DTIC AD1000339: On the Eigenstrain Application of Shot-Peened Residual Stresses Within a Crystal Plasticity Framework: Application to Ni-Base Superalloy Specimens (Postprint)

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WILLIAM D. MUSINSKI, Project Engineer DANIEL J. EVANS, Chief
Metals Branch Metals Branch
Structural Materials Division Structural Materials Division

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ROBERT T. MARSHALL, Deputy Chief
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Materials And Manufacturing Directorate

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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
ON THE EIGENSTRAIN APPLICATION OF SHOT-PEENED RESIDUAL In-House
STRESSES WITHIN A CRYSTAL PLASTICITY FRAMEWORK:
5b. GRANT NUMBER
APPLICATION TO NI-BASE SUPERALLOY SPECIMENS (POSTPRINT)
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62102F
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14. ABSTRACT
Shot-peening-induced compressive residual stresses are often introduced in Ni-base superalloy components to help prevent or
retard surface fatigue crack initiation and early growth at near surface inclusions. In certain cases these compressive residual
stresses can shift the fatigue crack initiation site from surface to sub-surface locations. However, the ability to computationally
predict the improvement in fatigue life response and scatter due to induced compressive residual stresses are lightly treated in the
literature. To address this issue, a method to incorporate shot-peened residual stresses within a 3D polycrystalline microstructure
is introduced in this work. These residual stresses are induced by a distribution of fictitious orquasi-thermal expansion eigenstrain
as a function of depth from the specimen surface. Two different material models are used, a J plasticity and a crystal plasticity
2
model. First, theJ plasticity model with combined isotropic and kinematic hardening is used to determine the distribution of
2
quasi-thermal expansion eigenstrain as a function of depth from the surface necessary to induce the target residual stress profile
within the microstructure. This distribution of quasi-thermal expansion eigenstrain is then used within a crystal plasticity
framework to model the effect of microstructure heterogeneity on the variability in residual stresses among multiple
instantiations. This model is verified with experimental X-ray diffraction (XRD) data for scatter in residual stresses for both the
initial microstructure and after a single load/unload cycle.
15. SUBJECT TERMS
crystal plasticity, Ni-base superalloys, IN100, residual stresses, eigenstrain
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6. AUTHOR(S)
William D. Musinski - Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB

David L. McDowell - Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta GA; School of
Materials; Science and Engineering, Georgia Institute of Technology, Atlanta GA
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Materials and Manufacturing Directorate
Air Force Research Laboratory
Wright-Patterson AFB OH 45433

Woodruff School of Mechanical Engineering
Georgia Institute of Technology
Atlanta GA 30332

School of Materials; Science and Engineering
Georgia Institute of Technology
Atlanta GA 30332
Standard Form 298 (Rev. 8-98)
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InternationalJournalofMechanicalSciences100(2015)195–208
ContentslistsavailableatScienceDirect
International Journal of Mechanical Sciences
journal homepage: www.elsevier.com/locate/ijmecsci
On the eigenstrain application of shot-peened residual stresses within
a crystal plasticity framework: Application to Ni-base
superalloy specimens
William D. Musinskia,b,n, David L. McDowellb,c
aMaterialsandManufacturingDirectorate,AirForceResearchLaboratory,Wright-PattersonAFB,OH45433,USA
bWoodruffSchoolofMechanicalEngineering,GeorgiaInstituteofTechnology,Atlanta,GA30332,USA
cSchoolofMaterialsScienceandEngineering,GeorgiaInstituteofTechnology,Atlanta,GA30332,USA
a r t i c l e i n f o a b s t r a c t
Articlehistory: Shot peening induced compressive residual stresses are often introduced in Ni base superalloy
Received20February2015 componentstohelppreventorretardsurfacefatiguecrackinitiationandearlygrowthatnearsurface
Receivedinrevisedform inclusions.Incertaincasesthesecompressiveresidualstressescanshiftthefatiguecrackinitiationsite
22May2015 fromsurfacetosub surfacelocations.However,theabilitytocomputationallypredicttheimprovement
Accepted29June2015
infatigueliferesponseandscatterduetoinducedcompressiveresidualstressesarelightlytreatedinthe
Availableonline8July2015
literature. To address this issue, a method to incorporate shot peened residual stresses within a 3D
Keywords: polycrystalline microstructure is introduced in this work. These residual stresses are induced by a
Crystalplasticity distribution of fictitious or quasi thermal expansion eigenstrain as a function of depth from the
Ni-basesuperalloys
specimensurface.Twodifferentmaterialmodelsareused,aJ plasticityandacrystalplasticitymodel.
