DTIC ADA621000: Constitutive Model for Anisotropic Creep Behaviors of Single-Crystal Ni-Base Superalloys in the Low-Temperature, High-Stress Regime (Postprint)

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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
CONSTITUTIVE MODEL FOR ANISOTROPIC CREEP BEHAVIORS OF In-house
SINGLE-CRYSTAL Ni-BASE SUPERALLOYS IN THE LOW-
5b. GRANT NUMBER
TEMPERATURE, HIGH-STRESS REGIME (POSTPRINT)
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62102F
6. AUTHOR(S) 5d. PROJECT NUMBER
Christopher Woodward, Dennis M. Dimiduk, and Michael D. Uchic - 4349
AFRL/RXCM
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5f. WORK UNIT NUMBER
Yoon Suk Choi and Triplicane A. Parthasarathy - UES, Inc.
X0W6
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Journal article published in Metallurgical and Materials Transactions A, Volume 43A, June 2012—1869. © 2012
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The final publication is available at DOI 10.1007/s11661-011-1047-7.
14. ABSTRACT
A crystallographic constitutive model is developed to capture orientation-sensitive primary and secondary creep
behaviors within approximately 20 deg from the [0 0 1] orientation in single crystal superalloys for the low-
temperature and high-stress regime. The crystal plasticity-based constitutive formulations phenomenologically
incorporate experimentally observed dislocation micromechanisms.
15. SUBJECT TERMS
creep, nickel-based superalloy, crystal plasticity
16. SECURITY CLASSIFICATION OF: 17. LIMITATION 18. NUMBER OF 19a. NAME OF RESPONSIBLE PERSON (Monitor)
a. REPORT b. ABSTRACT c. THIS PAGE OF ABSTRACT: PAGES Micheal E. Burba
Unclassified Unclassified Unclassified SAR 12 19b. TELEPHONE NUMBER (Include Area Code)
[phone]
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std. Z39-18
Constitutive Model for Anisotropic Creep Behaviors
of Single-Crystal Ni-Base Superalloys in the Low-Temperature,
High-Stress Regime
YOON SUK CHOI, TRIPLICANE A. PARTHASARATHY,
CHRISTOPHER WOODWARD, DENNIS M. DIMIDUK, and MICHAEL D. UCHIC
Acrystallographicconstitutivemodelisdevelopedtocaptureorientation-sensitiveprimaryand
secondary creep behaviors within approximately 20 deg from the [0 0 1] orientation in single-
crystal superalloys for the low-temperature and high-stress regime. The crystal plasticity-based
constitutive formulations phenomenologically incorporate experimentally observed dislocation
micromechanisms. Specifically, the model numerically delineates the nucleation, propagation,
and hardening of ah112i dislocations that shear multiple c0 precipitates by creating extended
stacking faults. Detailed numerical descriptions involve slip-system kinematics from a/2h110i
dislocations shearing the c-phase matrix, ah112i stacking fault dislocation ribbons shearing the
c0-phase precipitate, interactions between a/2h110i dislocations to nucleate ah112i dislocations,
and interactions between the two types of dislocations. The new constitutive model was
implemented in the finite-element method (FEM) framework and used to predict primary and
secondarycreepofasingle-crystalsuperalloyCMSX-4inthreeselectedorientationsnearthe[0
0 1] at 1023 K (750 (cid:2)C) and 750 MPa. Simulation results showed a reasonable, qualitative
agreement with the experimental data. The simulation results also indicated that a/2h110i
matrixdislocationsareimportanttolimitthepropagationofah112idislocations,whichleadsto
the transition to secondary creep.
DOI: 10.1007/s11661-011-1047-7
(cid:3) The Minerals, Metals & Materials Society and ASM International 2012
I. INTRODUCTION mentalstudies[3 5]identifiedthreedistinctivenear-[001]
creepbehaviorsdependingonrangesoftemperatureand
S -crystal
INGLE Ni-basesuperalloyshavebeenan
stress. Figure 1 illustrates schematically these creep
unbeatablechoiceforhot-sectionturbinebladesbecause
behaviors along with corresponding temperature–stress
of their superior thermomechanical performance at
regimesbasedonthecreepdataforasecond-generation
increased temperatures compared with other high-tem- Ni-base single crystal superalloy CMSX-4.[1] Other
perature materials. It is generally understood that a
second-generation, single-crystal superalloys, such as
large volume fraction of coherent cuboidal Ni Al-base
3 Rene N5 and PWA1484, exhibit similar creep behav-
(L12) c0-phase precipitates regularly distributed in a iors.[2] In Figure 1, the data points indicate stresses and
Ni-base solid solution c-phase matrix provides an
temperatures at which actual creep tests were taken.
