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

Modelling of high temperature

fatigue ra k growth in In onel

718 under hold time onditions

D. Gustafsson

a,

, E. Lundström

a

, K. Simonsson

a

a

Division of Solid Me hani s, Department of Management and Engineering, Linköping

University, SE-58183 Linköping, Sweden

Abstra t

In onel 718 is a frequently used material for gas turbine appli ations at temper-

atures up to 650◦C. The main load y le for su h omponents is typi ally de�ned

by the start-up and shut-down of the engine. It generally in ludes hold times

at high temperatures, whi h have been found to have a potential for greatly in-

reasing the fatigue ra k growth rate with respe t to the number of load y les.

However, these e�e ts may be totally or partly an elled by other load features,

su h as overloads or blo ks of ontinuous y li loading, and the a tual ra k

propagation rate will therefore depend on the totality of features en ompassed

by the load y le. It has previously been shown that the in reased ra k growth

rate found in hold time experiments an be asso iated with a damage evolution,

where the latter is not only responsible for the rapid intergranular ra k prop-

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

during the load reversals. In this paper, modelling of the hold time fatigue ra k

growth behaviour of In onel 718 has been arried out, using the on ept of a

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

and partly novel approa h, it is shown that good agreement with experimental

results an be found.

Email address: david.gustafsson�liu.se (D. Gustafsson)

Preprint submitted to Elsevier June 19, 2013

Keywords: ni kel-base superalloys, fatigue ra k propagation, In onel 718,

hold times, ra k propagation modelling

1. Introdu tion

In gas turbines it is important to design for as high gas temperatures as possible

in order to attain a high thermal e� ien y [1℄. For aeroengines, an in reased

temperature opens up for higher payloads, speed in rease and greater range of

operation. In the ase of power generating gas turbines, the in rease of tempera-

ture leads to lower fuel onsumption, redu ed pollution and thus lower osts [2℄.

The high-temperature load arrying ability of riti al omponents is therefore

one of the most important fa tors that set the limits in gas turbine design. Even

though high temperature resistant superalloys are used, hot omponents are usu-

ally designed to run near their temperature and load limits. Un ertainties in

models and methods used for fatigue life predi tion under these ir umstan es

are thus very problemati . Among the most important questions in gas turbine

design today is therefore how to predi t the life of su h omponents.

In onel 718 is a frequently used material for gas turbine appli ations at tem-

peratures up to 650◦C. For su h omponents, the main load y le is typi ally

de�ned by the start-up and shut-down of the engine. In this main loading

y le, hold times at high temperature are generally present for riti al ompo-

nents. These high temperature hold times may greatly in rease the fatigue ra k

growth rate with respe t to the number of y les, and it has been shown that

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

ausing the material to fail by intergranular fra ture [3�5℄. Between these hold

times di�erent types of loadings an o ur e.g. se tions/blo ks of ontinuous

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

also o ur on a more regular basis due to e.g. di�erent weather onditions and

engine vibrations. It has been shown previously that not only the ra k growth

rate during the hold times but also the y li ra k growth is a�e ted by the

2

material damage [6℄. Thus, it be omes important to understand the intera tion

between hold times at high temperature and y li loading onditions in order

to model the behaviour of e.g. real engine operation y les.

Fatigue ra k growth in In onel 718 with high temperature hold times has been

extensively studied previously, .f. [4, 5, 7�13℄, whi h all show that hold times

at high temperature may have devastating e�e ts on the ra k growth rate. The

modelling of hold time e�e ts is lassi ally handled by additive models with a

y li part, based on pure y li ra k growth, and a time dependent part, based

on pure time dependent ra k growth, see e.g. [14�18℄. However, this approa 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, it was found that signi� ant embrittlement of the grain boundaries

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

was on luded that not only the ra k growth during the hold time but also

the ra k growth during load reversal was a�e ted by the hold time period.

Thus, a signi� ant part of the ra king takes pla e during the unloading and

reloading of the test spe imen, see also [20℄. These e�e ts were also found in

thermome hani al fatigue ra k growth tests [22℄. Furthermore, in [6℄ a study

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

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

generally ontributes more than the time dependent part for shorter hold times,

while the opposite is the ase for longer hold times.

