International Journal of Fatigue 144 (2021) 106074
Available online 3 December 2020
0142-1123/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Modelling the crack growth behaviour of a single crystal nickel base
superalloy under TMF loading with long dwell times
Frans Palmert
a,b,*, Per Almroth
b, David Gustafsson
b, Jordi Loureiro-Homs
c, Ashok Saxena
d,
Johan Moverare
aaDivision of Engineering Materials, Dept. of Management and Engineering, Link¨oping University, 581 83 Link¨oping, Sweden bSiemens Energy AB, SE-61283 Finspång, Sweden
cDivision of Solid Mechanics, Dept. of Management and Engineering, Link¨oping University, 581 83 Link¨oping, Sweden dDepartment of Mechanical Engineering, University of Arkansas, Fayetteville, AR, USA
A R T I C L E I N F O Keywords:
Single crystal superalloy Creep Fatigue TMF Dwell Crack growth A B S T R A C T
The influence of hold time on the crack growth behaviour of a single crystal nickel base superalloy under in- phase thermomechanical fatigue is investigated. Two da/dt models were calibrated using creep crack growth tests in the temperature range 750–950 ◦C: one based on K and the other on (C
t)avg. The models were applied, in combination with a cycle dependent model, to predict da/dN of in-phase thermomechanical fatigue crack growth tests with 1–6 h hold time. The predictions based on K were inaccurate and generally non-conservative, whereas the predictions based on (Ct)avg were accurate. da/dt vs (Ct)avg followed a single trendline for all temperatures.
1. Introduction
Single crystal nickel base superalloys have an exceptional ability to withstand high temperatures in combination with high mechanical loads. Therefore, these materials are often used for gas turbine blades, where severe operating conditions are encountered, especially in the first stage of turbine blades. During steady-state operation of the gas turbine, the resistance towards creep and oxidation are of key impor-tance. The start-stop cycle of the gas turbine results in a cyclic variation of temperature and mechanical load, which may give rise to thermo-mechanical fatigue (TMF). Reliable predictions of the service life of the gas turbine blades requires knowledge of the material’s resistance to-wards the initiation and propagation of cracks under TMF loading, where the damage mechanisms of creep, oxidation and fatigue operate simultaneously. TMF crack initiation in single crystal nickel base alloys has been studied extensively [1–13]. However, less work has been published regarding TMF crack propagation [14–23]. Considering the influence of hold time on TMF crack growth, the published data is even more limited [16,17,21]. In general, both creep and oxidation may cause an increase of the TMF cyclic crack growth rate with increasing hold time. For the polycrystalline nickel-base alloy C1023, Kraemer et al. applied a TMF crack growth model incorporating the influence of hold time both via creep and oxidation. Bouvard et al. proposed a
phenomenological model for the single crystal nickel base superalloy AM1 to account for the interactions between creep, oxidation and fa-tigue, including overload effects. In both these studies the TMF crack growth testing used for validation of the models was limited to short hold times up to 5 min. A recent publication investigates the TMF crack growth behaviour of the same single crystal alloy studied in the present work, under the influence of hold times up to 6 h and temperatures up to 850 ◦C [23]. Under these conditions, a clear influence of hold time was observed on the crack growth rate. It was however shown that there was no significant crack growth during the actual hold time. Instead, the increase in crack growth rate with hold time could be attributed solely to a hold time influence on the crack closure behaviour [23]. In the present work, the TMF crack growth testing has been planned in such a way as to promote sustained load crack growth of sufficient magnitude to signif-icantly influence the overall crack growth rate. Isothermal creep crack growth tests and fatigue crack growth tests have also been performed in order to separate the influence of cycle dependent crack growth and time dependent crack growth for modelling purposes. Such a separation is of crucial importance, in order to enable the construction of crack growth rate models giving accurate predictions for hold times substan-tially longer than the hold times used in tests for model calibration. This is an important aspect especially for industrial gas turbines, with oper-ating cycles involving long times at high power output since testing with service-like hold times is prohibitively time consuming and expensive. * Corresponding author at: Division of Engineering Materials, Dept. of Management and Engineering, Link¨oping University, 581 83 Link¨oping, Sweden.
E-mail address: frans.palmert@siemens.com (F. Palmert).
Contents lists available at ScienceDirect
International Journal of Fatigue
journal homepage: www.elsevier.com/locate/ijfatigue
https://doi.org/10.1016/j.ijfatigue.2020.106074
The studied alloy was specifically developed as a turbine blade alloy for industrial gas turbines and therefore there is great technological interest in improving the understanding of its crack growth behaviour under TMF loading with long hold times [24]. The improvement of this un-derstanding is the primary aim of the present work. To this end two different crack growth models were evaluated against isothermal tests and then compared with respect to their ability to predict crack growth rates in IP-TMF crack growth tests with hold times of 1 h and 6 h.
2. Method
2.1. Material
The material tested is a single crystal nickel base superalloy devel-oped specifically as a turbine blade material for industrial gas turbines. The studied alloy is similar to the alloy described in a publication by Reed et al. [24]. Its main alloying elements, in order of decreasing wt%, are as follows: Ni-Cr-Ta-Co-Al-W-Mo-Si-Hf-C-Ce. The microstructure mainly consists of a γ-matrix precipitation strengthened by approxi-mately 50 vol% of γ′particles. The majority of the γ′is in the form of cuboidal particles, but smaller spheroidal γ′ particles are also present within the γ-channels. In addition, the microstructure also contains MC carbides. The fatigue crack growth behaviour of this alloy has been studied previously, but not under conditions that also include creep crack growth [22,23].
