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(1)

Modelling of high temperature fatigue crack

growth in Inconel 718 under hold time

conditions

David Gustafsson, Erik Lundström and Kjell Simonsson

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

David Gustafsson, Erik Lundström and Kjell Simonsson, Modelling of high temperature

fatigue crack growth in Inconel 718 under hold time conditions, 2013, International Journal of

Fatigue, (52), 124-130.

http://dx.doi.org/10.1016/j.ijfatigue.2013.03.004

Copyright: Elsevier

http://www.elsevier.com/

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-85933

(2)

fatigue ra k growth in In onel

718 under hold time onditions

D. Gustafsson a, ,E.Lundström a ,K.Simonsson a a

DivisionofSolidMe hani s,DepartmentofManagementandEngineering,Linköping University,SE-58183Linköping,Sweden

Abstra t

In onel718isafrequentlyusedmaterialforgasturbineappli ationsat

temper-aturesupto

650

C.Themainload y leforsu h omponentsistypi allydened

bythe start-upandshut-down ofthe engine. It generallyin ludes hold times

athightemperatures,whi hhavebeenfoundtohaveapotentialforgreatly

in- reasingthefatigue ra kgrowthratewithrespe ttothenumberofload y les.

However,theseee tsmaybetotallyorpartly an elledbyotherloadfeatures,

su h as overloads orblo ksof ontinuous y li loading, and thea tual ra k

propagationratewillthereforedependonthetotalityoffeaturesen ompassed

bytheload y le. Ithaspreviouslybeenshownthatthein reased ra kgrowth

ratefoundinholdtimeexperiments anbeasso iatedwithadamageevolution,

wherethelatterisnotonlyresponsible fortherapidintergranular ra k

prop-agation during the a tual hold times, but also for the in reased ra k growth

duringtheloadreversals. Inthispaper,modellingoftheholdtimefatigue ra k

growth behaviourof In onel 718 hasbeen arried out, using the on ept of a

damaged zone asthe basis for the treatment. With this on eptually simple

andpartlynovelapproa h,it isshownthatgoodagreementwithexperimental

results an befound.

(3)

holdtimes, ra kpropagationmodelling

1. Introdu tion

Ingasturbinesitisimportanttodesignforashighgastemperaturesaspossible

in order to attain ahigh thermal e ien y [1℄. Foraeroengines, anin reased

temperatureopensupforhigherpayloads,speedin reaseandgreaterrangeof

operation. Inthe aseofpowergeneratinggasturbines,thein reaseof

tempera-tureleadstolowerfuel onsumption,redu edpollutionandthuslower osts[2℄.

Thehigh-temperatureload arrying abilityof riti al omponents istherefore

oneofthemostimportantfa torsthatsetthelimitsingasturbinedesign. Even

thoughhightemperatureresistantsuperalloysareused,hot omponentsare

usu-allydesigned to run near their temperature and load limits. Un ertainties in

modelsand methodsused forfatigue lifepredi tionunder these ir umstan es

arethusveryproblemati . Amongthemostimportantquestionsingasturbine

designtodayisthereforehowtopredi tthelifeofsu h omponents.

In onel 718is a frequentlyused material for gasturbine appli ations at

tem-peratures upto

650

C. For su h omponents, themain load y leis typi ally

dened by the start-up and shut-down of the engine. In this main loading

y le,hold timesat hightemperaturearegenerallypresentfor riti al

ompo-nents. Thesehightemperatureholdtimesmaygreatlyin reasethefatigue ra k

growthrate with respe tto thenumberof y les, andit hasbeenshown that

this anomalous behaviour is due to material damage in the ra k tip vi inity

ausingthematerialtofailbyintergranularfra ture[35℄. Betweenthesehold

times dierent types of loadings an o ure.g. se tions/blo ks of ontinuous

y li loading. These an be aused by abnormal servi e onditions but an

alsoo uronamoreregularbasisdue toe.g. dierentweather onditionsand

enginevibrations. Ithasbeenshownpreviouslythatnotonlythe ra kgrowth

(4)

betweenholdtimesat hightemperatureand y li loading onditionsin order

tomodelthebehaviourofe.g. real engineoperation y les.

