Table Of ContentProgressinMaterialsSciencexxx(2011)xxx–xxx
ContentslistsavailableatScienceDirect
Progress in Materials Science
journal homepage: www.elsevier.com/locate/pmatsci
The elastic properties, elastic models and elastic
perspectives of metallic glasses
⇑
Wei Hua Wang
InstituteofPhysics,ChineseAcademyofSciences,Beijing100190,PRChina
a r t i c l e i n f o a b s t r a c t
Articlehistory: Bulkmetallicglass(BMG)providesplentifulpreciseknowledgeof
Received7September2010 fundamental parameters of elastic moduli, which offer a bench-
Accepted21July2011 mark reference point for understanding and applications of the
Availableonlinexxxx
glassymaterials.Thispapercomprehensivelyreviewsthecurrent
stateoftheartofthestudyofelasticproperties,theestablishments
ofcorrelationsbetweenelasticmoduliandproperties/features,and
theelasticmodelsandelasticperspectivesofmetallicglasses.The
goalistoshowthekeyrolesofelasticmoduliinstudy,formation,
and understanding of metallic glasses, and to present a compre-
hensiveelasticperspectivesonthemajorfundamentalissuesfrom
processingtostructuretopropertiesintherapidlymovingfield.
A plentiful of data and results involving in acoustic velocities,
elastic constants and their response to aging, relaxation, applied
press,pressureandtemperatureofthemetallicglasseshavebeen
compiled. The thermodynamic and kinetic parameters, stability,
mechanical and physical properties of various available metallic
glassesespeciallyBMGshavealsobeencollected.Asurveybased
on the plentiful experimental data reveals that the linear elastic
constants have striking systematic correlations with the micro-
structuralfeatures,glasstransitiontemperature,meltingtempera-
ture, relaxation behavior, boson peak, strength, hardness, plastic
yieldingoftheglass,andevenrheologicalpropertiesoftheglass
formingliquids.TheelasticconstantsofBMGsalsoshowacorrela-
tionwithaweightedaverageoftheelasticconstantsoftheconstit-
uentelements.Weshowthattheelasticmodulicorrelationscan
assistinselectingalloyingcomponentswithsuitableelasticmoduli
for controlling the elastic properties and glass-forming ability of
⇑
Address:InstituteofPhysics,ChineseAcademyofSciences,P.O.Box603(42-1),Beijing100190,PRChina.Tel.:+8610
82649198;fax:+861082640223.
E-mailaddress:[email protected]
URL:http://www.mmp.iphy.ac.cn
0079-6425/$-seefrontmatter(cid:2)2011ElsevierLtd.Allrightsreserved.
doi:10.1016/j.pmatsci.2011.07.001
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2 W.H.Wang/ProgressinMaterialsSciencexxx(2011)xxx–xxx
themetallicglasses,andthustheresultswouldenablethedesign,
control and tuning of the formation and properties of metallic
glasses.
Wedemonstratethattheglasstransition,theprimaryandsec-
ondaryrelaxations,plasticdeformationandyieldcanbeattributed
tothefreevolumeincreaseinducedflow,andtheflowcanbemod-
eledastheactivatedhoppingbetweentheinherentstatesinthe
potentialenergylandscape.Wethenproposeanextendedelastic
model to understand flow in metallic glass and glass-forming
supercooledliquid,andthemodelpresentsasimpleandquantita-
tivemathematicexpressionforflowactivationenergyofvarious
glasses.Theelasticperspectives,whichconsiderallmetallicglasses
exhibit universal behavior based on a small number of readily
measurableparametersofelasticmoduli,arepresentedforunder-
standingthenatureanddiversepropertiesofthemetallicglasses.
(cid:2)2011ElsevierLtd.Allrightsreserved.
