a generalized theory of elastodynamic homogenization for periodic media 2016 international journal...

7/26/2019 A Generalized Theory of Elastodynamic Homogenization for Periodic Media 2016 International Journal of Solids and

1/8

InternationalJournalofSolidsandStructures84(2016)139146

ContentslistsavailableatScienceDirect

International

Journal

of

Solids

and

Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

A generalized theory of elastodynamic homogenizationfor

periodic

media

H.Nassar,Q.-C.He,N.Auffray

ModlisationetSimulationMulti-Echelle,Universit Paris-Est,MSMEUMR8208CNRS,5bdDescartes,77454Marne-la-Valle,France

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received13August2015

Revised

18

December

2015

Availableonline4February2016

Keywords:

Homogenization

Willistheory

Generalizedcontinua

Optical

branches

Bandgaps

a

b

s

t

r

a

c

t

For

periodically

inhomogeneous

media,

a

generalized

theory

of

elastodynamic

homogenization

is

pro-

posed

so

that

even

the

long-wavelength

and

low-frequency

asymptotic

expansions

of

the

resulting

ef-

fective

(or

macroscopic)

motion

equation

can,

approximately

but

simultaneously,

capture

all

the

acoustic

and

some

of

the

optical

branches

of

the

microscopic

dispersion

curve.

The

key

to

constructing

the

gen-

eralized

theory

resides

in

incorporating

new

kinematical

degrees

of

freedom

in

conjunction

with

rapidly

oscillating

body

forces

as

microscopic

and

macroscopic

loadings

while

satisfying

an

energetical

consis-

tency

constraint

reminiscent

of

Hill

Mandel

lemma.

By

this

constraint,

an

effective

displacement

field

is

naturally

defined

as

the

projection

of

a

microscopic

one

onto

the

dual

to

the

space

of

body

forces.

To

illustrate

these

results,

a

two-phase

string

is

studied

in

detail.

2016

Elsevier

Ltd.

All

rights

reserved.

1.Introduction

Theelastodynamichomogenizationapproachesreportedupto

nowintheliteratureareobservedtorunintodifficultieswhenbe-ingusedtomodeldynamicaleffectsoverawidefrequencyrange.

1.The classical lowest-order Long-Wavelength (LW) Low-

Frequency(LF)homogenizationapproaches(Bensoussanetal.,

1978;Sanchez-Palencia,1980)yieldahomogeneoussubstitu-

tionCauchymediumwhichmissesalldispersiveeffectsandall

internalresonances,i.e.,allopticaloscillationmodes.

2.Thehigher-orderLW-LFasymptotichomogenizationapproaches

(Andrianovetal.,2008;BoutinandAuriault,1993)leadtoef-

fectivestrain-gradientmediawhichcanmodelwelldispersive

behaviorsandsizeeffectsbutarevalidonlyneartheacoustic

branchesindependentlyoftheorderoftheasymptoticapprox-

imationsused.

3.

The

high-frequency

asymptotic

approaches

(

Antonakakis

et

al.,

2014;Boutinetal.,2014;Colquittetal.,2014;Crasteretal.,

2010;Dayaetal.,2002;Noldeetal.,2011)aresuccessful in

capturinghigh-frequencyopticalmodesbutstillvalidonlyin

thevicinityofsomefinitefrequency.

4.The high-contrast asymptotic approaches (Auriault and Bon-

net,1985;AuriaultandBoutin,2012;Smyshlyaev,2009)havea

widefrequencyvaliditydomainenglobinganinfinitenumberof

Correspondingauthor.Tel.:+33160967794.

E-mailaddress:[email protected](H.Nassar).

opticalbranches.However,thecorrespondingeffectivebehavior

iscomplexandnonlocalintime.

5.Thenon-asymptotictheoryofWillis(1997,2011)yieldsexactly

thewholedispersioncurve.Nonetheless,thedescribedeffectivefieldsareonlyrelevantforlowfrequencies(Nassaretal.,2015b;

SrivastavaandNemat-Nasser,2014).

Themainpurposeofthepresentpaperistoconstructagener-

alizedtheoryofelastodynamichomogenizationforperiodicmedia

whichimprovesthequalityoftheWilliseffectivebehaviorasan

approximationtothemicroscopicbehaviorinawaythatLW-LF

asymptoticexpansionsbecomeabletocapture,approximatelybut

simultaneously,alltheacousticandsomeoftheopticalbranches

ofthemicroscopicdispersioncurve.Toachievethispurpose,new

kinematicalDegreesOfFreedom(DOFs)aretakenintoaccountso

astodescribesomeshort-wavelengthcomponentsofthemicro-

scopicdisplacementfieldwhichbecomedominantathighfrequen-

cies.