excellent barrier against the motion of dislocations on
Here, the data points with a same symbol indicate a
thermomechanical loading. Experimental studies[1 7]
similar creep behavior (or creep curves) as grouped by
show that the dislocation behavior in the c matrix and
shaded regions. In the low-temperature, high-stress
thec=c0 interface,andthestabilityofc0 precipitatesvary
regime (LH regime in Figure 1), the deformation is
dependingonthetemperatureandstress,andtheyresult
dominated by primary and secondary creep. The creep
in different creep behaviors for these alloys.
strain and time for the transition from primary to
Near-[0 0 1] orientation (<20 deg) creep behavior of
secondary creep is highly sensitive to the crystallo-
Ni-base, single-crystal superalloys has been subject to
graphic orientation. The magnitude of primary creep is
numerous experimental and numerical studies. Experi-
often tied qualitatively to the degree of single slip in a
particularslipsystemuntilitishamperedbylatenthard-
ening caused by activated additional slip systems.[3 7]
YOONSUKCHOI,SeniorResearchScientist,andTRIPLICANE Typically, the deformation-induced geometrical rota-
A.PARTHASARATHY,DirectorofMaterials&ProcessesDivision, tions are sufficient to activate additional slip systems.
arewiththeUES,Inc.,Dayton,OH45432.Contactemail:yoonsuk.
In the intermediate-temperature, intermediate-stress
[email protected] CHRISTOPHER WOODWARD, Principal
regime (MM regime in Figure 1), a typical creep curve
Materials Research Engineer, DENNIS M. DIMIDUK, Technical
Director, and MICHAEL D. UCHIC, Senior Materials Research shows an extended period of small creep strain (in the
Engineer,arewiththeMetalsCeramics&NondestructiveEvaluation absence of primary creep) until it reaches a tertiary-
Division, Air Force Research Laboratory, AFRL/RXLM, Wright creep-like stage.[3,8,9] In this regime, the gradual pro-
PattersonAFB,OH45433.
gression of directional coarsening (e.g., rafting) of
ManuscriptsubmittedAugust25,2011.
ArticlepublishedonlineJanuary19,2012 c0 precipitates is believed to be responsible for the
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME43A,JUNE2012 1861
representativemicrostructuralfeaturesthatthenewcreep
constitutive model accounts for are clarified. Finally,
creepat1023 K(750 (cid:2)C)and750 MPawassimulatedfor
selectedloadingdirections(within18 degfromthe[001]),
andtheresultsarecomparedwiththeexperimentalcreep
curvesforCMSX-4.
II. DISLOCATION MICROMECHANISMS
AND PREVIOUS MODELS FOR NEAR-[0 0 1]
CREEP IN THE LH REGIME
In the LH regime, the stress is high enough to induce
c0 shearing while the temperature is still too low to
facilitatedirectionalcoarseningofc0 precipitates.Exper-
imental observations indicated that dislocations with a
net Burgers vector ah112i cut c0 precipitates by creating
stacking faults (SF shearing) in a form of a superlattice
intrinsic stacking fault (SISF) or a superlattice extrinsic
stacking fault (SESF) in this temperature and stress
regime.[3 8] Here, a/2h112i, a/3h112i, and a/6h112i
Fig.1 Three temperature stress regimes grouped in a stress tem partial dislocations with a net Burgers vector ah112i
peratureplot:Eachregimerepresentsadistinctivecreepbehavior,as are formed in various combinations to shear c0 precip-
illustrated schematically a creep curve. The classification of creep itates.[7,8]However,currentconstitutivemodelshavenot
behaviors is based on the literature survey of near[0 0 1] creep of
been developed to a sufficient state as to represent
CMSX4.Allunitsintheschematiccreepcurvesarearbitrary.