2.2. Test specimens
Two different single edge notch (SEN) specimens were used for the testing, see Fig. 1. Specimen SEN-1 has a U-notch, whereas specimen SEN-2 only has a small slit introduced by electro discharge machining. The different notch geometries are not believed to have had any sig-nificant influence on the results reported in the present work, since fa-tigue pre-cracking was done before the generation of crack growth rate
data, as described in Section 2.3. The nominal crystal orientation was 〈001〉 in the axial direction and 〈010〉 in the crack depth direction.
2.3. Testing procedure
The testing was performed using 100kN servo-hydraulic testing machines, using the same test setup and post-processing procedure as in Ref. [23]. Additional post-processing steps were required in the present work to obtain the creep displacement rate and calculate the crack tip parameters C* and (Ct)avg, as described in Section 2.3.2. The gauge length of the extensometer was 12 mm. The specimen was heated by an induction coil and cooled by compressed air, see Fig. 2.
The temperature was controlled using one thermocouple spot welded in the axial center of the gauge length on the back surface of the spec-imen, opposite the notch. The heating and cooling rate of the TMF tests was 2 ◦C/s. The current set up has been calibrated for the specific specimen geometry and material. Prior to testing, the temperature uniformity was verified using a specimen subjected to the thermal cycles subsequently to be used in the testing, with at least 6 thermocouples distributed within the gauge section. The temperature variation was within ±20 ◦C, if the entire gauge section is considered and significantly smaller in the region where the crack propagates. The thermal strain vs temperature was characterized by evaluating thermal cycles at zero force for each specimen before starting the test. The thermal strain was described using two individual functions, one for the heating ramp and one for the cooling ramp, as described in [23]. All TMF tests were conducted in force control with In-phase (IP) thermal cycling, in other words the maximum temperature coincides with the maximum force. Tests were conducted with maximum temperatures of 750 ◦C, 850 ◦C and 950 ◦C and hold times of 1 h or 6 h. The minimum temperature of all TMF tests was 100 ◦C. The TMF cycle is intended to simulate the start/ stop cycle of the gas turbine, where the minimum metal temperature of the gas turbine blades is typically close to room temperature. Arrell et al. have demonstrated the importance of using a representative minimum
Nomenclature
A Calibration parameter in Model 1
a Crack length
B Specimen thickness
C Calibration parameter in Model 1 and Model 2
C* C* integral. Crack tip parameter for extensive creep
conditions (described in Section 2.3.2)
Ct Crack tip parameter, for non-steady-state creep conditions
(described in Section 2.3.2)
(Ct)avg Average value of Ct during a hold time (correlated with da/
dt in Model 2)
(Ct)SSC Ct, evaluated under the assumption of small-scale creep conditions
D Calibration parameter in Model 2
F Geometry factor for K
δ Crack mouth opening displacement
δc Crack mouth opening displacement due to creep
deformation
δe Crack mouth opening displacement due to elastic
deformation
δext Displacement measured by extensometer
ΔK Stress intensity factor range:Kmax− Kmin
ΔKeff Effective stress intensity factor range:Kmax− Kopen
ηp Geometry factor for Jp
ηc Geometry factor for C*
ηCt Geometry factor for(Ct)SSC
G Elastic energy release rate
Jp Fully plastic J integral. Crack tip parameter, described in Section 2.3.2
K Stress intensity factor
Kmax Maximum stress intensity factor of the cycle Kmin Minimum stress intensity factor of the cycle Kopen Stress intensity factor at crack opening
Lo Width of the notch mouth of an unloaded specimen
Lo,ext Extensometer gauge length
m Calibration parameter in Model 1 and Model 2
N Number of fatigue cycles
mRO Ramberg-Osgood strain hardening exponent for power law
plasticity
n Norton stress exponent for power law creep
p Calibration parameter in Model 1
P Applied force
q Calibration parameter in Model 2
R Stress ratio
t Time
T Temperature
th Hold time in fatigue cycle
tT Transition time, for the transition from short-term to long
term behaviour
V Force line displacement
Vc Force line displacement due to creep
temperature in TMF testing of gas turbine blade alloys [25]. The mini-mum temperature of 100 ◦C was chosen in the present work to obtain TMF cycles representative of service conditions, whilst avoiding the practical difficulties associated with cooling to temperatures close to the ambient temperature. The testing conditions are summarized in Table 1. Isothermal cycling at elevated temperature (750 ◦C, 850◦or 950 ◦C) was used to create an initial Mode I crack of suitable length to start the TMF crack growth test. Isothermal crack growth data can be obtained from the pre-cracking. The length of the pre-cracks was at least 0.8 mm, measured from the notch root. Isothermal fatigue crack growth tests and creep crack growth tests were also conducted using the same test setup.
