electromechanical fields in multi-coated long fiber composites Micromechanical method for effective piezoelectric properties and International Journal of Solids and Structures

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International Journal of Solids and Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

Micromechanical method for effective piezoelectric properties and electromechanical fields in multi-coated long fiber composites

George Chatzigeorgiou

^a

, Ali Javili

^b

, Fodil Meraghni

^a,∗

a Arts et Métiers ParisTech, LEM3-UMR 7239 CNRS, 4 Rue, Augustin Fresnel, Metz 57078, France

b Department of Mechanical Engineering, Bilkent University, Ankara 06800, Turkey

a rt i c l e i n f o

Article history:

Received 17 April 2018 Revised 13 September 2018 Available online 15 September 2018 Keywords:

Piezoelectricity Fiber composites Multi-coated fibers Composite cylinders method

a b s t r a c t

Thispaperproposesamicromechanicalframework foridentifying themacroscopic behavior ofmulti- coatedlongfibercomposites,aswellas theaverageelectromechanicalmicroscopicfieldsofallphases (matrix,fibers,coatinglayers),generateduponknownmacroscopicconditions.Theworkaimsatdevel- opingaunifiedmicromechanicalapproachthatprovidesananalyticalsolutionstandingfornon-coated andmulti-coatedlongfibercompositeswithtransverselyisotropicpiezoelectricbehavior.Theproposed methodsolvesspecificboundaryvalueproblemsandutilizestheMori-Tanakahomogenizationscheme, inwhichthedilutestrainandelectricfieldconcentrationtensorsareobtainedanalyticallywiththehelp ofanextendedcompositecylindersmethodthataccountsforcoupledelectromechanicalfields.Thecapa- bilitiesofthishomogenizationstrategyareillustratedwiththehelpofnumericalexamples,andcompar- isonswithknownsolutionsfromtheliteraturefornon-coatedandcoatedfiberpiezoelectriccomposites areprovided.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Piezoelectricmaterialsareveryattractiveinapplicationsinvolv- ing thedesignofsensors, actuators,transducers,etc. dueto their uniquecapabilitytoconvertelectricalintomechanicalenergy.Us- ing piezoelectric ceramics, like PZT, in bulk form is not always convenient, mainly due to their increased weight. To avoid such issues, an efficient solution is to combine these materials with non-piezoelectric polymersin the formof composites. These ad- vancedcompositematerials openednewhorizonsinthedevelop- mentofnewtransducersandsensorswithhighstrength,lowther- mal expansion coefficients, increased thermal conductivities and decreaseddielectricconstants.

During the last 30 years several models have been pro- posed in the literature to study the piezoelectric and the combined thermo-magneto-electro-elastic response of compos- ites. Dunn andTaya (1993) haveintroduced an Eshelby-type ap- proach byidentifying appropriate Eshelbyandconcentration ten- sors for the combined electromechanical response. This tech- nique later was extended to account for other phenomena, lin- ear (Li and Dunn, 1998) and nonlinear (Hossain et al., 2015), while Zou et al. (2011) identified Eshelby tensors for arbi-

∗ Corresponding author.

E-mail addresses: georges.chatzigeorgiou@ensam.eu (G. Chatzigeorgiou), ajavili@bilkent.edu.tr (A. Javili),fodil.meraghni@ensam.eu (F. Meraghni).

