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Acknowledgments
WewanttoexpressoursinceregratitudetoProfessorEtiennePatoorfor extensivediscussionsduringthelast15yearsandProfessorAndréEberhardtforassistance,encouragement,andforprovidingusanimagefor thecoverofthebook.
WealsothankProfessorDimitrisLagoudasforourfruitfuldiscussions aboutmicromechanicsandProfessorFrançoisMuratforourlongdiscussionsabouttheperiodichomogenization.Additionally,wehighlyappreciatethescientificexchangesandideaswehadovertheyearswithour nationalandinternationalpartnersProfessorYvesChemisky(University ofBordeaux),ProfessorAndréChrysochoos(UniversityofMontpellier), ProfessorFransisco(Paco)Chinesta(ENSAMParis),AssistantProfessor JosephFitoussi(ENSAMParis),Dr.GillesRobert(DomoChemicals),Dr. RenanLéon(Valeo),AssistantProfessorTheocharisBaxevanis(University ofHouston),AssistantProfessorDarrenHartl(TexasA&MUniversity), AssociateProfessorGaryDonSeidel(VirginiaTech),ProfessorPaulSteinmann(FAUErlangen–Nürnberg),AssistantProfessorAliJavili(Bilkent University),ProfessorBjörnKiefer(TUBergakademieFreiberg),ProfessorMarek–JerzyPindera(UniversityofVirginia)andProfessorAndreas Menzel(TUDortmund).
Itisofcourseimportantforustowarmlythankallthemembersofthe SMARTteamofLEM3laboratoryfortheirsupport.Specialthanksbelong toourcolleagues,AssistantProfessorsAdilBenaarbia,FrancisPraudand BorisPiotrowski,aswellasourPostdocQiangChenandourPh.D.student SoheilSatouri,fortheirvaluablecommentsandremarks.
Chatzigeorgiou,Meraghni,andCharalambakis May2021
Abouttheauthorsxi
Forewordxiii
Prefacexv Acknowledgmentsxvii
Tensorsandcontinuummechanicsconcepts
1.Tensors
1.1TensorsinCartesiancoordinates3
1.2Cartesiansystemsandtensorrotation5
1.3Tensorcalculus7
1.4Examplesintensoroperations8
1.5Voigtnotation:generalaspects13
1.6OperationsusingtheVoigtnotation16
1.7TensorrotationinVoigtnotation19
1.8ExamplesinVoigtnotationoperations24 References27
2.Continuummechanics
2.1Strain29
2.2Stress32
2.3Elasticity34
2.3.1.Generalaspects34
2.3.2.Specialsymmetries36
2.4Reductionto2–Dproblems46
2.4.1.Planestrain47
2.4.2.Planestress49
2.5Examples50 References53
3.Generalconceptsofmicromechanics
3.1Heterogeneousmedia57
3.2Homogenization59
3.3Homogenizationprinciples62
3.3.1.Averagetheorems63
3.3.2.Hill–Mandelprinciple66
3.3.3.Concentrationtensors69
3.4Boundsintheoverallresponse71
3.4.1.VoigtandReussbounds72
3.4.2.Hashin–Shtrikmanbounds73
3.5Examples74 References82
4.VoigtandReussbounds
4.1Theory83
4.1.1.Voigtupperbound84
4.1.2.Reusslowerbound84
4.2Simplemethodsforfibercomposites86
4.3Compositebeams90
4.3.1.Essentialelementsofbeambendingtheory90
4.3.2.Beammadeoftwomaterials92
4.4Examples96 References99
5.Eshelbysolution–basedmean–fieldmethods
5.1Inclusionproblems101
5.1.1.Eshelby’sinclusionproblem102
5.1.2.Inhomogeneityproblem105
5.2Eshelby–basedhomogenizationapproaches109
5.2.1.Eshelbydilute111
5.2.2.Mori–Tanaka112
5.2.3.Self–consistent114
5.3Examples116 References126
6.Periodichomogenization
6.1Preliminaries127
6.2Theoreticalbackground128
6.3Computationoftheoverallelasticitytensor130
6.4Particularcase:multilayeredcomposite132
6.5Examples135 References143
7.Classicallaminatetheory
7.1Introduction148
7.2Stress–strainrelationforanorthotropicmaterial149