The crack length in each cycle was measured using the compliance method, adapted for TMF conditions, as described in previous publica-tions [23,25]. To enable compliance-based crack length measurement
also in the creep crack growth tests, a 10% unloading/reloading was done every 6 h. This means that the crack length measurement fre-quency was limited to 1 measurement every 6 h and the crack length could not be monitored during the hold time. The stress intensity factor
K was calculated based on 3D finite element simulations for the two
specimen geometries, under the assumption of isotropic elastic behav-iour. The crack opening stress and effective stress intensity factor range were evaluated using the method previously used by Palmert et al. [23]. An example of a typical IP-TMF cycle is shown in Fig. 3. Table 1 provides an overview of the performed testing.
2.3.1. Evaluation of crack mouth opening displacement
In the present testing, the displacement was measured by an exten-someter of 12 mm gauge length, centered around the notch, see Fig. 2b. In order to estimate C* and (Ct)avg using the approach described in Section 2.3.2, it was necessary to evaluate the crack mouth opening displacement (CMOD). FE simulations were performed to obtain a factor to convert the displacement measured by the extensometer, δext, to a corresponding CMOD, here denoted as δ. The simulations were done under the assumption of isotropic elastic material behaviour. δ was obtained from the simulations as the displacement across the mouth of the notch. The definitions of δ, δext are illustrated in Fig. 3, where Lo is the width of the notch mouth of an unloaded specimen and Lo,ext is the extensometer gauge length. Fig. 3 corresponds to specimen SEN-1. The definitions of δ, δext were similar for specimen SEN-2. The results for the two specimens SEN-1 and SEN-2 are also shown in Fig. 3, where a is defined as the total crack length, including the depth of the notch.
2.3.2. Correlating parameters for creep crack growth
The Ct parameter was used for correlating creep crack growth, as
proposed by Saxena [26,27]. The treatment was simplified by consid-ering the average value of Ct during each hold time, (Ct)avg. A weighted average approach was used to estimate (Ct)avg under conditions ranging from small-scale creep to extensive creep. The following part of this section will describe this estimation scheme in detail and the back-ground upon which it is based. The starting point is the well-established
C* integral, which is a crack tip parameter proposed independently by
Landes and Begley [28], Ohji et al. [29] and Nikbin et al. [30] to characterize crack growth under steady-state creep conditions. The definition of C* is obtained by analogy with the J integral by replacing strains with strain rates and displacements with displacement rates. For the estimation of C* in the present work, the energy release rate defi-nition of Jp and its creep analogue C* are used, according to Eq. (1) and
Eq. (2) respectively.
Jp= Pδp
B(W − a)ηp (1)
In Eq. (1), P is the applied force, δpis the crack mouth opening
displacement (CMOD) assuming fully plastic material behaviour, B is the specimen thickness, W is the specimen width, a is the crack length and ηp
is the geometry correction factor for Jp that also includes the relationship
between load-line displacement and CMOD. Accordingly, C* is given by:
C∗= P˙δc
B(W − a)ηc (2)
ηc=ηp n
n + 1 (3)
In Eq. (2), ˙δc is the CMOD rate due to creep and ηc is the geometry
factor for C*. Assuming power law creep ηc is obtained from Eq. (3),
where n is the Norton stress exponent for creep. The value of n as a function of temperature was obtained from Siemens’ internal database. It is noted that the estimation of C* is not particularly sensitive to the Fig. 1. Specimen drawings. a: Specimen SEN-1, b: Specimen SEN-2.
value of n. For n values in the range 5–12, ηc varies from 0.83 to 0.92. In
the present work, ˙δc is taken as an average over each dwell time, see Eq. (4) where Δδ is the measured total change in CMOD displacement during the dwell, Δδe is the elastic contribution to the CMOD due to
crack growth during the dwell and th is the hold time.
˙δc=
Δδ − Δδe
th (4)
This averaging approach was applicable for the creep crack growth tests, since they were not completely static, but involved a 10% unloading/reloading every 6 h. For the specimens analysed in the pre-sent work, a crack length independent geometry factor of ηp =1 is used in accordance with the ASTM E1457-19 creep crack growth testing standard [31]. It is noted that the measured creep displacement rate, as defined by Eq. (4), is not just due to secondary creep but involves all contributions to the displacement except the elastic displacement due to crack growth. The C* integral is defined under the assumption of global steady-state creep conditions and thus characterizes the long-term behaviour of a crack which grows at a sufficiently low rate to enable global steady-state creep conditions to develop. Various approaches have been proposed to describe the transition from short-time behaviour to long-term behaviour, such as the C(t) parameter [32,33] and the Ct
parameter [26,27]. These approaches take into account the stress
relaxation that occurs with time within the crack tip creep zone. Consequently, both C(t) and Ct are greater than C* at short times and
decay with time to approach C* for long times as steady-state creep is approached. Schweitzer performed a detailed analysis of the transient behaviour and presented an evolution equation for C(t) for elastic- viscoplastic material behaviour [34]. For the purposes of the present work, the evolution of C(t) with time is not analysed in detail. It is deemed sufficient to consider the transition time tT for the transition
from short-term to long term behaviour, as defined by Riedel and Rice according to Eq. (5).
tT= G
(n + 1)C∗ (5)
In Eq. (5), G is the elastic energy release rate and n is the Norton stress exponent for creep. For the calculation of tT the value of C∗is
obtained according to Eq. (2), with ˙δc calculated based on a secondary
creep model obtained from Siemens’ internal database. The measured value of ˙δc of Eq. (4) is not appropriate to use in this context since it
includes a significant contribution from small-scale creep. If the time required for significant crack growth is ≫tT, then C* is the appropriate
crack driving force parameter. If the time required for significant crack growth is ≪tT, then C(t) or Ct is the appropriate crack driving force
parameter. In the ASTM E1457-19 creep crack growth testing standard, Fig. 2. a: Test set-up for TMF crack growth testing. b: Cracked specimen in test rig, showing the location of the extensometer rods. The controlling thermo couple is
spot welded in the axial center of the gauge length on the back surface of the specimen, opposite the notch.
