MCMC-Based Fatigue Crack Growth Prediction on 2024-T6 …downloads.hindawi.com/journals/mpe/2017/9409101.pdf · 2019. 7. 30. · 4 MathematicalProblemsinEngineering Table5:Estimationvalueofmodelparameters

Research ArticleMCMC-Based Fatigue Crack Growth Prediction on2024-T6 Aluminum Alloy

Xu Du1 Yu-ting He1 Chao Gao12 Kai Liu1 Teng Zhang1 and Sheng Zhang1

1Aeronautics and Astronautics Engineering College Air Force Engineering University Xirsquoan 710038 China2Beijing Aeronautical Technology Research Center Beijing 100076 China

Correspondence should be addressed to Yu-ting He heyut666126com

Received 31 May 2017 Accepted 22 August 2017 Published 3 October 2017

Academic Editor Yuqiang Wu

Copyright copy 2017 Xu Du et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Thiswork aims tomake the crack growth prediction on 2024-T6 aluminumalloy by usingMarkov chainMonte Carlo (MCMC)Thefatigue crack growth test is conducted on the 2024-T62 aluminum alloy standard specimens and the scatter of fatigue crack growthbehavior was analyzed by using experimental data based onmathematical statistics An empirical analytical solution of Parisrsquo crackgrowth model was introduced to describe the crack growth behavior of 2024-T62 aluminum alloy The crack growth test resultswere set as prior information and prior distributions of model parameters were obtained byMCMC using OpenBUGS package Inthe additional crack growth test the first test point data was regarded as experimental data and the posterior distribution of modelparameters was obtained based on prior distributions combined with experimental data by using the Bayesian updating At lastthe veracity and superiority of the proposed method were verified by additional crack growth test

1 Introduction

In aircraft structural engineering domain the concept ofdamage tolerance was introduced to design key componentsThe prediction of fatigue crack propagation is a main workfor aircraft structure serviceuse life management

Accurate prediction of fatigue crack growth determinedthe safety of service and usage Due to the scatter of materialthe fatigue crack growth behavior of some components stillcreates scatter though the components made by the samematerial and served under the same loading and environmen-tal conditions

The residual crack propagation life of the aircraft com-ponents is assessed based on the in-service observation andthe mathematical model which describes the crack growthbehavior of these components The input data provided tothese mathematical models are generally uncertain Thereare uncertainties in the material performance componentsdimensions the measurements and the degradation model[1] Probability methodology is widely used to make decisionby modelling these uncertainties with suitable probabilitydistribution function Scholars around the world have done

a lot of work on the scatter of fatigue crack growth [2ndash4] Inaddition some stochastic fatigue crack growth models havebeen proposed [5 6]

The experimental results used to characterize the modelparameters can be regarded as the prior information andit has an associated prior probability distribution functionThe in-service inspection data can be used as experimentalobservation which conjuncts the prior distribution of themodel parameters to reduce the uncertainty in the predictionof fatigue crack growth curve for one component [7] Theposterior distribution can be used to make updated estimatesof model parameter for the crack growth behavior

The combination of prior distribution of fatigue crackgrowth model parameters with experimental observationscan be carried out by applying the concepts of Bayesianupdating The results from updated distribution functionare also termed as posterior distribution The Markov chainMonte Carlo (MCMC) is one of the common probabilitysimulation methods through the Bayesian updating [8] Andthe Bayesian updating can be accomplished by OpenBUGSpackage

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 9409101 12 pageshttpsdoiorg10115520179409101

2 Mathematical Problems in Engineering

Table 1 Chemical compositions of 2024-T62 ()

Cu Mg Mn Si Fe Zn Ti Cr Other Al464 149 068 lt05 lt05 lt025 lt015 lt01 lt015 Balance

Table 2 Mechanical performance of 2024-T62 ()

EGPa 120590119887MPa 12059002MPa 120577571 451 400 72

In this study fatigue crack growth tests of 2024-T62aluminum alloy were conducted under the constant ampli-tude loading The prior distribution of model parameterswas obtained based on experimental a-N data Afterwards anew fatigue crack growth prediction approach was proposedbased on the prior distribution in-service observation andBayesian updating method Finally the proposed approachis validated by the additional fatigue crack growth under thesame test conditions

2 Fatigue Crack Propagation Test

The2024-T62 aluminum alloy is widely used as themain loadstructure in Chinese aviation industry

21 Material and Specimen Chemical compositions andmechanical performance of 2024-T62 aluminum alloy areshown in Tables 1 and 2 respectively

With a hole in the center throughout whole specimenit has been machined from the 2024-T62 aluminum alloypipematerial along the L-T directionThedimensional lengthof specimen is shown in Figure 1 the thickness is 25mmand the diameter of center hole is 4mm The specimen wasdesigned and conducted according to the ASTM standardE647 requirements [9]

22 Experimental Procedure All the tests were carried outwith the loading frequency of 20Hz using a servo hydraulicuniversal dynamical test machine (MTS-810-500KN) All thetests were conducted under the room temperature (20∘C) Inthe test themeasurements of the crack length use theQuestarQM1 long working distancemicroscope (limit of resolution is27 120583m)

When the crack approximately propagates to 05mmstart to count the number of loading cycle At the same timethe crack length was measured by per 9000 loading cycles Itis worth noting that 05mm is called initial crack length