Contents
1. Introduction......................................................................... 00
2. Thebriefdescriptionofthetheoryofelasticity............................................. 00
2.1. Thecharacterizationofelasticpropertiesofsolids..................................... 00
2.2. Physicaloriginfortheelasticity.................................................... 00
2.2.1. Basisforlinearelasticity .................................................. 00
2.2.2. Elasticpropertiesandenergylandscapes..................................... 00
2.3. Elasticpropertiesinequilibriumliquidsandsupercooledliquids......................... 00
2.3.1. Instantaneouselasticconstants............................................. 00
2.3.2. Elasticpropertiesintheequilibriumliquidsandsupercooledliquids .............. 00
2.4. Correlationsbetweenelasticmoduliandphysicalpropertiesincrystallinematerials......... 00
3. Experimentalmethodsfordeterminingelasticmoduliofglasses .............................. 00
3.1. Ultrasonicmethods.............................................................. 00
3.1.1. Thetheoryforultrasonicmeasurements ..................................... 00
3.1.2. Theultrasonicexperimentalmethods........................................ 00
3.1.3. Theinsitumeasurementofultrasonicwavevelocitiesunderhighpressure......... 00
3.1.4. Themeasurementofultrasonicwavevelocitiesupontemperature................ 00
3.2. Theinformationprovidedfromultrasonicstudy ...................................... 00
4. Elasticmoduliofmetallicglasses........................................................ 00
5. Pressuredependenceofelasticpropertiesofmetallicglasses ................................. 00
6. Temperaturedependenceofelasticpropertiesofmetallicglasses.............................. 00
6.1. TheelasticpropertyofBMGsuponannealingtemperatures............................. 00
6.2. InsitumeasurementoftemperaturedependenceofelasticmoduliofBMGs................ 00
6.2.1. Insitumeasurementoftemperaturedependenceofelasticmoduliduringglass
transition................................................................................. 00
6.2.2. Insitumeasurementoftemperaturedependenceofelasticmoduliatlowtemperatures 00
7. Timedependence(structuralrelaxation)ofelasticconstantsofmetallicglasses.................. 00
8. Correlationsbetweenelasticmoduliandmicrostructure,glassformation,glasstransition,fragility
andmelting........................................................................... 00
8.1. Thecorrelationsbetweenelasticpropertiesandmicrostructureofmetallicglasses.......... 00
8.2. Correlationbetweenelasticconstantsofmetallicglassesandelasticconstantsoftheir
constituents....................................................................... 00
8.3. Correlationbetweenelasticpropertiesandglassformation ............................. 00
8.4. Correlationbetweenelasticmoduliandglasstransition................................ 00
8.5. Correlationbetweenelasticmoduliandmelting ...................................... 00
8.6. Correlationbetweenelasticmoduliandfragility ...................................... 00
8.7. Correlationbetweenelasticpropertiesandbosonpeak................................. 00
8.8. Othercorrelations............................................................... 00
9. Correlationsbetweenelasticmoduliandmechanicalproperties............................... 00
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W.H.Wang/ProgressinMaterialsSciencexxx(2011)xxx–xxx 3
9.1. Correlationbetweenelasticmoduliandstrength...................................... 00
9.2. Correlationbetweenelasticmoduliandmicrohardness................................. 00
9.3. Correlationbetweenelasticmoduliandtoughness/plasticity ............................ 00
9.4. Correlationbetweenelasticmoduliandfracture ...................................... 00
10. Thesummaryforelasticcorrelations.................................................... 00
11. Searchingmetallicglassesbasedonelasticmodulicriterion................................. 00
12. Elasticmodelsofsupercooledliquidsandmetallicglasses................................... 00
12.1. Elasticmodelsforglass-formingsupercooledliquids.................................. 00
12.2. Elasticmodelsformetallicglasses................................................. 00
12.3. Extendedelasticmodelforflowinmetallicglass-formingliquidsandglasses ............. 00
13. Theelasticperspectivesonmetallicglasses............................................... 00
14. Summaryandoutlook................................................................ 00
Acknowledgments.................................................................... 00
References .......................................................................... 00
1.Introduction
Glassystate,whichisauniversalpropertyofsupercooledliquidsiftheyarecooledrapidlyenough,
isregardedasthefourthstateofmatter[1–5].Themysteriousglasstransitionphenomenon,which
connectstheliquidandglassystates,isrelatedwidelytodailylife,industry,materialspreparation,
organismpreservationandalotofnaturalphenomena.However,theexactandcomprehensivephys-
icalunderstandingoftheglasstransitionandglassnaturesisconsideredtobeoneofthemostchal-
lengingproblemsincondensedmatterphysicsandmaterialscience.Duetotherandomdisordered
structure,thecharacterizationoftheglassesareverydifficult,andthisleadstoproblemsforunder-
standing the formation, deformation, fracture, nature, and the structure–properties relationship of
theglasses[1–18].
Metallicglass,whichisanewcomeringlassyfamily(discoveredin1959)andatthecuttingedgeof
currentmetallicmaterialsresearch,isofcurrentinterestandsignificanceincondensedmatterphys-
ics, materials science and engineering because of its unique structural features and outstanding
mechanical, many novel, applicable physical and chemical properties [11,14]. Metallic glasses have
also been the focus of research advancing our fundamental understanding of liquids and glasses
andprovidemodelsystemsforstudyingsomelong-standingfundamentalissuesandhavepotential
engineering and functional applications. The recently developed bulk metallic glass (BMG) forming
systems are complicated multicomponent alloys that vitrify with remarkable ease during conven-
tionalsolidification.TheBMGsrepresentanovelandexcitinggroupofmetallicmaterialswithmany
favorableandapplicablepropertiesascomparedtotheircrystallinecounterparts.Theiruniquecom-
binations of mechanical, physical, tribological and chemical properties are of current interest and
importance[1–15].Itisknownthatthestructuredeterminesthepropertiesofacrystallinematerial,
whileinthemetallicglasses,duetotherandomatomicstructureinlong-range[16–18],thedifficult
forcharacterizingofthestructuraldisorderedglassesleadstoproblemforunderstandingtheforma-
tionandthestructure–propertiesrelationshipofthemetallicglasses[1–18].Therefore,thebirthofthe
novelmetallicglassesalsopresentsalotofunresolvedissues,outstandingquestionsandchallenges.