Table 1
Test matrix.
Test number Specimen Geometry Type of test Tmax [◦C] Stress ratio, R Hold time [h] Nominal maximum stress1 [MPa]
1 SEN-2 Isothermal fatigue 950 −1 0 100
2 SEN-1 Isothermal fatigue 950 −1 0 100
3 SEN-1 Isothermal fatigue 950 −2 0 75
4 SEN-1 Isothermal fatigue 950 −2 0 75
5 SEN-2 Isothermal fatigue 950 −2 0 75
6 SEN-2 Isothermal fatigue 950 0 0 75
7 SEN-1 Creep 950 0.9 6 100 8 SEN-1 Creep 950 0.9 6 150 9 SEN-1 Creep 850 0.9 6 150 10 SEN-1 Creep 850 0.9 6 250 11 SEN-1 Creep 750 0.9 6 250 12 SEN-2 IP-TMF 950 0 1 75 13 SEN-2 IP-TMF 950 −1 1 100 14 SEN-1 IP-TMF 950 −1 1 75 15 SEN-1 IP-TMF 950 −1 1 75 16 SEN-2 IP-TMF 850 −1 6 150 17 SEN-2 IP-TMF 850 −1 6 150 18 SEN-2 IP-TMF 850 0 1 75 19 SEN-2 IP-TMF 850 0 1 225 20 SEN-2 IP-TMF 850 −1 6 215 21 SEN-2 IP-TMF 750 0 1 200
the criteria below are given to determine when C* shall be used as crack driving force parameter to correlate with da/dt [31].
Validity criteria 1: t > tT (6) Validity criteria 2: ˙ Vc ˙ V ≥0.5 (7)
In Eq. (7), ˙V and ˙Vc denote the total force line displacement rate and
the force line displacement rate due to creep respectively. These validity criteria were obeyed in the analysis of the present creep crack growth tests. For the IP-TMF crack growth data presented, Validity criteria 2 is obeyed. As for Validity criteria 1, it is obeyed for the presented IP-TMF crack growth data only if one considers the accumulated dwell time. If one considers the time from the start of each dwell it is found that tT is
shorter than th in some tests but not in all tests. The influence of small-
scale creep was accounted for by using the Ct parameter proposed by
Saxena [26]. Under small scale creep conditions, Ct=(Ct)SSC, given by Eq. (8). (Ct)SSC= P˙δc B(W − a)ηCt (8) ηCt= F’ F W − a W∗˙δ/ ˙V (9) F =KB ̅̅̅̅̅ W √ P (10) F’ = dF d(a W) (11) In the above equations ˙V is the measured force line displacement rate and F is the geometry factor for K. The ratio ˙δ/˙δLL is a function of crack
length which was obtained via the h1, h2 and h3 functions for the SEN specimen, given in EPRI report NP-1931 [35], assuming a strain hard-ening exponent of mRO =20. The sensitivity to mRO was investigated by evaluating ˙δ/ ˙V for mRO =7 and the result was not significantly different from the ˙δ/ ˙V ratio evaluated with mRO =20. F’ was obtained as the
derivative of a polynomial fitted to F(a/W) from 3D finite element simulations. Since ˙δc is the average creep displacement rate during the
hold time, the value of (Ct)SSC obtained from Eq. (8) is the average value throughout the hold time. However, if there is a transition from small- scale creep conditions to steady state creep conditions during the hold time, Eq. (8) is only valid for the small-scale creep portion of the hold time, typically characterized by the transition time tT given by Eq. (5). In
the limit of steady state creep Ct =C* by definition of the two crack tip parameters [26]. For t ≈ tT there is a gradual transition from Ct = (Ct)SSC to Ct =C∗. Using the approximation of a sharp transition at t
T, the
average value (Ct)avgduring a hold time is given by Eq. (12). Eq. (12) was
used in the present work to estimate (Ct)avg, which is later used as a measure of the driving force for creep crack growth during the hold time. (Ct)avg= ⎧ ⎪ ⎨ ⎪ ⎩ (Ct)SSC for th≤tT tT(Ct)SSC+ (th− tT)C∗ th for th>tT (12) Eq. (12) estimates (Ct)avg for any length of hold time. For th≫tT, Eq. (12) predicts that (Ct)avg approaches C∗, which is a fundamental
requirement to obtain a reasonable interpolation from short-term to long-term behaviour. The value of tT was estimated from Eq. (5). This
should be regarded as a first order approximation, since the influence of the fatigue cycle is not accounted for. The influence of the fatigue cycle on the driving force for creep crack growth is discussed in Section 3.5.