23 Experimental Results The experimental fatigue crackpropagation length from number 1 to number 4 is listed inTable 3 It is well known that the crack growth propagationregion can be divided to three regions near-threshold (Δ119870th)region stable crack propagation region and unstable (fast)crack propagation region [10] In order to obtain more stabledata the stable crack propagation region was chosen (cracklength 05mmsim6mm)

80

825

165

330

36

16

16R328

05 A

L-T

Figure 1 Geometry length of specimen

Cycle (N)

Number 1Number 2

Number 3Number 4

00051015202530354045505560

Crac

k siz

e (m

m)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Figure 2 Experimental fatigue crack growth a-N curve

The experimental fatigue crack propagation a-N curvesfor four specimens are shown in Figure 2

As shown in Figure 2 at a given cycles crack lengthhas a wide range of scatter even under the same loadingcondition and test condition The scatter of crack growthmay be mainly attributed to the inhomogeneous materialproperties Therefore it is necessary to investigate the scatterof fatigue crack growth behavior

3 Statistical Analyses of Crack Growth Data

The scatter of 2024-T62 aluminum alloy is analyzed by sta-tistical analysis method And crack growth behavior of 2024-T62 aluminum alloy is described by mathematical model

31 Scatter Analysis of Crack Growth Behavior The meanvalue 119909 standard deviation 120590 and coefficients of variation(COV) of specimens sample are common statistical param-eters to describe the variation of statistics Mean value is usedto describe the centralized location of sample data but 120590 andCOV are used to describe the scatter

Mathematical Problems in Engineering 3

Table 3 Crack lengths for different cycles (mm)

Specimen Crack length0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000

No 1 05 062 08 102 127 156 189 23 268 305 369No 2 051 08 111 147 191 242 29 348 408 47 557No 3 054 076 103 141 178 223 269 323 374 423 527No 4 048 077 102 128 154 188 216 253 295 333 399

Table 4 Statistics variables of crack lengths for different cycles

Statistics variables Cycles9000 18000 27000 36000 45000 54000 63000 72000 81000 90000120583 07375 099 1295 1625 20225 241 2885 33625 38275 463120590 00870 01443 02168 03060 04134 05058 06080 07126 08350 10078

COV 01087 01343 01542 01735 01883 01934 01942 01953 02010 02005

00

02

04

06

08

10

Cycle (N)

of

crac

k siz

e

of crack size

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

(a)

010

012

014

016

018

020C

ov o

f cra

ck si

ze

Cov of crack sizeCycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

1times105

9times104

(b)

Figure 3 Change trend of statistical variables (a) standard deviation 120590 and (b) coefficients of variation COV

Mean value and standard deviation of observation1199091 1199092 119909119899 can be expressed as

119909 = 1119899 (1199091 + 1199092 + sdot sdot sdot + 119909119899)120590 = radicsum119899119894=1 1199091198942 minus 1198991199092119899 minus 1

(1)

The coefficients of variationCOV (simply asC) are shownas

119862 = 119904119909 (2)

The statistics 120583 120590 and COV were calculated based onexperimental results (in Table 4) The statistical calculatingformulas ((1) and (2)) are listed in Table 4

Figure 3 gives the trend of standard deviation and COVof crack length (under same loading cycles)

As listed in Table 4 and depicted in Figure 3 standarddeviation 120590 and coefficients of variation COV increase grad-ually with loading cycles increase In a word the scatter ofcrack growth process increases gradually

32 Statistics of Fatigue Crack Growth Behavior The fatiguecrack growth (FCG) is modelled using Parisrsquo law [11] Parisrsquolaw gives the crack growth rate per stress cycle as a functionof the range of stress intensity factor and material constant

119889119886119889119873 = 119862 (Δ119870)119898 (3)

where a is the crack length (mm) N is number of loadingcycles 119889119886119889119873 is the crack growth rate per loading cycledue to fatigue Δ119870 is the range of stress intensity factor (inMparadicm) and C and m are constant material propertiesobtained by testing standard specimensmade of thismaterial


Table 5 Estimation value of model parameters

Specimen Parameter estimate value1205791 1205792Number 1 24312 minus02598Number 2 37303 minus05435Number 3 32953 minus04592Number 4 30991 minus07331

0

00051015202530354045505560

Crac

k siz

e (m

m)

Number 1 testNumber 1 fittingNumber 2 testNumber 2 fitting


Cycles (N)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Figure 4 Fitting curves versus experimental results

The crack length after N number of loading cycles is119886(119873)The crack growth rate dadN determines the true cracklength 119886(119873) according to Parisrsquo law (3) which has the formof 1205791119896(119886)1205792+1 for some function 119896(119886) When 119896(119886) = a thesolution for 119886(119873) can be expressed by [12]

119886 (119873) = 1198860(1 minus 1198860120579212057911205792119873)11205792 (4)

where a0 is the initial crack length (in mm) 1205791 and 1205792 are theparameters of crack growth model for 2024-T62 aluminumalloy under constant amplitude loading So (4) is an empiricalanalytical solution form for Parisrsquo crack growth law

Firstly it is necessary to validate the feasibility of using(4) which can describe the crack growth behavior of 2024-T62 aluminum alloy The estimation value of 1205791 and 1205792 canbe obtained based on crack growth experimental a-N databy maximum likelihoodmethod (MLE)The values of modelparameters 1205791 and 1205792 for four specimens are listed in Table 5