accurately the spectrum of possible ah112i-type disloca-
tion reactions and the full fidelity of their effects on
tertiary-dominated creep behavior as illustrated in strain hardening during creep, nor has that challenge
Figure 1.[8] Creep in the high-temperature, low-stress been solved by the present model. In the current work,
regime(HLregimeinFigure 1)typicallyexhibitsasmall these partial dislocation reactions are referred to as
amount of primary creep followed by clear secondary ‘‘ah112i dislocation ribbons’’ without the subsequent
creep and catastrophic tertiary creep that immediately distinction between specific dislocation reactions. Rae
causes failure. In this regime, the completion of c0 et al.[4,5,7]proposedmicromechanismsforprimarycreep
directional coarsening is observed even in the primary caused by SF shearing of c0 precipitates by ah112i
creep stage, and the progression of secondary and dislocation ribbons and the transition to secondary
tertiary creep is believed to be controlled by the creep. Their primary creep mechanism involves three
destabilizationofthec=c0interfacialdislocationnetwork steps: the nucleation, propagation, and hardening of
and the heterogeneity of inherent defects.[10 14] Note ah112i dislocation ribbons. The nucleation of ah112i
thatthesethreedistinctivecreepregimesreflectdifferent dislocation ribbons requires a reaction of two active a/
microstructuralsensitivities and dislocationmicromech- 2h110i dislocations in the c-matrix channels or c=c0
anisms. Thus, it is imperative to incorporate those interfaces. Once ah112i dislocation ribbons are nucle-
micromechanisms into a constitutive model to capture ated,theirresolvedshearstressesshouldbehighenough
microstructure sensitivities of the creep behavior com- to percolate dislocations through c0 precipitates (in a
prehensively in all stress and temperature regimes. It is form of mixed cutting and looping). Shearing of c0
theauthors’ long-term goalto develop a comprehensive precipitatescanbedonebyapairofpartialdislocations
creep constitutive model for all stress and temperature (of a net Burgers vector ah112i) separated by SISF or
regimes. SESF, or two pairs of partial dislocations (still of a net
As a first step, in the current study, we focused on Burgers vector ah112i) separated by the complex stack-
developing a constitutive model that represents the ing fault and the antiphase boundary (APB).[3,5,7]
orientation-sensitiveprimaryandsecondarynear-[001] Generally, the a/2h110i dislocations are not observed
creepbehaviorintheLHregime.Significantprogresshas to cut c0 precipitates because an APB fault would have
been achieved in understanding dislocation micromech- to be formed and then annihilated by a second coupled
anisms in this creep regime,[3 7] although a mechanistic a/2h110i dislocation. Here, highly orientation-sensitive
strain-hardening law for creep has not been achieved. primary creep in the vicinity of the [0 0 1] orientation is
Creepmodelshavealsoimprovedprogressivelybythese believed to be caused by the propagation of single
pacingadvancesinunderstandingofcreepmechanisms. ah112i {1 1 1} slip. Suppose that orientations A and B
The first part of this paper summarizes the current are located in slightly different positions in the stereo-
understandingofdislocationmicromechanismsandcreep graphic projection near the [0 0 1], as shown as open
modelsfororientation-sensitivecreepintheLHregime. symbols in Figure 2, and ah112i{1 1 1} slip systems are
Distinctive features of the new creep model developed operative. After applying tension stress in A and B
in the current study are discussed subsequently along directions, different amounts of ½112(cid:2)ð111Þ slip (hence,
with detailed constitutive formulations. In particular, primary creep) are expected for orientations A and B
1862 VOLUME43A,JUNE2012 METALLURGICALANDMATERIALSTRANSACTIONSA
Earlymodelingoftheanisotropiccreepbehaviornear
the [0 0 1] was done by Matan et al.[3] They proposed a
phenomenological creep model, which was adopted
from Gilman’s dislocation density model,[17] by assum-
ingthatthemobiledislocationdensitydirectlytiestothe
creep strain. To capture the orientation dependence of
near-[0 0 1] creep, they set the mobile dislocation
attritioncoefficienttobeinverselyproportionaltoh(the
rotation required to reach a multislip orientation from
theoriginalloadingorientation).MacLachlanet al.[18 21]
proposed a series of creep models for anisotropic creep
of single-crystal superalloys. Their first model was a
continuum damage model, which incorporated slip
system kinematics to predict the anisotropic creep
behavior.[18] They introduced plastic shear rates and
resolved shear stresses for a/2h011i slip and ah112i slip,
and their effective values were linked to continuum
damage parameters. In addition, they used a kinematic
Fig.2 Stereographic projection near [0 0 1]. The arrows indicate
rotation paths of orientations A and B once the ½1(cid:2)12(cid:2)ð(cid:2)111Þ slip sys hardening term to account for hardening contributions
temisactivatedunderthe[001]tension. causedbyinteractionsbetweena/2h011iandah112islip.
