Multicoated elliptic fibrous composites of piezoelectric and piezomagnetic phases

15 Pages·2011·0.96 MB·English

InternationalJournalofEngineeringScience49(2011)561–575
ContentslistsavailableatScienceDirect
International Journal of Engineering Science
journal homepage: www.elsevier.com/locate/ijengsci
Multicoated elliptic fibrous composites of piezoelectric
and piezomagnetic phases
Hsin-Yi Kuo
DepartmentofCivilEngineering,NationalChiaoTungUniversity,Hsinchu30010,Taiwan
a r t i c l e i n f o a b s t r a c t
Articlehistory: Atheoreticalframeworkisdevelopedtoinvestigatethemagnetoelectroelasticpotentialin
Received8December2010 amulticoatedellipticfibrouscompositewithpiezoelectricandpiezomagneticphases.We
Receivedinrevisedform28January2011 generalize the classic work of Rayleigh (1892) to obtain the electrostatic potential in
Accepted17February2011
orderedconductivecompositesanditsextensiontoadisorderedsystem(Kuo,2010;Kuo
Availableonline12March2011
&Chen,2008)tothecurrentcoupledmagnetoelectroelasticmulticoatedellipticcompos-
ites.Wecombinethemethodsofcomplexpotentialswithare-expansionformulaeand
Keywords:
the generalized Rayleigh’s formulation to obtain a complete solution of the multi-field
Magnetoelectricity
many-inclusionproblem.Itisshownthatthecoefficientsoffieldexpansionscanbewritten
Piezoelectric
intheformofaninfinitesetoflinearalgebraicequations.Numericalresultsarepresented
Piezomagnetic
Ellipticfibrouscomposite forseveralconfigurations.WeusethismethodtostudyBaTiO3–CoFe2O4compositesand
Multicoated findthat,withappropriate coating,theeffectivemagnetoelectric voltagecoefficientcan
beenhancedwithoneorderofmagnitudecomparedtotheirnon-coatingcounterpart.
(cid:2)2011ElsevierLtd.Allrightsreserved.
1.Introduction
Magnetoelectricmaterials,whichinducethepolarizationbyamagneticfield,orconverselyinducethemagnetizationby
anelectricfield,havebeenthefocusofresearchduetotheirvarietiesofmicrostructuralphenomenaandmacroscopicprop-
erties.Thesemakethempromisingforawiderangeofapplications,suchasfour-statememories,magneticfieldsensors,and
magnetically controlled opto-electric devices (Eerenstein, Mathur, & Scott, 2006; Nan, Bichurin, Dong, Viehland, &
Srinivasan, 2008). The study of magneto–electric coupling can be traced back to 1957 when Landau and Lifshitz (1984)
showedthepossibilityofthecouplingbetweentheelectricandmagneticfieldsinasubstancewithacertainmagneticsym-
metryclass.ThiswassubsequentlyexperimentallyconfirmedinasinglecrystalCr O byAstrov(1960)andbyRadoand
2 3
Folen(1961)over50yearsago.However,thiscouplingisweakinsinglephasematerials,andthushasmotivatedthestudy
of composites of piezoelectric and piezomagnetic media. The basic idea is to couple a piezoelectric and a piezomagnetic
materialusingstrain:anappliedelectricfieldcreatesastraininthepiezoelectricmaterialwhichinturninducesadeforma-
tioninthepiezomagneticmaterial,resultinginamagneticfield.
Anumberofmicromechanicalmodelshencewereproposedtopredicttheeffectivemoduliofmultiferroiccomposites.
Forinstance,Green’sfunctionapproachwasusedbyNan(1994)andHuangandKuo(1997)tostudyafibrouscomposite
consisting of Barium Titanate and Cobalt Ferrite. For such transversely isotropic fibrous composites, Benveniste (1995)
derived exact connections among effective magnetoelectroelastic moduli based on a formalism discovered by Milgrom
andShtrikman(1989).ParticulatecompositeswereinvestigatedbyHarshé,Dougherty,andNewnham(1993)usingacubic
model,whileahomogenizationmicromechanicalmethodwasemployedbyAboudi(2001).Eshelby’sequivalentinclusion
approach and the mean field Mori–Tanaka model have been generalized to multiferroic composites by Li and Dunn
E-mailaddress:[email protected]
0020-7225/$-seefrontmatter(cid:2)2011ElsevierLtd.Allrightsreserved.
doi:10.1016/j.ijengsci.2011.02.002
562 H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575
(1998a,1998b),Huang(1998),Li(2000),WuandHuang(2000)andSrinivasetal.(2006).Atwo-scaleasymptotichomog-
enizationtheorywasadoptedbyCamacho-Montes,Sabina,Bravo-Castillero,Guinovart-Díaz,andRodríguez-Ramos(2009)
onthemagnetoelectriccouplingandcross-propertyconnectionsinatwo-phasemultiferroiccomposite.Foragoodoverview
ofthesubject,thereaderisreferredtothereviewarticlebyNanetal.(2008).