7.2.1.Fromtensortocontracted(Voigt)notation149
7.2.2.Hooke’slawfororthotropicmaterialinVoigtnotation152
7.3Hooke’slawforanorthotropiclaminaundertheassumptionofplanestress156
7.4Stress–strainrelationsforalaminaofarbitraryorientation:off–axisloading158
7.4.1.Stressandstraininglobalaxes (x y) 159
7.4.2.Off–axisstress–strainrelations160
7.4.3.Off–axisstrain–stressrelations162
7.4.4.Engineeringconstantsandinducedcoefficientsofshear–axialstrainmutual influenceinananglelamina163
7.4.5.Example168
7.5Macromechanicalresponseofalaminatecompositethinplate170
7.5.1.Laminatecodeandconvention171
7.5.2.LaminatedthinplatesandKirchhoff–Lovehypothesis173
7.5.3.Kinematicsofthinlaminatedplatesandstrain–displacementrelation174
7.5.4.Stressvariationinalaminate177
7.5.5.Forceandmomentresultantsrelatedtomidplanestrainsandcurvatures178
7.5.6.Physicalmeaningofsomecouplingcomponentsofthelaminatesstiffness matrices183
7.5.7.Workflowandsummary187
7.5.8.Example189 References195
III
Specialtopicsinhomogenization
8.Compositesphere/cylinderassemblage
8.1Compositesphereassemblage199
8.2Compositecylinderassemblage208
8.3Eshelby’senergyprinciple221
8.4Universalrelationsforfibercomposites226 8.5Examples229 References235
9.Green’stensor
9.1Preliminaries237
9.1.1.Fouriertransform237
9.1.2.Betti’sreciprocaltheorem238
9.2Definitionandproperties239
9.3ApplicationsofGreen’stensor244
9.3.1.Infinitehomogeneousbodywithvaryingeigenstresses244 9.3.2.Eshelby’sinclusionproblem246 9.4Examples247 References248
10.Hashin–Shtrikmanbounds
10.1Preliminaries251
10.1.1.Positiveandnegativedefinitematrices251
10.1.2.Calculusofvariations255
10.2Hashin–Shtrikmanvariationalprinciple256
10.3Boundsinabi–phasecomposite262
10.4Examples267 References269
11.Mathematicalhomogenizationtheory
11.1Preliminaries271
11.2Variationalformulation273
11.2.1.Functionalspaces273
11.2.2.Homogeneousbody275
11.2.3.Heterogeneousbodywithasurfaceofdiscontinuity277
11.2.4.Approximatingfunctions278
11.2.5.Finiteelementmethod279
11.3Convergenceoftheheterogeneousproblem280
11.3.1.Weakconvergence282
11.3.2.Mathematicalhomogenization283
11.4Asymptoticexpansionapproach286 11.5Examples288 References296
12.Nonlinearcomposites
12.1Introduction299
12.2Inelasticmechanismsinperiodichomogenization300
12.3Inelasticmechanismsinmean–fieldtheories302
12.3.1.Inhomogeneityproblemwithtwoeigenstrains303
12.3.2.Mori–Tanaka/TFAmethodforcompositeswithinelasticstrains307
12.4Examples308
References322
A.Fiberorientationincomposites
A.1Introduction325
A.2Reinforcementorientationinaplane325
A.3Reinforcementorientationin3–Dspace327
Foradditionalinformationonthetopicscoveredinthebook,visitthe companionsite: https://www.elsevier.com/books-and-journals/ book-companion/9780128231432
1.1TensorsinCartesiancoordinates
Atensorisdefinedasageometricobjectthatdescribeslinearrelations betweenscalars,vectors,orothertensors,andthusitcanbeseenasamultilinearmap.Inmathematicsandphysicsatensorisaverygeneralobject, intrinsicallydefinedfromavectorspacethatisindependentofcoordinate systems.Indepthanalysisabouttensors,tensoroperations,anddifferentialgeometryforcontinuummechanicsapplicationshasbeenpresented elsewhere[1,2].Inthischapter,wefocusonthosekeyaspectsthathelpthe readertofollowthediscussionthroughoutthebook.