Their first model was refined even more by enhancing
because the duration of ½112(cid:2)ð111Þ slip may be different softening and hardening terms for ah112i slip and, the
between the two orientations as differing amounts of modified model captured the near-[0 0 1] anisotropic
geometrical rotation (indicated by dotted arrows in creep behavior of CMSX-4 at 1023 K (750 (cid:2)C) and
Figure 2) are required to reach a multislip orientation 750 MPa.[19] In their most recent model, hardening
(orientations along the [0 0 1]-[0 1 1] symmetry bound- contributionsfroma/2h011islipinteractions,ah112islip
aryinFigure 2).Asimpledislocationpercolationbased interactions, and a/2h011i-ah112i slip interactions were
calculation suggested that SF shearing of c0 precipitates treatedseparately.[20]Thismodelwasusedtopredictthe
by h112i{1 1 1} slip produces much more shear strain creep behavior of a single-crystal turbine blade compo-
(by a factor of approximately 13) compared with the nent.[21] Recently, Ma et al.[22] proposed an analytical
caseforonlyc-matrixshearingbya/2h110i{111}slip.[4] creep model for the prediction of anisotropic creep for
That estimate indicates that a relatively small difference CMSX-4. They treated creep strain rates in c matrices
in single ah112i{1 1 1} slip duration could result in a and c0 precipitates by a/2h011i slip and ah112i slip,
large difference in primary creep strain. Here, single respectively. Their creep model was based on the
ah112i{1 1 1} slip is hardened by interactions between evolution of dislocation densities in c matrices and c0
ah112i dislocation ribbons and a/2h110i matrix disloca- precipitates, as well as corresponding dislocation veloc-
tions, as well as between ah112i dislocation ribbons ities.Majormicrostructuralparameters,suchasawidth
fromdifferentah112i{111}slipsystems.[7]Thisleadsto of c-matrix channels (a volume fraction of c0 precipi-
the transition to secondary creep. tates), the APB energy, and the c=c0 lattice misfit, were
Now,itseemstobeclearthattheorientation-sensitive incorporated in formulations of dislocation densities
primary creep for crystals oriented near [0 0 1] neces- and velocities but not tied explicitly to slip system
sitates relatively easy propagation of ah112i dislocation activation. They also incorporated an incubation effect,
ribbons throughout the entire c=c0 microstructure. In which is often observed in low-temperature, relatively
other words, the orientation sensitivity may be less low-stress creep, and the effect of void damage for the
pronounced if the propagation of ah112i dislocation better prediction of primary and tertiary creep behav-
ribbons is hampered effectively. This is probably the iors, respectively.
reason for different levels of orientation sensitivity of Generally,constitutivemodelsfornear-[001]creepin
near-[0 0 1] primary creep observed for a variety of the LH regime focused on the facilitation of single
single-crystal superalloys.[6,15,16] Different compositions ah112i slip in the primary creep stage and latent
and heat treatments result in different values of fault hardening of ah112i slip (a result of the single slip-
energies (such as SISF and APB energies), the lattice induced geometrical rotation and the interaction with
misfit between c and c0 phases, and different volume a/2h110i c-matrix dislocations) for the transition from
fractionsofc0precipitates.Allthesefactorsinfluencethe primary to secondary creep. In crystal plasticity (CP)
efficiency of ah112i dislocation ribbon propagation, FEM,microstructuralinfluencesonsingleslipandstrain
hence the orientation sensitivity of near-[0 0 1] primary hardeninginasinglecrystalmaybelesswellrepresented
creep. Unfortunately, state-of-the-art constitutive mod- becausethesingle-slipbehavioristreatedinacontinuum
eling approaches cannot capture fully such chemistry basis, which means no discrete representation of dislo-
and microstructure dependences. The current study is cation transmission or interaction in and across precip-
limited to representing primary and secondary creep itates in a single crystal. Because of this, constitutive
behaviorsphenomenologicallyforselectedsingle-crystal models often used ‘‘flags’’ (in a form of hyperbolic or
superalloys showing highly orientation-sensitive near-[0 sinusoidal functions) to facilitate and control the single
0 1] creep. slip behavior in CP-FEM effectively.[19 21] The current
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME43A,JUNE2012 1863
constitutive model also used those functional forms for
the similarpurpose.
III. NEW CONSTITUTIVE MODELING
APPROACHES FOR NEAR-[0 0 1] CREEP
IN THE LH REGIME
Beforedescribingindetailthenewconstitutivemodel,
it is important to clarify the representative microstruc-
tural features on which the new creep model is based.
Various microstructural and defect features at different
length scales may limit thermomechanical responses of
single-crystalsuperalloys.Figure 3showstypicalmicro-
structures ofasingle-crystal superalloyviewed atdiffer-
ent length scales (magnifications). At the macroscopic
scale (several millimeters to centimeters, Figure 3(a))
columnar dendrites, which are slightly misoriented
relativetoeachother,andacolonyofafewmisoriented
grains (fleckles) may be influential features. At the
intermediate scale (about a few hundred micrometers,
Figure 3(b)), the chemistry gradients between dendritic
cores and the interdendritic region that contains non-
metallic inclusions and microvoids are probably key
features that one needs to account for. At the micro-
scopicscale(severaltotensofmicrometers,Figure 3(c)),
regularly distributed cuboidal c0 precipitates and sur-
rounding c-matrix channels are microstuctural features
thatcontrolthermomechanicalresponses.Thenewcreep
constitutive model was developed by incorporating
dislocation micromechanisms identified from electron
microscopy analysis[3 7] together with certain assump-
tions about strain hardening. In this sense, Figure 3(c)
represents the microstructural features that are repre-
sentedbythecurrentcreep constitutivemodel.Thenew
modelhasnoexplicittreatmentoftheothermicrostruc-
tural and defect features shown in Figures 3(a) and (b).
However, dislocation behaviors in the c=c0 microstruc-
ture(Figure 3(c))seemtobeamajorfactorthatcontrols
anisotropic primary and secondary creep in the LH
regime,which isthefocusof thecurrent study.