Fatigue ra kgrowthinIn onel718withhightemperatureholdtimeshasbeen

extensivelystudied previously, .f. [4,5,713℄, whi h allshowthat holdtimes

athightemperaturemayhavedevastatingee tsonthe ra kgrowthrate. The

modelling ofhold time ee ts is lassi allyhandled byadditivemodelswith a

y li part,basedonpure y li ra kgrowth,andatimedependentpart,based

onpuretimedependent ra kgrowth,seee.g. [1418℄. However,thisapproa h

has been shown by e.g. Gustafsson et al. [6, 19℄ to be questionable from a

physi al point of view, see also [20℄. When the ra k growth during the hold

time is separated from the ra k growth o urring during the unloading and

reloading, itwasfound that signi ant embrittlement of thegrain boundaries

has o urred [3℄. This aspe t was further dis ussed in [6℄ and [21℄ where it

was on luded that notonly the ra k growth during the hold time but also

the ra k growth during load reversal was ae ted by the hold time period.

Thus, a signi ant partof the ra kingtakes pla e during the unloading and

reloadingof the test spe imen, see also [20℄. These ee ts were also found in

thermome hani alfatigue ra kgrowthtests[22℄. Furthermore,in [6℄astudy

of the relative ontributions of y li and time dependent ra k growth was

performed. It wasshown that, for the load y les onsidered, the y li part

generally ontributesmorethanthetimedependentpartforshorterholdtimes,

whiletheoppositeisthe aseforlongerholdtimes.

Whenit omes to des ribingthetime dependent behaviour, someauthorsuse

asimple phenomenologi almodelling approa h, basi ally modifying the Paris

law[23℄byin ludingparameterssu hasfrequen yandtemperaturetoa ount

forthetimedependen eofthehightemperature ra kgrowthpro ess,seee.g.

[13℄. Othersuseamorephysi allybasedapproa handtheirmodellingisusually

basedonanobservationofaspe i ra kgrowthordamageme hanism,su h

(5)

growthinhightemperatureappli ations,onlyafewhavedealtwithtransitions

between blo ks of hold times and blo ks of y li ra k growth, see e.g. [32℄

whereIn onel100wasinvestigated,and[33℄whereIn onel718wasinvestigated

and modelled using a linear umulative damage model. In these papers the

transientsbetweenhold time y lesand pure y li loadingwereinvestigated.

Thishasalsobeenstudiedine.g. [6,21℄and[34℄forIn onel718andIn onel783,

respe tively. However, the spe i modelling of the transient ee ts between

holdtimesand y li loadingisstillnot ompletely overedintheliterature.

Thehigh-temperaturehold timesgiverisetoan embrittlementthat auses

in-tergranularfra ture. Theme hanismsoftheholdtimeee tinuen eavolume

ofmaterial aroundthe ra ktip, herereferredto asthedamaged zone,whi h

gets a lowered resistan e to fatigue ra k propagation ompared to the

unaf-fe ted material, see e.g. [6℄ and also [7, 20, 3539℄ for further dis ussions on

theme hanismsbehindtheholdtimeee t. Measurementsofthelengthofthe

damaged zone by hangingload y le type, are dis ussed in [6℄ and [40℄, and

ithas been found that itslength varies withrespe t to temperatureand hold

timebutisusually,inastabilizedstate,tenthsofmillimetreswhi hisgenerally

mu hlargerthantheplasti zone.

In this paper, modelling of the hold time fatigue ra k growth behaviour of

In onel 718 in the time dependent region and at the temperature

550

C has

been arriedout byusing the on ept of adamaged zone, where s alefa tors

dependingonitslengthareusedfora eleratingboththe y li andholdtime

parts. This approa hof usingthe lengthof thedamaged zonein ombination

withs alingfa torshastotheauthorsknowledgenotpreviouslybeendis ussed

intheliterature. Inadditiontohavingareasonablysoundphysi alfoundation,

this modelling approa h is also shown to give good results for the onsidered

(6)

2.1. Materialdata

Thematerialusedin thistestserieswas,asmentionedpreviously,In onel718,

awroughtpoly rystallineni kelbasedsuperalloywithahigh ontentofFeand

Cr. Its omposition (in weight %) is presentedin Table 1. The materialwas

delivered in the form of bars with a diameter of 25.4 mm and was solution

annealedfor1hat

945

C,followedbyageingattwotemperatures,8hat

718

C

and 8 h at

621

C a ording to the AMS 5663 standard. The line inter ept

methodwasusedto estimatethegrainsize tobeapproximately10

µ

m,whi h is representative for ne-grained forged materials in gas turbine omponents,

e.g. turbinedis s.

