Bayesian framework can help designing coupontest campaigns when alloy characterization is still in its prelimi-nary stages by monitoring uncertainty levels.

Despite the many desirable features, the proposed methodologyfor using priors from similar alloys should be used with caution.As in any other Bayesian method, there is an inherent risk to gen-erate biased predictions when used to compensate for the lack ofdata. Unexpected features of an alloy can undesirably skew itsposterior distribution, invalidating its usability to generate priordistributions for other alloys. Furthermore, unanticipated effectsnot manifested in the observed experimental data may nullify anypredictions for the novel alloy. Additionally, in the presented stud-ies, the strain-life behavior of the selected alloys is remarkablysimilar, a scenario that may not always be the case. The mainassumption here is that engineering judgment can be used todefine scenarios in which such uncertainties “transferability” canbe carefully used. No studies concerning the limits of this“transferability” approach and the risks associated with it wereperformed, which opens the opportunity for future studies toaddress such aspects.

Moreover, several possible extensions deserve to be exploredfurther. For example, our formulation is based on the log-normaldistribution of fatigue life (normally distributed in the log-space);it would be useful to explore the use of other distributions. Wehave advocated in favor of using the uncertainty convergence to

Fig. 6 Posterior confidence and prediction intervals for strainamplitude versus fatigue life when Hast-X model is updatedwith data in which ea � 0:004 (remaining data shown only forillustration). (a) Hast-X CMBH calibration with uniform priors.(b) Hast-X CMBH calibration with IN-617 posterior distributionsas priors.

Fig. 7 Comparison of lower bands (2.5 percentile) of predic-tion intervals for Hast-X when model is updated with data inwhich �a � 0:004

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gauge whether or not more coupon tests are needed. It would beinteresting to see if there is a preferred strain range that wouldlead to most model improvement when performing new coupontests; or alternatively, if there is a limit to the reduction of predic-tion intervals. Finally, it is worth exploring the extension of theproposed approach to other alloys (maybe nickel–titanium alloysor even outside the family of nickel-based alloys).

Nomenclature

B ¼ fatigue strength exponentC ¼ fatigue ductility exponent

CMBH ¼ Coffin–Manson–Basquin–HafordE ¼ Young’s modulus

OLS ¼ ordinary least squareNf ¼ fatigue lifeK ¼ observation error model parametersD ¼ observation errorea ¼ strain amplitudee0f ¼ fatigue ductility coefficient

h ¼ set of calibration parametersrf ¼ fatigue strength coefficient

R ¼ covariance matrix

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