Thermomechanical Fatigue Behavior of a Directionally M. M

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M. M. Shenoy

A. P. Gordon

D. L. McDowell

R. W. Neu1

e-mail: [email protected]

The George W. Woodruff Schoolof Mechanical Engineering,

Georgia Institute of Technology,Atlanta, GA 30332-0405

Thermomechanical FatigueBehavior of a DirectionallySolidified Ni-Base SuperalloyA continuum crystal plasticity model is used to simulate the material behaviordirectionally solidified Ni-base superalloy, DS GTD-111, in the longitudinal and trverse orientations. Isothermal uniaxial fatigue tests with hold times and creep teconducted at temperatures ranging from room temperature (RT) to1038°C to charac-terize the deformation response. The constitutive model is implemented as a Userial subroutine (UMAT) in ABAQUS (2003, Hibbitt, Karlsson, and Sorensen, Inc., Pdence, RI, v6.3) and a parameter estimation scheme is developed to obtain the mconstants. Both in-phase and out-of-phase thermo-mechanical fatigue tests aducted. A physically based model is developed for correlating crack initiation life bon the experimental life data and predictions are made using the crack initmodel. fDOI: 10.1115/1.1924560g

Keywords: Constitutive Modeling, Crack Initiation, Directionally Solidified SuperaFatigue Life Prediction

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1 IntroductionNickel-base superalloys are technologically significant ma

als that are used extensively in applications that require a mawith high strength, good creep, and fatigue and corrosion rtance, even at elevated temperatures. Some of the applicinclude turbine blades, turbine discs, burner cans, and vaneoperating temperatures of these components can range fromtemperaturesRTd to very high temperaturess.1000°Cd and theyare exposed to complex mechanical loading superimposedthermal transients; for example, creep deformation interactsthermal and/or mechanical fatigue. Specialized processingniques are employed to enhance mechanical properties, suelimination of grain boundaries along the turbine blade axidirectional solidification. However, this introduces new challenin constitutive modeling of the deformation behavior andprediction.

The deformation mechanisms change with varying loads, otations, strain rates, and temperatures. A physically based ctutive relation that can simulate the structural response in ordfacilitate component life prediction should incorporate the acdeformation mechanisms as a function of stress level, strainand temperature. The crystal plasticity approach provides acompromise between manageability and modeling capabilitiepecially when modeling components where the size, morphoand crystallographic orientation of the grains control the antropic deformation behaviorf1g. A physically based constitutivmodel is outlined to capture the homogenized deformationsponse in a directionally solidified superalloy, DS GTD-111.material parameters are determined based on experimentacycle fatiguesLCFd and creep data in both the longitudinalsLdand transversesTd orientations. Fatigue tests with short hold timare also conducted to accurately capture the short-term prcreep behavior.

The combination of cyclic centrifugal forces experiencedblades at high temperatures in an aggressive environment, cowith the heterogeneity and inherent anisotropy of the micros

1Author to whom correspondence should be addressed.Contributed by the Materials Division for publication in theJOURNAL OF ENGINEER

ING MATERIALS AND TECHNOLOGY. Manuscript received October 5, 2004. Final ma
script received February 2, 2005. Review conducted by: R. Craig McClung.
Journal of Engineering Materials and TechnologyCopyright © 200

rom: http://materialstechnology.asmedigitalcollection.asme.org/ on 07/23/

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2 Material Microstructure and CompositionDS GTD-111 nickel-base superalloy has the chemical co

sition as shown in Table 1. The as-cast structure contains cohorderedg8 intermetallic precipitates dispersed uniformly inausteniticg phase as shown in Fig. 1. There is a bi-modal dibution of g8 precipitates, with cuboidal primary precipitas0.5–1mmd and spheroidal secondary precipitas0.05–0.2mmd, with an overall volume fraction of approximate46%. These precipitates are interspersed in the relatively sotrix and are separated by the thin matrix channels. The graincolumnar and are roughly 125 mm long and 2 mm wide, wlimits the role of grain boundaries in crack initiation. Primdendrite stems grow parallel to the solidification direction. Thstems are accompanied by secondary and tertiary dendritewhich grow outward alongf100g and f010g directions. MC carbide inclusions, rich in Ti and Ta, are located in the interdendregions without any preferential orientation. The carbides50–150mm long with a high aspect ratios,10:1 or greaterd. Inaddition to these carbides, M23C6 and g /g8 eutectic nodules alocated in the interdendritic channels. Theg /g8 eutectics are dutile and vary in size from 30 to 150mm.