Table1: CompositionofelementsforIn onel718

Element Ni Cr Mo Nb Al Ti Fe C

Weight% balan e 19.0 3.0 5.1 0.5 0.9 18.5 0.04

2.2. Experimental ra k growth pro edure

Cra kgrowthexperimentswere ondu tedon Kb-typespe imenswith

re tan-gular ross se tions of 4.3 x 10.2 mm, see Fig. 1. An initial starter not h

of nominal depth 0.075 mmand totalwidth of 0.15mm wasgenerated using

ele tro dis hargema hining (EDM). Before thehigh temperature testing was

arried out, the spe imens were fatigue pre ra ked at room temperature and

R = σ

min

max

= 0.05

, to obtain asharp semi ir ular ra k with a depth of about0.3mm. Alltestswhereinterruptedatanapproximate ra klengthof2.5

mm. Thefatigue ra kgrowthtestingwasthen arriedoutunderload ontrol

using an MTS servo hydrali ma hine with a maximum load apa ity of 160

kN, using an Instron 8800 ontrol systemand the software WaveMaker. The

(7)

ra k propagation was monitored by the dire t urrent Potential Drop (PD)

te hnique a ordingto ASTM E 647[41℄ using aMatele t DCM-1, 2 hannel

pulsedDCPDsystem. Theloadatea hholdtime orrespondedto

σ

hold

= 650

MPa. Furthermore,alltests weredonein laboratoryairandat aloadratioof

R = σ

min

hold

= 0.05

.

Figure1:InstrumentedKb-typetestspe imen

ThemeasuredPDratiowastranslatedto ra klength,assumingasemi- ir ular

ra kfront,whi hwas onrmed post-mortem, throughanexperimentally

ob-tained alibrationfun tion basedoninitialandnal ra klengthsmeasuredon

thefra turesurfa easwellasbymeasuredindu edbea hmarks. Thismethod

formonitoringthe ra klengthhasan a ura yof 0.01mm ra klength.

Fi-nally,theanalyti alsolutionforthestressintensityfa tor

K

wasobtainedusing apresolved asefor asemi-ellipti surfa e ra k [42℄. For adetailed re ordof

thematerialandexperimental onditionswealsoreferto[3, 6,21℄.

3. Hightemperature fatigue ra k growth modelling; physi al

moti-vation, adopted modelling framework and parameter alibration

pro edure

Inthisse tiontheadopted modellingframeworkanditsphysi albasisare

(8)

resultsarefoundinSe tion 3.3.

3.1. Physi al motivation; damagedzoneand ra kdriving me hanisms

As dis ussed previously, it has been found that high-temperature hold times

giverisetoanembrittlementthat ausesintergranularfra ture. However,ithas

alsobeenshownthattheratiobetweentransgranularandintergranularfra ture

depends onboth temperature and hold time length[13℄. Con eptually, three

fra ture type regions an be identied, representing i) fully y le dependent

transgranularfra ture, ii) fully time dependent intergranular fra ture and iii)

mixed type transgranularand intergranularfra ture, where the ranges of the

latterbe omeshorter forhigher temperatures. Furthermore,it is to be noted

thatevenforlongholdtimesand/or hightemperatures,mixed mode ra king

an be dominant in transient regions when a damaged zone is not yet fully

developed.

The underlying me hanisms of the hold time ee t are still not fully

under-stood. However,twodominatingtheories anbefound: stressa eleratedgrain

boundaryoxidationanddynami embrittlement[37℄,where theformerpro ess

involvesoxidation of grain boundaries ahead of the ra k tip and subsequent

ra kingof the oxide,exposing newsurfa es to oxygen. The dynami

embrit-tlementtheoryontheotherhand advo atesembrittlingof thegrainboundary

by oxygen diusion, separation of the embrittled boundaries and subsequent

oxidationofthefreshsurfa es[7,35,38,43℄.

Whateverme hanismis a tive, the damaged zone itself is probably a volume

of embrittled and partially ra ked material, .f. [6,35, 36, 38, 39℄. In these

worksitisshownthatintergranular ra ksgrowinpreferablegrainboundaries

whi h leadsto an uneven ra k front. Certain parts of the ra k grow faster

and unbroken mi ros opi ligaments are left behind the ra k front, see e.g.