2.3.3. Relevance of K
As described in Section 2.4, an attempt is made to correlate da/dt with the stress intensity factor during the dwell period. Since creep ductile behaviour is observed in the present tests, the physical basis for such a correlation is clearly questionable. Nevertheless, it is of interest to examine the potential correlation between da/dt and K from a purely empirical standpoint and compare the da/dt model based on K with the da/dt model based on (Ct)avg. Kraemer et al. used K to characterize creep crack growth in the conventionally cast superalloy C1023 and found the correlation with da/dt to be satisfactory, even though large scale in-elastic deformation was observed in the tests [16].
Fig. 3. Example of a typical IP-TMF cycle with 1 h hold time. The nominal stress is defined as the average stress on the original area of the smallest cross-section of
2.4. Crack growth models
The crack growth rate under IP-TMF loading is modelled using an additive model, as described by Eq. (13).
da dN= da dNcyclic+ da dNhold (13) da dNhold = ∫t h 0 da dtdt ≈ th⋅ da dtavg (14)
In Eq. (13), da/dN is the total crack growth rate per cycle, whereas da
dNcyclic represents the contribution form cyclic loading and dNhold da repre-sents the contribution to the crack growth during the hold time. For the model calibration, it is assumed that these two contributions can be treated as independent of each other. Furthermore, the approximation of Eq. (14) was used in all tests. This approximation was applicable to the creep crack growth tests since they were run with an unloading/ reloading every 6 h (th=6h). If crack growth during the hold time causes significant changes in crack driving force, then the approximation of Eq. (14) may not be able to capture the crack growth behavior of the ma-terial. In the present work, the crack growth during each hold time is in fact small enough for its influence on the crack driving force to be negligible in all tests. If this would not have been the case (e.g. in case of very long hold times), da
dNhold should be determined by integration over the hold time, without making the approximation of Eq. (14). da
dNcyclic is modelled using Paris’ law, with the effective stress intensity factor range
ΔKeff as as the correlating parameter, see Eq. (15), where C and m are
fitting parameters. ΔK and ΔKeff are defined in Eq. (16) and Eq. (17)
respectively, where Kmax and Kmin are the maximum and minimum stress
intensity factors of the cycle and Kopen is the stress intensity factor at
crack opening. Kopen is determined experimentally, using the
compliance-based method proposed by Palmert [23]. Since ΔKeff is used
as crack driving force parameter for the cycle dependent crack growth, the fitting parameters C and m are assumed to be independent of the stress ratio R, as defined by Eq. (18).
da dNcyclic
=C⋅ΔKeffm (15)
ΔK = Kmax− Kmin (16)
ΔKeff=Kmax− Kopen (17)
R =Kmin
Kmax (18)
Two different models are used to describe the hold time dependent crack growth, resulting in two different models to predict the overall crack growth rate da/dN. The overall crack growth rate models are referred to as Model 1 and Model 2, defined by Eq. (19) and Eq. (20) respectively. Model 1 assumes creep brittle behaviour and uses the stress intensity factor during the dwell, Kdwell, to characterize the hold time
crack growth. Model 2 assumes creep ductile behaviour and uses (Ct)avg, to characterize the hold time crack growth.
Model 1 da
dNModel1=
da
dNcyclic+th⋅A⋅Kdwell
p (19)
In Eq. (19) A and p are fitting parameters. In the present work, all tests are force controlled and Kdwell =Kmax in all tests.
Model 2 da dN= da dNcyclic +th⋅D⋅(Ct)avg q (20) da dtavg= 1 th [ da dN− da dNcyclic ] =D⋅(Ct)avgq (21)
In Eqs. (20) and (21), D and q are fitting parameters. The estimation of (Ct)avg in the present work is described in Section 2.3.2. It is assumed
that the crack growth during the hold time starts immediately as maximum load is reached. Similar models based on (Ct)avg have been successfully applied in previous work [36-38].
3. Results and discussion
3.1. Fatigue crack growth without hold time
The fatigue crack growth rate vs ΔKeff of the present alloy for no hold
conditions at 750 ◦C and 850 ◦C has already been reported in a previous publication [23]. It was shown that there was no significant difference in da/dN vs ΔKeff when comparing IP-TMF testing without hold time with
isothermal fatigue crack growth testing at the maximum temperature of the TMF cycle. The crack growth model from this previous publication is used to describe the cycle dependent crack growth, i.e. the first term of Eq. (13). For 950 ◦C, there is no previously published fatigue crack growth data for the present alloy. Isothermal fatigue crack growth tests were performed to generate data at 950 ◦C for no hold conditions. Three different stress ratios were used: R = 0, R = − 1 and R = − 2. The results and fitted Paris curve are presented in Fig. 4. When the crack growth rate is plotted against the full ΔK range, there is a clear influence of stress ratio, R. When plotting against ΔKeff all tests collapse along the same
trendline, as expected. ΔKeff was experimentally evaluated using the
compliance-based approach described in Ref. [23]. The trendline shown in Fig. 4 is the Paris curve used to describe the cycle dependent part of the crack growth rate in Model 1 and Model 2. The values of the fitting parameters C and m were obtained by manual optimization and are given in Table 2.