Inaclassicwork,LordRayleighcomputedtheelectrostaticpotentialforaconductingcompositeconsistingofaperiodic
arrayofcylindricalorsphericalinclusions.ThiswasextendedtoarbitraryarrangementsbyKuoandChen(2008)andtoellip-
ticcylindersbyKuo(2010).Theseworksconcernsinglefields.Later,KuoandBhattacharya(submittedforpublication)gen-
eralized this methodology to electrostatic, magnetostatic and mechanical coupled fields. In this paper, we extend this
Rayleigh’sformulationfurthertoamultiferroiccompositeconsistingofellipticcylinders,specificallymulticoatedellipses.
Coatingplaysanimportantroleinhigh-temperaturesystemsandinvariousengineeringapplications.Forinstance,tore-
duce heat or stress concentration along the interface, interphase layers between the inclusions and the matrix are often
introducedtoactasthermalbarrier.Gradedmaterialscanalsobemoredamage-resistantthantheirconventionalhomoge-
neous counterpart (Suresh, 2001). Such interphase layer may have constant properties or spatially varying properties.
Researchintogradedmultiferroicshasprimarilybeenconfinedtobilayerandmultilayerstructures.Amongthem,piezoelec-
tricorpiezomagneticcoefficientsareassumedlinearvariationinthethicknessdirectionbyChenandLee(2003),Petrov,
Srinivasan, and Galkina (2008) and Petrov and Srinivasan (2008), while exponentially graded assumption is adopted by
PanandHan(2005)andWangetal.(2009).Toourknowledge,thesubjectofpiezoelectric/piezomagneticfibrouscomposites
withmulticoatedellipticcylindershasnotbeenexaminedintheliteraturebefore.
Theplanofthisarticleisorganizedasfollows.Firstweconsideracompositemediummadeofpiezoelectricandpiezo-
magneticphasesarrangedinamicrostructureconsistingofparallelellipticcylindersinamatrixinSection2.Thephasesare
transverselyisotropicandunderanti-planeshearwithin-planeelectromagneticfields.Inthissituation,thefieldsaredecou-
pledintheinteriorofeveryphase,andthecouplingbetweenthefieldsoccursonlythroughtheinterfaceconditions(Kuo&
Bhattacharya,submittedforpublication).WeexploitthisinSection2.2toobtainarepresentationofthesolution.Thebasic
ideaistofollowKuo(2010)andexpandeachfieldineachmediuminaseries.WeconsiderperiodicarraysinSection2.3.In
Section3weconsiderthecaseofmulticoatedellipticcylinders.Weshowthata(6(cid:2)6)arrayalonecanmathematicallysim-
ulatetheeffectsofmultiplecoatings.WeobtaineffectivepropertiesinSection4,andsignificantlyshowthatthemacroscopic
propertiesdependsolelyonasingleexpansioncoefficient(amongsttheinfinite).ThismethodologyisillustratedinSection5
usingcompositesofBaTiO andCoFe O .Wechoosethismaterialpairforitspracticalpotentialandalsobecauseitenables
3 2 4
comparisonwithpreviouswork.Weobservethatthecompositemediumhasanon-trivialmagnetoelectriccouplingeven
throughtheindividualcomponentsdonot.Further,weshowthattheMEcoefficientcanbeenhancedwithanorderofmag-
nitudeiftheBaTiO fiberiscoatedwithTerfenol-D.
3
2.Multipleellipticcylinders
2.1.Basicformulations
LetusconsideraninfinitemediumR3 containingNarbitrarilydistributed,parallelandseparatedellipticcylinders.The
domainofthepthellipticcylinderisdenotedV ,p=1,2,...,N,andtheremainingmatrixisdenotedX .Weassumethatthe
p m
cylinders and the matrix are made of distinct phases. Further, we assume that each phase is either piezoelectric or
piezomagnetic with transversely isotropic symmetry (i.e. has 6mm symmetry) about the fiber axes. We introduce a
Cartesian coordinate system positioned at a selected point O of the plane with the x- and y-axes in the plane of the
cross-section and z- along the axes of the cylinders (Fig. 1). The centroids of the pth elliptic cylinders are designated as
O ,withO x andO y axesaredirectedalongthemajorandminoraxesoftheellipse.Eachofellipsehasthemajorand
p p p p p
minorsemi-axis,lðpÞandlðpÞ,andtheinter-focidistanceis2d ,whered2¼lðpÞ2(cid:3)lðpÞ2.Theellipsesarewellseparatedsothat
x y p p x y
anytwoinclusionswillnotgetintouchwitheachother.