IN100 2
First,theJ plasticitymodelwithcombinedisotropicandkinematichardeningisusedtodeterminethe
Residualstresses 2
distributionofquasi thermalexpansioneigenstrainasafunctionofdepthfromthesurfacenecessaryto
Eigenstrain
induce the target residual stress profile within the microstructure. This distribution of quasi thermal
expansion eigenstrain is then used within a crystal plasticity framework to model the effect of
microstructureheterogeneityonthevariabilityinresidualstressesamongmultipleinstantiations.This
modelisverifiedwithexperimentalXraydiffraction(XRD)dataforscatterinresidualstressesforboth
theinitialmicrostructureandafterasingleload/unloadcycle.
PublishedbyElsevierLtd.
1. Introduction fatiguecracks[5].Accordingly,surfacecompressiveresidualstres
sesareoftenintroducedinNi basesuperalloycomponentstohelp
1.1. Motivationformodelingresidualstresses prolong/retard fatigue crack initiation [14] and early growth at
nearsurfaceinclusionsandpotentiallyshiftfatiguecrackinitiation
The beneficial effects of compressive surface residual stresses sitesfromsurfacetosubsurfacelocations[1,2]toincreasefatigue
onhighcyclefatigue(HCF)responsehavebeenwelldocumented life.However,compressiveresidualstressesareusuallyonlyuseful
in the literature [1 4]. For Ni base superalloys, a shift from in the transition fatigue and HCF (N 45(cid:3)105 cycles) regimes
f
surface dominated to subsurface dominated fatigue crack initia since residual stress relaxation at higher applied stress/strain
tion sites exists for the transition from lowcycle fatigue (LCF) to values can eliminate the effectiveness of compressive residual
HCFregimes [5 9]. A similar surface to subsurface transition has stressesonfatiguelife[14 17].
beenreportedfortitaniumalloys[10,11]andhighstrengthsteels Compressivesurfaceresidualstressescanbeappliedviamulti
[12,13].Inthetransitionfatigueregime(cyclestofailure,N (cid:1)1(cid:3) ple techniques (shot/gravity peening, low plasticity burnishing,
f
104 5(cid:3)105 cycles), subsurface initiated fatigue cracks tend to lasershockpeening,etc.[18]).Shotpeeningisthemostcommonly
require more cycles to failure as compared to surface initiated usedtechniqueinindustrytoinducecompressiveresidualstresses
atthesurface,andisthefocusofthisresidualstressstudy.During
theshotpeeningprocessmultiplehigh velocityshotbeadsimpact
the surface forming multiple indentations and inhomogeneous
nCorrespondingauthorat:MaterialsandManufacturingDirectorate,AirForce
ResearchLaboratory,Wright-PattersonAFB,OH45433,USA.Tel.:þ19372550485. compressive elastic/plastic deformation of the near surface layer.
E-mailaddress:[email protected](W.D.Musinski). Assuch,theresultingresidualstressprofileisdictatedby(1)the
http://dx.doi.org/10.1016/j.ijmecsci.2015.06.020
0020-7403/PublishedbyElsevierLtd.
1
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196 W.D.Musinski,D.L.McDowell/InternationalJournalofMechanicalSciences100(2015)195–208
interaction between multiple indentations, (2) localized con and/or Almen intensityas a function of shot peening parameters
strainedcompression,(3)strainratesensitivityandelastic/plastic (shot size, speed, coverage, material properties, incident angles,
responseofthematerial,and(4)localmicrostructure.Inmodeling, etc.)[26 42]and(2)Simulationoftheoverallinducedmechanical
these discrete impingement events can either be modeled expli response due to shot peening through a deformation process
citly or idealized as a collective elastic/plastic constrained com [20,43,44]. Some examples of relevant works regarding these
pressiveloading/unloadingeventasschematicallyshowninFig.1. twomethodsarediscussedbelow.
The idealization of the multiple impingement shot peening pro
cessasasingledeformationeventisadoptedinthispaper.