The fatigue crack growth curve can be obtained bysubstituting the estimation value of model parameters 1205791 and1205792 to (4) And the obtained curve is the fitting curve of crackgrowth which is shown in Figure 4

0

minus7minus6minus5minus4minus3minus2minus1

012345

Cycles (N)Number 1Number 2

Number 3Number 4

Erro

r (

)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105

Figure 5 Error between experimental data and fitting curves

It can be seen from Figure 4 that the fitting curves (basedon (4)) are in good agreement with the experimental resultswhich can validate the feasibility of (4)

Relative error (crack growth fitting curves to experimen-tal data) of four specimens is listed in Figure 5 and Table 6

As listed in Table 6 and shown in Figure 5 the relativeerrors between the fitting curves and the experimental resultswere less than 7 Itmeans that the approach (include (4) andparameter estimating method) could be satisfactorily used inengineering As a result (4) is a feasible and accurate modelto describe the crack growth behavior of 2024-T62 aluminumalloy under constant amplitude loads

4 Prediction Method of Fatigue Crack Growth

In aircraft structural engineering the common use of themedia crack growth a-N experimental results represents thefatigue crack propagation behavior which cannot considerthe scatter and the distinguish of different specimens

41 Prior Distribution As we all know crack length 119886(119873) isa nonlinear function of loading cycles numbers N By takingnatural logarithms e in (4) we can obtain the transformedreal crack length ln[119886(119873)1198860] as

ln [119886 (119873)1198860 ] = minus 11205792 ln (1 minus 1198860120579212057911205792119873) (5)

For the experimental observed transformed crack lengthY it includes the measure error 120576 And every 119884119894119895 can beexpressed as

119884119894119895 = minus 11205792119894 ln (1 minus 1198860119894120579211989412057911198941205792119894119873119894119895) + 120576119894119895 (6)

where subscripts 119894 and 119895 represent sequence number ofspecimen and sequence number of experimental observa-tions point respectively So 119884119894119895 is the observed transformed


Table 6 Parameter Estimate of fatigue crack propagation curve

Specimen Error ()9000 18000 27000 36000 45000 54000 63000 72000 81000 90000

Number 1 391 215 022 minus053 minus104 minus116 minus260 minus057 314 minus004Number 2 minus395 minus194 minus079 minus172 minus316 minus149 minus184 minus146 minus065 minus372Number 3 minus553 minus495 minus891 minus815 minus890 minus799 minus816 minus625 minus319 minus1018Number 4 minus220 113 404 722 576 839 708 478 468 minus243

crack length for jth observation point of ith specimen And1198860119894 is the initial crack length of ith specimen 120576119894119895 is theobservation error for jth observation point of ith specimenwhich is subjected to normal distribution (119873(0 1205902)) Anderror 120576119894119895 for different specimens and different observations isindependent identically distribution (iid)

The scatter of fatigue crack growth behavior in the modelparameter is modelled by using a prior probability densityfunction (PDF) And the assumption model parameters997888120579 119894 = (1205791119894 1205792119894) subjected to multinormal distribution can beexpressed as

997888120579 119894 sim multi-Normal (997888120583 997888Σ) (7)

Likelihood of experimental transformed crack lengthincludes the PDF of 119910119894119895 and the PDF of model matrix

parameter997888120579 119894 = (1205791119894 1205792119894)

119910119894119895 sim Normal [minus( 11205792119894) ln (1 minus 1198860119894120579211989412057911198941205792119894119873119894119895) 1205902] (8)

In case of completely lack prior information the diffusedistribution can be used to describe the prior informationthat can also be called noninformative prior distribution [13]Assumption matrix parameter 997888120583 subjected to multinormaldistribution can be expressed as

997888120583 sim multi-Noraml (997888120583 997888Σ1205830) 997888120583 0 = (00) 997888Σ1205830 = (1000 00 1000)

(9)

Assumption parameter 997888Σ subjected to inverse Wishartdistribution can be expressed as

997888Σ sim InverseWishart (997888Σ 0 2) 997888Σ 0 = (10 00 10) (10)

And assumption parameter 1205902 subjected to inverseGamma distribution is expressed as

1205902 sim InverseGamma (3 0001) (11)

1205831 1205832 Σ11 Σ12 Σ21 Σ22 and 120590 are the parameters offatigue crack growth model The posterior PDF of the modelparameter is obtained using Bayesian updating implemented

using the MCMC algorithms MCMC algorithms are ageneral class of computational methods used to producesamples from posterior samples They are often easy toimplement and at least in principle can be used to simulatefrom very high-dimensional posterior distributions [12] Andthey have been successfully applied to literally thousandsof applications Metropolis-Hastings algorithms and Gibbssamplers are two general categories of MCMC simulation

Briefly OpenBUGS is a software package for Bayesiananalysis of complex statistical models with the Gibbs sam-pler which is a based implementation of BUGS (BayesianInference Using Gibbs Sample) The package contains flex-ible software for analyzing complex statistical models usingMCMC methods For more information about backgroundapplication and introduction to OpenBUGS refer to Amiriand Modarres [14] and Spiegelhalter et al [15] and CowlesThe model parameters statistics estimated from tested data(in Table 3) and above method are listed in Table 7

Oneway to assess the accuracy of the posterior estimationis by calculating the Monte Carlo error (MC-error) for eachparameter This is an estimate for the difference betweenthe mean of the sampled values (which we are using as ourestimate of the posterior mean for each parameter) and thetrue posterior mean