Thefollowingiskeyquestionsthatrepresentfundamentalissuescriticalformetallicglasses.These
issuesinclude:
(1) Whereandwhydoesmetallicliquidendandglassbegin?Howcouldliquidsuddenlystopmov-
ingwhileapproachingglasstransitiontemperatureT ?Howdoesstructureconnecttodynam-
g
icsastheygive risetodynamicalheterogeneity,fragility,agingand rejuvenation?Sofar, the
understanding of these common phenomena in glass is still fragmentary. The fundamental
studies on the rheology, dynamic, and nature of glasses is one of major drivers for research
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Nomenclature
BMG bulkmetallicglass
MG metallicglass
GFA glass-formingability
GFR glass-formingcompositionrange
EOS equationofstate
RT roomtemperature
STZ sheartransformationzone
BP bosonpeak
q densityofmatter
g(r) pairdistributionfunction
M masspermolar
C atomicpackingdensity
g
r stress
r fracturestrength
F
r yieldstrength
y
e strain
s shearstress
s yieldshearstrength
y
c shearstrain
K fracturetoughness
c
G fractureenergy
c
Hv Vicker’shardness
r positionvectorbeforethedeformation
u displacementvector
V volume
V averagemolarvolume
m
V atomicvolume
a
P pressure
T temperature
t time
g viscosity
s relaxationtime
R
x frequency
K bulkmodulus(ormodulusofcompression)
v compressibility
K instantaneousbulkmodulus
1
E Young’smodulus
Q(cid:2)1 internalfriction
G shearmodulus(modulusofrigidity)
G instantaneousshearmodulus
1
G0 storagemodulus
G00 lossmodulus
m Poisson’sratio
c Grueneisenconstant
G
v longitudinalacousticvelocity
l
v transverseacousticvelocity
s
DE activationenergy
q theflowactivationenergydensity
E
a thermalexpansioncoefficient
T glasstransitiontemperature
g
T crystallizationtemperature
x
T meltingtemperature
m
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T liqudustemperature
l
T fictivetemperature
f
T Kauzmanntemperature
K
T kineticidealglasstransitiontemperatureorVFTtemperature
0
T crossovertemperature
c
T reducedglasstransitiontemperature(=T/T)
rg g l
DT supercooledliquidtemperatureregion,DT=T (cid:2)T
x g
C orC specificheat
p V
h Debyetemperature
D
h effectiveEinsteintemperature
E
m fragility
k Boltzmannconstant
B
N theAvogadroconstant
A
h Planckconstant
DSC differentialscanningcalorimeter
DTA differentialthermalanalysis
XRD X-raydiffraction
TEM transmissionelectronmicroscope
SEM scanningelectronmicroscope
DMA thermalmechanicalanalyzer
onthemetallicglasses,whichareregardedtohavethesimplestatomicstructureinglassfam-
ily. The study will further the understanding of above subjects on glass dating from ancient
times[9].
(2) Whatphysicalfactorsdeterminetheglass-formingability(GFA)inanalloy?Whataretheuni-
versalaspectsofglassformation,andwhataccountsfortheseuniversalaspects?Whatfactors
suppressoraffectcrystalnucleationinmulti-componentalloysystems?Thestructuraltrans-
formuponchangesinthermodynamicandprocessingvariables,andcorrelationsoflocalatomic
structures,electronicstructures,and atomicmotionswithGFAstill remainunclear.How can
one predicts the composition with excellent GFA and search new metallic glass system and
compositioninsmartway?Manyempiricalcriteriacorrelatingwiththermodynamicparame-
terswereproposedtopredicttheGFAofthemetallicglass-formingalloys,whilethesecriteria
cannotpredicttheuniversalGFA,andaredifficulttobeappliedtosearchingnewmetallicglass
inpractice.Despitetheidentificationofavastrangeofmetallicglassesandapplications,the
developmentofmethodsandcriterionforrationallydesigningmetallicglasseswithexcellent
GFAreachingorapproachingthatofoxideglassesanddesirablephysicalandmechanicalprop-
ertiesisstillachallenge[8–10,17–20].
