creep–environment interactions in dwell-fatigue crack growth of nickel based superalloys

Creep–Environment Interactions in Dwell-Fatigue Crack Growthof Nickel Based Superalloys

KIMBERLY MACIEJEWSKI, JINESH DAHAL, YAOFENG SUN,and HAMOUDA GHONEM

A multi-scale, mechanistic model is developed to describe and predict the dwell-fatigue crackgrowth rate in the P/M disk superalloy, ME3, as a function of creep–environment interactions.In this model, the time-dependent cracking mechanisms involve grain boundary sliding anddynamic embrittlement, which are identified by the grain boundary activation energy, as well as,the slip/grain boundary interactions in both air and vacuum. Modeling of the damage events isachieved by adapting a cohesive zone (CZ) approach which considers the deformation behaviorof the grain boundary element at the crack tip. The deformation response of this element iscontrolled by the surrounding continuum in both far field (internal state variable model) andnear field (crystal plasticity model) regions and the intrinsic grain boundary viscosity whichdefines the mobility of the element by scaling up the motion of dislocations into a mesoscopicscale. This intergranular cracking process is characterized by the rate at which the grainboundary sliding reaches a critical displacement. A damage criterion is introduced by consid-ering the grain boundary mobility limit in the tangential direction leading to strain incompat-ibility and failure. Results of simulated intergranular crack growth rate using the CZ model aregenerated for temperatures ranging from 923 K to 1073 K (650 �C to 800 �C), in both air andvacuum. These results are compared with those experimentally obtained and analysis of themodel sensitivity to loading conditions, particularly temperature and oxygen partial pressure,are presented.

DOI: 10.1007/s11661-014-2199-z� The Minerals, Metals & Materials Society and ASM International 2014

I. INTRODUCTION

THE high temperature crack growth in nickel-basedsuperalloys is classified as cycle-dependent and time-dependent damage processes. The transition betweenthese two mechanisms is determined by a transgranular/intergranular transitional loading frequency. Parametersinfluencing this transition including temperature, grainsize, loading parameters (DK, R) and environment havebeen examined in several alloys including IN718,Waspaloy, IN100 and ME3.[1–7] The cycle-dependentcracking occurring at frequencies higher than thematerial’s transitional frequency, takes place along slipplanes and results in a transgranular fracture mode. Onthe other hand, time-dependent crack growth, achievedat loading frequencies below the transitional frequency,is a thermally activated process and is characterized byintergranular cracking. The determination of the trans-granular/intergranular transitional frequency and de-tails of mechanisms associated with each of these twofracture modes are studied by many authors including

Ghonem et al.[1–3] in their work on IN718 at 923 K(650 �C), Leo Prakash et al.[4] on the same alloy in thetemperature range of 823 K to 923 K (550 �C to650 �C), Lu et al.[5] on Haynes 230 alloy at 922 K to1200 K (649 �C to 927 �C), and Lynch et al.[6] onWaspaloy at 773 K to 973 K (500 �C to 700 �C), Dalbyand Tong[7] on Alloy X, at 923 K and 998 K (650 �Cand 725 �C). The time-dependent cracking has beenstudied by the introduction of a dwell at maximum load.The work of Lynch et al.[6] on Waspaloy has shown thata continuous loading frequency of 2 Hz results intransgranular fracture and the addition of a 60 secondshold time produced a higher crack growth rate andchanged the fracture mode to predominantly intergran-ular fracture. The work of Lu and coworkers[5] hasobserved that the transitional frequency in Haynes 230alloy increases with an increase in temperature in therange of 922 K to 1200 K (649 �C to 927 �C) undertriangular and trapezoidal waveforms. They have attrib-uted this to the changes in creep vs fatigue crack tip zonesizes. Similarly, Yang et al.[8] have discussed the relativesizes of the creep zone and cyclic plastic zone in a nickelbased superalloy at 1023 K (750 �C) under trapezoidalwaveform, with 1.5 seconds loading, 1.5 seconds unload-ing and with a hold time of 0, 90, 450 and 1500 secondsat maximum load. The creep zone becomes dominantwith the increase in hold time. In addition to creep, time-dependent cracking is influenced by environmentaldamage. Wei and Huang[9] have examined P/M nickel-based superalloys in high purity argon and oxygen at

KIMBERLY MACIEJEWSKI and YAOFENG SUN, Post Doc-torates, JINESH DAHAL, Graduate Student, and HAMOUDAGHONEM, Professor, are with the Mechanics of Materials ResearchLaboratory, Department of Mechanical, Industrial and SystemsEngineering, University of Rhode Island, Kingston, RI 02881. Contacte-mail: [email protected]

Manuscript submitted May 7, 2013.

METALLURGICAL AND MATERIALS TRANSACTIONS A

873 K to 973 K (1146 �C to 1246 �C) at a load ratio of0.1 under constant DK with a trapezoidal wave form of0.05 seconds loading, 0.05 seconds unloading and dwelltimes of 0, 3, 10 and 30 seconds at maximum load levels.Their results showed that the crack growth rateincreased linearly with the increase in the dwell time.This increase in crack growth rate in an argon environ-ment is enhanced with the introduction of oxygen.

A large amount of research has been devoted to thestudy of creepdamagemechanismswhich operate in dwellcrack tip in Nickel based superalloys. The general view isthat these mechanisms involve cavitation and/or grainboundary sliding. Cavitation has been observed in differ-ent alloys and is described by the nucleation, growth andcoalescence of voids along the grain boundaries.[10–12]

Nucleation occurs at a high stress concentration at slipband/grain boundary intersections. Void nucleation sitesare also found at triple junctions or on boundariesadjacent to hard particles such as oxides or carbides.[13–15]

For void growth to occur there must be vacancy sources,diffusion of vacancies to cavities, and the grains must beallowed to move apart. For a boundary to supplyvacancies there must be a tensile stress applied and thetemperature must be high enough to move vacan-cies.[10,12,16,17] On the other hand, the grain boundarysliding process is governed by the stress field present at thegrain boundary crack tip, which controls the slip/grainboundary interactions. The high stress at the crack tipcauses dissociation of lattice dislocations, present in a slipband, into a sessile dislocation and a mobile glissiledislocation, thus inducing grain boundary sliding.[18–24]

The second important time-dependent damage com-ponent acting in the intergranular crack tip is related toenvironment effects. In nickel based superalloys, envi-ronmental damage is described as a consequence of theoxygen diffusion at the crack tip. The diffusion processdepends on the alloy composition and the externalloading conditions and is classified as a short or longrange diffusion, see for example References 25 and 26.The short range diffusion is linked to the formation ofgrain boundary oxides.[25,26] This has been demon-strated in alloys such as IN718, in which NiO and FeOform at the grain boundary under atmospheric pressureof oxygen. These porous oxides cause a decrease in theoxygen partial pressure reaching the base metal. Whenthe oxygen partial pressure is not sufficient to form theseoxides, a dense oxide (Cr2O3) is formed. This type ofoxide is thermodynamically stable and prevents furtheroxygen penetration along the grain boundary. In thismechanism, an incubation time is required to form theseoxides. The long range diffusion of oxygen which occursas interstitial atoms, lowers the grain boundary cohesivestrength and results in grain boundary embrittlement.This mechanism which has been discussed in relation toalloys such as IN100, IN718 and ALLVAC718PLUS,[27–29] does not require an incubation time,see also References 30 and 31.

