strengthening

ConsiderationofOrowanstrengtheningeffectin

particulate-reinforcedmetalmatrixnanocomposites:

Amodelforpredictingtheiryieldstrength

,*

DepartmentofMechanicalandIndustrialEngineering,RyersonUniversity,350VictoriaStreet,Toronto,Ont.,CanadaM5B2K3

Received28October2005;receivedinrevidform17November2005;accepted8December2005

Availableonline18January2006

Abstract

Ananalyticalmodelforpredictingtheyieldstrengthof

strengtheningeffectsinvolving(i)Orowanstrengtheningeffect,(ii)enhanceddislocationdensityduetotheresidualplasticstraincaud

bythedifferenceinthecoefficientsofthermalexpansionbetweenthematrixandparticles,and(iii)load-bearingeffecthavebeentaken

dictionisingoodagreementwiththeexperimentaldatareportedintheliterature.

Óhtsrerved.

Keywords:Metalmatrixnanocomposites;Yieldstrength;Orowanstrengthening;Load-bearingeffect;Enhanceddislocationdensitystrengthening

uction

Nanocrystallinematerialsformanexcitingareaofmate-

rialsrearchbecaubulkmaterialswithgrainsizesofless

than100nmhavepropertiesthatarenotenintheir

microcrystallinecounterparts[1,2].However,nanostruc-

turedmaterialsgenerallysufferfrominsufficientductility

andreducedtoughnesscomparedwiththeconventional

therhand,metalmatrix

nanocomposites(MMNCs)aremostpromisinginproduc-

ingbalancedmechanicalpropertiesbetweennano-and

micro-structuredmaterials,i.e.,enhancedhardness,

Young’smodulus,0.2%yieldstrength,ultimatetensile

strengthandductility[3–9],duetotheadditionofnano-

sizedreinforcementparticlesintothematrix.

TofacilitatethedevelopmentofMMNCs,itisnecessary

todevelopconstitutiverelationshipsthatcanbeudto

predictthebulkmechanicalpropertiesofMMNCsasa

functionofthereinforcement,matrix,andprocessingcon-

astfewyears,somemodelingwork[10–13]

l.[10]propoda

generalizedlawofmixturebyusingarigorouscontinuum

mechanicsanalysisandanequivalentmicrostructural

.[11]andHoltzetal.

[12]qualitativelyexplainedtheirresultsusingFanetal.’s

tal.[13]developedacontinuummechanics

modelbyconsiderationofinteractionsbetweenthenano-

r,inordertouthecon-

tinuummechanicsapproachtheauthors[10–13]triedto

modifytheinterfacebetweenthematrixandreinforcement

fficultywiththecontinuumapproachis

thatitignorestheinfluenceofparticlesonthemicrome-

chanicsofdeformationandstrengtheningmechanisms,

suchasthelocationofparticles,grainsize,anddislocation

density[14].Thatistosay,thestrengtheningmechanisms

orthetypesofMMNCs,whicharethekeyfactorsindom-

inatingthemechanicalbehavior,especiallytheyield

strength,eantime,

Ramakrishnan[15]propodananalyticalmodelforpre-

dictingtheyieldstrengthofthemicrosizedparticulate-rein-

forcedmetalmatrixcomposite(MMCs),usingacomposite

spheremodelfortheintra-granulartypeofMMCsand

1359-6462/$-efrontmatterÓhtsrerved.

doi:10.1016/amat.2005.12.017

*.:+x6487;fax:+1416979

5265.

E-mailaddress:dchen@().

ScriptaMaterialia54(2006)1321–1326

incorporatingtwoimprovementparametersassociated

withthedislocationstrengtheningofthematrixandthe

load-bearingeffdel,repre-

ntinganincorporationofbothcontinuumandmicrome-

chanicsapproaches,hasbeenudtopredictthelow-cycle

fatiguelifeofdiscontinuousreinforcedMMCs[16,17].