The observed deformation response in DS GTD-111 is cotent with that of other nickel base superalloys and can be surized as followsf3g:

1. The yield strength increases with increasing temperatuto a peak temperature of 750°C, beyond which it decre

2. Slip is generally dictated by the resolved shear stressesthe favorable crystallographic slip planes. However, nibase superalloys do not obey Schmid’s law in all orietions and slip on any given plane can also depend onresolved stresses on other planesse.g., dislocation cross slplanesd due to dislocation core spreading effects in theg8
precipitates.
JULY 2005, Vol. 127 / 3255 by ASME

2015 Terms of Use: http://asme.org/terms

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3. There is tensionsTd-compressionsCd asymmetry of the flowstress. The nature of this asymmetry is orientation andperature dependentf4–8g.

4. The initial yield behavior is nearly temperature path hisindependent, i.e., if a sample is first deformed at atemperature at which the yield strength is high anddeformed at a low temperature, the material responsimilar to a virgin material deformed at the lowtemperaturef9g.

Any constitutive model should take these pointsconsideration.

3 Crystal Plasticity FormulationA homogeneous constitutive model is proposed in this pape

which no explicit distinction is made between the matrix andcipitate phases. A crystal plasticity framework is employed toture the orientation-dependent material response. As the temture increases, strain rate effects are increasingly importantrate-dependent method is employed. Crystal plasticity approhave been used extensively in modeling single crystal superaf1,7,10g and the general methodology can be briefly describefollows. A multiplicative rule is used for the deformation gradieF, given asf11,12g

F = Fe ·Fp s1dAs shown in Fig. 2, the deformation gradient is separated intoparts:sid the plastic deformation gradientsFpd formed due to dislocation motion, associated with a change in the shape of thebut not its crystal lattice, andsii d the elastic deformation gradiesFed, which essentially models reversible elastic stretch andbody rotation of the lattice. For any given lattice, the close-paplanes act as the slip planes with unit normal vectorm> o

a in thereference configuration for each of thea slip systems, alonwhich the dislocations move in the slip direction in the refereconfiguration with unit vectors>o

a. The shearing ratesga along eachof the active slip systemssad depend on the resolved shear ston the slip systems. The macroscopic plastic velocity gradieobtained by summing over all slip systems in the intermedrelaxed configuration according to

Table 1 Nominal composition of DS GTD111

Fig. 1 Two-phase microstructure of DS GTD-111 superalloy †2‡

326 / Vol. 127, JULY 2005


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o

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idd

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siste

L p = Dp + Wp = oa=1

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where the plastic rate of deformationsDpd and plastic spinsWpdare given by the symmetric and anti-symmetric parts of the pvelocity gradient, i.e.,

Dijp = 1

2sLijp + Lij

pd, Wijp = 1

2sLijp − Lij

pd s3d

It should be noted that the intermediate, relaxed configuratiFig. 2 is not unique since any superimposed rigid body rotatiothe deformation gradient will also satisfy Eq.s1d. Uniqueness othe intermediate configuration is obtained by assuming thasubstructure spin is co-rotational with the lattice.

3.1 Constitutive Model. A rate-dependent crystal plasticformulation is used. All slip systems are assumed active abovthreshold stress, with the shearing rate of theath system relatedits associated viscous overstress by

ga = goQsTdK tva

DaLn

expHBoK tva

DaLn+1Jsgnsta − xad s4d

where the slip system viscous overstresstva is given in terms o

the resolved shear stressta by

tva = uta − xau − ka m

mos5d

Here,xa is the backstress andka is the threshold stress on easlip system. Increase in the threshold stress leads to an incresize of the viscoplastic flow potential, while the back stress rein a shift of the potential surface. The threshold stress maviewed as the resistance to plastic flow arising from statisstrengthening mechanisms associated with an increase of dtion density, solid solution strengthening, etc. The back stresplays a directional dependence and is associated with severtures of the heterogeneous material at the microscale, inclinternal stresses that develop with deformation due to dislocpile-up at obstacles such as precipitate particles, grain orboundaries, differential yielding with strain in hardsdislocationwallsd and soft regions of the microstructure, and statistical iactions of dislocations bypassing barriers. Both back stressxa,and the threshold stress,ka, depend on the history of temperatand viscoplastic deformation.