Letthecompositebesubjectedtotheanti-planeshearstrain(cid:2)e ; (cid:2)e ;thein-planeelectricfieldsE ; E ;andthemagnetic
zx zy x y
fieldsH ; H atinfinity.Thustheheterogeneousmaterialisinastateofanti-planeshearproblem(Benveniste,1995;Chen,
x y
1993;Kuo&Bhattacharya,submittedforpublication)andcanbedescribedby
u ¼u ¼0; u ¼wðx;yÞ;
x y z
u¼uðx;yÞ; ð2:1Þ
w¼wðx;yÞ;
whereu,u ,u arethemechanicaldisplacementsalongthex-,y-,andz-axes,andu andwaretheelectricandmagnetic
x y z
potentials,respectively.
Theconstitutivelawsoftheconstituentsforthenon-vanishingfieldsbecome
0r 1 0C e q 10 e 1
zj 44 15 15 zj
BD C¼Be (cid:3)j (cid:3)k CB(cid:3)E C; ð2:2Þ
@ j A @ 15 11 11A@ jA
B q (cid:3)k (cid:3)l (cid:3)H
j 15 11 11 j
H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575 563
y
x
O y
p
x
O p
p
Fig.1. Thecross-sectionofthefibercomposite.
wherejdenotesthecomponentx,y.Wecanwritethiscompactlyas
RU ¼L ZV; U;V ¼w;u;w; j¼x;y; ð2:3Þ
j UV j
where
0r 1 0C e q 1 0 e 1
zj 44 15 15 zj
R ¼BD C; L¼Be (cid:3)j (cid:3)k C; Z ¼B(cid:3)E C: ð2:4Þ
j @ j A @ 15 11 11A j @ jA
B q (cid:3)k (cid:3)l (cid:3)H
j 15 11 11 j
Herer ,D,B,e ,E andH arethestress,electricdisplacement,magneticflux,strain,electricfield,andthemagneticfield,
zj j j zj j j
respectively.C ,j ,l andk aretheelasticmodulus,dielectricpermittivity,magneticpermeabilityandmagnetoelectric
44 11 11 11
coefficients.Theshearstrainse ande ,in-planeelectricfieldsE andE ,andin-planemagneticfieldsH andH canbede-
zx zy x y x y
rivedfromthegradientofelasticdisplacement,electricpotential,andmagneticpotentialasfollows:
@w @w
e ¼ ; e ¼ ;
zx @x zy @y
@u @u
E ¼(cid:3) ; E ¼(cid:3) ; ð2:5Þ
x @x y @y
@w @w
H ¼(cid:3) ; H ¼(cid:3) :
x @x y @y
Further,theequilibriumequations,intheabsenceofbodyforce,electricchargedensityandelectriccurrentdensity,are
givenby
@r @r
zxþ zy¼0;
@x @y
@D @D
xþ y¼0; ð2:6Þ
@x @y
@B @B
xþ y¼0:
@x @y
SubstitutionofEq.(2.3)intoEq.(2.6)yields
C r2wþe r2uþq r2w¼0;
44 15 15
e r2w(cid:3)j r2u(cid:3)k r2w¼0; ð2:7Þ
15 11 11
q r2w(cid:3)k r2u(cid:3)l r2w¼0;
15 11 11
564 H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575
wherer2=@2/@x2+@2/@y2representsthetwo-dimensionalLaplaceoperatorforthevariablexandy.SinceLisanonsingular
matrix,genericallywecancompletelydecouple(2.7)intothreeindependentLaplaceequations,
r2w¼0; r2u¼0; r2w¼0 ð2:8Þ
intheinteriorofeachphase.
Inadditiontothesedifferentialequations,wehavetouseinterfaceconditions.Weassumethattheinterfacesareper-
fectlybonded,andthereforethefieldssatisfy
sZtt¼0; s½Rnt¼sðLZÞnt¼0 ð2:9Þ
j j j j j j
wheres(cid:4)tdenotesthejumpinsomequantityacrosstheinterface,t istheunittangenttotheinterfaceandn istheunit
j j
outwardnormaltotheinterface,andtherepeatedindexjdenotessummingoverthecomponentsx,y.SinceLisdifferent
ineachphase,thefieldsw,uandwaregenerallycoupledbytheinterfaceequations.
2.2.Representationofthesolution
Westartbyconsideringthecasethatthecylindersarehomogeneous.Weshowedabovethatthefieldsaredecoupledin
theinteriorofeveryphase,butarecoupledattheinterfaces.Therefore,wemayfollowKuo(2010)anduseaseriesexpansion
for each field in the interior of each phase and then obtain the coefficients by enforcing the interface and boundary
conditions.
Sincew,uandwareharmonic,wecanconstructananalyticfunctionU(z)=U(z)+iU⁄(z),ofthecomplexvariablez=x+iy,
whereU⁄istheconjugateharmonicfunction,relatedtoUbytheCauchy-Riemannequation
@U @U(cid:5) @U @U(cid:5)
¼ ; ¼(cid:3) ; U¼w;u;w: ð2:10Þ
@x @y @y @x
Further,theshapeofthecrosssectionofthecylindersdefinesellipticcoordinates:(l>0,(cid:3)p<h6p)
z¼xþiy¼dcoshx¼dcoshðlþihÞ ð2:11Þ
arethemostappropriatesystemforthesolutionofLaplace’sequation.