As a rule of thumb the simulation should be run untilthe Monte Carlo error for each parameter of interest is lessthan about 5 of the sample standard deviation As listedin Table 7 the MC-error for each parameter is all less than5Therefore we can know that the posterior distribution ofmodel parameter is accurate

42 Convergence Diagnose Convergence diagnosis is animportant work in the using of MCMC algorithm If theconvergence feature of chains is poor the posterior distribu-tions from chains based on MCMC are not accurate Figurediagnosis method checking throughout mean method andcontrasting deviation method are common methods to diag-nose the convergence And contrasting deviationmethod hasa wide use in the convergence diagnosis analysis

Based on the contrasting deviation method proposed byGelman and Rubin [16] a more brief and convenient methodwas established by Brooks and Gelman [17] The basic idea isto generate multiple chains starting at overdispersed initialvalues and assess convergence by comparing within-chainand between-chain variability over the second half of thosechains We denote the number of chains generated by L andthe length of each chain by 2T We take as a measure ofposterior variability the width of the 100(1 minus 120572) credible



Parameter Mean Standard deviation MC-error () Different percentile0025 005 05 095 09751205831 31390 03344 009 24860 2638 31390 3641 379401205832 minus04979 01209 004 minus07369 minus06819 minus04977 minus03163 minus02605Σ11 04434 11000 037 00787 009277 02627 1245 18240Σ12 minus00881 03623 012 minus04487 minus0301 minus00468 00212 00507Σ21 minus00881 03623 012 minus04487 minus0301 minus00468 00212 00507Σ22 00581 01555 006 00095 001134 00341 01653 02420120590 00251 00030 001 00201 002073 00248 003045 00318

interval for the parameter of interest (in OpenBUGS 120572 =02) From the final T iterations we calculate the empiricalcredible interval for each chainWe then calculate the averagewidth of the intervals across the L chains and denote this byL Finally we calculate the width l of the empirical credibleinterval based on all LT samples pooled together The ratio = 119871119897 of pooled to average interval widths should be greaterthan 1 if the starting values are suitably overdispersed it willalso tend to 1 as convergence is approached and so we mightassume convergence for practical purposes if lt 105

Rather than calculating a single value of we canexamine the behavior of over iteration-time by performingthe above procedure repeatedly for an increasingly largefraction of the total iteration range ending with all of thefinal T iterations contributing to the calculation as describedabove

In the OpenBUGS software the bgr diagram can achieveabove approach OpenBUGS automatically chooses the num-ber of iterations between the ends of successive rangesmax (100 2119879100) It then plots in red 119871 (pooled) in greenand 119897 (average) in blue As shown in Figure 6 the red linerepresents the green line represents 119871 and 119897 correspondsto blue line

In this study three chains for each parameter were gen-erated by OpenBUGS software The convergence diagnosisschematic diagrams for seven model parameters were shownin Figure 6

As shown in Figure 6 the convergence feature of modelparameters is good and the trend of model parameter withiteration increase is convergent In other words the posteriordistributions of seven parameters are accurate and credible

43 Prediction Model of Fatigue Crack Growth The predic-tion of crack growth life (period) is a main thesis in theaircraft structural engineering In this study it is aimed toaccurately predict the fatigue crack growth behavior (curve)for one target specimen using least information

The Bayesian theorem can be written in model updatingcontext as

119875(997888120579 | 119863119872) prop 119875(119863 | 997888120579 119872)119875 (997888120579 | 119872) (12)

where 119875(997888120579 | 119863119872) corresponds to the PDF for the crackgrowth model M after updating with the crack growth test

observations D it is called the PDF of posterior distribution119875(997888120579 | 119872) is the PDF of model parameters997888120579 for the crack

growthmodelM before updating it is called the PDF of priordistributionAnd119875(119863 | 997888120579 119872) is the likelihood of occurrenceof the crack growth observation D given the vector of modelparameter

997888120579 and crack modelMThe fatigue crack growth experimental results can be

regarded as the prior information And the parameter priordistribution of crack growth model (4) can be obtained byabove approach According to the analysis and inference inthe Section 41 crack growth model parameters

997888120579 = (1205791 1205792)are subjected to multinormal (997888120583 997888Σ)

997888120583 = (31390 minus04979)997888Σ = ( 04434 minus00881minus00881 00581 )

(13)

Therefore the posterior likelihood function of modelparameters for one target specimen can be calculated as

119875(997888120579 | 119863119872)prop 119898prod119911=1

exp(minus12 (119910119911 minus (11205792) ln (1 minus 1198860120579212057911205792119873119911)120590 )2)

sdot 119867(997888120579) (14)

where m is the number of observation point used to updatethe posterior distribution of model parameters And if thefirst test data point was used to update the posterior distri-bution of parameters 1205791 and 1205792119898 = 1

The posterior distribution can be obtained via usingMarkov chain Monte Carlo (MCMC) simulation with thecombination of the likelihood function (14) with prior dis-tribution (13)

If the model parameters 1205791 and 1205792 for a specimen aredetermined the fatigue crack propagation curve of thisspecimen can be obtained by (4) As a result the residualgrowth life (period) of this specimen can be determined


Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

Figure 6 The bgr diagram of convergence diagnosis for fatigue crack growth model parameters (a) 1205791 (b) 1205792 (c) Σ11 (d) Σ12 (e) Σ21 (f)Σ22 and (g) 120590

