International Journal of Fatigue 57 (2013) 93–102

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Loading frequency and microstructure interactions in intergranular fatiguecrack growth in a disk Ni-based superalloy

Jinesh Dahal, Kimberly Maciejewski, Hamouda Ghonem ⇑Mechanics of Materials Research Laboratory, Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI 02881, USA

a r t i c l e i n f o

Article history:Received 9 May 2012Received in revised form 4 December 2012Accepted 17 December 2012Available online 27 December 2012

Keywords:Nickel-based superalloyCrack growth rateTransitional frequencySlip bandGrain boundary dislocations

0142-1123/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijfatigue.2012.12.009

⇑ Corresponding author. Tel.: +1 401 874 2909.E-mail address: ghonem@egr.uri.edu (H. Ghonem)

a b s t r a c t

The paper examines the role of the loading frequency on the dwell fatigue crack growth mechanism inthe super-solvus nickel-based superalloy, ME3. This is accomplished by carrying out a set of crack growthexperiments in air and vacuum at three temperatures; 650 �C, 704 �C and 760 �C using a dwell loadingcycle with hold time periods up to 7200 s imposed at the maximum load level. Results of these tests showthat the transitional transgranular/intergranular loading frequency is 0.1 Hz, and are used to determinethe apparent activation energy of the time-dependent crack growth process. Analysis of this energy inboth air and vacuum showed that the intergranular cracking is governed by a mechanism involving grainboundary sliding. This mechanism is explained in terms of absorption of dissociated lattice dislocationsinto grain boundary dislocations. The gliding of these dislocations under shear loading is assumed tocause grain boundary sliding. A condition for this mechanism to occur, is that a critical minimum distanceexists between slip bands impinging the affected grain boundary. This condition is examined by correlat-ing the slip band spacing (SBS) and loading frequency using a model based on minimum strain energyaccumulation within slip bands and that a unique configuration of number and spacing of bands existsfor a given plastic strain. The model outcome expressed in terms of SBS as a function of loading frequencyis supported by experimental measurements at both high and low loading frequencies. Results of themodel show that a saturation of SBS, signifying a condition for intergranualr cracking, is reached atapproximately 3 lm which is shown to coincide with the transitional loading frequency of 0.1 Hz.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

An important use of nickel-based superalloys, in jet engines, isin the manufacturing of the high pressure turbine disks. Thesecomponents are subjected to severe stresses (dwell-time loadingdue to normal start-flight-landing cycles, high cycle fatigue dueto vibrations, air flow and centrifugal stresses) at temperaturesreaching up to 760 �C (815 �C for military applications) [1]. As such,important design considerations of the disk are the selected mate-rial’s low cycle fatigue and cracking resistance parameters. Theselection of nickel-based superalloys as a disk material is basedon their high creep–fatigue resistance and the stability of theiryield strength over a wide range of temperatures as compared,for example, with high temperature titanium alloys. A large vol-ume of studies in literature have examined the damage mecha-nisms in nickel-based superalloys under different loadingprofiles, temperature levels and environmental conditions [2–10].In these, the loading frequency is considered to be a significantparameter that influences the crack growth rate. This is illustrated,for example, in the work of Clavel and Pineau [11] on Alloy 718 at

ll rights reserved.

.

550 �C over the range of 0.005–20 Hz, Ghonem and Zheng [12] onthe same alloy over the range 15–0.0167 Hz in both air and vac-uum environment, and in a recent work on IN100 alloy at 650 �Cand 700 �C and ME3 alloy at 650 �C, 704 �C and 760 �C over a widerange of loading frequencies [13–16]. The correlation betweenloading frequency and crack tip fracture mode was also extensivelystudied in references [12,17–19] with the conclusion for the sametest temperature, the crack tip fracture mode being transgranularor intergranular, depends primarily on the applied loading fre-quency. Several authors [2,3,20] have identified the role of theloading frequency with respect to a transitional loading frequency,ft, which defines the deformation response in the respective fatigueand creep damage process zones in the crack tip region. In this, thecycle-dependent damage process is correlated with loading fre-quencies higher than ft and occurs mainly along slip planes result-ing in transgranular fracture. On the other hand, time-dependentdamage operates at frequencies lower than ft and is therefore, athermally activated process characterized by intergranular fracture[13–16]. The transitional frequency was found to be material, load-ing and temperature dependent. For example, the work of Lu et al.[20] on Haynes 230 alloy in the range of 649–927 �C, has observedthat the transitional frequency increases with an increase in tem-perature under triangular and trapezoidal waveforms. In addition,

94 J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102

the work of Ghonem et al. [2,3], on alloy 718 at 650 �C, showed thatthis ft is a function of intrinsic material parameters includingchemical composition, precipitate character and grain boundarymorphology, load levels (DK, R), and environmental conditions(air and vacuum).

The role of loading frequency on the crack tip damage mode hasbeen interpreted in terms of the amount of slip dispersal occurringat a crack tip in a given cycle. This behavior has been studied byGhonem et al. [12,21] who considered the relationship betweenthe loading frequency, being a strain rate related parameter, andthe associated slip line density and resulting crack tip fracturemode. They showed that the slip line density increases proportion-ally to the loading frequency. At frequencies higher than the tran-sitional frequency, the increased slip density was assumed to leadto strain accommodation, as well as, stress relief along strainedgrain boundaries in the crack tip region. Cracking in this case pro-ceeds in a predominantly transgranular mode. This concept wasexamined by considering the influence of high frequency loadingon the subsequent low frequency crack growth behavior in nick-el-based alloy 718 at 650 �C through the use of a sequential high/low frequency load waveform. The parameters that have beenexamined include the crack growth rate, fracture surface morphol-ogy, and slip line density at and below the fracture surface. Resultsof this study show that prior application of high frequency loadingresults in reduction of the subsequent low frequency crack growthrate. This modification is interpreted as being a result of the cracktip conditioning through the increase in the slip line density duringthe high frequency part of the loading cycle. Tong and Byrne [22] intheir work on U720Li at 650 �C under variable triangular loading atfrequencies ranging from 0.0001 Hz to 5 Hz, have established amechanism map in terms of crack growth rate per cycle and load-ing frequency. Their conclusion is that the fracture mode is dic-tated by the degree of slip homogeneity at the crack tip; a highdegree of slip homogeneity results in a predominantly transgranu-lar fracture process and a lower degree in slip homogeneity resultsin an increase in intergranular cracking. In addition, as DK in-creases, the degree of slip homogeneity increases, thus, the transi-tional frequency decreases as the DK increases. Similarly,MacLachlan and Knowles [23] have characterized the deformationand failure mechanisms for the superalloy CMSX-4 as a function offrequency under conditions of creep and fatigue at three tempera-tures; 760 �C, 850 �C and 950 �C. They observed that as tempera-ture increases or frequency decreases, the crack tip deformationbecomes less heterogeneous and more similar to creep deforma-tion. Rho and Nam [24] have observed the changes in the fatiguefracture mode for Nb-modified A286 alloy at 550 �C as a functionof applied strain range. The fracture mode changed from transgran-ular to intergranular with the increase in the applied strain range.Under constant temperature, an increase in plastic strain range