Inthenew constitutivemodel,theplasticshearstrain
rate caused by creep deformation consists of two
contributions
c_a ¼f c_a þc_a ½1(cid:2)
c m SF
where c_a is the shear strain rate caused by the a/2h110i
m Fig.3 Microstructural and defect features that can influence ther
dislocationactivity(theslipsystemindexarangesfrom1
momechanical behaviors of singlecrystal superalloys at (a) macro
to12)inc-matrixchannelsandfcisthevolumefractionof scopic, (b) intermediate, and (c) microscopic length scales. All
the cmatrix. Also, c_a istheshear strain rate causedby micrographs were taken from a secondgeneration, singlecrystal
SF
shearing c0 precipitates and the c matrix by ah112i superalloyPWA1484.Thedirectionalsolidificationisalongtheverti
caldirectionin(a),whereas(b)and(c)arethecrosssectionalview.
dislocation ribbons (a ranges from 13 to 24). Details
abouthowconstitutiveformulationsweredevelopedfor
eachofthetwocontributionsaregiveninthesubsequent
sections. c_am ¼bqamvam ½2(cid:2)
where b, qa, and va are the magnitude of Burgers
A. c_am Resulting from a/2h110i Dislocations in the c vector, themdensity omf a/2h110i dislocations in the c
Matrix
matrix, and the corresponding average velocity, respec-
As noted previously, c_a reflects a/2h110i dislocation tively. In Eq. [2], the evolution of qa was assumed to
m m
behaviors in narrow c-matrix channels. The Orowan follow the Kocks–Mecking–Estrin equation (strain
equation was used to express c_a hardening law)[23,24] by
m
1864 VOLUME43A,JUNE2012 METALLURGICALANDMATERIALSTRANSACTIONSA
q_am ¼ðk1pqam(cid:3)k2qamÞ(cid:3)(cid:3)c_am(cid:3)(cid:3) ½3(cid:2) where qaSF and vaSF are the density of ah112i dislocation
ribbons and the corresponding average velocity,
where k and k are the constants that control the con- respectively. In Eq. [9], the qa evolution (q_a ) and va
1 2 SF SF SF
tributions of dislocation multiplication (hardening) were described numerically by adopting dislocation
and annihilation (recovery), respectively, for the evolv- evolution equations for the nucleation, propagation,
ing a/2h110i dislocation density. In Eq. [2], a power and hardening of ah112i SF dislocation ribbons, as
law was adopted to describe va proposed by Rae et al.[4,5,7] The evolution of qa was
m SF
expressed by
vam ¼vo(cid:3)(cid:3)(cid:3)(cid:3)gs^amm(cid:3)(cid:3)(cid:3)(cid:3)1=msignðsamÞ ½4(cid:2) q_aSF ¼(cid:4)kSFoqaSFMaSFþkSF11bl1 (cid:3)kSF2qaSFHaSF(cid:5)(cid:3)(cid:3)c_aSF(cid:3)(cid:3)
SF
and
½10(cid:2)
(cid:4) (cid:5)
Q
v ¼v0 exp (cid:3) m ½5(cid:2) where k , k , and k are the constants that control
o o kT SFo SF1 SF2
thecontributionofthecorrespondingterms.Equation [10]
where sa and g^ are the resolved shear stress and the basically incorporates the nucleation, propagation, and
slip resismtance fomr a/2h110i dislocations in the c matrix, immobilizationofah112idislocationribbons.
Thefirsttermoftheright-handside(RHS)ofEq. [10]
respectively, and m is the strain-rate sensitivity param-
eter. Also, in Eq. [5], v0 and Q are the preexponent accounts for the production of the ah112i dislocation
and a phenomenologicaol activamtion energy for time- density because of the nucleation of ah112i dislocation
dependent a/2h110i dislocation motion in the c matrix, ribbons from the reaction of two a/2h110i disloca-
tions in the c matrix and the c=c0 interface.[7] Table I
respectively. In this equation, the slip resistance g^ for
m
a/2h110i dislocations in the c matrix was set to consist shows a list of slip systems for a/2h110i slip and ah112i
slip along with the corresponding notations.
of three contributions
The reaction of two a/2h110i{1 1 1} slip systems can
g^m ¼g^oþg^Orowanþg^mSF ½6(cid:2) produce the a h112i{1 1 1} slip system if the resulting
h112i vector and h110i vector lie in the same {1 1 1}
where g^ ; g^ , and g^ are the slip resistances
o Orowan mSF plane. Based on this reaction criterion, Table II lists
caused by the intrinsic c-matrix strength, Orowan bow-
possibleah112i{111}slipsystemsfromreactionsoftwo
ing in the c-matrix channels, and the interaction of
a/2h110i{1 1 1} slip systems.
