Evaluation of Effective Elastic, Piezoelectric, and ...downloads.hindawi.com/journals/jnm/2017/1924651.pdf · ResearchArticle Evaluation of Effective Elastic, Piezoelectric, and Dielectric

Research ArticleEvaluation of Effective Elastic Piezoelectric and DielectricProperties of SU8ZnO Nanocomposite for Vertically IntegratedNanogenerators Using Finite Element Method

NeelamMishra Braj Krishna Randhir Singh and Kaushik Das

School of Minerals Metallurgical and Materials Engineering Indian Institute of Technology BhubaneswarToshali Bhawan Bhubaneswar Odisha 751007 India

Correspondence should be addressed to Kaushik Das kaushikiitbbsacin

Received 24 January 2017 Accepted 13 April 2017 Published 15 May 2017

Academic Editor R Torrecillas

Copyright copy 2017 Neelam Mishra et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nanogenerator is a nanodevice which converts ambient mechanical energy into electrical energy A piezoelectric nanocompositecomposed of vertical arrays of piezoelectric zinc oxide (ZnO) nanowires encapsulated in a compliant polymeric matrix is one ofmost common configurations of a nanogenerator Knowledge of the effective elastic piezoelectric and dielectricmaterial propertiesof the piezoelectric nanocomposite is critical in the design of a nanogenerator In this work the effective material properties of aunidirectional unimodal continuous piezoelectric composite consisting of SU8 photoresist as matrix and vertical array of ZnOnanowires as reinforcement are systematically evaluated using finite elementmethod (FEM)TheFEMsimulationswere carried outon cubic representative volume elements (RVEs) Four different types of arrangements of ZnO nanowires and three sizes of RVEshave been considered The volume fraction of ZnO nanowires is varied from 0 to a maximum of 07 Homogeneous displacementand electric potential are prescribed as boundary conditions The material properties are evaluated as functions of reinforcementvolume fraction The values obtained through FEM simulations are compared with the results obtained via the Eshelby-Mori-Tanaka micromechanics The results demonstrate the significant effects of ZnO arrangement ZnO volume fraction and size ofRVE on the material properties

1 Introduction

Mechanical energy is one of the most ubiquitous and abun-dant among sources of energy Mechanical energy can beharvested from wind flows [1] from ambient vibrations [2]and from the human movements [3] Advanced piezoelectricmaterials especially piezoelectric composites are central inharvestingmechanical energy and converting it into electricalenergy The mechanical strain in these material leads tothe development of electric potential which can be used tocharge a battery in portable electronic devices or be usedfor stand-alone self-powered nanodevices [4] A mechanicalenergy-harvesting nanodevice based on piezoelectric zincoxide (ZnO) nanowires was first demonstrated by Wangand Song in 2006 [5] Since then a new group of energy-harvesting nanosystem known as nanogenerator has been

developed [6ndash9] The most common configuration of ananogenerator involves piezoelectric ZnO nanowire grownvertically on a silicon substrate [10] The vertically grownarray of ZnO nanowires is encapsulated with a polymerwhich may be active or passive [11] Combination of pho-topatternable polymer SU8 with ZnO nanowires is expectedto create a nanocomposite that can be easily integrated withthe microfabrication process flow established for fabricationof microsystems and flexible electronics Design and opti-mization of a nanogenerator with nanocomposites as theactive material require knowledge of the effective elasticpiezoelectric and dielectric properties of the nanocomposite

Prediction of effective properties of a piezoelectric com-posite can be accomplished by numerous approaches SimpleVoigt and Reuss approaches [12] developed for uncoupledelastic composites have been extended to predict upper

HindawiJournal of NanomaterialsVolume 2017 Article ID 1924651 14 pageshttpsdoiorg10115520171924651

2 Journal of Nanomaterials

and lower bounds of effective dielectric and piezoelectricconstants of composites with specific reinforcement arrange-ments [13 14] Bisegna and Luciano have extended Hashin-Shtrikman type of variational approaches [15] to electroelasticproblems and have estimated bounds of effective materialproperties [14 16] Several authors have extended Eshelbyrsquosclassical solution [17] of an ellipsoidal inclusion in a homo-geneous matrix to the field of linear piezoelectricity andhave calculated the effective electroelastic constants [18ndash20]Odegard has proposed a newmodel by generalizing theMori-Tanaka [21 22] and self-consistent [23 24] approaches forpredicting the coupled electromechanical properties of piezo-electric composites [25] Using asymptotic homogenizationmethod Guinovart-Dıaz et al have derived exact expressionsfor binary piezoelectric composites where both componentshave 6mm symmetry for cases where the reinforcement is apiezoelectric ceramic and the polymeric matrix may or maynot be piezoelectrically active [26]

In addition to these analytical and semianalyticalapproaches computational methods based on finite elementmodeling have also been employed to evaluate effectiveproperties of piezoelectric composites The approachesbased on FEM are extensions of the representative volumeelement (RVE) approach commonly employed to solveproblems of linear elasticity in case of composites and foams[27 28] Gaudenzi employed a RVE-based FEM to evaluatethe electromechanical response of a 0ndash3 piezoelectriccomposite [29] Poizat and Sester have evaluated effectivelongitudinal and transverse piezoelectric constants of 0ndash3and 1ndash3 composites with embedded piezoelectric fibres usingFEM [30] Pettermann and Suresh have developed a RVE-based model and predicted the complete set of moduli forpiezoelectric composites with periodic hexagonal and squarearray of continuous fibres [31]The effect of fibre distributioneffect of fibre shape and geometric connectivity and effectof poling on the effective electromechanical propertiesof 1ndash3 piezoelectric composites have been reported byKar-Gupta and Venkatesh [32ndash34] Unit cell models ofunidirectional periodic 1ndash3 piezoelectric composites usingperiodic boundary conditions have also been reported byBerger et al [35 36] and by Medeiros et al [37] Recent yearshave seen an interest in nonlinear finite element modeling oflaminated piezoelectric composites [38 39]

Literature review revealed a gap in the experimentalas well computational study of process-structure-propertyrelationships of nanocomposites for applications in nanogen-erators Previous work on nanogenerators with piezoelectricZnOnanowires embedded in a PMMAmatrix used simplisticReuss and Voigt analyses to predict effective properties of thecomposite assuming both the matrix and reinforcement tobe isotropic [8]This work presents a rigorous and systematicanalysis based on FEMwhich can be applied for piezoelectricnanocomposites with anisotropic constituents as well aswith different geometries of the reinforcement phase aslong as the RVE has three mutually perpendicular planesof mirror symmetry passing through the centroid of theRVE To the best of our knowledge this work representsthe first systematic study of the effect of ZnO nanowirevolume fraction and ZnO nanowire arrangement on the

effective elastic piezoelectric and dielectric properties ofa unidirectional unimodal SU8ZnO nanocomposites Theeffective properties of the SU8ZnO nanocomposite wereobtained by FEMandby an approach based onEshelby-Mori-Tanaka (EMT) micromechanics