When it omes to des ribing the time dependent behaviour, some authors use

a simple phenomenologi al modelling approa h, basi ally modifying the Paris

law [23℄ by in luding parameters su h as frequen y and temperature to a ount

for the time dependen e of the high temperature ra k growth pro ess, see e.g.

[13℄. Others use a more physi ally based approa h and their modelling is usually

based on an observation of a spe i� ra k growth or damage me hanism, su h

as oxide penetration see e.g. [25�29℄, and more re ently [30℄ and [31℄.

3

Although a number of di�erent models already exist for predi ting fatigue ra k

growth in high temperature appli ations, only a few have dealt with transitions

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

where In onel 100 was investigated, and [33℄ where In onel 718 was investigated

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

transients between hold time y les and pure y li loading were investigated.

This has also been studied in e.g. [6, 21℄ and [34℄ for In onel 718 and In onel 783,

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

hold times and y li loading is still not ompletely overed in the literature.

The high-temperature hold times give rise to an embrittlement that auses in-

tergranular fra ture. The me hanisms of the hold time e�e t in�uen e a volume

of material around the ra k tip, here referred to as the damaged 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, 35�39℄ for further dis ussions on

the me hanisms behind the hold time e�e t. Measurements of the length of the

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

it has been found that its length varies with respe t to temperature and hold

time but is usually, in a stabilized state, tenths of millimetres whi h is generally

mu h larger than the plasti 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 arried out by using the on ept of a damaged zone, where s ale fa tors

depending on its length are used for a elerating both the y li and hold time

parts. This approa h of using the length of the damaged zone in ombination

with s aling fa tors has to the authors knowledge not previously been dis ussed

in the literature. In addition to having a reasonably sound physi al foundation,

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

low-frequen y onditions.

4

2. Material and experimental pro edure

2.1. Material data

The material used in this test series was, as mentioned previously, In onel 718,

a wrought poly rystalline ni kel based superalloy with a high ontent of Fe and

Cr. Its omposition (in weight %) is presented in Table 1. The material was

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

annealed for 1h at 945◦C, followed by ageing at two temperatures, 8h at 718◦C

and 8 h at 621◦C a ording to the AMS 5663 standard. The line inter ept

method was used to estimate the grain size to be approximately 10 µm, whi h

is representative for �ne-grained forged materials in gas turbine omponents,

e.g. turbine dis s.

Table 1: Composition of elements for In onel 718

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 k growth experiments were ondu ted on Kb-type spe imens with 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 mm and total width of 0.15 mm was generated using

ele tro dis harge ma hining (EDM). Before the high temperature testing was

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

R = σmin/σmax = 0.05, to obtain a sharp semi ir ular ra k with a depth of

about 0.3 mm. All tests where interrupted at an approximate ra k length of 2.5

mm. The fatigue ra k growth testing was then arried out under load ontrol

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

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

furna e used was an MTS high temperature furna e with three temperature

5

zones (model 652.01/MTS with a temperature ontroller of model 409.81). The

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

te hnique a ording to ASTM E 647 [41℄ using a Matele t DCM-1, 2 hannel

pulsed DCPD system. The load at ea h hold time orresponded to σhold = 650

MPa. Furthermore, all tests were done in laboratory air and at a load ratio of

R = σmin/σhold = 0.05.

Figure 1: Instrumented Kb-type test spe imen

The measured PD ratio was translated to ra k length, assuming a semi- ir ular

ra k front, whi h was on�rmed post-mortem, through an experimentally ob-

tained alibration fun tion based on initial and �nal ra k lengths measured on

the fra ture surfa e as well as by measured indu ed bea h marks. This method

for monitoring the ra k length has an a ura y of 0.01 mm ra k length. Fi-

nally, the analyti al solution for the stress intensity fa tor K was obtained using

a presolved ase for a semi-ellipti surfa e ra k [42℄. For a detailed re ord of

the material and experimental onditions we also refer to [3, 6, 21℄.

3. High temperature fatigue ra k growth modelling; physi al moti-

vation, adopted modelling framework and parameter alibration

pro edure

In this se tion the adopted modelling framework and its physi al basis are pre-

sented and dis ussed. Furthermore, also the parameter alibration pro edure is

6

dis ussed from a on eptual point of view. Detailed alibration and validation

results are found in Se tion 3.3.