Equationss4d and s5d contain several other additional paraeters defined as follows:go is a reference shearing rate,n governsthe power law creep regime,Bo andsn+1d collectively govern thpower law breakdown behavior at higher strain rates wherresponse becomes nearly rate independent,Da is the drag stres

Fig. 2 Elastoplastic decomposition of the deformation gradi-ent †11,12‡

that is weakly dependent on the history of temperature and visco-

Transactions of the ASME


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plastic deformation when the behavior of theg andg8 phases arhomogenized,m is the shear modulus, withmo being the value othe shear modulus at absolute zero, andQsTd is the diffusivityparameter given by

QsTd = expS−Qo

RTD for T ù

Tm

2;

s6d

QsTd = expH−2Qo

RTmFlnSTm

2TD + 1GJ for T ø

Tm

2

whereQo is the activation energy for thermally activated dislotion bypass of obstacles,R is the universal gas constant, andTm isthe absolute melting point temperature.

The effective strain rate sensitivity exponent on the stresssin-verse strain rate sensitivityd is given by

m=] ln ga

] ln ta =] ln ga

] ln tva = n +

Bosn + 1dtom/mo

K tva

tom/moLn

s7d

The anomalous yield behavior ofg-g8 nickel base superalloyi.e., the yield stress increases with temperature in the intermetemperature regime, is incorporated in the threshold stressanomalous behavior has been attributed to mechanisms invoh111j k110l octahedral slip within theg8 precipitates. Severmodels attempt to explain this mechanism, including the Takeand Karamoto modelf13,14g, the Paidar-Pope-ViteksPPVd modelf5g, and the model proposed by Hirschf15g. Though recent experimental evidencef16g seems to support the Hirsch theory,PPV model has perhaps been the most widely used modelf10g, atleast in the anomalous temperature regimesroom temperature t750°Cd.

According to the PPV modelf5g, in the anomalous behavi

temperature regimesroom temperature to 750°Cd k101l screwdislocations ons111d planes split into two super-Shockley partiseparated by an antiphase boundarysAPBd. There are two con

figurations of thea/2k101l screw dislocation: a glissile configration with its core spread in thes111d plane and a sessile co

figuration with a nonplanar core spread in thes111d plane. Thecore transformation is explained in three steps:sid constriction othe glissile core on thes111d plane, sii d movement of the constricted dislocation along thes010d plane, andsiii d spreading o

the dislocation on thes111d plane. The glissile core has to costrict first for this to happen; hence, any shear stress that athe constriction of the partial dislocations will assist cross-leading to formation of sessile locks which act as obstacles tglissile dislocations. Slip on the octahedral planes will be inenced by the stresses on the primarysstepsidd and secondary croslip planessstep siii dd, along the direction perpendicular to tpartial dislocations, as well as the stresses acting on theplanesstepsii dd. The additional dependence of the activationthalpy for the glissile to sessile transformation on the shear scomponentsstpe, tse, and tcb, i.e., the shear stresses on themary, secondary, and cube slip systems, respectivelyd leads to anon-Schmid effect. Since the enthalpy is also dependent osign of the stress components for stepssid andsii d, it will differ intension and compression. This leads to a tension-compreasymmetry that is orientation dependent. It should be notedstepsiii d is independent of the sense of loading.

To include the non-Schmid effects, the yield criterion for ohedral slip systems is modified based on the PPV theory asposed by Qin and Bassanif17g. Each octahedral slip systemactive only when the Schmid stress and a linear combinatiothe additional stress terms exceed a critical value, i.e.,

ta = hpetpea + hsetse

a + hcbutcba u = tcr s8d

wheretpe, tse, andtcb are the non-Schmid shear stresses andtcr is
the critical resolved shear stress. This formulation is embedded
Journal of Engineering Materials and Technology


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the current set of constitutive relations through a critical thresstressskad, i.e.,

ka = kca + ke

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where

kca = ko

asTd + hpetpea + hsetse

a + hcbutcba u

and

kea = hoo

b=1

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qabugbu − hkskao

b=1

Nslip

ugbu − hsQsTdkka − kthlrs

The back stress evolution for homogenizedg-g8 models can bwritten in the hardening-dynamic recovery plus static thermacovery format as

xa = hxugausgnsta − xad − hxdxaugau + S 1

Rx

]Rx

]T+

1

hxd

]hxd

]TDxaT

− Vxa s10d

where Vxa=hxssTduxaurxs is the static thermal recovery term,hx,

hxd, hxs, rxs are material parameters, andRx=hx /hxd. Thetemperature-rate-dependent term is necessary to properlyhysteresis behavior under thermomechanical fatiguesTMFd f18g,since the model parameters that govern the back stress evorates are themselves functions of temperature. Note that forphase models, the back stress evolution equation can likeneglected, but it is essential to describe the overall average bior of the crystallographic grains in homogenizedg-g8 models. Iis a very significant fraction of the flow stressstypically.30%dfor Ni-base superalloysf19g. The constitutive equations are sumarized in Table 2.