traryshaped piezoelectricinclusions. Benveniste(1994),based on the initial framework of Benveniste and Dvorak (1992), studied the macroscopic response of piezoelectric composites using par- tially the composite cylinders method. Aboudi (2001) developed a computational method for coupled electro-magneto-thermo- elastic composites and Lee et al. (2005) have proposed numeri- calandEshelby-basedanalyticalstrategiesforthreephaseelectro- magneto-elasticcomposites.Piezoelectriccomposites(Bergeretal., 2005;2006;Maruccioetal.,2015),electro-magneto-thermo-elastic composites (Bravo-Castillero et al., 2009) and piezoelastic plates (Kalamkarov andKolpakov,2001) havebeenstudiedby usingthe periodic homogenization theory. A homogenization approach for studying piezoelectric composites with periodic and random mi- crostructure was proposed by Spinelli and Lopez-Pamies (2014). Sharma et al.(2007) identified a theoretical framework that de- scribestheconditionsunderwhichnon-piezoelectricmaterialscan be used for the design ofpiezoelectric nanocomposites.Ray and Batra (2009) developed a micromechanical scheme for study- ingpiezoelectric composites withsquare crosssection fibers,and similar techniquewasdevelopedlater for magneto-electro-elastic withsquare cross section fibers (Pakam and Arockiarajan, 2014).

Koutsawa et al. (2010), using Hill’s interfacial operators, pro- posedaselfconsistentschemeforstudyingthermo-electro-elastic properties of composites with multi-coated ellipsoidal particles.

Wangetal.(2014)developedamicromechanicalmethodforpiezo- electric composites with imperfect interfaces between the ellip- https://doi.org/10.1016/j.ijsolstr.2018.09.018

0020-7683/© 2018 Elsevier Ltd. All rights reserved.

soidalparticlesandthematrixphase,usingtheconceptofequiv- alent particle. Piezoelectric composites with imperfect interfaces havebeenalsostudiedbyGuetal.(2014,2015).

The goal of this work is to develop a unified micromechani- calapproachaimedatprovidingananalyticalsolutionthatstands fornon-coatedandmulti-coatedlongfibercompositeswithtrans- versely isotropic piezoelectric behavior. Studying the coating be- tween a fiberand a matrixin composites is ofvital importance.

Alotofnonlinear deformationmechanisms, likeplasticityand/or martensitetransformation,occurfrequentlyatasmallregionclose tothefibers andlead toan interactionwiththe localdamage of thefiber/matrixinterface(Payandehetal.,2010;2012).Developing appropriate computationaltools that identifythe mechanicaland electricfieldsatthe proximity ofthe fibers canassist in thede- signofmoreaccurate damageandfailurecriteriaforpiezoelectric composites.

The developed approach is based on solving specific bound- ary value problems, extending the composite cylinders model of Hashin and Rosen (1964). This effort can be considered as a generalization of the Dvorak and Benveniste (1992) and Benveniste(1994)methodology,providinganalyticalexpressionsof thedilutestrain-electricfieldcoupledconcentrationtensors,which canbeutilizedinclassicalmicromechanicaltechniques,likeMori- Tanakaorselfconsistent.Theadvantageofsuchinformationisthat it permits to identify not only the overall response of the com- posite,but alsothe variousaverage electromechanicalfields gen- eratedat the matrix,the fiber andthe coating layers for known macroscopic electromechanical conditions. For non-coated fibers theobtaineddilutetensorsareequivalentwiththoseofDunnand Taya(1993).Tothebestoftheauthorsknowledge,theonlyavail- ableframeworkintheliteraturethatcomputesconcentration ten- sorsfor piezoelectriccomposites withcoatedfibers isthe one of Koutsawaetal.(2010),butit isbasedoncertain approximations.

Theapproach discussed hereindoesnot requiresuch approxima- tions, since the solution proposed in this work for the Eshelby’s inhomogeneityproblemisexact.

The organizationofthe manuscriptisasfollows: After thein- troduction,Section 2 startswith a smallrecall on the piezoelec- tricity concepts, and then it describes the Mori-Tanaka type mi- cromechanicalframeworkandthegeneralprocedureforobtaining thedilute concentration tensors forcomposites withtransversely isotropicpiezoelectricmaterial constituents(matrix, fiber,coating layers).Section 3 presents the caseof non-coated fiber compos- itesanddiscussestheconsistencyoftheapproachwithpublished results from the open literature. Section 4 discusses the case of coatedfiberswithonecoatinglayer.Numericalexamplesandcom- parisons with existing finite element calculations and other mi- cromechanicalapproachesaredemonstratedinSection5.Thesec- tionalso includes a studyinwhich, forgiven macroscopicstrain andelectric field and known volume fractions of the phases, all theelectromechanicalfieldsateveryphaseandattheoverallcom- posite are computed (Table 2). The paper finishes with a sec- tion giving some concluding remarks and future developments.