1.2.1. Simulationofsingleandmultipleimpactevents
While the influence of residual stress on fatigue life has been
Thefirstmeanstomodelthehigh velocityimpactshotpeening
reportedextensivelyintheliterature,theabilitytocomputation
processfocusedonsingleimpacteventsonanelastic plastictarget
allypredict the improvement in fatigue resistance and change of
substrate.ChenandHutchinson[27,28]andBoyceetal.[26]found
scatterduetoinducedcompressiveresidualstressesarelackingin
that explicit dynamic simulations incorporating effects of strain
the literature. A significant effort in modeling inclusion and
ratesensitivity,inertia,andelasticwavepropagationresultedina
residual stress effects in shot peened martensitic gear steels was
better prediction of residual stresses than quasi static analyses.
undertakenbyPrasannavenkatesanetal.[19 22].Theyconsidered
Frija et al. [29] used an energy equivalence expression and 3D
separate 3D finite element method (FEM) models at discrete
quasi static FEM analysis to find good correlation between the
depths subjected to the required amount of compressive load/
predicted residual stress along the shot bead impact centerline
unloadstraintoinducetherequiredresidualstressprofileateach
and experiments. Zion and Johnson [34] studied the impact of a
particular depth. The simulated inclusion sizes (o10μm) were
hardandsoftshotbeadandahigh strengthsteelandfoundthat
small compared tothe overall 3D FEM model dimensions, sothe
the most highly significant input factor was the value of friction
gradient in applied stress and residual stress over the inclusion
coefficient assumed between the shot bead and target material.
was considered to be negligible in their analysis. Alternatively,
Whilethesesingleimpactsimulationscanunveilkeyrelationships
inclusion sizes in powder metallurgy polycrystalline Ni base
between shot peen input (e.g. shot diameter, impact velocity,
superalloy IN100 are on the order of 10 100μm [6,23,24]. For
incident angle) and residual stress output [32], multiple impact
inclusions of these sizes, the gradientin residualstress(RS) field
events should be used to simulate more realistic peening
duetoshotpeening(Ref.RSprofilein[25])canhaveasignificant
conditions.
effect on stress/strain response and should be considered when
One means to model multiple impacts is to combine discrete
analyzing inclusion and RS effects in Ni base superalloys. There
element modeling (DEM) and FEM to determine RS profile. DEM
fore,a simulation approach that accounts for the entire distribu
simulates spatial interactions/collisions among multiple discrete
tion of residual stress, and not just at discrete surface depths, is
particles within the shot stream. DEM exports particle/substrate
warranted. Hence, we aim to develop a framework to assess
impact velocity,location, and contact forcesintoaFEM model to
(1)theeffectofmicrostructureontheentireresidualstressprofile
calculate resulting plastic strains and residual stresses. Multiple
duetoshotpeeningand(2)theeffectofcyclicloadingonresidual
researchers[39 42]haveusedthiscombinedDEM FEMapproach
stressrelaxationinpolycrystallineNi basesuperalloycomponents.
to link the effect of complex part geometry on the overall shot
Thissectionbeginswithabriefoverviewofpreviousmethods
peening process, including the effects of impact angle, impact
to impose residual stresses within components. Next, the eigen
density, and coverage on location specific residual stress
strainapplication of residualstresseswithin a FEM framework is
formation.
discussedandaJ plasticitymodelispresentedforcalibratingthis
2 These combined DEM FEM simulations of multiple impinge
model.Finally,themethodbywhichcrystalplasticityisincorpo
ment events are certainly noteworthy. However, it is hard to
rated is presented with results for variability of initial residual
discernhowmuchvariability in localresidual stressisattributed
stress and retained residual stress due to a single load/unload
to (1) the localized inhomogeneous deformation due to the
sequence.
stochastic impingement events inherent in the shot peening
processand(2)theeffectoflocalmicrostructure,especiallygrain
1.2. Previousmethodsforsimulatedapplicationofresidualstresses orientation.Therefore,toisolatetheeffectoflocalmicrostructure,
we propose to use a uniform average “amount of impingement”
Techniquestosimulatetheshotpeeningprocesscanbedivided within a crystalplasticity finite element model. In this approach,
intotwomethods:(1)Simulationoftheimpactresponsebetween theindividualshotpeeneventsthatweremodeledintheprevious
the shot bead and the shot peened surface by quasi static or method(s)arenotmodeled.Instead,theresultingmaterialdefor
explicit dynamic analyses to predict the resulting residual stress mationandhardeningstatesduetoshotpeeningare(1)induced
explicitlywithinanimplicitorexplicitFEMmodeland(2)usedas
initial conditions for subsequent relaxation/fatigue analysis
simulations.