(9)

me hanism a ting in front of the ra k fully ontrols the ra k growth rate.

Wheneveraloadreversalisapplied,themain ra kwillpropagateintomaterial

with lower ra k growth resistan e and, as long as the main ra k does not

propagatetoofarintothedamagedzone,su haloadreversalwillnotae tthe

overall ra kgrowthratepertimein rement(d

a

/d

t

).

3.2. Adoptedmodellingframework

Basedontheobservationsdis ussedinSe tion3.1,itisobviousthattobeable

to model the orre t ra kgrowthbehaviourin a high temperaturehold time

(HT) ontextitisimperativetobeabletomodeltheevolutionofthedamaged

zone. Furthermore,sin eithasbeenshowninpreviousworkthatthedamaged

zoneae tsthe ra kgrowthofboththeloadreversalsandtheholdtimepart

oftheload y leitbe omesnaturaltouseanadditivedes ription. Finally,sin e

ourmodellingframeworkissetupusingthe on eptofadamagedzoneasthe

foundation,its main domainof validity willbethetime dependentregion, see

Se tion3.1.

Indetailthefollowingadditivelawisproposed

 da

dN



total

=

 da

dN



cyclic

+

 da

dN



time dependent

(1) where

 da

dN



cyclic

= S

c

(D) ·

 da

dN



baseline

= S

c

(D) · C

c

(∆K)

n

c

(2) and

 da

dN



time dependent

=

Z

t

hold

S

t

(D) ·

 da

dt



s

dt

=

Z

t

hold

S

t

(D) · C

t

(K

hold

)

n

t

dt

(3)

(10)

respe tively and where the label

s

indi ates stabilized time dependent ra k growth. Furthermore, the labelling baseline refer to behaviour under

stan-dardtesting,i.e. y li loadingwithasu ientlylargefrequen yforwhi hno

damagedzonehastimetoevolve. Finally,

S

c

and

S

t

aremonotoni ally in reas-ings alingfun tions ofthe urrentlengthofthedamagedzone

D

and,nally,

C

c

,

n

c

,

C

t

and

n

t

arepositive onstantsin theParislawexpressions.

It is to be noted that the use of a s aled baseline-term impli itly implies the

assumptionofsu ientlyrapidloadreversals. However,althoughthedamaged

zone is ontinuously growing during the hold times the ra k is also growing,

butusuallyat amu hslowerrate, seeFig. 2. Furthermore,ifablo kof y li

loadingshouldbeimposed,thedamagedzonewouldbeprogressivelydestroyed,

see Fig. 3. Thus, the total rate of the evolution of the damaged zone will

dependontwoparts,

m

˙

and

˙a

,forminga ombinedratewhere

m

˙

representsthe me hanismbasedgrowthrateof thedamagedzone and

˙a

representsthe ra k growthrate.

(11)

ǻa

D

n

D

n+1

Time

Load

n

n+1

(12)

Time

Load

ǻa

D

n

D

n+1

n n+1

Figure3:Destru tion ofthedamagedzone

Basedontheadopted timedependent ra kpropagation des ription,itishere

proposedthat

˙

D = ˙

m − ˙a

(4)

˙

m = C

t

K

n

t

hold

(5)

Withthis hoi eofevolutionequationfor

D

, in ombinationwiththe assump-tionof amonotoni ally in reasings aling fun tion

S

t

, theevolution of

D

will be stable in thesense that

D

will neverbe largerthanthe valueforwhi h

S

t

rea hesunity.

(13)

based ra kgrowthduringtheloadreversals anbenegle ted,thusimplying

dD

dN

= −

da

dN

(6)

Finally, thelaws ontrolling

S

t

and

S

c

areto besetup. Therst,

S

t

, ontrols to what extentthedamaged zone ae tsthe hold timepart, seeEq. (7), and

shoulde.g. beabletodes ribethetransientfrompure y li loadingtoablo k

ofholdtimes,seeFig. 4.

ǻK

da/dN

Time

Load

Figure4:A elerationofthefatigue ra kgrowthwithin reaseddamagedzonelength

Sin e

S

t

istobeamonotoni allyin reasingfun tion ofD, andsin eitfor the aseofnodamagedzoneshouldbezero,itmaybegiventheformfoundinEq.