Boldlettersareusedtodenotetensorstodistinguishthemfromscalars. Thischapterfrequentlyadoptstheindexnotation.Whenthetensorsappearintheirindicialnotation,theyhavenormalform(noboldfont).A
tensor A oforder n isexpressedas
Ai1 i2 i3 ...in .
In3–Dspace,theindices i1 , i2 ,etc.takethevalues1,2,or3.Themost commontensorsare:
•Atensoroforder0,whichdenotesascalar(noboldletterisusedinthis case).
•Atensoroforder1,whichrepresentsa 3 ˆ 1 vector.
•Atensoroforder2,whichisrepresentedbya 3 ˆ 3 matrix.1
Intheindexnotation,theEinsteinsummationconventionforrepeated indicesisused.Forexample:
•Theproduct ai bi oftwovectors a and b implies ai bi “ a1 b1 ` a2 b2 ` a3 b3 .
•Theproduct Aij Bjk oftwosecond–ordertensors A and B implies Aij Bjk “ Ai 1
•Theproduct Aijkl Bkl ofafourth–ordertensor A andasecond–ordertensor B implies Aijkl Bkl “ Aij 11 B11 ` Aij 22 B22 ` Aij 33 B33 ` Aij 12 B12 ` Aij 21 B
Therepeatedindicesare“dummy”inthesensethatwecanchangethe letterusedforthemwithanotherletter.Thus ai bi and aj bj representthe samesummation.
Thesymbol I denotesthesecond–orderidentitytensorwithcomponentsgivenbytheKroneckerdelta δij : Iij “ δij ,δij “ δji “ " 1 ,i “ j, 0 ,i ‰ j. (1.1)
Clearly, δii “ δ11 ` δ22 ` δ33 “ 3. Thethird–orderCartesianpermutationtensor isdefinedas ijk “ $ & % 1 if pi,j,k q is
,
, 1 if pi,j,k q is p1, 3, 2q, p2, 1, 3q, or p3, 2, 1q, 0 if i “ j,j “ k, or k “ i, (1.2)
1 Forsymmetricsecond–ordertensors,wecanutilizetheVoigtnotationandwritethemas 6 ˆ 1 vectors;seelaterinthischapter.
andhastheproperty ijk pqk “ δip δjq δiq δjp . Thesymmetricfourth–orderidentitytensor I isdefinedas
Fromitsstructureitisclearthat I exhibitstheminorsymmetries Iijkl “ Ijikl “ Iijlk andthemajorsymmetry Iijkl “ Iklij . Usingtheindexnotationleadsveryoftentolargeexpressions.Toreducethesizeoftheexpressions,thefollowingsymbolsarefrequently adopted(especially,inthemicromechanicspartofthisbook):
1. Atensor A oforder n,atensor B oforder n,andascalar c producethe nth–ordertensors
C “ A ` B with Ci1 i2 ...in “ Ai1 i2 ...in ` Bi1 i2 ...in , C “ c A with Ci1 i2 ...in “ cAi1 i2 ...in
2. Atensor A oforder n ` 1 andatensor B oforder m ` 1 producethe pn ` mqth–ordertensor
C “ A · B with Ci1 i2 ...in`m “ Ai1 i2 ...in q Bqin`1 in`2 ...in`m .
Thisoperationiscalledthesinglecontractionproduct.
3. Atensor A oforder n ` 2 andatensor B oforder m ` 2 producethe pn ` mqth–ordertensor
C “ A : B with Ci1 i2 ...in`m “ Ai1 i2 ...in pq Bpqin`1 in`2 ...in`m .
Thisoperationiscalledthedoublecontractionproduct.
4. Atensor A oforder n andatensor B oforder m producethe pn ` mqth–ordertensor
C “ A b B with Ci1 i2 ...in`m “ Ai1 i2 ...in Bin`1 in`2 ...in`m .
Thisoperationiscalledthedyadicproduct.
5. Atensor A oforder n andatensor B oforder 1 producethe nth–order tensors
C “ A ˆ B with A ˆ B “rA b B s : ,
C “ B ˆ A with B ˆ A “ : rB b As.