1.2.2. Techniquestoinduceoverallmechanicalresponseduetoshot
peening
One way to induce residual stresses explicitly within an FEM
model is to deform the FEM model in displacement controlled
constrained compression. This method mimics the collective
mechanical meansinwhichbiaxialresidualstressesareimposed
within a component during the shot peening process and traces
the evolution of the material state throughout the deformation
process. For example, Prasannavenkatesan et al. [20,21] used a
simple displacement controlled method in conjunction with iso
tropic plasticity [20] and polycrystal plasticity [21] to induce
Fig. 1. Schematic showing idealization of shot peening process as a single
deformationevent. residual stresses [20] and reproduce experimental trends in
2
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W.D.Musinski,D.L.McDowell/InternationalJournalofMechanicalSciences100(2015)195–208 197
residual stress relaxation [21] due to HCF cyclic bending of a whether the given eigenstrain distribution is able to reconstruct
martensitic gear steel. For these analyses, individual 3D FEM the desired residual stress profile. For example, Korsunsky [44]
modelswereusedatdiscretedepthsandsubjectedtotherequired used axisymmetric plate theory to analytically find stresses and
amountofcompressiveload/unloadstraintoinducetherequired deformations arising due to peening and found the necessary
residualstressprofileateachparticulardepth.Thisdisplacement eigenstrainasafunctionofplatedepthusingpolynomialfunctions
controlled method presented by Refs. [20,21] is best used for asbasisfunctions.
simulated application of residual stresses to the surface of a Methodsthatincorporateboththedynamicmaterialresponse
smoothplanarspecimenwithwell definedboundaryandloading during residual stress application and the subsequent effect that
conditions. theseresidualstresseshaveonmaterialbehaviortypicallyrequire
An alternative means to initialize residual stresses within a separateanalysesduetodifferenttimescales(dynamicshot/laser
FEMmodelistousetheresidualstressandmaterialhardening peen process versus quasi static LCF/HCF/creep loading). For
statesfortargetinitialconditions.Forexample,Buchananetal. example, Achintha et al. [55 57] combined explicit dynamic and
[25,45] initialized their material model with an initial material implicit FEM analyses to model the laser shock peening in a Ti
hardeningstate(effectiveplasticstrain,backstress)andresidual 6Al 4V alloy using an eigenstrain approach and an assumed
stressstatebasedonexperimentalXRDandcoldworkmeasure elastic perfectlyplasticmaterialmodel.Theyfoundthatalthough
mentsatdifferentdepthsinaNi basesuperalloyIN100material eigenstrain distributions were similar for different specimen
[45].Similarly,Benedettietal.[46,47]imposedinitiallocalyield thicknesses, the resulting residual stresses were quite different
strength and hardening variables at different depths based on amongdifferentspecimenthicknesses.
the experimental microhardness measurement at that depth.
Additionally, the initial residual stress profile was induced by
introducing an experimentally fit eigenstrain distribution
within the FEMmodel.The methodof introducingeigenstrains 1.3. Objectives,scopeandlimitationsofcurrentwork
withinanFEMmodeltoproduceresidualstressesiscoveredin
moredetailnext. Thepreviouseigenstrainmethodsmentionedintheforegoing
all suffer from the same shortcomings: they used simple elastic
plastic models that are unable to determine the effect of local/
1.2.3. Eigenstrainmethodofimposingresidualstresses
random microstructure (e.g. grain size/orientation) on residual
Eigenstrains have long been used to model micromechanical
stressprofilevariabilityandresidualstressrelaxation.Hence,the
misfit strains that are not associated with globally/externally
objective of this paper is to develop a combined eigenstrain and
applied mechanical loading. Some examples where eigenstrains
crystal plasticity finite element simulation approach to address
are used include internal stresses due to inclusions/particles or
thiscriticalneed.
fibers [48], differences in coefficient of thermal expansion of
Therearemanyapproximationsmadewithregardtotheactual
differentphases/layups[49],phasetransformations[50],andheat
shot peening process in our simulations. The process of shot
treatment effects [51]. Several authors have used the eigenstrain
peening involves multiple random shot indentations inducing
method to model/reconstruct residual stresses induced by shot
surface roughness (cf. [15,47,70]). As stated previously, we do
peening [44,52 54], laser shock peening [55 61], and welding
not model individual shot peening events. Rather, the collective
[62 66]. For shot peening analysis, the amount of residual stress
shot peening process is modeled as a uniform, quasi static
induced as a function of depth can be controlled by specifying
displacement event (cf. Fig.1) that spatially varies as a function
spatial distributions of thermal eigenstrains. The actual residual
ofsurfacedepth.Thevariabilityinresidualstresscomesintoplay
stressapplicationprocess isnominallyisothermal,sothe applied
due to the microstructure, not due to incomplete coverage or
temperature change and thermal expansion eigenstrains are
random/sporadic shot locations. In practice, the XRD residual
fictitious; they are merely introduced as a means to induce
stressmeasurementsareaveragedovertheirradiatedareadeter
residualstresseswithinacomponentunderisothermalconditions.
