Uncertainty quantification and model validation of fatigue crack growth prediction

Engineering Fracture Mechanics 78 (2011) 1487–1504

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Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

Uncertainty quantification and model validation of fatigue crackgrowth prediction

Shankar Sankararaman, You Ling, Sankaran Mahadevan ⇑Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, United States

a r t i c l e i n f o

Article history:Received 27 September 2010Received in revised form 14 February 2011Accepted 21 February 2011Available online 1 March 2011

Keywords:Uncertainty quantificationModel validationBayes factorCrack growthModel uncertainty

0013-7944/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.engfracmech.2011.02.017

⇑ Corresponding author. Tel.: +1 615 322 3040.E-mail address: sankaran.mahadevan@vanderbil

a b s t r a c t

This paper presents a methodology for uncertainty quantification and model validation infatigue crack growth analysis. Several models – finite element model, crack growth model,surrogate model, etc. – are connected through a Bayes network that aids in model calibra-tion, uncertainty quantification, and model validation. Three types of uncertainty areincluded in both uncertainty quantification and model validation: (1) natural variabilityin loading and material properties; (2) data uncertainty due to measurement errors, sparsedata, and different inspection results (crack not detected, crack detected but size not mea-sured, and crack detected with size measurement); and (3) modeling uncertainty anderrors during crack growth analysis, numerical approximations, and finite element discret-ization. Global sensitivity analysis is used to quantify the contribution of each source ofuncertainty to the overall prediction uncertainty and to identify the important parametersthat need to be calibrated. Bayesian hypothesis testing is used for model validation and theBayes factor metric is used to quantify the confidence in the model prediction. The pro-posed methodology is illustrated using a numerical example of surface cracking in a cylin-drical component.

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1. Motivation

The process of fatigue crack growth is affected by many sources of variability, such as loading, material properties, geom-etry and boundary conditions. Therefore it is appropriate to describe the crack size after a certain number of load cyclesthrough a probability distribution. Probabilistic fracture mechanics is an extensive area of research and numerous studieshave addressed both model-based [1–22] and data driven [23–26] techniques for probabilistic crack growth and life predic-tion. This paper focuses on model-based methods which include both probabilistic techniques [1–3] and advanced compu-tational mechanics techniques [4,5].

Probabilistic crack growth analysis has been applied to both metals (e.g. Johnson and Cook [6], Maymon [7]) and compos-ites [8–12]). Practical applications of these methods include nuclear structural components [13], helicopter gears [14], gasturbine engines [15], and aircraft components [16–17]. These developments have led to software for probabilistic fracturemechanics [18], and several commercial software tools such as DARWIN [19], GENOA [20], and other software tools thatbuild probabilistic analysis around well-established codes such as AFGROW [21] and FASTRAN [22].

The aforementioned studies on probabilistic crack growth prediction have primarily included only the effects of naturalvariability in loading conditions, material and geometric properties, and crack growth model parameters. However, many ofthem have not explicitly accounted for other sources of uncertainty such as data uncertainty and other types of modeluncertainty and error. The different types of error and uncertainty appear at different stages of analysis, and may combine

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Nomenclature

a flaw (crack) sizeN number of cyclesaN flaw size after N cyclesur retardation parameterC, n, and m model parameters of modified Paris lawDK stress intensity factorDKth threshold stress intensity factorDKeqv equivalent stress intensity factoram measured crack sizeem measurement erroreg crack growth law model erroreh discretization errorX calibration (inference) quantitiesL(X) likelihood function of calibration quantitiesDrf fatigue limitPOD probability of detectionB Bayes factor

1488 S. Sankararaman et al. / Engineering Fracture Mechanics 78 (2011) 1487–1504

in nonlinear, nested, or iterative manner. Further, some of the errors are deterministic (e.g., finite element discretization er-ror) while some others are stochastic (e.g., material properties). It is essential to systematically account for all these sourcesof error and uncertainty in order to ensure robustness in model calibration, model validation, and uncertainty quantification.This paper includes the different sources of variability, data uncertainty, model uncertainty and errors that affect fatiguecrack growth analysis, and develops a methodology to quantify the uncertainty and assess the validity in the overall crackgrowth model prediction (or prognosis).

Saxena et al. [27] discussed several metrics to assess the performance of prognosis algorithms. These metrics are based on(1) observed error, (2) standard deviation of the observed quantity, (3) sensitivity, (4) reliability, and (5) cost-benefit anal-ysis. These metrics do not directly quantify the confidence in the model prediction and do not delineate the contributions ofthe various sources of uncertainty. A rigorous approach to model validation needs to explicitly account for the differentsources of uncertainty and quantitatively judge the performance of the model [28].

There are several existing approaches for model validation [29–37] that not only include different sources of uncertaintyand but also quantify the extent of agreement between the model prediction and experimental observation. Both classicaland Bayesian statistics have been pursued for this purpose. While the classical statistics-based methods [28–32] are basedon computing the statistics of the error (difference between model prediction and observation), Bayesian statistics-basedmethods [33–37] quantify the extent to which the model supports the data and directly computes the probability of themodel being correct.

Most of the aforementioned approaches for model validation have primarily dealt with physical variability and includedmeasurement errors and some model errors. However, they have several limitations: (1) only a single computational modelwas considered for validation whereas fatigue crack prediction requires a combination of multiple models for prediction; (2)time-dependent models and the fact that validation data may be available at different points in time were not considered;and (3) it was not clear how to account for different sources of error and uncertainty in multiple models and assess the con-fidence in the overall model prediction.

The current paper extends the Bayesian hypothesis testing methodology [35] to overcome these limitations. One impor-tant advantage in the proposed methodology is that the various quantities of interest can be connected through a Bayes net-work, which allows for the addition of more models and facilitates systematic inclusion of different sources of error anduncertainty. Further, for use in time-dependent problems, a dynamic Bayesian network can be constructed and this canbe useful in two ways: (1) in an inverse problem [38] where the uncertainty in the model parameters is quantified usingexperimental evidence; and (2) in a forward problem [39] that aids in probabilistic prediction.

In the inverse problem of calibration, previous techniques [38] have been used to calibrate either a single parameter[40,41] or the parameters of a single model [42]. Further, these studies [40–42] have not explicitly included the effect of dif-ferent types of uncertainty – physical variability, data uncertainty, and model uncertainty and errors – on calibration. As aresult, all the difference between the model prediction and experimental observation gets attributed to the calibrationparameters without acknowledging or accounting for the other sources of error and uncertainty. The authors have previouslyproposed a systematic procedure for the treatment of these different types of uncertainty in calibration [43]. However, onlyone parameter, namely the equivalent initial flaw size, was calibrated; calibration of multiple parameters and error terms inmultiple models was not considered.