3.1. Physi al motivation; damaged zone and ra k driving me hanisms

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

give rise to an embrittlement that auses intergranular fra ture. However, it has

also been shown that the ratio between transgranular and intergranular fra ture

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

fra ture type regions an be identi�ed, representing i) fully y le dependent

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

mixed type transgranular and intergranular fra ture, where the ranges of the

latter be ome shorter for higher temperatures. Furthermore, it is to be noted

that even for long hold times and/or high temperatures, 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 e�e t are still not fully under-

stood. However, two dominating theories an be found: stress a elerated grain

boundary oxidation and dynami embrittlement [37℄, where the former pro ess

involves oxidation of grain boundaries ahead of the ra k tip and subsequent

ra king of the oxide, exposing new surfa es to oxygen. The dynami embrit-

tlement theory on the other hand advo ates embrittling of the grain boundary

by oxygen di�usion, separation of the embrittled boundaries and subsequent

oxidation of the fresh surfa es [7, 35, 38, 43℄.

Whatever me hanism is 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

works it is shown that intergranular ra ks grow in preferable grain boundaries

whi h leads to 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.

[6, 36, 43℄.

7

In [19℄ it was shown that, for su� iently long hold times, the ra k driving

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

Whenever a load reversal is applied, the main ra k will propagate into material

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

propagate too far into the damaged zone, su h a load reversal will not a�e t the

overall ra k growth rate per time in rement (da/dt).

3.2. Adopted modelling framework

Based on the observations dis ussed in Se tion 3.1, it is obvious that to be able

to model the orre t ra k growth behaviour in a high temperature hold time

(HT) ontext it is imperative to be able to model the evolution of the damaged

zone. Furthermore, sin e it has been shown in previous work that the damaged

zone a�e ts the ra k growth of both the load reversals and the hold time part

of the load y le it be omes natural to use an additive des ription. Finally, sin e

our modelling frame work is set up using the on ept of a damaged zone as the

foundation, its main domain of validity will be the time dependent region, see

Se tion 3.1.

In detail the following additive law is proposed

(

da

dN

)

total

=

(

da

dN

)

cyclic

+

(

da

dN

)

time dependent

(1)

where

(

da

dN

)

cyclic

= Sc (D) ·

(

da

dN

)

baseline

= Sc (D) · Cc (∆K)nc

(2)

and

(

da

dN

)

time dependent

=

∫

thold

St (D) ·

(

da

dt

)

s

dt

=

∫

thold

St (D) · Ct (Khold)nt dt

(3)

8

where the labelling � � and �t� refer to y li - and time-dependent quantities,

respe tively and where the label s indi ates stabilized time dependent ra k

growth. Furthermore, the labelling �baseline� refer to behaviour under stan-

dard testing, i.e. y li loading with a su� iently large frequen y for whi h no

damaged zone has time to evolve. Finally, Sc and St are monotoni ally in reas-

ing s aling fun tions of the urrent length of the damaged zone D and, �nally,

Cc, nc, Ct and nt are positive onstants in the Paris law expressions.

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

assumption of su� iently rapid load reversals. However, although the damaged

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

but usually at a mu h slower rate, see Fig. 2. Furthermore, if a blo k of y li

loading should be imposed, the damaged zone would be progressively destroyed,

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

depend on two parts, m and a, forming a ombined rate where m represents the

me hanism based growth rate of the damaged zone and a represents the ra k

growth rate.

9

a

Dn

Dn+1

Time

Load n n+1

Figure 2: Build up of the damaged zone

10

Time

Load

a

Dn

Dn+1

n n+1

Figure 3: Destru tion of the damaged zone

Based on the adopted time dependent ra k propagation des ription, it is here

proposed that

D = m− a (4)

m = CtKnt

hold (5)

With this hoi e of evolution equation for D, in ombination with the assump-

tion of a monotoni ally in reasing s aling fun tion St, the evolution of D will

be stable in the sense that D will never be larger than the value for whi h St

rea hes unity.

11

Sin e the frequen y for the pure y li loading is assumed high, the me hanisms

based ra k growth during the load reversals an be negle ted, thus implying

dD

dN= −

da

dN(6)

Finally, the laws ontrolling St and Sc are to be set up. The �rst, St, ontrols

to what extent the damaged zone a�e ts the hold time part, see Eq. (7), and

should e.g. be able to des ribe the transient from pure y li loading to a blo k

of hold times, see Fig. 4.