25μm

(b)(a)

1μm

GB

Fig. 1. (a) Optical micrograph of the ME3 material, etched with waterless kallings for 15planar GB morphology. Secondary electron image of the ME3 material, etched with AGmatrix.

results in a decrease in the slip band spacing thus suggesting thatthe strain is accommodated at grain boundaries resulting in inter-granular damage.

In summary, studies in literature show that in nickel-basedsuperalloys, the loading frequency is an important parameterdetermining the high temperature fatigue crack growth and thecorresponding fracture mode. In addition, the slip band density isviewed as a measure of the localized plastic strain in the cracktip region and is found to be proportional to the loading frequency.These observations are used in the current work to examine thecrack growth mechanism in the disk superalloy, ME3. This willbe studied by performing high temperature crack growth experi-ments under a range of loading frequencies in order to identifythe transgranular/intergranular transitional frequency. Focusingon the intergranular fracture mechanism, a relationship betweenloading frequency and slip band density will be examined andthe coupling of this relationship with the transitional frequencywill be employed to interpret the observed time-dependent crack-ing path.

2. ME3 superalloy material

The material in this study is the nickel based superalloy, ME3.The average composition of this alloy (weight %) is 3.5Al, 3.7Ti,20.6Cr, 3.5Mo, 0.05Zr, 0.03B, 0.04C, 2.1W, 2.4Ta, 0.9Nb. The heattreatment of the as received material is a three stage process con-sisting of: (i) super-solvus solutioning stage at 1171 �C for one hourfollowed by cooling at 111 �C/min to room temperature, (ii) stabil-ization at 843 �C for four hours followed by air cooling, and (iii)aging at 760 �C for eight hours followed by air cooling. The averageRockwell C hardness of the as received material is 46.4 HRC. Itsmicrostructure (Fig. 1a), is an equiaxed grain structure, with anaverage grain size of 44 lm (ASTM 6) and planar grain boundarymorphology (see insert in Fig. 1a). The shape and distribution ofthe c0 precipitates exhibit a bimodal size distribution, where thelarger precipitate (Fig. 1b), secondary c0 (c0s), appear cubical, whilethe smaller ones (Fig. 1c), tertiary c0 (c0T ), seem to be spherical.Sizes and volume fractions of the c0s are 230 nm and 41%, respec-tively, while for c0T , the measurements are 42 nm and 10%, respec-tively. Additional details of the microstructure are given inreferences [14–16].

3. Crack growth rate and transitional frequency

Dwell-fatigue crack growth tests were performed on standardcompact tension (CT) specimens with dimensions following ASTME647, using servohydraulic fatigue testing machines with the crackgrowth measurements being monitored using the potential drop

500nm 200nm

(c)

s, showing the typical grain size and a secondary electron image insert showing the-21 for 15 s, showing (b) c0s size and distribution and (c) c0t particles within in the

Frequency (cycle/sec)10-5 10-4 10-3 10-2 10-1 100 101

da/d

N (m

/cyc

le)

10-7

10-6

10-5

10-4

10-3

30 MPa m40 MPa m50 MPa m60 MPa m

Region I

II

III

√√√√

Fig. 3. Typical crack growth rate, da/dN, versus frequency, f, for various values ofDK for ME3 at 704 �C in an air environment.

J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102 95

technique. Pre-cracking was first performed at room temperaturebetween 5 and 10 Hz. The loading cycle, during the test, consistsof 1.5 s loading, 1.5 s unloading and a dwell time ranging from0 s to 7200 s superimposed at the maximum load level with an ini-tial starting DK in the range of 28–32 MPa

pm. All tests were per-

formed at a stress ratio of 0.1 at three temperature levels; 650, 704and 760 �C in air and vacuum environments. For each vacuum test,the vacuum chamber pressure is maintained in the range of 10�7 -torr during the entire test. Results of these tests in terms of da/dNversus DK, as shown in Fig. 2, show that da/dN increases with bothtemperature and hold time duration. In addition, in air tests, thecrack growth rate is higher than that in vacuum.

The relationship between the crack growth rate and the loadingfrequency, f, is plotted in Fig. 3 which shows three regions, a time-dependent region (Region I), a transitional region (Region II), and atime-independent region (Region III). The transitional region oc-curs at approximately 0.1 Hz, termed a transitional frequency ft,and identifies the transition from transgranular fracture modeoccurring at f > ft, see Fig. 4c, to intergranular mode at f < ft, seeFig. 4a. The fracture at or near ft, occurs as a mixed mode, seeFig. 4b. In addition to air environment, intergranular fracture hasbeen observed in a vacuum environment for a loading cycle of1.5 s–300 s–1.5 s, at both 704 �C and 760 �C. Identifying the transi-tional frequency allows the selection of the crack growth curves inFig. 2 which correspond to time-dependent, intergranular fracture.These curves can then be expressed in terms of the crack growthspeed, da/dt, versus Kmax, as shown in Fig. 5. This figure shows thatwhile da/dt is temperature dependent, the crack growth curves atdifferent frequencies, for each test temperature, consolidate into asingle line. This points to the fact that da/dt for the same Kmax isindependent of the length of the hold time, indicating that thecracking process is continuous in nature and does not involve dam-age events that require an incubation time. As will be detailed laterin the paper, this conclusion precludes the crack tip oxide forma-tion as an active environmental damage mechanism. Fig. 5b showsthat the consolidation of da/dt for different hold times at 760 �C isnot complete, this may be due to variation in crack growth damagesensitivity at this temperature to environment (low DK) and creep(high DK). This high variation may not exist at the two lower tem-peratures; 650 �C and 704 �C. In addition, this temperature hasbeen shown in previous work [25] to be microstructurally unsta-ble. Since this is the aging temperature for this alloy, evidence ofcoarsening and dissolution of the tertiary c0 particles has been ob-served. Analysis of the crack growth process, presented in the nextsection, will only consider the average position of the data shownin Fig. 5b (the solid line), representing the typical behavior of ME3at 760 �C in air.