The forward problem involves the use of the calibrated model for probabilistic crack growth prediction. Probabilisticdamage prediction aids in effective prognosis under uncertainty, and helps to forecast the remaining useful life of the struc-tural/mechanical system under study. All three types of uncertainty need to be included in this prediction.

S. Sankararaman et al. / Engineering Fracture Mechanics 78 (2011) 1487–1504 1489

In this paper, several different models are combined in crack growth analysis (finite element analysis (FEA), surrogatemodel analysis, crack growth model) and the effect of the three different types of uncertainty (physical variability, datauncertainty, and model uncertainty and errors) are explicitly included in both the inverse problem of model calibrationand the forward problem of probabilistic crack growth prediction.

Different types of data situations are considered – (1) data available for constructing the probability distributions of inputvariables may be sparse, leading to uncertainty regarding their distribution types and parameters; (2) inspection data usedfor calibration and/or validation may be of three different types: crack not detected, crack detected but size not measured, orcrack detected with size measurement; and (3) measurement errors in inspection data.

The steps of the proposed methodology can be summarized as follows: (1) connect different models – finite element mod-el, surrogate model, crack growth law, etc. efficiently through a Bayes network; (2) quantify the errors in each model, treat-ing deterministic and stochastic errors differently; (3) include the different sources of uncertainty – physical variability, datauncertainty, and model uncertainty in the Bayes network; (4) use global sensitivity analysis to select calibration parameters;(5) collect inspection data and update the parameters in multiple models; (6) include different cases of inspection such ascrack not detected, crack detected but size not measured, and crack detected with size measurement; and (7) use a Bayesianmetric to assess the validity of the model prediction. (Note that the data for model calibration and model validation shouldbe different and independent.)

The above steps are implemented in this paper for planar crack growth analysis, in the context of linear-elastic fracturemechanics (LEFM) with small-scale plasticity. Different types of model uncertainty and errors – crack growth model uncer-tainty, surrogate model uncertainty and finite element discretization error – are considered explicitly. Deterministic errorsare addressed by correcting where they arise and stochastic errors are included through sampling. The effects of modelassumptions such as planar crack, LEFM, equivalent stress intensity factor for multi-modal fracture, and crack retardationmodel are not individually quantified. However, in the model validation step, the difference between the model predictionand the experimental observation obviously includes the contribution of these sources.

The rest of the paper is organized as follows: Section 2 discusses the crack growth modeling procedure used in this paper.Section 3 discusses several sources of uncertainty and proposes methods to handle them. Section 4 discusses global sensi-tivity analysis to screen the model parameters for calibration, and outlines the proposed Bayesian inference technique forcalibrating these parameters. Section 5 extends Bayesian hypothesis testing to time-dependent problems, in order to assessthe validity and the confidence in the crack growth model prediction. Section 6 illustrates the proposed methods using anumerical example, surface cracking in a cylindrical component.

2. Crack growth modeling

Different models such as the modified Paris’ law (for long crack growth analysis), Wheeler’s retardation model (for var-iable amplitude loading), characteristic plane approach (to calculate an equivalent stress intensity factor in the presence ofmulti-axial loading) are connected together to predict the crack growth as a function of number of load cycles. Note thatthese models are used only for the purpose of illustration and any other combination of appropriate models may be used.The focus of the paper is on uncertainty quantification and model validation and these techniques are applicable to differentcrack growth analysis procedures.

Consider any long crack growth law used to describe the relationship between da/dN and DK, where N represents thenumber of cycles, a represents the crack size and DK represents the stress intensity factor range. This paper uses a modifiedParis’ law [44] for illustration purposes and includes Wheeler’s retardation model [45] as:

dadN¼ urCðDKÞn 1� DKth

DK

� �m

ð1Þ

In Eq. (1), DKth refers to the threshold stress intensity factor range and ur refers to the retardation parameter which can bedefined as:

ur ¼rp;i

aOLþrp;OL�ai

h ikif ai þ rp;i < aOL þ rp;OL

1 if ai þ rp;i P aOL þ rp;OL

8<: ð2Þ

In Eq. (2), aOL is the crack length at which the overload is applied, ai is the current crack length, rp,OL is the size of the plasticzone produced by the overload at aOL, rp,i is the size of the plastic zone produced at the current crack length ai, and k is thecurve fitting parameter for the original Wheeler model termed the shape exponent [46]. Sheu et al. [47] and Song et al. [48]observed that crack growth retardation actually takes place within an effective plastic zone. Hence the size of the plasticzone can be calculated in terms of the applied stress intensity factor (K) and yield strength (r) as:

rp ¼ aKr

� �2

ð3Þ

where a is known as the effective plastic zone size constant which is obtained experimentally [47]. Eqs. (2) and (3) can beused in combination with Eq. (1) under the assumption of small-scale plasticity, where the plastic zone size is estimatedusing linear-elastic fracture mechanics (LEFM).


The expressions in Eqs. (2) and (3) can be combined with Eq. (1) and used to calculate the crack growth as a function ofnumber of cycles. In each cycle, the stress intensity factor can be expressed as a function of the crack size (a), loading (L) andangle of orientation (v). Hence, the crack growth law in Eq. (1) can be rewritten as

dadN¼ gða; L;vÞ ð4Þ

Note that the long crack growth model is not applicable to the short crack growth regime and the concept of an equivalentinitial flaw size (EIFS) was proposed to bypass short crack growth analysis and make direct use of a long crack growth law forfatigue life prediction. The equivalent initial flaw size h, i.e., the initial condition of the differential equation in Eq. (1), can becalculated from material properties (DKth, the threshold stress intensity factor and Drf, the fatigue limit) and geometricshape factor (Y) as derived by Liu and Mahadevan [49]:

h ¼ 1p

DKth

YDrf

� �2

ð5Þ

By integrating the expression in Eq. (1) starting from h, the number of cycles (N) to reach a particular crack size aN can becalculated as:

N ¼Z

dN ¼Z aN

h

1

urCðDKÞn 1� DKthDK

� �m da ð6Þ

The stress intensity factor range DK in Eq. (6) can be expressed as a closed form function of the crack size for specimens withsimple geometry subjected to constant amplitude loading. However, this is not the case in many mechanical components,where DK depends on the loading conditions, geometry and the crack size. Further, if the loading is multi-axial (for example,simultaneous tension, torsion and bending), then the stress intensity factors corresponding to three modes need to be takeninto account. This can be accomplished using an equivalent stress intensity factor. If KI, KII, KIII represent the mode-I, mode-IIand mode-III stress intensity factors respectively, then the equivalent stress intensity factor Keqv can be calculated using acharacteristic plane approach proposed by Liu and Mahadevan [50] as:

Keqv ¼1B

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1Þ2 þ

k2

s

� �2

þ k3

s

� �2

þ AkH

s

� �2s

ð7Þ

In Eq. (7), k1, k2, k3 are the parameters associated with mode-I, II, and III loading, respectively. kH is related to hydrostaticstress. s is the ratio of Modes II and I stress intensity factor at a specific crack growth rate (da/dN). A and B are materialparameters. The characteristic plane approach is applicable only when the crack surface can be approximated to be planar.The use of the characteristic plane approach for crack growth prediction under multi-axial variable amplitude loading hasbeen applied to cracks in railroad wheels [50] and validated earlier with several data sets [51,52].