The relative significance of creep and environment onthe time-dependent crack tip damage has been treatedby many models[32–37] ranging from linear summation ofdamage components to models which acknowledge theinteractive nature between these time-dependent events.

Few studies have attempted the coupling of the damageevents along the affected grain boundary path with theevolving deformation fields surrounding the crack tipgrain boundary elements. The work in this paper appliesthis coupling concept in order to study the creep–environment interactions in the nickel based superalloy,ME3, in the temperature range 923 K to 1073 K(650 �C to 800 �C). This is achieved by describing thegrain boundary cracking mechanisms in terms of grainboundary sliding and dynamic embrittlement processes.These mechanisms are implemented in a cohesive zone(CZ) model in which the deformation behavior of thecontinuum material surrounding the grain boundaryfracture path is described by multi-scale constitutiveequations. The interaction of creep and environment isconsidered through a fracture criterion in which thegrain boundary sliding limit is reduced by the grainboundary mobility due to dynamic embrittlement. Thevalidity of the CZ model is determined by comparing thesimulated and experimental crack growth rates at thetemperatures mentioned above.

II. TIME-DEPENDENT CRACK GROWTH RATE:EXPERIMENTAL RESULTS

The material being examined is the powder metal-lurgy, nickel based superalloy ME3. The average com-position of this alloy (wt pct) is 3.5Al, 3.7Ti, 20.6Cr,3.5Mo, 0.05Zr, 0.03B, 0.04C, 2.1W, 2.4Ta, 0.9Nb. Thematerial has been subjected to a three stage heattreatment process; super-solvus solutioning at 1444 K(1171 �C) for one hour followed by cooling at 384 K/min (111 �C/min) to room temperature, stabilization at1116 K (843 �C) for 4 hours followed by air cooling,and aging at 1033 K (760 �C) for 8 hours followed byair cooling. The average Rockwell C hardness of the asreceived material is 46.4 HRC. Its microstructure is anequiaxed grain structure, with an average grain size of44 lm (ASTM 6) and planar grain boundary morphol-ogy, as shown in Figures 1(a) and (b), respectively. Theshape and distribution of the c¢ precipitates exhibit abimodal size distribution, where the larger precipitate,secondary c¢ (c0s), appear cubical, while the smaller ones,tertiary c¢ (c0t), seem to be spherical, as shown inFigure 1(c). Sizes and volume fractions of the c0s are230 nm and 41 pct, respectively, while for c0t, themeasurements are 42 nm and 10.4 pct, respectively.The precipitate sizes are reported as the equivalentparticle diameter. Figures 2(a) and (b) show the particledistributions of the c0s and c0t precipitates, respectively.Dwell-fatigue crack growth tests were performed on

standard compact tension (CT) specimens with dimen-sions following ASTM E647. The test loading cycleconsists of 1.5 seconds loading, 1.5 seconds unloadingand a dwell time ranging from 0 to 7200 secondssuperimposed at the maximum load level with an initialstarting DK in the range of 28 to 32 MPa�m. All testswere performed at a stress ratio of 0.1 at threetemperature levels, 923 K, 977 K and 1033 K (650 �C,704 �C and 760 �C) in air and vacuum environments.For each vacuum test, the test chamber pressure is


maintained in the range of 10�6 to 10�7 torr during theentire test. Results of these tests in terms of da/dN vs DK,detailed in Reference 19, show that da/dN increases withboth temperature and hold time duration. In addition,in the air environment tests, the crack growth rate ishigher that that in vacuum. In order to examine the timedependency of the crack growth curves, the relationshipbetween the crack growth rate and the loading fre-quency, f, is plotted showing that the transition fromtime-independent to time-dependent growth rate isoccurring at a transitional frequency of 0.01 Hz. Thisfrequency also identifies the transition from transgran-ular fracture mode occurring at f> ft, to intergranularmode at f< ft. Details of this transitional frequencyanalysis are given in Reference 19. Identifying thetransitional frequency allows the selection of the crackgrowth curves which correspond to time-dependent,intergranular fracture. Typical intergranular fracturesurfaces in both air and vacuum environment are shownin Figure 3. These curves can then be expressed in termsof the crack growth speed, da/dt, vs Kmax, as shown in

Figure 4. This figure shows that the crack speed istemperature dependent; an increase in temperatureproduces an increase in the crack growth rate. Further-more, for each of the test temperatures, the crackgrowth curves corresponding to different hold timeperiods, are consolidated into a single line pointing outthat for the same Kmax, da/dt is independent of thelength of the hold time. This result shows that the crackgrowth is a continuous process (no discrete jumps) andthat the associated crack tip damage events do notrequire incubation times. In addition, for the sametemperature, the crack growth rate in air is higher thanthat in vacuum, thus, the introduction of oxygenaccelerates the intergranular cracking process. It shouldalso be mentioned here that intergranular cracking isobserved in both air and vacuum environment. Fur-thermore, Figure 4(b), shows that the consolidation ofthe crack speed at 1033 K (760 �C) for different holdtime is not complete. This behavior may be due to otherdamage mechanisms operating at this temperature andmay not exist at 923 K or 977 K (650 �C or 704 �C).One possibility is that the microstructure is unstable at1033 K (760 �C), since this is the aging temperature ofthe ME3 alloy which, combined with high stresses, couldresult in dissolution of the tertiary c¢ particles in theimmediate crack tip region.[38]

25 µm

1 µm

GB(b)(a)

500 nm

(c)

25 µm25 µm

1 µm

GB(b)

1 µm1 µm

GB(b)(a)

500 nm500 nm500 nm

(c)

Fig. 1—(a) Secondary electron image of the ME3 material, electroetched at 3 V in 50/50 etchant, showing the typical grain size of44 lm. (b) Secondary electron image of the ME3 material etchedwith AG-21 etchant for 15 s, showing the planar grain boundarymorphology. (c) Secondary electron image of the ME3 materialshowing the c0s and c0t precipitates distributed within in the c matrix,with volume fractions of 41 and 10.4 pct, respectively.