ΔK (MPa√m)20 30 40 50 60 75 90

da/d

N (m

/cyc

le)

10-7

10-6

10-5

10-4 650°C, Air, 1.5-1.5650°C, Air, 1.5-300-1.5650°C, Air, 1.5-600-1.5704°C, Air, 1.5-1.5704°C, Air, 1.5-100-1.5704C, Air, 1.5-300-1.5704°C, Air, 0.5-600-0.5760°C, Air, 1.5-1.5760°C, Air, 1.5-100-1.5760C, Air, 1.5-300-1.5760C, Air, 1.5-600-1.5704°C, Vac, 1.5-300-1.5760°C, Vac, 1.5-300-1.5

Fig. 2. da/dN versus DK at 650, 704 and 760 �C for various loading frequencies inboth air and vacuum environments.

4. Intergranular crack growth mechanisms

An important factor in identifying the cracking mechanisms intime-dependent fracture processes is the knowledge of the relatedapparent activation energy, Q. This parameter can be obtained firstby fitting the intergranular crack growth curves, as shown inshown in Fig. 5, being a result of a thermally activated damage pro-cess into the Arrhenius type equation:

dadt¼ A exp �Q

R1000

T

� �ð1Þ

where A is the frequency factor, Q is the apparent activation energyand is a function of Kmax, R is the universal gas constant and T is theabsolute temperature. Thus, for each of the different temperatures,the crack growth rate is calculated at different Kmax and plotted inFig. 6. The calculated values of Q in both air and vacuum environ-ment are plotted versus Kmax in Fig. 7. In this figure, the Q versusKmax relationship can be fit into a polynomial equation of the form:

Q ¼ Q0 þ Q aðKmaxÞ þ QbðKmaxÞ2 ð2Þ

where Q is in kJ/mol and the three constants Q0, Qa, and Qb are224.2, �1.6, and 7.39E�3 in air and 144.1, �0.2 and 0 in vacuum;respectively.

Results in Fig. 7 show that in the Kmax range 30–80 MPap

m, Qranges from 180 to 140 kJ/mol (air), and 140 to 130 kJ/mol (vac-uum), respectively. In general, the extreme cases of Q correspondto damage processes leading to one of two failure modes; thermaland athermal [26]. The thermal mode of damage occurs when thethermal activation, in the absence of stress, is sufficient to causefailure. In this case, the stress free energy corresponds to Kmax = 0in Eq. (2), thus yielding Q = Q0. On the other hand, athermal failureoccurs when the stress, in the absence of thermal activation, is suf-ficient to cause failure. This failure mode could occur as a single ora multiple mechanism process. An approach to identify the natureof this process is by examining the value of Q corresponding to theasymptote dQ/dKmax = 0. For a single athermal process, Kmax ap-proaches infinity or a threshold value (Kath

max) at which Q = dQ/dKmax = 0. On the other hand, a multiple damage process wouldcorrespond to a condition where at dQ/dKmax = 0, the value ofQ(Kath

max) approaches a finite value [27]. The athermal thresholdstress, Kath

max, is determined by equating the derivative of Eq. (2)with respect to Kmax, to zero which yields a value of 108 MPa

pm

in air at which the activation energy is a finite value of 138 kJ/mol indicating the presence of multiple damage processes. In vac-uum, since Q is idealized as a linear function of Kmax, Kath

max is takento be equal to that at Q = 0 which in turn identifies a single fracture

10μm 10μm 10μm

(b)(a) (c)

Fig. 4. Typical (a) intergranular, (b) mixed mode and (c) transgranular fracture surfaces corresponding to Regions I, II and III in Fig. 3; respectively.

Kmax (MPa√m)

25 30 40 50 60 70 85100

da/d

t (m

/sec

)

10-10

10-9

10-8

10-7

10-6

760°C / Air1.5-100-1.51.5-300-1.51.5-600-1.51.5-3000-1.5

760°C / Vac1.5-300-1.5

760°C / Air760°C / Vac

760°C / Vac

760°C / Air

Kmax (MPa√m)

25 30 40 50 60 70 85100

da/d

t (m

/sec

)

10-10

10-9

10-8

10-7

10-6

650°C / Air1.5-300-1.51.5-600-1.5

704°C / Air1.5-100-1.51.5-300-1.50.5-600-0.51.5-7200-1.5

704°C / Vac1.5-300-1.5

650°C / Air704°C / Air704°C / Vac

704°C / Vac

704°C / Air

650°C / Air

(b)(a)

√

760°C / Air1.5-100-1.51.5-300-1.51.5-600-1.51.5-3000-1.5

760°C / Vac1.5-300-1.5

760°C / Air760°C / Vac

760°C / Vac

760°C / Air

√

650°C / Air1.5-300-1.51.5-600-1.5

704°C / Air1.5-100-1.51.5-300-1.50.5-600-0.51.5-7200-1.5

704°C / Vac1.5-300-1.5

650°C / Air704°C / Air704°C / Vac

704°C / Vac

704°C / Air

650°C / Air

(b)(a)

Fig. 5. Crack growth rate in terms of da/dt versus Kmax at (a) 650 �C, 704 �C and (b) 760 �C in air and vacuum environments.

1000/T (1000/K)0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10

da/d

t (m

/sec

)

10-10

10-9

10-8

10-7

10-6

35 MPa√m45 MPa√m55 MPa√m65 MPa√m

1000/T (1000/K)0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03

da/d

t (m

/sec

)

10-10

10-9

10-8

10-7

35 MPa√m45 MPa√m55 MPa√m65 MPa√m

(a) (b)

Fig. 6. Crack growth rate, da/dt, versus the inverse of absolute temperature for different Kmax values for ME3 in (a) air and (b) vacuum environments.

