Inference of equivalent initial flaw size under multiple sources of uncertainty

International Journal of Fatigue 33 (2011) 75–89

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Inference of equivalent initial flaw size under multiple sources of uncertainty

Shankar Sankararaman, You Ling, Chris Shantz, Sankaran Mahadevan *

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA

a r t i c l e i n f o

Article history:Received 1 April 2010Received in revised form 16 June 2010Accepted 30 June 2010Available online 6 July 2010

Keywords:Fatigue life predictionInitial flaw sizeCrack growthUncertainty quantificationBayesian inference

0142-1123/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2010.06.008

* Corresponding author. Tel.: +1 615 322 3040.E-mail address: [email protected]

a b s t r a c t

A probabilistic methodology is proposed in this paper to estimate the equivalent initial flaw size (EIFS)distribution accounting for various sources of variability, uncertainty and error, for mechanical compo-nents with complicated geometry and multi-axial variable amplitude loading conditions. A Bayesianapproach is used to calibrate the distribution of EIFS, where the likelihood function is constructed frommodel-based fatigue crack growth analysis and inspection results. The variability, uncertainties anderrors in the above procedures are quantified, and the distribution of EIFS is calibrated by explicitlyaccounting for the various sources of uncertainty. Three types of uncertainty are considered: (1) naturalvariability in loading and material properties; (2) data uncertainty, due to crack detection uncertainty,measurement errors, and sparse data; (3) modeling uncertainty and errors during crack growth analysis,numerical approximations, and finite element discretization. A Monte Carlo simulation-based approach isdeveloped for uncertainty quantification in the crack growth analysis and for constructing the likelihoodfunction of EIFS. The proposed methodology is illustrated by a numerical example.

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1. Introduction

Various sources of uncertainty, such as loading, material prop-erties and geometry of mechanical components, contribute to thenon-deterministic behavior of fatigue crack growth. Fracturemechanics-based models, describing how the crack size increasesunder cyclic loading, are widely used to predict the fatigue lifeand evaluate the reliability of mechanical components. A prerequi-site of using the crack growth models is to determine the initialcrack size. Two problems are encountered when the initial cracksize is small. First it is difficult to measure the initial crack size ifthe limit of the inspection technique is reached. The second prob-lem is that the commonly used crack growth models are applicableonly when the initial crack size is beyond micro-scale. In practice,empirically assumed initial crack in the long size region is used[1–3]. However, this leads to a very conservative fatigue life pre-diction if the actual initial crack size is small. The concept of equiv-alent initial flaw size (EIFS) was developed about 30 years ago toaddress the two problems above [4]. It is defined such that the pre-diction of a long crack growth model which starts from EIFS is inagreement with the actual fatigue crack growth [5]. Note that EIFSis not a physical quantity pertaining to the component, but a hypo-thetical flaw size that can be used as an initial crack size in longcrack growth models.

One approach to calculate EIFS is the back-extrapolation meth-od. Given some sets of experimental data (observed crack sizes at a

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du (S. Mahadevan).

certain time point), back-extrapolation from the given time instantto initial time instant is conducted based on fatigue crack growthmodels [5–7] or observed crack growth curves [8,9]. The cracksizes corresponding to the initial time instant are used to estimatethe distribution of EIFS. The application of the back-extrapolationmethod is limited by its requirement for large amount of fatiguecrack growth data. Insufficient test data will cause more uncer-tainty in the estimate of EIFS.

A probabilistic methodology for the calculation of EIFS withoutback-extrapolation was proposed by Liu and Mahadevan [4] basedon matching the infinite fatigue life predicted by the model withthe experimental data from S–N tests. In this approach, somematerial properties (fatigue limit and threshold stress intensityfactor) are treated as random variables. The distribution of EIFS isderived from the distribution of material properties, and hencecrack growth analysis is avoided. Reasonable agreement was ob-served between this EIFS-based fatigue life prediction and theexperimental results for specimens with simple geometry underuni-axial and multi-axial loading [4,10,11]. Components with com-plicated geometry under variable amplitude loading conditionhave not been considered yet in this line of research.

Makeev et al. [12] and Cross et al. [13] applied statistical tech-niques, namely the maximum likelihood approach and the Bayesianapproach, to calibrate the distribution of EIFS based on inspectiondata. In this approach, EIFS is treated as a model parameter that iscalibrated through the observation of crack sizes at different num-bers of cycles. This approach is significantly different from earliermethods used for EIFS calculation using inspection data [5–9];these are not based on statistics and use back-extrapolation of the

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Nomenclature

Symbola flaw sizea0 equivalent initial flaw sizeN number of cyclesaN flaw size after N cyclesur retardation parameterC, n, m model parameters of modified Paris lawDK stress intensity factorDKN stress intensity factor at the Nth cycle

DKth threshold stress intensity factorDKeqv equivalent stress intensity factoram measured crack sizeem measurement erroreg crack growth law model form erroreh discretization errorL(a0) likelihood function of equivalent initial flaw sizeDrf fatigue limitPOD probability of detection

76 S. Sankararaman et al. / International Journal of Fatigue 33 (2011) 75–89

crack growth curve to calculate the EIFS. Makeev et al. [12] andCross et al. [13] illustrated the calibration methodology for simplestructural elements subjected to constant amplitude loading.Sankararaman et al. [14] extended this methodology for structureswith complicated geometry and multi-axial variable amplitudeloading, by combining finite element analysis, crack growth analy-sis, surrogate model development, Bayesian inference, and proba-bility integration.

In [12–14], the uncertainty in crack growth prediction was rep-resented by adding a single noise term to the output of the crackgrowth analysis. The noise term is the result of contributions frommany sources, such as variability in loading and material proper-ties, model uncertainty and errors in structural stress analysisand crack growth model, and data uncertainty and errors in labo-ratory experiments and field inspections. A systematic and rigor-ous procedure for the quantification of different sources ofuncertainty is necessary for the accurate calibration of EIFS, insteadof lumping all contributions into a single noise term with an arbi-trarily assumed distribution (e.g., lognormal [12,13] or normal[14]).

This paper develops a methodology to implement the Bayesianapproach for the statistical inference of EIFS by explicitly consid-ering various sources of variability, uncertainty and error inmodel-based fatigue crack growth analysis, for components withcomplicated geometry and multi-axial variable amplitude loadingconditions. The most important step in this procedure is the con-struction of a likelihood function, which is proportional to theprobability of observing the given data conditioned on EIFS. In[12–14], the likelihood function construction was very simple,since it was based on an assumed noise term. In the current paper,the likelihood function needs to consider the contribution of dif-ferent sources of error and uncertainty. The combination of differ-ent sources of error and uncertainty is complicated, and this paperdevelops a systematic procedure to account for these sources inthe construction of the likelihood function.