Each cycle in the integration of Eq. (6) involves the computation of DKeqv using a finite element analysis owing to (1) com-plicated geometry, and (2) variable amplitude, multi-axial loading. Repeated evaluation of this finite element analysis ren-ders this integration extremely expensive. Hence, it is computationally more affordable to substitute the finite elementmodel with an inexpensive surrogate model (also known as response surface model). Different kinds of surrogate models(polynomial chaos [53], support vector regression [54], relevance vector regression [55], and Gaussian Process interpolation[56]) have been explored and the Gaussian process (GP) modeling technique has been found to have the least error for thetype of problem considered in this paper [41]. A few runs of the finite element analysis are used to train the GP surrogatemodel and then, this GP model is used to predict the stress intensity factor for other crack sizes and loading cases (for whichfinite element analysis has not been carried out).

Suppose that there are m training points, x1, x2, x3, . . . , xm of a d-dimensional input variable vector (the input variablesbeing the crack size and loading conditions in this case), yielding the resultant observed random vector Y(x1), Y(x2),Y(x3), . . . , Y(xm) (the equivalent stress intensity factor, in this case). R is the m �m matrix of correlations among input vari-ables at the training points. Under the assumption that the parameters governing both the trend function (fT(xi), at eachtraining point) and the covariance (k) are known, the expected value and the variance of the Gaussian process at an untestedlocation x⁄ is calculated as in Eqs. (8) and (9) respectively.

Y� ¼ E½Yðx�ÞjY � ¼ f Tðx�Þbþ rTðx�ÞR�1ðY � FbÞ ð8Þr2

Y� ¼ Var½Yðx�ÞjY� ¼ kð1� rT R�1rÞ ð9Þ

In Eqs. (8) and (9), F is a matrix containing the rows of trend functions fT(xi) (i = 1 to m), r is the vector of correlations betweenx⁄ and each of the training points, b represents the coefficients of the regression trend. This paper uses a constant trend func-tion (fT(xi)) and an exponential covariance function (k) that decays as a function of the squared distance between two points.See McFarland [57] for details of the Gaussian process modeling technique. Sankararaman et al. [41,43] describe in detailhow the Gaussian process surrogate model can be constructed and used to predict the equivalent stress intensity factorin cycle-by-cycle integration of the crack growth law, thereby calculating the crack size (A) as a function of number of loadcycles (N). The entire procedure for the adopted crack growth analysis is summarized in Fig. 1.

Fig. 1. Deterministic crack propagation analysis.


In Fig. 1, note that the finite element analysis and the construction of surrogate model are performed offline, i.e. before thestart of crack growth analysis. Crack propagation analysis is done only with the surrogate model.

The algorithm shown in Fig. 1 for crack propagation analysis is deterministic and does not account for errors and sourcesof uncertainty. The next section discusses different sources of uncertainty associated with each of the blocks in Fig. 1.

3. Sources of uncertainty

This section proposes methods to include the different sources of uncertainty in the crack growth analysis algorithmshown in Fig. 1. These sources of uncertainty can be classified into three different types – physical variability, data uncer-tainty and model uncertainty – as shown in Fig. 2.

Fig. 2 shows the different sources of error and uncertainty considered in this paper for the sake of illustration of the pro-posed methodology. There are several other sources of uncertainty that are not considered here. The physical variability ingeometry, boundary conditions and some material properties (modulus of elasticity, friction coefficient, Poisson ratio, yieldstress, ultimate stress, etc.) is assumed to less significant and hence these quantities are treated to be deterministic. How-ever, these can be included in the proposed framework by constructing different FEA models (for different geometry, bound-ary conditions, and material properties) and use multiple runs of these FEA models to train the Gaussian process surrogatemodel. Then, these parameters can also be treated as inputs to the surrogate model and sampled randomly in the uncertaintyquantification procedure explained later in Sections 4 and 5. Further, there are several other types of model errors that arisedue to (1) the assumption of a planar crack; (2) the assumption of linear elastic behavior and use of LEFM; (3) the use of acharacteristic plane approach (to calculate an equivalent stress intensity factor) and parameters of the characteristic planemodel in Eq. (7); (4) errors in the estimation of retardation coefficient ur in Wheeler’s model, etc. These errors are not con-sidered in this paper and the quantification of these errors is not trivial; these errors will be considered in future work. Thefocus of this paper is not the quantification of all the error sources but to develop a methodology to assess the validity of thecrack growth prediction by systematically accounting for the various sources of uncertainty and errors. The error and uncer-tainty terms considered in this paper adequately illustrate the various techniques for model calibration, validation anduncertainty quantification.

The following subsections discuss each source of uncertainty considered in this paper and propose methods to handlethem.

3.1. Physical or natural variability

Physical variability refers to the natural variability or fluctuations in the environment, test procedure, instruments, ob-server, etc. Hence, repeated observations of the same physical quantity do not yield identical results. This paper considersthe physical variability in (i) loading conditions, and (ii) material properties. As mentioned earlier, the variability in othermaterial properties such as modulus of elasticity, Poisson ratio, etc. is not considered.

3.2. Data uncertainty

As mentioned earlier in Section 1, field inspections are used to collect crack growth data and this information is used tocalibrate the distribution of EIFS. The different types of uncertainty in this procedure – crack detection uncertainty and

Fig. 2. Sources of uncertainty in crack growth prediction.


measurement errors – are discussed in Section 3.2.1. Also, the probability distributions of some material properties are in-ferred using data from laboratory experiments. These data may be sparse and cause uncertainty regarding the probabilitydistribution type and parameters. This type of uncertainty is discussed in Section 3.2.2.