(3) Themetastablenatureofthemetallicglassesimposesabarriertobroadcommercialandindus-
tryapplications,andoneofthemainchallengesistoavoidcrystallizationandagingduringpro-
cessing,annealingandapplications.Hence,theunderstandingofthecrystallizationandaging
processesisnecessarytodevelopfabricationprocessesandtoprocessandapplybulkmetallic
glasses.Whatisthenatureofthecrystallizationandageingmechanisms,andhowdotheeffects
ofheterogeneitiesonthecrystallizationkinetics,andthethermodynamicanddynamicaspects
ofaging?Whatfactorssuppressoraffectcrystalnucleationandgrowthinglassalloysystems?
Alloftheissueshavetobedeeplystudied.Theresearchonthecrystallizationofmetallicglasses
canuncoverthenucleationandgrowthinsupercooledliquidandfindnovelprocessingoppor-
tunities for the metallic glasses. The aging and crystallization studies are important both for
applicationsandunderstandingofmetallicglasses.
(4) Howtocharacterizeandmodelthemetallicglassystructure,andhowdotheatomsandclusters
packinmetallicglasseswithliquid-likemetastablestructure?Whatarethestructuralandelec-
tronicfeaturesandcharacteristicsofmetallicglasses?Andhowdoesthestructuredetermine
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the properties of glasses? How to characterize and model the structural heterogeneity or
‘‘defects’’?Andwhatistherelationbetweenthestaticheterogeneityanddynamicheterogene-
ity?Whatismicrostructureoriginofstructuralrelaxationandaging,physicalandmechanical
propertiesofmetallicglasses?Allofthemarekeytopicinthisfield[16].Manymetallicliquids
can be deeply supercooled without crystallizing and subsequently to form metallic glasses,
whilefundamentalunderstandingsoftheatomicarrangementsinsupercooledmetallicliquids
arestilllacking.Tounderstandthestructureofthesupercooledliquid,oneneedstoidentifydif-
ferenttypesoftopologicalstructuralordering,andmonitorhierarchicalstructuralorderingof
supercooled liquids temporally and spatially, and uncover the correlations between atomic
packingschemesandthedynamicslowing-down,liquidfragility,glassformingability,etc.of
the supercooled liquids. To reveal the details of atomic-level structural evolution and local
atomic rearrangement in metallic liquids would advance the understanding of the structural
underpinningsofthevitrificationprocessofsupercooledmetallicliquids.
(5) Compared with conventional crystalline metals and alloys, non-crystalline metallic glasses
show higher strength and yet can still sustain plastic flow (permanent strains and shape
changes).Theplasticityofcrystallinemetalsiswellknowntobecarriedbystructuraldefects
such as dislocations or grain boundaries. How does metallic glass respond to the application
ofmechanicalstressesorotherexternalappliedenergybothatandfarbelowtheglasstransi-
tion?Incontrast,thecorrespondingplasticflowmechanisminmetallicglassesremainslargely
unresolved. Under imposed stresses, there should be preferential ‘‘flow defects’’ or ‘‘plastic
units’’generatedwhichcarriestheplasticstrain,andtheremustbestructuraloriginsresponsi-
bleforthelocalizedbasicflowevents.Thelocationsoffertileandliquid-likesites,theinitiation
and percolation for the cooperative flow events under stresses or other external activations
havenotbeenidentifiedbasedonspecificlocalstructuralanddynamicalproperties,including
thedegreeoflocalorderandsymmetry,atomic-sitestressesandfreevolumecontent.Howdo
weidentifythe‘‘flowdefects’’?Theatomicflowmechanisms(thetriggeringeventsandcoop-
erativeatomicshear/shuffling),thelocalizedflowsize(numberofatomsinvolved)andthefer-
tilityorpropensityoflocalatomsforflow,theevolutionoftheshort-to-mediumrangeorder
duringtheflowandintheflowstate,andthepercolatingbehavioroflocalizedfloweventslead-
ing to later shear banding still remain unclear. The flow barrier and its dependence on local
structure and properties is a key issue for deformation in metallic glasses. The deformation
eventsinmetallicglassesinvolvesinlargerangeoflengthscalefromatomicscaleoffreevol-
umes, to nano-scale of plastic units (or shear transformation zone) and medium-range order
change,tomicrometerscaleshearbands.Howtocharacterizethechangesoflocalstructures
indifferentlengthscalesuponplasticdeformationbothatandfarbelowtheglasstransition,
theconnectionandcorrelationamongthesefloweventsindifferentlengthscales,thenature,
operationandpropagationoftheshearbands,andhowtoovercometheshearbandproblem
and makethemetallicglassesductileand tough,and theintrinsicmechanisms oftheplastic
deformationandrecordhighstrengthinmetallicglassesarestillchallenges[1,2,21–24].