The

new

DOFs

are

excited

by

incorporating

various

rapidly

oscillatingbodyforcesonthemicroscaleandonthemacroscale

underanenergetical consistency constrainthereaftercalledEn-

ergyEquivalencyPrinciple(EEP).TheEEPisabalancebetweenthe

microscopicandmacroscopicvirtualworksandislaterprovento

yieldageneralizedversionofthewell-knownHillMandellemma.

WithrespecttoWillistheory,weunderlinetwomajordifferences.

First,theincorporatedloadingsaremuchricherthanthoseem-

ployedbyWillis(1997,2011).Thishastheconsequenceofreduc-

ingtheerrorcommittedduringtheupscalingprocessandprovid-

inganextendedfrequencyvaliditydomain.Second,theEEPcon-

cernsvirtualworksandnottheirexpectancies.Fromthephysical

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140 H.Nassaretal./InternationalJournalofSolidsandStructures84(2016)139146

standpoint,thisleadstoacleardistinctionbetweenthemacroscale

andthemicroscaleintermsofwavelengths.Nevertheless,itshould

bepointedoutthatthegeneralizedtheorypresentedhereisby

construction limited toperiodically inhomogeneousmediawhile

Willistheoryisformallyvalidbothforperiodicallyandrandomly

inhomogeneousmedia.

Thepaperisorganizedasfollows.InSection2,werecallsome

geometricalelementsusefulfordescribingperiodicmedia,summa-

rize

the

equations

governing

the

kinematics

and

dynamics

of

them,

andsimplifytheseequationsbyusingBloch-waveexpansions.The

mainbodyofthegeneralizedtheoryispresentedinSection3.The

EEPisfirstpostulated;thespaceofadmissiblebodyforcesisthen

definedasthesetofmacroscopicallyappliedloadings;theeffec-

tivedisplacementfieldassociatedtoamicroscopicdisplacementis

obtainedbytheEEPandproventobeanimprovementoverthe

onedefinedbyWillis;theeffectivemotionequationisfinallyde-

rivedinaformalwayandaHilMandelrelationisdemonstrated.

InSection4,ananalyticalLW-LFasymptoticapproximationtothe

effectivemotionequationisgivenforaparticular1Dtwo-phase

string.Exactandapproximatedispersioncurvesareplottedand

compared. Itappears thenhow the resultingasymptoticmodel,

thoughbasedonLFexpansions,cansimultaneouslycaptureacous-

ticandopticalbrancheswhileconservingalow-orderlocalmotion

equation.

2.Preliminaries

Inthissection,somegeometricalelementsusefulforthestudy

ofperiodicmediaarerecalled.Thegoverningequationsoflinear

elasticityare recapitulated.Bloch-waveexpansionsoffieldsand

workarealsointroduced.

2.1.Geometryandperiodicity

Let bead-dimensionalinfinitebody.DefineE asthevector

spaceoftranslationsactingonthepointsof.Givendindepen-

dent

translations

(b

j

)

j=1

...

d

,

denote

byR

the

subset

ofE

obtained

byintegercombinationsofthesevectors.ThesubsetR iscalled

alattice.Then,ascalar,vectorortensorfieldhdefinedover is

saidtobeR-periodicifandonlyifitsatisfiesh(x+r)=h(x)forallpointsx andalltranslationsrR.Accordingly,hneedsbeingdefinedonlyoveraunitcell

T=

xo+r

r= dj=1

rjbj,1/2rj


3/8

H.Nassaretal./InternationalJournalofSolidsandStructures84(2016)139146 141

WenowassumethatCand areR-periodicsothatthesolutionuk

(x)canbewrittenas

uk

(x)=uk

(x)eikx

withanR-periodicamplitude uk

(x

)(Gazaletetal.,2013).Inother

words,totheexpansion(2.2),therecorrespondsasimilarBloch-

waveexpansionofthesolutionto(2.1):

u(x)=

Tuk(x)dd

k=

Tuk(x)ei

kxdd

k, (2.3)

withuk

(x)beingthedisplacementfieldover subjectedtofk

(x).