K

da/dN

Time

Load

Figure 4: A eleration of the fatigue ra k growth with in reased damaged zone length

Sin e St is to be a monotoni ally in reasing fun tion of D, and sin e it for the

ase of no damaged zone should be zero, it may be given the form found in Eq.

(7) below, where Bt is a �tting parameter

St =

(

D

Dmax

)Bt

Bt ≥ 0 (7)

More omplex expressions involving more parameters are of ourse possible to

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

order to introdu e as few parameters as possible, the above expression was

12

hosen. Note that the quantity Dmax orresponds to the measurable length of

the damaged zone under stabilized time dependent ra k growth.

The se ond fa tor, Sc, ontrols to what extent the damaged zone in�uen es the

y li part, and should e.g. be able to des ribe the transient from a hold time

blo k to pure y li loading, see Fig. 5.

K

da/dN

Time

Load

Figure 5: De eleration of the fatigue ra k growth due to redu ed damaged zone

Sin e Sc is to be a monotoni ally in reasing fun tion of D, and sin e it is to

take the value one for the ase of an �undamaged� ra k-tip material, it may be

given the form shown in Eq. (8), where the two �tting parameters Ac and Bc

have been introdu ed

Sc = 1 +Ac

(

D

Dmax

)Bc

Bc ≥ 0 (8)

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

model as simple as possible.

With the hosen evolution laws for Sc and St, all ra k growth rates will be

between the pure y li fatigue ra k growth rate and the pure time dependent

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

will only rea h its maximum value of one for su� iently long hold times, whi h

orresponds to pure time dependent ra k growth, i.e. D = Dmax. The opposite

13

applies for Sc, starting at a high value depending on how lose D is to Dmax,

and slowly de reasing towards its minimum value of one as the damaged zone is

ompletely � onsumed�. Thus, our model is robust in the sense that it will only

predi t ra k growth within two well de�ned and lassi ally well-known rates.

3.3. Parameter alibration pro edure

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

found from basi experiments. A tually, as will be dis ussed in the last se tion

below, all parameters an be determined from one single test type.

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

pro edure; a pure time dependent ra k growth test and a blo k test with hold

times of 2160 s. All tests were performed at 550◦C. In the blo k tests y li

and hold time loadings were alternated in separate blo ks. In detail, they start

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

hold time ra k growth periods up to a spe i� ra k length and then both of

these steps again, thus ending with a hold time ra k growth period, see Fig.

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

ra k length. It may be noted that su h blo k tests were also used in [6, 21℄ to

determine the length of the damaged zone for di�erent hold times.

Time

Load

Figure 6: Blo k test load urve

14

4. Results and dis ussion

In this se tion the proposed model is alibrated and validated against previously

reported experimental results [3, 6, 19, 21℄. The model has been implemented

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

used to solve the ra k propagation equations. Finally, the optimisation of the

model parameters was done by using the ommersial software LS-OPT [45℄, by

minimisation of a least square based error fun tion.

4.1. Calibration

The pure time dependent ra k growth test is used to determine the parameters

Bt, Ct and nt, where the last two are found from the stabilised part of the urve

i.e. when the transition zone has ended. The values were found to be 5.76·10−12

and 4.57, respe tively. The remaining parameter for the test, Bt, is optimised

to �t the transient at the beginning of the test, see Fig. 7, and was found to

be 1.62. The stabilised damaged zone length, Dmax was estimated to 0.5 mm

based on the observations found in [21℄.

15 20 25 30 35 40 4510

−7

10−6

10−5

10−4

10−3

da/d

t[m

m/s]

Kmax [MPa√

m]

Time dependent crack growth test

Model

Figure 7: Parameter determination of Ct, nt and Bt

15

The blo k test is used to determine all remaining parameters in the y li part

of the additive law. To start with, Cc and nc are dire tly taken from the �rst

y li part of the test and were found to be 1.26 · 10−7and 2.39, respe tively.

The parameters Ac and Bc in Sc were optimised to �t the �rst transient between

hold time loading to y li loading, see Fig. 8, and were found to be 823 and

2.92, respe tively.