Forthe purpose of the paper’s completeness, two appendices at the end of the article provide the piezoelectric Eshelby tensor of Dunn and Taya (1993) and explain briefly the framework of Koutsawaetal.(2010).

2. Micromechanicalapproachforacoatedfiber/matrix piezoelectriccomposite

2.1.Generalconceptsfrompiezoelectricityandnotations

In alinearpiezoelectric material,theconstitutive lawthat de- scribesthe relationbetweenthe stress tensor

σ

^{,theelectric dis-}

placementvectorD,thestraintensor

ε

^{andtheelectricfieldvector}

Eis writteninthefollowing indicialform(double indicesdenote summation):

σ

i j=C_{i jkl}

ε

kl− ei jmEm, D_i=e_imn

ε

^mn+

κ

i jE_j. Intensorialnotation,theaboverelationsareexpressedas

σ

=C:

ε

− e· E, D=e^T:

ε

+

κ

· E. (1) Inthe above relations Cisthe fourthorderelasticity tensor,

κ

^is

thesecondorderpermittivitymodulitensorandeisthethirdor- der piezoelectric moduli tensor. The strain tensor

ε

^{and electric}

field vectorEare relatedwiththedisplacementvector uandthe electricpotential

φ

respectivelythroughtherelations

ε

=1 2



gradu+[gradu]^T



, E=−grad

φ

. (2)

Moreover,theequilibriumandelectrostaticequationsread

div

σ

^{=0, divD=0. (3)}

Unlessstatedotherwise,theVoigtnotationwillbeadoptedinthe sequel to represent second, third and fourth order tensors. The conventionusedinstrainsandstressestoreplacetheindicesijto asingleindexkisthefollowing:

i j=11 → k=1, i j=22 → k=2, i j=33 → k=3, i j=12 → k=4, i j=13 → k=5, i j=23 → k=6. 2.2. ApplicationoftheMori-Tanakamethod

Let’sconsideraN+1-phasecomposite,consistingofamatrix, denoted as 0,and infinitelylong, multi-coated, cylindrical fibers.

The fibersare denoted withtheindex1, whilethe N− 1 coating layersaredenotedwiththeindices2,3,...,N(Fig.1).Allphasesare assumedtobemadebylinearpiezoelectric materials.Thephases are characterizedby their piezoelectricmoduli C_q,e_q and

κ

q,q= 0,1,2,3,...,N.

Accordingtotheusualmicromechanicsarguments,aRepresen- tative VolumeElement (RVE)withtotal volume V issufficient to describetheoverall responseofthecomposite.Eachphaseinthe RVEoccupiesthespaceq(Fig.2),hasitsownvolumeVqandvol- umefractionc_qforq=0,1,2,3,...,N.Theclassical volumesum- mationrulestatesthat

N

q=0

Vq=V,

N

q=0

cq=

N

q=0

Vq

V =1.

Themacroscopicstrain

ε

¯,stress

σ

¯,electricfield ¯Eandelectricdis- placement ¯DcorrespondingtotheRVEarecomputedfromthevol- umeaveragesoftheirmicroscopiccounterpartsatallphases,i.e.