Incontrast,exceptforahomogeneousbody,therearenosimilar

resultsforplanewaves.

Given theBloch-wave formofuk

,andsimplifying thephase

factoreik x,themotionequationbecomes,intermsof uk(x)andfk

(x),

(+ik) {

C(x):[

(+ik)suk

(x)]

}

+fk

(x)=2(x) uk

(x).

(2.4)

Last, thex-dependenciesofBloch amplitudes fordisplacements

and body forces were explicitly annotated. In what follows, all

fieldsareunderstoodtobex-dependentunlessotherwisestated.

2.4.Externalwork

Define the virtualworkofexternal body forcesf associated

withavirtualdisplacementfielduby

f

u

whereasuperscripted denotescomplexconjugation.Also,intro-ducetheaveragingoperatorby

h

= 1|

T|

T

h(x)ddx

for all R-periodic functions h. Note that is independent ofthe choiceofT and is ill-defined fornon-R-periodic functions.

PlancherelsidentityforFouriertransformandParsevalsidentity

forFourierseriesdeliverthenasimilaridentityforBloch-waveex-

pansions:

f u=(2 )d

T

fk

uk

d

dk

=(2 )d

|

T|

T

T

fk

(x) uk

(x)ddxddk. (2.5)

Consequently,indefiningtheenergeticallyequivalenteffectivebe-

havior,wecanworkwithfieldsofasingleBlochwavenumberkT andthenapplythesuperpositionprincipleeventhoughworkis

quadratic

and

not

linear.

3.

A

general

theory

Inthissection,theenergyequivalencyprinciple(EEP),thecor-

nerstoneofthepresentapproach,isfirstpostulated,givenasim-

pleformandexploitedtodefinetheeffectivedisplacementfield.

Aformalderivationoftheeffectivemotionequationisthenpre-

sented.Theeffectiveconstitutivebehaviorisnonlocalinbothspace

andtimewhichraisesquestionsaboutitsuniqueness(Fietzand

Shvets,2010;Willis,2011). Inorder toavoid thisdifficulty,we

willbeinterestedonlyintheeffectivemotionequationwhichis

unique.Nonetheless,wewillderiveexpressions forthegeneral-

izedstress,momentum,velocityandstrainmeasureswhichare,in

particular,neededfordetermininganeffectiveconstitutivelaw.

Fig.

1.

The

effective

displacement

field

D

associated

to

a

given

microscopic

one

u

is

geometrically

interpreted

as

the

projection

of

the

latter

onto

the

space

of

admissible

displacements.SpacesF andF areisomorphicand,here,aretakentobeequaluptoachangeinunits.

3.1.Energyequivalency

In classical static or quasi-static homogenization, an energy

equivalencyrelation,knownasHillMandellemma,isprovenfora

familyofboundaryconditionsprescribedonarepresentativevol-

umeelementasmacroscopic loadings.Once theboundarycon-

ditionshavebeenspecified,HillMandel lemmacanbeusedto

define, by duality, the macroscopic stress in a strain-based ap-

proachorthemacroscopicstraininastress-basedapproach.Inthe

presentformulation,admissiblebodyforcesappliedgloballytoinsteadofboundaryconditionsaretakentobemacroscopicload-

ing.Then,anEEPispostulatedsoastodualizebodyforcesand

displacements.Thisdualitywillallowustodefinethemacroscopic

displacementfield,calledD, in termsofthemicroscopiconeu,

onceadmissiblebodyforceshavebeenimposed.

LetF bethespaceofBlochamplitudes fk,involvedin(2.2),

ofadmissiblebodyforces.TheelementsofFareseenasexternal

loadingslikelytobeappliedto.Notethattheywillremainthe

sameafterthescaletransition.ThespaceFactsasaparameterof

theapproachtobeelaboratedandneedstobechosenadequately.

Next,letF,thespacedualtoF,bethespaceofBlochampli-tudes Dk

ofadmissibleeffectivedisplacementfields.Foragiven

microscopicdisplacementfieldu,thecorrespondingeffective(or

macroscopic)displacementfieldisdefinedastheuniqueadmissi-bledisplacementsuchthat

fD=

f u (3.1)

foralladmissiblevirtualbodyforcesf(i.e., fkF,forallkT).Physically,theEEP(3.1)canbeinterpretedasrequiringthattheef-

fectivedisplacementfieldDassociatedtoagivenmicroscopicfield

ubesuchthattheworkdonebyeveryadmissiblevirtualbody

forcefinthecourseofD isequaltotheonedonebyfinthe

courseofu.Geometrically,theEEP(3.1)simplymeansthatDisthe

projectionofuontothespaceofadmissibledisplacements(Fig.1).