mined by the size of the X ray beam. The experimental XRD
Themainchallengeofthisapproachisdeterminingthecorrect
averageresidualstressdatausedforcomparisoninthisstudywas
eigenstraindistributionrequiredtoproduceagivenresidualstress
found over an irradiated area of 3mm(cid:3)5mm [45], which
profile. General solutions/frameworks to the so called “inverse
encompasses over 10,000 grains. Therefore, the highly localized
eigenstrain problem” have been developed/presented by many
variations in residual stresses due to surface undulations are
authors [44,52,53,67 69]. Universally, these approaches assume
averagedoutforthereportedXRDresidualstressdata.Therefore,
thattheeigenstraindistributioncanbereconstructedasasuper
we do not simulate the random surface undulations due to the
positionofatruncatedseriesofbasisfunctions,i.e.,
multiple surface impingements induced during the shot peening
XN process.Instead,weimposeauniform“amountofimpingement”
εnðxÞ¼ c ξðxÞ ð1Þ
i i ineachsurfacelayerbymeansofanappliedeigenstrain.
i 1 Freesurfaceeffectswithrespecttoin planestresscomponents
were neglected/avoided by assuming that the finite element
Inthisequation,Nisthenumberofbasisfunctions,ξðxÞ,andc material is fromthe centerof the specimen. The increased hard
i i
arethecoefficientmultipliersforeachbasisfunction.Thebenefit eningofthesurfacelayeroftheshot peenedmaterialisaccounted
ofthismodelisthatoneisatlibertytochoosethetotalnumberof forinthecrystalplasticityfiniteelementmodelviaanincreasein
basis functions and the form of each basis function; as a result, dislocationdensity(andthesubsequentincreaseinyieldstrength
therearemultiplesetsofbasisfunctionsthatcandescribeagiven [71])atthesurface.
eigenstrain distribution. For example, one could use a series of Itshouldalsobenotedthattheintenseamountofnearsurface
smooth basis functions [67 69], multiple polynomials [44], or coldworkinganddensedislocationnetworkproducedduringshot
evenasuperpositionofmultiplekernel densityfunctionsinclud peening caninvoke nearsurface plasticity induced refinementof
ingtriangularfunctions[53]ornormaldistributionfunctions[52] the microstructure [45,70]. This microstructure refinement is not
astheoveralleigenstraindistributionestimator.Regardlessofthe accounted for in this work. Thus, the purpose of our model is to
functional form, the most important factor that should be con simulate the overall induced mechanical response due to shot
sidered is how to solve for the basis function coefficients and peening,ratherthanindividualimpacteventsorgrainrefinement.
3
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198 W.D.Musinski,D.L.McDowell/InternationalJournalofMechanicalSciences100(2015)195–208
2. Methodologyforimposingresidualstressesusing for by the evolution of isotropic hardening via
(cid:5) (cid:6)
eigenstrainapproach σ ¼σ þκ 1 expð bεpÞ [76,77],whereσ istheyieldstressat
ys o s o
zero plastic strain, κ is the maximum change in the size of the
s
The methodology used to impose residual stresses within an yield surface, and b defines the rate at which cyclic hardening
eigenstrain framework and its extension to the crystal plasticity occurs.Theevolutionofthebackstresstensorischaracterizedby
(cid:2) (cid:3)
fithneirtemaellemexepnatnsmioenthoddistarirbeutcioovnerecadninbethiosptsiemctizioend. Ttohefiqtuaasniy α_k¼Foσcrykstσhisαstuε_dpy,rwkαekeε_mp [p2l0oy,75k]¼. 2 backstress terms (c ¼280,900,
targeted/measured residual stress profile. The actual process is c ¼10,178, r ¼1163.8, r ¼55.65) to facilitate better s1tress strain
isothermal, so the thermal expansion fields are simply used to fi2ttingatlow1er(k¼1)and2 higher(k¼2)strains.Thiscomputational
induce plasticstraingradients andresidualstresses.Additionally,
model was matched to experimental stress strain data [78] for a
thistechniquehastheaddedbenefitthatitcanbeusedformore
coarse grainedIN100Ni basesuperalloymicrostructure.Theresult
complex material response (e.g., crystal plasticity) and more
ingstress strainplotcomparingthecomputationalandexperimen
complexgeometries(e.g.,notches).