5 Experimental Validations

This section presents the validation of the proposed method-ology via using crack propagation data from additional crackgrowth test

51 Additional Crack Growth Test Additional crack growthtest employs the same experimental method environmental

condition and measurement technology as the test in Sec-tion 2 Similarly when the crack approximately propagatesto 05mm start counting the number of loading cycles Inother words the crack length is 05mm when the number ofloading cycles is zero

The specific initial crack length and the crack lengthafter 9000 cycles (number 5 number 6 and number 7target specimen) are listed in Table 8 It is noted that the


Table 8 Crack lengths of initialization and first test point (mm)

Specimen Initial crack length Crack length after 9000 cyclesNumber 5 05 066Number 6 053 074Number 7 05 069

experimental data (in Table 8) can represent the in-serviceobservation during the service for aircraft structures

52 Prediction of Crack Growth The prediction of crackgrowth curve (number 5 specimen) combines different testdata information (different data sample length and differenttest data) Mark different information source as Case I CaseII and Case III A particular introduction of Case IndashCase IIIis presented as follows

For Case I the prior distribution of matrix parameter997888120579 = (1205791 1205792) is the only information source for the predictionof crack growth curve So the posterior distribution mean ofparameter

997888120579 = (1205791 1205792) is (31390 minus04979)For Case II the information source includes prior dis-

tribution of matrix parameter997888120579 = (1205791 1205792) and the first

observation data point (N = 9000 119886(119873) = 066mm) Andthe predictionmodel of Case II is the proposedmodel whichcorresponds to the Section 43 Based on (14) the likelihoodof Case II is expressed as

119875(997888120579 | 119863119872)prop exp(minus12 (11991051 minus (ln (1 minus 11988650

1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)

Based on the above likelihood function of posteriordistribution and the OpenBUGS package the mean valueof posterior distribution of matrix parameters

997888120579 = (1205791 1205792)can be obtained Through the MC-error control and theconvergence diagnose of the posterior distribution of thematrix parameter

997888120579 the mean value is (26035 minus03172)For Case III the first observation data (119873 = 9000 119886(119873) =066mm) is the only information source for the prediction of

crack growth curveThe posterior distribution mean value of997888120579 = (1205791 1205792) is (08838 minus24941) which can be estimated byusing the method of Section 32

For number 6 and number 7 specimen they are alsodeviated to Case I Case II and Case III The posteriordistribution mean values of matrix parameters

997888120579 = (1205791 1205792)for the additional three target specimens are listed in Table 10

53 Additional Experimental Observation The additionalfatigue crack growth tests to 90000 loading cycles wereconducted The crack growth lengths of number 5 number6 and number 7 specimens are listed in Table 9 (include thedata of Table 8)

00051015202530354045505560

Experimental resultsCase ImdashPrior distribution with no test data Case IImdashPrior distribution with first test point Case IIImdashFirst test point (no prior information) Case IVmdashTen test points (no prior information)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)

Figure 7 Crack propagation prediction curves of number 5 speci-men

The parameters are fitted with the same method ofSection 32 And the information source is called Case IVTheposterior distribution of fitting parameters (1205791 1205792) is listed inTable 10

The model parameter posterior distribution mean valuesof Case IndashCase IV for number 5 number 6 and number 7specimens are listed in Table 10

The fatigue crack growth curve can be obtained bysubstituting the parameter of posterior distribution meanvalue to crack growth model (4) In this paper for onetarget specimen we can obtain four fatigue crack growthcurves corresponding to the above Case I Case II Case IIIand Case IV respectively For example four fatigue crackgrowth curves and experimental a-N observations of number5 specimen are shown in Figure 7

As shown in Figure 7 the fatigue crack growth curve ofCase III is far away from the experimental a-N observationsAnd the absolute error between fatigue crack growth curveof Case II and experimental a-N observations is smaller thanthat of Case III Compare Case II with Case III we can obtainthe prior information that is very important for the predictionof fatigue crack growth behavior

The fatigue crack growth curves for number 6 andnumber 7 specimens are obtained by the same method andanalysis steps which are shown in Figures 8 and 9 And theexperimental results of number 6 and number 7 specimen areshown in Figures 8 and 9 respectively

As shown in Figures 7 8 and 9 we can know that theprediction of fatigue crack growth of Case III is far awayfrom the experimental a-N observations so the error cannotsatisfy the requirements of aircraft engineering



Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000

Number 5 05 066 092 11 139 174 21 245 288 347 425Number 6 055 077 102 12 151 2 249 296 359 43 541Number 7 05 070 093 116 153 196 242 274 33 412 484

00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105

Figure 8 Crack propagation curves prediction of number 6 speci-men

Comparative analysis of the prediction for the fatiguecrack growth of Figures 7 8 and 9 is based on the crackgrowth model parameters prior distribution and observationdata (Case II) andwe can obtain the crack growth curve accu-ratelyTherefore the predictionmethod (Case II) proposed inthis study is an advanced technology

54 Error Analysis Due to the high error of Case III it willnot be considered in the following analysis The absoluteerrors (prediction value to experimental observations value)of Case I Case II and Case IV are analyzed in the followingsteps And the absolute error is defined as

absolute error

= 100381610038161003816100381610038161003816100381610038161003816(pridiction value) minus (experiment value)

experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)

Figure 10 shows the prediction crack life (Figure 7)versus the experimental results (Table 9) of number 5 targetspecimen