a/2h110i dislocations with ah112i dislocation ribbons,
Because the reaction matrix in Table II is symmetric,
respectively. The Orowan bowing contribution g^
Orowan only a diagonal half of the reaction matrix needs to be
was expressed by g¢lb/l, where g¢, l, and l are the geo-
considered.Here,onecannoticefromTable IIthateach
metric constant, the shear modulus, and the width of
ah112i{1 1 1} slip system can be produced by three
the c-matrix channel in the {111} plane (l = (3/2)1/2l ,
c different pairs of a/2h110i{1 1 1} slip systems. This
where l is the channel thickness), respectively. In
c findingimpliesthattheavailabilityofnucleatingeachof
Eq. [6], the evolution of g^ was expressed by
mSF ah112i{1 1 1} slip system can be checked by examining
g^_ ¼X12 X24 hab c_b ½7(cid:2) activities of the corresponding three pairs of a/2h110i{1
mSF a¼1 b¼13 mSF SF 11}slipsystems.Thus,thenucleationterminEq. [10]is
completed by introducing Ma to quantify the avail-
and SF
ability of the ah112i{1 1 1} nucleii
!!
hambSF ¼hsm(cid:4) 1þX2b4¼13fambSF gs^bSbSFF ½8(cid:2) MaSF ¼ im¼1a;2x;3((cid:3)(cid:3)(cid:3)(cid:3)(cid:3)ssmmmmainxððaaii;;bbiiÞÞ(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:3)(cid:3)smavgðai;biÞ(cid:3)(cid:3))j^s1SFj(cid:3)Mo!R1M
where h and fab are the initial hardening modulus ½11(cid:2)
sm mSF
and the coefficient for the a/2h110i - ah112i interaction,
respectively. Also, c_b , sb , and g^b are the shear rate,
SF SF SF TableI. Twelve a/2h110i{11 1}andTwelve ah112i{111}
the resolved shear stress and the slip resistance for the Slip Systems andTheir Notations
h112i{1 1 1} slip system b, respectively. Details about
thesethreestatevariableswillbediscussedinthefollow- h110i{111} h112i{11 1}
(cid:6) (cid:7) (cid:6) (cid:7)
ing section. Equations [7] and [8] were introduced to A1 011 ð111Þ AI 112 ð111Þ
(cid:6) (cid:7) (cid:6) (cid:7)
A2 101 ð111Þ AII 121 ð111Þ
account for strengthening of the a/2h110i slip systems (cid:6) (cid:7) (cid:6) (cid:7)
A3 110 ð111Þ AIII 211 ð111Þ
because of the interaction with ah112i dislocation (cid:6) (cid:7)(cid:8) (cid:9) (cid:6) (cid:7)(cid:8) (cid:9)
B1 011 111 BI 112 111
ribbons. (cid:6) (cid:7)(cid:8) (cid:9) (cid:6) (cid:7)(cid:8) (cid:9)
B2 101 111 BII 121 111
(cid:8) (cid:9) (cid:8) (cid:9)
B3½110(cid:2) 111 BIII ½211(cid:2) 111
(cid:8) (cid:9) (cid:6) (cid:7)(cid:8) (cid:9)
C1½011(cid:2) 111 CI 112 111
B. The ah112i Dislocation Shear Rate, c_a (cid:6) (cid:7)(cid:8) (cid:9) (cid:8) (cid:9)
SF C2 101 111 CII ½121(cid:2) 111
(cid:6) (cid:7)(cid:8) (cid:9) (cid:6) (cid:7)(cid:8) (cid:9)
Similar to Eq. [2], c_a was expressed using the C3 110 111 CIII 211 111
Orowan equation SF D1(cid:6)011(cid:7)(cid:8)111(cid:9) DI ½112(cid:2)(cid:8)111(cid:9)
(cid:8) (cid:9) (cid:6) (cid:7)(cid:8) (cid:9)
D2½101(cid:2) 111 DII 121 111
c_a ¼bqa va ½9(cid:2) D3(cid:6)110(cid:7)(cid:8)111(cid:9) DIII (cid:6)211(cid:7)(cid:8)111(cid:9)
SF SF SF
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME43A,JUNE2012 1865
TableII. TheMatrix Showing Output ah112i{11 1}Slip Systems from Reactionsof Twoa/2h110i{11 1}Slip Systems
A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3
A1 AI AII BI BII AI AII
A2 AI AIII AI CI CIII AIII
A3 AII AIII AII AIII DII DIII
B1 AI AII BI BII BII BI
B2 BI BI BIII BIII DI DIII
B3 BII BII BIII CII CIII BIII
C1 CI CII CI CII DI DII
C2 AI AIII CIII CI CIII CI
C3 CIII BII BIII CII CIII CII
D1 DII DI CI CII DI DII
D2 DIII BI BIII DI DI DIII
D3 AII AIII DIII DII DII DIII
where for each of ah112i{1 1 1} slip systems (a = 13 to Returning to Eq. [9], the average velocity of ah112i
24 in Eq. [11]), the index i indicates three available dislocation ribbons was expressed using a power law as
pairs of a/2h110i{1 1 1} slip systems and sminðai;biÞ,
sammveaxrðaagi;beiÞ,resaonldvedsmasvhgðeaai;briÞstraersesesthfeorsemacahllear/,2hl1a1r0giem{r1, 1an1d} vaSF ¼vSFo(cid:3)(cid:3)(cid:3)(cid:3)gs^aSF(cid:3)(cid:3)(cid:3)(cid:3)1=mSFsignðsaSFÞ ½13(cid:2)
SF
pair, respectively. Also, ^s is the stress required to
SF
create a stacking fault (^s ¼g E =b, where g and and
SF SF SF SF
ESF are the constant and the stacking fault energy in (cid:4) Q (cid:5)
the c matrix, respectively), and M and R are v ¼v0 exp (cid:3) SF ½14(cid:2)
o M SFo SFo kT
constants. The first term of the RHS of Eq. [11] was
formulated such that the nucleation of the ah112i{1 1
where m is the strain rate sensitivity parameter.