3.2. Creep crack growth
The results of creep crack growth testing are shown in Fig. 5 and Fig. 6, where da/dt is plotted against Kdwell and Ct respectively. For the
creep crack growth tests, the test duration is several orders of magnitude longer than the transition time tT and therefore Ct is assumed to be
identical to C*. The solid trendlines in these figures represent the da/dt models used to predict the contribution of hold time to the overall crack growth rate during IP-TMF as per Eq. (19) and Eq. (20). The dashed lines correspond to a factor 3 scatter band in da/dt. When plotted against
Kdwell, da/dt increases strongly with temperature, as seen in Fig. 5. This
is to be expected when the crack growth is governed by thermally activated processes such as creep and/or oxidation. When the same data is plotted against Ct, da/dt collapses along one trendline for all
tem-peratures, as seen in Fig. 6. This is regarded as an indication that the temperature dependence seen when correlating da/dt with Kdwell is a
direct consequence of the temperature dependence of the material’s creep resistance. When correlating da/dt with Ct, the temperature
dependence of the creep resistance is accounted for and thus the tem-perature dependence of da/dt vanishes. This leads to the conclusion that creep crack growth is dominant, and that oxidation does not have a significant influence on da/dt. The values of the fitting parameters D and q were determined by least-squares fitting, whereas A and p were found by manual optimization, using the same value of p for all temperatures. Parameters D, q and p are described as temperature independent, whereas A is temperature dependent.
3.3. Fitted model parameters
Two different crack growth rate models were evaluated in the pre-sent work, as described in Section 2.4. The calibration of the model parameters is described in Sections 3.1 and 3.2, with regard to cycle- dependent and time-dependent crack growth respectively. The fitted model parameters are given in Table 2, where the following units are
used: [MPa m1/2] for the stress intensity factor, [MJ m−2 h−1] for Ct, [mm/cycle] for da/dN and [mm/h] for da/dt.
3.4. IP-TMF crack growth
The models utilized to predict the crack growth rate under IP-TMF with hold times are described in Section 2.4, whereas the calibration
of fitting parameters is described in Sections 3.1 and 3.2. The experi-mental evaluation of the creep crack growth rate was done by separating
da
dtavg from the overall crack growth rate, in accordance with Eq. (21). The
transition time was evaluated using Eq. (5) and the result is shown in Table 3. Since the transition time varies throughout the test, an interval is given for each test. (Ct)avg was estimated using Eq. (12). For 6 out of the 10 IP-TMF crack growth tests, th≤tT throughout the entire test and thus (Ct)avg= (Ct)SSC.
In Fig. 7, da
dtavg is plotted against Kdwelland compared with the
pre-diction of Model 1, calibrated against isothermal creep crack growth tests. The observed creep crack growth rate is severely underpredicted for several of the tests, whereas the crack growth rate is slightly over-predicted for two of the tests. The inaccuracy of Model 1 is not sur-prising, since creep ductile behaviour was observed in the tests and thus
K is not an appropriate crack tip parameter to correlate with the creep
crack growth rate [31]. For the creep crack growth tests, plotted in Fig. 6, there may appear to be a useful correlation between Kdwell and da
dtavg. This correlation is however coincidental and specific to the
particular testing configuration, specimen geometry and loading Fig. 4. Ratio between the displacement measured by the extensometer (δext) and the simulated crack mouth opening displacement (δ), plotted vs crack length. Table 2
Fitted model parameters. The following units are used: [MPa m1/2] for the stress intensity factor, [MJ m−2 h−1] for C
t, [mm/cycle] for da/dN and [mm/h] for da/dt.
da dNcyclic
da dt
Eq. (15) Model 1: Eq. (19) Model 2: Eq. (20)
Temperature [◦C] C m A p D q
750 6.98E− 08 3 5.00E− 30 15 64.4 1.32 850 1.40E− 07 3.00E− 27
950 7.00E− 06 2 8.00E− 23
Fig. 5. Crack growth rates of isothermal fatigue crack growth tests at 950 ◦C. a: da/dN vs ΔK. b: da/dN vs ΔK eff.
conditions. In the creep crack growth tests, there is only a small varia-tion in the average stress on the remaining ligament and creep rate for data points at the same Kdwell and temperature. In the IP-TMF crack
growth tests, there is a larger variation in creep rate between tests at the same Kdwell and thus it becomes apparent that it is not Kdwell that governs da
dtavg. The (Ct)avg parameter is expected to be an appropriate parameter to correlate with da
dtavg for creep ductile materials [27]. In the present work,
(Ct)avg is evaluated based on measured creep rates, which means that the influence of load reversal on the creep rate is accounted for. In light of this, it is not surprising that (Ct)avg shows a good correlation with dadtavg of the IP-TMF tests.