Wenowconsiderasituationwherethecompositeissubjectedtoamacroscopicallyuniaxialloading
w ¼(cid:2)e x; u ¼(cid:3)E x; w ¼(cid:3)H x; ð2:12Þ
ext zx ext x ext x
forconstant(cid:2)e ;E andH .Wemayrewritethisinshortas
zx x x
U ¼CUz; ð2:13Þ
ext
whereUrepresentstheappropriatefield–theanti-planedeformationw,electricpotentialu,ormagneticpotentialw–and
CU¼CUþiCU thecorrespondingappliedfield–(cid:2)e ; (cid:3)E or(cid:3)H .
R I zx x x
Werewritethegoverningequation,Eq.(2.8),inellipticcoordinates(l,h),
!
1 @2U @2U
r2U¼ þ ¼0: ð2:14Þ
d2ðcosh2l(cid:3)cos2hÞ @l2 @h2
ThepotentialfieldforthepthellipticcylinderanditssurroundingmatrixcanbeexpandedwithrespecttoitscentroidO as
p
UðipÞðzpÞ¼ X1 CnUðpÞe(cid:3)nxp; CnUðpÞ¼C(cid:3)UðnpÞ ð2:15Þ
n¼(cid:3)1
fortheinclusion,and
UðmpÞðzpÞ¼ X1 AnUðpÞe(cid:3)nxpþX1 BnUðpÞe(cid:3)nxp; AnUðpÞ¼A(cid:3)UnðpÞ ð2:16Þ
n¼(cid:3)1 n¼1
forthematrix.Herex =l +ih isthelocalellipticcoordinatecenteredattheoriginofthepthellipse,thesubscriptsiandm
p p p
denote the inclusion and matrix, respectively. The coefficients AUðpÞ¼AUðpÞþiAUðpÞ; BUðpÞ¼BUðpÞþiBUðpÞ and
n nR nI n nR nI
CUðpÞ¼CUðpÞþiCUðpÞ aresomecomplexunknownstobedetermined.Thesuperscriptspappearingin(2.15)and(2.16)indi-
n nR nI
catethatthefieldsareexpandedwithrespecttothepthellipsecentroid.
Werecalltheinterfaceconditions(2.9)whichwerewriteas
ReUðipÞ(cid:2)(cid:2)(cid:2)@Vp ¼ReUðmpÞ(cid:2)(cid:2)@Vp; ðRUÞðmpÞ(cid:4)np(cid:2)(cid:2)(cid:2)@Vp ¼ðRUÞðipÞ(cid:4)np(cid:2)(cid:2)(cid:2)@Vp ð2:17Þ
where
Rw¼ðr ;r Þ; Ru¼ðD ;D Þ; Rw¼ðB ;B Þ; ð2:18Þ
zx zy x y x y
H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575 565
@V :l =a denotestheinterfacebetweenthematrixandthepthellipticcylinder,andn istheunitoutwardnormalofthe
p p p p
interface@V .
p
Usingtheorthogonalitypropertiesoftrigonometricfunctions,theinterfaceconditions(2.17)provide
aðpÞ¼TðpÞ bðpÞ; cðpÞ¼TðpÞbðpÞ; ð2:19Þ
nK anK nK nK cnK nK
andAUðpÞ¼CUðpÞ;where
0 0
0AwðpÞ1 0BwðpÞ1 0CwðpÞ1
nK nK nK
aðpÞ¼BAuðpÞC; bðpÞ¼BBuðpÞC; cðpÞ¼BCuðpÞC; ð2:20Þ
nK @ nK A nK @ nK A nK @ nK A
AwðpÞ BwðpÞ CwðpÞ
nK nK nK
TðpÞ ¼(cid:3)LðmÞ(cid:3)LðpÞ(cid:4)(cid:3)1(cid:3)coshna LðmÞþsinhna LðpÞ(cid:4) e(cid:3)nap ;
anR p p sinh2na
p
TðapnÞI¼(cid:3)LðmÞ(cid:3)LðpÞ(cid:4)(cid:3)1(cid:3)sinhnapLðmÞþcoshnapLðpÞ(cid:4)si(cid:3)nhe(cid:3)2nnapa ; ð2:21Þ
p
TðpÞ ¼TðpÞ þ e(cid:3)nap I; TðpÞ¼TðpÞ(cid:3) e(cid:3)nap I;
cnR anR 2coshna cnI anI 2sinhna
p p
KrepresentsR,therealpart,orI,theimaginarypartofthecoefficients,andIisthe3(cid:2)3identitytensor.
