modelling of high temperature fatigue crack growth in ... · fatigue crack wth gro in inconel 718...
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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