tal data is illustrated in Fig. 2. The optimized isotropic/kinematic
hardening parameters for this J plasticity model are listed in the
2
2.1. Materialmodel upperlefthandcornerinsetoftheFigure.AsshownisthisFigure,
thisJ plasticitymodelmimicsthecyclicstress strainbehaviorwell
2
2.1.1. Experimentsusedtocalibratemodel foraverycomplexloadinghistory.Thus,thisJ plasticitymodelis
2
During the shot peening process, equibiaxial compressive deemed sufficient for the calibration of the thermal expansion
residual stresses are introduced near the surface due to con eigenstrainmethodtoimposeresidualstressesinsmoothspecimen
strained plastic deformation. To satisfyequilibrium, tensile stres components.Theseresultsarepresentedlater.
ses form within the subsurface of the material. These tensile
stresses may decay with increasing depth for thicker specimens
[72 74] or approach a steady state value within the bulk of the 2.1.3. Polycrystalplasticityframeworkwithquasi thermalexpansion
material for thinner specimens [25,45]. For example, Buchanan eigenstrain
et al. [25] performed residual stress relaxation studies of 2 mm This section describes how the concept of quasi thermal
thick powder metallurgy (PM) Ni base superalloy IN100 (25μm expansioneigenstrainfiniteelementmethod(coveredinthenext
averagegrainsize)specimensthatwereshotpeenedtoanAlmen section) was extended in the context of polycrystal plasticity for
intensityof 6A. The residual stressprofile given in [25] contains purposesofimposingatargetsubsurfaceresidualstressfielddue
average residual stress values measured over a 3mm(cid:3)5mm to shot peening. The benefitof crystalplasticity relative tothe J
2
irradiated X ray region at given depths from the surface. These plasticity model is that it can characterize microstructure varia
measuredaverageresidualstressvalueswerefittoacurveofthe bility; a number of different statistically representative micro
form[45] structure instantiations can be simulated to address the
σ ðxÞ¼½ðσ σ ÞþC x(cid:5)expð C xÞþσ ð2Þ probabilisticfatiguestrain lifedistribution.Thegoalofthisstudy
RS s int 1 2 int wastodevelopaframeworktoinduceafullresidualstressprofile
where the least square values of σs¼ 879.0, σint¼205.7, withinacrystalplasticityfiniteelementframeworktoaccountfor
C1¼ 67028, and C2¼20.89 were determined by Buchanan [45] this microstructure variability. Such a framework can be used to
tofittheexperimentalresidualstressdataasafunctionofdepth assess the effectiveness of shot peening in suppressing near
(x) from the surface. Eq. (2) assumes that the residual stress surfacecrackinitiationfrominclusionslocatedneartothesurface
approaches a steady state internal value of σint with increasing ofsmoothspecimens[79].
depth(x)andisapplicablefordepthvaluesfromthesurface(x¼0) Toincorporatethermalexpansioneigenstrainwithinthecrystal
tohalfdepth(x¼1mm),wheretheconditionofhalf symmetryis plasticity finite element method under ostensibly isothermal
assumed. loadingconditions,a“quasi” thermalexpansioncontributionmust
SincetheworkofBuchanan[45]containsinformationonboth beincluded within the deformation gradient. As shownin Fig. 3,
theinitialresidualstresscurveandrelaxationofresidualstresses the total deformation gradient is assumed to be multiplicatively
withfatigueloadingandisappliedtothinner(2mm)specimens,
wewilluseEq.(2)fortheinitialresidualstresscurve.Additionally,
welimitourcomputationalspecimenthicknessto2mmtoensure
1500
thatitisconsistentwithexperiments. k = 1100
s
b = 0.9
2.1.2. IsotropicJ plasticitymodel 1000 c = 280900
2 1
A rate independent J2 plasticity finite element model with r1 = 1163.8
c = 10178
combined isotropic/kinematic hardening was used to calibrate a) 500 r2 = 55.65
thequasi thermalresidualstressapplicationapproach.Theresult P 2
M
ifnorgacaclriybsrtaatledplaesigtiecnitsytrfiainnitediestlerimbuetniotnmwetahsodlat(eCrPFuEsMed)amsoadneli.nTphuet ss ( 0
e
mJ2opdlae(cid:2)slticittyh(cid:3)amtodeelmepmlopylsoyedthiesanVeoxnistinMgAisBeAsQUySie[7ld5]msautrefaricael Str −500
F¼f σ α σ ¼0,whereF¼0duringplasticflow,σ andα are
ys
thestressandbackstresstensors,respectively,andσys istheVon −1000
Mises equivalent yield strength. The function f is given by Experimental
(cid:2) (cid:3) q (cid:2) (cid:3) (cid:2) (cid:3) Computational
f σ α ¼ 32 S α : S α where S is the deviatoric stress −1500
−0.02 0 0.02 0.04 0.06 0.08
tensor. An associated flow rule is assumed as ε_¼ε_p∂F. Here, ε_p is
q ∂σ Strain
the plastic strain rate tensor and ε_p¼ 2ε_p:ε_p is the equivalent
3 Fig. 2. Comparison of J2 plasticity model in ABAQUS to experimental data of a
plasticstrainrate.Cyclichardeningoftheyieldstressisaccounted coarsegrainIN100cycledat6501C.ExperimentaldataarefromRef.[78].