Particle Size (nm)10 20 30 40 50 60 70 80

Fre

quen

cy (

%)

0

10

20

30

40

50

60

Particle Size (nm)0 100 200 300 400 500

Fre

quen

cy (

%)

0

10

20

30

40(a)

(b)

(a)

(b)

Fig. 2—(a) The c0s particle distribution of the ME3 material with amean equivalent particle diameter of 230 nm. (b) Similarly, the c0tparticle distribution with a mean size of 42 nm. In each figure, aGaussian distribution (solid line) is superimposed over the experi-mental data (bar graphs).


III. INTERGRANULAR CRACKINGMECHANISMS

The time-dependent crack growth rate, being athermally activated process, can be described by theArrhenius form:

da

dt¼ A exp �Q

R

1000

T

� �; ½1�

where A is the frequency factor, Q is the apparent acti-vation energy which is a function of Kmax, R is theuniversal gas constant and T is the absolute tempera-ture. The parameter Q is determined by fitting the da/dt vs 1000/T to Eq. [1]. The calculated values of theapparent activation energy in both air and vacuumenvironment are plotted vs Kmax in Figure 5. Theresulting Q vs Kmax relationship is fit into a polynomialequation of the form:

Q ¼ Q0 þQa Kmaxð Þ þQb Kmaxð Þ2; ½2�

where Q is in kJ/mol and the three constants Q0, Qa,and Qb are 224.2, �1.6, and 7.39E�3 in air and 144.1,�0.2 and 0 in vacuum; respectively. The parameter Q

defines the minimum energy path connecting two localminima on the potential energy vs distance curve. Thisenergy is a function of the local stress r, or Kmax inthe case of crack tip activation. Q can be associatedwith a single and multiple activation process for whichthe Q–r relationships can be schematically representedby the trends shown in Figure 6, see References 39and 40. The limits of Q are Q(0) = Q0 andQ(rath) = 0 which correspond to damage processesinvolving fully thermal and fully athermal effects;respectively. Furthermore, the slope, �dQ/dr, of thecurve in Figure 6 is a measure of the activation vol-ume, Xa which is proportional to the total volume ofactivated atoms involved in the process. When Q(rath)reaches a finite value at Xa(rath) = 0, an additionalthermally activated rate process must be acting in ser-ies with the main process and the finite value repre-sents the activation energy associated with thesecondary process.[40] This concept is applied to resultsin Figure 5 where Kmax is the driving force of the acti-vation process and the limits of Q are bounded byKmax = 0 and Kmax = Kmax

ath . In this figure, for theKmax between 30 and 80 MPa�m, Q in air ranges from180 to 140 kJ/mol and 140 to 130 kJ/mol for vacuumconditions. With reference to Eq. [2], the activationvolume Xa is defined by taking the derivative of Eq.[2] as:

Xa ¼ �dQ

dKmax¼ � Qa þ 2QbðKmaxÞ½ �: ½3�

Figure 5 shows that Xa in air is higher than that invacuum. Also, by setting Xa = 0 in Eq. [2], the athermalthreshold stress, Kmax

ath , is determined to be 108 MPa�min air which corresponds to a finite value of 138 kJ/mol.This indicates, with reference to the discussion above,the presence of multiple damage processes. In vacuum,since Q is idealized as a linear function of Kmax, Kmax

ath istaken to be equal to that at Q = 0 which in turnidentifies a single fracture process operating at the cracktip. The nature of the operating damage process isdetermined by comparing the results of Q to knownmechanisms identified in several studies on Nickel-basedsuperalloys, see for example Starink and Reed.[41] Thiscomparison shows that the activation energy in vacuum(140 to 130 kJ/mol) corresponds to a single damagemechanism involving grain boundary creep, while Q inair (180 to 140 kJ/mol) identifies a multi damagemechanism involving grain boundary creep enhancedby oxidation processes that, as mentioned earlier couldbe in the form of oxide formation or dynamic embrit-tlement. The experimental observations that da/dt ishold time independent at a constant temperature, andthat the crack length and the crack opening displace-ment as a function of number of cycles, are continuousprocess without jumps, suggests that the crack tipdamage event does not require incubation time. There-fore, one could exclude oxide formation as a possibledamage mechanism, thus suggesting that the environ-mental damage process in this alloy mainly occurs bydynamic embrittlement associated with oxygen diffusion.

(a)

(b)

50 µm

50 µm

(a)

(b)

50 µm50 µm

50 µm50 µm

Fig. 3—Typical intergranular fracture surfaces for crack growth testscarried out at 1033 K (760 �C), with a loading frequency of 1.5 to300 to 1.5 s, in an (a) air showing an oxide layer covering the grainboundary surfaces, and (b) vacuum environment.


Furthermore, creep damage is generally described interms of cavitation and/or grain boundary sliding.Examination of fracture surfaces has shown no evidenceof cavities in this alloy and therefore, crack tip creep is

considered to be related to grain boundary sliding. Inorder to provide a rationale for the grain boundarysliding, Dahal et al.[19] proposed that for frequencieslower than the transitional frequency (f< ft), the disso-ciated lattice dislocations along the intersection of a slipband and grain boundary are absorbed in the grainboundaries, see also Reference 20. The gliding of thesedislocations under shear loading, causes grain boundarysliding. It is then suggested that a condition for thismechanism to operate, is that a critical minimumdistance between pinning points exists and is equal tothe spacing between slip bands. Since sliding occurs forf< ft, it is assumed that the slip band spacing is afunction of the loading frequency. This relationship isdeveloped by Dahal et al.[19] on the basis of the work ofVenkataraman et al.,[42] employing the concept ofminimum strain energy accumulation within slip bandsduring reversal loadings. The model outcome in terms ofslip band spacing vs the loading frequency is supportedby experimental measurements at both high and lowloading frequencies.[19] Results of the model show that asaturation of slip band spacing, signifying a conditionfor intergranular cracking mode, is reached at approx-imately 3 lm which is shown to coincide with thetransgranular/intergranular transitional loading fre-quency of 0.01 Hz. Efforts have been carried out tomeasure the grain boundary sliding in ME3 crackgrowth CT specimens subjected loadings with dwellperiods ranging from 300 to 7200 seconds at 977 K and1033 K (704 �C and 760 �C) using a pre-test scribingmethod. Results of these experiments show that theaverage sliding distance, under the loading conditionsmentioned above is 4 to 6 lm.