OnthebasisoftheEEP(3.1),ageneralizedHillMandellemmawill

beproveninSection3.4.

UsingtheBlochdecomposition(2.5),theEEP(3.1)canbeequiv-

alentlywrittenintermsofBlochamplitudesas

kT, fk

F,fk

Dk

=fk

uk

. (3.2)

3.2.Effectivedisplacementfield

BearinginmindtheEEP(3.1),choosingthespaceFofadmis-

siblebodyforcesbecomesakeysteptowardelaboratingageneral-

izedtheory.ThechoiceofFdependsultimatelyonthedegreeof

accuracywithwhichDisrequiredtoapproximateu.ThebiggerF

is,thecloserDistou.Whenallbodyforcesareconsideredasad-

missible,therelation(3.1)impliesD=uandtheeffectivemediumistriviallytheoriginalone.Inwhatfollows,westudytherather

generalcaseofpracticalimportancewhereFisfinite-dimensional

andshowhowDderivesfrom(3.1)correspondingly.


4/8


5/8

H.Nassaretal./InternationalJournalofSolidsandStructures84(2016)139146 143

Fig.2.Plotsoftherealpartsofthemicroscopic(u),Willis(U),andgeneralized(D)Blochdisplacementamplitudesoveroneperiodfor3eigenmodes:(k=0,=1

(0))

(top),(k=/2a,=2

(/2a))(middle),(k=0,=2

(0))(bottom).Twoshapefunctionshavebeenused:aconstantandasinewave.DetailsaregiveninSection4.

3.2.4.Effectivedisplacementfieldunderinfinitescaleseparation

ItisofinteresttoexaminewhatDbecomesunderthehypoth-

esisofinfinitescaleseparation,namely,whenk

0and

0.

Itisknownthatinthiscase,tothelowestorder,thedisplacementfielddependsonlyontheslowvariable (refertoBoutinandAu-

riault,1993,forinstance).IntermsofBlochamplitudes,thismeans

that ukisconstant.Consequently,

Uk

=

uk

=uk

,

u

k

=

uk= uk=0.Therefore,thetranslationalDOFUistheonlynon-nullcomponent

ofD,tothelowestorder.Theuseofageneralizedkinematicsis

hencejustifiedonlyunderweakscaleseparationorforhighfre-

quencieswhenmicroscopicdeformationmodesbecomesignificant.

Otherwise,itisenoughtokeeptrackofUexclusivelyasintheun-

weightedWillistheory.Asamatteroffact,ithasbeenobserved

that

a

periodic

medium

was

homogenizable in

the

Willis

sense

overtheacousticandthefirstopticalbranchesonly(Nassaretal.,

2015b;SrivastavaandNemat-Nasser,2014).Forhigherfrequencies,

oneneedstousenon-uniformshapefunctions.

3.3.Effectivemotionequation

Havingspecifiedbodyforces,themotionEq.(2.4)becomes

(

+

i

k)

{

C(x

)

:[

(

+

i

k)suk

(x

)]

}

+

Fk

+

f

k

(x

)(x)

=2(x) uk

(x

),

whichneeds tobe solvedover.Since uk

isR-periodic, it is

enoughtosolvetheaboveequationoveraunitcellTunderpe-

riodicboundaryconditions.Letgkbethecorrespondingperiodic

secondorderGreenoperator.Then,

uk

(y)

=

1

T

T

gk

(y,x)

fk

(x)ddx

=

1

T

Tgk

(y,x)ddx

Fk

+

1

T

Tgk

(y,x) ddx

f

k

(3.7)

which,combinedwith(3.5),deliversthefollowingexpressionsfor

thecomponentsofthemacroscopicdisplacementfield:

Uk

=gk

(y,x) Fk

+gk

(y,x) (x) fk

,

u

k

=(y)gk

(y,x)

Fk

+(y)gk

(y,x) (x)

f

k

,(3.8)

wheremeansaveragingwithrespecttobothxandy.TheGreenoperatoroftheeffectivemediumGkisgivenbythe

lasttwoequalitieswhichcanbewrittenconciselyas

u

j

k

=G

ji

k

f

i

k

,

wherenodistinctionismadebetweentheclassicalandgeneralized

DOFs(recallthat Uk

=

dj=1

ujk

j).Invertingtheprecedingequa-tiondeliverstheeffectivemotionequationinFourierdomain:

Z

i jk

u

jk

= fik

, (3.9)

whereZk,theinverseofGk,iscalledtheeffectiveimpedance.It

dependsimplicitlyonthefrequency.BysummingoverkTandover,weobtaintheeffectivemotionequationinxandtas

Z

i j(x,t) uj(x,t)=fi(x,t),

whereZ(x,t)isanintegro-differentialoperatorand denotescon-volutionproductwithrespecttospaceandtime.Theeffectivemo-

tionequationishencenonlocal inbothspaceandtimeand in-


6/8


volveslongwavelengths(kT)only.Thisgeneralizesequation(3.28)derivedbyWillis(1997)forperiodicmedia.

3.4.Internalwork

Asmentionedearlier,itisnotessentialforachievingthemain

purposeofthepresentworktoderiveanexplicitexpressionfor

theunderlyingeffectiveconstitutive lawwhichisnotuniqueas

in thetheoryofWillis.However, it isof interesttospecifythemacroscopicstress,momentum,strainandvelocitymeasuresthat

aneffectiveconstitutivelawinvolves.Inaddition,thesegeneralized

macroscopicmeasureswillbeshowntoberelatedtotheirmicro-

scopiccounterpartsthroughanextendedHillMandelrelation.

3.4.1.Generalizedstressandmomentummeasures

Saidmeasuresaretakentobetheonesinvolvedintheeffective

motionequationwrittenasaconservationlawequivalentto(3.9).

Startingwiththemicroscopicmotionequation

(+ik) k

+Fk

+fk

=i pk

, (3.10)

where kand pkaretheBlochamplitudesofstressandmomen-tum,wetakeitsvolumeaverageoveraunitcelltoobtain,with

thehelpofthedivergencetheorem,

ik k

+Fk

=i Pk

. (3.11)

Thisisthefirsteffectivemotionequationinvolvingtheclassical

macroscopicstressandmomentummeasures:

k

k

, Pk

pk

.

Further,projectingEq.(3.10)ontothespacespannedbytheother

shapefunctions givesriseto

ik k

s

:k

+f

k

=i pk

,

where,forsimplicity,wehaveassumedthecontinuityof sothattheboundarytermvanishes.Thegeneralizedstressandmo-

mentummeasurescanbeidentifiedas

k

k

, s

k

s

:

k

, p

k

pk

.

Theadditionalmotionequationbecomesthensimply

s

k

+

i

k

k

+

f

k

=

i

p

k

. (3.12)

NotethatEqs.(3.11)and(3.12)ononehand,and(3.9)ontheother,

arerelatedtooneanotherthroughanon-uniqueeffectiveconsti-

tutive lawwhosecharacterization isbeyond thepurposeof the

presentwork(seethediscussionbyWillis,2011,2012).

In summary, the motion equations in the spacedomain are

givenby

+F

=

iP,

s

+

+f =ip . (3.13)

Theseequationsareamicromechanicalversionoftheequations

ofequilibrium phenomenologicallyderivedbyGermain(1973).

3.4.2.Generalizedstrainandvelocitymeasures

Saidmeasuresareobtainedbyduality.Thevirtualworktheo-

remcombinedwiththeEEPyieldsk

:

k

pk

vk

=

fk

u

k

=

Fk

U

k

+

f

k

u

k

,

where kandvkaretheBlochamplitudesofthestrainandveloc-ityfields,respectively,givenby

k

=(

+

ik)

s

uk

, vk

=

i uk

.

Fig.3.Unitcell.

Substitutingbodyforcesbythecorrespondingstressandmo-

mentummeasuresaccordingto(3.12)delivers

k:

k

p

k

vk

= k:(iksUk) Pk (i Uk)+

k

(i

k

u

k

)

p

k

(i

u

k

)

s

k

u

k

.

Summingoverk,andusingPlacherelsidentity,weobtainagener-

alizedversionoftheHillMandellemma:

{

:p v}

=

{

:(sU)P (iU)

+ (u)p(iu) su}

.(3.14)

Thisresult isvalid forallvirtualcouples (,p)equilibratedbyanadmissiblebodyforcefieldandforallcouples(,v)derivedfromanarbitrarydisplacementfieldu.Fromtheaboverelation,

weidentifytheclassicalmacroscopicmeasuresofstrainandve-

locityassUandiUwhilethegeneralizedonesareu ,uandiu .

