Bayesian analysis of low cycle fatigue failure in printed wiring boards Case Studies in Engineering Failure Analysis 7 (2016) 65–70 Bayesian analysis of low cycle fatigue failure in printed wiring boa[.]
Trang 1Bayesian analysis of low-cycle fatigue failure in printed wiring
boards
a
Arizona State University, United States
b
Honeywell Inc., United States
A R T I C L E I N F O
Article history:
Received 28 July 2016
Received in revised form 28 October 2016
Accepted 7 November 2016
Available online 12 November 2016
Keywords:
Reliability
Circuit board
Thermal cycling test
Weibull regression
Bayesian analysis
A B S T R A C T
Inthisstudy,alow-cyclefatigueexperimentwasconductedonprintedwiringboards (PWB).TheWeibullregressionmodelandcomputationalBayesiananalysismethodwere appliedtoanalyzefailuretimedataandtoidentifyimportantfactorsthatinfluencethe PWB lifetime.Theanalysisshowsthat bothshapeparameterand scale parameterof WeibulldistributionareaffectedbythesupplierfactorandpreconditioningmethodsBased
ontheenergyequivalenceapproach,a6-cyclereflowpreconditioncanbereplacedbya 5-cycleISTprecondition,thusthetotaltestingtimecanbegreatlyreduced.Thisconclusion wasvalidatedbythelikelihoodratiotestoftwodatasetscollectedundertwodifferent preconditioning methods Therefore, the Weibull regression modeling approach is an effectiveapproachforaccountingforthevariationofexperimentalsettinginthePWB lifetimeprediction
ã2016TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCC
BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/)
1.Background
Acceleratedlifetesting(ALT)ofprintedwiringboards(PWB)isanessentialtoolforpredictingcircuitboardlifetimeinthe electronicindustry.A standardpracticeof usingALTsis tosimulatethermally inducedfailureorlow-cycle fatigueby subjectingacircuitboardcoupontoaprescribednumberofspecificthermalcyclesthatrepresentsin-serviceuseofthe product[1].Forexample,thestandardpracticeintheavionicindustryemploysinterconnectstresstest(IST)perIPC-TM-650
[2]withallcouponsinalotpassing350thermalcyclesastheacceptancetestcriteria.Inourexperiment,thetestcoupons weredrivenbeyondthenormaltestlimitsof350cyclesassuggestedin[3,4]toprecipitatefailuresandtostudydifferencesin preconditioningprocesses.Thegoalofthisstudyistri-folded:First,wedevelopanenergy-equivalentmodelforestablishing theISTsetup.Second,wecomparetheresultsfromcouponsfabricatedbyfoursuppliers.Lastly,thiscasestudydemonstrates theeffectivenessofusingWeibullregressionandcomputationalBayesiananalysistechniquesforelectroniccomponent failureanalysis
TheISTcouponsaremanufacturedalongthesideofacircuitboardprototypeandmultipleviabarrelsareproducedonit (seeFig.1(a)).Thefailuremodeofthedatapresentedinthispaperisthermallyinducedfatigueduetotheexpansionand contractionofviabarrels(aviaisthemechanismbywhichdifferentcircuitlayersareconnected).Theselowcyclefatigueson interconnectshavedrawnalotofattentionsfromacademicresearchersandindustrialpractitioners[5–7];however,mostof themdiscussedthefatiguesonleadorlead-freesolders,notonviabarrels.Fig.2(b)illustratesthefracturesinaviabarrel
* Corresponding author.
E-mail address: rong.pan@asu.edu (R Pan).