4 ExperimentsTest sections were sectioned from two batches of cast p

Solid specimens with a uniform gage length of 12.7 mms0.5 in.dand a diameter of 6.35 mms0.25 in.d were machined from setions oriented in the longitudinalsLd and transversesTd directionsThe specimen was prepared according to ASTM Standardf20g. Tests were performed on a 45 kNs10 kipd axial servohydraulic MTS testing machine with dual-channel controllersTestStar softwaresTestware SX 4.0Dd. Load, strain, and temperture data were recorded digitally. The tests were conductedmechanical strain control using an extensometer with ceramiin direct contact with the specimen to obtain the total strain.mechanical strain ratioR«=«min/«max was held constant atR«=−1. An Ameriterm 2 kW RF induction heater provided heat tospecimens. Five K-type thermocouples were directly attachthe specimensthree within the gage sectiond to achieve a neauniform temperature distribution within the gage section, andinduction coil was iteratively modified to achieve a maximgage section temperature variation of either 1% ofTmax or ±3 K.For both LCF and TMF tests, the temperature was maintausing a PID temperature controller. In TMF tests, the temperreading was transmitted to the TestStar controller and uscalculate the components of thermal and mechanical strainfollowing the procedural guidelines in ASTM E2368f21g, TMFtesting can accurately simulate damage caused by the comeffects of thermal cycling and mechanical fatigue. Prior to etest, a relationship for thermal strains«thsTdd is determined usinthe free response under unloaded thermal cycles. This correis replayed during TMF cycling. Thus, all TMF tests were cducted using closed-loop, temperature-based thermal straintrol. A total of 36 strain-controlled isothermal LCF tests anTMF tests were conducted, as shown in Table 3. Three typ
incycles were used: continuous cycling withsad no hold timesCCd,
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sbd with 2 min hold time in tensionsHTd, andscd with 2 min holdtime in compressionsHCd. A constant strain rate of 0.005 s−1 wasused during the loading and unloading ramps.

The cycles to crack initiation in Table 3 is defined as the nber of cycles at which a 20% drop in the maximum axial loccurred. In many instances, the cycle with the critical load doff coincided with complete fracture of the gage section.

5 Implementation and Determination of Material Pa-rameters

The constitutive model was implemented as User MATesUMAT d subroutine in ABAQUSf22g, based on the implicit integration scheme of McGintyf23g. A hyperelastic formulation waused, valid for arbitrary finite strain. A time step subincremetion scheme and a linear search algorithm were employed i

Table 2 Summary o

constitutive subroutine to ensure convergence.

328 / Vol. 127, JULY 2005


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It is assumed that the flow rule given in Eq.s4d is suitable fomodeling the homogenizedg-g8 phases, but with a much highintrinsic lattice friction necessary to model the higher activaenergy for slip compared to modeling only a single phase, sug. It is commonly observed that the creep exponent of Ni-superalloys is more or less invariant with respect tog8 volumefraction and morphology, suggesting that slip is indeed restrin the precipitates, which provide primary elevated temperastrength.

The 12 octahedralh111jk110l slip systems are appropriateFCC single crystals and are active for the entire temperrange. The six cube slip systemsh100jk110l typesthree planes antwo directions=six systemsd may be active at high homologotemperatures and high resolved shear stresssin g8 phased, but theirrole is less well understood and characterized. A precise m

onstitutive model
f c
nism is still largely conjectural, since the identification of the role



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of cube slip is challenging. For homogenizedg-g8 single crystalsthe role of cube slip is usually assessed by considering the bdown of the capability of octahedral slip to model the strstrain-time behavior as a function of temperature, which of codepends on other elements of the constitutive framework.

Table 3 Test matrix for isothermal LCF and TMF.transverse „T… orientations using in-phase „IPing histories. Cycle type CC—continuous cycC—continuous cycling with 120 s hold time at peT—continuous cycling with 120 s hold time at pethermal LCF in longitudinal orientation and „

orientation.

work of Bettge and Osterlef24g on SC16 has revealed that mac-



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roscopic cube slip is a manifestation of octahedral slip andplace by multiple cross slip events on octahedral planes, pring zig-zag dislocation lines. This enables the dislocationtravel large distances along matrix channels, thus effectshearing material along the cube planes. In addition, actual

„a… Test matrix for TMF in longitudinal „L… andd out-of-phase „OP… thermomechanical load-

without hold time, Cycle type 2 minstrain in compression, Cycle type 2 minstrain in tension. „b… Test matrix for iso-test matrix for isothermal LCF in transverse

… anlingakakc…

slip has been observedf16g in the g8 precipitates at higher tem-

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peratures and along orientations closer tof111g. Experimentallysoftening occurs as the orientation is rotated from thef100g ori-entationf25g. When only octahedral slip is active, hardeningpredicted as the loading axis is rotated from thef100g direction,which is inconsistent with experimental observation. Both souof cube slip, zig-zag cross slip in theg phase and motion odislocation segments on cube planes in theg8 phase, are lumpetogether in the present model, treated via the activation ocube slip systems. Both the octahedral and cube slip systemassumed to be active in this model at all times.