The proposed method can be used with any fracture mechanics-based crack growth model and this paper uses a modified Paris lawonly for the purpose of illustration. The associated model uncer-tainty and errors are considered. A block loading with randomblock length and amplitudes is used as an illustration to representthe natural variability in fatigue loading. An equivalent stressintensity factor (SIF) is calculated using a characteristic plane ap-proach proposed by Liu and Mahadevan [15] to combine SIFs inmultiple modes. In the presence of variable amplitude loading,Wheeler’s retardation model is employed as an illustration to ac-count for the effect of overload, and SIF is calculated in a cycle-by-cycle manner. Finite element analysis (FEA) is required for SIFcalculation due to the complicated component geometry. It is com-putationally expensive to use a finite element model in every load-ing cycle; therefore, an accurate surrogate modeling approach isused in this paper. Sankararaman et al. [14] investigated differenttypes of surrogate modeling techniques for this purpose and found

the Gaussian process (GP) modeling technique to be most effective.A limited number of FEA runs are used to build an accurate GPmodel. The use of the surrogate model introduces additional uncer-tainty, and FEA results are subjected to discretization errors. Theseare addressed in the proposed methodology. The physics-basedEIFS distribution proposed by Liu and Mahadevan [4] is treatedas the prior distribution for Bayesian updating, and the materialproperties of components used to calculate this prior distributionare considered as random variables. The statistics of material prop-erties are estimated from experimental data and the estimationcould be inaccurate due to insufficient data. Hence, input datauncertainty is taken into account to reflect this issue. Since theinference of EIFS requires experimental crack growth data, suchas the crack size measured after N load cycles, the uncertainty inthe crack inspection technique is also included in the analysis.

In summary, the following sources of uncertainty are explicitlyconsidered and quantified during the calibration of EIFS in thispaper:

1. Natural variability in loading and material properties.2. Data uncertainty due to crack detection uncertainty, measure-

ment errors, and sparse data.3. Model uncertainty and errors in crack growth analysis, numer-

ical approximations, and finite element discretization.

Section 2 first presents the deterministic crack growth analysisfor components with complicated geometry and multi-axial vari-able amplitude loading conditions, and then different sources ofuncertainty and errors are discussed. Section 3 describes theBayesian approach for updating the prior distribution of EIFS. Anumerical example – cracking in a two-radius hollow cylindricalcomponent – is used in Section 4 to illustrate the proposedmethodology.

2. Fatigue crack growth analysis and the associated uncertaintyand errors

This section discusses the quantification of various sources oferror and uncertainty in fatigue crack growth analysis. Section2.1 presents the deterministic crack growth analysis for compo-nents with complicated geometry and multi-axial variable ampli-tude loading conditions. Sections 2.2–2.4 discuss the differentsources of uncertainty and errors involved in the fatigue crackgrowth analysis and structural inspection and propose methodsto handle them. The different types of uncertainty are classifiedinto three major categories: natural variability, data uncertainty,and model uncertainty.

2.1. Deterministic fatigue crack growth analysis

Consider any long crack growth law used to describe the rela-tionship between da/dN and DK, where N represents the number

S. Sankararaman et al. / International Journal of Fatigue 33 (2011) 75–89 77

of cycles, a represents the crack size and DK represents the stressintensity factor. This paper uses a modified Paris’ law [16] with aWheeler’s model [17] to include retardation effects as:

dadN¼ urCðDKÞn 1� DKth

DK

� �m

ð1Þ

Note that several models [18–20] have been proposed to tacklevariable amplitude loading conditions. This paper uses the afore-mentioned model only to illustrate the proposed uncertainty quan-tification methodology, and other appropriate models can also beused instead.

In Eq. (1), C (rate), n and m (exponents) are model parameters ofthe modified Paris law, and DKth represents the threshold stressintensity factor. ur refers to the retardation parameter, and is equalto unity if ai + rp,i > aOL + rp,OL, where aOL is the crack length at whichthe overload is applied, ai is the current crack length, rp,OL is the sizeof the plastic zone produced by the overload at aOL, and rp,i is thesize of the plastic zone produced at the current crack length ai.Otherwise, ur is calculated as [17]:

ur ¼ rp;i

aOL þ rp;OL � ai

� �k

ð2Þ

where k is the curve fitting parameter for the original Wheeler mod-el, termed the shape exponent. The size of the plastic zone can becalculated in terms of the applied stress intensity factor (K) andyield strength (r) as:

rp ¼ aKr

� �2

ð3Þ

where a is known as the effective plastic zone size constant which iscalculated experimentally.

By integrating the expression in Eq. (1), the number of cycles(N) to reach a particular crack size aN can be calculated as shownin Eq. (4).

N ¼Z

dN ¼Z aN

a0

1urCðDKÞnð1� DKth

DK Þm da ð4Þ

The stress intensity factor DK in Eq. (4) can be expressed as aclosed form function of the crack size for specimens with simplegeometry subjected to constant amplitude loading. However, thisis not the case in many mechanical components, where DK de-pends on the loading conditions, geometry and the crack size. Fur-ther, for multi-axial loading condition (for example, simultaneoustension, torsion and bending), the stress intensity factors corre-sponding to the three modes need to be taken into account. Thiscan be accomplished using an equivalent stress intensity factor[15]. If KI, KII, KIII represent the mode-I, mode-II and mode-III stressintensity factors respectively, then the equivalent stress intensityfactor Keqv, which combines the effect of multiple modes, is ob-tained through a characteristic plane approach proposed by Liuand Mahadevan [15], as follows:

Keqv ¼1B

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1Þ2 þ

k2

s

� �2

þ k3

s

� �2

þ AkH

s

!2vuut ð5Þ

where k1, k2, k3 are the parameters associated with modes I, II, andIII loading, respectively. kH is related to hydrostatic stress. s is theratio of Modes II and I stress intensity factor at a specific crackgrowth rate (da/dN). A and B are material parameters. The use ofthe characteristic plane approach for crack growth prediction undermulti-axial variable amplitude loading has been validated earlierwith several data sets [15].