http://dx.doi.org/10.1016/j.csefa.2016.11.001
2213-2902/ã 2016 The Author(s) Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/
Case Studies in Engineering Failure Analysis 7 (2016) 65–70
ContentslistsavailableatScienceDirect
j o u r n a lh o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c s e fa
Trang 2Failureisdeterminedwhencouponresistanceisatgreaterthan10%resistancechangefromtheoriginalresistanceatthe initialcycleatthehighestpointofthetesttemperatureafterpreconditioning.Theresistanceincreasesbecausewhenacrack formslessmaterialislefttoconductcurrent
A preconditioning thermal cycle step is required beforethe designed thermal cycling test that simulates the life experienceofcircuitboard.Thispreconditioningstepistoaccountforthethermalstressduringthecircuitboard’ssoldering process.Twomodesofheattransfercanbeusedtoreproducetheproductionthermalstresstheresistiveheattransferas usedbyISTortheconvectionheattransferbyreflowoven.Althoughthelatteronecanmorerealisticallyrepresentthe manufacturingprocess,itdemandsinvaluablemanufacturingresource(reflowoven)andiscostlyandtimeconsuming.In contrast,ISTcanheattheinternalenvironmentoftestedcouponsbyresistiveheattransferinaveryshorttime.Therefore,it
isimportanttoknowwhatISTsettingscanbeusedtoreplacethereflowpreconditioning
2.Experiment
PWBcouponsofsize5”0.70.1”wereusedinthisexperiment.Eachcouponwasmadeof14layersofcircuitrywithan electricalcircuitdaisychain.Thesecouponscame fromfourdifferentsuppliersanda batchofsixcouponswastested togetheratonetime.Therewereatotalof134couponsbeingtested.Thefailuretimes(numberofthermalcycles)ofthese testcouponsaregiveninAppendixA.Ifacoupondidnotfail,itssurvivaltimeismarkedwith“+”
Atestcouponmayexperience5or6ISTpreconditioningcycles(IST5orIST6)or6reflowpreconditioningcycles(RFO6) Theexperimentalsettingsofthesepreconditioningmethodsaredescribedbelow:
UseISTtoheattestcouponsforthreeminutesuntilitreachesthemaximumtemperatureof230C,andthencoolthe couponintheroomtemperature(25C)environmentfortwominutes.ThismakesonecycletimefortheISTtesttobefive minutes.However,thisexperimentalsettingwasmodifiedforthecouponsfromonesupplier,inwhichthemaximum temperaturewasincreasedto240Cand245C
Fig 1 (a) A typical PWB coupon used in this study; (b) Failure mode is a cracked via barrel at arrow points due to thermal induced fatigue.
66 R Pan et al / Case Studies in Engineering Failure Analysis 7 (2016) 65–70
Trang 3Passtestcouponsthroughthereflowovenfor12minunderthetemperatureof250C.Thisway,couponsareheated directlyviaconvectionheattransfer.Thetestcouponthenstaysintheroomtemperatureenvironmentfor8mintocool down.Thus,onecycletimeforthereflowsystemis20minutes
3.Engineeringanalysis
PriortotheselectionoftheWeibulldistributionastheappropriatelifetimedistributionforthedata,allthedatasetswere fittedbyWeibull,normal,logistic,lognormalandloglogisticdistributions.Werankedthesedistributionsbytheir Anderson-Darlingstatistics.ItwasfoundthatbothWeibullandlognormaldistributionshavethebestgoodness-of-fit;however,the Weibulldistributionwaschosen,becauseatthehighestcycles-to-failuretheWeibulldistributiontendedtohaveabetterfit whenweexaminedtheindividualprobabilityplotofeachdataset
3.1.Weibullregressionmodel
Weibulldistributionhastwoparameters–theshapeparameterv>0andthescaleparameter(thecharacteristiclife)
h>0,anditsprobabilitydensityfunctionisgivenby
fðtÞ¼hv hx
v1
eðt=hÞ v
Accordingly,itscumulativefailuredistributionfunctionisgivenby
FðxÞ¼1eðt=hÞ v