1} comes from a most active a/2h110i{1 1 1} pair by Also, in SEFq. [14], v0 and Q are the preexponent
checking the corresponding interactions in Table II. In SFo SF
and the activation energy for ah112i dislocations,
Eq. [11], the second term Mo was intended to cut off respectively. Here, the slip resistance g^a for ah112i
the value range of the first term before the normaliza- SF
dislocation ribbons was set to originate from two con-
tion by R .
M tributions
ThesecondtermoftheRHSofEq. [10]isintendedto
incorporate the multiplication of ah112i dislocation g^_a ¼ha c_maxþX12 habc_b ½15(cid:2)
ribbons.Here, l istheaverage c0 SF-shearingdistance SF SF SF b¼1 m m
SF
(l = (L2/(3)1/2)1/2, where L is the size of the cuboidal
SF The first and second terms of the RHS of Eq. [15] are
c0 precipitate[25]). The thirdterm ofthe RHS ofEq. [10]
hardening contributions from ah112i - ah112i interac-
accounts for the immobilization of ah112i dislocation
tions and from ah112i - a/2h110i interactions, respec-
ribbons by interactions between different ah112i dislo- tively. The initial value of g^a was determined by
cation ribbons. The law governing the immobilization g^a ¼E =b, where E is SthFe APB energy for c0
follows a modified Kock, Argon, and Ashby form that pSrFecipitaAtPeBs. In Eq. [15]A,PcB_max is the maximum shear
accounts for the maximally stressed ah112i{1 1 1} glide rate of ah112i{1 1 1} slipSFsystems. Here, ha is the
system. Here, Ha was expressed by SF
SF hardening modulus from ah112i - ah112i interactions
2 0 ( (cid:3) (cid:3) ) and was expressed by
(cid:3)sb (cid:3) (cid:3) (cid:3)
HaSF ¼1(cid:3)exp4(cid:3)@b¼mb16¼3a.a.x.24 faSbF(cid:3)(cid:3)(cid:3)smSSFaFx(cid:3)(cid:3)(cid:3)(cid:4)(cid:3)(cid:3)saSvFgða;bÞ(cid:3)(cid:3) haSF ¼(cid:4)hs(cid:4)sech1=2(cid:4)ccaSF(cid:5)(cid:5)(cid:4)(cid:8)1þnSFHaSF(cid:9) ½16(cid:2)
1
1 !qSF# where h and c are the initial hardening modulus and
(cid:3)(cid:3)smSFax(cid:3)(cid:3)þpSF ½12(cid:2) the norms alizati1on constant, respectively. Also, nSF is
the scaling constant that links Ha of Eq. [12] to ha .
SF SF
For the second term in Eq. [15], hab is the hardening
In Eq. [12], faSbF is the interaction coefficient, and pSF modulus stemming from ah112i - a/m2h110i interactions
and qSF are constants. Also, smSFax is the maximum that was expressed by
resolved shear stress of all ah112i{1 1 1} slip systems,
and sSavFgða;bÞ is the average resolved shear stress of the hab ¼h (cid:4)(cid:4)1þX12 fabtanh(cid:4) cbm (cid:5)(cid:5) ½17(cid:2)
two ah112i{1 1 1} slip systems. Equation [12] was m SFm b¼1 m cmax
SF
formulated in such a fashion so that rapid immobiliza-
tion (in a form of an S-type curve) of the primary slip where h and fab are the initial hardening modulus
SFm m
systemtakesplaceasthesecondaryslipsystembecomes and the coefficient for ah112i - a/2h110i interactions,
active. respectively.
1866 VOLUME43A,JUNE2012 METALLURGICALANDMATERIALSTRANSACTIONSA
IV. SIMULATION OF ANISOTROPIC CREEP that the deformed geometry is locally nonuniform,
FOR CMSX-4 which is typical for single crystals deformed in low
symmetry (viz., single slip oriented) orientations.