4
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W.D.Musinski,D.L.McDowell/InternationalJournalofMechanicalSciences100(2015)195–208 199
decomposedvia 2.2. Finiteelementmodelimpositionofeigenstrainfield
F¼FeUFpUFθ ð3Þ To simulate the material stress state after the shot peening
process, a finite element model is used and calibrated to the
whereFe denoteselasticdistortionandrigidbodyrotationofthe
experimentalresidualstressXRDdatareportedbyBuchananetal.
crystal lattice, Fp accounts for plastic deformation through dis
location glide along crystallographic planes, and Fθ constitutes [14,25,45,83,84] on a supersolvus PM Ni base superalloy IN100
with an average grain size of 25μm that was shot peened to a
thermal expansion. It is noted that temperature change is intro
Almenintensityof6A.Aquasi thermal methodofapplication of
ducedonlytoproduceaneigenstrainfieldtoinduceinitialresidual
residualstressesischoseninthisworkbecauseitiseasytoapply
stresses from associated elastic plastic deformation. No other
andcalibrateanditiseasilyextendedtomorecomplexmodelsfor
properties need be temperature dependent. Isotropic, linearized
material response (e.g., crystal plasticity) and more complex
thermalexpansionisassumed,i.e.
geometries(e.g.,notches).
p
Fθ¼ 1þ2αΔθ I ð4Þ
where α is the isotropic thermal expansion coefficient, Δθ is the 2.2.1. ApplicationtoJ2plasticitymodel
change in temperature, and I is the second rank identity tensor. The general methodology for quasi thermal application of
TheformofEqn.(4)waschosensothatthelinearizedformofthe residual stressesto asmoothspecimen is shownin Figs. 4 and5
Greenfinite straintensorduetothermalexpansion,Eθ,is[80] andcanbesummarizedasfollows:
Eθ¼1(cid:7)(cid:5)Fθ(cid:6)TUFθ I(cid:8)¼αΔθ I ð5Þ 1. Initialconfiguration:Fig.4showsthefiniteelementmodelused
2 to simulate residual stress application to a smooth specimen.
The experimental smooth specimen [45] had a nominal gage
Therestofthemicrostructure sensitivecrystalplasticityequations
section length, width, and thickness of y¼20mm, z¼10mm
for modeling deformation behavior of a coarse grain IN100 at
6501C follow those outlined by Przybyla and McDowell [70]. A andx¼2mm,respectively(seeFig.4forx,y,andzdirections).
Similar to the finite element model employed by Buchanan
detaileddescriptionoftheCPFEMmodelisomittedinthispaper
etal.[25,45],asmallportioninthecenterofthisgagesection
to maintain brevity. We only highlight the incorporation of a
quasi thermalexpansiondeformationgradient(Fθ)inthissection was used for the finite element model. Half symmetry within
thedepth(x dimension)ofthematerialwasemployedsothat
to distinguish what is new/unique in the current CPFEM model
the overall dimensions of the finite element model for J
relative to the previous IN100 model. More details on the devel 2
plasticity simulations were xdim¼1mm, ydim¼34μm, and
opment of the IN100 model and the numerical implementation
zdim¼34μm.Foreigenstraindistributioncalibration,thefinite
technique in ABAQUS can be found in the work of Shenoy et al.
element model was divided into many 2.5μm thick finite
[78,81]andMcGinty[82],respectively.
elements,asshownintherightmostimageinFig.4.Itshould
be noted that since the compressive region of residual stress
field is very thin (x depth (cid:1)200μm) and there is a high
gradientofresidualstresswithdepth,averyfinefiniteelement
thickness of 5μm is required near the surface to provide
convergence of the FEM response. Buchanan [45] reported a
similar FEM mesh size requirement for convergence. In this
work, a finer mesh of 2.5μm was used to provide more
eigenstrain distribution data points (black dots in Fig. 5) to
improve functional form fitting; this fitting is covered later.