3.2.1. Crack detection uncertainty and output measurement errorsFor an in-service component, non-destructive inspection (NDI) is commonly used for damage detection. Several metrics

could be used to evaluate the performance of NDI, such as probability of detection (POD), flaw size measurement accuracy,and false call probability (FCP). These criteria are developed from different methods, and they are used to evaluate differentaspects of NDI performance. However, Zhang and Mahadevan [58] showed that these quantities can be mathematically re-lated. POD and FCP can be derived from size measurement accuracy, which measures the difference between actual valuesand observed values of the crack size.

In the context of calibration, inspection results may be of three different types: (a) crack is not detected; (b) crack is de-tected but size not measured; and (c) crack is detected and size is measured. The concept of POD can be employed for thefirst two cases and for the case of detecting a crack and also measuring its size, size measurement accuracy can be employed.Based on the above consideration, size measurement accuracy can be used to quantify the uncertainty in experimental crackgrowth data, with the following expression determined by regression analysis [58]:

am ¼ b0 þ A� b1 þ em ð10Þ

In Eq. (10), am is the measured flaw size; A is the actual flaw size; b0 and b1 are the regression coefficients; em represents theunbiased measurement error, assumed as a normal random variable with zero mean and standard deviation re. The value ofre is different for each inspection technique used.

3.2.2. Sparse data for characterizing input random variables (DKth and Drf)This section proposes a general methodology to characterize uncertainty in the input data, from which statistical distri-

butions need to be inferred. In the numerical example in Section 6, this method is implemented using experimental dataavailable in the literature to characterize the distribution of threshold stress intensity factor (DKth) and fatigue limit (Drf).

Consider a random variable X whose statistics are to be determined from experimental data, given by x = {x1, x2, . . . , xn}.The likelihood of the parameters P can be calculated from first principles as [59]:

LðPÞ /Yn

i¼1

fXðxijðPÞÞ ð11Þ


where fX(x|P) is the probability density function of the random variable X conditioned on the parameters P. For example, sup-pose that the random variable X follows a normal distribution, then the parameters (P) of this distribution, i.e. mean andvariance of X can be estimated from the entire data set x.

Typically, the maximum likelihood estimates (by maximizing of Eq. (11)) of the parameters (P) are alone considered. Inthe aforementioned example of a normal distribution, the maximum likelihood estimates of the mean and the standard devi-ation are calculated. However, due to sparseness of data, these estimates of these parameters (P) may not be accurate. In thispaper, the likelihood function is used to construct the entire PDF of the distribution parameters P as in Eq. (12). In fact, Bar-nard et al. [60] emphasize that the entire likelihood function needs to be used for inference and not its maximizer alone. LetfP(p) denote the joint probability density of the parameters P; by choosing a uniform prior density f 0P(p) = h, Bayes’ theoremreduces to:

fPðpÞ ¼LðPÞhRhLðPÞdP

¼ LðPÞRLðPÞdP

ð12Þ

Note that the uniform prior density function can be defined over the entire admissible range of the parameters P. Forexample, the mean of a normal distribution can vary in (�1,1) while the standard deviation can vary in (0,1). The prob-ability distribution of the parameters (P) can be calculated by solving Eq. (12), either by using Markov Chain Monte Carlo-based methods [57] or by using advanced numerical integration methods [41].

For each instance of a set of parameters (P), X is defined by a realization of the chosen distribution type whose densityfunction is given by fX(x|P). However, because the parameters (P) themselves are stochastic and have a joint probability den-sity function fP(P), X is defined by a family of distributions. In the aforementioned example of the normal distribution, forevery given value of the mean and the standard deviation, there exists a unique normal probability distribution. Hence,by varying the mean and the standard deviation, the random variable X can be represented using a family of normaldistributions.

The unconditional probability density function of X (fX(x)) can be calculated by multiplying the conditional probabilitydensity function fX(x|P) of the variable X, and the joint probability density function fP(P), of the parameters P as [61]:

fXðxÞ ¼Z

fXðxjPÞfPðPÞdP ð13Þ

The integral in Eq. (13) can be evaluated through quadrature techniques or through Monte Carlo sampling. This can beachieved by selecting a sample of P, and then selecting one sample of X corresponding to the previously sampled P, andrepeating the entire procedure to obtain multiple samples of X from which the unconditional probability density functionof X (fX(x)) can be calculated. This unconditional density function of X accounts for the sparse data as it includes the uncer-tainty in the estimate of the distribution parameters. In this paper, this method has been used to characterize the uncertaintyin the threshold stress intensity factor (DKth) and fatigue limit (Drf).

3.3. Model uncertainty and errors

This paper uses several models (finite element model, characteristic plane model, surrogate model, crack growth model,retardation model, etc.), and each of these models has its own error/uncertainty. In the discussion below, all the errors exceptthe error due to the use of the characteristic plane approach and the uncertainty in the calculation of the retardation coef-ficient are considered. Some of them are deterministic while others are stochastic; the two types of errors need to be treatedin different ways.

3.3.1. Uncertainty in crack growth modelMore than 20 different crack growth laws (e.g., Paris law, Foreman’s equation, Weertman’s equation) have been proposed

in the literature. The mere presence of many such different models explains that none of these models can be applied uni-versally to all fatigue crack growth problems. Each of these models has its own limitations and uncertainty. In this paper, amodified Paris law has been used only for illustration; however, the proposed methodology for uncertainty quantificationand model validation can be implemented using any crack growth model.

The uncertainty in crack growth model can be subdivided into two different types: crack growth model error and uncer-tainty in model coefficients. If ecg is used to denote the crack growth model fitting error, then the modified Paris law (includingthe Wheeler retardation term) can be rewritten as in Eq. (14). Note that this error term is assumed to represent the differencebetween the model prediction and the experimental observation. Conventionally, in statistical model fitting, a normal errorterm is added to the model. This paper uses a lognormal and hence multiplicative error term as shown in Eq. (14).

dadN¼ urCðDKÞn 1� DKth

DK

� �m

ecg ð14Þ

An estimate of the variance of ecg can be obtained while calibrating the model parameters based on the available inspectiondata. The model coefficients in Paris law are C, m and n, and the uncertainty in these parameters can be represented throughprobability distributions.


3.3.2. Discretization error in finite element analysisTheoretically, an infinitesimally small mesh size will lead to the exact solution but this is difficult to implement in prac-

tice. Hence, finite element analyses are carried out at a particular mesh size and the error in the solution, caused due to dis-cretization needs to be quantified. Several methods are available in literature [62–64] but many of them quantify somesurrogate measure of error to facilitate adaptive mesh refinement. The Richardson extrapolation (RE) method has been foundto come closest to quantifying the actual discretization error [34,65]. (However, note that the use of Richardson extrapola-tion requires the model solution to be convergent and the domain to be discretized uniformly (uniform meshing) [35]. Some-times, in the case of models with large mesh size, the assumption of monotone truncation error convergence may not bevalid.)