96 J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102

process operating at the crack tip. The nature of the operatingdamage process is determined by comparing the current resultsof Q for ME3 to those identified in relation to known mechanismsby Starink and Reed [28]. The activation energy in vacuum in therange of 130–140 kJ/mol, corresponds to a single damage mecha-nism involving grain boundary (GB) creep. On the other hand,the values of Q in air are in the range of 140–180 kJ/mol whichidentifies a multi damage mechanism involving GB creep enhancedby oxidation processes. The environmental damage process in thisalloy has been mainly attributed to dynamic embrittlement associ-ated with oxygen diffusion. This is supported by the experimentalobservations that the da/dt is hold time independent at a constanttemperature, and the crack length and crack opening displacementas a function of number of cycles, are continuous in nature [16].This suggests, as mentioned earlier, that the cracking process doesnot involve damage events that require incubation time and that

environmental damage due to oxide formation can be precludedas a possible damage mechanism. Furthermore, creep damagecould be attributed to cavitation and/or GB sliding. Examinationof fracture surfaces has shown no evidence of cavities in this alloy,thus, the sliding mechanism is considered to be dominant. Thismechanism will be discussed further in the following sections.

5. Grain boundary sliding mechanism

Grain boundaries allow slip propagation from one grain to an-other by several modes including absorption of a dislocation intothe GB, absorption and emission and absorption–emission–reflec-tion. The stress required to transmit a dislocation across a bound-ary is shown by Shen et al. [29] to be higher than the macroscopicyield stress. On the other hand, Pond and Smith [30] showed that

Kmax (MPa√m)30 40 50 60 70 80

Q (k

J/m

ol)

120

130

140

150

160

170

180

190

Air

Vacuum

Fig. 7. Apparent activation energy as a function of Kmax in an air and vacuumenvironment.

J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102 97

the absorption of lattice dislocations by high angle grain bound-aries can occur by dissociation, at a rate limited by climb, intoGB dislocations with Burgers vectors characteristic of the natureof the GB. At high temperature, where the mobility of GB disloca-tions is high, dissociation of a trapped lattice dislocation is favoredover transmission across the GB [31]. Observations of stable dislo-cations parallel to the GB at high temperature by Kegg et al. [32]indicates that shear of the GB is due to the gliding of thesedislocations.

Expanding the disassociation concept into slip bands where mul-tiple dislocations interact with grain boundaries, the concept of localGB sliding is introduced by Sheikh-Ali and Szpunar [33]. This con-cept indicates that the GB sliding displacement is a summation ob-tained from the same oriented dislocation movement betweensections of two sessile dislocations. The dislocations entering theGB from two different grains would simply annihilate and nullifyGB sliding, whereas the sliding is observed only when dislocationsof the same sign move along the GB. Fig. 8a shows the case of trans-mission which leaves a sessile dislocation and does not contribute toGB sliding. Transmission is more favorable in symmetric grainboundaries which lack the incompatibility needed to produce glis-sile dislocations during sliding. Fig. 8b represents a case of compat-ible deformation in both the grains which is analogous to an equalnumber of slip lines in each grain. Due to annihilation of oppositesign dislocations, GB sliding is not observed. Fig. 8c (when w < wc)represents the case in which the slip band spacing, w, is small andall the glissile dislocations are obstructed, thus, does not allow

Lattice D Sessile G

Grain 1

Grain 2

Grain 1

Grain 2

GB GB

(a) (b)

Fig. 8. Two grains exhibiting no GB sliding for various cases depicting (a) transmission obands in the two grains meet at the GB [34]. (c) Two grains corresponding to two configurglissile dislocations are obstructed [34] and the second, when w > wc, in which the gliss

gliding displacement and hence no sliding is observed. In summary,all the three cases (when w < wc) presented in Fig. 8 lead to a similarcondition which has no free gliding dislocations or the strain in-duced by gliding dislocations nullifies due to the dislocation motionfrom the two neighboring grains. Thus, GB sliding is only observed ifa plastic incompatibility exits between the adjacent grains, produc-ing dislocations of similar signs with no obstructions [34].

The above argument indicates that GB sliding can occur by glid-ing of extrinsic glissile dislocations along the non equilibrium GBplane of two grains which have plastic deformation incompatibilityamong them. Furthermore, GB sliding requires low slip density orlarge slip band spacing. This condition provides larger mean freepath for the extrinsic gliding dislocations and insures that theobstruction of sessile dislocations is reduced. Vailev et al. [36] pro-posed a linear function between GB sliding and dislocation densityor dislocation mean free path. Another correlation presented byReading and Smith [37], shows that the chemical potential gradi-ent mean free path is inversely proportional to strain rate. Thus,a higher strain rate minimizes the GB sliding as the distance forannihilation or gliding becomes very small. This also supportsthe arguments presented in references [38,39] where an upperbound strain rate for the GB sliding mechanism exists and thespacing, being inversely proportional to the strain rate, definesthe sliding criteria. Thus, the length scale for GB sliding is the slipband spacing rather than the entire grain diameter as traditionallyused. This is experimentally observed by Sheikh-Ali and Szpunar[33] who showed that the sliding displacements on both sides ofa single GB, in zinc bicrystals, were different indicating a differentcharacteristic length scale for both grains. Although no correlationis made with slip band spacing, the explanation given is that thelocal structure influences GB sliding and is complementary to theabove stated concept which can be represented schematically inFig. 8c (when w > wc). A critical value of slip band spacing repre-sents a suitable space for slide hardening and mobility of the GB.

In order to examine the presence of GB sliding in ME3, a pre-testscribing method was used, as shown in Fig. 9a, in which surfacemeasurements of post-test GB sliding are taken as the amount ofshift in the scribe line in the direction parallel to the GB. These posttest measurements are carried out on crack growth CT specimenstested in air and vacuum at 760 �C. Typical results of the shift inscribe lines, i.e. GB sliding, are shown in Fig. 9b in air and Fig. 9cin vacuum. These measurements show that the average GB slidingdisplacement is in the range of 1–6 lm. Evidence of GB sliding hasalso been observed in specimens tested at 704 �C in a vacuum envi-ronment. While 650 �C results in air are intergranular, no attemptwas made to measure grain boundary sliding.

BD Glissile GBD

Grain 1

Grain 2

GB

w

(c)

f the lattice dislocation and (b) annihilation of the glissile dislocation when the slipations, the first is when w < wc, resulting in compatible deformation in which all theile dislocations are free to move resulting in GB sliding.