The article is arranged in the following fashion First thelinear theory of piezoelectricity is introduced in Section 2Next two approaches of prediction of effective properties ofa piezoelectric composite are discussed Section 21 discussesthe approach based on EMT micromechanics while Sec-tion 22 presents the FEM study on idealised RVE Section 3presents the different types of RVEs considered in this studyThematerial properties of the constituent phases are providedin Section 4 The boundary conditions applied to the RVEsto calculate the effective material properties are presented inSection 5 Finally the results of the finite element analysesand EMTmicromechanics-based approach are presented andcompared in Section 6 The results are discussed in Section 7and final conclusions are drawn in Section 8

2 Theory

The theory of linear piezoelectricity couples the interactionbetween the mechanical (elastic) and electric variables by thefollowing constitutive equations [40]

120590119894119895 = 119862119894119895119896119897120576119896119897 minus 119890119897119894119895119864l119863i = 119890119894119896119897120576119896119897 + 120581119894119896119864119896 (1)

where the indices 119894 119895 119896 and 119897 range from 1 to 3 Here 120576119896119897 arethe components of the elastic strain119864l are the components ofthe electric field 120590119894119895 are the components of the stress tensorand 119863i are the components of the electric displacement Inthis stress-charge form of piezoelectric constitutive equa-tions the elastic strain tensor and the electric field vector arethe independent variables119862119894119895119896119897 are components of the fourth-order elastic stiffness tensor measured in the absence of anapplied electric field 119890119897119894119895 are components of the piezoelectricmodulus tensor measured in the absence of an applied strain120581119894119896 are components of the dielectric modulus measured inthe absence of an applied strain In addition solution ofany problem in the domain of linear piezoelectricity requiresthat the equations of elastic equilibrium and Gaussrsquo law ofelectrostatics be satisfied In the absence of body forces theequation for elastic equilibrium is given by120590119894119895119895 = 0 (2)In (2) and in the following the Einstein summation conven-tion is used The comma signifies partial differentiation withrespect to the coordinate following it The Gauss equation ofelectrostatics in the absence of free charges can be expressedas

119863119894119894 = 0 (3)The components of the strain tensor 120576119894119895 and the componentsof the electric field 119864119894 are related to displacement (119906119894) andelectric potential (120593) by the following equations

120576119894119895 = 12 (119906119894119895 + 119906119895119894) 119864119894 = minus120593119894(4)

Journal of Nanomaterials 3

In modeling of piezoelectric behaviour it is convenient totreat the elastic and the electric variables in a similar fashionThis is accomplished by employing a notation introduced byBarnett and Lothe [41] which involves uppercase indices thatrange from 1 to 4 as well as lowercase indices that rangefrom 1 to 3 According to this notation the stress and electricdisplacement are represented by

Θ119894119869 = 120590119894119895 for 119869 = 1 2 3119863119894 for 119869 = 4 (5)

And the elastic strain and the electric field are represented by

119885119870119897 = 120576119896119897 for119870 = 1 2 3minus120601119897 for119870 = 4 (6)

The electroelastic moduli relating Θ119894119869 and 119885119870119897 can be repre-sented as

Σ119894119869119870119897 =

119862119894119895119896119897 for 119869 = 1 2 3 119870 = 1 2 3119890119897119894119895 for 119869 = 1 2 3 119870 = 4119890119894119896119897 for 119869 = 4 119870 = 1 2 3minus120581119894119897 for 119869 = 4 119870 = 4

(7)

In short-hand notation

Θ119894119869 = Σ119894119869119870119897119885119870119897 (8)

It becomes convenient to represent the constitutive equationsin a generalized Voigt two-index notation (with two indiceseach ranging from 1 to 9) as [19 42] The generalizedVoigt notation involves first representing Θ119894119869 and 119885119870119897 as9 times 1 matrices and transforming the (119894119869119870119897) index of theelectroelastic moduli to a two-index form The followingtransformation of subscripts is as follows

(11) 997888rarr 1(22) 997888rarr 2(33) 997888rarr 3

(23) (32) 997888rarr 4(13) (31) 997888rarr 5(12) (21) 997888rarr 6(14) (41) 997888rarr 7(24) (42) 997888rarr 8(34) (43) 997888rarr 9

(9)

Thus Σ1122 would transform to Σ12 with a value of 11986212 whileΣ3443 would transform to Σ99 with a value of minus12058133 Thereforethe constitutive equations can now be represented as

[120590D] = [C eT

e minus120581][120576

minusE] (10)

The electroelastic modulus matrix (Σ) can now be defined as

Σ = [C eT

e minus120581] (11)

For an isotropic nonpiezoelectric solid (i) there are twononzero independent constants for the stiffness matrix thatis 11986211 and 11986212 (ii) all components of the piezoelectricmodulus matrix of this type of solid are zero and (iii) thereis only one independent constant for the dielectric modulusmatrix The electroelastic modulus for an isotropic passivematrix phase is given by

Σm

=

[[[[[[[[[[[[[[[[[[[[

11986211 11986212 11986212 0 0 0 0 0 011986212 11986211 11986212 0 0 0 0 0 011986212 11986212 11986211 0 0 0 0 0 00 0 0 11986244 0 0 0 0 00 0 0 0 11986244 0 0 0 00 0 0 0 0 11986244 0 0 00 0 0 0 0 0 minus12058111 0 00 0 0 0 0 0 0 minus12058111 00 0 0 0 0 0 0 0 minus12058111

]]]]]]]]]]]]]]]]]]]]

(12)

For a transversely isotropic piezoelectric solid where 1-2plane is the plane of isotropy there are (i) five independentconstants for the stiffness matrix that is 11986211 11986212 11986213 11986233and11986244 (ii) three independent constants for the piezoelectricmodulus that is 11989013 11989015 and 11989033 and (iii) two independentconstants for the dielectric modulus that is 12058111 and 12058133 Theelectroelastic modulus for the reinforcement phase is givenby

Σf

=

[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]

(13)

For a 1ndash3 piezoelectric composite composed of an isotropicnonpiezoelectric matrix and with transversely isotropicpiezoelectric reinforcements the effective values of the com-ponents of the electroelastic matrix can be evaluated using(i) a micromechanics-based semianalytical approach and (ii)finite element method as discussed in detail in the followingsubsections


21 Micromechanics-Based Approach The effective electroe-lastic modulus of a two-phase piezoelectric composite can beexpressed in terms of the electroelastic moduli and the vol-ume fractions of the constituent phases Under homogeneouselastic displacement and electric potential boundary condi-tions generalizing the average strain theorem of elasticity[43] for piezoelectricity the effective electroelastic modulusof a piezoelectric composite is obtained as follows [19]

Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)A (14)

Here the subscripts 119898 and 119891 refer to the matrix andthe reinforcement respectively V119891 is the volume fractionof the reinforcement and A is the strain-potential gradi-ent concentration matrix A relates the average strain andpotential gradient in the reinforcement phase to the appliedhomogeneous displacement and electric potential in thecomposite Estimation of the concentration tensor A canbe achieved through several micromechanical schemes forexample by using the dilute approximation scheme [17] theself-consistent scheme [23] the differential scheme [24] andthe EMT mean field approach [21 22] In this work wehave considered EMT mean field approach which will bediscussed in detail

Consider the case of an ellipsoidal inclusion in an infinitematrix wherein the electroelastic fields of the reinforcementphases do not interact The electromechanical response ofsuch ldquodiluterdquo composites can be obtained using the Eshelbymethod which is also known as the dilute approximationmethod For dilute approximation the concentration tensorAdil is given by

Adil = [I + SΣminus1m (Σf minus Σm)]minus1 (15)

where I is the 9 times 9 identity matrix S is analogous to Eshelbytensor [17] and is represented by a 9 times 9 matrix Similar toEshelby tensor components of S also known as the constrainttensor are functions of the geometry of the reinforcementphase and the electroelastic moduli of the matrix phaseExplicit expressions of S are available in the literature [18 1944] Now for the nondilute approximation when there aremultiple inclusions present in the matrix the electroelasticfields of the inclusions interact Mori-Tanaka approach takesthese interactions into account by modeling the interactingelectroelastic field as a background electroelastic field Themodified concentration factor can be expressed as follows[22]

AMT = Adil [(1 minus V119891) I + V119891Adil]minus1 (16)

Combining (14) and (16) the effective electroelastic modulusof a piezoelectric composite can be estimated as follows

Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)AMT (17)

The inverse of the effective electroelastic modulus matrix isrepresented by F that is

F = Σminus1 (18)

The components of the piezoelectric strain coefficientsmatrixare defined as follows

11988931 = minus1198659111986599 11988933 = minus1198659311986599 11988915 = minus1198657511986577

(19)

22 FEM Approach The first step towards prediction ofeffective properties of a composite using FEM is to select anRVE or a unit cell The RVEmust be small enough comparedto the macroscopic body but it should be large enough tocapture themajor features of themicrostructure [37] AnRVEmay be of several geometries for example cubic hexagonaland spherical A cubic RVE in spite of several deficiencieslike accommodation of low volume fraction of reinforcementis most commonly used due to the relative ease of applyingboundary conditions A cubic RVE can be further reduced insize by taking into account the symmetry in the arrangementof the reinforcement phase(s) The micromechanics-basedmethod discussed in the previous section assumes the stressand strain fields inside the inclusions to be constant whereasfinite element approach predicts a more-realistic picture ofnonuniform stress and strain fields at each nodal point Theevaluation of the effective properties of a composite is thenaccomplished by employing the homogenization methodThis method relates the volume average stress strain electricdisplacement and electric field to the effective propertiesof the composite The composite is thus modeled as ahomogenized medium Using FEM the volume averages canbe calculated as follows

120590119894119895 = 1119881 int120590119894119895119889119881 = 1119881nelsum119899=1

120590(119899)119894119895 119881(119899)

120576119894119895 = 1119881 int 120576119894119895119889119881 = 1119881nelsum119899=1

120576(119899)119894119895 119881(119899)

119863119894 = 1119881 int119863119894119889119881 = 1119881nelsum119899=1

119863(119899)119894 119881(119899)

119864119894 = 1119881 int119864119894119889119881 = 1119881nelsum119899=1

119864(119899)119894 119881(119899)

(20)

In (20) 119881 is the volume of the RVE 120590119894119895 120576119894119895 119863119894 and 119864119894are the volume-averaged values of stress strain electricdisplacement and electric field respectively nel is totalnumber of finite elements in the RVE 119881(119899) is the volume ofthe 119899th element 120590(119899)119894119895 is 119894119895th component of the stress tensorcalculated in the 119899th element and 120576(119899)119894119895 is 119894119895th component ofthe strain tensor calculated in the 119899th element Similarly119863(119899)119894and 119864(119899)119894 are the 119894th component of the electric displacementfield and of the electric field respectively each calculated


Table 1 Characteristics of the RVEs with regard to their size and the density and arrangement of ZnO nanowires

RVE name Arrangement of nanowiresin 119909-119910 plane

Edge length of full cubicRVE (2h)

Edge length of (18)th of thefull RVE (h)

Effective number of nanowiresin full RVE

RVE1 Square array 120 nm 60 nm 1RVE2 Square array 240 nm 120 nm 4RVE3 Face-centred rectangle array 240 nm 120 nm 4RVE4 Face-centred square array 240 nm 120 nm 8RVE5 Face-centred square array 120 nm 60 nm 2RVE6 Quasi-random 800 nm 400 nm 64

in the 119899th element Consider a unidirectional compositewhere thematrix is passive and isotropic and the piezoelectricreinforcement is transversely isotropic The local 3-axis ofthe piezoelectric reinforcement is aligned with the global119911-axis In terms of these average values discussed earlierthe constitutive equations of linear piezoelectricity for thismaterial can be expressed in the matrix form as follows

120590111205902212059033120590231205901312059012119863111986321198633

=

[[[[[[[[[[[[[[[[[[[[[[

119862eff11 119862eff12 119862eff13 0 0 0 0 0 119890eff13

119862eff12 119862eff11 119862eff13 0 0 0 0 0 119890eff13

119862eff13 119862eff13 119862eff33 0 0 0 0 0 119890eff33

0 0 0 119862eff44 0 0 0 119890eff15 0

0 0 0 0 119862eff44 0 119890eff15 0 0

0 0 0 0 0 119862eff66 0 0 0

0 0 0 0 119890eff15 0 minus120581eff11 0 00 0 0 119890eff15 0 0 0 minus120581eff11 0119890eff13 119890eff13 119890eff33 0 0 0 0 0 minus120581eff33

]]]]]]]]]]]]]]]]]]]]]]

120576111205762212057633212057623212057613212057612minus1198641minus1198642minus1198643

(21)

where the superscript eff refers to the effective propertiesof the composite Equation (21) represents the basis onwhich the boundary conditions are applied on the RVE Forhomogeneous applied strain (1205760119894119895) and homogeneous appliedelectric field (1198640119894 ) the boundary conditions applied on thesurfaces of the cubic RVE are of the following form

119906119894 = 1205760119894119895119909119895120593 = minus1198640119894 119909119894

(22)

where 119909119894 refers to the coordinates of the surfaces of the RVE

3 Types of RVE

TheRVEs considered are cubes of fixed dimensions (2ℎtimes2ℎtimes2ℎ)with centroid at the origin and with the boundaries givenby the following