ð2Þ andthereliabilityfunction
RðxÞ¼eð t =hÞ v
ð3Þ Theshapeparametervisofteninfluencedbythesupplierfactorandthepreconditioningmethod,becausetheyhavean impactonthematerialbeingtested.Thus,wemodeltheshapeparameterbythefollowinglinearfunction:
wheres1;s2ands3areindicatorvariablesforidentifyingsuppliersandrrepresentsthepreconditioningmethod.Whens1¼1 ands2¼s3¼0,thefirstsupplier’scouponisinuse.Similarly,thesecondandthirdsuppliersareidentifiedbys2¼1and
s3¼1,respectively,andthelastsupplierisidentifiedbys1¼s2¼s3¼0.ThereflowandISTpreconditioningmethodsare indicatedbyr¼1andr¼0,respectively.Usingthisregressionmodel,wecanpoolallavailabledataformodelparameter estimation
Forthescaleparameter,ourpreviousstudysuggeststhatitcanbeinfluencedbytheenergyabsorbedbythecoupon duringthepreconditioningstep[8].Aseachpreconditioningmethodhasdifferenttargetedtemperature,rampingtimeand cycletime,wecalculatetheirjouleequivalentenergyusingthefollowingequation:
Energy¼PCCDTRT
where PCC represents the number of preconditioning cycles, DT represents the temperature gap between ramping temperatureandcoolingdowntemperature,RTistherampingtime,andCTisthetotalcycletime.Accordingto[9],coupons reachsteady statetemperaturessofastthat itis reasonabletoassume thatthese couponsarealways atthereadout temperature.Basedontheinversepowerlaw,alog-linearmodelfortheWeibullcharacteristiclifeisgivenby
wherevariableedenotestheenergyabsorbedbycoupon
3.2.Bayesianinference
InordertointegratepriorknowledgeofWeibullparametersintoourdataanalysis,wechosetheBayesianinference method.AWeibullregressionanalysiswasconductedinWinBUGSenvironment[10]usingthefollowingmodel:
t½iweibullðv½i;l½iÞ
v½i ¼a þas ½i þa s½i þas ½i þa r½i
R Pan et al / Case Studies in Engineering Failure Analysis 7 (2016) 65–70 67
Trang 4l½i¼h½iv½ i
logh½i¼b0þb1s1½iþb2s2½iþb3s3½iþb4loge½iþb5r½i
whereirepresentsithtestdata,i2N;Nisthetotalnumberoftestdata
Ourpriorknowledgeabouttheseparameterswasderivedfromasimilartestconductedin[4].Intheirtest,therewere
54couponsundertheISTpreconditioningprocessforsixcyclesthathasthejouleequivalentenergyof738,andanother
58couponsunderreflowpreconditioningprocessforsixcycleswiththejouleequivalentenergyof615.Fittingtheirdata resultedintheequation,logh¼263loge.Therefore,wesetthepriordistribution,b4Nð3;1Þ.Otherpriordistributions arespecifiedasb0Nð20;1Þ,biNð0;1Þ,i¼1;2;3;5.Inaddition,thepriordistributionsfora0saresetasa0Nð2:5;1Þ,
aiNð0;1Þ,i¼1;2;3
MarkovchainMonteCarlo(MCMC)methodwasimplementedbytheGibbssampler,whichiterativelydrewsamplesofa parameterfromitscorrespondingconditionaldistributionmodel(see[11]forthedetailsofGibbssampler)
4.Numericalanalysis
TwoMarkovchainswithdifferentinitialvalueswererun,with100,000iterationsand10,000burn-initerationsforeach chain.Tovalidatethemodelandparametersettings,theGelman-Rubinconvergencediagnosiswasperformed.Wehad foundthehighautocorrelationsamongb0andb4samples.Thereasonofhighautocorrelationsintheseparameterscanbe explainedbythelowvarietyofequivalentenergyvalues.Therefore,weconductedsamplethinningwith20thinninginterval beingsetforeachparameter
Table1 givestheposterior estimationsof modelparameters.Using theestimatedvalueof each parameterand its correspondingstandarddeviationvalue,wecanperformattesttoshowwhetherornottheparameteris statistically significant.Thep-valuesofthesetestsarelistedinthetable.Wenoticethat,fortheWeibullshapeparameter,onlysupplier