(7)below,where

B

t

isattingparameter

S

t

=



D

D

max



B

t

B

t

0

(7)

More omplexexpressionsinvolvingmoreparametersareof oursepossibleto

on eive, but in order to keep the des ription as simple as possible, and in

(14)

hosen. Notethat thequantity

D

max

orrespondstothemeasurable lengthof thedamagedzoneunder stabilizedtimedependent ra kgrowth.

These ondfa tor,

S

c

, ontrolstowhatextentthedamagedzoneinuen esthe y li part,and shoulde.g. beableto des ribethetransientfrom aholdtime

blo ktopure y li loading,seeFig. 5.

ǻK

da/dN

Time

Load

Figure5:De elerationofthefatigue ra kgrowthduetoredu eddamagedzone

Sin e

S

c

is to be a monotoni ally in reasing fun tion of

D

, and sin e it is to takethevalueoneforthe aseofanundamaged ra k-tipmaterial,itmaybe

giventheform shown in Eq. (8), where thetwotting parameters

A

c

and

B

c

havebeenintrodu ed

S

c

= 1 + A

c



D

D

max



B

c

B

c

0

(8)

Again, the simplest possible expression has been hosen in order to keep the

modelassimpleaspossible.

With the hosen evolution laws for

S

c

and

S

t

, all ra k growth rates will be betweenthepure y li fatigue ra kgrowthrateandthepuretimedependent

ra k growthrate. As one an see from the model expression in Eq. (7),

S

t

willonlyrea hitsmaximumvalueofoneforsu ientlylongholdtimes,whi h

(15)

appliesfor

S

c

, startingat ahigh value depending onhow lose

D

is to

D

max

, andslowlyde reasingtowardsitsminimumvalueofoneasthedamagedzoneis

ompletely onsumed. Thus,ourmodelisrobustinthesensethatitwillonly

predi t ra kgrowthwithintwowelldenedand lassi allywell-knownrates.

3.3. Parameter alibrationpro edure

The proposed model ontains a small set of tting parameters whi h an be

foundfrombasi experiments. A tually,aswillbedis ussedinthelastse tion

below,allparameters anbedeterminedfromonesingletesttype.

In this work, twotypes of tests have been used in the parameter alibration

pro edure;apuretimedependent ra kgrowthtestandablo ktestwithhold

times of 2160 s. All tests were performed at

550

C. In the blo k tests y li

andholdtimeloadingswerealternatedin separateblo ks. Indetail,theystart

with a y li loading (without hold time) up to a spe i ra k length, then

holdtime ra k growthperiodsupto aspe i ra klengthand thenbothof

these stepsagain, thus ending with ahold time ra k growthperiod, see Fig.

6. Furthermore, ea h of these four blo ks are run over an equal amount of

ra klength. Itmaybenotedthatsu hblo ktestswerealsousedin[6,21℄to

determinethelengthofthedamaged zonefordierentholdtimes.

Time

Load

(16)

Inthisse tiontheproposedmodelis alibratedandvalidatedagainstpreviously

reportedexperimental results[3,6, 19,21℄. Themodel hasbeenimplemented

in MATLAB [44℄ and FORTRAN and an Euler Forward algorithm has been

usedtosolvethe ra kpropagationequations. Finally, theoptimisationof the

modelparameterswasdonebyusingthe ommersialsoftwareLS-OPT[45℄,by

minimisationofaleastsquarebasederrorfun tion.

4.1. Calibration

Thepuretimedependent ra kgrowthtestisusedtodeterminetheparameters

B

t

,

C

t

and

n

t

,wherethelasttwoarefoundfromthestabilisedpartofthe urve i.e. whenthetransitionzonehasended. Thevalueswerefoundtobe

5.76·10

−12

and

4.57

, respe tively. Theremaining parameterforthetest,

B

t

, isoptimised to t thetransient at thebeginning of thetest, see Fig. 7, and wasfound to

be 1.62. The stabiliseddamaged zone length,

D

max

wasestimatedto 0.5mm basedontheobservationsfoundin[21℄.