IV. INTERGRANULAR CRACK GROWTHMODEL COMPONENTS

Under dwell-fatigue loading at elevated tempera-ture conditions promoting intergranular cracking in a

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

1.5-100-1.51.5-300-1.51.5-600-1.51.5-3000-1.5

1.5-300-1.51033K (760°C) / Air1033K (760°C) / Vac

1033K(760°C) / Air

1033K(760°C) / Vac

1033K (760°C) / Air

1033K (760°C) / Vac

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

1.5-300-1.51.5-600-1.5

1.5-100-1.51.5-300-1.50.5-600-0.51.5-7200-1.5

1.5-300-1.5923K (650°C) / Air977K (704°C) / Air977K (704°C) / Vac

977K(704°C) / Air

977K(704°C) / Vac

923K (650°C) / Air

923K (650°C) / Air

977K (704°C) / Air

977K (704°C) / Vac

(a)

(b) 1.5-100-1.51.5-300-1.51.5-600-1.51.5-3000-1.5

1.5-300-1.51033K (760°C) / Air1033K (760°C) / Vac

1033K(760°C) / Air

1033K(760°C) / Vac

1033K (760°C) / Air

1033K (760°C) / Vac

1.5-300-1.51.5-600-1.5

1.5-100-1.51.5-300-1.50.5-600-0.51.5-7200-1.5

1.5-300-1.5923K (650°C) / Air977K (704°C) / Air977K (704°C) / Vac

977K(704°C) / Air

977K(704°C) / Vac

923K (650°C) / Air

923K (650°C) / Air

977K (704°C) / Air

977K (704°C) / Vac

(a)

(b)

Fig. 4—Crack growth rate in terms of da/dt vs Kmax at (a) 923 Kand 977 K (650 �C and 704 �C) and (b) 1033 K (760 �C) in air andvacuum environments indicating the independence of hold timeduration.

Kmax (MPa√m)

30 40 50 60 70 80

Q (

kJ/m

ol)

120

130

140

150

160

170

180

190

Vacuum

Air

Vacuum

Air

Fig. 5—Apparent activation energy as a function of Kmax in an airand vacuum environments. These trends are represented by Eq. [2].

Q(σ)

σσath

Q01

-Ωa1

-Ωa2

Q02

Q(σath) = Finite

Single ProcessQ(σath) = 0

σ

σ

-Ωa2

Multi Processassistance

Fig. 6—Schematic of stress dependent activation energy, Q(r), as afunction of stress, r, for a single and multiple activation process.


vacuum environment, a crack tip element is subjected toa stress field arising from two continuum domains; a far-field macroscopic viscoplastic continuum acting in theentire domain located several grains away from thecrack path and a near-field microstructure-sensitivedomain acting on a limited number of grains in theimmediate vicinity of the crack tip. The evolution of thestress field in these two domains depends on the cyclicstress–strain behavior, strain rate dependence, creep/relaxation, as well as, diffusional and dynamic recoveryof the alloy microstructure. The deformation of thecrack tip grain boundary element is controlled bythe evolving stress in the two continuum domains andthe intrinsic grain boundary viscous behavior whichdefines the mobility of the element in both tangentialand normal directions. Details of the grain boundarymobility are influenced by the surrounding continuummaterial and the microstructure of the grain boundary,particularly the grain boundary morphology, distribu-tion and geometry of precipitates and the presence of agrain boundary precipitate free zone. The creep com-ponent of the loading cycle affects time-dependentevolution of the stress field, as well as, the grainboundary migration, sliding or nucleation and growthof cavities. The fatigue component, on the other hand, isrelated to development of persistent slip bands that willintersect the affected grain boundary thus allowingabsorption of lattice dislocations causing grain bound-ary sliding. These interactive processes are limited by therate at which sliding reaches a critical displacement. Adamage criterion can then be formulated by consideringthe grain boundary mobility limit in the tangentialdirection leading to strain incompatibility and failure.Furthermore, the separation of the grain boundary iscontrolled by the rate of cavity growth reaching acritical size and spacing which, while dependent on thegrain boundary normal tractions, is coupled with thegrain boundary shear and sliding. These interactivecreep-fatigue deformation and damage events will beexperimentally investigated and modeled.

This section describes the main features of a CZmodel that has been developed to simulate intergranularcracking, a schematic of the model components andtheir relative positions are shown in Figure 7. Thismodel is built using a multi-scale approach whichutilizes the knowledge of the grain boundary externaland internal deformation fields. The external field isgenerated by developing and coupling two continuumconstitutive models including (i) a macroscopic internalstate variable (ISV) model for the purpose of modelingthe response of the far-field region located several grainsaway from the crack path and (ii) a near-field, micro-structure explicit coarse scale crystal plasticity modelwhich is appropriate for the representation of thecontinuum region at the immediate crack tip. Thesecond requirement in the implementation of the CZmodel is a grain boundary deformation model which hasbeen developed on the basis of viscous flow rules of theboundary material. These rules correlate the rate of thegrain boundary sliding displacement to material and

load dependent parameters. A damage criterion isintroduced as a critical sliding limit which, as mentionedabove, is degraded by the effects of oxygen diffusionleading to failure along the grain boundary. The basiccomponents of this CZ model are described below.

A. Continuum Deformation Models

The deformation of the continuum surrounding thecrack tip grain boundary fracture path is developed attwo length scales; a macroscopic level described by anISV type model and a smaller scale described by a coarsecrystal plasticity type model (XP). The ISV model isbased on non linear kinematic hardening laws and willbe used to describe the far field material region,approximately 10 grains removed from the crack tip.The ISV model has fewer variables as compared to theXP model, thus, it improves computational efficiency.The near field region, which is comprised of severalgrains in direct contact with the grain boundary fracturepath, is described by a coarse scale crystal plasticitymodel, which considers the hardening of the 12 slipsystems of the FCC crystal. This region consists of anarray of 100 grains, each with an orientation within±15 deg of its neighboring grains. This scale is neces-sary, since it considers grain orientations, which allowthe displacement and tractions across the grain bound-ary (along the crack path) to be dissociated into bothnormal and tangential directions. Both scale modelsconsider isotropic and kinematic hardening compo-nents, in which a two term back stress variable includesthe effects of dynamic and static recovery. Details of theconstitutive equations pertaining to the ISV and the XPmodels as well as procedures and values of their materialconstants are fully described in Reference 43.

ISV

θ

Tn

Ttun

ut GB

Single Grain

XP

ISV

θ

Tn

Ttun

Single Grain

XP

Fig. 7—Schematic of the cohesive zone model components consistingof the multi-scale continuum modeled by a coarse crystal plasticity(XP) model in the near crack tip region and by an internal state var-iable (ISV) model in the far field region. The deformation and dam-age within the grain boundary element is governed by tractiondisplacement laws, which consider grain boundary sliding and dy-namic embrittlement mechanisms.