Wenowneedtorelatethesolutionstotheappliedboundaryconditions.WedosobyapplyingtheGreen’ssecondiden-
tity(Arfken&Weber,2001)tothematrixdomainX .Thisgives
m
Z hGðx;x0Þr02U ðx0Þ(cid:3)U ðx0Þr02Gðx;x0ÞidA0¼Z (cid:5)Gðx;x0Þr0U ðx0Þ(cid:3)U ðx0Þr0Gðx;x0Þ(cid:6)(cid:4)n0ds0; ð2:22Þ
m m m m
Xm @Xm
where the prime 0 denotes the operation in reference to the x0 coordinate, n0 is the outward unit normal to the matrix’s
boundary@X , dA0 representstheareaelementforthex0 coordinate,ds0 isthedifferentialarclength.HereG(x;x0)isthe
m
free-spaceGreen’sfunctionforLaplaceoperatorsatisfyingr2G(x;x0)=(cid:3)d(x(cid:3)x0),whered(x(cid:3)x0)istheDirac-deltafunction.
FollowingtheprocedureinKuo(2010),itcanbeshownthatEq.(2.22)yields
UmðzÞ¼UextðzÞþXN X1 BmUðlÞe(cid:3)nxl: ð2:23Þ
l¼1 n¼1
Thisistheconsistencyequationwhichrelatestheexternalappliedfieldstothelocalpotentialexpansions.Notethatthefield
identity(2.23)iswrittenbasedondifferentcoordinates.
Toproceed,weshifttheoriginoftheexpansions(2.23)toafixedpoint,sayZ ,thecentroidofthepthellipse,byexpand-
p
ingtheterme(cid:3)mxl asKushch,Shmegera,andBuryachenko(2005)
e(cid:3)nxl ¼ X1 glp e(cid:3)mxp; ð2:24Þ
nm
m¼(cid:3)1
with
glp ¼ð(cid:3)1Þmn(cid:7)dl(cid:8)nX1 v(cid:3)ðnþmþ2sÞXs ð(cid:3)1Þs(cid:3)t(cid:7)dp(cid:8)mþ2tM ðd;d Þðnþmþtþs(cid:3)1Þ!; ð2:25Þ
nm d lp ðs(cid:3)tÞ! d nmt l p ðs(cid:3)tÞ!
lp s¼0 t¼0 lp
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
whered (cid:6)d þd ; v (cid:6)Z =d þ ðZ =d Þ2(cid:3)1and
lp l p lp lp lp lp lp
M ðd;d Þ¼Xt ðdl=dpÞ2k : ð2:26Þ
nmt l p k!ðt(cid:3)kÞ!ðkþnÞ!ðmþt(cid:3)kÞ!
k¼0
Introducing(2.24)into(2.23),wehavetheexpansion
Uðmp;ÞnearðzÞ¼CUZpþ X1 (cid:3)BnUðpÞþbnUðpÞ(cid:4)e(cid:3)mxp; ð2:27Þ
n¼(cid:3)1
where
bUðpÞ¼CUdpd þXN X1 BUðlÞglp ð2:28Þ
n 2 n;(cid:7)1 m mn
l–p m¼1
validforthedomainwithinanellipsecenteredinZ withinter-focidistance2d andpassingthepoleoflthellipticcoordi-
p lp
natesystemsclosesttoZ (Kushchetal.,2005).Further,sincezliesinthematrixdomain,Eqs.(2.27)and(2.16)shouldbe
p
identical.Thisprovidesthecondition
X1 AnUðpÞe(cid:3)nxp ¼CUZpþ X1 bnUðpÞe(cid:3)nxp: ð2:29Þ
n¼(cid:3)1 n¼(cid:3)1
566 H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575
Takingtherealpartandtheimaginarypartof(2.29),wefindthetwoconditions
AUðpÞ¼(cid:10)CUReZ (cid:3)CUImZ (cid:11)d þbUðpÞ; ð2:30Þ
nR R p I p n;0 nR
and
AUðpÞ¼(cid:10)CUReZ þCUImZ (cid:11)d þbUðpÞ: ð2:31Þ
nI I p R p n;0 nI
Eqs.(2.30),(2.31)and(2.19) constituteaninfinitesetoflinearalgebraicequations.Uponappropriatetruncationsofthe
1
expansionsterms,wecandeterminetheexpansioncoefficientsAUðpÞ; BUðpÞ; CUðpÞ.Herewemakeonefurtherremark.
n n n
Remark. The essential step of the framework is to establish the generalized Rayleigh’s identities, (2.30) and (2.31). We
observe, however, that the derivation of the identities is independent of inclusions’ expansions. In other words, these
identities can be applicable to inclusions with inhomogeneous constituents provided that the admissible fields in the
inclusionsandthetransitionrelations,similarto(2.19)canbeconstructed.