Experimental resultsCase ImdashPrior distribution with no test data Case IImdashPrior distribution with first test point Case IIImdashFirst test point (no prior information)Case IVmdashTen test points (no prior information)

00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18

Case I Case II Case IV

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Figure 10 Prediction error of crack propagation curve for number5 specimen


Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2


As shown in Figure 10 the absolute error of Case IVis the smallest for the three cases In other words theprediction curve (life) of Case III is in good agreementwith experimental observations Based on the absolute errorcondition of Case I it can be known that using the average119886-119873 curve of number 1 number 2 number 3 and number4 specimens to represent crack growth curve of number 5specimen may cause bigger error

Crack life prediction (Figures 8 and 9) versus theexperimental results (Table 8) for number 6 and number 7specimens is shown in Figures 11 and 12 respectively

As shown in Figures 10 11 and 12 the absolute error ofCase II can satisfy the aircraft engineering accurate needs

It is worth noting that Case I represents the mean ofexperimental results In some time the prediction of Case Imay be more concise than Case II but it will not happen allthe time In other words it is a special circumstance and it isdetermined by the stochastic and the scatter of fatigue crackgrowth behavior For instance the prediction fatigue crackgrowth curve of Case I (number 7 specimen in Figure 12) isin good agreement with the experimental results

6 Conclusion and Discussion

The paper presents a method of fatigue crack propagation byusing the crack test information and in-service inspectioninformation The fatigue crack growth curve of 2024-T62aluminum alloy is predicted by Bayesian theory with MCMCalgorithm and OpenBUGS package Based on the currentinvestigation four conclusions are drawn(1) The introduced model (4) can describe the fatiguecrack growth behavior of 2024-T62 aluminum alloy accu-rately

0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1



(2) The scatter of crack length (under same loadingcycles) gradually increases with crack propagation process(3) Prior information of crack growthmodel parameter isan important information source to predict the crack growthbehavior And the MCMC is a good method to resolve thestatistics estimation issue of multiparameter(4) The proposed approach provides an insight intofatigue crack growth curve prediction and associates theMCMC with OpenBUGS package The predicted fatiguecrack growth curve and residual life are in good agreementwith the experimental observations And the performance ofproposed approach shows the superior results to the meanvalue of a-N data

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Project no 51505495) The authorsgratefully appreciate the support

References

[1] W F Wu and C C Ni ldquoProbabilistic models of fatigue crackpropagation and their experimental verificationrdquo ProbabilisticEngineering Mechanics vol 19 no 3 pp 247ndash257 2004

[2] X L Zou ldquoStatistical moments of fatigue crack growth underrandom loadingrdquo Theoretical and Applied Fracture Mechanicsvol 39 no 1 pp 1ndash5 2003


[3] W Li T Sakai Q Li and PWang ldquoStatistical analysis of fatiguecrack growth behavior for grade B cast steelrdquo Materials andDesign vol 32 no 3 pp 1262ndash1272 2011

[4] H Y Liou C C Ni and W F Wu ldquoScatter and statisticalanalysis of fatigue crack growth datardquo Journal of Chinese Societyof Mechanical Engineers vol 22 no 5 pp 399ndash407 2001

[5] J-K KimandD-S Shim ldquoVariation in fatigue crack growth dueto the thickness effectrdquo International Journal of Fatigue vol 22no 7 pp 611ndash618 2000

[6] W F Wu and C C Ni ldquoA study of stochastic fatigue crackgrowth modeling through experimental datardquo ProbabilisticEngineering Mechanics vol 18 no 2 pp 107ndash118 2003

[7] R Rastogi S Ghosh A K Ghosh K K Vaze and P KSingh ldquoFatigue crack growth prediction in nuclear piping usingMarkov chain Monte Carlo simulationrdquo Fatigue and Fracture ofEngineering Materials and Structures vol 40 no 1 pp 145ndash1562017

[8] WRGilks S Richardson andD J SpiegelhalterMarkovChainMonte Carlo in Practice Interdisciplinary Statistics ChapmanampHallCRC London UK 1996

[9] Standard A S T M ldquoE647 Standard test method for mea-surement of fatigue crack growth ratesrdquo Annual Book of ASTMStandards Section Three Metals Test Methods and AnalyticalProcedures vol 3 pp 628ndash670 2002

[10] N E Dowling ldquoMechanical behavior of materials engineeringmethods for deformationrdquo Fracture and Fatigue Pearson 2012

[11] N Pugno M Ciavarella P Cornetti and A Carpinteri ldquoAgeneralized Parisrsquo law for fatigue crack growthrdquo Journal of theMechanics and Physics of Solids vol 54 no 7 pp 1333ndash13492006

[12] M S Hamada A Wilson C S Reese and H Martz BayesianReliability Springer Science amp BusinessMedia NewYork 2008

[13] K Mosegaard and A Tarantola ldquo16 Probabilistic approach toinverse problemsrdquo International Geophysics vol 81 pp 237ndash2652002

[14] M Amiri and M Modarres ldquoShort fatigue crack initiation andgrowth modeling in aluminum 7075-T6rdquo Journal of MechanicalEngineering Science vol 229 no 7 pp 1206ndash1214 2015

[15] D Spiegelhalter A Thomas N Best et al OpenBUGS usermanual version 302 MRCBiostatistics Unit Cambridge 2007

[16] A G Gelman and D B Rubin ldquoInference from iterativesimulation using multiple sequencesrdquo Statistical Science vol 7no 4 pp 457ndash472 1992