B. Grain Boundary Interface Traction-DisplacementLaws

The grain boundary deformation analysis is based theassumption that the boundary dislocation network canbe simulated as a Newtonian viscous fluid element. Therelationship between the shear stress and the velocitygradient can be obtained by representing the boundaryas two planes closely spaced at a distance y, separated bya grain boundary material. The applied shear stress, s,can then be related to the boundary velocity, _us and theseparation distance d, as:

s ¼ g @u=@yð Þ ¼ g=dð Þ _us: ½4�

Rearranging this equation yields the relative velocityof two sliding surface written as a viscous flow law interms of the tangential stress, Tt, taken to be equivalentto the shear stress s[44]:

_ust ¼ ðd=gÞs ¼ ðd=gÞTt: ½5�

Tt can be calculated from the traction-displacementlaws of the grain boundary phase in both shear andnormal directions can be described as:

Tt ¼ kt ut � usð Þ; ½6�

Tn ¼ kn un � ucð Þ; ½7�

where the subscripts t and n represent tangential andnormal directions, respectively, Tt,n is the grain bound-ary traction, kt,n is the cohesive stiffness, ut,n is the totalopening displacement, uc is the normal opening dis-placement due to cavities and us is the tangentialdisplacement due to grain boundary sliding. Extensivemicroscopy examination of the fracture surfaces of alltest specimens at all temperature conditions in air andvacuum did confirm the absence of cavities in thismaterial under the specified loading conditions. There-fore, in the term uc in Eq. [7] is taken equal to zero.

The viscosity term in Eq. [5], can be derived followingthe approach of Ke,[45] where the rate of relativedisplacements of the sliding surfaces can be describedby the distance slipped along the grain boundary duringthe relaxation time, tc which is the time required by adislocation to climb a distance k (0.3 nm) at a velocity, vc.This velocity is given by the Einstein mobility relation asthe product of the mobility of grain boundary disloca-tions, Md, and the force per unit length acting on thedislocation.[46–48] This approach yields the expression:

g ¼ dEGB

wctc ¼

dEGB

wc

kMdTtb

: ½8�

EGB is the unrelaxed shear modulus, wc is thecharacteristic length over which the grain boundarysliding occurs and b is the Burger’s vector (0.452 nm).The intrinsic mobility term, Md, is a function of thegrain boundary diffusivity, Dgb, Boltzmann’s constant,k, and temperature, T, and is expressed as:

Md ¼ Dgb=kT: ½9�

Substituting Eq. [8] into Eq. [5] yields the sliding ratein terms of dislocation mobility, given as:

_us ¼ wcMdb

EGBkT2t : ½10�

The grain boundary sliding strain rate is obtained bydividing Eq. [10] by the characteristic length wc. Thisrate is written as:

_es ¼_uswc¼ Mdb

EGBkT2t : ½11�

The temperature dependent material constants for thegrain boundary interface model are given in Table I.The grain boundary interface model is implemented intoan Abaqus[49] UINTER subroutine in order to describethe traction displacement laws of the intergranular grainboundary crack path.The sliding rate described in Eqs. [10] and [11] are a

function of the critical sliding length, wc which is theminimum length required between two pinning pointsover which the grain boundary sliding occurs. Thislength has been experimentally determined by analyzingthe interactions between lattice dislocations within theslip band and the grain boundary. It is assumed in thework of Dahal et al.,[19] that the macroscopic strain,during cyclic loading, is distributed along slipbands,[42,50] the optimum number and spacing of whichare calculated based on a minimum energy configura-tion.[51] Based on previous work,[18,19] the slip bandspacing, w, is proportional to e�f=2, where f is theloading frequency. This relationship has been validatedexperimentally for both high and loading frequencyshowing that the minimum slip band spacing associatedwith intergranular cracking, wc is 3 lm.

Table I. Grain Boundary Interface Material Constants for ME3 at 923 K, 977 K and 1033 K (6510 �C, 704 �C and 760 �C)

Parameter 923 K (650 �C) 977 K (704 �C) 1033 K (760 �C) Units

kn 2.32E+07 2.20E+07 2.00E+07 MPa/mmkt 9.25E+06 8.77E+06 8.00E+06 MPa/mmDgb 9.86E�08 4.71E�07 2.00E�06 mm2/sEgb 2.07E+12 1.97E+12 1.80E+12 MPa


C. Grain Boundary Fracture Criterion

As discussed earlier, intergranular cracking, in theME3 alloy in the temperature range studied, occurs bycoupled grain boundary sliding and dynamic embrittle-ment. It has been shown that intergranular fractureoccurs in both air and vacuum environments, thus, afracture criterion is considered here to be a critical grainboundary sliding limit modified by environment. Theaccelerated damage in the crack tip grain boundary pathin air environment is considered to be a result of grainboundary dynamic embrittlement by oxygen diffusion,the effect of which is to pin the grain boundarydislocations and reduce their effective mobility. Fractureoccurs once the accumulated oxygen at the crack tip issufficient to result in grain boundary immobility leadingto a localized boundary decohesion. This embrittlementmechanism is the basis of a grain boundary fracturecriterion formulated in terms of the critical sliding strainin vacuum reduced by a grain boundary mobilityparameter which is a function of the oxygen concentra-tion.[18] This critical strain is written as:

ecrit ¼ ecrit0 � z=MO2; ½12�

where ecrit0 is the critical strain for grain boundarysliding in vacuum obtained as a function of tempera-ture. This strain limit is degraded by the factor z=MO2

,where z is a temperature dependent, oxygen diffusionrelated constant. The temperature dependence of theecrit0 and z parameters will be discussed in Section Vof the paper. The dislocation mobility in the presenceof impurity atoms, MO2

, decreases due to the dragcaused by segregated impurities[46,47] and is a functionof diffusivity of oxygen in the grain boundary, DO2

,and oxygen concentration in the bulk material, CO2

.The DO2

is expressed by an Arrhenius type of law andCO2

can be derived from Fick’s first and second law.The mobility, diffusivity and concentration terms aredescribed by:

MO2¼ DO2

XkTb2CO2

; ½13�

DO2¼ D0e

�QO2=RT; ½14�

CO2¼ C0 þ CS � C0ð Þ 1� erf

r

2ffiffiffiffiffiffiffiffiffiffiDO2

tp

!" #; ½15�

where X is the atomic volume (0.0327 nm3), D0 is thefrequency factor (20 mm2/s), QO2

is the stress depen-dent activation energy for diffusion of oxygen in thegrain boundary, R is the universal gas constant, T isabsolute temperature, C0 is the initial concentration ofoxygen in the bulk material (0 pct), CS is the concen-tration of oxygen in air (21 pct), r is the distance fromthe crack tip and t is time. The stress dependence ofQO2

is established by correlating it with the equivalentgrain boundary traction, Ttot

[43]:

QO2¼ 226:2� 0:05 Ttotð Þ � 1:26E� 06 Ttotð Þ2; ½16�

Ttot ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:5T2

n þ 0:5T2t

q: ½17�

This critical sliding criterion, in combination with themodel described above, is utilized to simulate theintergranular crack growth process due to grain bound-ary sliding and dynamic embrittlement.