15 20 25 30 35 40 4510

−5

10−4

10−3

10−2

10−1

100

da/d

N[m

m/cycle]

∆K [MPa√

m]

2160 s block testModel

Figure 8: Parameter determination of Cc, nc, Ac and Bc

A summary of all material parameters is shown in Table 2 below.

Table 2: Optimised model parameters

Parameter Value

Ct 5.76 · 10−12

nt 4.57

Cc 1.26 · 10−7

nc 2.39

Bt 1.62

Ac 823

Bc 2.92

16

4.2. Validation

The model was validated for �ve di�erent tests, see Table 3. For a omplete

des ription of these tests see [3, 6, 19, 21℄.

Table 3: Tests used for model validation

Hold time [s℄ Cy le type

2160 Hold time test

21600 Hold time test

21600 Blo k test

90 Blo k test

- Long pre- ra k time dependent ra k growth

As an be seen in Fig. 9 the 2160 s and 21600 s hold time tests are aptured

reasonably well.

10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

100

101

da/d

N[m

m/cycle]

∆K [MPa√

m]

2160 s HT test2160 s HT model21600 s HT test21600 s HT model

Figure 9: Model vs hold time tests

As an example of the model output for ra k length vs. time, the result for the

21600 s hold time test is shown in Fig. 10. As an be seen the model aptures

17

the overall behaviour of the test satisfa torily. However, in this ontext the

results for the 2160 s hold time test is not aptured well, whi h is due to the

very small initial transient found in that test. The reason for this is not lear,

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

pre- ra k growth.

0 0.5 1 1.5 2 2.5 3x 10

5

0

0.5

1

1.5

2

2.5

a[m

m]

Time [s]

21600 s HT test

Model

Figure 10: Cra k length vs time for the 21600 s hold time test

In Fig. 11 a omparison between blo k test results with 21600 s hold time is

shown. As an be seen, the transitions between blo ks of hold times and pure

y li loading are aptured reasonably well. However, it is to be noted that the

model predi ts ra k length as a fun tion of time. To obtain results in a da/dN

ontext, for the hold time y les, the evaluation is based on the mean value of

the ra k length for ea h y le. Thus, this explains why it seems that there is

a part missing between the end of the �rst hold time blo k and the start of the

se ond y li blo k.

18

15 20 25 30 35 40 4510

−5

10−4

10−3

10−2

10−1

100

da/d

N[m

m/cycle]

∆K [MPa√

m]

21600 s block test

Model

Figure 11: Model vs 21600 s blo k test

In Fig. 12 a omparison between blo k test results with 90 s hold time is shown.

As an be seen, the model overpredi ts the hold time parts. The reason for this

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 e�e ts spe i� ally

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

growth rate for this short hold time.

19

15 20 25 30 35 40 4510

−5

10−4

10−3

10−2

10−1

da/d

N[m

m/cycle]

∆K [MPa√

m]

90 s block test

Model

Figure 12: Model vs 90 s blo k test

In Fig. 13 the results for time dependent test with a long pre- ra k of approx-

imately 1.31 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 as well. As an be seen, the model does apture the overall behaviour

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

di ted, indi ating that the equations for D and/or St may need more omplex

expressions with more parameters, in order to handle the nonlinearities found

in di�erent transients.

20

20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

da/dt[m

m/s]

Kmax [MPa√

m]

Time dependent crack growth test, long pre-crack

Model, long pre-crack

Model, short pre-crack

Figure 13: Model vs time dependent ra k growth test with a long pre- ra k. Also shown is

the model output for the time dependent ra k growth test found with shorter pre- ra k

For typi al results of the evolution of the damaged zone, see Fig. 14, where the

damaged zone size for the 2160 s and the 21600 s blo k test is shown. First a

y li ra k growth blo k is applied and thus we see no growth of the damaged

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 be noted 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 k growth blo k begins. Here we see a progressive redu tion of the

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

a while the test �nishes. As noted in Fig. 14, the two maximum values of the

damaged zone are marked, and these are seen to be in good agreement with the

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

blo k test, respe tively.

21

0 1 2 3 4 5 6x 10

4

0

0.1

0.2

0.270.3

0.40.43

0.5

D[m

m]

Time [s]

2160 s

21600 s

Figure 14: Build up and destru tion of the damaged zone during the 2160 s and the 21600 s

blo k test

5. Summary and on lusions

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

using the on ept of a damaged zone as its basi foundation has been presented.

