Journal of ASTM International, Vol. 3, No. 10
Paper ID JAI100574
Available online at www.astm.org
Richard J Cross, M.Sc.,1 Andrew Makeev, Ph.D.,1 and Erian Armanios, Ph.D.1
A Comparison of Predictions From Probabilistic Crack
Growth Models Inferred From Virkler’s Data
ABSTRACT: The significant variability observed in fatigue crack growth experiments requires application of
probabilistic modeling techniques to fatigue analyses. In this work, philosophical and practical concerns for
probabilistic modeling are discussed. Several probabilistic fatigue models are derived from the Virkler data
and then used to make predictions under a different loading condition. This comparison of predictions
demonstrates their sensitivity to the choice of model. In this effort a method for inference of random process
parameters and corresponding confidence limits from highly statistically dependent observations is developed.
KEYWORDS: fatigue, probabilistic modeling, reliability
Introduction
The random nature of fatigue crack growth has been shown in numerous experimental studies. For example, the results of Virkler et al. 关1兴 and Ghonem and Dore 关2兴 demonstrate that significant variability in
crack propagation occurs even after crack initiation. This fact and the need for certain important analysis
tasks, such as structural risk assessment 关3兴, require stochastic models of fatigue crack growth that not only
fit the observed experimental results well but also possess good predictive capabilities.
Significant effort has been devoted to the development of probabilistic formulations for modeling
fatigue life distributions. Two main modeling classes have emerged from these studies: random variable
共RV兲 models and random process 共RP兲 models 关4兴. RV models capture the variability in crack growth by
propagating parameter uncertainty through deterministic crack growth models. Often RV models are derived by assuming all uncertainty is due to one or a couple of dominant sources. For example, RV models
have been employed that assume all uncertainty comes from equivalent initial flaw size 关5,6兴, crack
initiation times 关7,8兴 or Paris Equation constants 关9兴. Alternatively, RP models assume each crack history
is a stochastic process indexed by load cycles, time, or crack length. Several RP models have been derived
for load cycle or time indexed stochastic processes 关10,11兴 as well as crack length indexed processes
关12,13兴. The ability of these models to fit existing data sets well has been demonstrated in several studies
关11,13–15兴.
While much effort has been devoted to the development of probabilistic models, relatively little work
has been done to investigate their prognostic abilities beyond the data used to infer them. It is important to
determine how model choice influences predictions if they are to be used in design and analysis. This need
has motivated a round-robin exercise in which participants are provided with the Virkler data and asked to
predict the distribution of fatigue life under slightly different conditions. In this study, stochastic crack
growth models from different classes are inferred, and then used to make predictions for the new loading
condition. Comparison of these prognostic results demonstrates that significant differences in predictions
can be obtained from different stochastic fatigue models even if all fit the data well.
Manuscript received March 31, 2006; accepted for publication September 12, 2006; published online October 2006.
1
School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332-0150.
Copyright © 2006 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.
2 JOURNAL OF ASTM INTERNATIONAL
FIG. 1—Virkler data set [1].
Theoretical and Practical Considerations
Random Variable Models
Random variable models introduce uncertainty into fatigue crack growth analysis by assigning a joint
probability distribution to one or more relevant parameters. The distribution can be assumed a priori,
inferred from experiment, or determined through supplemental investigations. A fatigue crack is then
assumed to grow deterministically given a realization of the parameter vector.
Philosophical Considerations—RV models reflect some important aspects of the physical reality of
fatigue crack growth. Many influential parameters such as initial flaw size, residual stresses, and environmental conditions can contain significant variability. However, the assumption of deterministic crack
growth may not be sufficiently accurate as many experimental results 共such as that of Virkler et al., Fig. 1兲
exhibit notable irregularity in individual crack histories. This irregularity can be due to a number of
sources, such as microstructural variability, damage, and geometric variations. Thus, RV models may fail
to capture some potentially important aspects of reality. Further, the deterministic growth assumption also
limits the probabilistic modeling capability of random variable models because they do not possess any
parameters that directly govern the stochastic evolution of the crack history.
Practical Considerations—Computationally, RV models are simple to implement. Once a joint distribution for the parameters is obtained, the uncertainty is propagated through the model by any number of
established methods such as Monte Carlo or advanced mean value 关16兴. Inferences are usually simple to
perform as well. Often the inference can be done by estimating the random parameter vector once for each
crack in the data set and then fitting a multivariate distribution to the set of estimators.
Random Process Models
Random process 共RP兲 models assume that each crack history is a single realization of a stochastic process.