GBScribe Line

1μm10μm

(c)

GB

Scribe Line

5μm10μm

(b)

200μm

(a)

Fig. 9. (a) Optical micrograph showing the global scribing of a CT specimen withlines approximately 1 lm thickness. Post-test GB sliding observations in hold timetests at 760 �C in an (b) air and (c) a vacuum environment.

98 J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102

As discussed above, a condition for GB sliding is that the slipband spacing, w, intersecting the relative boundary must be equalto or exceed a critical spacing, wc. The limit of the GB length overwhich sliding would occur could be as long as the grain size whichwould provide a crack growth pattern in a discrete step-wise form.From crack length and crack opening displacement measurements,it is observed that the crack growth is a continuous process. This isalso supported by the general observation that in low cycle fatigue(LCF) tests, see Maciejewski [16], in which surface cracks are prop-agating by creep mechanisms, the GB openings are limited tolengths smaller than the grain size, Fig. 10a. Similar observations,indicating that GB sliding occurs along distances smaller than thegrain size, are obtained from surfaces of post-test crack growthspecimens as shown in Fig. 10b.

6. Slip band spacing as a criterion for intergranular cracking

The previous section showed that GB sliding occurs only when acritical sliding distance, wc, exists between slip bands intersectingthe affected grain boundary. This distance, as shown in Fig. 10, can

(a)

1μm

Fig. 10. Secondary electron micrograph showing surface GB openings (arrows) during (growth test at 760 �C in a vacuum environment. Both images indicate that GB sliding o

be determined by considering the interactions between lattice dis-locations within the slip band and GB using the assumption thatmacroscopic strain, during cyclic loading, is distributed eitherhomogenously or heterogeneously along a collection of slip bands[17,40]. The plastic deformation is concentrated locally along thesebands, which generally extends between two grain boundaries andthe optimum number and spacing of slip bands can be estimatedbased on a minimum energy configuration which indicates the ab-sence of long range stresses in the bulk [41]. This section will focuson extending the work of Venkataraman et al. [40] to model theslip band spacing as a function of loading frequency. This modelis based on the criterion of minimum strain energy accumulationwithin a slip band. This requires that a unique configuration ofnumber and spacing of slip bands exists within a grain for a givenplastic strain.

A series of slip bands within a grain are characterized by theirspacing, w, the width of a slip band, h, and the pile-up length, a,as schematically illustrated in Fig. 11. The spacing is inversely pro-portional to the number of slip bands and is a function of loadingand microstructure parameters. In this configuration, it is assumedthat each band contains two slip planes, one containing m positivedislocation dipoles and one containing m negative dislocationdipoles.

Following the work of Venkataraman et al. [40] the total slipdisplacement on all bands within the grain shown in the above fig-ure is calculated as Nl, where N is the total number of dipoles on allof the slip bands and l is the average distance moved by a single di-pole (multiple of the Burger’s vector b) and is expressed as:

Nl ¼ enadc að1� vÞ

2ðDs� 2kÞ

G�c ð3Þ

where e is the effective slip irreversibility factor, nc is the number ofcycles, and ad is an exponent between 0.5 and 1, a is the pile-uplength, �c is normalized plastic displacement, v is Poisson’s ratio, Gis shear modulus, b is Burger’s vector, Ds is the applied shear stressrange, and 2k is the effective frictional resistance. The factor of 2 ac-counts for the frictional stress from the two layers of neighboringplanes consisting of positive and negative dislocation dipoles. Theexponent ad relates the creation of dipoles (can be seen as the effectof dislocation density) with the number of cycles. The number ofnew dipoles in the slip band decreases as the number of cycles in-creases since the back stress in the slip band encourages reversibil-ity of the slip.

The total cumulative plastic shear strain averaged over m slipbands gives the average plastic shear strain, Cav, stored in eachband as:

Cav ¼ enadc cp ¼

Nlwm

ð4Þ

where cp, is the applied plastic shear strain amplitude. From above,the plastic shear strain in the slip bands, cband, is calculated as:

5μm

(b)

a) a LCF test (760 �C, 1.5% strain range, 1 � 10�4 s�1) and (b) a dwell-fatigue crackccurs over a distance comparable with the slip band spacing.

w

h

aGB

Fig. 11. Schematic of an idealized grain, showing m slip bands with accumulateddislocation dipoles, where w is the slip band spacing, h is the slip band width and ais the pile up length.

Strain (mm/mm)-0.04 -0.02 0.00 0.02 0.04

Stre

ss (M

Pa)

-1500

-1000

-500

0

500

1000

1500

Δσ2σF

-σm

σy

- σy

σm

Fig. 12. Cyclic stress strain curve for an LCF test on ME3 at 760 �C under straincontrol at a strain rate of 1e�4 s�1 where ry is the onset of plastic deformation, rm

is the maximum stress, Dr is the stress range and rF is the frictional stress.

J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102 99

cband ¼Cav

enadc¼ Nl

enadc wm

¼ ð1� vÞ awðDs� 2kÞ

2G�cm

ð5Þ

For a grain consisting of a matrix material with a volume fraction ofslip bands, vf, the average plastic shear strain (applied plastic shearstrain) within a grain can be described using the law of mixtures fora two phase material accounting for cband and cmatrix, the averageshear strain in the slip bands and matrix; respectively. The plasticshear strain is assumed to be localized in the slip bands since cband

is much greater then cmatrix, thus cp is reduced to:

cp ¼ v f cband ¼ v f ð1� vÞ awðDs� 2kÞ

2l�cm

¼ ð1� vÞ ahw2

ðDs� 2kÞ2G

�cm

ð6Þ

where the volume fraction of slip bands, vf, is approximated by h/w.Solving the above equation for the slip band spacing, w, yields:

w2 ¼ ð1� vÞ ahcp

ðDs� 2kÞ2G

�cm

ð7Þ

The total applied strain in the material can be accommodatedby different combinations of m and w. A unique configuration ofm and w is the optimum combination that has the minimum inter-nal energy stored within the bands or the lowest energy disloca-tion structure (LEDS) to maintain a stable thermodynamicsystem. The basic characteristic of LEDS is that there is no longrange stress field in the slip band which stratifies the laws of ther-modynamics [41]. The presence of a long range stress field evolvesthe dislocation structure during cyclic loading which has been cat-egorized in various stages [42]. The current model, which assumesa saturated dislocation structure of parallel slip bands with no newslip bands forming, is described as a Taylor lattice structure[41,43]. Furthermore, Venkataraman et al. [40] concluded thatthe optimum number of slip bands per grain, m, is linearly depen-dent on normalized plastic displacement, �c; by a factor of 0.36. Thework of Venkataraman et al. imply that the factor 0.36 is indepen-dent of the alloy under consideration, they have compared theirtheoretical work to that obtained experimentally for polycrystal-line copper and results agree well. Although this work has not con-sidered the effect of loading rate or variation of time duringdeformation, the current paper attempts to include these effects.The introduction of time-dependent deformation would result in

an increase in the strain accompanied with a reduction of disloca-tion dipole density by recovery since time dependent deformationtends to increase the homogeneity. This would lead to a lowernumber of dislocation dipoles in the slip bands and a reductionof the back stress, which depends on the length of dipole pile-up,thus resulting in a reduction of the number slip bands formed inthe grain. It is then proposed that a lower number of slip bandscan accommodate larger strains with fewer dislocation dipoleswhile maintaining the stress equilibrium. As a result, a modifiedrelation for the optimum number of slip bands is derived as [14]:

m ¼ �cB expðB0f Þ ð8Þ

where f is the loading frequency, B and B0 are scaling parameters.Substituting Eq. (8) into Eq. (7) yields the optimum slip band spac-ing as a function of loading frequency as:

w2 ¼ ð1� vÞB expðB0f Þ

ahcp

ðDs� 2kÞ2G

ð9Þ

Using the Taylor factor, M, the local parameters can be converted toglobal parameters by the following relationships:

cp ¼ Mep; Ds ¼ Dr=M; k ¼ rF=M ð10Þ

where ep is the applied tensile plastic strain, Dr is the remote ten-sile stress, and rF is the frictional stress measured from macroscopichysteresis loops. Substituting the relationships given above in Eq.(9) yields an equation describing the slip band spacing as a functionof global loading parameters as:

w2 ¼ ð1� vÞah

2M2GB

ðDr� 2rFÞep expðB0f Þ

ð11Þ

For the same applied stress and plastic strain, the expression gi-ven by Eq. (11) shows that slip band spacing is smaller with higherfrequency. The material constants in this equation are determinedfollowing the procedure outlined in reference [14]. Values for theapplied tensile plastic strain, ep, tensile stress range, Dr, and fric-tional stress, rF, were determined from LCF tests performed onsmooth cylindrical specimens at 760 �C, see the work of Maciejew-ski [16]. A Ramberg–Osgood relationship [44] was fit to stressstrain data. This relationship is described as:

ep ¼ 2Dr2Y

� �1=p

ð12Þ

where p and Y are material constants. The tensile stress range, Dr,is taken to be 2000 MPa and the corresponding plastic strain is

Table 1Material constants used to define the slip band spacing given in Eq. (11).

Parameter Value Units Description References

h 0.2 lm Slip band width [45]d 44 lm Grain sizev 0.30 Poisson’s ratioM 3.06 Taylor factor [46]Dr 2000 MPa Applied stress [14]rF 750 MPa Frictional stress [14]Y 2571 MPa Cyclic plastic modulus in Eq. (12) [14]p 0.236 Stress sensitivity parameter in

Eq. (12)[14]

G 66142 MPa Shear modulus [16]B0 2.80 S/

cycleConstant for m in Eq. (8) [14]

B 0.006 Constant for m in Eq. (8) [14]

Frequency (cycle/sec)10-4 10-3 10-2 10-1 100 101

Slip

Ban

d Sp

acin

g (m

icro

n)

0

1

2

3

4

NumericalExperimental( ft, wc )

II IIIRegion I

Fig. 13. Slip band spacing as a function of loading frequency with Regions I, II andIII corresponding to those in Fig. 3. Note that the solid line represents Eq. (11), thesolid circles correspond to experimentally measured points and the X represents theposition on the solid line corresponding to the transitional frequency and criticalslip band spacing.

100 J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102

determined from Eq. (12). The value of frictional stress was ob-tained from saturated stress strain curves using the guidelines fromreference [45] as:

2rF ¼ ry þ rm ð13Þ

where ry is the onset of plastic deformation or the yield stress at thesaturated cyclic loop (measured as the stress value at the onsetdeviation from elastic linearity of the stress–strain curve) and rm

is the maximum stress level during loading. These stress valuesare indicated in Fig. 12 for a saturated cyclic stress strain curve.The value of rF used is 750 MPa and the constants B and B0 weredetermined from comparison with experimental data. A list of thematerial constants used in this study is given in Table 1. Using

10μm

(a)

Fig. 14. Secondary electron micrographs of unetched specimens post-testing showing tyvacuum corresponding to Region I in Fig. 13 and (b) a high frequency test of 0.5 s–0.5 s

the material constants given in this table, Eq. (11) is plotted as afunction of loading frequency. The resulting relationship which isshown in Fig. 13, has been verified by comparing it with experimen-tal results. For this, two measurements of slip band spacing are car-ried out. The first corresponds to a high frequency test of 0.5 s–0.5 sat 704 �C in air, see Fig. 14a, which measures 0.65 lm, and the sec-ond belongs to a low frequency test of 1.5 s–300 s–1.5 s at 760 �C invacuum, see Fig. 14b, which measures 3.31 lm. It should be notedhere that these measurements were taken for multiple micrographsand the average is reported. These two experimental measurementsas well as the transitional frequency are superimposed on the the-oretical w versus f curve in Fig. 14. This figure indicates that a min-imum slip band spacing of 3 lm is required in order to insure theGB sliding and, therefore, below this distance, the fracture processis trangranular while above it, the fracture mode is intergranular.Furthermore, the environment, being a localized phenomena, onlyaffects the grain boundaries in the immediate vicinity of the cracktip, whereas the slip bands are formed in the continuum withinthe grains. As such, environment is considered to have no markedinfluence on the slip band spacing.