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

time tests, time dependent ra k growth tests and blo k tests an be handled

satisfa torily. The model has few �tting parameters and the material parameters

an easily be obtained from fatigue ra k growth testing. However, the ra k

growth behaviour within the mixed transgranular/intergranular fra ture region

is not aptured. Furthermore, only one temperature is onsidered. If the model

were to be alibrated for higher temperatures, it would probably show a better

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

with in reasing temperature.

During the alibration pro edure, di�erent parts of the tests were used to ali-

22

brate di�erent parameters. This indi ates that only one arefully planned test

type needs to be used for determining all eight parameters (in luding Dmax)

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

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

whi h, when the Paris urve has been stabilised, is followed by a blo k of on-

tinuous y ling. From the transient whi h will be found during the y li blo k

the stabilised damaged zone (Dmax) will be found, see [6℄. The rest of the pa-

rameters an then be obtained from well spe i�ed parts of the test, i.e. Bt from

the transient at the beginning, Ct and nt from the stabilised part of the time

dependent test, Ac and Bc from the transient during the start of the ontinuous

y ling and �nally Cc and nc from the stabilised level at the �nal ontinuous

y ling, for whi h the damaged zone has been ompletely onsumed, see Fig.

15.

KKhold

da/dNda/dt

Dmax

Bt

Ctnt

Ccnc

AcBc

Figure 15: Proposed me hani al test for parameter determination

It should be pointed out that, at present we do not have a ess to a statisti ally

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

ourse will make the presented parameter determination only qualitative.

23

Finally, this modelling paradigm should also be appropriate for in luding the

e�e ts of overloads. Su h an addition may give a model able to predi t many

relevant omponent load y les in a gas turbine ontext. Work on erning this

is in progress.

A knowledgements

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

the laboratory work, and, further, the proje t teams at Linköping University,

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-

ergy Agen y, Siemens Industrial Turboma hinery AB, GKN Aerospa e Engine

Systems, and the Royal Institute of Te hnology through the Swedish resear h

program TURBO POWER, the support of whi h is gratefully a knowledged.

Referen es

[1℄ R.C. Reed, The Superalloys - Fundamentals and Appli ations, Cambridge

University Press, Cambridge, 2006.

[2℄ A. Pineau, S.D. Antolovi h, High temperature of ni kel-base superalloys -

A review with spe ial emphasis on deformation modes and oxidation, Engi-

neering Failure Analysis, 16 (2009) 2668-2697.

[3℄ D. Gustafsson, J.J. Moverare, S. Johansson, M. Hörnqvist, K. Simonsson, S.

Sjöström, B. Shari�majd, Fatigue ra k growth behaviour of In onel 718 with

high temperature hold times, Pro edia Engineering, 2(1) (2010) 1095-1104.

[4℄ P. Heuler, E. A�eldt, R.J.H. Wanhill, E�e ts of Loading Waveform and

Stress Field on High Temperature Fatigue Cra k Growth of Alloy 718, Mate-

rialwissens haft und Werksto�te hnik, 34(9) (2003) 790-796.

24

[5℄ J.P. Pédron, A. Pineau, The e�e t of mi rostru ture and environment on the

ra k growth behaviour of In onel 718 alloy at 650◦C under fatigue, reep and

ombined loading, Materials S ien e and Engineering, 56(2) (1982) 143-156.

[6℄ D. Gustafsson, J.J. Moverare, S. Johansson, K. Simonsson, M. Hörnqvist,

T. Månsson, S. Sjöström, In�uen e of high temperature hold times on the

fatigue ra k propagation in In onel 718, International Journal of Fatigue,

33(11) (2011) 1461-1469.

[7℄ H. Ghonem, T. Ni holas, A. Pineau, Elevated Temperature Fatigue Cra k

Growth in Alloy 718-Part II: E�e ts of Environmental and Material Variables,

Fatigue and Fra ture of Engineering Materials and Stru tures, 16(6) (1993)

577-590.

[8℄ E. Andrieu, Intergranular ra k tip oxidation me hanisms in ni kel-based

superalloy, Materials S ien e and Engineering 154(1) (1992) 21-8.

[9℄ R. Molins, G. Ho hstetter, J.C. Chassaigne, E. Andrieu, Oxidation e�e ts

on the fatigue ra k growth behavior of alloy 718 at high temperature, A ta

Materialia, 45(2) (1997) 663-674.