2.3.Periodicarrays
Theanalysiscarriedoutintheprevioussectionforthearbitrarysystemwithafinitenumberofcylindersmayalsobe
adaptedforthecaseofaperiodicarrayofcylinders.Hereweconcentrateontherectangularlattice,andwesketchtheout-
lineofthederivationfocussingonthedifferencesfromtheprevioussituation.
LetusfirstintroduceaCartesiancoordinatesystem(x,y)positionedatthecentroidOofoneoftheellipsesinarectangular
array,asshowninFig.2.Thesidesoftherectangularcellparalleltothexandycoordinatesare,respectively,denotedbya
andb.Theellipticcylindersareofthesameorientation,ellipticradiusl=aandinter-focidistance2d.UniformintensitiesE
x
andH areappliedalongthepositivexaxis,andananti-planesheardeformation(cid:2)e isappliedoutofthexyplane.Intermsof
x zx
ellipticcoordinates,thegeneralsolutionhastheadmissibleform
U ¼ X1 CUe(cid:3)nx; CU¼CU ð2:32Þ
i n n (cid:3)n
n¼(cid:3)1
forl<a,and
U ¼ X1 AUe(cid:3)nxþX1 BUe(cid:3)nx; AU¼AU ; ð2:33Þ
m n n n (cid:3)n
n¼(cid:3)1 n¼1
forl>a.ThecoefficientsAU; BU;andCUareunknowncomplexconstantstobedeterminedfromtheinterfaceandboundary
n n n
conditions.
Analogousto(2.19),thecontinuityconditionsattheinterfacewillgiveconstraints(2.19)betweenthecoefficients.Next,
imposingtheperiodicityconditionsanalogoustotheboundaryconditionwedidtoderive(2.30)and(2.31),leadtogeneral-
izedRayleigh’sidentities
AU ¼bU ; K¼R;I: ð2:34Þ
nK nK
Fig.2. Aschematicrepresentationofaunitcell.
H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575 567
Herethequantities
bU¼(cid:3)CUdd þX1 BUX1 glo ; ð2:35Þ
n 2 n;(cid:7)1 m mn
m¼1 l–o
X1 glo ¼ð(cid:3)1ÞmnX1 (cid:7)d(cid:8)nþmþ2tM ðd;dÞðnþmþ2t(cid:3)1Þ!S ; ð2:36Þ
mn 2 nmt nþmþ2t
l–o t¼0
whereM ((cid:4))isdefinedin(2.26),and
nmt
S ¼XZ(cid:3)m; ð2:37Þ
m l
l–o
arethelatticesumscharacterizingthegeometryoftheperiodicstructure,andZ isthecentroidofthelthcylinderwhenmea-
l
suredinthecomplexplanecenteredatthecentralpointO.Theindexlrunsoverallcylinders’centersunderlyingtheperiodic
arrayexceptthecentralone.Previousstudies(Rayleigh,1892)havereportedthatthesumS isconditionallyconvergentand
2
itsvaluedependsupontheshapeoftheexteriorboundaryofthearray.AlistofS fordifferentvaluesofa/bcanbefoundin
2
NicroviciandMcPhedran(1996).
Eqs.(2.34)and(2.19) constituteaninfinitesetoflinearalgebraicequations.Uponappropriatetruncationsoftheexpan-
1
siontermsatafiniteorderM,wecandeterminetheexpansioncoefficientsAU; BU,andCU.
n n n
3.Confocallymulticoatedellipticcylinders
Fromthepreviousremark,wenowconsiderthattheinclusionsareconfocallymulticoatedellipticcylinderswiththeout-
er elliptic radius að1Þ; p¼1;2;...;N; where N is the number of inclusions. We denote the matrix as phase 0, with trans-
p
versely isotropic material parameters Cð0Þ; eð0Þ; qð0Þ; jð0Þ; lð0Þ and kð0Þ. The multicoated cylinder consists of a core, with
44 15 15 11 11 11
radius l ¼aðMÞ, surrounded by ðM(cid:3)1Þ layers of coating. The jth layer of the coatings occupies the annulus
p p
VðjÞ:aðjþ1Þ6l 6aðjÞ; j¼1;2;...;M;inwhichV ¼Vð1Þ[Vð2Þ[(cid:4)(cid:4)(cid:4)[VðMÞ.Heretheinnermostcoreissolidsothatwehave
p p p p p p p p
aðMþ1Þ¼0.WeassumethatthematerialpropertiesofjthconstituentlayerofthepthmulticoatedcylinderareCðp;jÞ; eðp;jÞ;
p 44 15
qðp;jÞ; jðp;jÞ; lðp;jÞ andkðp;jÞ.