ε

¯=

N

q=0

cq

ε

q,

σ

¯ =

N

q=0

cq

σ

q, ¯E=

N

q=0

cqEq, ¯D=

N

q=0

cqDq. (4)

Inthese expressions

ε

^q,

σ

^{q,Eq andDq denotetheaverage strain,}

stress,electricfieldandelectricdisplacementintheq_th phasere- spectively:

ε

^q=_V¹

q



q

ε (

^x

)

^dV,

σ

^q=_V¹

q



q

σ (

^x

)

^dV, Eq= 1

Vq



q

E

(

^x

)

^dV, Dq= 1 Vq



q

D

(

^x

)

^dV. (5)

Theconstitutivelawofeachphasestatesthat

σ

^q=Cq:

ε

^{q− eq· Eq, Dq=eT}q:

ε

^q+

κ

^{q· Eq, q=0,1,2,3,...,N. (6)}

Thegoalofhomogenizationistoidentifyasimilartypeofconsti- tutivelawforthemacroscopicquantities,i.e.

σ

¯ ^=¯C:

ε

^{¯− ¯e· ¯E, ¯D=¯eT:}

ε

^¯+

κ

^{¯· ¯E, (7)}

Fig. 1. Unidirectional multi-coated fiber composite. The coating has N − 1 distinct layers.

Fig. 2. Typical RVE of a unidirectional multi-coated fiber composite.

where ¯C,

κ

¯ ^{and ¯eare the}macroscopicelasticity, permittivity and piezoelectrictensorsrespectively.¹Thelaw(7)representstheover- all behavior of the composite. According to the Mori-Tanaka ap- proach,theaveragefieldsinthefiberorthecoatingsandthema- trixphase (q=0) are connectedtoeach other throughthedilute concentrationtensors:

ε

^q=Tmmq :

ε

^0+Tmeq · E0, Eq=T^em_q :

ε

^0+Teeq · E0,

q=1,2,3,...,N. (8)

T^mm_q are fourthorder tensors, written as6× 6 matrices, T^me_q are third ordertensors,writtenas6× 3 matrices,T^em_q arethirdorder tensors, written as 3× 6 matrices andT^ee_q are second orderten- sors,writtenas3× 3matrices.Combining(8)with(4)_1,3andafter somealgebra,thefollowingrelationsareobtained:

ε

^q=Ammq :

ε

¯^+Ameq · ¯E,Eq=A^em_q :

ε

¯^+Aeeq · ¯E,q=0,1,2,3,...,N, (9)

1 Contrarily to the electromechanical fields, the macroscopic moduli are gener- ally not equal to the volume averages of their microscopic counterparts. The Voigt bound, which considers such relation, often provides a poor estimate of the real response.

where

A^mm₀ =





^mm−



^me·





^ee



−1

·



^em



−1

, A^me₀ = −A^mm0 :



^me·





^ee



−1

, A^em₀ = −





^ee



−1

·



^em:Amm0 , A^ee₀ =





^ee



−1

−





^ee



−1

:



^em:Ame0 ,

(10)

and

A^mm_q =T^mm_q :A^mm₀ +T^me_q · A^em0 , A^me_q =T^mm_q :A^me₀ +T^me_q · A^ee0, A^em_q =T^em_q :A^mm₀ +T^ee_q · A^em₀ , A^ee_q =T^em_q :A^me₀ +T^ee_q · A^ee₀, (11) forq=1,2,3,...,N.Intheaboverelations,



^mm =c0I+

N

q=1

cqT^mm_q ,



^me=

N

q=1

cqT^me_q ,



^em=

N

q=1

cqT^em_q ,



^ee=c0I+

N

q=1

cqT^ee_q, (12)

whileIandI,with [I]_{i jkl}= 1

2[

δ

ik

δ

jl+

δ

il

δ

jk], [I]i j=

δ

^{i j,}

δ

^{i j:Kronecker delta,}

denotethesymmetricfourthorderandsecondorderidentityten- sorsrespectively.Substituting(9)in(6)gives

σ

^q=



Cq:A^mm_q −

ε

^{q· Aem}q

 ε

¯⁻



eq· A^ee_q − Cq:A^me_q



¯E, Dq=



e^T_q:A^mm_q +

κ

^{q· Aem}q

 ε

¯⁺



e^T_q:A^me_q +

κ

^{q· Aee}q



¯E, (13)

for q=0,1,2,3,...,N. Implementing these results in (4)_2,4 and comparingwith(7)eventuallyyields

¯C=

N

q=0

cq



Cq:A^mm_q −

ε

^q· A^emq



, ¯e=

N

q=0

cq



eq· A^eeq − Cq:A^me_q



,

¯e^T=

N

q=0

cq



e^T_q:A^mm_q +

κ

^q· A^emq



,

κ

¯=

N

q=0

cq



e^T_q:A^me_q +

κ

^q· A^eeq



.