[10℄ F.V. Antunes, High temperature fatigue ra k growth in In onel 718, Ma-

terials at High Temperatures, 17(4) (2000) 439-448.

[11℄ F. V. Antunes, J. M. Ferreira, C. M. Bran o, J. Byrne, In�uen e of stress

state on high temperature fatigue ra k growth in In onel 718, Journal of

Materials Engineering and Performan e, 24(2) (2008) 127-135.

[12℄ C.M. Bran o, Fatigue behaviour of the ni kel-based superalloy IN718 at

elevated temperature, Materials at High Temperatures, 12(4) (1994) 261-267.

[13℄ T. Weerasooriya, E�e t of Frequen y on Fatigue Cra k Growth Rate of

In onel 718 at High Temperature, Fra ture Me hani s: 19th Symp. ASTM,

969 (1988) 907-923.

25

[14℄ J. Gayda, T.P. Gabb, R.V. Miner, Fatigue ra k propagation of ni kel-

base superalloys at 650oC, Ameri an So iety for Testing and Material, (1988)

293-309.

[15℄ A. Saxena, A model for predi ting the e�e t of frequen y on fatigue ra k

growth behaviour at elevated temperature, Fatigue and Fra ture of Engineer-

ing Materials and Stru tures, 3(3) (1980) 247-255.

[16℄ A. Saxena, R.S. Williams, T.T. Shih, A model for representing and predi t-

ing the in�uen e of hold time on Fatigue ra k growth behaviour at elevated

temperature, ASTM Spe ial Te hni al Publi ation, (1981) 86-99.

[17℄ H. Ghonem, D. Zheng, Depth of intergranular oxygen di�usion during

environment-dependent fatigue ra k propagation in alloy 718, Materials S i-

en e and Engineering, 150 (1992) 151-160.

[18℄ R.H. Van Stone, D.C. Slavik, Predi tion of Time Dependent Cra k Growth

with Retardation E�e ts in Ni kel Base Alloys, ASTM STP 1389, (1999)

405-426.

[19℄ D. Gustafsson, E. Lundström, High temperature fatigue ra k growth be-

haviour of In onel 718 under hold time and overload onditions, International

Journal of Fatigue, 48 (2013) 178-186.

[20℄ K. Wa kermann, U. Krupp, H-J. Christ, E�e ts of the Environment on

the Cra k Propagation Behavior of IN718 in the Temperature Range of the

Dynami Embrittlement, Journal of ASTM International, 8(5) (2011) 297-

312.

[21℄ D. Gustafsson, J.J. Moverare, K. Simonsson, S. Johansson, M. Hörnqvist,

T. Månsson, S. Sjöström, Fatigue ra k growth behaviour of In onel 718 - The

on ept of a damaged zone aused by high temperature hold times, Pro edia

Engineering, 10 (2011) 2821-2826.

26

[22℄ J.J. Moverare, D. Gustafsson, Hold-time e�e t on the thermo-me hani al

fatigue ra k growth behaviour of In onel 718, Materials S ien e and Engi-

neering A, 528(29-30) (2011) 8660-8670.

[23℄ P. Paris, F. Erdogan, A riti al analysis of ra k propagation laws, Journal

of Basi Engineering, 85(4) (1963) 528-534.

[24℄ J.R. Haigh, R.P. Skelton, C.E. Ri hards, Oxidation-assisted ra k growth

during high y le fatigue of a 1% Cr-Mo-V steel at 550◦C, Materials S ien e

and Engineering, 26(2) (1976) 167-174.

[25℄ H. Ghonem, D. Zheng, Oxidation-assisted fatigue ra k growth behavior

in alloy 718-part I. quantitative modeling, Fatigue & Fra ture of Engineering

Materials & Stru tures, 14(7) (1991) 749-760.

[26℄ R.P. Skelton, J.I. Bu klow, Cy li oxidation and ra k growth during high

strain fatigue of low alloy steel, Metal S ien e, 12(2) (1978) 64-70.

[27℄ J.J. M Gowan, H.W. Liu, A kineti model of high temperature fatigue

ra k growth, Environment and temperature e�e ts, (1983) 377-390.