[17] S P Brooks and A Gelman ldquoGeneral methods for monitoringconvergence of iterative simulationsrdquo Journal of Computationaland Graphical Statistics vol 7 no 4 pp 434ndash455 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Chemical compositions of 2024-T62 ()

Cu Mg Mn Si Fe Zn Ti Cr Other Al464 149 068 lt05 lt05 lt025 lt015 lt01 lt015 Balance

Table 2 Mechanical performance of 2024-T62 ()

EGPa 120590119887MPa 12059002MPa 120577571 451 400 72

In this study fatigue crack growth tests of 2024-T62aluminum alloy were conducted under the constant ampli-tude loading The prior distribution of model parameterswas obtained based on experimental a-N data Afterwards anew fatigue crack growth prediction approach was proposedbased on the prior distribution in-service observation andBayesian updating method Finally the proposed approachis validated by the additional fatigue crack growth under thesame test conditions

2 Fatigue Crack Propagation Test

The2024-T62 aluminum alloy is widely used as themain loadstructure in Chinese aviation industry

21 Material and Specimen Chemical compositions andmechanical performance of 2024-T62 aluminum alloy areshown in Tables 1 and 2 respectively

With a hole in the center throughout whole specimenit has been machined from the 2024-T62 aluminum alloypipematerial along the L-T directionThedimensional lengthof specimen is shown in Figure 1 the thickness is 25mmand the diameter of center hole is 4mm The specimen wasdesigned and conducted according to the ASTM standardE647 requirements [9]

22 Experimental Procedure All the tests were carried outwith the loading frequency of 20Hz using a servo hydraulicuniversal dynamical test machine (MTS-810-500KN) All thetests were conducted under the room temperature (20∘C) Inthe test themeasurements of the crack length use theQuestarQM1 long working distancemicroscope (limit of resolution is27 120583m)

When the crack approximately propagates to 05mmstart to count the number of loading cycle At the same timethe crack length was measured by per 9000 loading cycles Itis worth noting that 05mm is called initial crack length

23 Experimental Results The experimental fatigue crackpropagation length from number 1 to number 4 is listed inTable 3 It is well known that the crack growth propagationregion can be divided to three regions near-threshold (Δ119870th)region stable crack propagation region and unstable (fast)crack propagation region [10] In order to obtain more stabledata the stable crack propagation region was chosen (cracklength 05mmsim6mm)

80

825

165

330

36

16

16R328

05 A

L-T

Figure 1 Geometry length of specimen

Cycle (N)

Number 1Number 2

Number 3Number 4

00051015202530354045505560

Crac

k siz

e (m

m)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Figure 2 Experimental fatigue crack growth a-N curve

The experimental fatigue crack propagation a-N curvesfor four specimens are shown in Figure 2

As shown in Figure 2 at a given cycles crack lengthhas a wide range of scatter even under the same loadingcondition and test condition The scatter of crack growthmay be mainly attributed to the inhomogeneous materialproperties Therefore it is necessary to investigate the scatterof fatigue crack growth behavior

3 Statistical Analyses of Crack Growth Data

The scatter of 2024-T62 aluminum alloy is analyzed by sta-tistical analysis method And crack growth behavior of 2024-T62 aluminum alloy is described by mathematical model

31 Scatter Analysis of Crack Growth Behavior The meanvalue 119909 standard deviation 120590 and coefficients of variation(COV) of specimens sample are common statistical param-eters to describe the variation of statistics Mean value is usedto describe the centralized location of sample data but 120590 andCOV are used to describe the scatter




No 1 05 062 08 102 127 156 189 23 268 305 369No 2 051 08 111 147 191 242 29 348 408 47 557No 3 054 076 103 141 178 223 269 323 374 423 527No 4 048 077 102 128 154 188 216 253 295 333 399



COV 01087 01343 01542 01735 01883 01934 01942 01953 02010 02005

00

02

04

06

08

10

Cycle (N)

of

crac

k siz

e

of crack size

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

(a)

010

012

014

016

018

020C

ov o

f cra

ck si

ze


0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

1times105

9times104

(b)




(1)


119862 = 119904119909 (2)





119889119886119889119873 = 119862 (Δ119870)119898 (3)





0

00051015202530354045505560

Crac

k siz

e (m

m)



Cycles (N)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



119886 (119873) = 1198860(1 minus 1198860120579212057911205792119873)11205792 (4)




0


012345


Number 3Number 4

Erro

r (

)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105








ln [119886 (119873)1198860 ] = minus 11205792 ln (1 minus 1198860120579212057911205792119873) (5)


119884119894119895 = minus 11205792119894 ln (1 minus 1198860119894120579211989412057911198941205792119894119873119894119895) + 120576119894119895 (6)




Specimen Error ()9000 18000 27000 36000 45000 54000 63000 72000 81000 90000






parameter997888120579 119894 = (1205791119894 1205792119894)




(9)






















119875(997888120579 | 119863119872) prop 119875(119863 | 997888120579 119872)119875 (997888120579 | 119872) (12)








(13)


119875(997888120579 | 119863119872)prop 119898prod119911=1


sdot 119867(997888120579) (14)





Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

















1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







No 1 05 062 08 102 127 156 189 23 268 305 369No 2 051 08 111 147 191 242 29 348 408 47 557No 3 054 076 103 141 178 223 269 323 374 423 527No 4 048 077 102 128 154 188 216 253 295 333 399