15

20

25

30

35

40

45

10

−7

10

−6

10

−5

10

−4

10

−3

d

a/d

t

[m

m

/s

]

K

max

[MPa√m]

Time dependent crack growth test

Model

(17)

oftheadditivelaw. Tostartwith,

C

c

and

n

c

aredire tly takenfrom therst y li partof thetest and were foundto be

1.26 · 10

−7

and

2.39

, respe tively. Theparameters

A

c

and

B

c

in

S

c

wereoptimisedtotthersttransientbetween hold timeloadingto y li loading,see Fig. 8, andwere foundto be823and

2.92,respe tively.

15

20

25

30

35

40

45

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

d

a/d

N

[m

m

/c

y

cl

e]

∆K [MPa√m]

2160 s block test

Model

Figure8:Parameterdeterminationof

C

c

, n

c

, A

c

and

B

c

AsummaryofallmaterialparametersisshowninTable2below.

Table2:Optimisedmodelparameters Parameter Value

C

t

5.76 · 10

−12

n

t

4.57

C

c

1.26 · 10

−7

n

c

2.39

B

t

1.62

A

c

823

B

c

2.92

(18)

The model was validated for ve dierent tests, see Table 3. Fora omplete

des riptionofthesetestssee [3,6,19,21℄.

Table3: Testsusedformodelvalidation Holdtime [s℄ Cy letype

2160 Holdtimetest

21600 Holdtimetest

21600 Blo ktest

90 Blo ktest

- Longpre- ra ktimedependent ra kgrowth

As an be seenin Fig. 9the 2160s and21600s hold time testsare aptured

reasonablywell.

10

15

20

25

30

35

40

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

10

1

d

a/d

N

[m

m

/c

y

cl

e]

∆K [MPa√m]

2160 s HT test

2160 s HT model

21600 s HT test

21600 s HT model

Figure9:Modelvsholdtimetests

Asanexampleofthemodeloutputfor ra klengthvs. time,theresultforthe

(19)

resultsfor the2160 s hold time test is not aptured well, whi h is due to the

verysmallinitial transientfoundin that test. Thereasonfor thisisnot lear,

but is likely to be asso iated with theheating of the furna e after the initial

pre- ra kgrowth.

0

0.5

1

1.5

2

2.5

3

x 10

5

0

0.5

1

1.5

2

2.5

a

[m

m

]

Time [s]

21600 s HT test

Model

Figure10: Cra klengthvstimeforthe21600sholdtimetest

InFig. 11 a omparison between blo ktest resultswith 21600s hold time is

shown. As anbeseen,the transitionsbetweenblo ksof holdtimes andpure

y li loadingare apturedreasonablywell. However,itistobenotedthat the

modelpredi ts ra klengthasafun tionoftime. Toobtainresultsinad

a

/d

N

ontext,for thehold time y les, theevaluation isbasedonthemeanvalueof

the ra klengthfor ea h y le. Thus,this explainswhyit seemsthatthere is

apartmissingbetweentheendofthersthold timeblo kandthestartofthe

(20)

15

20

25

30

35

40

45

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

d

a/d

N

[m

m

/c

y

cl

e]

∆K [MPa

√m]

21600 s block test

Model

Figure11:Modelvs21600sblo ktest

InFig.12a omparisonbetweenblo ktestresultswith90sholdtimeisshown.

As anbeseen,themodeloverpredi tstheholdtimeparts. Thereasonforthis

is, most likely, that the 90 s hold time is too short to be in the fully time

dependent region [13℄. Sin e our modell does not onsider ee ts spe i ally

related to the region of mixed fra ture, it fails to predi t the orre t ra k

(21)

15

20

25

30

35

40

45

10

−5

10

−4

10

−3

10

−2

10

−1

d

a/d

N

[m

m

/c

y

cl

e]

∆K [MPa

√m]

90 s block test

Model

Figure12: Modelvs90sblo ktest

InFig. 13theresultsfortimedependenttestwithalongpre- ra kof

approx-imately1.31 mm are shown. For omparison, the model output for the time

dependent ra k growth test with shorter pre- ra k, found in Se tion 4.2, is

plotted aswell. As anbeseen, themodel does apturethe overall behaviour

of the experiment, but the transient in the beginning of the test is

overpre-di ted,indi atingthat theequationsfor

D

˙

and/or

S

t

mayneedmore omplex expressionswith moreparameters,in orderto handle thenonlinearities found

(22)

20

25

30

35

40

10

−7

10

−6

10

−5

10

−4

10

−3

d

a/d

t

[m

m

/s

]

K

max

[MPa√m]