1198781 119909 = minusℎ minusℎ le 119910 le ℎ minusℎ le 119911 le ℎ 119899 = (minus1 0 0)1198782 119909 = +ℎ minusℎ le 119910 le ℎ minusℎ le 119911 le ℎ 119899 = (1 0 0)1198783 119910 = minusℎ minusℎ le 119909 le ℎ minusℎ le 119911 le ℎ 119899 = (0 minus1 0)1198784 119910 = +ℎ minusℎ le 119909 le ℎ minusℎ le 119911 le ℎ 119899 = (0 1 0)1198785 119911 = minusℎ minusℎ le 119909 le ℎ minusℎ le 119910 le ℎ 119899 = (0 0 minus1)1198786 119911 = +ℎ minusℎ le 119909 le ℎ minusℎ le 119910 le ℎ 119899 = (0 0 1)1198781ndash1198786 are the six bounding surfaces of the representativecubic volume element The nanowires are oriented alongthe 119911-direction The RVEs are chosen such that reflectionalsymmetries are present about 119909- 119910- and 119911-axes and henceonly (18)th of the cubes have been considered for finiteelement analysis The boundaries (1198611ndash1198616) of the reducedstructures are as follows

1198611 119909 = 0 0 le 119910 le ℎ 0 le 119911 le ℎ n = (minus1 0 0)1198612 119909 = +ℎ 0 le 119910 le ℎ 0 le 119911 le ℎ n = (1 0 0)1198613 119910 = 0 0 le 119909 le ℎ 0 le 119911 le ℎ n = (0 minus1 0)1198614 119910 = +ℎ 0 le 119909 le ℎ 0 le 119911 le ℎ n = (0 1 0)1198615 119911 = 0 0 le 119909 le ℎ 0 le 119910 le ℎ n = (0 0 minus1)1198616 119911 = +ℎ 0 le 119909 le ℎ 0 le 119910 le ℎ n = (0 0 1)Here n represents the outward unit normal vector to thebounding surface Six types of cubic RVEs are considered inthis study The RVEs differ from one another in terms of sizeof the unit cube and the arrangement of the nanowires on the119909-119910 plane The sizes of the RVEs are chosen such that themaximum diameter of the ZnO nanowires correspondingto the maximum volume fraction for each RVE is in therange of 80ndash110 nm Three different sizes and four differentarrangements are considered with an aim to study the effectof RVE-size as well as the effect of the distribution of the ZnOnanowires on the effective properties of the nanocompositeDetails of these RVEs are provided in Table 1 RVE1 and RVE2both have the identical square array in terms of distributionof the nanowires on the 119909-119910 plane Images of RVE1 andRVE2 are provided in Figures 1(a) and 1(b) respectivelyIn RVE3 the nanowires are distributed in a face-centred


y

z

x

(a)

y

z

x

(b)

y

z

x

(c)

y

z

x

(d)

y

z

x

(e)

y

z

x

(f)

Figure 1 Images of full RVEs showing (a) RVE1 with cube-edge length of 120 nm (b) RVE2 with cube-edge length of 240 nm (c) RVE3with cube-edge length of 240 nm (d) RVE4 with cube-edge length of 240 nm (e) RVE5 with cube-edge length of 120 nm and (f) RVE6 withcube-edge length of 800 nm The lighter regions represent the matrix whereas the darker regions represent the cylindrical ZnO nanowiresThe traces of the mirror planes that is 119909 = 0 119910 = 0 and 119911 = 0 are also shown

rectangular array as shown in Figure 1(c) In RVE4 and inRVE5 the nanowires are distributed in a face-centred squarearray as shown in Figures 1(d) and 1(e) respectively InRVE1ndashRVE5 the maximum reinforcement volume fractionconsidered is 07 since the maximum theoretical volumefraction of cylindrical reinforcements in a cubic volume is0785 In RVE6 as shown in Figure 1(f) the nanowires aredistributed in a quasi-random fashion while maintaining thereflectional symmetries The maximum volume fraction ofthe reinforcement in RVE6 is limited to sim052 to ensure thatthe fibre geometries do not intersect each other

4 Material Properties

The matrix SU8 is a negative photoresist commonly em-ployed in microfabrication and nanocomposites with SU8as matrix have been used to fabricate structural componentsof MEMS sensors [48ndash52] The mechanical properties of thispolymer depend on the processing conditions The materialproperties of SU8 considered in this study assume that thephotoresist has been hard-baked at sim95∘C SU8 is consideredas isotropic in terms of elastic and dielectric properties andit does not exhibit piezoelectricity The reinforcement ZnOis assumed to have a wurtzite structure which is transverselyisotropicThe elastic piezoelectric and dielectric constants ofthe electroelastic moduli of the matrix and the reinforcementare provided in Table 2

Table 2 Material properties of SU8 [45 46] and ZnO [47]

Property Units SU8 ZnO11986211 GPa 502 2097111986212 GPa 142 1211411986213 GPa 142 1053611986233 GPa 502 2111911986244 GPa 180 423711986266 GPa 180 442811989013 Cm2 0 minus056700511989033 Cm2 0 13204411989015 Cm2 0 minus048050812058111 nFm 00354 0075712058133 nFm 00354 00903

5 Boundary Conditions andSolution Procedure

The finite element modeling was performed using COMSOLMultiphysics 50 wherein all the steps from model geometrycreation to meshing solution and postprocessing were per-formed The physics interface ldquopiezoelectric devicesrdquo in thestructural mechanics module of COMSOL Multiphysics wasused For three-dimensionalmodels tetrahedral elements areused for meshing A direct solver with a relative tolerance


Table 3 Boundary conditions to evaluate the effective properties of the composite

Effective property 1198611 1198612 1198613 1198614 1198615 1198616 Formula

119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810

119890eff13 = minus120590111198643119890eff33 = minus120590331198643120581eff33 = 11986331198643

119890eff15 119906 = 0 119906 = 0 V = 0120601 = 0 V = 0120601 = 1198810 119908 = 0 119908 = 0 119890eff15 = minus120590231198642119884eff3 ]eff31

119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23

119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ1205760120601 = 0 119908 = 0120601 = 0 119908 = ℎ12057633120601 = 0119884eff2 = 1205902212057622]eff23 = minus1205763312057622

of 0001 was used The boundary conditions applied onthe six surfaces of the cube are in the form of prescribeddisplacements and prescribed electric potentials Boundaryconditions applied and postprocessing steps for evaluatingthe effective electroelastic constants are presented in Table 3In addition the boundary conditions for evaluating someof the commonly used elastic properties like axial modulus(119884eff2 ) transverse modulus (119884eff

3 ) and Poissonrsquos ratios are alsodescribed In Table 3 119906 V and 119908 are the displacement com-ponents along the 119909- 119910- and 119911-coordinate axes respectively1205760 and1198810 are the applied strain and applied electric potentialrespectively