Let h denote the mesh size used in the finite element analysis and W the corresponding finite element prediction. Let WT

denote the ‘‘true’’ solution of the finite element analysis which is obtained as h tends to zero. According to the generalizedRichardson extrapolation [65], the relation between h and W can be expressed as:

WT ¼ Wþ Ahp ð15Þ

In Eq. (15), p is the order of convergence, and A is the polynomial coefficient. In order to estimate the true solution WT, threedifferent mesh sizes (h1 < h2 < h3) are considered and the corresponding finite element solutions (W1, W2, W3) are calculated.Eq. (15) has three unknowns p, A, and WT, which can be estimated based on the three mesh solutions. Mesh doubling/halvingis commonly done to simplify the equations. If r = h3/h2 = h2/h1, then the discretization error (eh) and the true solution can becalculated as:

WT ¼ W1 � eh; eh ¼W2 �W1

rp � 1; p ¼

log W3�W1W2�W1

� �logðrÞ ð16Þ

The solutions W1, W2, W3 are dependent on the inputs (loading, current crack size, aspect ratio and angle of orientation) tothe finite element analysis and hence the error estimates are also functions of these input variables. For each set of inputs, acorresponding discretization error (eh) is calculated and this error is subtracted from the finite element solution (using thefinest mesh h1) to estimate the ‘‘true’’ solution WT. The corrected solutions and the corresponding sets of inputs are used astraining points for the surrogate model.

3.3.3. Uncertainty in the surrogate model outputSeveral finite element runs for some combination of input–output variable values are used to train the Gaussian process

surrogate model in this paper. Then, this surrogate model is used to evaluate the stress intensity factor for other combina-tions of input variable values. The residuals of the GP model form a Gaussian random field [57]. In each cycle, the error in thepredicted DKeqv is sampled using this random normal field and added to the GP model prediction.

(Note: The GP model is used as a surrogate for a deterministic finite element model, and the variance of the residual fieldaccounts only for the uncertainty in approximating the original finite element model output with a surrogate model).

The stress intensity factor calculated by the surrogate model (mean and variance of the prediction in Eqs. (8) and (9)respectively) is used in the crack growth equation to predict the crack size as a function of number of cycles as explainedearlier in Section 2. The following section incorporates all these sources of uncertainty into the calibration methodology,and Section 5 assesses the validity of the model by comparing the model prediction with validation data.

4. Uncertainty quantification in model parameters

This section explains the Bayesian calibration technique used to infer the probability distributions of the model param-eters and modeling errors using experimental data. The calibration parameters are assigned prior distributions, and thesedistributions are updated after collecting experimental evidence corresponding to the model output (crack inspection aftera particular number of loading cycles).

4.1. Bayes network

There are several possible quantities that can be updated using the given procedure. These include: (1) equivalent initialflaw size; (2) parameters of modified Paris’s law (C, n, m); (3) error (ecg) of the modified Paris law; (4) material propertiessuch as threshold stress intensity factor (DKth), and fatigue limit (Drf); (5) variance of the output (crack size after N cycles)measurement error (em), etc. All of these parameters can be connected in a Bayes network as shown in Fig. 3.

Note that this is a dynamic Bayes network, which connects the variables in one load cycle (i) to the next load cycle (i + 1).Though it is theoretically possible to calibrate all the parameters simultaneously, a multi-dimensional calibration proce-

dure is computationally intensive. Also, in several situations, only a few parameters contribute effectively to the overalluncertainty in the final model prediction. It is advantageous to identify only such ‘‘important’’ parameters and calibratethem. This paper uses global sensitivity analysis to quantify the contribution of the various sources of uncertainty to theoverall uncertainty in the crack growth prediction, and uses the results of global sensitivity analysis to identify the param-eters that need to be calibrated.

Fig. 3. Bayes network.


4.2. Global sensitivity analysis

Sankararaman et al. [39] used a reduction of variance-based approach to identify the important contributors of uncer-tainty in crack growth analysis. However, this involves the ‘‘freezing’’ of one source of uncertainty and this approach is localwith respect to where the uncertain variable is fixed. Saltelli et al. [66] argue that local sensitivities are not sufficient to studythe contributions of multiple sources of uncertainty to the overall prediction uncertainty and it is necessary to purse a globalsensitivity analysis approach for this purpose. Note that this approach is based only on second-moment calculations, andcalculates the effect of the variance of an input quantity on the variance of the output quantity. Consider a model given by:

Y ¼ GðX1;X2;X3 . . . XnÞ ð17Þ

Note that the X’s and the Y are input–output pairs for a generic model. In the context of this paper, the output is the crack sizeat the end of a particular number of cycles, and the inputs include all quantities that affect that crack size prediction (equiv-alent initial flaw size, loading, crack growth parameters, model errors, etc.).

The first-order effect index of a variance Xi is given by

S1i ¼

VXiðEX�iðYjXiÞÞ

VðYÞ ð18Þ

In Eq. (18), EX�iðYjXiÞ denotes the expectation of the output Y by fixing the variable Xi at a particular value and considering

random variations of all other variables (denoted by X�i). The output of this calculation is conditioned on where the variableXi is fixed. Hence, Eq. (18) calculates the variance of this expectation by considering the random variation in Xi and fixing it atrandom values. The calculation of Eq. (18) requires an outer Monte carlo loop to calculate the variance and an inner Montecarlo loop to calculate the expectation. However, Saltelli et al. [66] provide single loop Monte carlo methods to evaluate Eq.(18).

The first-order effect calculated in Eq. (18) gives an estimate of the contribution of the variable Xi to the uncertainty in Y,without considering the effects of the other variables X�i, as their contribution is averaged. The sum of first order indices ofall variables is always less than or equal to unity; equality holds when the model G is additive with respect to all the vari-ables. The contribution of the variable Xi in combination with all other variables is known as the total effects index and canbe calculated as:

STi ¼ 1� EX�i

ðVXiðY jX�iÞÞ

VðYÞ ð19Þ

The terms in Eq. (19) are similar to that in Eq. (18); there are two notable differences though. First, the variance is calculatedin the inner loop and the expectation of the variance is calculated in the outer loop. Second, in the inner loop, all quantities


other than Xi (denoted by X�i) are fixed at a particular quantity to calculate the variance by considering random variations inXi., and then the outer loop considers random variations in all other quantities (X�i). There are single loop approaches avail-able in the literature [66] to evaluate Eq. (19) as well.

The sum of the total effects indices of all variables is always greater than or equal to unity; equality holds when the modelG is additive with respect to all variables. (In this case, the first-order effects indices are equal to the total effects indices).