[28℄ S. Krush, P. Prigent, J.L. Chabo he, A fra ture me hani s based fatigue-

reep-environment ra k growth model for high temperature, International

Journal of Pressure Vessels and Piping, 59 (1994) 141-148.

[29℄ F. Gallerneau, S. Kru h, P. Kanouté, A New Modelling of Cra k Propaga-

tion with Fatigue-Creep-Oxidation Intera tion under Non Isothermal Load-

ing, Symposium on Ageing Me hanisms and Control: Part B Monitoring and

Management of Gas Turbine Fleets for Extended Life and Redu ed Costs,

Man herster UK, 8-11 O tober 2001.

[30℄ L.G. Zhao, J. Tong, M.C. Hardy, Predi tion of ra k growth in a ni kel-

based superalloy under fatigue-oxidation onditions, Engineering fra ture me-

hani s, 77 (2010) 925-938.

27

[31℄ K.O. Findley, J.L. Evans, A. Saxena, A riti al assessment of fatigue ra k

nu leation and growth models for Ni- and Ni, Fe-based superalloys, Interna-

tional Materials Reviews, 56(1) (2005) 49-71.

[32℄ J.M. Larsen and T. Ni holas, Load sequen e ra k growth transients in a

superalloy at elevated temperature, Fra ture Me hani s: Fourteenth Sympo-

sium - Volume II: Testing and Appli ations, ASTM STP 791, (1983) 536-552.

[33℄ T. Ni holas, T. Weerasooriya, Hold-time e�e ts in elevated temperature

fatigue ra k propagation, Fra ture Me hani s: Seventeenth Volume, ASTM

STP 905, (1986) 155-168.

[34℄ L. Ma, K.-M.Chang, Identi� ation of SAGBO-indu ed damage zone ahead

of ra k tip to hara terize sustained loading ra k growth in alloy 783, S ripta

Materialia, 48(9) (2003) 1271-1276.

[35℄ U. Krupp, Dynami embrittlement - time-dependent quasi-brittle inter-

granular fra ture at high temperatures, International Materials Reviews, 50

(2005) 83-97.

[36℄ U. Krupp, W. Kane, X. Liu, O. Dueber, C. Laird, C.J. M Mahon Jr.,

The e�e t of grain-boundary-engineering-type pro essing on oxygen indu ed

ra king of IN718, Materials S ien e and Engineering A, 349(1-2) (2002) 213-

217.

[37℄ D.A. Woodford, Gas phase embrittlement and time dependent ra king of

Ni kel based superalloys, Energy Materials, 1 (2006) 59-79.

[38℄ J.A. Pfaendtner, C.J. Jr M Mahon, Oxygen-indu ed intergranular ra k-

ing of a Ni-based alloy at elevated temperatures - an example of dynami

embrittlement, A ta Materialia, 49 (2001) 3369-3377.

[39℄ D. Bika, J.A. Pfaendtner, M. Menyhard, C.J. Jr M Mahon, Sulfur-indu ed

dynami embrittlement in a low-alloy steel, A ta Metallurgi a et Materialia,

43(5) (1995) 1895-1908.

28

[40℄ X.B. Liu, L.Z. Ma, K.M. Chang, E. Barbero, Fatigue Cra k Propagation

of Ni-based Superalloys, A ta Metallurgi a Sini a, 18(1) (2005) 55-64.

[41℄ ASME E647-08., Standard test method for measurement of fatigue ra k

growth rates, Annual Book of ASME Standards, Volume 03.01, West Con-

shoho ken (PA): ASM International.

[42℄ ASTM E740-03., Standard pra ti e for fra ture testing with surfa e- ra k

tension spe imens, Annual Book of ASME Standards, Volume 03.01, West

Conshoho ken (PA): ASM International.

[43℄ U. Krupp, C.J. M Mahon Jr, Dynami embrittlement-time-dependent brit-

tle fra ture, Journal of Alloys and Compounds, 378 (2004) 79-84.

[44℄ MATLAB version 7.14.0.739 (R2012a), The MathWorks In ., Nati k, Mas-

sa husetts, (2012).

[45℄ N. Stander, W. Roux, T. Goel, T. Eggleston, K. Craig, LS-OPT User's

manual, Version 4.2, Livermore Software Te hnology Corporation, Livermore,

CA (2011).

29