Withthedevelopmentofmoreandmorebulkmetallicglasssystems,theacousticmeasurements
arewidelyappliedtostudytheiracousticandelasticpropertiesanddeterminetheelasticmodulias
wellastheircomposition,processing,coolingrate,temperature,pressure,agingtime,creepandexter-
nalstressdependencesofthemetallicglasses.Plentifuldataontheelasticpropertiesofthemetallic
glasseswereaccumulatedandalotofinterestingexperimentalphenomenawereobserved.Combin-
ingwiththeplentifulelasticmodulidataofnon-metallicglasses,remarkablelinksamongtheelastic
moduli and thermodynamic, kinetic, and property parameters are found. These elastic correlations
should have common structural and physical origins, and the key challenge now is to understand
these correlations and exploit such understanding to develop new glass systems and compositions
that combine excellent glass-forming ability with desirable properties. The correlations with other
properties and features of metallic glasses, and the elastic models provide insight into above long-
standingissuesoftheglassesandarealsousefulfornewmetallicglassesdesignandsearching.Some
elasticmodelshavebeenproposedbasedontheelasticdataandcorrelationstounderstandthenature
oftheglasses.Moreandmoreevidencesshowthattheelasticmoduli,whichcanbeeasilymeasured
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byexperimentsduetothemacroscopicallyisotropicfeatureofthemetallicglasses,arekeyparame-
tersfordeveloping,characterizing,tougheningandunderstandingthemetallicglasses,andtheyare
thekeyphysicalquantityforcontrollingthemainthermodynamicandkinetic,intrinsicstaticstruc-
turalheterogeneity,mechanicalanddynamicproperties.Elasticmodelandelasticperspectivespro-
videdeepinsightintoabovelistedissuesinthemetallicglasses.
Thepurposesofthisreviewpaperaretocomprehensivelyreviewandsummarizetherecentelastic
propertiesstudiesofthemetallicglassesandtheestablishedelasticmodulicorrelationsandelastic
models.Basedontheelasticmodulidata,thefoundcorrelations,theelasticmodelsonsupercooled
liquidandglassesandotherexperimentalobservations,wetrytounderstand,characterizeanddis-
cusstheissuesontheformation,glasstransition,crystallization,stability,relaxation,structuraland
dynamic features, physical and mechanical properties and other found phenomena of the metallic
glasses by using the elastic moduli as the key parameter. We propose an extended elastic model
andelasticperspectivesonaboveissuesinmetallicglassesandarguethatthevariousmetallicglasses
exhibituniversalbehaviorbasedonasmallnumberofmeasurableparametersofelasticmoduli.We
showthatagenuineunderstandingoftheseissuesbasedonelasticproperties,elasticmodelcanassist
inselectingalloyingcomponentsforcontrollingtheelasticpropertiesandGFAofthemetallicglasses,
andthuscanguidethenewmetallicglassesdesignandprocess,andshouldalsomakeitpossibleto
optimizepropertiesofmetallicglasses.
Thepaperconsistsof14parts.Thefirstpartisabriefintroductiononthemainissuesinthisfield
andonthepurposeandthestructureofthepaper.Thesecondpartbrieflyintroducesthegeneralthe-
oryofelasticityforsolid,thephysicaloriginfortheelasticityanditsrelationshipwithenergyland-
scape, and describes how to characterize the elasticity of a solid especially the homogeneous and
isotropicsolid.Thedescriptionofhowtodeterminetheelasticmoduliinanisotropicsolidisalsopre-
sented.Todescribetheelasticityofsupercooledliquidorliquid,theconceptofinstantaneouselastic
moduliisintroduced.Theinstantaneouselasticmoduliaremeasuredonshorttimescalesorahigh-
frequencydisturbancewheretheliquiddoesnothaveenoughtimetoflow.Then,webrieflyintroduce
thetheoryofelasticpropertiesintheequilibriumliquidsandsupercooledliquids.Thesetheoriesare
basis for characterizing and understanding the elastic properties and elastic models of the metallic
glass-formingliquidsandglasses.
Part3describestheexperimentalmeasurementsoftheelasticmoduliinmetallicglassesandother
non-metallic glasses with isotropic structure. In particularly, the ultrasonic methods for the elastic
moduli measurement in ambient conditions, under high pressure, under high or low temperature
are systematically introduced. And the significanceinformationon the metallicglasses one can get
fromtheultrasonicstudyisalsosummarized.
Inpart4,theelasticmodulidatacompilationforavailablemetallicglassessofarispresented.The
elasticmoduliofvariousmetallicglassesarecarefullyanalyzed,classifiedandcharacterized,andare
comparedwiththatoftheircrystallinecounterpartsandothertypicalnon-metallicglasses.Thesedata
providethebasisforestablishingtheelasticcorrelationandelasticmodel.
