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
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.
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
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
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
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
/σ
max
= 0.05
, to obtain asharp semi ir ular ra k with a depth of about0.3mm. Alltestswhereinterruptedatanapproximate ra klengthof2.5mm. 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
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 aloadratioofR = σ
min
/σ
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 ordofthematerialandexperimental 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
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.
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
/dt
).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) whereda
dN
cyclic
= S
c
(D) ·
da
dN
baseline
= S
c
(D) · C
c
(∆K)
n
c
(2) andda
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)respe tively and where the label
s
indi ates stabilized time dependent ra k growth. Furthermore, the labelling baseline refer to behaviour understan-dardtesting,i.e. y li loadingwithasu ientlylargefrequen yforwhi hno
damagedzonehastimetoevolve. Finally,
S
c
andS
t
aremonotoni ally in reas-ings alingfun tions ofthe urrentlengthofthedamagedzoneD
and,nally,C
c
,n
c
,C
t
andn
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 ombinedratewherem
˙
representsthe me hanismbasedgrowthrateof thedamagedzone and˙a
representsthe ra k growthrate.ǻa
D
n
D
n+1
Time
Load
n
n+1
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 tionS
t
, theevolution ofD
will be stable in thesense thatD
will neverbe largerthanthe valueforwhi hS
t
rea hesunity.based ra kgrowthduringtheloadreversals anbenegle ted,thusimplying
dD
dN
= −
da
dN
(6)Finally, thelaws ontrolling
S
t
andS
c
areto besetup. Therst,S
t
, ontrols to what extentthedamaged zone ae tsthe hold timepart, seeEq. (7), andshoulde.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
isattingparameterS
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
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 aholdtimeblo 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 ofD
, and sin e it is to takethevalueoneforthe aseofanundamaged ra k-tipmaterial,itmaybegiventheform shown in Eq. (8), where thetwotting parameters
A
c
andB
c
havebeenintrodu edS
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
andS
t
, all ra k growth rates will be betweenthepure y li fatigue ra kgrowthrateandthepuretimedependentra k growthrate. As one an see from the model expression in Eq. (7),
S
t
willonlyrea hitsmaximumvalueofoneforsu ientlylongholdtimes,whi happliesfor
S
c
, startingat ahigh value depending onhow loseD
is toD
max
, andslowlyde reasingtowardsitsminimumvalueofoneasthedamagedzoneisompletely 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
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
andn
t
,wherethelasttwoarefoundfromthestabilisedpartofthe urve i.e. whenthetransitionzonehasended. Thevalueswerefoundtobe5.76·10
−12
and
4.57
, respe tively. Theremaining parameterforthetest,B
t
, isoptimised to t thetransient at thebeginning of thetest, see Fig. 7, and wasfound tobe 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
oftheadditivelaw. Tostartwith,
C
c
andn
c
aredire tly takenfrom therst y li partof thetest and were foundto be1.26 · 10
−7
and
2.39
, respe tively. TheparametersA
c
andB
c
inS
c
wereoptimisedtotthersttransientbetween hold timeloadingto y li loading,see Fig. 8, andwere foundto be823and2.92,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
andB
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
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
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
/dN
ontext,for thehold time y les, theevaluation isbasedonthemeanvalueofthe ra klengthfor ea h y le. Thus,this explainswhyit seemsthatthere is
apartmissingbetweentheendofthersthold timeblo kandthestartofthe
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
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/orS
t
mayneedmore omplex expressionswith moreparameters,in orderto handle thenonlinearities found20
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
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.
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 besubje 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
andn
t
from thestabilised partof thetime dependenttest,A
c
andB
c
fromthetransientduringthestartofthe ontinuous y ling and nallyC
c
andn
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
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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