This is often accomplished by multiplying the deterministic crack growth rate by a suitable stochastic
process indexed by either load cycles or crack length. In the case of a process indexed by load cycles, the
resulting equation can be written as
da
= X共N;␼兲f„⌬K共a兲,␪…
dN
共1兲
where X共N , ␼兲 denotes the stochastic process, f(⌬K共a兲 , ␪) is the analytical crack velocity expression, and
␪ denotes the vector of material and geometric parameters. Equation 1 is a separable differential equation
that can be solved by integrating the stochastic process over the desired time interval. Following Yang and
Manning 关11兴,
冕
a共N兲
a0
da
=
f„⌬K共a兲,␪…
冕 X共n;␼兲dn = W共N;␼兲
N
0
共2兲
CROSS ET AL. ON PROBABILISTIC CRACK GROWTH 3
From Eq 2 the cumulative distribution function 共CDF兲 of a共N兲 can be found from the distribution of
W共N ; ␼兲 关11兴 such that
冉
P„a共N兲 ⬎ a f … = P W共N;␼兲 ⬎
冕
af
a0
da
f„⌬K共a兲,␪…
冊
共3兲
Alternatively, for a crack length indexed process, the growth rate can be written as
da
= Y共a;␼兲−1 f„⌬K共a兲,␪…
dN
共4兲
where Y共a , ␼兲 represents a crack length indexed random process.
Solving Eq 4 shows that the number of cycles to a crack of length a f is a random variable equal to the
integral of a stochastic process such that
N共a f 兲 =
冕
af
a0
Y共a;␼兲
da
f„⌬K共a兲,␪…
共5兲
Philosophical Considerations—RP models can capture the nondeterministic aspects of the individual
growing crack because each history is assumed to be a realization of a stochastic process. This gives the
model more flexibility and fidelity than possible with RV models. Further, the characteristics of the random
process can evolve with crack length or service time to capture potentially important nonstationary effects.
The choice of index for the stochastic process also has conceptual significance. Ortiz 关13兴 noted that
a random process indexed by crack length has more physical relevance as the spatial irregularity of the
material is the sole contributor to component life variability given a known loading history and initial
crack length. Thus, a spatially indexed stochastic process inferred from one data set is more likely to be
applicable to other applications of the same material.
Practical Considerations—This modeling approach can create significant practical difficulty in calculating the distributions of crack sizes and cycle times as well as inferring parameters from experimental
data. To compute the distribution of crack size at a given time, or life to a given crack size, one must
determine the distribution of the integral of the appropriate stochastic process. Except in the case of
Gaussian random processes, few closed form expressions exist for distributions of stochastic integrals.
Some methods have been proposed to overcome this difficulty. Yang and Manning 关11兴 derived a
second-order approximation for the distribution of the integral in Eq 2 when X共N ; ␼兲 is a log-normal
process with a median of 1 and exponentially decaying auto-covariance. Likewise, a second-order approximation for the distribution of N共a f 兲 in Eq 5 will be developed and employed in this study.
Random process models also defy the application of conventional inference techniques to determine
their parameters from experimental data. Multiple observations of a given crack can make the likelihood
function computationally intractable because of the usually strong statistical dependence between observations. The aforementioned second-order approximations do not provide a practical method for computing the likelihood of data obtained when a crack is observed more than once.
Random Variable Modeling of Virkler Data
Data for performing the numerical examples come from the experiments of Virkler et al. 关1兴. The specimens for these tests were 2.54-mm-thick and 152.4-mm-wide center cracked sheets of 2024-T3 aluminum
with a 9-mm pre-crack. The specimens were subjected to a cyclic loading with a maximum load of 23 kN
and a stress ratio of R = 0.2. Each of the 68 specimens tested was observed 163 times.
The first step in selecting the appropriate RV model is to determine an appropriate functional relationship between the logarithm of stress intensity range and the logarithm of crack velocity. Candidate models
can be selected by visual inspection of the log-log plot of the crack velocity versus stress intensity
appearing in Fig. 2. Virkler’s crack velocity data suggest that a curvilinear relationship with an inflection
point should be used to adequately describe the crack growth rate.
A commonly used curvilinear crack growth model that meets the requirements is the hyperbolic sine
model 关17兴, written as
4 JOURNAL OF ASTM INTERNATIONAL
FIG. 2—Crack velocities from Virkler data set [1].
ln
da
= C1 sinh兵C2关ln ⌬K共a兲 + C3兴其 + C4
dN
共6兲
where Ci are considered random variables. However, C1 and C2 become confounded when the linear
region dominates the growth rate data, as seen in Fig. 2. This can be shown by expanding Eq 6 in a Taylor
series as
ln
da
= C4 + C1C2关ln ⌬K共a兲 + C3兴 + C1O兵C32关ln ⌬K共a兲 + C3兴3其
dN
共7兲
where the cubic term is negligible by the knowledge that the linear behavior is dominant. Since C1 and C2
modify the dominant term in the same manner, their effects are practically interchangeable for the Virkler
data. Therefore, C1 and C2 are not identifiable.