In parts 5–7, the temperature, pressure and relaxation (or ageing) time dependences of elastic
modulioftypicalbulkmetallicglassesarepresentedandarecomparedwiththatoftheircrystalline
counterparts and other typical non-metallic glasses. These elastic moduli data demonstrate unique
featuresofthemetallicglassesunderpressureandloworhightemperatures,andshowtheinfluence
ofageingeffectonthephysicalandmechanicalpropertiesofmetallicglasses.Thestudiesalsoprovide
theinformationaboutequationofstate,atomicconfigurations,excitationsinthemetallicglassysolids
includingtheanharmonicityoftheatomicvibrations,thedensityofelectronstatesattheFermilevel
andtheshapeoftheFermisurface,andtheactivationbarriersforthermallyactivatedrelaxations.
Aftergivinganoverviewofthebasicexperimentalfactsoftheelasticpropertystudy,thepaperin
parts8–9attemptstocorrelatetheelasticmoduliofthemetallicglassestotheirmicrostructure,for-
mation,glass-formingability,stability,crystallizationbehavior,glasstransition,thermodynamicand
kineticfeatures,mechanicalandphysicalproperties,andrehologyoftheglass-formingliquids.Itis
showninthesepartsthatthelinearelasticmodulihavecorrelationswiththemicrostructure,glass
transition temperature, relaxation, GFA, melting temperature, strength, hardness, plasticity and
toughness, and even the fragility of the glass forming liquids. Meanwhile, the elastic constants of
available BMGs show a rough correlation with a weighted average of the elastic constants for the
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constituentelements.Inpart10,wesummarizevariouscorrelationsinmetallicglassesamongelastic
moduliandfeaturesandproperties.Thephysicaloriginsforthesecorrelationsarediscussed.
Basedonthesefoundelasticcorrelations,inpart11,theplausibleelasticmodulicriterionispro-
posedforsearchingmetallicglasses.Itisdemonstratedthattheelasticcorrelationsaswellastheelas-
ticcriterioncanassistinBMGdesignandexploration.
Inpart12,theprevailingelasticmodelsforsupercooledliquidsandglassesareintroduced,andan
extendedelasticmodelforflowinmetallicglassesisproposed.Itisshownthattheextendedelastic
modelisinremarkableagreementwithavarietyofexperimentalobservationsandoffersasimplesce-
narioforexplainingsomecorrelationsbetweenelasticconstantsandpropertiesandforunderstanding
thenatureofthemetallicglasses.
Inpart13,attemptismadetounderstandsomepuzzlesandfundamentalissuesinmetallicglasses
basedontheirelasticproperties,theelasticcorrelationsandtheelasticmodels.Therolesofelastic
moduli as the key parameters for developing, characterizing, toughening and understanding the
metallicglassesandtheirformingliquidsareemphasized.Theelasticperspectivesontheformation,
GFA,thethermodynamicandkineticaspects,intrinsicstaticstructuralanddynamicheterogeneities,
mechanicalandphysicalpropertiesofthemetallicglassesaresuggested.Thereviewpaperendswith
summaryandoutlookinpart14.
2.Thebriefdescriptionofthetheoryofelasticity
2.1.Thecharacterizationofelasticpropertiesofsolids
Themechanicsofsolids,regardedascontinuousmedia,formsthecontentofthetheoryofelastic-
ity.Themacroscopicbehaviorofasolidisdescribedbyacontinuumfieldtheory,thetheoryofelas-
ticity, which describes the way a solid deforms when external stresses are applied [25]. Under the
actionofappliedstress,solidbodyexhibitsshapeandvolumechangestosomeextent,andeverypoint
inthesolidbodyisingeneraldisplaced.Letthepositionvectorbeforethedeformationber,andafter
thedeformationhasavaluer0withcomponentx.Thedisplacementofthispointduetodeformation
i
thengivenbythedisplacementvectoru=r(cid:2)r0 oru ¼x0(cid:2)x.Ifu (x ,x ,x )isthejthcomponentof
i I i ij 1 2 3
thedisplacementatpoint(x ,x ,x ),thestraintensorforsmalldeformationsis
1 2 3
1(cid:2)@u @u(cid:3)
u ¼ iþ j ð2:1Þ
ij 2 @x @x
j i
Whena deformation happens, thebody ceasesto bein its original state ofequilibrium,and the
forces,whicharecalledinternalstresses,thereforearisewhichtendtoreturnthebodytoitsequilib-
riumstate.Ifthedeformationofthebodyisfairlysmall,itreturnstoitsoriginalundeformedstate
whentheexternalforcesceasetoact.Suchdeformationsareelastic.Forlargedeformations,there-
movaloftheexternalforcesdoesnotresultinthefullyrecoveryofthedeformation.Suchdeforma-
tionsareplastic[25].Foranisotropicsolid,inelasticdeformationcase,thestresstensorintermsof
thestraintensorcanbeexpressedas[25]:
r ¼Ku d þ2Gðu (cid:2)d u =3Þ ð2:2Þ
ij ll ij ij ij ll
where K (in many text books, a different notation B is used) is bulk modulus, and G (in many text
books,adifferentnotationlisused)isshearmodulus.Thebulkmodulusdescribesthestrainresponse
ofabodytohydrostaticstressinvolvingchangeinvolumewithoutchangeofshape.Theshearmod-
ulusrelatestostrainresponseofabodytoshearortorsionalstress.Itinvolveschangeofshapewith-
outchangeofvolume.