(14) Of course, the forms of the dilute concentration tensors T^mm_q , T^me_q , T^em_q and T^ee_q should respect the compatibility between the Eqs.(14)2and(14)3.

Itisworth mentioningthatitispossibletoconstructone sin- gle diluteconcentration tensor that combinesthe fourtensors of

Fig. 3. Coated fiber inside a matrix, subjected to linear displacement and electric potential at far distance. All phases are piezoelectric materials.

the expressions (8) (see for instance Dunn andTaya, 1993). The advantageofusingseparatedilutetensorsforeachfield(mechan- icalandelectrical)andthecouplingsarisingfromthemisthatthe schemecanbe extendedmorenaturallytoaccount foradditional mechanisms.Indeed, one can obtain dilute concentration tensors for thermomechanical response (Benveniste et al., 1991; Chatzi- georgiou et al., 2018), or for inelastic response, like in the case of the Transformation Field Analysis approach (Dvorak and Ben- veniste,1992;Dvorak,1992).

2.3.Dilutepiezoelectricconcentrationtensors

Fortheidentificationofthediluteconcentrationtensors,asin- gle linear piezoelectric coatedfiber is assumed to be embedded insidethelinearpiezoelectricmatrix,asshowninFig.3.Thefiber occupiesthespace1withvolumeV₁,itscoatingsthespacesq

withvolumesVq,q=2,3,...,N andthematrixoccupiesthespace

0, whichis extended tofar distancefrom thefiber. The matrix issubjected to linear displacement u₀=

ε

0· x andlinear electric potential

φ

0=−E0· x atfar distancefrom thefiber (r→∞). The interfacebetweeneach phase isconsidered perfect.The interface betweenthe phase q andthe phase q+1 for q=1,...,N− 1 is denotedas

∂

q,whiletheinterfacebetweenthelastcoatinglayer andthematrixisdenotedas

∂

N.

Thevariouselectromechanicalfieldsgeneratedateveryphaseq dependonthespatialposition,i.e.

u^(q)

(

^x

)

,

ε

^(q)

(

^x

)

,

σ

^(q)

(

^x

)

,

φ

^(q)

(

^x

)

,

E^(q)

(

^x

)

^{, D(q)}

(

^x

)

^,

∀

^x∈



^{q. (15)}

Ininfinitelylongfibercomposites,itismoreconvenienttode- scribeallthenecessaryequationsincylindricalcoordinates,byde- scribingtheposition vector interms ofthe radius r, theangle

θ

andthelongitudinal position z (Fig. 4). Fortransversely isotropic piezoelectricmatrix,fiberandcoatings,theconstitutivelaw(6)for

Fig. 4. Cylindrical coordinate system.

eachphaseiswritteninVoigtnotationas

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣ σ

rr^(q)

σ

_θθ^(q)

σ

zz^(q)

σ

r⁽θ^q)

σ

rz^(q)

σ

_θ^(qz⁾

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

K_q^tr+

μ

^trq K_q^tr−

μ

^trq lq 0 0 0 K_q^tr−

μ

^trq K_q^tr+

μ

^trq lq 0 0 0

lq lq nq 0 0 0

0 0 0

μ

^trq 0 0

0 0 0 0

μ

^axq 0

0 0 0 0 0

μ

^axq

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

·

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣ ε

rr^(q)

ε

_θθ^(q)