Another possibility is to perform a polynomial regression in the ln ⌬K − ln da / dN space, which gives
a growth rate equation of the form
冉兺
da
= exp
dN
N
j=0
c j关ln ⌬K共a兲兴 j
冊
共8兲
where the c j’s are assumed to be a jointly distributed random vector.
Cubic Polynomial Model of Virker Data
Assuming a model of the form in Eq 8 with N = 3, a series of polynomial regressions was performed on the
logarithm of the crack velocity versus the logarithm of ⌬K to obtain 68 samples of the random vector
兵c0 , . . . , c3其. A multivariate normal model was fit using the resulting vectors to obtain a joint distribution of
the parameters. The uncertainty was propagated through the model using a Monte Carlo simulation. A set
of 10 000 sample vectors of the ci were drawn from the inferred multivariate normal model. Each sampled
parameter vector was substituted in Eq 8, which was then integrated to generate 10 000 sample crack
histories from which the distributions of cycles to given crack lengths were estimated. Figure 3 presents
FIG. 3—Comparison of cubic RV model predicted moments to Virkler data moments.
CROSS ET AL. ON PROBABILISTIC CRACK GROWTH 5
FIG. 4—Comparison of predicted and empirical service time CDFs for a f = 15, 25, and 35 mm using cubic
RV model.
the error relative to the Virkler data in the predicted mean and standard deviation of component life as a
function of final crack size. The comparison shows that the cubic RV model fits the data well as the mean
and standard deviation errors are less than 2 % and 10 %, respectively, for most final crack lengths.
Figure 4 compares the predicted CDFs of cycles to final crack lengths of 15, 25, and 35 mm to the
corresponding empirical CDFs. The slight mean error noted earlier is apparent in the small leftward shift
of the predicted CDFs. The standard deviation error previously noted is due to the discrepancy for
cumulative probabilities greater than 0.8. However, this is not worrisome because the lower tails 共low
cumulative probabilities兲 are more important for risk assessment purposes.
Random Process Modeling of Virkler Data
Yang and Manning’s Cycle Indexed RP Model
Following Yang and Manning 关11兴, the random process X共N , ␼兲 in Eq 1 is assumed to be a lognormal
process with unit median and auto-covariance given by
cov„X共N1 ;␼兲,X共N2 ;␼兲… = ␴2X exp共− ␵兩N1 − N2兩兲
共9兲
The parameters to be inferred from the Virkler data are the correlation decay rate ␵ and log-normal
variance ␴Z2 = var共ln X共N ; ␼兲兲. Using the second-order approximation of Yang and Manning, the CDF for
cycle time can be written as
FN共a兲共N兲 = ⌽
冉
ln N + ln ␭共N兲 − ln N̄共a兲
␴共N兲
冊
共10兲
where
N̄共a兲 =
冕
a
a0
da
f„⌬K共a兲;␪…
共11兲
␾ = 冑2关exp共− ␵N兲 + ␵N兴 − 1/␵N
共12兲
␴ = 冑ln„1 + ␾2 exp共␴Z2 兲 − ␾2…
共13兲
␭=
冑
exp共␴Z2 兲
1 + ␾2 exp共␴Z2 兲 − ␾2
共14兲
Because of the assumed auto-covariance functional form, the model implies statistical dependence
between separate observations of a single crack. Since each crack is observed 163 times, the true likelihood function becomes computationally infeasible to calculate. However, it is observed that a good-fitting
model can be inferred by using the likelihood function obtained from a second-order approximation and
assuming independence. This likelihood function is written as
6 JOURNAL OF ASTM INTERNATIONAL
FIG. 5—Comparison of cycle-indexed RP model predicted moments to Virkler data moments.
68 163
ln L共D兩␵,␴Z兲 = 兺
兺
c=1 k=1
冉
ln
dFN共a兲共Nck兲
dN
冊
共15兲
where Nck is the number of cycles sustained by crack c at observation k. A cubic polynomial in the
ln ⌬K − ln da / dN space is used to model the median growth rate. Maximizing Eq 15 gives the point
estimates ␴ˆ Z = 0.1077 and ␵ˆ = 1.359⫻ 10−5.