Foranisotropicsolid,itselasticbehaviorisfullydescribedbythelongitudinalmodulus(L=c )
11
andshearmodulus(G=c ).Betweenthecomponentsoftherelatedelasticstiffnesstensor,theiso-
44
tropicrelation[26]:c =c +2c .Ifinadditiontheatomsinteractthroughacentralpotential,the
11 12 44
Cauchyidentityc =c mayhold,andthen,oneobtainsareductiontoonlyoneindependentelastic
12 44
constantc =3c (orL=3G).TheCauchy’sidentity,validforanisotropicsolidcomposedofmolecular
11 44
interactingwithtwo-bodycentralforce,canalsobestatedas:
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W.H.Wang/ProgressinMaterialsSciencexxx(2011)xxx–xxx 9
5
K¼ G ð2:3Þ
3
whichimpliesthatLandKareequalto0whenG=0.FortheLennardJonesinteractionpotential,this
identitychangesto,L=a+bG,whereaandbareconstants.Thisleadstotherelation,
K¼aþðb(cid:2)1:333ÞG ð2:4Þ
which is known as the generalized Cauchy relation. Its parameters a and b remain constant with
changingtemperatureandpressure,butaresensitivetosmallchangesinthepotential.Thisrelation
isvalidforbothliquidsandglasses[25].Recently,ageneralizedCauchyrelationwasobservedtohold
insupercooledliquidsandcanbeexpressedas:c (T)=Bc (T)+A,whereBandAareconstantsinde-
11 44
pendentofTandPwithintheirrangeinvolvingtheglasstransition[26].
Forminordeformation,theu isalinearfunctionofther .Thatis,thedeformationisproportional
ij ij
totheappliedforces.ThislawiscalledHooke’slaw,whichisactuallyapplicabletoalmostallelastic
deformationsinsolids[25].Insimplecaseofhomogenousdeformationsinwhichthestraintensoris
constantthroughoutthevolumeofthebody,thestresstensorisgivenintermsofthetraintensorby
E (cid:4) m (cid:5)
r ¼ u þ u d ð2:5Þ
ij 1þm ij 1(cid:2)2m ll ij
whereEisthemodulusofextensionorYoung’s modulus.Young’smodulusisnamedafter Thomas
Young(the19thcenturyBritishscientist).However,theconceptwasdevelopedin1727byLeonhard
Euler,andthefirstexperimentthatusedtheconceptofYoung’smoduluswasperformedbytheItalian
scientistGiordanoRiccatiin1782[25].Young’smodulus,alsoknownasthetensilemodulus,isamea-
sureofthestiffnessofanisotropicelasticmaterial.Itisdefinedastheratiooftheuniaxialstressover
theuniaxialstrainintheelasticregime.Thiscanbeexperimentallydeterminedfromtheslopeofa
stress–straincurvecreatedduringtensileorcompressiontestsconductedonamaterial.Young’smod-
ulusis alsocommonly,but incorrectly,calledtheelasticmodulusor modulusofelasticity,because
Young’smodulusEisthebest-knownelasticconstant,whichismostcommonlyusedinengineering
design.Thereareotherelasticmoduli,suchasthebulkmodulusandtheshearmodulus.TheEcanbe
givenintermsofGandKby:
E¼9KG=ð3KþGÞ ð2:6Þ
The ratioof thetransverse compressiontothe longitudinal extensionis calledPoisson’s ratio,m,
namedafterSiméonPoisson.Poisson’sratioisanimportantmaterialpropertyusedinelasticanalysis
ofmaterial.Whenamaterialiscompressedinonedirection,itusuallytendstoexpandintheother
twodirectionsperpendiculartothedirectionofcompression.ThisphenomenoniscalledthePoisson
effect.Poisson’sratiomisameasureofthePoissoneffect.Themistheratioofthefractionofexpansion
dividedbythefractionofcompression(forsmallvaluesofthesechanges).Instretchcaseofasolid
rather than the compression, the Poisson ratio will be the ratio of relative contraction to relative
stretching,andwillhavethesamevalueascompressioncase.TherelationofPoisson’sratiowithother
elasticconstantsis(forisotropicsolids):
1
m¼ ð3K(cid:2)2GÞ=ð3KþGÞ ð2:7Þ
2
Since K and G are always positive, the Poisson’s ratio can vary between (cid:2)1 (for K=0) and 0.5 (for
G=0),thatis(cid:2)16m60.5.OnemightseePoisson’sratioslargerthan0.5reportedintheliterature,
however, this implies that the material was stressed to cracking. Generally, ‘‘stiffer’’ materials will
havelowerPoisson’sratiosthan‘‘softer’’materials.Whileonlyinveryrarecases,amaterialwillactu-
allyshrinkinthetransversedirectionwhencompressed(orexpandwhenstretched)whichwillyielda
negativevalueofthePoisson’sratio.