Time dependent crack growth test, long pre-crack

Model, long pre-crack

Model, short pre-crack

Figure13: Modelvstimedependent ra kgrowthtestwithalongpre- ra k. Alsoshownis themodeloutputforthetimedependent ra kgrowthtestfoundwithshorterpre- ra k

Fortypi alresultsoftheevolutionofthedamagedzone,seeFig. 14,where the

damagedzone size forthe2160 sand the21600sblo ktest isshown. First a

y li ra kgrowthblo kisappliedandthusweseenogrowthofthedamaged

zone. After this a hold time blo k starts and we an now see a progressive

build up of the damaged zone. It is to benoted that the saw tooth pattern

appearing is due to the partial destru tion of the damaged zone during ea h

load reversal in between the hold times. After the hold time blo k another

y li ra kgrowth blo k begins. Here we see a progressiveredu tion of the

zone until nothing remains. Finally, another hold time blo k starts and after

awhile thetest nishes. As notedinFig. 14,thetwomaximumvaluesof the

damagedzonearemarked,andtheseareseentobeingoodagreementwiththe

ones measured in [21℄, 0.27mm and 0.42 mm forthe 2160 s and the 21600s

(23)

0

1

2

3

4

5

6

x 10

4

0

0.1

0.2

0.27

0.3

0.4

0.43

0.5

D

[m

m

]

Time [s]

2160 s

21600 s

Figure14: Buildupanddestru tionofthedamagedzoneduringthe2160sandthe21600s blo ktest

5. Summaryand on lusions

In this papermodelling of the hold time fatigue ra k growth in In onel 718

usingthe on eptofadamagedzoneasitsbasi foundationhasbeenpresented.

Using the evolution model of the damaged zone and the s ale fa tors, hold

timetests, timedependent ra kgrowthtests and blo k tests anbe handled

satisfa torily. Themodelhasfewttingparametersandthematerialparameters

aneasily beobtained from fatigue ra kgrowthtesting. However, the ra k

growthbehaviourwithinthemixed transgranular/intergranularfra tureregion

isnot aptured. Furthermore,onlyonetemperatureis onsidered. Ifthemodel

weretobe alibratedforhighertemperatures,itwouldprobablyshowabetter

agreement for shorter hold times sin e the time dependent region is growing

within reasingtemperature.

(24)

ali-type needs to be used for determining all eight parameters (in luding

D

max

) for one temperature. This an be done by rst letting the test spe imen be

subje ted to a onstantloading i.e. a pure time dependent ra k growth test

whi h, whentheParis urvehasbeenstabilised, isfollowedbyablo k of

on-tinuous y ling. Fromthetransientwhi hwillbefoundduringthe y li blo k

thestabiliseddamagedzone (

D

max

)will befound, see[6℄. Therest ofthe pa-rameters anthenbeobtainedfromwellspe iedpartsofthetest,i.e.

B

t

from thetransient at thebeginning,

C

t

and

n

t

from thestabilised partof thetime dependenttest,

A

c

and

B

c

fromthetransientduringthestartofthe ontinuous y ling and nally

C

c

and

n

c

from the stabilisedlevelat thenal ontinuous y ling, for whi h the damaged zone hasbeen ompletely onsumed, see Fig.

15.

ǻK

K

hold

da/dN

da/dt

D

max

B

t

C

t

n

t

C

c

n

c

A

c

B

c

Figure15: Proposedme hani altestforparameterdetermination

Itshouldbepointedoutthat,atpresentwedonothavea esstoastatisti ally

relevanttest population, only one experiment for ea h loading ase, whi h of

(25)

ee ts ofoverloads. Su h an additionmaygiveamodel ableto predi t many

relevant omponentload y lesin agasturbine ontext. Work on erningthis

isinprogress.

A knowledgements

The authors would like to thank Mr. Bo Skoog, Linköping University, for

thelaboratory work,and, further, the proje tteams at LinköpingUniversity,

Siemens Industrial Turboma hinery AB and GKN Aerospa e Engine Systems

for valuable dis ussions. This resear h has been funded by the Swedish

En-ergyAgen y,Siemens IndustrialTurboma hineryAB,GKNAerospa eEngine

Systems, andthe RoyalInstitute of Te hnology throughthe Swedish resear h

programTURBO POWER, thesupportofwhi his gratefullya knowledged.

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