Figure 5 compares the predicted mean and standard deviation for component life from the random
process model to those of the Virkler data. The predicted mean lifetimes remain within 2 % of the observed
mean lifetimes. The standard deviation prediction has large error until the crack reaches about 13.7 mm in
length. From that point, the absolute error remains less than 10 %.
Figure 6 compares the cycle time CDFs predicted by the random process model to the empirical
distributions for final crack lengths of 15, 25, and 35 mm. As with the cubic RV model, the predicted
CDFs of cycles to a given crack size fit the observed data well.
Confidence Intervals for RP Parameters
It must be emphasized that performing a maximum likelihood estimation with Eq 14 can provide only a
point estimate of the RP model parameters. This is because the likelihood expression in Eq 14 does not
capture the statistical dependence of observations, and thus confidence bounds obtained by calculating the
observed Fisher information will be much too narrow. Fortunately, it is still possible to infer confidence
intervals for the estimators by using the bootstrap method 关18兴. The bootstrap assumes that the empirical
distribution obtained from the data approximates the data’s true distribution well. Under this assumption,
the true sampling distribution for the estimators can be estimated by drawing new data set samples from
the empirical data distribution and calculating the estimators’ value for each new data set. The set of
estimators calculated from the resampled data sets can then be used to approximate their sampling distribution.
FIG. 6—Comparison of predicted and empirical service time CDFs for a f = 15, 25, and 35 mm using
cycle-indexed RP model.
CROSS ET AL. ON PROBABILISTIC CRACK GROWTH 7
FIG. 7—Bootstrap histogram and estimated sampling distribution CDF for ␴Z estimator.
For the Virkler data, each resample consists of 68 cracks chosen from the original data set with
replacement. The resampling is performed on the set of cracks, rather than the set of individual measurements, so that the approximate sampling distribution captures the statistical dependence between separate
observations of the same crack. This choice is logical since each crack history corresponds to a single
realization of a random process. Each resampled set of cracks was used to compute the estimators of the
two random process parameters, and the set resampled estimators were then used to approximate the
sampling distributions. Figures 7 and 8 depict the bootstrap sample histograms for the resampled estimators and estimated marginal sampling distribution CDFs for the estimators ␴ˆ Z and ␵ˆ , respectively. From
these estimated sampling distributions, the 95 % confidence intervals for ␴ˆ Z and ␵ˆ were estimated to be
关0.0847, 0.1564兴 and 关1.3439⫻ 10−6, 5.6621⫻ 10−5兴, respectively.
Spatial-index RP Model
The spatial random process Y共a ; ␼兲 defined in Eq 4 is assumed to be a covariant stationary log-normal
process with unit median and exponentially decaying auto-covariance written as
cov„Y共a;␼兲,Y共a + ⌬a;␼兲… = ␴2Y exp共− ␩兩⌬a兩兲
共16兲
Because of the complexity of the definition of the random variable N in Eq 5, a second-order lognormal approximation is used to describe the distribution of N共a f 兲. First, using the knowledge that N共a f 兲
increases monotonically with Y共a ; ␼兲, the median of N共a f 兲 is obtained as
N̄共a f 兲 =
冕
af
a0
med„Y共a;␼兲…
da =
f„⌬K共a兲,␪…
冕
af
a0
da
f„⌬K共a兲,␪…
共17兲
Next, the variance N共a f 兲 is calculated as
var N共a f 兲 =
冕冕
af
af
a0
a0
cov„Y共a;␼…,Y„a⬘ ;␼兲…
dada⬘
f„⌬K共a兲,␪…f„⌬K共a⬘兲,␪…
2
The variance, sln
N共a f 兲 of ln N共a f 兲 is found by solving the nonlinear equation
FIG. 8—Bootstrap histogram and estimated sampling distribution CDF for ␵ estimator.
共18兲
8 JOURNAL OF ASTM INTERNATIONAL
FIG. 9—Comparison of spatial-indexed RP model predicted moments to Virkler data moments.