The material parameters for the octahedral slip systems atermined from data obtained from uniaxial cyclic loading of lgitudinal specimens along thef100g orientation, for which the thloading axis is oriented along the grain growth direction. Cslip is not activated in this orientation for small deformations.material parameters for the cube slip systems are obtaineddata from transverse specimenssloading perpendicular to thgrain growth directiond due to lack of data in other orientationsshould be noted that the response of directionally solidified mrial in the transverse orientation is an average response coming several grains. A total of 30 grains with random transvorientations are used to obtain this average behavior alonloading axis. The Taylor constraint for intergranular interactionassumed in this averaging process, such that each grainswithdifferent orientations along the transverse directiond is subjectedto the same deformation gradient; this approach is conveniethat a single element can be used to obtain the initial estimathe material parameters, and is sufficiently accurate for slpolycrystals with high symmetry cubic structures. A more rigous finite element analysis is subsequently run with 30 hexagrains, each containing 96 elements, to optimize the fit of maparameters.

Under cyclic loading, lattice rotation may be insignificant,in the presence of creep or cyclic ratcheting, lattice rotationbuild up at sites of stress concentration or near interfaces sugrain boundaries. However, the creep and ratcheting strainsmall in the cyclic loading experiments conducted to determthe material parameters, so these lattice rotations are neglecthe purpose of estimating the material parameters. Severalsimplifications can be made regarding the deformation behavthis material at this temperature. In the temperature rangeconsideration, the precipitate structuresssize, shape, morphologdare stable, so higher-scale microstructural changes such asening of precipitates can be neglected in the evolution equa

From the cyclic deformation data, it is observed that the inyield strength does not change significantly with cyclic loadThis indicates that the threshold and drag stress are nearlystant and that the backstress is the primary internal state vathat evolves with deformation. The threshold stress is mainlyited to the critical valuekc

a that incorporates non-Schmid termspreviously explained, and the drag stresssDad is kept constanThe remaining material parameters must be physically admisThis calls for a systematic procedure for determining thestants. First, certain physical parameters such as elastic propand activation energies of thermally activated diffusionalcesses are obtained from literature. The activation energy isQ=309 kJ/mol, based on the data from Daleo and WilsonGTD-111 f26g. Estimating the remaining material parametersusing graphical methods in the analysis of deformation daoften not possible with viscoplasticity models because therestrong coupling between creep and cyclic plasticity. Approximvalues of the material parameters can often be obtained,trial-and-error approach almost always must be employed to rthe parameters to accurately capture the experimental deformresponse. A more robust and efficient approach is to integraconstitutive model, simulating the actual loading historusing an optimization scheme to iteratively evaluate the mparametersf27g.

The general optimization scheme involves running sever

330 / Vol. 127, JULY 2005


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simulations involving different thermomechanical historiescluding cyclic deformation with and without hold periods, as was creep deformation, using a set of parameters suggestedoptimization code. Then the model response is compared tactual experimental response generating a value of an objfunction defined on the basis of the norm of the error betweeexperimental and predicted deformation responses. The Ecode by Synapsf28g is used to minimize the objective functioniteratively modifying the material parameters based on minimtion of the objective function. Epogy employs four widely usearch methodsslinear simplex, downhill simplex, gradient, agenetic algorithmsd to assure stability and enhance the probabthat an optimum set of material parameters will be obtaamong various local minima. The rate of convergence and etiveness of the optimization is highly dependent on the objefunction, which in turn depends on the type of loading history.uniaxial cyclic loading, the error function is defined as

Error =oi

ssexpi − spre

i d2Wi s11d

wheresexpi is the experimentally measured stress along the

ing direction,sprei is the stress predicted by the model at the s

strain, andWi is a weighting parameter for theith strain valueThe weighting parameter is varied from 0.8 to 1.2, dependinwhich region of the stress-strain response is being simulatedweight is placed on the elastic portions of the deformationsWi

=0.8d while Wi =1.2 is used during inelastic flow. During strhold periods in which inelastic deformation and stress relaxoccurs,Wi =1.0.