15 11 11 11
Theadmissiblepotentialsineachconstituentlayerofthemulticoatedinclusioncanbeexpressedas
Uðp;jÞ¼ X1 AUðp;jÞe(cid:3)nxpþX1 BUðp;jÞe(cid:3)nxp; AUðp;jÞ¼AUðp;jÞ; ð3:1Þ
n n n (cid:3)n
n¼(cid:3)1 n¼1
whereAUðp;jÞ¼AUðp;jÞþiAUðp;jÞandBUðp;jÞ¼BUðp;jÞþiBUðp;jÞareunknowncomplexconstantstobedetermined.Notethatthepo-
n nR nI n nR nI
tentialatl?0shouldbefiniteandthuswecanset
BUðp;MÞ¼0: ð3:2Þ
n
We consider that the interfaces are perfectlybonded, the potential and the normal componentof flux are continuous
acrosstheinterfaces,
(cid:2) (cid:2)
ReUðp;j(cid:3)1Þ(cid:2) ¼ReUðp;jÞ(cid:2) ;
(cid:10)RU(cid:11)ðp;j(cid:3)1Þ(cid:2)l(cid:4)pn¼ðajðpÞjÞ(cid:2)(cid:2) ¼ðR(cid:2)UlpÞ¼ðpa;jðpÞjÞ(cid:4)nðjÞ(cid:2)(cid:2) : ð3:3Þ
p (cid:2)lp¼aðpjÞ p (cid:2)lp¼aðpjÞ
Thesecontinuityconditionsleadto
aðp;j(cid:3)1Þ! aðp;jÞ!
nK ¼kðp;jÞ nK ; K¼R;I; j¼1;2;...;M; ð3:4Þ
bðp;j(cid:3)1Þ nK bðp;jÞ
nK nK
where
0 1 0 1
Awðp;jÞ Bwðp;jÞ
nK nK
aðnpK;jÞ¼BB@AnuKðp;jÞCCA; bðnpK;jÞ¼BB@BunKðp;jÞCCA; ð3:5Þ
Awðp;jÞ Bwðp;jÞ
nK nK
kðp;jÞ(cid:6)0@ 2coshnapðjÞI e(cid:3)naðpjÞI 1A(cid:3)10@ 2coshnapðjÞI e(cid:3)naðpjÞI 1A;
nR (cid:3)2sinhnaðpjÞLðj(cid:3)1Þ e(cid:3)naðpjÞLðj(cid:3)1Þ (cid:3)2sinhnaðpjÞLðjÞ e(cid:3)napðjÞLðjÞ
ð3:6Þ
kðp;jÞ(cid:6)0@ (cid:3)2sinhnapðjÞI e(cid:3)naðpjÞI 1A(cid:3)10@(cid:3)2sinhnaðpjÞI e(cid:3)naðpjÞI 1A;
nI 2coshnaðpjÞLðj(cid:3)1Þ e(cid:3)naðpjÞLðj(cid:3)1Þ 2coshnaðpjÞLðjÞ e(cid:3)naðpjÞLðjÞ
568 H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575
andIisthe3(cid:2)3identitymatrix.Now,repeateduseof(3.4)gives
aðp;0Þ! aðp;jÞ!
nK ¼Kðj;nÞ nK ; j¼1;2;...;M; ð3:7Þ
bðp;0Þ pK bðp;jÞ
nK nK
where
Kðj;nÞ(cid:6)kðp;1Þkðp;2Þ(cid:4)(cid:4)(cid:4)kðp;jÞ: ð3:8Þ
pK nK nK nK
Forj¼M,wehave
aðp;0Þ! aðp;MÞ!
nK ¼KðM;nÞ nK : ð3:9Þ
bðp;0Þ pK bðp;MÞ
nK nK
Further,accordingto(3.2),(3.9),impliesthat
aðp;0Þ¼hKðM;nÞi hKðM;nÞi(cid:3)1bðp;0Þ: ð3:10Þ
nK pK pK nK
11 21
Here½KðM;nÞ(cid:8) representstheupper-left(3(cid:2)3)submatrixofKðM;nÞand½KðM;nÞ(cid:8) isthelower-left(3(cid:2)3)submatrixofKðM;nÞ.
pK 11 pK pK 21 pK
Theformulationimpliesthattheeffectofthemultiplecoatingscanbeincorporatedthrougharecurrenceprocedureandis
solelyrepresentedbya(6(cid:2)6)arrayalone.Wementionthatonceweconstructtheadmissiblefield(3.1)andthetransition
relation(3.4)intheinhomogneneousinclusions,wecanfollowtheremaininggeneralizedRayleigh’sframeworkproposedin
previoussectiontodeterminethepotentialdistribution.