2hasasignificanteffect,whiletheeffects fromothersuppliersarenot statisticallydifferent.Forthescale parameter (characteristiclife),allsuppliersaresignificant.Inaddition,thelargemagnitudeofb4(thecoefficientoftheequivalent energyfactor) indicatesthat theenergy equivalencevariablecan explaina largeportionof variabilityin theWeibull distribution’scharacteristiclife.Meanwhile,thenegativevalueofb4indicatesthatthelifetimeofPWBcouponisinversely proportiontotheenergyitabsorbed.ThemethodofpreconditioninghasanimpactonthelifetimeofPWBcoupononly throughtheWeibullcharacteristiclife,notthroughitsshapeparameter,asthecoefficienta4isnotstatisticallysignificant Using the regression model for the Weibull characteristic life, we may consider replacing the traditional reflow preconditionbyaproperISTprecondition.Asstatedin[8],a6-cyclereflowpreconditionwiththetemperaturerangefrom
25Cto250C,12minramptimeand20mincycletimescanproduce782jouleequivalentenergy,thusthelasttwotermsof therighthandsideofEq.(6)iscalculatedas1:755log782þ0:2382¼4:84.ByusinganISTpreconditionwiththe temperaturerangefrom25Cto245C,3minramptimeand5mincycletime,a5-cycleISThas660jouleequivalentenergy andthelasttwotermsofEq.(6)is1:755log660¼4:94,whichisclosetothepreviousreflowcalculation.Thus,thisIST preconditioningmethodcanbeusedtoreplacethetraditionalreflowpreconditioningmethodsoastoavoidtheuseofreflow oveninthetestandtoreducethetotaltestingtime.Furthermore,weperformedalikelihoodratiotestonthe5-cycleISTdata (245C)andthe6-cycleRFOdatafromSupplier4andconcludedthattheirfailuretimedistributionsarenotstatistically different.ThisconclusionisalsoconfirmedbyFig.2,wherethefittedWeibulldistributionsforthesetwodatasetsoverlap eachother
Table 1
Posterior estimation of Weibull regression parameters.
alpha0 3.353 0.3618 <0.0001 2.668 3.342 4.076 alpha1 0.3456 0.5693 0.6636 0.7581 0.3476 1.488 alpha2 2.039 0.4438 <0.0001 2.884 2.045 1.147 alpha3 0.3183 0.5573 0.6778 0.7455 0.3054 1.44 alpha4 0.9209 0.4466 0.0952 0.06004 0.9201 1.806 beta0 19.12 0.9538 <0.0001 17.25 19.07 21.1 beta1 1.104 0.06232 <0.0001 1.225 1.104 0.9813 beta2 2.906 0.1896 <0.0001 3.27 2.912 2.513 beta3 0.6304 0.05737 <0.0001 0.7407 0.6305 0.5184 beta4 1.755 0.1462 <0.0001 2.06 1.748 1.467 beta5 0.2382 0.05425 <0.0001 0.1332 0.2379 0.3475
68 R Pan et al / Case Studies in Engineering Failure Analysis 7 (2016) 65–70
Trang 5The low-cycle fatigue tests wereconducted on the PWB coupons from four differentsuppliers In this paper we demonstratetheuseofWeibullregressionmodelandcomputationalBayesiananalysismethodforidentifyingimportant factorsonPWBlifetime.Our resultshowsthatasthelifetimesofcouponsfromfoursuppliersaredifferentingeneral, couponsfromsupplier2possesssignificantlylowerlifecharacteristicthanothers.Furthermore,wedemonstratethatthe energyequivalenceapproachisaneffectiveapproachforaccountingforthevariationinlifetimeestimationduetodifferent preconditioningmethodsandforsetting ISTparameters.Based onthis approach,a6-cyclereflow preconditioncanbe replacedbya5-cycleISTprecondition,thusthetotaltestingtimecanbegreatlyreduced
Acknowledgements
Wethanktheanonymousrefereeforhis/herconstructivecommentsandsuggestions.Thisworkwaspartiallysupported
bytheNSFGrantCMMI1301075
AppendixA.Thermalcycletestdataset
Thermalcycletestdataset
Supplier 1 Supplier 2 Supplier 3 Supplier 4
IST5
(230C)
IST6
(230C)
RFO6 IST5
(230C)
RFO6 IST5
(230C)
IST6 (230C)
RFO6 IST6
(240C)
IST5 (245C)
1500+ 1500+ 2437 2821+
2733 1800+
1800+
1800+
1800+
1800+
1800+
1800+
2800+
2800+
2800+
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