V. INTERGRANULAR CRACK GROWTH RATESIMULATIONS

Validation of the CZ model described above has beencarried out by comparing the simulated crack growthdata with that obtained experimentally. This compari-son is used to optimize the different model componentswhich determine the sliding displacement and concen-tration of oxygen at the crack tip. Additionally, thisprovides a route to assess the relative significance ofeach of these components to the intergranular damageassociated with dwell fatigue crack growth in the ME3alloy.

A. Finite Element Model

The crack growth simulations were carried out on ahalf model of CT12.5 specimen geometry, see Fig-ure 8(a). The small rectangular zone in Figure 8(a),which is modeled by the crystal plasticity model, is anarray of 100 hexagonal grains; each grain has a size of44 lm. This zone is surrounded by the ISV material.The crack tip is located 2 grains within the crystalplasticity zone. The finite element model, with itsboundary conditions, is shown in Figure 8(b). The grainboundary crack path is modeled as a straight slab with anode to node contact. Force controlled loading isapplied on the pin hole of the specimen at the desiredramp and hold time condition. There are a total of 3601plane strain elements. These are four and three nodebilinear type elements (CPE4 and CPE3). The elementsize is approximately 5 lm to 1.5 mm in length, forelements located away from the crack path. Along thegrain boundary crack path in the grains immediatelysurrounding the crack tip, the element length is 2 lm inlength. This length is smaller than the critical slidinglength, wc, over which sliding occurs. Details of thisnumerical model are given in Reference 43.

B. Grain Boundary Sliding Fracture Criterion

In order to predict the crack growth rate as a functionof Kmax, fracture is assumed to occur when the grainboundary sliding displacement measured over a charac-teristic length reaches a critical limit and is considered tobe a function of both temperature and environment, seeEq. [12]. In order to identify this criterion, the timecorresponding to the critical sliding displacement in avacuum, ucrit0, is calculated as:

tbreak ¼ wc=v; ½18�


where wc is the element length at the crack tip, in thiscase 2 lm and v is the crack growth rate at a givenKmax and temperature condition. Model simulationswere carried out to determine the evolution of thegrain boundary sliding, us, with time for three differentKmax values at 977 K and 1033 K (704 �C and760 �C). Results of these simulations shown in Fig-ure 9 identify tbreak and the corresponding us fromwhich ucrit0 values are calculated (represented by thehorizontal lines in Figure 9). The average values ofucrit0 normalized by element size (ucrit0/2 lm) are plot-ted in Figure 10 and are fit into a closed form solutionfor the critical grain boundary sliding strain in vac-uum, which is given as:

ecrit0 ¼ ucrit0=wc ¼ A0eb0T ½19�

where the constants A¢ and b¢ are 9.25e�5 and 7.7e�3respectively, and T is the absolute temperature.

As discussed previously, the influence of environmenton the grain boundary sliding limit in air environment isintroduced by considering the mobility of oxygen, MO2

,in the grain boundary. This type of interaction isdescribed by Eq. [12]. The factor z=MO2

addresses thecomplex stress–environment interactions where, for a

hold time condition, the crack tip stress relaxation isaccompanied with both a decrease of the diffusivity ofoxygen in the grain boundary, DgbO2

, and an increase inthe concentration of oxygen, CO2

. This combination ofeffects result in a decrease in the mobility leading toaccelerated failure of the CZ element. The temperature

Rigid Plane

ISV Zone

XP Zone Crack Tip

W = 25.4

Ø = 6.35

15.24 0.37 a = 8.088

0.42

(b)

ISV Zone

(a)

Fig. 8—(a) Schematic of specimen dimensions (in units of mm) for aCT12.5 specimen. The specimen thickness is 6.5 mm. (b) Crackgrowth model consisting of a half CT12.5 specimen linked to a rigidplane.

us

us(tbreak)

ucrit0

50 MPa√m

55 MPa√m60 MPa√m

us

us(tbreak)

ucrit0

50 MPa√m

55 MPa√m

60 MPa√m

(a)

(b)

Time (sec)

u s (m

icro

n)

0.0

0.1

0.2

0.3

0.4

0.5

us

us(tbreak)

ucrit0

50 MPa√m

55 MPa√m60 MPa√m

Time (sec)

0 200 400 600 800 1000 1200

0 100 200 300 400 500 600

u s (m

icro

n)

0.0

0.2

0.4

0.6

0.8

us

us(tbreak)

ucrit0

50 MPa√m

55 MPa√m

60 MPa√m

(a)

(b)

Fig. 9—Evolution of grain boundary sliding displacement as a func-tion of time with corresponding critical displacement values at fail-ure for (a) 977 K (704 �C) and (b) 1033 K (760 �C) under creepcrack growth conditions in a vacuum environment. Note the solidlines represent us obtained from the simulation at the indicated Kmax,the open circles correspond to us at time tbreak and the horizontaldashed line is the average of the open circles corresponding to theucrit0.

Temperature (K)

850 900 950 1000 1050 1100

ε crit0

(un

itles

s)

0.0

0.1

0.2

0.3

0.4

0.5

Fig. 10—Critical grain boundary sliding strain in vacuum, ecrit0, as afunction of temperature as described by Eq. [19].


dependent parameter z can then be determined byrearranging Eq. [12]:

z ¼ ecrit0 � ecritAð ÞMO2; ½20�

where ecritA and MO2are the critical sliding strain and

dislocation mobility in the presence of oxygen (in anair environment) at time tbreak, respectively. The timetbreak is calculated by Eq. [18] using the experimentalcrack growth rate in air. Since MO2

is stress dependent,an average value of z is approximated for the differentKmax simulations. These average values of z are plottedin Figure 11 as a function of temperature and are fitto a closed form solution:

z ¼ z1 þ z2ez3T; ½21�

where z1 = �0.112, z2 = 2.8e�6, and z3 = 0.012 andT is temperature in Kelvin. Equations [19] and [21] arecombined with Eq. [12] to fully define the critical slid-ing strain as a function of both temperature and envi-ronment. This limit is written as:

ecrit ¼ A0eb0T

� �� z1 þ z2e

z3T� �

=MO2: ½22�

Integrating the above described fracture criterion intothe grain deformation model provides a tool to deter-mine the time for the crack tip to advance the charac-teristic length (wc, the cohesive element size) which, inturn, provides the crack growth rate under the pre-scribed loading and environment conditions.