COV 01087 01343 01542 01735 01883 01934 01942 01953 02010 02005

00

02

04

06

08

10

Cycle (N)

of

crac

k siz

e

of crack size

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

(a)

010

012

014

016

018

020C

ov o

f cra

ck si

ze


0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

1times105

9times104

(b)




(1)


119862 = 119904119909 (2)





119889119886119889119873 = 119862 (Δ119870)119898 (3)





0

00051015202530354045505560

Crac

k siz

e (m

m)



Cycles (N)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



119886 (119873) = 1198860(1 minus 1198860120579212057911205792119873)11205792 (4)




0


012345


Number 3Number 4

Erro

r (

)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105








ln [119886 (119873)1198860 ] = minus 11205792 ln (1 minus 1198860120579212057911205792119873) (5)


119884119894119895 = minus 11205792119894 ln (1 minus 1198860119894120579211989412057911198941205792119894119873119894119895) + 120576119894119895 (6)




Specimen Error ()9000 18000 27000 36000 45000 54000 63000 72000 81000 90000






parameter997888120579 119894 = (1205791119894 1205792119894)




(9)






















119875(997888120579 | 119863119872) prop 119875(119863 | 997888120579 119872)119875 (997888120579 | 119872) (12)








(13)


119875(997888120579 | 119863119872)prop 119898prod119911=1


sdot 119867(997888120579) (14)





Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

















1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0

00051015202530354045505560

Crac

k siz

e (m

m)



Cycles (N)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



119886 (119873) = 1198860(1 minus 1198860120579212057911205792119873)11205792 (4)




0


012345


Number 3Number 4

Erro

r (

)

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105








ln [119886 (119873)1198860 ] = minus 11205792 ln (1 minus 1198860120579212057911205792119873) (5)


119884119894119895 = minus 11205792119894 ln (1 minus 1198860119894120579211989412057911198941205792119894119873119894119895) + 120576119894119895 (6)




Specimen Error ()9000 18000 27000 36000 45000 54000 63000 72000 81000 90000






parameter997888120579 119894 = (1205791119894 1205792119894)




(9)






















119875(997888120579 | 119863119872) prop 119875(119863 | 997888120579 119872)119875 (997888120579 | 119872) (12)








(13)


119875(997888120579 | 119863119872)prop 119898prod119911=1


sdot 119867(997888120579) (14)





Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

















1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Specimen Error ()9000 18000 27000 36000 45000 54000 63000 72000 81000 90000






parameter997888120579 119894 = (1205791119894 1205792119894)




(9)






















119875(997888120579 | 119863119872) prop 119875(119863 | 997888120579 119872)119875 (997888120579 | 119872) (12)








(13)


119875(997888120579 | 119863119872)prop 119898prod119911=1


sdot 119867(997888120579) (14)





Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

















1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














119875(997888120579 | 119863119872) prop 119875(119863 | 997888120579 119872)119875 (997888120579 | 119872) (12)








(13)


119875(997888120579 | 119863119872)prop 119898prod119911=1


sdot 119867(997888120579) (14)





Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

















1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Iteration2241 10000 20000

00

05

10

1 chains 1

(a)

00

05

10

2 chains 1

10000 200002241Iteration(b)

00

05

10

Iteration2241 10000 20000

Σ11 chains 1

(c)

00

05

10

Iteration2241 10000 20000

Σ12 chains 1

(d)

00

05

10

Iteration2241 10000 20000

Σ21 chains 1

(e)

00

05

10

Iteration2241 10000 20000

Σ22 chains 1

(f)

00

05

10

Iteration2241 10000 20000

chains 1

(g)

















1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














1205795212057951120579521198731) 12057952)120590 )2)sdot 119867(997888120579)

(15)








00051015202530354045505560


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

Crac

k siz

e (m

m)










Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Specimen Cycle0 9000 18000 27000 36000 45000 54000 63000 72000 81000 90000


00051015202530354045505560


Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

1times105




absolute error


experiment value

100381610038161003816100381610038161003816100381610038161003816times 100(16)



00051015202530354045505560

Crac

k siz

e (m

m)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104


0

2

4

6

8

10

12

14

16

18


Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104



Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Table10Param

eter

estim

ateo

ffatigue

crackprop

agationcurve

Specim

enBa

sedon

priord

istrib

utionandtestdata

Onlybasedon

testdata

Case

Ino

testdata

Case

IIfirsttestp

oint

Case

IIIon

lyfirsttestpo

int

CaseIV

tentestpo

int

120579 1120579 2

120579 1120579 2

120579 1120579 2

120579 1120579 2

Num

ber5

31390

minus04979

26035

minus03172

08838

minus24941

26938

minus03901

Num

ber6

31390

minus04979

30037

minus04595

23531

minus13182

29778

minus03122

Num

ber7

31390

minus04979

28857

minus04128

3119

9minus02

654

29766

minus03517


02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





02468

1012141618202224

Erro

r (

)


Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus2








0123456789

10111213

Erro

r (

)

Cycle (N)

0

1times104

2times104

3times104

4times104

5times104

6times104

7times104

8times104

9times104

minus1






Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

MCMC-Based Fatigue Crack Growth Prediction on 2024-T6 …downloads.hindawi.com/journals/mpe/2017/9409101.pdf · 2019. 7. 30. · 4 MathematicalProblemsinEngineering Table5:Estimationvalueofmodelparameters