One can then summarize the outcome of this study by statingthat the transition between transgranular and intergranular frac-ture modes is viewed here in terms of slip band spacing, which isa measure of the degree of deformation compatibility along theGB path in the crack tip region [14]. During high loading fre-quency where the slip band density is high and the slip bandspacing is smaller than wc, a state of deformation compatibilitywould exist along the GB, thus allowing slip transmission. In thiscondition, no stress build up would develop along the GB, and fa-tigue failure is expected to proceed along slip planes resulting intransgranular fracture. On the other hand, during low loading fre-quency accompanied by a larger slip band spacing approachingwc, an incompatibility would develop along the GB making trans-mission of dislocations difficult. In this condition, a stress concen-tration is built up at the GB causing the lattice dislocation todissociate into two extrinsic GB dislocations [31,32,35–37]; oneparallel to the GB plane and the other perpendicular to it. Undershear traction and providing the distances between slip bands areequal or larger than wc the gliding of the parallel dislocationscauses GB sliding. GB failure would occur when the glissile dislo-cations reach a GB obstruction [32] thus resulting in intergranularfracture. With fewer slip bands, the obstruction of sessile disloca-tions is reduced, thus, increasing the available sliding distance. InFig. 13, the critical slip band spacing, above which GB slidingoccurs (intergranular fracture), corresponds to the spacing mea-sured at the transitional frequency, w(ft), taken as approximately3 lm. This critical slip band spacing, wc, represents a minimumlength for mobility of the gliding GB dislocations in order tocreate an incompatibility between the two neighboring grainsleading to GB sliding.

5μm

(b)

pical slip band spacing for (a) a low frequency test of 1.5 s–300 s–1.5 s at 760 �C inat 704 �C in air corresponding to Region III in Fig. 13.

J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102 101

7. Conclusions

The work described in this paper has examined the role of theloading frequency on the dwell fatigue crack growth mechanismin the supersolvus nickel-based superalloy, ME3. This is accom-plished by carrying out a set of dwell crack growth experimentsin air and vacuum at three high temperature levels. Results of thesetests show that an increase in temperature and/or dwell timeaccelerates the crack growth rate. Furthermore, a transgranular/intergranular transitional frequency, ft, is determined to be in therange 0.1 Hz for the three temperature conditions. The crackgrowth mode is found to be predominately intergranular for fre-quencies below ft, whereas above this frequency, the fracture modeis transgranular. A mixed mode of failure with characteristics ofboth intergranular and transgranular fracture is observed in the re-gion near the transitional frequency.

Analysis of the intergranular crack growth rates for the threetemperatures provided a means of measuring the apparent activa-tion energy, Q, for the intergranular cracking process. Q, is found tobe in the range of 130–140 kJ/mole in vacuum reaching 140–180 kJ/mole in air environment. This apparent activation energyis correlated in a nonlinear form with Kmax, the analysis of whichshows that the crack growth kinetics is a dual mechanism involv-ing creep by GB sliding and environmental degradation due to dy-namic embrittlement through oxygen diffusion.

In order to provide a rationale for the GB sliding, it is proposedthat for frequencies lower than ft, the dissociated lattice disloca-tions along the intersection of a slip band and GB are absorbed inthe grain boundaries. The gliding of these dislocations under shearloading, causes GB sliding. It is then suggested that a condition forthis mechanism to operate, is that a critical minimum distance be-tween pinning points exists, corresponding to the spacing betweenslip bands. These concepts assume that a unique relationship existsbetween loading frequency and corresponding slip band spacing.This relationship is verified through the development of an analyt-ical model based on minimum strain energy accumulation withinslip bands during reversal loadings. The model outcome in termsof slip band spacing versus the loading frequency is supported byexperimental measurements at both high and low loading frequen-cies. Results of the model show that a saturation of SBS, signifyinga condition for intergranular cracking mode, is reached at approx-imately 3 lm which is shown to coincide with the transgranular/intergranular transitional loading frequency of 0.1 Hz.

Acknowledgements

The authors acknowledge the support from the MAI (MetalsAffordability Initiative) program (2008-2011), in collaborationwith the Air Force Research Lab, Pratt & Whitney, GE Aviation,Georgia Institute of Technology and Ohio State University.

References

[1] Furrer D, Fecht H. Ni-based superalloys for turbine discs. JOM 1999:14–7.[2] Ghonem H, Nicholas T, Pineau A. Elevated temperature fatigue crack growth in

alloy 718 – Part I: effects of mechanical variables. Fatigue Fract Eng MaterStruct 1993;5:565–76.

[3] Ghonem H, Nicholas T, Pineau A. Elevated temperature fatigue crack growth inalloy 718 – Part II: effects of environmental and material variables. FatigueFract Eng Mater Struct 1993;16:577–90.

[4] Pineau A, Antolovich S. High temperature fatigue of nickel-base superalloys – areview with special emphasis on deformation modes and oxidation. EngFailure Anal 2009;16:2668–97.

[5] Garimella L, Liaw PK, Klarstrom DL. Fatigue behavior in nickel-basedsuperalloys: a literature review. J Minerals 1997:67–71.

[6] Onofrio G, Osinkolu GA, Marchionni M. Fatigue crack growth of UDIMET 720 Lisuperalloys at elevated temperature. Int J Fatigue 2001;23:887–95.

[7] Dalby S, Tong J. Crack growth in a new nickel base superalloys at elevatedtemperature – Part 1: effects of loading waveform and frequency on crackgrowth. J Mater Sci 2005;40:1217–28.

[8] Djavanroodi F. Creep–fatigue crack growth interaction in nickel basesuperalloys. Am J Appl Sci 2008;5:454–60.

[9] Liu X, Bruce K, Chang KM. The effect of hold-time on fatigue crack growthbehavior of Washpaloy alloy at elevated temperature. Mater Sci Eng2003;340:8–14.

[10] James LA. The effect of grain size upon the fatigue-crack propagation behaviorof alloy 718 under hold-time cycling at elevated temperature. Eng Fract Mech1986;25(3):305–14.

[11] Clavel M, Pineau A. Frequency and wave-form effects on the fatigue crackgrowth behavior of alloy 718 at 298 K and 823 K. Metall Transas A1978;9A:471–80.

[12] Ghonem H, Zheng D. Frequency interaction in high temperature fatigue crackgrowth in superalloys. Metall Mater Trans A 1992;23:3067–72.

[13] Kirchhoff S. Effects of microstructure and environment on intergranular crackgrowth behavior of a powder metallurgy nickel based superalloy. Maste’sthesis. University of Rhode Island; 2008.

[14] Dahal J. Grain boundary deformation and damage mechanisms inintergranular crack growth of a nickel based superalloy. Master’s thesis.University of Rhode Island; 2011.