C. Simulated Crack Growth Rate 923 K to 1073 K(650 �C to 800 �C)

Results obtained from the simulation of crack growthwithout environment effects are validated by comparingwith the crack growth experiments performed in vac-uum. The effect of environment is nullified by puttingthe concentration of oxygen at the surface (Cs) equal tozero and the fracture limit is defined solely by ecrit0.Figure 12(a) [977 K (704 �C) Vac] and Figure 12(b)[1033 K (760 �C) Vac] shows the simulated curvescompared with the experimentally obtained crackgrowth rates in vacuum. The numerical crack growth

curves for 977 K and 1033 K (704 �C and 760 �C) invacuum are shown to follow a trend similar to that ofthe experimental data. The simulated data shows anunder prediction of the crack growth rates at lower Kmax

values, while the opposite is observed at high Kmax. Thisis possibly due to the stress sensitivity of the grainboundary sliding rate to the stress exponent which isused in the calculation of grain boundary slidingdisplacement, i.e., Tt in Eq. [10] is raised to a power of2. In addition to the 923 K to 1033 K (650 �C to760 �C) temperature range, the crack growth processwas also examined at 1073 K (800 �C) in an airenvironment. The experimental crack growth rate atthis temperature is slightly higher than that at 1033 K(760 �C). The crack growth rates for temperaturesranging from 923 K to 1073 K (650 �C to 800 �C) inair environment have been predicted and the results areshown also in Figure 12. The numerical results of thefour temperatures studied are in agreement with therange and trends of the experimental crack growth data.Similar to the vacuum results, the numerically simulatedcrack growth rate curves in air can be represented by apower law form. However, in contrast to vacuumresults, air simulated curves show a slope slightly lowerthan that obtained in the experimental curves. This ispossibly due to the uncertainty in determining thefrequency term of the oxygen diffusivity function (D0 inEq. [14]). Another possibility for this deviation is thatthe model does not consider variation in the oxidestructure which could differ as a function of tempera-ture. The low Kmax region can be identified as a moretemperature dependent region, with an increased effectof environment. The impact of both temperature andenvironment is reduced in the high crack growth rateregion.

VI. RELATIVE SIGNIFICANCE OFTEMPERATURE AND OXYGEN PARTIAL

PRESSURE

Simulation cases, performed at temperatures of923 K, 977 K and 1033 K (650 �C, 704 �C and760 �C) at a Kmax of 55 MPa�m, under sustained loadcrack growth subjected to different partial pressures ofoxygen are carried out in order to investigate the relativesignificance of these two parameters on da/dt. Results ofthese simulations are presented in this section.The first simulation deals with the influence of

temperature on the sliding displacement and the grainboundary traction. Figure 13(a) illustrates that thesliding displacement as a function of time shows a rapidincrease followed by saturation. In addition, this dis-placement increases rapidly with the increase in temper-ature. At 1000 seconds, the sliding displacement at977 K and 1033 K (704 �C and 760 �C) are 2 and 3.5times higher, respectively, than that at 923 K (650 �C).This behavior can be attributed to the rapid relaxationin the tangential traction with the increase in tempera-ture, as seen in Figure 13(b). Initially, the tangentialtraction is higher at 1033 K (760 �C), but the rapiddecrease in traction is enhanced by the large sliding

Temperature (K)850 900 950 1000 1050 1100

z (1

/MP

a*se

c)

0.0

0.5

1.0

1.5

2.0

2.5

Fig. 11—Material parameter, z, as a function of temperature as de-scribed by Eq. [21].


displacement. As the sliding displacement becomescomparable in value to the total tangential displace-ment, the elastic component vanishes and as timeincreases, the tangential traction approaches zero.

The partial pressure of oxygen has been studied byvarying the surface concentration of oxygen, Cs, fromCs = 0 pct (vacuum condition) to Cs = 21 pct (airenvironment). The difference between the surface con-centration of oxygen and the concentration of oxygen inthe bulk material, Cs � C0, controls the oxygen flux inthe grain boundary which in turn controls the criticalsliding strain. The latter term is calculated as a functionof the grain boundary mobility,MO2

, and is proportionalto T and CO2

. The effect ofCs on the critical sliding strainat a Kmax of 55 MPa�m is shown in Figure 14. Thisfigure shows that the time to fail of the crack tip elementwhich corresponds to the intersection of the criticalsliding strain with the developed grain boundary slidingstrain, increases with the decrease in both temperature

and surface concentration of oxygen. The GB slidingstrain is sensitive to the reference distance over which thestrain is calculated. In this study, es and ecrit values shownin Figure 14 are numerically calculated with respect to aGB element size of 2 lmwhich is proportional to the slipband spacing, wc. The maximum strain obtained usingthis reference length is 26.4 pct at 1033 K (760 �C). Thiscorresponds to a GB sliding displacement of 0.53 lmwhich would have resulted in a strain of 1.2 pct if thereference distance is taken equal to the grain size of44 lm. The latter reference distance would have indi-cated that GB sliding failure occurs as a jump over thegrain size, which is not experimentally observed.[19]

Furthermore, the strain values, es and ecrit, are sensitiveto the selection of the numerical fitting parameters kt, kn,EGB, and Dgb in Eqs. [6] through [10].The influence of temperature alone on the critical

sliding displacement is shown in Figure 10, illustratingthat ecrit0 increases with temperature. Thus, the grainboundary can accommodate more sliding at highertemperatures. The addition of environment degradesthis value, as shown in Figure 14. The relative influenceof both temperature and environment can be examinedby plotting the ecrit as a function of Cs, as shown inFigure 15. This figure shows that the percent of degra-dation of the critical grain boundary sliding strain in aircompared to that in vacuum is 50, 40 and 28 pct for the

1.5-300-1.51.5-600-1.51.5-100-1.51.5-300-1.50.5-600-0.51.5-7200-1.51.5-300-1.5

977K(704°C)

Air

923K(650°C)

Air977K

(704°C)Vac

1.5-100-1.51.5-300-1.51.5-600-1.51.5-3000-1.51.5-300-1.51.5-300-1.5

1033K (760°C)Vac

1073K(800°C)

Air

1033K (760°C)Air

(a)

(b)

Kmax (MPa√m)

da/d

t (m

/sec

)

10-10

10-9

10-8

10-7

10-6

1.5-300-1.51.5-600-1.51.5-100-1.51.5-300-1.50.5-600-0.51.5-7200-1.51.5-300-1.5

977K(704°C)

Air

923K(650°C)

Air977K

(704°C)Vac

Kmax (MPa√m)

25 30 40 50 60 70 85 100

25 30 40 50 60 70 85 100

da/d

t (m

/sec

)

10-9

10-8

10-7

10-6

1.5-100-1.51.5-300-1.51.5-600-1.51.5-3000-1.51.5-300-1.51.5-300-1.5

1033K (760°C)Vac

1073K(800°C)

Air

1033K (760°C)Air

(a)

(b)

Fig. 12—Experimental and numerical crack growth rate as a func-tion of Kmax at (a) 923 K and 977 K (650 �C and 704 �C) and (b)1033 K and 1073 K (760 �C and 800 �C), in both air and vacuumenvironments. Note that the symbols are the experimental datapoints and the lines represent the numerically obtained crack growthrate from the cohesive zone simulation.