4.Effectivemoduli
Weareinterestedintheeffectivebehaviorforasituationwherewehavealargenumberofcylinders.Theeffectivemate-
rialpropertiesaredefinedintermsaveragedfields,
hRi(cid:6)L(cid:5)hZi; nosummation; ð4:1Þ
j j j
wheretheangularbracketsdenotetheareaaveragesovertherepresentativevolumeelement(unitcellinthecaseofperi-
odiccomposites)
1 Z 1 Z
hRi¼ Rdx; hZi¼ Zdx; ð4:2Þ
j V j j V j
V V
andL(cid:5)denotestheeffectivemagnetoelectroelasticparametersofthecomposite.Notethatsincethemicrostructureisnon-
j
symmetric,theresponsesofthecompositealongxandy-axesaredifferent:
0C(cid:5) e(cid:5) q(cid:5) 1 0C(cid:5) e(cid:5) q(cid:5) 1
55 15 15 44 24 24
L(cid:5)¼BBe(cid:5) (cid:3)j(cid:5) (cid:3)k(cid:5) CC; L(cid:5)¼BBe(cid:5) (cid:3)j(cid:5) (cid:3)k(cid:5) CC: ð4:3Þ
x @ 15 11 11A y @ 24 22 22A
q(cid:5) (cid:3)k(cid:5) (cid:3)l(cid:5) q(cid:5) (cid:3)k(cid:5) (cid:3)l(cid:5)
15 11 11 24 22 22
Letthecompositebesubjectedtouniformintensities(cid:2)e ; (cid:3)E ;and(cid:3)H alongthepositivex-axis.Wecancomputethe
zx x x
averageZ bynotingthateachcomponentisagradientandapplyingthedivergencetheorem.Weobtain:
x
hZUi¼CU: ð4:4Þ
x R
Next,tofindhRUi,weagainusethedivergencetheoremandtheequilibriumcondition(includingtheinterfacecondi-
x
tions)toobtain
1 Z 1 Z 1 Z
hRUi¼ RUdx¼ r(cid:4)ðxRUÞdx¼ xðRUÞ (cid:4)nds; ð4:5Þ
x V x V V m
V V @V
whereRUisdefinedin(2.18).Wethenusetheexpansions(2.16)forthefieldstoobtain
1 Z pdBU
xðZUÞ (cid:4)nds¼CU(cid:3) 1R; ð4:6Þ
V m R ab
@V
where
Zw¼ðe ;e Þ; Zu¼(cid:3)ðE ;E Þ; Zw¼(cid:3)ðH ;H Þ: ð4:7Þ
zx zy x y x y
Putting(4.5)and(4.6)together,andrecallingtheconstitutiverelation(2.2)forthematrix,weobtain
H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575 569
Table1
MaterialparametersofBaTiO3,CoFe2O4(Li&Dunn,1998a),andTerfenol-D(Liuetal.,2003;Liuetal.,
2004).
Property BaTiO3 CoFe2O4 Terfenol-D
C44(N/m2) 43(cid:2)109 45.3(cid:2)109 13.6(cid:2)109
e15(C/m2) 11.6 0 0
q15(N/Am) 0 550 108.3
j11(C2/Nm2) 11.2(cid:2)10(cid:3)9 0.08(cid:2)10(cid:3)9 0.05(cid:2)10(cid:3)9
l11(Ns2/C2) 5(cid:2)10(cid:3)6 590(cid:2)10(cid:3)6 5.4(cid:2)10(cid:3)6
k11(Ns/VC) 0 0 0
a d
b e
c f
Fig.3. Potentialcontoursforasquarearraycompositeðf¼0:2; (cid:2)ezx¼0; Ex¼0; Hx¼1C=msÞ(a)–(c)BTOfibersembeddedinaCFOmatrix,(d)–(f)BTO
fiberscoatedTerfenol-DembeddedinaCFOmatrix,(a),(d)Verticaldisplacement(m),(b),(e)Electricpotential(V),(c),(f)Magneticpotential(C/s).
570 H.-Y.Kuo/InternationalJournalofEngineeringScience49(2011)561–575
0hr i1 0C e q 1ðmÞ0 (cid:2)ezx(cid:3)pdaBbw1R 1
zx 44 15 15 B C
B@hDxiCA¼B@e15 (cid:3)j11 (cid:3)k11CA BBB(cid:3)Ex(cid:3)pdaBbu1RCCC: ð4:8Þ
hBxi q15 (cid:3)k11 (cid:3)l11 @(cid:3)Hx(cid:3)pdaBbw1RA
Puttingtogether(4.1)and(4.8)andnotingthatthecoefficientsBU dependlinearlyontheappliedfieldCU;weobtainsetof
1R R
equationsfortheeffectivepropertyL(cid:5).Wecandeterminethisbyapplyingdifferentloadingcombinationsbetween(cid:2)e ; E
x zx x
andH .SimilarlywecandetermineL(cid:5) byapplyingdifferentloadingcombinationsbetween(cid:2)e ; E andH .
x y zy y y
a d
e
b
c f
Fig.4. EffectivemoduliofacompositeofBTOfibersinaCFOmatrixversusfibervolumefractions(a)effectiveelasticmodulus,(b)effectivedielectric
permittivity,(c)effectivemagneticpermeability,(d)effectivepiezoelectricmodulus,(e)effectivepiezomagneticmodulus,(f)effectivemagnetoelectric
coefficient.