Eachfiniteelementwasassignedagivenquasi thermalexpan
sion coefficient α and J plasticity material properties. The
j 2
initial distribution of α values was assigned so that the
j
resulting residual stress values were within 7250MPa from
thetargetresidualstressprofileatagivendepthtoavoidany
numerical instabilities in the numerical optimization scheme
describedbelow.
2. Eigenstrain application: with all surfaces constrained from
Fig.3. Multiplicativedecompositionofthedeformationgradient,includingquasi- normal displacement, an eigenstrain (εntherm;j¼αjΔT) is intro
thermalexpansion. ducedwithinthemodelasshownintheupperlefthandcorner
Fig.4. Finiteelementmodelusedtoapplyresidualstresstoasmoothspecimen.
5
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200 W.D.Musinski,D.L.McDowell/InternationalJournalofMechanicalSciences100(2015)195–208
Fig.5. Methodologyforquasi-thermalapplicationofresidualstressestoasmoothspecimen.
of Fig. 5. It should be noted that an arbitrary value of ΔT¼1 the secant root finding method and the red and green solid
wasapplied uniformlythroughoutthewhole specimen, with linesshowthefunctionalformfittingofthesecomputationally
outlossofgenerality,sincetemperatureisnotaphysicalfield optimized thermal expansion coefficients. The top middle plot
variableduringtheimpositionofresidualstress. in Fig. 5 displays the entire distribution of thermal expansion
3. Release surface constraint: the surface constraint at x¼0 is coefficient as a function of x distance from the surface. The
removedtosimulatethespringbackofthematerialaftershot bottomrighttwoplotsinFig.5illustratezoomed inversionsof
beadimpact(cf.lowerlefthandportionofFig.4).Thisstepis thefittingof the required thermal expansion coefficienttothe
requiredtorelaxthestresscomponentnormaltothesurfaceso piecewise functional forms described below. A piecewise
that σ (cid:1)0, which is representative of the nearsurface stress smooth functional form was required so that the thermal
xx
state atthe end of the shotpeening process [20]. Thisrelaxa expansion coefficients can be defined independent of mesh
tionstepisanalogoustothe“unload”stepinthedisplacement sizeandsothatthisfunctionalformcouldbe usedas aninput
controlled method of residual stress application presented by for the crystal plasticity model. The thermal expansion func
Prasannavenkatesanetal.[20]. tion,αðxÞ,was splitinto4sections:
4. Optimization of thermal expansion coefficients: a secant root
findingmethod[85]isusedinconjunctionwiththeFEMmodel 1. xo0.058mm: this section of the curvewas fit using a super
to optimize the spatial distribution of thermal expansion positionoftwoGaussianprobabilitydensityfunctions(PDFs).A
parameterstofittheexperimentalresidualstressprofile. similar description of using two Gaussian PDFs was demon
5. Fit thermal expansion coefficients to functional form: Gaussian strated in [52] to describe the eigenstrains induced by shot
probability density functions and polynomials are used to fit peening a GW103 magnesium alloy. The functional form used
the optimized thermal expansion coefficient as a function of tofitthethermalexpansioncoefficientsatxo0.058mmis
depth from the surface. More details on this functional form " (cid:2) (cid:3) # " (cid:2) (cid:3) #
x μ 2 x μ 2
andhowitisusedwithinthecrystalplasticityframeworkare αðxÞ¼Aexp B þDexp E þGxþH ð6Þ
coveredinthefollowing. 2σ2C 2σ2F
2.2.2. Fittingthermalexpansioncoefficienttofunctionalform 2. 0.058mmrxo0.082mm: this section was fit using a 5th
The right hand side of Fig. 5 shows the required thermal
orderpolynomial:
expansion as a function of x distance (depth) from the surface
thatisusedtoreplicatethetargetresidualstressprofileshown
intheupperrighthandportionofFig.5.TheblackdotsinFig.5
showtheoptimizedthermalexpansioncoefficientfoundusing αðxÞ¼a21x5þa22x4þa23x3þa24x2þa25xþa26 ð7Þ
6
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