It is important to note that the finite element model andthe associated boundary conditions presented in this sectioncan only be applied to symmetric RVEs These boundaryconditions cannot be applied to evaluate effective proper-ties of a composite considering nonsymmetric RVEs suchas those with completely random geometries and randomarrangement of reinforcements Another assumption implicit

in this analysis is that the interface between SU8 and ZnO isperfect

6 Results

In this section effective material properties of a SU8 pho-toresist reinforced with vertically aligned ZnO nanowiresare presented as a function of volume fraction of the ZnOnanowires The volume fraction of ZnO nanowires rangesfrom zero to a maximum of 07 The volume fraction ofthe nanowires is increased by increasing the radius of thecylindrical nanowires while keeping the length and numberdensity of the nanowires constant The effective materialproperties are evaluated as a function of ZnO volume fractionfor all six types of RVEs described in Section 3 using the FEMand the analytical EMT micromechanics-based approachCalculated data based on EMT appears in each plot as a solidcurve or line

Figure 2 shows that all components of the stiffness matrixexhibit a monotonic increase with increasing ZnO volume


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)

01 02 03 04 05 06 0700ZnO volume fraction

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)

Figure 2 Coefficients of the effective elastic stiffness matrix of SU8ZnO unidirectional composite as a function of ZnO volume fraction (a)119862eff11 (b) 119862eff

12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)

Figure 3 Coefficients of the effective piezoelectric modulus matrix of SU8ZnO unidirectional composite as a function of ZnO volumefraction (a) 119890eff13 (b) 119890eff33 and (c) 119890eff15

fraction In addition it is observed that the doubling thesize of the RVE does not have any effect on the effectivemechanical properties For example the values obtained foreffective mechanical properties for RVE1 and RVE2 coincideSimilarly the effective mechanical properties obtained forRVE4 and for RVE5 are identical It is also observed thatvalues of coefficients of effective stiffness matrix obtainedfrom the FEMmatch very well with the values obtained fromthe EMT-based approach at low volume fractions and in thecase of 119862eff

33 at all volume fractionsFrom Figure 2(a) it is observed that predictions of 119862eff

11

from EMT approach match best the results of FEM derived

for RVE4 and for RVE5 From Figure 2(b) it can be seen thatthe values of 119862eff

12 predicted by EMT-based approach matchmost closely the values obtained for RVE6 up to volumefraction of sim05 Figure 2(c) shows that 119862eff

13 is less structure-sensitive compared to 119862eff

11 and 119862eff12 EMT-based approach

predicts lower values of 119862eff13 compared to those predicted

by RVE-based finite element approach especially at ZnOvolume fractions higher than sim05 Close match betweenEMT-based approach and FEM approach is obtained for 119862eff

13

for RVE1 REV2 RVE4 andRVE5 Figure 2(d) shows that119862eff33

exhibits the least structure sensitivity among the coefficients


RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)

Figure 4 Coefficients of the effective dielectric modulus matrix of SU8ZnO unidirectional composite as a function of ZnO volume fraction(a) 120581eff11 and (b) 120581eff33

of elastic stiffness matrix The predicted values of all RVEsmatchwith each other andwith the values predicted by EMT-based approach Figures 2(e) and 2(f) show that for both119862eff44 and 119862eff

66 EMT-based approach predicts values whichare greater than those predicted by FEM approach For 119862eff

44 the arrangement of the ZnO nanowires in square array andin face-centred square array produces similar values Valuesof 119862eff44 obtained for RVE6 most closely match the values

obtained from EMT approach For 119862eff66 the values obtained

for RVE4 and for RVE5 are in closest agreement with thevalues predicted by EMT approach

Figure 3 shows the predicted values of the coefficients ofthe piezoelectric modulus matrix 119890eff13 and 119890eff15 decrease while119890eff33 increases monotonically with increasing volume fractionof ZnO nanowires 119890eff33 is least sensitive to the arrangement ofZnO nanowires while 119890eff15 is most sensitive to arrangementof the ZnO nanowires From Figure 3(a) it is observed thatpredictions of 119890eff13 from the EMT-based approachmatchmostclosely the predictions from FEM for RVE4 and RVE5

In case of 119890eff13 the values for RVEs with square arraymatch the predictions of RVEs with face-centred squarearray Figure 3(b) shows that predictions from EMT-basedapproach match the predictions of the values of 119890eff33 fromFEM Figure 3(c) shows that the values of 119890eff15 predicted byEMT-based approach are close to the values predicted forRVE2 and for RVE3

Figure 4 shows that the nonzero independent compo-nents of the effective dielectric modulus matrix namely 120581eff13and 120581eff33 increase with increasing volume fraction of ZnO

nanowires These coefficients are not affected strongly by thearrangement of the nanowires and predictions of the valuesof these coefficients by EMT-based approach match closelythe values predicted by FEM

The components of the effective piezoelectric strain coef-ficient matrix 119889eff31 119889eff33 and 119889eff15 are calculated using (15)(16) and (17) respectively 119889eff31 decreases and 119889eff33 increaseswith increasing volume fraction of ZnO nanowires as shownin Figures 5(a) and 5(b) respectively These components arenot sensitive to the arrangement of the nanowires and alsomatch the predictions based onEMTapproach In contrast asshown in Figure 5(c) 119889eff15 is found to be extremely sensitive tothe arrangement of the ZnO nanowires and shows a complexbehaviour on increasing volume fraction of ZnO where thevalue of 119889eff15 first decreases then reaching a minimum for theZnO volume fraction of 05ndash06 thereafter it finally increasescontinually up to the maximum ZnO volume fraction testedThe values of 119889eff15 predicted by the approach based on EMTmethod best match the predictions for RVE4

Figure 6 shows that both axial and transverse moduliincrease monotonically with increasing volume fraction ofZnO The axial modulus is not sensitive to the arrangementof the ZnO nanowires and the predictions from EMT-basedapproach match the values predicted using the FEM Thetransverse modulus on the other hand is more sensitive tothe arrangement of nanowires and the values obtained usingthe EMT approach most closely match the values predictedfor RVE3

Figure 7(a) shows that Poissonrsquos ratio ]eff23 decreasesrapidly on increasing ZnO volume fraction reaches a min-imum at ZnO volume fraction of sim03ndash04 depending on the


RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)

Figure 5 Components of the effective piezoelectric strain coefficient matrix of SU8ZnO unidirectional composite as a function of ZnOvolume fraction (a) 119889eff13 (b) 119889eff33 and (c) 119889eff15

RVE type and then increases as volume fraction increasesRVE1 and RVE2 show similar values of ]eff23 while RVE3RVE4 and RVE5 match closely with each other and withthe values predicted by an approach based on EMT methodRVE6 with quasi-random arrangement of nanowires gener-ates slightly higher value of ]eff23 compared to other RVEs