[15] Ghonem H, Maciejewski K. Mechanistic modeling of dwell fatigue crackgrowth in P/M nickel-based superalloys using a cohesive zone approach.Report, Metals Affordability Initiative Program. AFRL; 2012.

[16] Maciejewski K. The role of microstructure on deformation and damagemechanisms in a Ni-based superalloy at elevated temperatures. PhD thesis.University of Rhode Island; 2012.

[17] Gell M, Leverent GR. Mechanisms of high temperature fatigue. FatigueElevated Temp 1973:37–66.

[18] Liu XB, Ma LZ, Chang KM, Barbero E. Fatigue crack growth of Ni basesuperalloys. Acta Metall 2005;18:55–64.

[19] Pelloux RM, Huang JS. Creep–fatigue–environment interactions in Astroloy, increep–fatigue–environment interactions. In: Pelloux RM, Stoloff NS, editors.Conference proceedings. 1980. p. 151–164.

[20] Lu YL, Chen LJ, Liaw PK, Wang GY, Brooks CR, Thompson SA, et al. Effects oftemperature and hold time on creep–fatigue crack-growth behavior ofHAYNES� 230� alloy. Mater Sci Eng A 2006;429:1–10.

[21] Zheng D, Rosenberger A, Ghonem H. Influence of prestraining on hightemperature, low frequency fatigue crack growth in superalloys. Mater SciEng A 1993;161(1):13–21.

[22] Tong J, Byrne J. Effects of frequency on fatigue crack growth at elevatedtemperatures, Blackwell Science Ltd. Fatigue Fract Engng Mater Struct1999;22:185–93.

[23] MacLachlan DW, Knowles DM. Fatigue behaviour and lifing of two singlecrystal superalloys, Blackwell Science Ltd. Fatigue Fract Engng Mater Struct2001;24:503–21.

[24] Rho BS, Nam SW. The effect of applied strain range on the fatigue cracking inNb-A286 iron-base superalloy. Mater Lett 2001;48:49–55.

[25] Dahal J, Maciejewski K, Ghonem H. In: Huron ES et al., editors. Grain boundarydeformation and damage mechanisms in dwell fatigue crack growth in turbinedisk superalloy ME3, superalloys. Warrendale (PA): The Minerals, Metals andMaterials Society; 2012. p. 149–58.

[26] Li J. The mechanics and physics of defect nucleation. MRS Bull 2007;32:151–9.[27] Hall Jr. MM. Thermally activated dislocation creep model for primary water

stress cracking of NiCrFe alloys. In: Paper A-I-09, proceedings of theinternational symposium on plant aging and life prediction of corrodiblestructures, Sapporo, Japan, May; 1995.

[28] Starink MJ, Reed PAS. Thermal activation of fatigue crack growth: analysingthe mechanisms of fatigue crack propagation in superalloys. Mater Sci Eng A2008;491:279–89.

[29] Shen Z, Wagoner RH, Clark WAT. Dislocation and grain boundary interactionsin metals. Acta Metall 1988;36(12):3231–42.

[30] Pond RC, Smith DA. On the absorption of dislocations by grain boundaries.Philos Mag 1977;36(2):353–66.

[31] Yoshida H, Yokoyama K, Shibata N, Ikuhara Y, Sakuma T. High temperaturegrain boundary sliding behaviour and grain boundary energy in cubic Zirconiabicrystals. Acta Metall 2004;52:2349–57.

[32] Kegg GR, Horton CAP, Silcock JM. Grain boundary dislocations in aluminumbicrystals after high temperature deformation. Philos Mag 1973;27:1041–55.

[33] Sheikh-Ali AD, Szpunar JA. Stimulation and suppression of grain boundarysliding by intragranular slip in Zinc bicrystals. Interface Sci 2003;11:439–50.

[34] Sheikh-Ali AD. On the contribution of extrinsic grain boundary dislocations tograin boundary sliding in bicrystals. Acta Metall 1997;45:3109–14.

[35] Sheikh-Ali AD. On the influence of intragranular slip on grain boundary slidingin bicrystals. Scripta Metall et Mater 1995;33(5):795–801.

[36] Valiev RZ, Khairullin VG, Sheikh-Ali AD. Phenomenology and mechanisms ofgrain boundary sliding. Russ Phys J 1991;34(3):253–61.

[37] Reading K, Smith D. A dynamic lattice dislocation grain boundary slidingmechanism. Philos Mag A 1985;51:71–8.

[38] Gandhi C, Raj R. An upper bound on strain rate for wedge type of fracture innickel during creep. Metall Trans A 1977;12A:515–20.

[39] Cosandey F. Grain boundary internal friction and relationship to intergranularfracture. J De Phys 1988;C5(10):581–6.

[40] Venkataraman G, Chung YW, Mura T. Application of minimum energyformalism in a multiple slip band model for fatigue I – calculation of slipband spacing. Acta Metall 1991;39:2621–9.

[41] Kuhlmaan-Wilsdorf D. Theory of plastic deformation: properties of low energydislocation structures, in low-energy dislocation structures II. In: Bassim MN,

102 J. Dahal et al. / International Journal of Fatigue 57 (2013) 93–102

Jesser WA, Kuhlmann-Wilsdorf D, Shiflet GJ, editors. 2nd Internationalconference on low energy dislocation structures. Material Science andEngineering. p. 1–41.

[42] Clavel M, Pineau A. Intergranular fracture associated with heterogeneousdeformation modes during low cycle fatigue in Ni-base superalloy. ScriptaMetall 1982;16:361–4.

[43] Shyam A, Milligan WW. Effects of deformation behavior on fatigue fracturesurface morphology in a nickel-base superalloys. Acta Mater 2004;52:1503–13.

[44] Skelton RP, Maier HJ, Christ HJ. The bauschinger effect, masing model and theramberg–osgood relation for cyclic deformation in metals. Mater Sci Eng A1997;A238:377–90.

[45] Kuhlmaan-Wilsdorf D, Laird C. Dislocation behavior in fatigue II. Friction stressand back stress as inferred from an analysis of hysteresis loops. Mater Sci Eng1979;37:111–20.

[46] Del Valle JA, Romero R, Picasso AC. Bauschinger effect in age-hardened InconelX-750 alloy. Mater Sci Eng 2001;A311:100–7.