During each cycle of loading, the crack grows and hence, thestress intensity factor needs to be reevaluated at the new crack sizefor the loading in the next cycle. Hence, it becomes necessary to

integrate the expression in Eq. (4) through a cycle-by-cycle proce-dure. Each cycle involves the computation of DK using a finite ele-ment analysis. Repeated evaluation of the finite element analysisrenders the aforementioned cycle-by-cycle integration computa-tionally expensive, perhaps impossible in some cases. Hence, it be-comes advantageous to substitute the finite element analysis by aninexpensive surrogate model. Different types of surrogate models –polynomial chaos expansion [21], support vector regression [22]and Gaussian Process interpolation [23] – have been explored;the Gaussian process modeling technique was found to be the mostaccurate method for the current problem [14] and has been em-ployed in this paper. A few runs of the finite element analysis areused to train this surrogate model and then, this model is usedto predict the stress intensity factor for other crack sizes and load-ing cases (for which finite element analysis has not been carriedout). The procedure of constructing a GP model for the calculationof the stress intensity factor is described by Sankararaman et al.[14].

It is important to note that replacing a deterministic finite ele-ment analysis with a Gaussian process surrogate model impartsadditional uncertainty to the crack growth analysis (discussed laterin Section 2.4.2). The Gaussian process model predicts the stressintensity factor as normal distribution with mean and variance.Higher the number of training points used in the construction ofthe surrogate model, lower is the uncertainty (variance) in the sur-rogate model prediction.

The choice of training points (i.e. finite element runs) can beguided using the greedy point algorithm developed by McFarland[23]. The greedy point algorithm is an adaptive method for selec-tion of training points in which (1) an optimization is used to esti-mate where the prediction variance is maximum and (2) a newtraining point is added at this particular location. The Gaussianprocess methodology is an interpolation technique and predictszero variance at all the training points. Hence, the maximum pre-diction variance over the entire domain decreases through theuse of the greedy point algorithm. This process is continued untilthe maximum variance is below a desirable threshold value.

Once the surrogate model is constructed, it can be used in cycle-by-cycle integration of the crack growth law. Eqs. (1)–(5) describedabove are used to calculate the final crack size A after N loading cy-cles. This entire procedure is summarized in Fig. 1.

The framework shown in Fig. 1 for crack growth analysis isdeterministic and does not account for errors and uncertainty.Uncertainty can be associated with each of the blocks in Fig. 1and accounted for in crack growth prediction. The following sec-tion investigates these sources of uncertainty and errors.

2.2. Natural variability

Because of numerous fluctuations in the environment, test pro-cedure, instruments, observer, etc., repeated observations of thesame physical quantity do not yield identical results. This is re-ferred to as natural or inherent variability [24].

2.2.1. Natural variability in loadingDue to the natural randomness in the environment and operat-

ing conditions, cyclic fatigue loading in real engineering problemsusually has non-deterministic amplitudes, and the sequences ofthese amplitudes are also uncertain. Some researchers have inves-tigated methods to quantify the variability in loading [25]. For thepurpose of illustration, this paper uses a block loading with ran-dom block length (number of load cycles within one block) andrandom amplitude in the crack growth simulation. A samplerealization of loading history is shown in Fig. 2. This loading historyis generated by considering a number of successive loading blocks.For each block, the block length is randomly selected, and the

Fatigue Crack Growth Analysis

Predict Final Crack Size

(aN) as a function of

number of load cycles (N)

Surrogate Model

Finite Element Analysis (Generate training points)

Loading

ΔKeqv

Characteristic Plane

KI, KII, KIII

Keqv

(Training Points)

EIFS (Initial Value )

Fig. 1. Fatigue crack growth analysis.

Fig. 2. Sample loading history.


loading amplitudes (maximum and minimum) corresponding tothat block are randomly generated from the assumed statistics.

2.2.2. Natural variability in material propertiesMaterial properties could be affected by many factors, such as

sample composition and micro-structure, and hence are randomin nature. Fatigue limit and threshold stress intensity factor, twomaterial parameters which will be used to derive the prior estima-tion of EIFS, are treated as random variables to account for the nat-ural variability in material properties.

2.3. Data uncertainty

As mentioned earlier in Section 1, field inspections are used tocollect crack growth data and this information is used to calibratethe distribution of EIFS. The different types of uncertainty in thisprocedure – crack detection uncertainty and measurement errors– are discussed in Section 2.3.1. Also, the probability distributionsof certain material properties are inferred from laboratory experi-ments. This data may be sparse and insufficient to construct aprobability distribution. This type of uncertainty is discussed inSection 2.3.2.

2.3.1. Crack detection uncertainty and output measurement errorsCrack growth data measured from laboratory experiments is di-

rectly used for the inference of EIFS in the approach followed inthis paper. These measurements are affected by several factors,such as environment, equipment and operator skills and it is nec-essary to account for these factors and include the measurementerrors in the inference of EIFS. For an in-service component,non-destructive inspection (NDI) technique is commonly used fordamage detection. Several metrics could be used to evaluate theperformance of NDI, such as probability of detection (POD), flawsize measurement accuracy, and false call probability (FCP). Thesecriteria are developed from different methods, and they are used toevaluate different aspects of NDI performance. However, Zhangand Mahadevan [26] showed that these quantities are mathemat-ically related. POD and FCP can be derived from size measurementaccuracy, which measures the difference between actual valuesand observed values of the crack size.

In the context of EIFS inference, POD could be used in two casesof inspection results: (a) crack is detected; (b) crack is not detected.For the case of detecting a crack and also measuring its size, sizemeasurement accuracy could be employed. Based on the aboveconsideration, size measurement accuracy can be used to quantify


the uncertainty in experimental crack growth data, with the fol-lowing expression determined by regression analysis [26]:

am ¼ b0 þ b1aþ em ð6Þ

In Eq. (6), am is the measured flaw size; a is the actual flaw size;b0 and b1 are the regression coefficients; em represents the unbi-ased measurement error, assumed as a normal random variablewith zero mean and standard deviation re. The value of re is differ-ent for each inspection technique used.

2.3.2. Sparse data for characterizing input random variablesSome important parameters used in crack growth analysis, such

as the threshold stress intensity factor and the fatigue limit aretreated as random variables [4], whose distributions are estimatedfrom available experimental and inspection data. The statistics ofloading as mentioned in Section 2.2 are also estimated from exper-imental or actual observed data. The estimation of these statisticsfrom limited data causes additional uncertainty, i.e., the estimatedstatistics are not necessarily accurate. McDonald et al. [27] pro-posed a methodology to account for sparse data, in which thequantity of interest can be represented using flexible probabilitydistributions, whose parameters are also treated as random vari-ables with flexible distributions. The uncertainty due to data sizeis taken into account while estimating the probability distributionof the threshold stress intensity factor and the fatigue limit, whichare in turn used to calculate the prior distribution of EIFS in Section3.2; similarly, the distribution parameters of loading amplitude arealso treated as random variables.