However,Ramakrishnan’smodelwasapplicableonlyfor

MMCscontainingmicrosizedparticles.

Theobjectiveofthisinvestigationwastomodeland

predicttheyieldstrengthoftheintra-granulartypeof

MMNCs,whichreprentsoneofthemostimportant

aspectsofthenanocompositestrengtheningmechanisms

andeffideringthestrengtheningmechanisms

ofMMNCs,andincorporatingRamakrishnan’smodel

andtheOrowanstrengtheningeffect,ananalyticalmodel

forpredictingtheyieldstrengthofparticulate-reinforced

oreticalpredictions

badonthismodelwerefoundtobeingoodagreement

withtheexperimentaldatareportedintheliterature.

evelopment

Duetotheexcellentmechanicalproperties,MMNCs

workhasbeendoneinvolvingdifferentsynthesismethods,

structures,mechanicalproperties,andstrengtheningmech-

hestrengtheningmechanismsof

MMNCsarefundamentaltothedevelopmentofthepres-

entmodel,theyarefirstsummarizedasfollows.

strengtheningmechanism

Orowanstrengthening,caudbytheresistanceof

clolyspacedhardparticlestothepassingofdislocations,

delyacknowl-

edged,however,thatOrowanstrengtheningisnotsignifi-

cantinthemicrosizedparticulate-reinforcedMMCs,

becauthereinforcementparticlesarecoarandthe

rmore,sincetherein-

forcementisoftenfoundtolieonthegrainboundariesof

thematrix,itisunclearwhethertheOrowanmechanism

canoperateatallunderthecircumstances[18].Formelt

procesdMMCswiththeusually-udparticlesof5lmor

larger,Orowanstrengtheninghasindeedbeenpointedout

tobenotamajorfactor[14].Incontrast,duetothepres-

enceofhighly-disperdnanosizedreinforcementparticles

(smallerthan$100nm)inametalmatrix,Orowan

hasbeenwellestablishedthattheprenceofadispersion

offine($100nm)insolubleparticlesinametalcanconsid-

erablyraithecreepresistance,evenforonlyasmall

volumefraction(<1%),duetothefactthatOrowanbowing

isnecessaryfordislocationstobypasstheparticles[18].For

compositescontainingfineparticles,strengtheningisoften

explainedbytheOrowanmechanism[7,19–21].Shao

etal.[7]explainedtheimprovedhardnessinthenanocom-

positeNi/Al

2

O

3

filmsbyusingtheOrowandislocation

etal.[21]obrvedOrowanloop

mechanismandudittosimulatethegoodmechanical

propertiesofCu/tedthatther-

malstressaroundthenanoparticlesarelargeenoughto

cauplasticdeformationinthematrixanddislocation

loopsaroundthevicinityofthenanoparticles[22,23].In

addition,condaryprocessing,suchaxtrusion,isud

earthatplasticdeformation

hasoccurredduringsynthesisofMMNCsandOrowan

loopsareexpectedtoexertabackstressondislocation

sources[24].Therefore,itisnecessarytotakeintoconsider-

ationtheOrowanstrengtheninginthemodelingof

MMNCs.

eddislocationdensitystrengthening

mechanism

InMMNCs,theincreadinterfacialareabetweenthe

reinforcementandmatrixcontributestotheenhanced

becauofthethermalmismatchbetweenthereinforce-

mentandthematrix,whichareinthethermalequilibrium

onlyatthetemperatureatwhichtheyarebroughtinto

contactduringtheprocess,oncoolingfromtheprocessing

temperaturethermalstressaroundthenanoparticles

largeenoughtocauplasticdeformationaregenerated

inthematrix,especiallyintheinterfaceregion[25].The

stressreducequicklywithincreasingdistancefromthe

boundary,whichcangeneratesmalldefectssuchasdislo-

cationsintheclovicinityofnanosizedparticles[23].