In Fig. 8, da
dtavg is plotted against (Ct)avg and compared with the pre-diction of Model 2, calibrated against isothermal creep crack growth tests. The predicted da
dtavg based on Model 2 shows a good agreement with
the measured rates in the IP-TMF crack growth tests. Most of the test data falls within the same factor 3 scatter-band observed when corre-lating da
dtavg with Ct for the creep crack growth tests, see Fig. 6. However,
test 15, 17 and 21 fall outside the scatter band. In these tests, da dNcyclic is of equal or greater magnitude than da
dNhold and therefore the uncertainty of da
dNcyclic has a significant contribution to the uncertainty of dadtavg, evaluated
using Eq. (21). The value of da
dNcyclic in Eq. (21) is predicted based on the
cycle dependent crack growth model of Eq. (15). It is reasonable to as-sume that the actual value of da
dNcyclic may be up to a factor 3 higher or lower than the predicted value [23]. This uncertainty explains the larger scatter of da
dtavg vs (Ct)avgseen in Fig. 8, as compared to the scatter of dadt vs
Ct seen in Fig. 6.
The measured and predicted overall crack growth rates of the IP-TMF crack growth tests at 950 ◦C, 850 ◦C and 750 ◦C are plotted in Fig. 9, Fig. 10 and Fig. 11 respectively. The predicted crack growth rates are plotted as open markers whereas the measured crack growth rates are plotted as solid markers. Blue markers correspond to 1 h hold time whereas red markers correspond to 6 h hold time. As previously described, the crack growth models used for the predictions were cali-brated against isothermal fatigue crack growth tests without hold time and isothermal creep crack growth tests.
The observed da/dN vs predicted da/dN for the two models is plotted in Fig. 12. Model 1 yields inaccurate predictions for several of the tests. This is not surprising since Model 1 relies on assumptions of linear elasticity, whereas creep ductile behaviour is observed in the tests. Model 2 is accurate within a factor 3 except for Test 21 at 750 ◦C, for which da/dN is overpredicted by a factor 4, see Fig. 11. This is regarded as good accuracy and quantitatively in line with the expected scatter associated with crack propagation testing. For correlations with C*, Webster and Ainsworth recommend that test data in the range 0.5 ≤
˙
Vc ˙
V ≤0.8 shall be excluded if it does not follow the same general trend as
the data for which V˙c ˙
V≥0.8 [39]. Since Model 2 is based on (Ct)avg this
recommendation is not applicable, but for reference it has been indi-cated in Fig. 12 which of the IP-TMF data points that correspond to 0.5 ≤V˙c
˙
V≤0.8. In Fig. 13, tests with 1 h hold time are plotted in blue,
whereas tests with 6 h hold time are plotted in red. As the hold time is increased from 1 h to 6 h there are no signs that the accuracy of Model 2 decreases. The accuracy of the model for long hold times (<10 h) is of key importance from an industrial gas turbine blade lifing perspective. Even though Model 2 was calibrated using only isothermal tests, it is able to accurately predict the crack growth rate under IP-TMF with 1–6 h hold times. For the alloy studied, it was previously shown that there is no significant difference in cycle dependent fatigue crack growth rate under Fig. 6. da
dtvs Kdwell of isothermal creep crack growth tests, including fitted lines corresponding to Model 1. Dashed lines indicate a factor 3 scatter band in da/dt.
Table 3
Transition times of the IP-TMF crack growth tests.
Test number Tmax [◦C] Stress ratio, R Hold time [h] Transition time [h]
12 950 0 1 1.1–2.8 13 950 − 1 1 6,8 14 950 − 1 1 2.4–11.5 15 950 − 1 1 3–11.4 16 850 − 1 6 4.3–6.7 17 850 − 1 6 1.9–5.6 18 850 0 1 2.8–3.8 19 850 0 1 1.5–4.8 20 850 − 1 6 5.5–14.6 21 750 0 1 0.07–0.15
IP-TMF and isothermal fatigue testing at the maximum temperature, provided that crack closure effects are accounted for by correlating the data with ΔKeff [23]. The present results indicate that the way in which
Model 2 accounts for both crack closure effects and creep crack growth, enables it to capture the main differences between IP-TMF and isothermal crack growth behaviour. During IP-TMF, both the cycle dependent and the time dependent crack advance occur at the maximum temperature, which also corresponds to the maximum applied stress. In light of this, the difference between IP-TMF and isothermal testing is limited to a difference in the loading history preceding the crack advance. By using measured crack opening forces and creep rates as input to the model, the difference in loading history for each specific test is accounted for. If Model 2 were to be applied in fracture mechanics
analysis of a gas turbine component, measured creep rates and crack opening forces would not be available and the accuracy of the model would then rely heavily on the accurate prediction of crack opening forces and creep rates. An approach to predict the crack opening force under the influence of creep, using finite element simulations, has recently been demonstrated by Loureiro et al. [40].
3.5. Influence of the fatigue cycle on (Ct)avg
Grover and Saxena studied creep-fatigue interactions in creep fatigue crack growth tests on a service exposed steam turbine casing [41]. They showed that the creep strains induced by the hold time were partially reversed during unloading in creep-fatigue crack growth tests at R = 0.1. Fig. 7. da
dtvs Ct of isothermal creep crack growth tests, including fitted line corresponding to Model 2. Dashed lines indicate a factor 3 scatter band in da/dt.
Fig. 8. da
dt avgvs Kdwell of IP-TMF crack growth tests, compared with Model 1 calibrated against isothermal creep crack growth tests. Dashed lines indicate a factor 3
Fig. 9. da
dt avgvs (Ct)avg of IP-TMF crack growth tests, compared with Model 2 calibrated against isothermal creep crack growth tests. Dashed lines indicate a factor 3
scatter band in da/dt.