The new constitutive model described in the previous
UsingthenewconstitutivemodeldescribedinSection
section was used to capture orientation sensitivity of
III and the sample geometry shown in Figure 4(a), the
near-[001]creepofasingle-crystalsuperalloyCMSX-4.
creep simulations were performed in three different
All constitutive formulations (Eqs. [1] to [17]) were
crystallographic directions near the [0 0 1] at 1023 K
implemented in the commercial FEM package ABA-
(750 (cid:2)C) and 750 MPa, and the resulting creep curves
QUS (Simulia, Providence, RI) through the user mate- were compared with those from CMSX-4[3] in Figure 5.
rial subroutine (UMAT). Here, a tangent modulus
Here, three creep orientations K, L, and M are
method[26] was adopted for the time integration.
misoriented from the [0 0 1] by 5.3, 8.0, and 17.5 deg,
Preliminary parametric studies were performed to iden-
respectively. The prediction result (Figure 5) confirms
tify reasonable ranges for values of major input param-
that the new creep constitutive model reasonably the
eters. Table III lists values of input parameters for the
orientation sensitivity of primary and secondary creep
description of c_a and c_a .
m SF captures for near-[0 0 1] oriented single-crystal superal-
In Table III, the values of parameters marked with
loys. In Figure 6, the shear strain rates from ah112i slip
the asterisk weredetermined byfittingto thecreep data
(c_ ) and a/2h110i matrix slip (c_ ) were averaged over
inFigure 5.[3]Thevaluesforactivationenergies(Q and SF m
m an entire sample volume and plotted as a function of
Q ) and fault energies (E and E ) were approxi-
SF APB SF creep time. As expected, ah112i slip dominates for all
mated from the experimental and modeling data.[27 30]
three orientations, and the degree of primary creep is
The values for l , f , L and b were directly taken from
c c showntobeinverselyproportionaltothelevelofthec_
the input microstructure (cuboidal c0 precipitates with SF
suppression, which is highest in M and lowest in K.
an edge length of 0.52 lm and the volume fraction of
Here,itisinterestingtonotethatc_ showstheopposite
68 pct). The constants in Eqs. [3] and [10], and the m
trend, the highest suppression in K and the lowest
parameters in Eqs. [4] and [13] were determined from a
suppression in M. Importantly, this result implies that
range of values used in other models.[31 33]The geomet-
a/2h110i matrix slip cannot be ignored, and it plays an
ric constants g¢ and g were set as 1 for the current
SF important role in blocking ah112i dislocation ribbons
simulations. Also, the coefficients (f , f , and f ) for
mSF m SF slip activities effectively, leading to the transition to
the interaction of different slip mechanisms were
secondary creep. The current simulation results seem to
assumed to be 1 because their values are not informed.
The shear modulus l was calculated from l =
[0.5ÆC Æ(C – C )]1/2, where the temperature depen-
44 11 12
denceofelasticconstants(C ,C ,andC )forCMSX-
11 12 44
4 was determined by experimental fits from Allan.[34]
Figure 4(a) shows the sample geometry used for FEM
simulations. A cylindrical sample was meshed by the
combination of linear wedge (C3D6) and brick (C3D8)
elements. Boundary conditions for creep deformation
were applied to the meshed sample by restraining the
bottom face vertically and applying appropriate body
forces to thetopmesh layer ofthecylindrical sampleto
induce an axial stress comparable with the creep stress.
Figure 4(b) shows the contour of accumulated shear in
the deformed sample geometry after 15 pct creep strain
alongthesingleslip-orienteddirection.Onecanobserve
TableIII. Values ofInput Parameters Usedfor theCurrent
Simulations
c_a c_a
m SF
Q 323kJ/mol Q 280kJ/mol M 0.4*
m SF o
g^ 300MPa* E 0.12 J/m2 R 0.3*
o APB M
k 1.29108/m E 0.11 J/m2 h 40GPa*
1 SF s
k 30 k 4 c 0.07*
2 SFo 1
v0 19109/s k 2 910(cid:3)4 n 24*
o SF1 SF
m 0.13 k 2 h 10GPa*
SF2 SFm
g¢ 1 v¢ 1 9109/s f 1
SFo m
l 0.068910(cid:3)6 m m 0.25 f 1
c SF SF
h 10GPa* g 1 p 0.028*
sm SF SF
f 1 L 0.52 910(cid:3)6m q 250* Fig.4 (a) A cylindrical sample geometry meshed by linear wedge
mSF SF
f 0.32 k 4 (C3D6)and brick (C3D8)elements.(b) The contour of accumulated
c SFo
b 2.489A˚ shearinthedeformedsamplegeometryafter15pctcreepstrainina
singlesliporienteddirection.
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME43A,JUNE2012 1867