2.4. Model uncertainty and errors

Computational models are only approximate representations ofactual system behavior. The difference between model predictionand true behavior is referred to as model error. This section dis-cusses the different types of model uncertainty and error in thedeterministic crack growth analysis procedure described earlierin Section 2.1.

2.4.1. Model uncertainty and errors in fatigue crack growth modelThe model uncertainty existing in the fatigue crack growth

model can be divided into two parts: model form uncertaintyand model parameter uncertainty.

Several empirical equations are available in the literature to de-scribe the fatigue crack growth behavior, including Paris law [28],modified Paris law [16], Forman’s equation [29], Weertman’sequation [30], etc. These crack growth models may be used for dif-ferent purposes. For example, Paris law is appropriate for theintermediate region of crack growth [31]; the modified Paris lawextends to the near-threshold region [4]; and both Forman’s andWeertman’s equations include the influence of the stress ratio[32]. Hence, the use of a particular empirical crack growth lawbrings in model form uncertainty, since it is difficult to tellwhether this selection is correct or not. The uncertainty in the pre-diction due to the model form error is addressed by introducing arandom error term eg.

Once the form of the fatigue crack growth model is selected, thestatistics of the corresponding parameters, denoted as P, are cali-brated from experimental data. To account for the uncertaintydue to calibration, the parameters in the crack growth model aretreated as random variables.

The general expression of fatigue crack growth law with ran-dom parameters P and model form error eg (treated as a lognormalrandom variable) could be written as:

dadN¼ gðDK;PÞ � eg ð7Þ

In Eq. (7), note that the error term (eg) is a lognormal variableand hence is multiplied. Using an additive normal error term isidentical to using a multiplicative lognormal error term [14].

2.4.2. Model uncertainty due to surrogate modelSeveral finite element runs for some combinations of input–

output variable values are used to train the Gaussian process(GP) surrogate model in this paper. Then, these surrogate modelsare used to evaluate the stress intensity factor for other combina-tions of input variable values. GP models treat the output (stressintensity factor) as a Gaussian process indexed by the input vari-ables (loading, current crack size, aspect ratio and angle of crackorientation, etc.) and hence, produce a normally distributed outputfor given inputs. The value for DKeq is randomly sampled from thedistribution of this output, thus accounting for the surrogate modeluncertainty. (Note: The GP model is used as a surrogate for thedeterministic finite element model and the variance of the GP out-put accounts only for the uncertainty in replacing the originalmodel with a Gaussian process and does not account for the uncer-tainty in the inputs to the model. The variance of the output is onlydependant on the ‘‘form” of the surrogate model. For example, alinear surrogate model will lead to constant variance at untrainedlocations but unknown distribution type [33].)

2.4.3. Model error due to FEA discretizationMesh density is an important factor affecting the accuracy of

FEA. Generally, better results will be obtained as the mesh size de-creases. However, smaller mesh size could lead to very time-con-suming analysis especially when the geometry of component iscomplicated. Therefore, it is desired to quantify the error due tothe coarse mesh, and then the corrected results from the coarsemesh could be used. Among several possible approaches, the Rich-ardson extrapolation (RE) method has been found to be appropri-ate for quantifying the actual discretization error [34]. Noted thatthe use of Richardson extrapolation is based on the assumptionof monotone convergence, i.e., the model solution is convergentand the domain is discretized uniformly (uniform meshing) [35].Solutions from three meshes with different densities are used tocalculate the discretization error, as follows:

eh ¼f1 � f2

rp � 1ð8Þ

In Eq. (8), f1 and f2 are solutions for a coarse mesh and a finemesh respectively. If the corresponding element sizes are denotedby h1 and h2, then the grid refinement ratio, denoted by r is calcu-lated as h2/h1. The exponent p in Eq. (8) is calculated as:

p ¼log f3�f2

f2�f1

� �logðrÞ ð9Þ

where f3 represents the solution for a third (finest) mesh with ele-ment size h3, with the same grid refinement ratio, i.e. r = h3/h2.

The solutions f1, f2, f3 are dependent on the inputs (loading, cur-rent crack size, aspect ratio and angle of crack orientation, etc.) tothe finite element analysis and hence the error estimates are alsofunctions of these input variables. For each set of inputs, a corre-sponding error is calculated and this error is added to the coarsemesh solution f1 from finite element analysis to calculate the cor-rected solution. Hence the corrected solution is associated to eachset of inputs and these values are used as training points for thesurrogate model.

Note that the Richardson extrapolation method relies only onsolutions from multiple mesh densities and does not consider theeffects of shape and distribution of elements. Hence, in reality, thisis a simple method that gives an estimate of the discretizationerror. Other sophisticated methods may be used to calculate the


discretization error. The focus of this paper is not to develop accu-rate methods for the calculation of discretization error but to dem-onstrate how the discretization error can be included in theconstruction of surrogate model and hence in the inference of EIFS.

3. Statistical inference of EIFS distribution

This section describes the Bayesian statistical inference approachfor updating the EIFS distribution. A major focus of this section is toconstruct the likelihood function of EIFS, i.e., the conditional proba-bility of inspection data given the value of EIFS.

3.1. Bayes theorem

The estimation of EIFS, in the presence of the various sources ofuncertainty discussed in Section 2, is based on Bayes’ theorem,which is a widely used approach that combines prior knowledgeand available experimental evidence to obtain the distribution ofEIFS.

Let a0 denote EIFS. Suppose fa0 ða0Þ represents the prior distribu-tion of EIFS, the likelihood function of EIFS L(a0) contains the infor-mation from available experimental crack growth data, andfa0 jDataða0jDataÞ denotes the updated EIFS distribution. The basicformula of Bayes theorem applied for EIFS estimation is:

fa0 jDataða0jDataÞ ¼ Lða0Þfa0 ða0ÞRLða0Þfa0 ða0Þda0

ð10Þ

3.2. Prior distribution of EIFS

As mentioned in Section 1, Liu and Mahadevan [4] proposed aphysics-based model for computing the EIFS based on materialproperties and structural geometry. Two parameters – fatigue limitDrf and threshold stress intensity factor DKth – are used to esti-mate the EIFS. Fatigue limit Drf is defined such that the componentfailure will not occur when the applied loading is under this limit.Threshold stress intensity factor defines a load criterion underwhich no significant crack growth would occur. These two con-cepts are connected using the Kitagawa–Takahashi diagram [36]and the El Haddad model [37].