Fig. 10. Crack growth rate vs ΔKeff for 100–950 ◦C IP-TMF tests with 1 h hold time. Solid markers: measured da/dN. Open markers: Predicted da/dN. a: prediction according to Model 1. b: prediction according to Model 2.
b
a
Fig. 11. Crack growth rate vs ΔKeff for 100–850 ◦C IP-TMF tests with hold times of 1 h (blue) and 6 h (red). Solid markers: measured da/dN. Open markers: Predicted da/dN. a: prediction according to Model 1. b: prediction according to Model 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
By taking the measured creep reversal into account they could obtain a more accurate estimation of (Ct)avg. They also proposed a method to predict the extent of creep reversal based on secondary creep and cyclic plasticity models for the material. In the present work Eq. (5), is used to assess the transition time under creep fatigue. This implies the assumption that the combined influence of creep reversal and crack growth during the load reversal results in a transition time equal to that of the static loading case corresponding to Eq. (5). This assumption is reasonable if the cyclic plastic zone is much larger than the creep zone. On the other extreme, if it is assumed that the combination of creep reversal and crack growth during the load reversal has no influence at all on the stress redistribution necessary for the transition from small-scale creep to steady-state creep, then the transition time is not related to the time from the start of the hold time but instead related to the accumu-lated hold time of the creep-fatigue crack growth test. Under this assumption (Ct)avg is obtained from Eq.(22), where N is the cycle number. (Ct)avg= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (Ct)SSC for Nth≤tT (tT− (N − 1)⋅th)⋅(Ct)SSC+ (Nth− tT)⋅C∗ th for (N − 1)th<tT<Nth C∗ for tT≤ (N − 1)th (22)
This assumption is reasonable if the creep zone is much larger than the cyclic plastic zone. Applying this definition of (Ct)avg to the IP-TMF
test data of the present work results in (Ct)avg=C∗, except for the first cycle of some of the tests. When (Ct)avg estimated by Eq. (22) is used in conjunction with Model 2 to predict the overall crack growth rate of the IP-TMF crack growth tests, this yields a similar result as the previously described Model 2 prediction based on (Ct)avg estimated by Eq. (12). Therefore, no conclusion can be drawn as to which of the two (Ct)avg estimation is most appropriate in a more general case. Further testing with other component geometries would be required to provide more insight into the creep-fatigue interactions. The results for the two different (Ct)avg estimations are shown in Fig. 13 (see Fig. 14).
4. Conclusions
The present work investigates the influence of hold time on the crack growth behaviour of a single crystal nickel base superalloy under IP- TMF loading. Two different time dependent crack growth models were calibrated against isothermal creep crack growth tests: one which was based on the stress intensity factor Kdwell (Model 1) and one which was
based on (Ct)avg (Model 2). (Ct)avg was estimated using a weighted average approach in order to account for the transition from small-scale creep to extensive creep conditions. By superposition of the two Fig. 12. Crack growth rate vs ΔKeff for 100–750 ◦C IP-TMF tests with 1 h hold time. Solid markers: measured da/dN. Open markers: Predicted da/dN. a: prediction according to Model 1. b: prediction according to Model 2.
Fig. 13. Observed crack growth rate vs model predictions. a: Model 1, b: Model 2. Solid markers: Vc˙ ˙ V≥0.8. Open markers: 0.5 ≤ ˙ Vc ˙ V≤0.8.
different time dependent models with the same cycle dependent model, two different models were obtained to predict the total crack growth rate of IP-TMF crack growth tests with hold time. The following con-clusions were drawn.
•A strong temperature dependence was seen on da/dt vs Kdwell, whereas no significant temperature dependence was seen on da/dt vs (Ct)avg. This is regarded as an indication that the crack propagation during static loading occurred in the presence of creep deformation and the effects of oxidation, if any, were secondary.
•Model 1 yielded inaccurate and generally non-conservative pre-dictions of da/dN in the IP-TMF crack growth tests with hold time. This was not surprising, since creep ductile behaviour was observed in the tests and thus Kdwell is not expected to be an appropriate
parameter to correlate with da/dt.
•Model 2 yielded accurate predictions of da/dN in the IP-TMF crack growth tests with hold time. These results confirm the strong cor-relation between (Ct)avg and dadtavg reported in previous work [36-38]. It is encouraging that the crack growth rate under IP-TMF loading could be successfully predicted using a relatively simple model calibrated using only isothermal test data. By incorporating crack closure effects and creep crack growth, using measured crack open-ing forces and measured creep rates, Model 2 is able to describe both IP-TMF tests and isothermal tests with the same set of calibration parameters.
•It was shown that the result of the (Ct)avg evaluation for the present IP-TMF tests was rather insensitive to the assumptions made regarding the influence of the fatigue cycle on the crack tip stress field. This observation indicates that additional testing would be required to improve the understanding of creep-fatigue interactions in the studied alloy.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The work has been supported financially by Siemens Energy AB in Finspång, Sweden and the Swedish Energy Agency, via the Research
Consortium of Materials Technology for Thermal Energy Processes, Grant No. KME-702. The authors would also like to thank Claes Isaksson and Daniel Ewest at Siemens Energy AB for valuable input and discussions.
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