ε

zz^(q)

2

ε

r^(qθ⁾ 2

ε

rz^(q)

2

ε

_θ^(q_z⁾

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

−

⎡

⎢ ⎢

⎢ ⎢

⎣

0 0 eq31

0 0 eq31

0 0 eq33

0 0 0

eq15 0 0 0 eq15 0

⎤

⎥ ⎥

⎥ ⎥

⎦

^·

⎡

⎢ ⎣

E_r^(q) E_θ^(q) E_z^(q)

⎤

⎥ ⎦

^{, (16)}

⎡

⎢ ⎣

D⁽_r^q) D⁽_θ^q) D⁽_z^q)

⎤

⎥ ⎦

⁼



_{0 0 0 0 e}

q15 0

0 0 0 0 0 eq15

eq31 eq31 eq33 0 0 0



·

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣ ε

rr^(q)

ε

_θθ^(q)

ε

zz^(q)

2

ε

_r^(q_θ⁾

2

ε

rz^(q)

2

ε

_θ^(q_z⁾

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

+

 κ

q11 0 0 0

κ

^q11 0 0 0

κ

^q33



·

⎡

⎢ ⎣

E_r^(q) E_θ^(q) Ez^(q)

⎤

⎥ ⎦

^{. (17)}

The strain tensor and the electricfield vector at each phase are givenbytheexpressions

ε

rr^(q)=

∂

^u(r^q)

∂

^{r ,}

ε

_θθ^(q)=1_r

∂

^u(_θ^q)

∂θ

⁺

u⁽_r^q)

r ,

ε

zz^(q)=

∂

^u(z^q)

∂

^{z ,}

2

ε

rz^(q)=

∂

^u(z^q)

∂

^{r +}

∂

^u(r^q)

∂

^{z ,2}

ε

_θ^(qz⁾=1 r

∂

^u(z^q)

∂θ

⁺

∂

^u(_θ^q)

∂

^{z ,}

2

ε

⁽rθ^q)=

∂

^u(_θ^q)

∂

^{r +}

1 r

∂

^u(r^q)

∂θ

⁻

u⁽_θ^q) r , E_r^(q)=−

∂ φ

^(q)

∂

^{r ,Eθ(q)=−1r}

∂ φ

^(q)

∂θ

^,Ez(q)=−

∂ φ

^(q)

∂

^{z , (18)}

whiletheequilibriumandelectrostaticequationsarewrittenas

∂ σ

rr^(q)

∂

^{r +}

1 r

∂ σ

_r⁽_θ^q)

∂θ

⁺

σ

rr^(q)−

σ

_θθ^(q)

r +

∂ σ

rz^(q)

∂

^{z =0,}

∂ σ

r⁽θ^q)

∂

^{r +1r}

∂ σ

_θθ^(q)

∂θ

⁺

2

σ

r⁽θ^q)

r +

∂ σ

_θ^(qz⁾

∂

^{z =0,}

∂ σ

rz^(q)

∂

^{r +1r}

∂ σ

_θ^(qz⁾

∂θ

⁺

σ

rz^(q)

r +

∂ σ

zz^(q)

∂

^{z =0,}

∂

^D(r^q)

∂

^{r +}

1 r

∂

^D(_θ^q)

∂θ

⁺

D⁽_r^q) r +

∂

^D(z^q)

∂

^{z =0. (19)}

The fiberis considered tohave radiusr=r1 andevery coating q hasexternalradiusrq(Fig.3).Theinterfaceconditionsbetweenthe phaseqandthephaseq+1forq=1,2,3,...,N− 1areexpressed as

u_r^(q)

(

^rq,

θ

^,z

)

^=u(r^q+1)

(

^rq,

θ

^,z

)

^{, u(}_θ^q)

(

^rq,

θ

^,z

)

^=u_θ^(q+1)

(

^rq,

θ

^,z

)

^, u_z^(q)

(

^rq,

θ

,z

)