TheGandKcanalsobegivenintermsofEandmby:
G¼E=2ð1þmÞ ð2:8Þ
K¼E=3ð1(cid:2)2mÞ ð2:9Þ
Pleasecitethisarticleinpressas:WangWH.Theelasticproperties,elasticmodelsandelasticperspectivesof
metallicglasses.ProgMaterSci(2011),doi:10.1016/j.pmatsci.2011.07.001
10 W.H.Wang/ProgressinMaterialsSciencexxx(2011)xxx–xxx
Thequantity1/Kiscalledcoefficientofcompressionv:
v¼1¼(cid:2)1(cid:2)@V(cid:3) ¼(cid:2)@ðlnVÞ(cid:6)(cid:6)(cid:6) ð2:10Þ
K V @P @P (cid:6)
T T
Theelasticpropertiesofasolidbodyaretemperaturedependent.Deformationscanoccuraccom-
paniedbyachangeintemperatureofthebodyeitherasaresultofthedeformationprocessitselfor
fromexternalcauses.Among thevarioustypesof deformations, isothermaland adiabaticdeforma-
tionsareofimportance.Inisothermaldeformation,thetemperatureTofthebodydoesnotchange,
theK,E,G,andmthereforecanbecalledisothermalmoduli.Ifthereisnoexchangeofheatbetween
thevariouspartsofthebody,thedeformationiscalledadiabatic,andtheadiabaticmoduliarelabeled
tobeE ,K ,G ,andm .Therelationsbetweenadiabaticandisothermalmoduliare[25]:
ad ad ad ad
1=K ¼1=K(cid:2)Ta2=C ð2:11Þ
ad p
G ¼G ð2:12Þ
ad
E
E ¼ ð2:13Þ
ad 1(cid:2)ETa2=9C
p
mþETa2=9C
m ¼ p ð2:14Þ
ad 1(cid:2)ETa2=9C
p
whereaisthethermalexpansioncoefficientofthesolid.ThisistheDebye–Grüneiseneffectofther-
malexpansiononelasticconstantsforanisolatedconfigurationalstate,inwhichTdependentelastic
behaviorarisesfromtheanharmonicityofthevibrationalpartsofthemotionofatoms[27–29].For
systemswithpositivethermalexpansiontheadiabaticKishigherthantheisothermalK.Theadiabatic
andisothermalshearmoduliarealwaysidentical.Thetimedependentelasticityistermedasviscoelas-
ticity[25,30–32].
Inisotropicsolids,basedontheHooke’slaw:
(1) Underthetensileor compressive state,E=r/e, wheree isstrain.TheEreflectsthetensileor
compressionstrainresistanceofthematerials.
(2) Underthetensileorcompressivestate,G=s/c,wheresisshearstress,andcisshearstrain.The
Greflectsthetensileorcompressionshearstrainresistanceofthematerials.
(3) TheKreflectstheresistancetodilatationcausedbythehydrostaticstressstate:
K¼(cid:2)P=ð(cid:2)DV=VÞ¼E=3ð1(cid:2)2mÞ ð2:15Þ
(4) ThePoisson’sratio,mcharacterizestherelativevalueofthecompressiveandsheardeformations
ofasolid.Formostcrystallinemetalsandalloys,m=1/3[32].
Fortheisotropicsolidsormediums,thereareonlytwoelasticconstantsareindependent,andthe
fourelasticconstantscorrelateas[25,32]:
E¼3Kð1(cid:2)2mÞ ð2:16Þ
E
G¼ ð2:17Þ
2ðmþ1Þ
Themetallicglassesaremacroscopicallyisotropicinstructureandhomogeneousinphysicaland
mechanicalproperties,andtherefore,aboveequationscanbeappliedtometallicglasses.
2.2.Physicaloriginfortheelasticity
2.2.1.Basisforlinearelasticity
Thelinearelasticbehaviorismacroscopicmanifestationofatomicbonding.Theelasticconstantsor
elasticmoduliareoffundamentalimportanceforamaterial.TheKisdirectlyrelatedtotheexternal
forcerequiredtocompressorextendinteratomicdistancesinoppositiontotheinternalforcesthat
seektoestablishequilibriuminteratomicdistance.TheGsimilarlyrepresentsadistortionorbending
Pleasecitethisarticleinpressas:WangWH.Theelasticproperties,elasticmodelsandelasticperspectivesof
metallicglasses.ProgMaterSci(2011),doi:10.1016/j.pmatsci.2011.07.001