As shown in Fig. 3, the fatigue limit of specimen can be consid-ered as a constant when the crack size is smaller than a certain va-lue. This value is affected by the fatigue crack threshold intensityfactor. The El Haddad model [37] further describes the relationshipbetween the fatigue limit and the threshold stress intensity factor:

Fig. 3. Schematic representation of the Kitagawa–Takahashi diagram.

DKth ¼ Drf

ffiffiffiffiffiffipap

Y ð11Þ

where Y is a geometry correction factor representing the crack con-figuration; a is a fictional crack length. The combination of Fig. 3and Eq. (11) derives the expression of EIFS (a0):

a0 ¼1p

DKth

Drf Y

� �2

ð12Þ

The EIFS calculated from Eq. (12) renders the infinite fatigue lifepredicted by the model match with the material S–N curve, i.e., thefatigue crack growth model which starts from EIFS will give infinitelife prediction while loading is under the fatigue limit. Fatigueanalysis based on this EIFS estimation has been validated usingexperimental data under constant amplitude, uni-axial and mul-ti-axial loading [4,10,11]. However, its accuracy for variable ampli-tude loading is not clear so far. Meanwhile, approximation isneeded while calculating the geometry factor for components withcomplicated geometry. Given these considerations, the distributionof EIFS based on Eq. (12) is used as a prior distribution, and a pos-terior distribution of EIFS is derived using experimental crackgrowth data of the actual components. Then the calibrated EIFScould be used to predict the fatigue life and reliability of similarcomponents.

Notice that the statistics of DKth and Drf are fitted from labora-tory experimental data, which introduce additional uncertaintyinto the prior estimation of EIFS distribution. As discussed inSection 2.3, the parameters of EIFS distribution k are treated as ran-dom quantities to account for this uncertainty. Hence, the Bayestheorem is also applied to update the distribution of k:

fkjDataðkjDataÞ ¼ LðkÞfkðkÞRLðkÞfkðkÞdk

ð13Þ

where L(k), the likelihood function of k, could be calculated fromthe likelihood function of EIFS through a conditional probabilityrelationship, as follows:

LðkÞ ¼Z

Lða0Þfa0 jkða0jkÞ�da0 ð14Þ

Once the updated statistics of the distribution parameters areobtained, the distribution of EIFS could also be updated using con-ditional probability:

fa0 jDataða0jDataÞ ¼Z

fa0 jkða0jkÞfkjDataðkjDataÞdk ð15Þ

3.3. Likelihood function of EIFS

In Bayesian inference, the likelihood function of EIFS L(a0) is theprobability of observing the available data given EIFS, i.e.,PDataja0 ðDataja0Þ, thus representing the information provided byexperimental data. The quantity Data is specified as two possibleinspection results after the Nth load cycle: crack with a certain sizeis detected and no crack is detected. Three steps are followed hereto explain how to calculate the likelihood function of EIFS:

3.3.1. Step 1: Calculate distribution of predicted crack sizeThis step calculates the statistics of the predicted crack size aN

after the Nth load cycle, if the crack starts growing from an EIFS va-lue. This is an uncertainty propagation problem, as shown in Fig. 4.

Assume a model represents the fatigue crack growth as:

dadN¼ gðDK;PÞ ð16Þ

where P is the vector of model parameters.The three equations below represent the crack growth analysis

within the Nth load cycle:

Fig. 4. Calculation of distribution of predicted crack size aN.


DKN ¼ f ðSN; aN�1;YÞ ð17Þ

DaN ¼ gðDKN;PÞ þ eg ð18Þ

aN ¼ aN�1 þ DaN ð19Þ

where SN denotes the loading condition in the Nth load cycle; Y is afactor representing the effect of geometry of the crack; DaN is theincrement of crack size in the Nth load cycle; aN�1 is the crack sizebefore the Nth load cycle; eg is the error in the crack growth model.

Considering the uncertainty sources discussed in Section 2, thevariables in Eqs. (17)–(19) are all random variables. Using theseequations, the following conditional probability density functionscan be obtained:

fDKN jSN ;aN�1 ;Y ðDKN jSN ; aN�1;YÞ ð20Þ

fDaN jaN�1 ;eg ;DKN ;PðDaNjaN�1; eg ;DKN ;PÞ ð21Þ

faN ja0 ;aN�1 ;DaN ðaN ja0; aN�1;DaNÞ ð22Þ

Suppose the statistics of loading SN, crack geometry factor Y,crack growth model parameter vector P, and crack growth modelerror eg have been quantified beforehand. Then the uncertaintypropagation within the Nth load cycle can be described as follows:

fDKN jaN�1 ðDKNjaN�1Þ ¼Z

fDKN jaN�1 ;SN ;YðDKNjSN; aN�1; YÞfSN

� ðSNÞfY ðYÞdSN dY ð23Þ

fDaN jaN�1 ðDaNjaN�1Þ ¼Z

fDaN jaN�1 ;eg ;DKN ;PðDaNjaN�1; eg ;DKN;PÞfeg

� ðegÞfDKN jaN�1 ðDKN jaN�1ÞfPðPÞdeg dDKN dPð24Þ

faN ja0 ðaN ja0Þ ¼Z

faN ja0 ;aN�1 ;DaN ðaN ja0; aN�1;DaNÞfDaN jaN�1

� ðDaN jaN�1ÞfaN�1 ja0ðaN�1ja0ÞdDaN daN�1 ð25Þ

Probability Distribution of Predicted Crack Size aN

(Section 3.3)

Yes Likelihood

function of EIFSwith Probability

of Detection

Is Crack Detected?

Likelihood function of EIFS with Size Measurement

Uncertainty

Prior Distriof EIF

(Section

Fig. 5. Bayesian approach for the

The statistics of crack size after the Nth load cycle given EIFS isgiven by Eq. (25). This is a critical step in constructing the likeli-hood function of EIFS.

The implementation of the above uncertainty propagation ap-proach requires a large amount of nested numerical integrationsin each load cycle. Maintaining the accuracy of these numericalintegrations is computationally expensive. Therefore, it is de-scribed only for the purpose of developing the concept. As a rela-tively more convenient and simpler method, Monte Carlosimulation is used here to numerically evaluate Eqs. (23)–(25).