Finally,theconstructedmacroscopicfieldsmeetthemostfun-damentalrequirementsforthemtobeinterpretedasstresses,mo-

menta,strainsandvelocitiessincetheyrigorouslysatisfylocalbal-

anceandcompatibilityequations.Nonetheless,howtophysically

interpretandmeasurethesequantitiesinaprecisewayultimately

dependsonthechosensetofshape functions.Moredetailsare

giveninSection4.4regardingthisaspect.

4.

An

application:

high-frequency

behavior

through

low-frequencyasymptotics

At this point, we have generalized Willis theory by using

enrichedkinematics to improve thequalityofapproximationof

amicroscopicdisplacementubyamacroscopiconeD.Thecost

is

however

the

increasing

complexity

of

the

resulting

effective

motionequation.Anumericalprocedurededicatedtotheimple-

mentationofWillis theoryorourpreviousone isquiteheavy,

theeffectivebehaviorbeingnonlocalinbothspaceandtimewith

infiniteradiiofinfluenceingeneral.

Taylorasymptoticexpansionsprovideanefficientwaytoap-

proximate thenonlocalbehaviorwitha localoneunderappro-

priateassumptionsonkand.LW-LFexpansionshavethemainadvantageofonly requiring the solutionof staticproblemsbut

presentthedisadvantageofbeinglimitedtotheLFbehavior.The

purpose of this section is to show explicitly how generalizing

Willistheorymakesitpossibletoextendthevaliditydomainof

LW-LFexpansionstohigh-frequencybehavioroverasimple1Dex-

ample.

4.1.Setting

Considertheperiodicallyinhomogeneousstringwhoseunitcell

isdepictedinFig.3,anddefinetheshapefunction

(x

)=

2

sin(

x/a)

whichdescribestherapidlyoscillatingbodyforce

f

=

F

+

q

andwillcarrythenewDOF .ThemacroscopicdisplacementDthenreads

D=U+with

U=u, =u.


7/8


8/8


homogenization theorywiththehigh-frequencyhomogeniza-

tiontheorysuggestedbyDayaetal.(2002)andCrasteretal.

(2010)andco-workers.

5.Concludingremarks

ThroughincorporatingnewkinematicalDOFs,thepresentwork

hasproposedanelastodynamichomogenizationtheorygeneraliz-

ing

the

one

of

Willis

in

the

case

of

periodic

media

and

reduc-

ingtheerrorcommittedduringtheupscalingprocess,especially

athighfrequencies.Inordertoillustratethepotentialofthepre-

sentedtheory,ithasbeenshownthattheLW-LFasymptoticex-

pansionoftheeffectivemotionequationiscapableofsimultane-

ouslycapturing theacousticand thefirstopticalbranchof the

microscopicdispersioncurveforasimple1Dmedium.

Two problems remain open. The first concerns the effective

elastodynamicconstitutivelawproducedbythegeneralizedtheory

proposed.Inthispaper,toavoidthedifficultyrelatedtoitsnon-

uniqueness,theeffectivemotionequationhasbeendirectlytreated

andexploited.However, innumeroussituations,itisusefuland

importanttoexplicitlyknowtheeffectiveelastodynamicconstitu-

tivelaw.Thesecondproblemregardstheoptimalchoiceofshape

functionsforwhichguidingcriteriaexistbutremainincomplete.

References

Andrianov,I.V.,Bolshakov,V.I.,Danishevskyy,V.V.,Weichert,D.,2008.Higherorderasymptotichomogenizationandwavepropagationinperiodiccompositemate-rials.ProceedingsoftheRoyalSocietyA:Mathematical,PhysicalandEngineer-ingSciences464,11811201.

Antonakakis,T.,Craster,R.V.,Guenneau,S.,2014.Homogenisationforelasticpho-toniccrystalsanddynamicanisotropy.J.MechanicsPhys.Solids71,8496.

Auriault,

J.-L.

,

Bonnet,

G.

,

1985.

Dynamics

of

periodic

elastic

composites

(Dynamique

descompositeselastiquesperiodiques).ArchiwumMechanikiStosowanej37(4),269284.

Auriault,J.-L.,Boutin,C.,2012.Longwavelengthinner-resonancecut-off frequenciesinelasticcompositematerials.Int.J.SolidsStruct.49(23-24),32693281.