Since the static thermal recovery terms are influential onllong hold periods and into the steady state creep regime,terms can be neglected in the preliminary parameter estimexercise that uses only continuous cycling data withouttimes. The results of this preliminary exercise provides tigbounds on those parameters that are less sensitive to rate fposes of later optimization exercises that include cyclic datastrain holds and creep data intended to refine the estimamaterial parameters. The error norm used for estimating paeters associated with creep behavior is

Errorcreep= oi

s«expi − «pre

i d2Eav2 s12d

where «expi −«pre

i is the difference between experimentally msured strain and predicted strain at a given time instant. Thisnorm employs the average value of the Young’s modulus,Eav, inthe specified orientation to scale the errors in terms of psestresses. For this final optimization step, the total error is theof the errors for the cyclic and creep deformation data,

Error = Errorcyclic + Errorcreep s13dThe material parameters obtained are listed in Table 4. Somthe material parameters are not listed due to unavailability operimental data for fitting parameters. Static thermal recoverfects in evolution ofk are omitted, for example. Accordingdashed entries in Table 4 signify zero values of the paramet

The most sensitive material parameters for this data set atermined by the optimization scheme arehxd andhxs, which con-trol the backstress evolution. It should be noted that backsplays an important role in Ni-base superalloys. Based on theby Ferney et al.f29g and Latif et al.f30g, it could be to the ordeof 50%–60% of the flow stress. Experimental data usedparameters and correlated simulations based on the constmodel are compared in Figs. 3–9.

6 Thermomechanical Fatigue (TMF)For TMF, a deformation gradientFu based on the thermal e

alpansion is included in the multiplicative decomposition of the



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deformation gradient. An intermediate, thermally expandedunstressed configuration and a hot plastically deformed reconfiguration are introducedf31g. Thus the total deformation grdient is given byF=Fe·Fp·Fu, where the linearized elastic defmation is described byFe, the plastic deformation byFp, and thethermal expansion/contraction byFu. Assuming isotropic thermexpansion, the velocity gradient associated with thermal exsion effects is given as

Table 4 Material parameters: „a… octahedral slislip system constants xa

„0…=0, kea„0…=0; and

Fig. 3 Stress-strain response: experimental data and corre-
lated simulations at 427°C „longitudinal, CC, first cycle …


oted

n-

L ou = Fu · sFud−1 = auI s14d

wherea is the thermal expansion coefficientsassumed constandand I is the second rank identity tensor. Solving the differenequation over a time step gives

Ft+Dtu = expsL o

uDtdFtu s15d

Material parameters are assigned temperature dependencethird-order interpolative polynomial fits of values determinedisothermal cyclic loading cases. The temperature is update

ystem constants xa„0…=0, ke

a„0…=0; „b… cube

common constants


p s„c…

lated simulations at 760°C „longitudinal, CC, first cycle …

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held constant for each Newton Raphson step and the maparameters relevant to that temperature are used. Thesesimulations can be considered as true predictions which correasonably well with the experimental results for loading in bthe longitudinal and transverse orientations, as shown in10–12.

Fig. 5 Stress-strain response: experimental data and corre-lated simulations at 871°C „longitudinal, HC, first cycle …

Fig. 6 Stress-strain response: experimental data and corre-lated simulations at 982°C „longitudinal, CC, first cycle …

lated simulations at 982°C „transverse, CC, first cycle …
332 / Vol. 127, JULY 2005


rialMFte

hs.

7 Fatigue Crack InitiationThe crack initiation life is defined as the number of cycle

which the load drops below 20% of the initial maximum stabilitensile load. The constitutive model described above is uspredict the cyclic material response under imposed isotherma

Fig. 8 Stress-strain response: experimental data and corre-lated simulations at 1038°C „transverse, CC, first cycle …

Fig. 9 Primary and secondary creep responses: experimentaldata and correlated simulations at 871°C „longitudinal …

Fig. 10 Stress-strain response: comparison of experimentaldata with model predictions for in-phase „IP… TMF
538°C–1038°C in the l ongitudinal orientation „first cycle …


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non-isothermal fatigue conditions for purposes of predictingtigue crack initiation life. Several isothermal fatiguesIFd and TMFlife prediction models have been developed to predict the se

Fig. 11 Stress-strain response: comparison of experimentaldata with model predictions for out-of-phase „OP… TMF538°C–927C in the longitudinal orientation

Fig. 12 Stress-strain response: comparison of experimentaldata with model predictions for out-of-phase „OP… TMF538°C–927°C in the transverse orientation