In the Monte Carlo simulation approach, the random variablesor stochastic process with known statistics are sampled first, suchas load history, crack geometry factor Y, crack growth modelparameter vector P, and model error eg. Based on the samples ofthese variables, the samples of other downstream variables couldbe obtained using the relationships in Eqs. (17)–(19). This samplingof the crack growth process is repeated until sufficient samples ofcrack size after the Nth load cycle for given EIFS are collected. Thena probability distribution is derived from these samples. A signifi-cant advantage of the Monte Carlo simulation approach is the inde-pendence between different samples of the crack growth process,which makes it easy to implement parallel computation for effi-cient output uncertainty quantification.

3.3.2. Step 2: Include size measurement accuracyThis step calculates the statistics of observed crack size after the

Nth load cycle, by combining the result from Step 1 with size mea-surement accuracy.

Let am be the crack size detected, and aN be the predicted cracksize. Then the conditional probability density function of measuredcrack size fam ja;em ðamjaN; emÞ is calculated using the relation specifiedin Eq. (6). Assuming the distribution of residual error em is known,

fam jaN ðamjaNÞ ¼Z

fam jaN ;em ðamjaN ; emÞfem ðemÞdem ð26Þ

Since the statistics of aN have been obtained in Step 1, thus

bution S 3.2)

Bayes Theorem

(Section 3.1)

Updated Distribution

of EIFS

inference of EIFS distribution.


fam ja0 ðamja0Þ ¼Z

fam jaN ðamjaNÞfaN ja0 ðaNja0ÞdaN ð27Þ

3.3.3. Step 3: Construct likelihood based on inspection dataIf the inspection on a component after the Nth load cycle shows

that a crack with size am is detected, then the likelihood function ofEIFS is the same as in Eq. (27), that is,

Lða0Þ ¼ fam ja0 ðamja0Þ ð28Þ

Fig. 6. Surface crack model.

Table 1Material properties of Aluminum 7075-T6.

Aluminum 7075-T6

Modulus of elasticity 71.7 GPaPoisson ratio 0.33Yield stress 691 MPaUltimate stress 764 MPa

Table 2Geometrical properties.

Cylinder properties

Length 0.152 mInside radius 8.76 mmOutside radius (narrow sect) 14.4 mmOutside radius (wide sect) 17.8 mm

Table 3Uncertainty quantification and associated statistics.

Uncertainty classification Uncertainty source Random variable

Physical variability Material properties DKth (MPa m0.5)Drf (MPa)

Loading Block length (CycleMin amplitude (kNMax amplitude (kN

Measurement em

Data uncertainty Prior EIFS kf

Loading amplitude lmax (kN m)rmax (kN m)lmin (kN m)rmin (kN m)

Model uncertainty Crack growth model C (m/cycle)eg

If the inspection on a component after the Nth load cycle showsthat no crack is detected, then the likelihood function of EIFS is

Lða0Þ ¼ 1� Pam ja0 ðam > 0ja0Þ ¼ 1�Z

am>0fam ja0ðamja0Þdam ð29Þ

It is easy to extend the above derivation of likelihood function ofEIFS to the inspection of multiple similar components, as below.

Suppose inspections are conducted on M similar components,and one crack is detected on each component. The crack size incomponent i is recorded as ai

m (i = 1, 2, . . . , M), then

Lða0Þ ¼YM

i

fam ja0 ðaimja0Þ ð30Þ

Suppose there is no crack detected on any of the M components,then

Lða0Þ ¼ 1�Z

am>0fam ja0 ðamja0Þdam

� �M

ð31Þ

If there are X components out of M with no crack detected, andY components with detected crack size ai

m (i = 1, 2, . . . , Y), then

Lða0Þ ¼ 1�Z

am>0fam ja0 ðamja0Þdam

� �X

�YY

i

fam ja0ðaimja0Þ ð32Þ

Note that ‘‘similar” refers to similar material property, struc-tural geometry, and loading environment. The EIFS distributionsof these similar components are assumed to be identical.

3.4. Summary

This section presented the overall procedure of EIFS inference.First, the prior distribution of EIFS is calculated based on theKitagawa–Takahashi diagram and the El Haddad model [4]. Thenthe likelihood function of EIFS is constructed based on uncertaintypropagation through model-based fatigue crack growth analysisand the inspection data. The implementation of uncertainty prop-agation is carried out using conditional probabilities and MonteCarlo simulation. Ultimately, the updated EIFS distribution is ob-tained using the Bayes theorem. A flowchart describing the wholeprocedure is provided in Fig. 5.

4. Numerical example

This section illustrates the application of the proposed statisti-cal inference of EIFS to mechanical components with complicatedgeometry and multi-axial variable amplitude loading, in the pres-ence of various sources of uncertainty and errors. The estimation

Distribution type Mean Standard deviation

Lognormal 5.66 0.268Lognormal 201 20.1

s) Uniform 250 144m) Normal lmax rmax

m) Normal lmin rmin

Normal 0 0.01

Normal �7.60 0.50Normal 0.22 0.10Uniform 1.355 0.263Uniform 0.455 0.130Uniform 2.710 0.260Uniform 0.455 0.130

Lognormal 6.54E�13 4.01E�13Lognormal 1 0.1


of EIFS combines the prior knowledge of EIFS and the informationfrom inspection data.

4.1. Description of problem

A two-radius hollow cylinder with an elliptical surface crack inthe fillet radius region is considered, and is assumed to be sub-jected to bending and torsion simultaneously. The commercial

Fig. 7. Likelihood functio


finite element software ANSYS (version 11.0) is used to calculateKI, KII, and KIII, representing the mode-I, mode-II, and mode-IIIstress intensity factors respectively. Then the equivalent stressintensity factor Keqv is obtained through a characteristic plane ap-proach proposed as described in Section 2.

Note that a sub-model technique is used to facilitate the finiteelement analysis, as shown in Fig. 6. First the entire componentis modeled with a coarse mesh; this is referred to as the full model.

n of EIFS for Case 1.



Then the region surrounding the crack is modeled using a refinedmesh, which is called the sub-model. The boundary conditions ofthe sub-model are obtained from the solution of the full model.This technique facilitates computational efficiency in finite ele-ment analysis.

Tables 1 and 2 list the material and geometrical properties ofthe component under study.

In reality, the parameters in Tables 1 and 2 may be random andmight require probabilistic treatment. However, as mentioned ear-lier, the geometry of the component, Young’s modulus, Poisson’sratio, boundary conditions, surface roughness, etc. are assumedto be deterministic in this paper.