The method of global sensitivity analysis has previously been applied only to naturally varying random inputs; this paperextends this method to include data uncertainty and model uncertainty as well. In this paper, Xi’s denote the various sourcesof uncertainty and stochastic error terms; deterministic errors are corrected before sensitivity analysis. The quantities thathave a high influence on the variance of the final crack growth prediction are alone selected for calibration. While thisscreening procedure is based on second-moment calculation, the model calibration and validation techniques explainedin the following sections consider the entire probability distribution for all quantities.

4.3. Model calibration

Let X denote the vector of quantities that are selected for calibration. Assume that there is a set of m experimental datapoints, i.e. crack inspection after N loading cycles. Each inspection may produce three possible results: (1) no crack detected;(2) crack detected but size not measured; and (3) crack detected and size measured to be A. This is the first set of data, re-ferred to as the calibration data.

Bayesian updating is a three step procedure:

1. Prior probability distributions are assumed for each of the parameters. The prior distribution of EIFS is calculated usingEq. (5). The crack growth model error (ecg) has zero mean and the standard deviation is assumed to have a non-informa-tive prior. The prior distributions of crack growth law parameters are obtained from Liu and Mahadevan [49].

2. The likelihood of X is defined as being proportional to the probability of observing the given data conditioned on X.(Edwards [67] states that the likelihood is meaningful only up to a proportionality constant. The mass of a likelihoodfunction, in general, is not equal to unity and hence, ‘‘the likelihood of X’’ of must not be confused with ‘‘the probabilityof X’’ or the ‘‘probability density of X’’.)

3. The prior and likelihood of X are multiplied and normalized to calculate the posterior probability distribution. Finally, thejoint distribution of X is used to calculate the marginal distributions of the individual parameters.

In step 2, the likelihood of X is calculated as the probability of observing the data conditioned on X and hence is a func-tion of X. For every given X, a Monte Carlo analysis is required for the calculation of likelihood along the following steps:

I. Construct the Gaussian process surrogate model as explained in Section 2. Include the sources of uncertainty asexplained in Section 3.

II. For a given X

III. Generate a loading history (N cycles) as explained in Section 3.IV. Use the deterministic prognosis methodology in Section 2 to calculate the final crack size at the end of Ni cycles.V. Repeat steps II and III and calculate the probability distribution of crack size at the end of Ni (for i=1 to m) cycles. Let

this distribution be denoted by f(a). Use Eq. (10) to calculate f(am|a). This probability density function can be used tocalculate the likelihood of X.

If no crack is detected, then the likelihood function can be calculated as:

LðXÞ / 1�Z Z

f ðamjaÞf ðajNi;XÞdadam ð20Þ

If a crack is detected but size not measured, the likelihood function can be calculated as:

LðXÞ /Z Z

f ðamjaÞf ðajNi;XÞdadam ð21Þ

If a crack is detected and the size is measured to be Ai, then the likelihood function can be calculated as:

LðXÞ /Z

f ðam ¼ AijaÞf ðajNi;XÞda ð22Þ

Note that am and a vary in the interval (0, 1).If different inspections give any of the three types of crack information, the corresponding likelihood function for each

data point is calculated using Eqs. (17), (18) or Eq. (19), and then all the likelihood functions are multiplied to get the overalllikelihood function for the entire inspection data set. Finally, the likelihood is multiplied with the prior and normalized tocalculate the joint posterior distribution of X [57], and this joint distribution can be marginalized to calculate the individualdistributions [61].


5. Model validation

This section proposes a methodology to assess the validity and the confidence in the prediction of the fatigue crackgrowth models. Consider the crack-growth algorithm discussed in Section 2. The probability distribution of the crack size(a) can be calculated as a function of number of load cycles (N) after accounting for the various sources of uncertainty ina systematic manner using the methods in Sections 2 and 3. Let f(a|N) denote the corresponding probability density function.The aim of this section is to assess the validity of the underlying crack growth model and propose a metric to quantify theconfidence in the model prediction. It is assumed that a second set of data (D), referred to as validation data, is available forthis purpose, and is independent of the calibration data.

Rebba et al. [35] used Bayesian hypothesis testing for model validation and confidence assessment. Suppose two compet-ing hypothesis (H0 and H1) are considered; the ratio of the likelihoods of the two hypotheses is:

B ¼ PðDjH0ÞPðDjH1Þ

ð23Þ

The likelihood ratio B is referred to as Bayes factor [68]. If B > 1, then the data D favors the hypothesis H0 more than thehypothesis H1. In the context of model validation, the two hypotheses H0 and H1 may be chosen as ‘‘Model is correct’’ (nullhypothesis) and ‘‘Model is incorrect’’ (alternate hypothesis) respectively. Hence, the Bayes factor is a measure of the extent towhich the data supports the model.

Consider Eq. (23). The calculation of the numerator is based on the available data (D), and the prediction of the crack sizef(a|N) and f(am|a) is calculated based on Eq. (10). For a particular inspection after Ni cycles, there are three possible cases:

Case 1: No crack detected

PðDjH0Þ / 1�Z Z

f ðamjaÞf ðajNiÞdadam ð24Þ

Case 2: Crack detected but size not measured

PðDjH0Þ /Z Z

f ðamjaÞf ðajNiÞdadam ð25Þ

Case 3: Crack detected and size measured to be equal to Ai

PðDjH0Þ /Z

f ðam ¼ AijaÞf ðajNiÞda ð26Þ

Note that Eqs. (24)–(26) are different from Eqs. (20)–(22). Previously, f(a|N, X) was calculated for a given realization of cal-ibration parameters X. This is not the case in Eqs. (24)–(26) where the uncertainty in the parameters X (sampled using pos-terior probability distributions of X) is included in f(a|N).

The calculation of the denominator is related to the hypothesis that the model is not correct. Hence, it is necessary toassume a distribution for f(a|N) under this hypothesis. As there is no information available for this hypothesis, a non-infor-mative uniform prior distribution for f(a|N) is used and the denominator term P(D|H1) is calculated based on Eqs. (24)–(26).