Fig. 13 Initial stress-mechanical strain responses ospecimens. „c… Inelastic strain versus fatigue crack
871°C


-

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lives of turbine blade materials subjected to high homolotemperaturesf32–39g. The approaches that have been develoare categorized as energy-based, strain-based, and damageCoffin f32g modified the Coffin-Manson law to include enviromental effects via a cycle frequency factor. Neu and Sehif33g extended Coffin-Manson concepts in a strain-based strto address coupled cyclic plastic strain, creep, and oxidationage mechanisms. An energy-based approach was presentedtergrenf34g and later modified by Zamrik and Renauldf35g andincluded the monotonic tensile elongation and ultimate stredata as functions of temperature. The model introduced by Ret al. f37g is restricted to a microstructurally small volume ement adjacent to an oxidized crack tip of a CT-type specimethis case, the critical fracture stress of the alloy and the oxidalloy are required. Miller et al.f38g extended damage rate eqtions discussed by McDowell et al.f39g for small fatigue cracpropagation under thermo mechanical loading to address cofatigue, creep, and oxidation effects in Ni-base superalloys.of these models were developed for isotropic polycrystallinesPCdmaterials in which creep damage mechanisms such as void gand grain boundary slip were dominant. In this paper, a cinitiation model is presented based on the observed damechanisms for DS GTD-111.

7.1 Observations.The stress response of DS GTD-111 unTMF is plotted with respect to the mechanical strain«m stotalstrain less thermal straind for several test cases in Figs. 13sad and13sbd. Figure 13scd plots the inelastic strain rangesmechanicastrain less elastic straind versus the number of cycles to crainitiation, D«in versusNi, for in-phasesIPd and out-of-phasesOPdTMF cases. The mechanisms for crack initiation changeloading history; however, no significant change in the initiamechanism is observed with change in orientationslongitudinaversus transversed.

Three primary damage mechanisms are identified: fatcreep-fatigue, and oxidation-fatigue damage. A single stage Ijor crack is observed in the isothermal fatigue tests withouttimes at all temperatures, with very little secondary crackinglower temperaturessø650°Cd, cracks initiate at cracked carbiinclusions near the surface as shown in Fig. 14sad. Initiation athigher temperatures occurs at the specimen surface from anspike along the interdendritic region, as shown in Fig. 14sbd. Car-bide inclusions near the surface act as diffusion short circuitsserving as preferential sites for oxidation. A softerg

8depleted

layer exists below the oxide layer further from the surface dudiffusion of aluminum. For the isothermal fatigue tests with h

F cycled „a… longitudinal and „b… transversetiation lives for TMF and LCF experiments at

f TMini

JULY 2005, Vol. 127 / 333


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timess2 min tension or compressiond, multiple voids are observearoundg /g8 eutectic nodules where crack initiation takes plathus, indicating a creep degradation mechanism. From Tablecrack initiation life is shorter in longitudinal oriented specimethe tests with hold time as compared to those without hold tdue to the contribution from a creep component in addition tofatigue and oxidation components. This is not true in the trverse orientation at higher temperaturess.871°Cd where the holdtimes in compression give higher crack initiation lives. Moreperimental tests will be conducted and fatigue specimens wanalyzed as a part of future work to understand this anombehavior in the transverse orientation at higher temperaturethe TMF tests, the material microstructure plays a much mimportant role in the IP tests as compared to the OP testscarbide inclusion cracking is observed near the specimen suin the IP tests and some voids are also observed around theg /g8eutectic nodules. Oxide spiking is the primary mode of cracktiation in the OP TMF tests and oxide spallation is also obsebut to a much smaller extent.

7.2 Creep-Fatigue-Environment Crack Initiation Model.The model in this section is based on application of the Tmodel developed by Neu and Sehitogluf33g. Assuming a uniqurelationship between the damage fraction and cycle fractionrespect to cycles to crack initiation for each damage modetotal crack initiation life can be represented in terms of the ividual damage componentssfatigue, creep-fatigue, and oxidatiofatigue, respectivelyd, and can be written asf33g

1

Ntot=

1

Nfat+

1

Ncr+

1

Noxs16d

Relationships for each of these components are determineconducting LCF and TMF tests to isolate the damage menisms. Mathematical formulations are developed to relatemechanism to temperature, grain orientation, hold times, etcfatigue damage mechanism is governed byf33g

Fig. 14 „a… Crack initiation at a subsurface carbide and „b…crack initiation at the surface due to an oxide spike

