2
Mesoscalelevelinthemechanicsof
materials
2.1Onthedefinitionsofscalelevels:micro-andmesomechanics
Thefollowingscalelevelsareusuallyrecognizedintheanalysisofthematerialbehavior:
•Macroscale(oraspecimenscale),oftheorderofmorethan1mm.Materialbehavior
atthisscalelevelisanalyzedusingcontinuummechanicsmethods.•Micro-andmesoscale(oramicrostructurescale),between1mand1mm.Material
behavioratthisscalelevelfallsintotheareaofmaterialsscience,andisanalyzed
usingmethodsofbothphysicsandmechanicsofmaterials,includingmicromechanics
andfracturemechanics.•Nano-andatomisticscales,lessthan1m.Materialbehavioratthisscalelevelfalls
intotheareaofthephysicsofmaterials.
Thequestionarisesastowhatisthedifferencebetween‘mesolevel’and‘microlevel’in
themechanicsofmaterials.
Theterm‘mesomechanics’hasgainedrelativelywideacceptanceafteritwasintro-
ducedbyPaninandco-workers(Panin,1998)asthenameforanewareaofresearch
(‘physicalmesomechanics’),themainpurposeofwhichhasbeendefinedasthedevelop-
mentofthetheoreticalbasisofmaterialimprovement,usingexperimentalandtheoretical
studiesofphysicalprocessesinloadedmaterialsatthemesolevel.
Needleman(Needleman,2000)defined‘mesoscalecontinuummechanics’as‘interme-
diatebetweendirectatomisticandanunstructuredcontinuumdescriptionofdeformation
processes’.Asacharacteristicfeatureofthisscalelevel,itwasnotedthat‘sizematters’
atthemesoscalelevel.
Panin(Panin,1998)suggestedamoredetailedclassification,withmesoIandmesoII
levels(definedasrelatedtotherotationmodesinsidestructuralelementsofdeformation,
ComputationalMesomechanicsofComposites: Numerical analysis of the effect of microstructures of composites on their strength and damage resistance L. Mishnaevsky Jr ©2007JohnWiley&Sons,Ltd. ISBN: 978-0-470-02764-614Sizeeffects
andtotheself-correlatedrotationsofmanystructuralelements,respectively).ThemesoI
levelcorrespondstothelevel‘insidemicrostructuralelements’(grains),whilethemesoII
levelisrelatedtothe‘conglomerates’ofmicrostructuralelements.Actually,Needleman’s
‘mesoscalecontinuummechanics’correspondstobothPanin’smesoIandIIlevels,and
tobothmicromechanicsandmesomechanicsintheaboveclassification.
Kocks(citedbyEstrin,1999)statedthat‘“mesoscopic”shouldrefertocaseswherethe
structuralscaleis,say,oftheorderof100m’(scalebetween‘microscopic’,associated
withthemicroscopeandmicrometer,and‘macroscopic’).
MishnaevskyJrandSchmauder(MishnaevskyJrandSchmauder,2001)definedthe
mesolevelinthematerialstructureasarangeofscalelevelswhicharetwotothreeorders
ofmagnitudegreaterthandefectsofstructure(whichvariedinthe10−9–10−5mscale
range)andonetothreeordersofmagnitudesmallerthanaspecimenorworkpieceinthe
following.Thelevelsofthedescriptioncanberelatedtothemethodsofthecontrolof
materialproperties:whereastheimprovementatmacrolevelcanbedonebymodifying
thespecimenconstruction,theimprovementofthestrengthandreliabilityofmaterials
atmicro-andmesoleveliscarriedoutbyheattreatment,metalworking,impregnation,
powdermetallurgymethods,etc.
FollowingMishnaevskyJr(MishnaevskyJr,2005b),onemaystatethatthemesome-
chanicsofmaterialsstudiesquantitativelytheinteractionandsynergisticeffectsofmany
elementsofmicrostructures(asinclusions,voids,shearbands,microcracks,etc.,or
generallytheheterogeneityofmaterials)onthestrengthandmechanicalpropertiesof
materials,whereasmicromechanicsdealswiththeeffectsofsingleelementsandaveraged
parametersofmicrostructuresonthemechanicalbehaviorofmaterials.Computational
mesomechanicsseekstocreatethenecessarynumericaltoolsfortheanalysisofthe
mechanicalbehavioranddegradationofmaterials,whichallowsthecomputationalanal-
ysisoftheinteractionbetweenmanyelementsofmicrostructuresandmicrostructure
evolution,andcanserveasabasisforthematerialdesign.
2.2Sizeeffects
Needleman(Needleman,2000)defined‘mesoscale’asascalelevelatwhich‘size
matters’.Theinfluenceofgeometricalsizeofcomponentsonthemechanicalbehavior
andstrengthofmaterialswasobservedinnumerousexperiments,carriedoutonmetals
(Flecketal.,1994),ceramics(XuandRowcliffe,2002),concretes(BazantandYavari,
2005)andpolymers(Tjernlund,2005).Thecommonobservationisthat‘smalleris
stronger’:thesmalleraspecimenoranelementofthemicrostructure,thehigherits
strength.
2.2.1Brittleandquasi-brittlematerials
Theinterestoftheresearchcommunityinthesizeeffectinmaterialswasfirstaroused
inconnectionwiththetestingofmaterialsforcivilengineeringuse.Thenecessityto
predictthestrengthofconcretesforlarge-scalestructuresonthebasisoftestingmuch
smallerspecimensledtointensivescientificeffortsinthisarea.
Severalexplanationsforthesizeeffectinconcretes,ceramicsandotherquasi-brittle
materialshavebeensuggested.TheoldestmodelofthesizeeffectisbasedonWeibull’s