Note that the proposed methodology can account for these vari-ations as well. This can be accomplished by constructing probabil-ity distributions for the aforementioned quantities and includingthem as inputs to the surrogate model used to calculate the stressintensity factor. In future work, these quantities will also be trea-ted as random variables and their effects will be considered inthe inference of EIFS.

The following sub-section discusses the numerical implementa-tion of the uncertainty quantification procedure.

4.2. Uncertainty and error quantification

The numerical details of the different sources of uncertainty anderrors are presented in this sub-section. Table 3 gives the statistics

Table 4Estimated statistics of EIFS for Case 1.

Bayesian inference of EIFS 10 inspections

Mean (mm) Std. (mm)

Prior (w/o data uncertainty): EIFS � logn(�7.60, 0.22); mean = 0.513 mm; Std. = 0.114 mm0.437 0.053

Prior (with data uncertainty): EIFS � logn(k, f); k � norm(�7.60, 0.50); f � norm(0.22, 0.0.422 0.117

Fig. 9. Prior and posterior PDF of EIFS withou

of random variables involved. Note that all these sources of uncer-tainty are equally weighted in the inference of EIFS and there is nosufficient basis to give more importance to any particular type ofuncertainty.

(I) The loading history is generated with random block lengthand amplitude. The block length is assumed to follow a uni-form distribution. The amplitudes of bending moment andtorsion are assumed related, i.e., the value of torsiondepends on the value of bending moment. Thus, the maxi-mum amplitude (Smax) and minimum amplitude (Smin) ofthe block loading could be represented by the bendingmoment, both of which are treated as normal random vari-ables (Smax � N(lmax, rmax), Smin � N(lmin, rmin)). To accountfor data uncertainty, lmax, rmax, lmin, and rmin are alsoassumed to be random variables.

(II) Material properties – fatigue limit Drf and threshold stressintensity factor DKth – are treated as lognormal random vari-ables [4].

(III) The modified Paris law is used for crack growth analysis,combined with Wheeler’s retardation model, as describedin Section 2.1. The parameter C in the crack growth law isassumed to have lognormal distributions, whereas n (=3.9)and p (=0.75) are treated as deterministic quantities. Theestimation of these parameters is explained in [4]. The

20 inspections 40 inspections

Mean (mm) Std. (mm) Mean (mm) Std. (mm)

0.390 0.040 0.406 0.030

10)0.398 0.110 0.414 0.111

t considering data uncertainty (Case 1).


uncertainty in the crack growth model (eg) is treated as alognormal random variable.

(IV) Measurement uncertainty is represented by size measure-ment accuracy discussed in Section 2.3.1. The regressioncoefficients b0 and b1 are assumed to be zero, and the resid-ual random error em is treated as a normal random variablewith zero mean.

Fig. 10. Prior and posterior PDF of EIFS witho

Fig. 11. Prior and posterior PDF of EIFS co

(V) Five hundred and forty runs of FEA are conducted, account-ing for the combinations of 10 different crack sizes, nine dif-ferent loading cases, two angles of orientation and threedifferent aspect ratios. (This does not imply that differentweights are placed on these variables. It means that therange and the variability of some quantities are high andhence, it is necessary to consider more training points for

ut considering data uncertainty (Case 2).

nsidering data uncertainty (Case 1).


those quantities. The crack size and the load values have lar-ger range and variability than the other quantities. Thoughthe crack is elliptical, the semi-major axis and the semi-minor axis crack lengths are alone computed; hence, twoangles of orientation, 0� and 90� are sufficient. Three differ-ent aspect ratios of 1, 1.25 and 1.5 are considered because itis not necessary to consider higher aspect ratios under thegiven crack size, loading, geometry, etc.) Though the greedypoint algorithm is not implemented for this example, adesired level of accuracy (the maximum coefficient of varia-tion in the surrogate model prediction is less than 0.01) isachieved using the aforementioned finite element runs. Foreach FEA solution, three different meshes are consideredand the discretization error is quantified as explained in Sec-tion 2.4.3. The discretization errors are added to the finiteelement solutions and the Gaussian process model is trainedto predict the stress intensity factor.

(VI) The uncertainty due to Gaussian process model is quantifiedusing the expected output and the corresponding variance.

(VII) The prior distribution of EIFS is characterized as discussed inSection 3.2, using the data given in [4]. A lognormal distribu-tion of EIFS with parameters k and f is assumed. To accountfor data uncertainty, k and f are also considered as randomvariables. Their prior distributions can be estimated usinga re-sampling technique. For the purpose of illustration, itis assumed that k and f are both normal random variables.

Table 5Estimated statistics of EIFS given Case 2.

Bayesian inference of EIFS 10 inspections 9 cracks detected 20 ins

Mean (mm) Std. (mm) Mean

Prior (w/o data uncertainty): EIFS � logn(�7.60, 0.22); mean = 0.513 mm; Std. = 0.114 mm0.433 0.059 0.412

Prior (with data uncertainty): EIFS � logn(k, f); k � norm(�7.60, 0.50); f � norm(0.22, 0.0.415 0.117 0.409


4.3. Inference of EIFS

Three cases of inspection results are considered in this problem:

(1) M1 (=10, 20, or 40) similar components are inspected after10,000 load cycles, and one crack is detected in each compo-nent. The sizes of cracks are measured.

(2) M2 (=10, 20, or 40) similar components are inspected after1000 load cycles. Cracks are detected in some components,while no crack is detected in the other components. Thesizes of detected cracks are measured.

(3) M3 (=1, 2, or 5) similar components are inspected after 1000load cycles, and no crack is detected in any of thesecomponents.

4.3.1. Inference of EIFS for Cases 1 and 2For the purpose of illustration, the measured crack sizes in

Cases 1 and 2 are simulated using an assumed ‘‘true” EIFS distribu-tion and crack growth according to the modified Paris law dis-cussed in Eq. (29). Suppose the ‘‘true” EIFS follows a normaldistribution with l = 0.381 and r = 0.0381. In order to investigatethe influence from the number of data points to the EIFS inferenceresults, three different sets of data are simulated – 10 ‘‘measure-ments”, 20 ‘‘measurements”, and 40 ‘‘measurements”. These mea-surements are assumed as the inspected crack sizes after 10,000loading cycles (Case 1) or 1000 loading cycles (Case 2).

pections 18 cracks detected 40 inspections 36 cracks detected

(mm) Std. (mm) Mean (mm) Std. (mm)

0.046 0.388 0.035

10)0.113 0.400 0.110



Given these data sets, the likelihood function of EIFS could beobtained following the procedure discussed in Section 3.3.