Thesetwooperationsarecalledthecrossproducts.
1.2Cartesiansystemsandtensorrotation
ACartesiancoordinatesystemisdescribedbythreebasisvectors e 1 , e 2 , and e 3 .Eachoneofthesevectorshasunitlength.ACartesiansystemis
FIGURE1.1 TransformationofaCartesiancoordinatesystemthrougharotation. orthogonal,whichmeansthat
LetusconsidertwoCartesiancoordinatesystemswithbasisvectors(e 1 , e 2 , e 3 )and(e 1 1 , e 1 2 , e 1 3 )havingthesamepointoforigin(Fig. 1.1).Therelation betweenthemcanbeexpressedwiththehelpofasecond–ordertensor R as
Thistensor R iscalledarotatorandhasthefollowingproperties: Rij Rik “ Rji Rki “ δjk anddet
wheredetdenotesthedeterminantof R ,
Duetoproperty(1.5)oftherotators,thevectors e i canbeexpressedin termsof e 1 i as
Thecomponentsof R arethecosinesoftheanglesbetweenthevectorsof thetwobases,
Rij “ cos ´e i , e 1 j ¯ “ e i e 1 j
Fortheelementarybasisvectors
weobtainthatthe i thcolumnofthetensor R representsthebasisvector e 1 i
Therotatortensorrepresentsatransformationfromonecoordinatesystemtoanotherthroughrotation.Theintroductionof R allowsustoprovideaproperdefinitionforthetensors:
BOX1.1Tensor
Atensor A oforder n isaquantitythatcanbetransformed,withthehelp ofarotator R ,fromonecoordinatesystemwithbasis e i toanothercoordinate systemwithbasis e 1 i accordingtotheformulas
1.3Tensorcalculus
ConsideringCartesiancoordinatesandthepositionvector x ,differentialgeometryprovidesthefollowingdefinitionsforan nth–ordertensor A [3,4]:
1. Thegradientof A isatensoroforder n ` 1 definedas
B Ai1 i2 ...in B xin`1
2. Thedivergenceof A isatensoroforder n 1 definedas
B Ai1 i2 ...in 1 in B xin .
3. Thecurlof A isatensoroforder n definedas
B Ai1 i2 ...in 1 j B xk jkin .
Moreover,foratensor A oforder n ą 0 andatensor B oforder m ą 0,the derivativeof A withrespectto B isthetensoroforder n ` m definedas
B Ai1 i2 ...in
B Bin`1 in`2 ...in`m
Clearly,thetypicalchainruleforderivativesholdsalsointhecaseoftensors.Forexample,fortwotensors A and B oforder2,thegradientoftheir product Aij Bjk iswrittenas
1.4Examplesintensoroperations
Example1
Provethefollowingtensorialidentities:
1. Forasecond–ordertensor A, Aij δjk “ Aij δkj “ Aik .
2. Forasecond–ordertensor A, Aij δij “ Aii .
3. If A issymmetricsecond–ordertensor(Aij “ Aji ),then Akl Iklij “ Iijkl Akl “ Aij
4. Ifthefourth–ordertensor Aijkl exhibitstheminorsymmetries Aijkl “ Ajikl “ Aijlk ,then Aijmn Imnkl “ Iijmn Amnkl “ Aijkl .
5. Foranarbitraryvector a , B ai B aj “ δij
6. If A isasymmetricsecond–ordertensor(Aij “ Aji ),then B Aij B Akl “ Iijkl .
Solution:
1. Forthefirstidentity,wehave
Therearethreecases:
k “ 1.Then
k “ 2.Then
• k “ 3.Then
Clearly,thisresultcanbeextendedfor nth–ordertensors.Wecaneasily showthatatanyindex ik ofan nth–ordertensor,multiplicationwiththe Kroneckerdeltayields
2. Usingthesymmetryof δij andthepreviousidentity,weobtain
3. Usingthesymmetryof A yields
Since Iijkl “ Iklij ,itbecomesevidentthatthesecondrelation
ijkl Akl “ Aij
alsoholds.
4. Usingtheextensionofthefirstidentitytofourth–ordertensorsyields
Notethat I itselfalsopossessestheminorsymmetries,andthusthe sameidentityholdsforittoo.