Theprenceofahighdislocationdensityneartheinter-

facebetweenthematrixandreinforcementparticleshas

beenexperimentallyobrved[26,27].

-bearingeffectofthereinforcement

strengtheningmechanism

Duetothenanosizeofthereinforcementparticlesand

thesoundsynthesizingmethods,thereisastrongcohesion

attheatomiclevelbetweenthematrixandparticles,i.e.,

thenanosizedparticlesaredirectlybondedtothematrix

[28–30].

Ingeneral,theyieldstrengthofacompositematerialis

thestressrequiredtooperatedislocationsourcesandis

governedbytheprenceandmagnitudeofalltheobsta-

clesthatrestrictthemotionofdislocationsinthematrix.

ForMMCs,Ramakrishnan[15]propodananalytical

modeltopredicttheyieldstrengthbyincorporatingamod-

ifiedshearlagmodel(continuummechanicsapproach)and

anenhanceddislocationdensitymodel(micromechanics

strengtheningapproach),

rm

yc

¼r

ym

ð1þf

l

Þð1þf

d

Þ;ð1Þ

whererm

yc

istheyieldstrengthoftheMMCs,r

ym

istheyield

strengthofthemonolithicmatrix,f

l

istheimprovement

factorassociatedwiththeload-bearingeffectoftherein-

forcement,f

d

istheimprovementfactorrelatedtothe

,/ScriptaMaterialia54(2006)1321–1326

dislocationdensityinthematrix,caudbythethermal

mismatchbetweenthematrixandthereinforcement

particles.

Asstatedabove,forMMNCsOrowanstrengthening

veralstrengtheningeffectsaresimultaneouslyprent,

onewaywouldbetoutherulesofadditionofthe

strengtheningcontributions,e.g.,byLilholt[31].Inthis

investigationRamakrishnan’sapproach[15]isconsidered,

sinceitwasshownthatbothadditiveandsynergisticeffects

,theyieldstrengthof

particulate-reinforcedMMNCs,r

yc

,maybeexpresdas,

r

yc

¼r

ym

ð1þf

l

Þð1þf

d

Þð1þf

Orowan

Þ;ð2Þ

wheref

Orowan

istheimprovementfactorassociatedwith

ticu-

late-reinforcedcompositesthegeneralexpressionforf

l

is

[15,16,32],

f

l

¼0:5V

p

;ð3Þ

whereV

p

isthevolumefractionofthereinforcementnano-

particles.f

d

hasbeenexpresdtobe[33],

f

d

¼kG

m

b

ffiffiffi

q

p

=r

ym

;ð4Þ

whereG

m

istheshearmodulusofthematrix,bistheBur-

gersvectorofthematrix,kisaconstant,approximately

equalto1.25,qistheenhanceddislocationdensitywhich

isassumedtobeentirelyduetotheresidualplasticstrain

developedduetothedifferenceinthecoefficientsof

thermalexpansion(DCTE)betweenthereinforcement

phaandthematrixduringthepost-fabricationcooling.

Forequiaxedparticulatesthefollowingexpressionwas

reported[34],

q¼12

DaDTV

p

bd

p

ð1ÀV

p

Þ

;ð5Þ

whered

p

istheparticlesize,Daisthedifferenceinthe

coefficientsofthethermalexpansion,DTisthedifference

betweentheprocessingandtesttemperatures.

Theimprovementfactorf

Orowan

relatedtotheOrowan

strengtheningofnanoparticlesintroducedinEq.(2)can

beexpresdas,

f

Orowan

¼Dr

Orowan

=r

ym

;ð6Þ

whereDr

Orowan

hasbeendescribedbytheOrowan–Ashby

equation[24],

Dr

Orowan

¼

0:13G

m

b

k

ln

r

b

;ð7Þ

whereristheparticleradius,r=d

p

/2,andkistheinterpar-

ticlespacing,expresdas[16,35],

k%d

p

1

2V

p