2
2
exp„2 ln N̄共a f 兲 + sln
N共a f 兲… · 关exp„sln N共a f 兲… − 1兴 = var N共a f 兲
共19兲
After numerically solving Eqs 17–19, the distribution of N共a f 兲 can be approximated as
P兵N共a f 兲 ⬍ n其 = ⌽
冉
ln n − ln N̄共a f 兲
sln N共a f 兲
冊
共20兲
Using this second-order approximation and assuming independence, the likelihood function for the
process parameters is calculated as
68 163
ln
L共D兩␴2Y ,␩兲
=兺
兺
c=1 k=1
ln
d
兩P兵N共a f,k兲 ⬍ n其兩n=Nc
k
dn
共21兲
A cubic polynomial in the ln ⌬K − ln da / dN space is used to model the median growth rate. Maximizing Eq 21 gives the point estimates ␴ˆ Y = 0.1249 and ␩ˆ = 0.2911 mm−1. Figure 9 compares the predicted
mean and standard deviation to those of the Virkler data as a function of final crack length. The fit is less
good than the previous two models with an error in the mean just larger than 2 % and standard deviation
errors reaching 18 %. However, the predicted component life CDFs still match the corresponding empirical cycle time CDFs. This can be seen in Fig. 10 where predicted distributions of cycles to cracks of length
15, 25, and 35 mm are compared to the observed distributions.
As before, bootstrap sampling was performed to characterize the random process parameter sampling
distributions. The bootstrapped sample histograms and estimated marginal sampling distribution CDFs for
the estimators ␴ˆ Y and ␩ˆ appear in Figs. 11 and 12, respectively. From these estimated sampling distributions, the 95 % confidence intervals for ␴ˆ Y and ␩ˆ were estimated to be 关0.0932, 0.1791兴 and 关0.1013 mm−1,
0.9043 mm−1兴, respectively.
FIG. 10—Comparison of predicted and empirical service time CDFs for a f = 15, 25, and 35 mm using
spatial-indexed RP model.
CROSS ET AL. ON PROBABILISTIC CRACK GROWTH 9
FIG. 11—Estimated marginal sampling distribution for growth rate standard deviation ␴Y estimator.
Application to New Loading Condition
For further comparison, the cubic RV model, cycle-indexed RP model, and spatial-indexed RP model were
applied to find the distributions of life to crack sizes under a different loading condition. The maximum
alternating stress was reduced by 27 % to 17 kN as specified in the round robin exercise. Table 1 compares
the mean cycles to final crack length predicted by the three models. The predictions are all acceptably
close with agreement to within 1–2 %.
Table 2 presents the standard deviations in component life predicted by the three probabilistic models.
Significant discrepancies can be seen between the models. For a f = 25 mm and a f = 30 mm, the standard
deviations predicted by the cubic RV model and the cycle-indexed RP model differ by 18 % and 11 %,
respectively. This difference decreases to around 6–8 % for larger final crack sizes.
For smaller final crack lengths, the standard deviations predicted by the cubic RV model and spatialindexed RP model agree to within 3 %. The difference grows to 5 % and then to 8 % for the larger final
FIG. 12—Estimated marginal sampling distribution for spatial correlation rate ␩ estimator.
TABLE 1—Comparison of predicted mean cycles under new loading.
a f , mm
25
30
35
40
45
Cubic RV
187 833
236 894
271 916
298 229
318 162
Cycle-indexed RP
186 671
235 437
270 146
296 266
315 845
Spatial-indexed RP
187 650
236 589
271 475
297 645
317 423
TABLE 2—Comparison of predicted standard deviation in cycles under new loading.
a f , mm
25
30
35
40
45
Cubic RV
17 003
18 549
19 697
20 777
21 865
Cycle-indexed RP
14 344
16 662
18 458
19 626
20 427
Spatial-indexed RP
16 605
18 548
19 450
19 925
20 188
10 JOURNAL OF ASTM INTERNATIONAL
crack lengths. Based on the trend in standard deviation error shown in Fig. 9, this difference should grow
as the final crack size increases beyond 45 mm.
Conclusions
Both the RV and RP modeling approaches are viable methods for obtaining good-fitting models of real
fatigue crack growth experimental results, as demonstrated by their application to the Virkler data. Tradeoffs between philosophical correctness and practical implementation were identified for the model classes.
RV models are more easily inferred at the cost of assuming deterministic crack growth. RP models capture
the randomness of individual crack histories but introduce difficulties in inferring parameters due to the
strong dependence between measurements of the same crack. In spite of this, it was demonstrated that a
good fitting RP model can be found by assuming independence, and valid confidence intervals can be
found using bootstrap resampling.
The choice of model was found to have a significant influence on the predictions of fatigue crack
growth beyond the data used to infer the model. The cubic RV model, cycle-indexed RP model, and
spatial-indexed RP model predicted noticeably different variances in service time when applied to a new
loading condition. This result indicates that further study into the predictive capabilities of the various
stochastic fatigue models is important.
Acknowledgments
The authors would like to acknowledge Dr. Eric Tuegel for initiating the ASTM Task Group E08.04.04
round-robin exercise on probabilistic crack growth analysis, and providing the Virkler data.
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关1兴
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关3兴
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关5兴
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
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