Table 5 Constants for th

334 / Vol. 127, JULY 2005


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F

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Nifat = C1f1sD«indC2 s17d

whereC1 andC2 are constants, andf1 is a factor that accounts fmaterial orientation with respect to the applied loading direcsi.e., L or Td. To characterize this relation, data are neededlow temperatures, high strain rates, and CC tests. Tests condat and below 871°C are used to correlate Eq.s17d. The creepfatigue interaction mechanism is found to correlate to

Nicr = C3QcrsT,tdS tc + th

tc + 2thD f2

sD«indC4 s18d

whereC3, C4, andf2 are constantsf33g. The inelastic strain rangD«in, is used here since it captures orientation and time-depedeformation. The diffusion coefficient,Qcr, along with the cycland dwell times, given bytc andth, respectively, account for timand temperature-dependent degradation mechanisms such anal void nucleation and growth that are not captured by Eq.s17d.The diffusion term is given by

QcrsT,td =1

tcE

0

tc

expS−Qcr

RTstdDdt s19d

HereQcr is the activation energy,Tstd is the temperature historandR is the universal gas constant. Specimens subjected totimes experience the most significant stress relaxation inducviscoplastic deformation. The correlation of Eq.s18d involves HTand HC tests at or above 871°C.

The term relating oxidation-fatigue interaction to crack intion is motivated by the oxide spiking mechanism. The interacis related to cycle-dependent repeated fracture of the oxideThis process is controlled by the elastic part of the mechastrain range,D«el, the oxidation growth kinetics, and the phasbetween the mechanical and thermal loading captured throphasing factor,Fox, i.e.,

Niox = C5fFoxQoxstc + thdgC6D«el

C7 s20d

HereC5, C6, andC7 are constantsf33g. The diffusion factor,Qox,in analogy to Eq.s19d for creep, captures the oxide film growkinetics. The phasing factor is given by

Fox = expf− C8s«th/«m + 1d2g s21d

This factor has a maximum value of unity for linear OP TMF,condition most susceptible to the oxide spiking mechanism.teresis data from isothermal tests above 871°C, along with dand TMF tests, are used to determine the constants for Eqss20dand s21d. Constants are summarized in Table 5.

7.3 Model Correlation and Predictions. Some trends predicted by the model are shown in Fig. 15 for continuously cylongitudinally oriented DS GTD-111 specimens. For both isomal sFig. 15sadd and nonisothermalsFig. 15sbdd cases, the cracinitiation life is characterized by the stress-strain response atmechanical strain ranges since the fatigue term dominates.

Under isothermal conditions, the change in slope of predlife below the mechanical strain range of 1.0% relates to a trtion from predominantly fatigue to a coupling of oxidation a

rack initiation model
e c


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fatigue mechanisms. As the temperature is decreased871°C, there is insufficient thermodynamic driving force for odation and the role of environment is limited as a result. UTMF conditions at lower mechanical strain rangessFig. 15sbdd,the OP cycle is predicted to be more damaging than the IP cThis prediction is consistent with experiments since the ospiking, which is more detrimental to life than oxide spallationexperienced during OP cycling. During IP cycling withD«m,1.0%, life is dominated by the fatigue mechanism. Withincrease ofDT from 389°C to 500°C, the constitutive modpredictsD«in to increase. Tests conducted at same temperaturtotal strain amplitudes on longitudinal and transverse orienteGTD-111 specimens lead to larger inelastic strain ranges for tverse specimens since transverse specimens have a largerstiffnesssFigs. 6 and 7d. Consequently, the lives of longitudinspecimens typically outlast those of transverse specimenstested at the same total strain range. The crystallographic ortion dependence in the constitutive model captures this pheenon, predicting the trends especially at the higher mechastrain ranges.

8 SummaryThe deformation and failure mechanisms occurring in a d

tionally solidified Ni-base superalloy, DS GTD-111, have bcharacterized as a function of temperature and loading histocontinuum crystal plasticity model is developed to charactethe material behavior in the longitudinal and transverse orietions. Isothermal and thermomechanical uniaxial fatigue testshold times and creep tests are conducted at temperatures rfrom room temperaturesRTd to 1038°C to characterize the defmation response. Using isothermal test data to fit constants, IOP TMF responses are adequately predicted. The fatiguemens are studied to identify the damage mechanisms and acally based model for predicting crack initiation life is develobased on the experimental stress-strain data. Three primaryage mechanisms are identified: fatigue, creep-fatigue,oxidation-fatigue induced damage. Crack initiation life pretions, based on the life models for each damage mechanismlow the experimental trends.

AcknowledgmentThe author acknowledges the help from Matt Trexler, G

Donaghy, and Tam Nguyen.

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Fig. 15 Predicted trends and selected experimental resuDS GTD111 under CC

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