The likelihood function of EIFS is the probability of observingthe given data conditioned on a given EIFS value. The number ofdata points affects the uncertainty, and this effect is reflected inthe scatter of the likelihood function. As shown in Figs. 7 and 8,the likelihood function based on 40 inspections has the least scat-ter, i.e., the variance is the smallest. On the contrary, 10 inspectionslead to the largest scatter in the likelihood function.

With the calculated likelihood function, the prior distribution ofEIFS without considering data uncertainty is updated using Eq.(10); if data uncertainty is considered, the updated EIFS distribu-tion is obtained by Eqs. (13)–(15). The inference results are shownin Table 4, Figs. 9 and 11 (Case 1), and Table 5, Figs. 10 and 12 (Case2).

From Tables 4 and 5, it is seen that the inference results tend tobe in agreement with the ‘‘true” EIFS distribution. Also note thatconsidering data uncertainty adds more variance to the estimatedEIFS distributions.

Figs. 9 and 10 each show four probability distribution functioncurves, namely the prior distribution and the three updated distri-butions. The updated distributions are calculated by normalizingthe product of the prior distributions and the likelihood functions.Therefore, the shapes of updated PDF curves should depend on the


Table 6Estimated statistics of EIFS for Case 3.

Bayesian inference of EIFS 1 inspection

Mean (mm) Std. (mm)

Prior: EIFS � logn(�7.60, 0.22); Mean = 0.513 mm; Std. = 0.114 mm0.425 0.082

Prior: EIFS � logn(k, f); k � norm (�7.60, 0.50); f � norm (0.22, 0.10)0.388 0.117

curves of corresponding prior PDF and likelihood functions. This isreflected in Figs. 9 and 10.

In Figs. 11 and 12, where data uncertainty is included, the priorknowledge has more influence on the updated PDF functions, com-pared with Figs. 9 and 10 (where data uncertainty in the distribu-tion of EIFS is not considered). This is because the prior andposterior EIFS share the same type of probability distribution (log-normal distribution). The data uncertainty in this example is repre-sented by randomness in the parameters of EIFS distribution. Thetype of EIFS probability distribution is assumed as lognormal (forboth prior and posterior). Instead of updating the distribution ofEIFS directly, the statistics of EIFS distribution parameters are up-dated using Eqs. (13)–(15). Therefore the posterior PDF curvesshow more similarity to the prior PDF curve.

4.3.2. Inference of EIFS for Case 3The likelihood function of EIFS for Case 3 could also be calcu-

lated following the steps in Section 3.3, the results are shown inFig. 13.

The likelihood function curves in Fig. 13 demonstrate that asmaller initial crack size has a higher likelihood than a larger cracksize. This is easy to understand since crack with larger size will bemore likely to be detected than small crack. If the crack is not de-tected, it implies a smaller initial crack size. If there are more com-


2 inspections 5 inspections

Mean (mm) Std. (mm) Mean (mm) Std. (mm)

0.379 0.068 0.304 0.050

0.366 0.108 0.344 0.102


ponents where cracks are not detected, there is more evidence for asmaller crack size. Hence, the likelihood function shifts to thesmaller crack side (left) when cracks are not detected in severalcomponents.

Fig. 14. Prior and posterior PDF of EIFS witho


Once the likelihood function has been calculated, the prior dis-tribution of EIFS is updated using Eq. (13) (excludes data uncer-tainty), or using Eqs. (13)–(15) (includes data uncertainty). Theinference results are showed in Table 6, Figs. 14 and 15.

ut considering data uncertainty (Case 3).



Fig. 14 shows a trend similar to that in Fig. 13. More inspectionresults reduce the variance in the estimated EIFS distribution. Inboth Figs. 14 and 15, the probability density function of the EIFStends to move leftward as more components are detected withoutcrack. This is in agreement with the likelihood function of EIFS.

5. Conclusion

A Bayesian approach for statistical inference of the EIFS distri-bution is developed in this paper for components with complicatedgeometry and multi-axial, variable amplitude loading. Varioussources of variability, uncertainties and errors are included in theinference, including natural variability in loading and materialproperties, model uncertainty and errors in structural stress anal-ysis and crack growth model, and data uncertainty in experimentsand inspection. A methodology based on conditional probabilityand Monte Carlo simulation is developed for the uncertainty prop-agation in fatigue crack growth and used to calculate the likelihoodfunction of EIFS, which is essential to combine prior knowledgewith available experimental data in the Bayes theorem. For thepurpose of illustration, the proposed methodology is used to esti-mate the distribution of EIFS for a two-radius hollow cylinder,which contains an elliptical surface crack in the fillet radius regionand is subjected to bending and torsion simultaneously.

The inference results of EIFS distribution are in good agreementwith the ‘‘true” EIFS distribution as shown in Section 4.3.1. Withmore inspection data collected, the estimation of EIFS tends tohave less variance, i.e., higher confidence in the inference resultsis obtained with more inspection data. It is convenient to imple-ment the proposed methodology of uncertainty propagation byparallel computing, which could significantly reduce the computa-tional cost.

The focus of this paper has been to develop a framework for thecalibration of the EIFS distribution by explicitly including differentsources of variability, uncertainty, and error. Future work couldconsider several directions for improvement: (1) extend the pro-posed calibration approach to include other parameters of interestin crack growth analysis, (2) incorporate more sophisticated errorquantification procedures in this framework, and (3) include addi-tional sources of variability, uncertainty, and error, such as vari-ability in the coefficient of friction between crack surface,variability in surface roughness, randomness in material and geo-metric properties, model uncertainty in the treatment of mixed-mode cracking, and uncertainty introduced by using samplingmethods.

Acknowledgements

The research reported in this paper was supported in partby the NASA ARMD/AvSP IVHM project under NRA AwardNNX09AY54A (Project Monitor: Dr. K. Goebel, NASA AMESResearch Center) through subcontract to Clarkson University (No.375-32531, Principal Investigator: Dr. Y. Liu), and by the FederalAviation Administration William J. Hughes Technical Centerthrough RCDT Project No. DTFACT-06-R-BAAVAN1 (Project Moni-tors: Dr. John Bakuckas, Ms. Traci Stadtmueller). The support isgratefully acknowledged.

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Inference of equivalent initial flaw size under multiple sources of uncertainty