=u⁽_z^q+1)

(

^rq,

θ

,z

)

,

σ

rr^(q)

(

^rq,

θ

,z

)

=

σ

rr^(q+1)

(

^rq,

θ

,z

)

,

σ

_r⁽_θ^q)

(

^rq,

θ

,z

)

=

σ

_r⁽_θ^q+1)

(

^rq,

θ

,z

)

,

σ

rz^(q)

(

^rq,

θ

,z

)

=

σ

rz^(q+1)

(

^rq,

θ

,z

)

,

φ

^(q)

(

^rq,

θ

^,z

)

=

φ

^(q+1)

(

^rq,

θ

^,z

)

, D⁽_r^q)

(

^rq,

θ

^,z

)

=D_r^(q+1)

(

^rq,

θ

^,z

)

. (20) Additionally,theinterface conditionsbetweentheN_thcoating and thematrixarewrittenas

u_r^(N)

(

^rN,

θ

^,z

)

^=u(r⁰⁾

(

^rN,

θ

^,z

)

^{, u(}_θ^N)

(

^rN,

θ

^,z

)

^=u(_θ⁰⁾

(

^rN,

θ

^,z

)

^, uz^(N)

(

^rN,

θ

,z

)

=u⁽z⁰⁾

(

^rN,

θ

,z

)

,

σ

rr^(N)

(

^rN,

θ

,z

)

=

σ

rr⁽⁰⁾

(

^rN,

θ

,z

)

,

σ

_r⁽_θ^N)

(

^rN,

θ

,z

)

=

σ

_r⁽_θ⁰⁾

(

^rN,

θ

,z

)

,

σ

rz^(N)

(

^rN,

θ

,z

)

=

σ

rz⁽⁰⁾

(

^rN,

θ

,z

)

,

φ

^(N)

(

^rN,

θ

^,z

)

=

φ

⁽⁰⁾

(

^rN,

θ

^,z

)

, D⁽_r^N)

(

^rN,

θ

^,z

)

=D⁽_r⁰⁾

(

^rN,

θ

^,z

)

. (21) Following the Eshelby’smethodology, the dilute concentration tensors provide the relationship between the average strain

ε

^q

andelectricfieldE_qinsidethephaseq(q=1,2,3,...,N),andthe strain

ε

0 andelectricfieldE₀atthefarfield:

ε

q=T^mm_q :

ε

0+T^me_q · E0, Eq=T^em_q :

ε

0+T^ee_q · E0. (22) TheMori-Tanakaapproachassumesthatthestrain

ε

0andtheelec- tric field E₀ applied inthe far field ofthisEshelby-type problem correspond tothe averagestrain

ε

0 andtheaverage electricfield E₀ofthematrixintheRVEofFig.2,whichareusedinEq.(8).Us- ing thedivergencetheorem,oneobtains forthefibertheaverage fields

ε

1= 1 V1



1

ε

^(1)dV= 1 V1



∂1

1

2[u⁽¹⁾n+nu⁽¹⁾]dS, E1= 1

V1



1

E⁽¹⁾dV= 1 V1



∂1

[−

φ

^{(1)]ndS, (23)}

where n is the unit vector ofthe interface

∂

1. Due to theho- mothetic topologyofthecoatedfiber,theunit vectoristhesame atall interfaces.Forall thecoating layers, theaverage strain and electricfieldaredefinedas

ε

^q=_V¹

q



q

ε

^(q)dV=_V¹

q



∂q

1

2[u^(q)_n+n_u^(q)]dS

−1 Vq



∂q−1

1

2[u^(q−1)_n+n_u^(q−1)]dS,Eq= 1 Vq



q

E^(q)d

V= 1 Vq



_

∂q

[−

φ

^(q)]ndS−

∂q−1

[−

φ

^(q−1)]ndS



. (24)

Obtaining the dilute piezoelectric tensors is not an easy task.