Bensoussan,A.,Lions,J.L.,Papanicolaou,G.,1978.Asymptoticanalysisforperiodicstructures.North-HollandPublishingCompany.

Boutin,C.,Auriault,J.-L.,1993.Rayleighscatteringinelasticcompositematerials.Int.J.Eng.Sci.31(12),16691689.

Boutin,C.,Rallu,A.,Hans,S.,2014.Largescalemodulationofhighfrequencywavesinperiodicelasticcomposites.J.Mech.Phys.Solids70,362381.

Colquitt,D.J.,Craster,R.V.,Antonakakis,T.,Guenneau,S.,2014.Rayleigh-Blochwavesalongelasticdiffractiongratings.Proc.RoyalSoc.A:Math.Phys.Eng.Sci.471,20140465.

Craster,R.V.,Kaplunov,J.,Pichugin,A.V., 2010.High-frequencyhomogenizationforperiodic

media.

Proc.

Royal

Soc.

A:

Math.

Phys.

Eng.

Sci.

466,

23412362

.

Daya,E.,Braikat,B.,Damil,N.,Potier-Ferry,M.,2002.Continuummodelingforthemodulatedvibrationmodesoflargerepetitivestructures.ComptesRendusM-canique330,333338.

Fietz,C.,Shvets,G.,2010.Current-drivenmetamaterialhomogenization.PhysicaB:

Condensed

Matt.

405

(14),

29302934

.

Forest,S.,2006.Milieuxcontinusgnralissetmatriauxhtrognes.CollectionSciencesdeLaMatire.PressesdelEcoledesMines.

Forest,S.,Sab,K.,1998.Cosseratoverallmodelingofheterogeneousmaterials.Me-chanicsRes.Commun.25(4),449454.

Gazalet,J.,Dupont,S.,Kastelik,J.C.,Rolland,Q.,Djafari-Rouhani,B.,2013.Atuto-rialsurveyonwavespropagatinginperiodicmedia:electronic,photonicandphononiccrystals.PerceptionoftheBlochtheoreminbothrealandFourierdo-mains.

Wave

Motion

50

(3),

619654

.

Germain,P.,1973.Themethodofvirtualpowerincontinuummechanics.Part2:Microstructure.SIAMJ.Appl.Math.25(3),556575.

Nassar,H.,He,Q.-C.,Auffray,N.,2015a.Onasymptoticelastodynamichomogeniza-tionapproachesforperiodicmedia.J.Mech.Phys.Solids(Submitted).

Nassar,

H.

,

He,

Q.-C.

,

Auffray,

N.

,

2015b.

Willis

elastodynamic

homogenization

theory

revisitedforperiodicmedia.J.Mech.Phys.Solids77,158178.Nolde,E.,Craster,R.V.,Kaplunov,J.,2011.Highfrequencyhomogenizationforstruc-

turalmechanics.J.Mech.Phys.Solids59(3),651671.Qur,Y.,1988.Physiquedesmatriaux:coursetproblmes.XEcolePolytechnique.

EditionMarketing.Sanchez-Palencia,

E.

,

1980.

Non-homogeneous

media

and

vibration

theory.

Springer-

Verlag.Smyshlyaev,V.P.,2009.Propagationand localizationofelasticwaves inhighly

anisotropicperiodiccompositesviatwo-scalehomogenization.Mech.Mater.41(4),434447.

Smyshlyaev,V.P.,Cherednichenko,K.D.,2000.Onrigorousderivationofstraingra-dienteffectsintheoverallbehaviourofperiodicheterogeneousmedia.J.Me-chanicsPhys.Solids48,13251357.

Srivastava,A.,Nemat-Nasser,S., 2014.Onthelimitandapplicabilityofdynamicho-mogenization.WaveMotion51(7),10451054.

Willis,J.R.,1997.Dynamicsofcomposites.In:Suquet,P.(Ed.),ContinuumMicrome-chanics.Springer-VerlagNewYork,Inc.,pp.265290.

Willis,

J.R.

,

2011.

Effective

constitutive

relations

for

waves

in

composites

and

meta-

materials.Proc.RoyalSoc.A:Math.Phys.Eng.Sci.467(2131),18651879.Willis,J.R.,2012.Theconstructionofeffectiverelationsforwavesinacomposite.

ComptesRendusMcanique340(4-5),181192.
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