Poissonrsquos ratio ]eff31 increases monotonically from a valueof 022 for pristine SU8 to a value of sim030 as the ZnOvolume fraction increases to 07 as shown in Figure 7(b)Thepredicted values of ]eff31 do not depend significantly on thearrangement of the ZnO nanowires EMT-based approachunderestimates ]eff31 for ZnO volume fractions greater than

sim04 Close match between the two approaches is obtainedfor RVE1 RVE2 RVE3 and RVE4

7 Discussions

The trends observed in this work for variation of effectivematerial properties as a function of volume fraction areconsistent with trends observed in other finite element anal-yses [25] andmicromechanics-based analyses [19] for similarpiezoelectric composites All coefficients of the electroelasticmatrix as well as the derived material properties for thecomposite are found to have values that are intermediate


RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)

Figure 6 Effective elastic moduli of SU8ZnO unidirectional composite as a function of ZnO volume fraction (a) axial modulus (119884eff3 ) and

(b) transverse modulus (119884eff2 )

000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)

Figure 7 Effective Poissonrsquos ratio of SU8ZnO unidirectional composite as a function of ZnO volume fraction (a) ]eff23 and (b) ]eff31

between the values of SU8 and ZnO As the volume fractionof ZnO increases the values of the coefficients change andtend to get closer to the value of the reinforcement phaseThischange happens in a monotonic fashion Experimental dataon elastic dielectric and piezoelectric properties SU8ZnOnanowire nanocomposites are not available for comparisonwith the predictions from FEM

The observation that 119890eff15 has a strong dependence on thearrangement of the reinforcement phase is corroborated by

Berger et al [53] and by Moreno et al [42] where significantdifferences in values of 119890eff15 were reported when comparingreinforcement arrangements in square array and in hexagonalarray The reason for this strong dependence of 119890eff15 onarrangement is not understood Analyticalsemianalyticaltechniques for example the one based on EMT microme-chanics presented in this study are insensitive to arrangementof the reinforcement phase A comparison of the results froma semianalytical method with the results obtained from FEM


for different RVEs sheds light on the range of volume frac-tions especially at low concentrations of ZnO where compu-tationally simpler analyticalsemianalytical techniques can beused without sacrificing accuracy

8 Conclusions

Finite element method was used to evaluate effective elasticpiezoelectric and dielectric properties of SU8 photoresistreinforced with vertically aligned ZnO nanowires as a func-tion of volume fraction of ZnO nanowires up to a volumefraction of 07 for all RVEs except the one with quasi-randomarrangement of nanowires where the maximum volumefraction is sim05 Effective material properties like axial andtransverse elastic moduli Poissonrsquos ratios and componentsof the effective piezoelectric strain coefficient matrix werealso derived for the SU8ZnO nanocomposite The valuesobtained from FEM for the effective properties are in closeagreement with values obtained through a semianalyticalEMT method especially at low volume fractions of ZnO

Four different arrangements of ZnO nanowires werestudied The arrangement of the ZnO nanowires leads todifferences in values obtained for transverse effective elas-tic and piezoelectric coefficients However these arrange-ments did not lead to significant differences in the effectiveaxiallongitudinal properties of the composite

For RVEs with square array and with face-centred squarearray of arrangement of ZnO nanowires two sizes of RVEwere studied Doubling the size of the RVE did not have anysignificant effect on the coupled electromechanical propertiesof the piezoelectric composite except for 119890eff15

These results represent the first step towards design ofefficient nanogenerators using polymer nanocompositesThemethodology presented in this work can be extended to stud-ies that take into consideration the complex microstructuraland interfacial features of the nanocomposite

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Financial support from the Department of Science andTechnology-Science and Engineering Research Board (DST-SERB) Government of India under Young Scientist Scheme(Grant no YSS2014000830) and from Seed Grant IITBhubaneswar (Grant no SP062) is gratefully acknowledged

References

[1] S Lee S Bae L Lin et al ldquoEnergy harvesting materials super-flexible nanogenerator for energy harvesting from gentle windand as an active deformation sensorrdquo Advanced FunctionalMaterials vol 23 no 19 pp 2445ndash2449 2013

[2] X D Wang ldquoPiezoelectric nanogenerators-harvesting ambientmechanical energy at the nanometer scalerdquo Nano Energy vol 1no 285 pp 13ndash24 2012

[3] R Yang Y Qin C Li G Zhu and Z L Wang ldquoConvertingbiomechanical energy into electricity by a muscle-movement-driven nanogeneratorrdquoNano Letters vol 9 no 3 pp 1201ndash12052009

[4] G Zhu R Yang S Wang and Z L Wang ldquoFlexible high-output nanogenerator based on lateral ZnO nanowire arrayrdquoNano Letters vol 10 no 8 pp 3151ndash3155 2010

[5] Z L Wang and J Song ldquoPiezoelectric nanogenerators based onzinc oxide nanowire arraysrdquo Science vol 312 no 5771 pp 243ndash246 2006

[6] Z L Wang R Yang J Zhou et al ldquoLateral nanowirenanobeltbased nanogenerators piezotronics and piezo-phototronicsrdquoMaterials Science and Engineering R Reports vol 70 no 3ndash6pp 320ndash329 2010

[7] C-Y Chen T-H Liu Y Zhou et al ldquoElectricity generationbased on vertically aligned PbZr02Ti08O3 nanowire arraysrdquoNano Energy vol 1 no 3 pp 424ndash428 2012

[8] R Hinchet S Lee G Ardila L Montes M Mouis and ZL Wang ldquoDesign and guideline rules for the performanceimprovement of vertically integrated nanogeneratorrdquo in Pro-ceedings of PowerMEMS pp 38ndash41 Atlanta Ga USA 2012

[9] Y-F Lin J Song Y Ding S-Y Lu and Z L Wang ldquoPiezo-electric nanogenerator using CdS nanowiresrdquo Applied PhysicsLetters vol 92 Article ID 022105 pp 2ndash5 2008

[10] A Yu H Li H Tang T Liu P Jiang and Z L WangldquoVertically integrated nanogenerator based on ZnO nanowirearraysrdquo Physica Status SolidimdashRapid Research Letters vol 5 no4 pp 162ndash164 2011

[11] C Oshman C Opoku A S Dahiya D Alquier N Camaraand G Poulin-Vittrant ldquoMeasurement of Spurious Voltages inZnO Piezoelectric Nanogeneratorsrdquo Journal of Microelectrome-chanical Systems vol 25 no 3 Article ID 7437376 pp 533ndash5412016

[12] R F Gibson Principles of Composite Material Mechanics CRCPress Boca Raton Fla USA Third edition 2012

[13] H Banno ldquoRecent Developments of piezoelectric ceramicproducts and composites of synthetic rubber and piezoelectricceramicparticlesrdquo Ferroelectrics vol 50 no 1 pp 3ndash12 1983