For certain, ellipsoidal-type, forms of inclusions without coating, Dunn andTaya (1993)havecomputedtheEshelbytensors forthe combinedstrain/electricfieldsystem(seeAppendixAforinfinitely

longfibers).Inthe presentwork, thediluteconcentrationtensors arecomputeddirectlyforlong coatedfiberpiezoelectric compos- iteswithtransverselyisotropicbehaviorateveryphase.Toachieve such goal, analytical solutions are utilized of similar boundary valueproblemswiththoseofthecompositecylindersmethodpro- posedby Hashin(1990),takingintoaccounttheeffectscausedby the combined presence of mechanical and electric fields. In the puremechanicalproblem,similartechniqueshavebeenutilizedin theliterature to obtain dilute(Benvenisteetal., 1989) and semi- dilute(Chatzigeorgiouetal.,2012)stressconcentrationtensors,as well asdilute strain concentrationtensors (Wangetal., 2016) for coatedfibercomposites.

In cylindricalcoordinates, the surfaceelement ina surface of constantradius r(averticalcylinder) is dsr=rd

θ

^{dz and thesur-}

faceelementinasurfaceofconstantzis dsz=rdrd

θ

^.Foranarbi-

trarytensorQ(r,

θ

^{,z)andacylinderofradiusrqandlength2L,the}

sumofsurfaceintegralswiththegeneralform

F = 1

2L

π

^rq

 _L

−L

_2π

0

Q

(

^rq,

θ

,z

)

^d

θ

^dz

+ 1 2L

π

^r2q

 _2π

0

_rq

0

[Q

(

r,

θ

,L

)

− Q

(

r,

θ

,−L

)

^]rdrd

θ

, (25)

isrequiredforthecomputationsoftheaveragequantities(23)and (24).² Thethree normalvectorsin cylindricalcoordinatesare ex- pressedas

n1=



_cos

θ

sin

θ

0



,n2=



_{− sin}

θ

cos

θ

0



,n3=



₀

0 1



. (26)

The displacements of the phasesare represented in matrix form as

u^(q)=u⁽_r^q)n1+u⁽_θ^q)n2+u⁽_z^q) n3. (27) Anotherimportantpointtobementionedisthatinlongfibercom- positeswith, at most, transverselyisotropic phases(axis of sym- metry:thedirectionoffibers),thedilutestrainconcentrationten- sors presenttransverse isotropy. InVoigt notation, they take the form

T^mm_q =

⎡

⎢ ⎢

⎢ ⎢

⎣

T_q^mm₁₁ T_q^mm₁₁ − Tq^mm44 T_q^mm₁₃ 0 0 0 T_q^mm₁₁ − Tq^mm44 T_q^mm₁₁ T_q^mm₁₃ 0 0 0

0 0 1 0 0 0

0 0 0 T_q^mm₄₄ 0 0

0 0 0 0 T_q^mm₅₅ 0

0 0 0 0 0 T_q^mm₅₅

⎤

⎥ ⎥

⎥ ⎥

⎦

^,

T^me_q =

⎡

⎢ ⎢

⎢ ⎢

⎣

0 0 T_q^me₃₁ 0 0 T_q^me₃₁

0 0 0

0 0 0

T_q^me₁₅ 0 0 0 T_q^me

15 0

⎤

⎥ ⎥

⎥ ⎥

⎦

^,

T^em_q =



_{0 0 0 0 T}em q15 0 0 0 0 0 0 T_q^em₁₅

0 0 0 0 0 0



,

T^ee_q =



_Tee

q11 0 0

0 T_q^ee₁₁ 0

0 0 1



. (28)

To obtain the unknown terms of these tensors, four types of boundaryvalue problemsshould be examined. Inthesequel two casesarestudied:afiber/matrixcompositewithoutcoatinganda fiber/matrixcompositewithonlyonecoatinglayer.

2 For infinitely long cylinder, L → ∞ . To avoid infinite values, the division by vol- ume takes care of L .