If there are multiple inspections at multiple time instants, and if the inspections can be assumed to be independent ofeach other, the overall Bayes factor (that accounts for data from multiple time instants) will be equal to the product ofthe individual Bayes factors. Assume that the data D is composed of data from several independent inspections D1, D2,D3, . . . , Dn and the Bayes factor corresponding to data Di is Bi. Then, the event of observing D is equivalent to the intersectionof the events of observing Di. Since the individual inspection data are independent, the overall Bayes factor can be calculatedas:

B ¼ PðDjH0ÞPðDjH1Þ

¼ PðD1 \ D2 . . . \ DnjH0ÞPðD1 \ D2 . . . \ DnjH1Þ

¼ PðD1jH0ÞPðD2jH0Þ . . . PðDnjH0ÞPðD1jH1ÞPðD2jH1Þ . . . PðDnjH1Þ

¼ B1B2 . . . Bn ð27Þ

Having calculated the Bayes factor, the posterior probabilities (after observing the inspection data) of the hypotheses H0 andH1 can be calculated using the Bayes theorem as follows:

PðH0jDÞ ¼PðDjH0ÞPðH0Þ

PðDjH0ÞPðH0Þ þ PðDjH1ÞPðH1Þð28Þ

PðH1jDÞ ¼PðDjH1ÞPðH1Þ

PðDjH0ÞPðH0Þ þ PðDjH1ÞPðH1Þð29Þ

Prior to model validation, the hypotheses H0 and H1 can be considered to be equally likely and hence, the posterior proba-bility of H0, i.e. the model being correct reduces to:

PðH0jDÞ ¼B

Bþ 1ð30Þ

This quantity can be used to express the amount of confidence in the model prediction. If the Bayes factor is equal to one,then the associated confidence is equal to 50%, implying that the two hypotheses ‘‘the model is correct’’ and ‘‘the model is


incorrect’’ are both equally likely. Note that the term ‘‘confidence’’ used in this paper refers only to the posterior probabilityof the ‘‘model being correct’’ and is not related to confidence intervals in statistical hypothesis testing.

The confidence associated with the prediction is thus calculated as B/ (B + 1). Higher the Bayes factor, higher is the con-fidence associated with the model prediction. Jiang and Mahadevan [69] derived a threshold Bayes factor value for modelacceptance, based on the costs associated with type-I and type-II errors in hypothesis testing.

In summary, Sections 4 and 5 proposed a Bayesian methodology for (1) calibration and uncertainty quantification of mod-el parameters using inspection data (crack size measurements after a number of cycles, including cases where no crackswere detected, and crack detections without measurements), and (2) assessing the validity of the crack growth predictionby comparing against a second independent set of validation data. The following section illustrates the proposed methodol-ogy using a numerical example.

6. Numerical example

A two-radius hollow cylinder with an elliptical surface crack in fillet radius region is chosen for illustrating the proposedmethodology, as shown in Fig. 4.

The commercial finite element software ANSYS [70] is used to build and analyze the finite element model. A sub-mod-eling technique is used near the region of the crack for accurate calculation of the stress intensity factor. As mentioned earlierin Section 3, the material and geometric properties of the specimen are assumed to be deterministic and are listed in Tables 1and 2 respectively.

A variable amplitude multi-axial (bending and torsion) loading is considered here for the sake of illustration. A block load-ing history is considered (sample shown in Fig. 5). The block length is assumed to be a random variable and the maximumand minimum amplitudes in each block are also treated as random variables. To generate each block of loading, first theblock length is randomly sampled and then the maximum and minimum amplitudes for that block are randomly sampled.The entire loading history is generated by repeating this process and creating several successive blocks to occupy the dura-tion of interest.

All the different types of uncertainty in Fig. 2 are included in this numerical example, as shown in Table 3.

Refined sub-model

Coarse full model

Fig. 4. Surface crack in a cylindrical component.

Table 1Material properties.

Aluminium 7075-T6

Modulus of elasticity 72 GPaPoisson ratio 0.32Yield stress 450 MPaUltimate stress 510 MPa

Table 2Geometric Properties.

Cylinder properties

Length 152.4 mmInside radius 8.76 mmOutside radius 14.43 mmOutside radius 17.78 mm

Fig. 5. Sample loading history.

Table 3Types of Uncertainty.

Type of uncertainty Quantities Details

Natural variability Loading Random block loadingBlock length � Uniform (0, 500)Max. Amplitude � N(lmax,rmax)Max. Amplitude � N(lmin,rmin)lmax � Uniform (1.355, 0.263) kNmrmax � Uniform (0.455, 0.130) kNmlmin � Uniform (2.710, 0.260) kNmrmin � Uniform (0.455, 0.130) kNm

Material properties (used to calculate priordistribution of EIFS)

(DKth and Drf) – Distribution type, mean, andstandard deviation based on [49]

Data uncertainty Sparse data Distribution parameters of (DKth and Drf) areuncertain ? Assumed coefficient of variation formean = 0.1 and standard deviation = 0.1 ? Use Eq.(13) to calculate PDFs of DKth and Drf ? Use Eq. (5)to calculate PDF of EIFS (h)

Inspection data (crack not detected, crack detectedbut size not measured, and crack size measured)

Considered all three cases both for calibration dataset and validation data set

Model uncertainty Crack growth model Statistics of parameters based on [49] andlognormal error term (ecg) with unit mean andstandard deviation equal to 0.05; in the numericalexample, EIFS (h), parameter C and error ecg arecalibrated with above distributions as priors.

Discretization error Calculated using Richardson extrapolation(Section 3.3.2)

Surrogate model error Calculated based on Eq. (9) and included in theuncertainty propagation (Section 3.3.3)


This finite element model is run for 10 different crack sizes and six different loading cases and these results are used totrain the Gaussian process surrogate model for the calculation of the stress intensity factor. In each run of the finite elementanalysis, the discretization error is calculated and the corrected solution in turn is used to train the Gaussian process surro-gate model. The maximum discretization error was found to be about 2% of the finest mesh solution. If the discretizationerrors were found to be very large, then it indicates a coarsely resolved calculation, which may not be suitable in the contextof model calibration.

Global sensitivity analysis was used to calculate the first order (individual) and total (combined) effects of the variouscrack growth parameters such as EIFS (h), model parameters (C, m), crack growth model error (ecg), threshold stress intensityfactor (DKth), loading, etc. It was observed that the sensitivity of the parameters changed with time and hence, it was nec-essary to calculate the sensitivity indices as a function of number of cycles. After a significant number of loading cycles, the

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

No. Cycles

Sens

itivi

ty In

dex

First Order Effect Sensitivity IndexTotal Effect Sensitivity Index

Fig. 6. Sensitivity indices of equivalent initial flaw size h.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.05

0.1

0.15

0.2

0.25

No. Cycles

Sens

itivi

ty In

dex


Fig. 7. Sensitivity indices of model parameter C.


parameters EIFS h, model parameter C, and crack growth model error (ecg) were found to be the most significant terms. Thefirst-order and total effects sensitivities of these quantities are shown in Figs. 6–8. Recall that the first-order effects describethe contribution of a particular variable to the uncertainty in the crack size, by itself, whereas the total effects describe theoverall contribution of a particular variable to the uncertainty in the crack size, in combination with all other variables.