u s (

mic

ron)

923K (650°C)

977K (704°C)

1033K (760°C)

923K (650°C)

977K (704°C)

1033K(760°C)

(a)

(b)

Time (sec)

0.0

0.2

0.4

0.6

0.8

1.0

923K (650°C)

977K (704°C)

1033K (760°C)

Time (sec)

0

0 200 400 600 800 1000 1200

200 400 600 800 1000 1200

Tt (

MP

a)

0

200

400

600

800

923K (650°C)

977K (704°C)

1033K(760°C)

(a)

(b)

Fig. 13—(a) Grain boundary sliding displacement, us, and (b) tan-gential traction, Tt, across the grain boundary as a function of timeand temperature at a Kmax of 55 MPa�m under sustained load crackgrowth conditions.


temperatures 923 K, 977 K and 1033 K (650 �C, 704 �Cand 760 �C) respectively. It is interesting, however, toobserve that the influence of Cs on the critical slidingstrain is relatively higher at 923 K (650 �C) than at theother two higher temperatures. The da/dt at 923 K(650 �C) is lower than da/dt at 977 K or 1033 K (704 �Cor 760 �C), thus, for the same Kmax and the same cracklength increment, it will take a longer time to reachfailure at 923 K (650 �C). This additional time will allow

the diffusion of oxygen to play a more significant role onthe ecrit. Similarly, environment will have a moredominant role at lower Kmax values, where the time tofailure is longer, as opposed to higher Kmax values.

VII. CONCLUSIONS

Creep–environment interactions in the intergranularcrack tip region of the ME3 alloy have been examinedby performing dwell-fatigue crack growth experimentsat four temperatures, 923 K, 977 K, 1033 K and 1073 K(650 �C, 704 �C, 760 �C, and 800 �C), in both air andvacuum environment. Results of these experimentsrepresented the basis for the development of a mecha-nistic based intergranular crack growth model whichconsiders time-dependent damage mechanisms. Theoutputs of the numerical analysis are compared withthose experimentally obtained and are used to identifythe sensitivity of the crack growth process to variationsin temperature and surrounding oxygen partial pressure.The main conclusions of this study are summarized asfollows:

– Dwell-fatigue crack growth tests are carried out atloading frequencies below the transitional transgra-nular/intergranular frequency which has been identi-fied as 0.01 Hz. Results of these tests as da/dt vsKmax are used to determine the apparent activationenergy (Q) of the crack growth process. This energyis calculated as 130 to 140 kJ/mol in vacuum and140 to 180 kJ/mol in air environment. The Q param-eter is correlated in a nonlinear form as a functionof Kmax. The analysis of this relationship classifiesthe crack growth kinetics as a dual mechanisminvolving creep by grain boundary sliding and envi-ronmental degradation due to dynamic embrittle-ment by oxygen diffusion.

– The governing crack tip damage mechanisms havebeen integrated in a mechanistic based time-depen-dent crack growth model which considers creep–environment interactions in both the bulk and thecrack tip region. Modeling of these mechanisms is

923K(650°C)

977K (704°C)

1033K(760°C)

CS (%)

0 5 10 15 20 25

ε crit

(%)

0

5

10

15

20

25

30

923K(650°C)

977K (704°C)

1033K(760°C)

Fig. 15—Critical grain boundary sliding strain, ecrit, as a function ofoxygen partial pressure in terms of the parameter Cs for 923 K,977 K and 1033 K (650 �C, 704 �C and 760 �C) at a Kmax of55 MPa�m.

1

2

3

45

6

1

2

3

4

56

1

2

3

4

5

6

(a)

(b)

(c)

Time (sec)

ε s %

0

2

4

6

8

10

12

14

1

2

3

45

6

Time (sec)

ε s %

0

5

10

15

20

1

2

3

4

56

Time (sec)

0 400 800 1200 1600

0 100 200 300 400 500 600 700

0 50 100 150 200 250 300

ε s %

0

5

10

15

20

25

30

1

2

3

4

5

6

(a)

(b)

(c)

Fig. 14—Grain boundary sliding strain, es, as a function of time for(a) 923 K (650 �C), (b) 977 K (704 �C) and (c) 1033 K (760 �C) un-der sustained load crack growth conditions at a Kmax of 55 MPa�mwith the corresponding critical sliding strains, ecrit, for different envi-ronments. Note that the number label for each curve corresponds tothe following conditions; 1: es at Kmax = 55 MPa�m, 2: ecrit atCs = 0 pct (vacuum), 3: ecrit at Cs = 5 pct, 4: ecrit at Cs = 10 pct, 5:ecrit at Cs = 15 pct, 6: ecrit at Cs = 21 pct (air).


achieved by adapting a CZ approach in which thegrain boundary dislocation network is smeared intoa Newtonian fluid element. The deformation behav-ior of the grain boundary element is controlled bythe continuum deformation and the intrinsic grainboundary viscosity. The continuum deformation isdescribed by a multi-scale model which includes thecoupling of a coarse crystal plasticity model simulat-ing the near crack tip region and an ISV model forthe far field response several grains removed fromthe crack tip. The grain boundary cracking processis controlled by the rate at which the sliding reachesa critical displacement and as such, a damage crite-rion is introduced by considering the mobility limitin the tangential direction leading to strain incom-patibility and failure. This limit is diminished byenvironmental effects which are introduced as adynamic embrittlement process that hinders grainboundary mobility due to oxygen diffusion.

– The CZ model has been applied to simulate thecrack tip element’s deformation and fracture as afunction of loading conditions and environment.Results of this simulation show the crack growthrates for temperatures ranging from 923 K to1073 K (650 �C to 800 �C) in both vacuum and airhave been predicted and results are in agreementwith the range and trends of the experimental data.The increase in the crack growth rate at 977 K and1033 K (704 �C to 760 �C) compared to the rate at923 K (650 �C) is analyzed and explained in termsthe rapid relaxation in the crack tip tangential trac-tion with the increase in temperature.

– The influence of oxygen partial pressure was studiedby simulating the crack growth rate at 923 K, 977 Kand 1033 K (650 �C, 704 �C and 760 �C) at a Kmax

of 55 MPa�m, under sustained load crack growthconditions. Results indicate that the crack growthrate is enhanced by the oxygen diffusion and theinfluence of environment is relatively higher as thetemperature decreases. A similar influence is observedwith the Kmax range.

ACKNOWLEDGMENTS

The authors acknowledge the support from theMAI (Metals Affordability Initiative) ProgramFA8650-08-2-5247, in collaboration with the Air ForceResearch Lab, Pratt & Whitney, GE Aviation, Geor-gia Institute of Technology and Ohio State University.

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