5. Thepartialderivative B ai B aj takesthevalue1when i “ j andiszerowhen i ‰ j .ThisissimilartothedefinitionofKroneckerdelta.
6. Thepartialderivative B Aij B Akl foranarbitrarytensor A isequalto1when i “ k and j “ l andzerootherwise.So,
Aij B Akl “ δik δjl
Thelatterresultthoughdoesnotaccountforsymmetries.Toinclude thesymmetryof A,wecanwrite
Thetensor I issymmetricandthusrepresentsthepartialderivativeof symmetricsecond–ordertensorwithitself.
Example2:Hillnotation
Thefourth–ordertensors A and B ,expressedintermsofthescalars Ab , As , B b ,and B s intheform
Aijkl “ 3Ab Ihyd ijkl ` 2As Idev ijkl , Bijkl “ 3B b Ihyd ijkl ` 2B s Idev ijkl , where Ihyd ijkl “ 1 3 δij δkl , Idev ijkl “ Iijkl Ihyd ijkl , arecalledisotropictensors.Provethefollowingidentities:
1. Aijmn Bmnkl “ Bijmn Amnkl “ 9Ab B b Ihyd ijkl ` 4As B s Idev ijkl . 2. If Aijmn Bmnkl “ Bijmn Amnkl “ Iijkl ,then Bijkl “ 1 3Ab Ihyd ijkl ` 1 2As Idev ijkl .
Remark:
Duetotheseinterestingidentities,aspecialnotationforisotropic fourth–ordertensorsisproposedin[5].Accordingtothisnotation,an isotropictensor A canberepresentedintheform A “p3Ab , 2As q.This allowscomputationsofscalartypewhendealingwithisotropictensoralgebra.
Solution:
Beforepassingtothemainproofs,weneedthefollowingidentities:
Ihyd ijmn Ihyd mnkl “ Ihyd ijkl ,
Ihyd ijmn Idev mnkl “ Idev ijmn Ihyd mnkl “ 0ijkl , Idev ijmn Idev mnkl “ Idev ijkl ,
where 0ijkl isthefourth–ordernulltensorwithallzeroterms.Forthefirst expression, I
ijmn I
δkl “ Ihyd ijkl
Forthesecondandthirdexpressions,notethat I hyd exhibitstheminor symmetries Ihyd ijkl “ Ihyd jikl “ Ihyd ijlk ,andthusthefourthidentityofExample1 holds.So Ihyd ijmn Idev mnkl “ Ihyd ijmn rImnkl Ihyd mnkl s“ Ihyd ijkl Ihyd ijkl “ 0ijkl ,
Idev ijmn Ihyd mnkl “rIijmn Ihyd ijmn sIhyd mnkl “ Ihyd ijkl Ihyd ijkl “ 0ijkl ,
Idev ijmn Idev mnkl “rIijmn Ihyd ijmn srImnkl Ihyd mnkl s “ Iijkl Ihyd ijkl Ihyd ijkl ` Ihyd ijkl “ Idev ijkl .
Withtheseexpressionswecannowpasstothemainproofs.
1. Forthefirstidentity,
Aijmn Bmnkl “r3Ab Ihyd ijmn ` 2As Idev ijmn sr3B b Ihyd mnkl ` 2B s Idev mnkl s
“ 9Ab B b Ihyd ijmn Ihyd mnkl ` 4As B s Idev ijmn Idev mnkl
` 6Ab B s Ihyd ijmn Idev mnkl ` 6As B b Idev ijmn Ihyd mnkl
“ 9Ab B b Ihyd ijkl ` 4As B s Idev ijkl .
Bysimilarcomputationswealsogettheidentity
Bijmn Amnkl “ 9Ab B b Ihyd ijkl ` 4As B s Idev ijkl .
2. Forthesecondidentity,usingthefirstyields
9Ab B b Ihyd ijkl ` 4As B s Idev ijkl “ Ihyd ijkl ` Idev ijkl
Fromthelastexpressionitbecomesevidentthat 9Ab B b “ 1 and 4As B s “ 1, or 3B b “ 1 3
Example3
Asymmetricsecond–ordertensor aij “ aji canbesplitintodeviatoric andvolumetric/hydrostaticparts,whicharealsosymmetric:
Fortwosymmetricsecond–ordertensors a and b ,provethefollowing identities:
1. a hyd ij b dev ij “ 0.