The sensitivity indices of the other parameters were insignificant (less than 0.001) and hence only three parameters (EIFSh, model parameter C, and crack growth model error ecg) are considered to have a dominant influence on the uncertainty inthe crack growth prediction in comparison with the other parameters. Hence, these three are chosen to be the calibrationparameters.

Experimental evidence of 100 data points (inspection) was simulated (based on assumed true distributions for the infer-ence quantities) and used for calibration as explained in Section 4. 100 components were inspected at different time in-stants; no crack was detected in one component, the crack size was measured in 49 components, and the crack size wasnot measured in 50 components. This experimental data is used to calculate the statistics of the parameters using the like-lihood function (Eqs. (20)–(22)) and Bayes theorem.

Note that Eqs. (20)–(22) calculate the joint likelihood of the inference quantities (EIFS h, model parameter C, and crackgrowth model error ecg) and hence, Bayes theorem gives the joint probability density function of these quantities. Hence,in addition to the marginal posterior distributions, the correlations between these quantities can also be estimated usingthe joint density function.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.005

0.01

0.015

No. Cycles

Sens

itivi

ty In

dex


Fig. 8. Sensitivity indices of crack growth model error ecg.

Table 4Results of Calibration.

Quantity True statistics Prior statistics Posterior statistics

Mean Std. dev. Mean Std. dev. Mean Std. dev.

EIFS (h) (mm) 0.3810 0.0254 0.5126 0.1146 0.3788 0.0222C (m/cycle) 4.0 E�13 0.4 E�13 6.5 E�13 4.0 E�13 5.7 E�13 3.1 E�13ecg (mm) 1 0.05 1 0.1 1.0 0.05

Table 5Results of Model Validation.

No. cycles Crack size (mm) Bayes factor

10,008 0.42 3.1613,852 0.31 2.0614,141 0.63 0.8014,368 0.41 2.3514,659 0.43 2.8015,155 1.54 0.7015,277 0.55 2.1017,746 0.43 2.5617,844 0.62 0.7919,912 0.35 3.00


The Bayesian calibration methodology is applicable to any combination of prior PDF and likelihood function; there are norestrictions on the distribution type. However, if the calculation of the likelihood function is non-trivial, as seen in thisnumerical example, the posterior distributions are non-parametric and may not conform to any particular distribution type.Hence, it is necessary to calculate the posterior distributions using numerical techniques; the mean and the standard devi-ation of the prior distribution and the posterior distribution of the inferred quantities are given in Table 4.

The Bayesian approach will be applicable even if the mean of the prior distribution is one or two orders of magnitudeaway from the truth as long as the uncertainty in the prior is large enough to contain the truth. In other words, the Bayesianapproach will work if there is any overlap between the prior distribution and the truth.

(If there is no overlap, then the posterior PDF will be zero everywhere. Note that the posterior distribution is proportionalto the prior times the likelihood. In the ‘‘region of the truth’’, the prior will be zero as there is no overlap. Outside the ‘‘regionof the truth’’, the likelihood will be zero.)

Having calibrated the parameters, the posterior distributions can be used to predict the distribution of crack size as afunction of number of loading cycles. This PDF was denoted by f(a|N) in Section 5. Additional independent validation datais assumed to be available and the Bayes factor is calculated for each data point individually. Recall that the Bayes factor


is the ratio of the likelihoods of the null and the alternate hypothesis. Eqs. (24)–(26) are used to calculate the P(D|H0) whereH0 represents the hypothesis that the model is correct. Similarly, the probability of observing the data under the alternatehypothesis is calculated by assuming a uniform distribution for the crack size. Then, the ratio of these two likelihoods is cal-culated as the Bayes factor for each experimental data available for validation. The results of validation are tabulated inTable 5.

From Table 5, it can be seen that the Bayes factor is greater than unity in most of the cases and lowest value is 0.70. Thecombined Bayes factor corresponding to all the data can be calculated by multiplying the individual Bayes factors as in Eq.(27) and is found to be equal to 306. This corresponds to a 99.6% confidence in the model prediction.

7. Conclusion

This paper presented a methodology for uncertainty quantification and model validation in fatigue crack growth analysis.Structures with complicated geometry and multi-axial variable amplitude loading conditions were considered. The finiteelement analysis used for calculation of stress intensity factor was replaced using a Gaussian process surrogate model, there-by reducing the computational effort. Different sources of uncertainty – physical variability, data uncertainty, and model er-ror/uncertainty – were included in the crack growth analysis. Different types of model errors – discretization error, crackgrowth model error, surrogate model error – were considered explicitly. Deterministic errors were corrected where they ar-ose and stochastic errors were included by using random samples during uncertainty propagation.

Bayesian inference was used to calibrate the parameters of different models using inspection results (crack size afternumber of cycles, including cases where no crack was detected and crack was detected but size not measured). A Bayesianconfidence metric was developed to assess the validity and quantify the confidence in fatigue crack growth prediction.

Note that this paper used a particular set of models – modified Paris’ law, Wheeler’s retardation model, characteristicplane approach for the calculation of an equivalent stress intensity factor – only for the purpose of illustration. The crackgrowth analysis was limited to linear-elastic fracture mechanics and planar cracks. Also, the effects of model assumptionssuch as planar crack, LEFM, equivalent stress intensity factor for multi-modal fracture, and crack retardation model werenot individually quantified. However, the Bayesian framework for uncertainty quantification and model validation is generaland can easily accommodate other advanced analysis models and corresponding model errors/uncertainty.

Also, the proposed approach for uncertainty quantification and model validation is applicable to several engineering dis-ciplines and the domain of crack growth analysis was used only as an illustration to develop the methodology. In general, theproposed methodology provides a fundamental framework in which multiple models can be connected through a Bayes net-work and the confidence in the overall model prediction can be assessed quantitatively. The use of a Bayes network helpswith uncertainty propagation between multiple models and also facilitates the systematic treatment of various sources ofuncertainty for the purpose of uncertainty quantification, model calibration and validation.

Acknowledgements

The research reported in this paper was supported in part by the Federal Aviation Administration William J. Hughes Tech-nical Center through RCDT Project No. DTFACT-06-R-BAAVAN1 (Project Monitors: Dr. John Bakuckas, Ms. Traci Stadtmueller)and by the NASA ARMD/AvSP IVHM project under NRA Award NNX09AY54A (Project Monitor: Dr. K. Goebel, NASA AMESResearch Center) through subcontract to Clarkson University (No. 375-32531, Principal Investigator: Dr. Y. Liu). The supportis gratefully acknowledged.

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Uncertainty quantification and model validation of fatigue crack growth prediction