[14] P Bisegna and R Luciano ldquoVariational bounds for the overallproperties of piezoelectric compositesrdquo Journal of theMechanicsand Physics of Solids vol 44 no 4 pp 583ndash602 1996

[15] Z Hashin and S Shtrikman ldquoA variational approach to thetheory of the elastic behaviour of multiphase materialsrdquo Journalof the Mechanics and Physics of Solids vol 11 pp 127ndash140 1963

[16] P Bisegna and R Luciano ldquoOn methods for bounding theoverall properties of periodic piezoelectric fibrous compositesrdquoJournal of the Mechanics and Physics of Solids vol 45 no 8 pp1329ndash1356 1997

[17] J D Eshelby ldquoThe determination of the elastic field of anellipsoidal inclusion and related problemsrdquo Proc R Soc A MathPhys Eng Sci vol 241 pp 376ndash396 1957

[18] M L Dunn andM Taya ldquoAn analysis of piezoelectric compos-itematerials containing ellipsoidal inhomogeneitiesrdquo Proc R SocA Math Phys Eng Sci vol 443 no 1918 pp 265ndash287 1993

[19] M L Dunn and M Taya ldquoMicromechanics predictions ofthe effective electroelastic moduli of piezoelectric compositesrdquoInternational Journal of Solids and Structures vol 30 no 2 pp161ndash175 1993

[20] Y Benveniste ldquoThe determination of the elastic and electricfields in a piezoelectric inhomogeneityrdquo Journal of AppliedPhysics vol 72 no 3 pp 1086ndash1095 1992


[21] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973

[22] Y Benveniste ldquoA new approach to the application of Mori-Tanakarsquos theory in composite materialsrdquoMechanics of Materialsvol 6 no 2 pp 147ndash157 1987

[23] B Budiansky ldquoOn the elastic moduli of some heterogeneousmaterialsrdquo Journal of the Mechanics and Physics of Solids vol13 no 4 pp 223ndash227 1965

[24] RMclaughlin ldquoA study of the differential scheme for compositematerialsrdquo International Journal of Engineering Science vol 15no 4 pp 237ndash244 1977

[25] G M Odegard ldquoConstitutive modeling of piezoelectric poly-mer compositesrdquo Acta Materialia vol 52 no 18 pp 5315ndash53302004

[26] R Guinovart-Dıaz J Bravo-Castillero R Rodrıguez-RamosF J Sabina and R Martınez-Rosado ldquoOverall properties ofpiezocomposite materials 1ndash3rdquo Materials Letters vol 48 no 2pp 93ndash98 2001

[27] R Singh P D Lee T C Lindley et al ldquoCharacterization of thedeformation behavior of intermediate porosity interconnectedTi foams using micro-computed tomography and direct finiteelement modelingrdquo Acta Biomaterialia vol 6 no 6 pp 2342ndash2351 2010

[28] C T Sun and R S Vaidya ldquoPrediction of composite propertiesfrom a representative volume elementrdquo Composites Science andTechnology vol 56 no 2 pp 171ndash179 1996

[29] P Gaudenzi ldquoOn the electromechanical response of activecomposite materials with piezoelectric inclusionsrdquo Computersand Structures vol 65 no 2 pp 157ndash168 1997

[30] C Poizat and M Sester ldquoEffective properties of compositeswith embedded piezoelectric fibresrdquo Computational MaterialsScience vol 16 no 1-4 pp 89ndash97 1999

[31] H E Pettermann and S Suresh ldquoA comprehensive unit cellmodel a study of coupled effects in piezoelectric 1ndash3 compos-itesrdquo International Journal of Solids and Structures vol 37 no39 pp 5447ndash5464 2000

[32] R Kar-Gupta and T A Venkatesh ldquoElectromechanical re-sponse of 1ndash3 piezoelectric composites a numerical model toassess the effects of fiber distributionrdquo Acta Materialia vol 55no 4 pp 1275ndash1292 2007

[33] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of piezoelectric composites Effects of geometric con-nectivity and grain sizerdquo Acta Materialia vol 56 no 15 pp3810ndash3823 2008

[34] R Kar-Gupta and T A Venkatesh ldquoElectromechanicalresponse of 1ndash3 piezoelectric composites effect of polingcharacteristicsrdquo Journal of Applied Physics vol 98 no 5 ArticleID 54102 2005

[35] H Berger S Kari U Gabbert et al ldquoUnit cell models ofpiezoelectric fiber composites for numerical and analytical cal-culation of effective propertiesrdquo SmartMaterials and Structuresvol 15 no 2 pp 451ndash458 2006

[36] H Berger S Kari U Gabbert R Rodriguez-Ramos J Bravo-Castillero and R Guinovart-Diaz ldquoCalculation of effectivecoefficients for piezoelectric fiber composites based on a generalnumerical homogenization techniquerdquo Composite Structuresvol 71 no 3-4 pp 397ndash400 2005

[37] R d Medeiros M E Moreno F D Marques and V TitaldquoEffective properties evaluation for smart composite materialsrdquoJournal of the Brazilian Society of Mechanical Sciences andEngineering vol 34 no spe pp 362ndash370 2012

[38] M N Rao S Tarun R Schmidt and K-U Schroder ldquoFiniteelement modeling and analysis of piezo-integrated compositestructures under large applied electric fieldsrdquo Smart Materialsand Structures vol 25 no 5 Article ID 055044 2016

[39] M C Ray and A K Pradhan ldquoThe performance of verticallyreinforced 1ndash3 piezoelectric composites in active damping ofsmart structuresrdquo Smart Materials and Structures vol 15 no 2pp 631ndash641 2006

[40] T Ikeda Fundamentals of Piezolectricity Oxford UniversityPress Oxford UK 1990

[41] D M Barnett and J Lothe ldquoDislocations and line charges inanisotropic piezoelectric insulatorsrdquo Physica Status Solidi B vol67 no 1 pp 105ndash111 1975

[42] M E Moreno V Tita and F D Marques ldquoInfluence ofboundary conditions on the determination of effective materialproperties for active fiber compositesrdquo in Proceedings of the 11thPan-American Congress of Applied Mechanics Foz do IguacuPR Brazil 2010

[43] J Aboudi Mechanics of Composite Materials A UnifiedMicromechanical Approach Elsevier Science Publishers Ams-terdam The Netherlands 1991

[44] M L Dunn ldquoElectroelastic greenrsquos functions for transverselyisotropic piezoelectric media and their application to the solu-tion of inclusion and inhomogeneity problemsrdquo InternationalJournal of Engineering Science vol 32 no 1 pp 119ndash131 1994

[45] F Chollet ldquoSU-8 thick photo-resist for MEMSrdquo 2013 httpmemscyclopediaorgsu8html

[46] V Wang Photoacoustic Imaging and Spectroscopy CRC PressBoca Raton Fla USA First edition 2009