2. aij b dev ij “ a dev ij b dev ij .
3. Ihyd ijkl akl “ akl Ihyd klij “ a hyd ij and Idev ijkl akl “ akl Idev klij “ a dev ij ,where Ihyd ijkl and Idev ijkl aredefinedinExample2.
4. B a ndv B aij “ a dev ij a ndv ,where a ndv “ ba dev kl a dev kl .
Solution:
1. Forthefirstidentity,
2. Usingthefirstidentity,weeasilyprovethesecond: a dev ij b dev ij “ ”aij a hyd ij ı b dev ij “ aij b dev ij
3. Wehave
“ ”Iijkl Ihyd ijkl ı akl “ aij a hyd ij “ a dev ij .
Since Ihyd ijkl “ Ihyd klij and Idev ijkl “ Idev klij ,wealsogettherelations
4. Wesplittheproofofthefourthidentityintotwoparts:
Since a dev ij issymmetric,thesecondidentityofidentities3holds,thatis,
1.5Voigtnotation:generalaspects
Untilnow,thenotionoftensorshasbeenpresented.Numericalcomputationsusingtensorialorindicialnotationcanbequitecumbersome,especiallywhendealingwithoperationsrelatedtofourth–ordertensors.Inthe presentchapter,wediscussthewell–knownVoigtnotation,whichallows ustotransformtensorialoperationstomatrixoperations.Thereaderisrequiredtohavesomebasicknowledgeaboutvectorandmatrixoperations.
TheVoigtnotationisapracticalwaytorepresentsecond–andfourth–ordersymmetrictensorsinvectorandmatrixform,respectively.TheVoigt representationsubstitutestwoindices i , j withoneindex I accordingto
I “ " i,i “ j, 1 `ri ` j s,i ‰ j.
LetuswritetheVoigtrepresentationindetail:
firstsecondresultant tensorialtensorialVoigt indexindexindex
Withthisinterchangerulewehavethefollowing:
•Asymmetricsecond–ordertensor a ischaracterizedbytheproperty aij “ aji .Itisusuallyrepresentedasthe 3 ˆ 3 matrix
Since a hasonlysixindependentcomponents,wecanalsoexpressitas a 6 ˆ 1 vector.TherearetwotypesofvectorsconsideredintheVoigt notation,the“s”typeandthe“e”type,
respectively.Thefactor2appearingonthe“e”typeisveryusefulwhen performingvarioustensoroperations,asitwillbecomeclearfurther.
2 Intheclassicalcontinuummechanicsstudiestheindices i “ 2,j “ 3 or i “ 3,j “ 2 correspondto I “ 4,andtheindices i “ 1,j “ 2 or i “ 2,j “ 1 correspondto I “ 6.Hereweadopt aslightmodificationintheVoigtconvention.
•Afourth–ordertensorthathasminorsymmetries(i.e., Aijkl “ Ajikl “ Aijlk )representsalinearrelationbetweensymmetricsecond–ordertensors;ithasonly36independentcomponentsandcanberepresentedasa 6 ˆ 6 matrix.ThetwoVoigtrepresentationsofsecond–ordertensorsnecessitatetheidentificationoffourVoigtrepresentationsoffourth–order tensors.Thestandard 6 ˆ 6 matrixformis
writtenforsimplicityas
wherethefourindices i,j,k,l aresubstitutedwithtwoindices I,J followingthenotation:
Thethreeadditionalmatrixformsthatcanbeidentifiedare[6]
1.6OperationsusingtheVoigtnotation
Usingtherepresentationsdescribedaboveforsecond–andfourth–ordertensors,wecansimplifyvarioustensorialoperations. Consideringthetwotypesofsecond–ordertensors,wecanwritethe scalartensorialproduct
wherethesymbol p · q intheVoigtnotationdenotestheclassicalmatrix multiplication,andthesuperscript T istheusualtransposeoperator.With regardtofourth–ordertensors,weeasilyshowthatthetensorialproducts