[47] C Jagadish and S J Pearton Eds Zinc Oxide Bulk Thin Filmsand Nanostructures Processing Properties and ApplicationsElsevier Hong Kong First edition 2006

[48] P RayV Seena RAKhare A R Bhattacharyya P RApte andR Rao ldquoSU8modified MWNT composite for piezoresistivesensor applicationrdquoMaterials Research Society Proceedings vol1299 pp 135ndash140 2011

[49] V Seena A Fernandes P Pant S Mukherji and V RamgopalRao ldquoPolymer nanocomposite nanomechanical cantilever sen-sors material characterization device development and appli-cation in explosive vapour detectionrdquo Nanotechnology vol 22no 29 Article ID 295501 2011

[50] K Prashanthi M Naresh V Seena T Thundat and V Ram-gopal Rao ldquoA novel photoplastic piezoelectric nanocompositefor MEMS applicationsrdquo Journal of MicroelectromechanicalSystems vol 21 no 2 Article ID 6118300 pp 259ndash261 2012

[51] K Prashanthi N Miriyala R D Gaikwad W Moussa V RRao and T Thundat ldquoVibtrational energy harvesting usingphoto-patternable piezoelectric nanocomposite cantileversrdquoNano Energy vol 2 no 5 pp 923ndash932 2013

[52] M Kandpal C Sharan P Poddar K Prashanthi P R Apteand V Ramgopal Rao ldquoPhotopatternable nano-composite (SU-8ZnO) thin films for piezo-electric applicationsrdquo AppliedPhysics Letters vol 101 no 10 Article ID 104102 2012

[53] H Berger S Kari U Gabbert et al ldquoAn analytical and numer-ical approach for calculating effective material coefficients ofpiezoelectric fiber compositesrdquo International Journal of Solidsand Structures vol 42 no 21-22 pp 5692ndash5714 2005

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of



and lower bounds of effective dielectric and piezoelectricconstants of composites with specific reinforcement arrange-ments [13 14] Bisegna and Luciano have extended Hashin-Shtrikman type of variational approaches [15] to electroelasticproblems and have estimated bounds of effective materialproperties [14 16] Several authors have extended Eshelbyrsquosclassical solution [17] of an ellipsoidal inclusion in a homo-geneous matrix to the field of linear piezoelectricity andhave calculated the effective electroelastic constants [18ndash20]Odegard has proposed a newmodel by generalizing theMori-Tanaka [21 22] and self-consistent [23 24] approaches forpredicting the coupled electromechanical properties of piezo-electric composites [25] Using asymptotic homogenizationmethod Guinovart-Dıaz et al have derived exact expressionsfor binary piezoelectric composites where both componentshave 6mm symmetry for cases where the reinforcement is apiezoelectric ceramic and the polymeric matrix may or maynot be piezoelectrically active [26]

In addition to these analytical and semianalyticalapproaches computational methods based on finite elementmodeling have also been employed to evaluate effectiveproperties of piezoelectric composites The approachesbased on FEM are extensions of the representative volumeelement (RVE) approach commonly employed to solveproblems of linear elasticity in case of composites and foams[27 28] Gaudenzi employed a RVE-based FEM to evaluatethe electromechanical response of a 0ndash3 piezoelectriccomposite [29] Poizat and Sester have evaluated effectivelongitudinal and transverse piezoelectric constants of 0ndash3and 1ndash3 composites with embedded piezoelectric fibres usingFEM [30] Pettermann and Suresh have developed a RVE-based model and predicted the complete set of moduli forpiezoelectric composites with periodic hexagonal and squarearray of continuous fibres [31]The effect of fibre distributioneffect of fibre shape and geometric connectivity and effectof poling on the effective electromechanical propertiesof 1ndash3 piezoelectric composites have been reported byKar-Gupta and Venkatesh [32ndash34] Unit cell models ofunidirectional periodic 1ndash3 piezoelectric composites usingperiodic boundary conditions have also been reported byBerger et al [35 36] and by Medeiros et al [37] Recent yearshave seen an interest in nonlinear finite element modeling oflaminated piezoelectric composites [38 39]

Literature review revealed a gap in the experimentalas well computational study of process-structure-propertyrelationships of nanocomposites for applications in nanogen-erators Previous work on nanogenerators with piezoelectricZnOnanowires embedded in a PMMAmatrix used simplisticReuss and Voigt analyses to predict effective properties of thecomposite assuming both the matrix and reinforcement tobe isotropic [8]This work presents a rigorous and systematicanalysis based on FEMwhich can be applied for piezoelectricnanocomposites with anisotropic constituents as well aswith different geometries of the reinforcement phase aslong as the RVE has three mutually perpendicular planesof mirror symmetry passing through the centroid of theRVE To the best of our knowledge this work representsthe first systematic study of the effect of ZnO nanowirevolume fraction and ZnO nanowire arrangement on the

effective elastic piezoelectric and dielectric properties ofa unidirectional unimodal SU8ZnO nanocomposites Theeffective properties of the SU8ZnO nanocomposite wereobtained by FEMandby an approach based onEshelby-Mori-Tanaka (EMT) micromechanics

The article is arranged in the following fashion First thelinear theory of piezoelectricity is introduced in Section 2Next two approaches of prediction of effective properties ofa piezoelectric composite are discussed Section 21 discussesthe approach based on EMT micromechanics while Sec-tion 22 presents the FEM study on idealised RVE Section 3presents the different types of RVEs considered in this studyThematerial properties of the constituent phases are providedin Section 4 The boundary conditions applied to the RVEsto calculate the effective material properties are presented inSection 5 Finally the results of the finite element analysesand EMTmicromechanics-based approach are presented andcompared in Section 6 The results are discussed in Section 7and final conclusions are drawn in Section 8

2 Theory

The theory of linear piezoelectricity couples the interactionbetween the mechanical (elastic) and electric variables by thefollowing constitutive equations [40]

120590119894119895 = 119862119894119895119896119897120576119896119897 minus 119890119897119894119895119864l119863i = 119890119894119896119897120576119896119897 + 120581119894119896119864119896 (1)

where the indices 119894 119895 119896 and 119897 range from 1 to 3 Here 120576119896119897 arethe components of the elastic strain119864l are the components ofthe electric field 120590119894119895 are the components of the stress tensorand 119863i are the components of the electric displacement Inthis stress-charge form of piezoelectric constitutive equa-tions the elastic strain tensor and the electric field vector arethe independent variables119862119894119895119896119897 are components of the fourth-order elastic stiffness tensor measured in the absence of anapplied electric field 119890119897119894119895 are components of the piezoelectricmodulus tensor measured in the absence of an applied strain120581119894119896 are components of the dielectric modulus measured inthe absence of an applied strain In addition solution ofany problem in the domain of linear piezoelectricity requiresthat the equations of elastic equilibrium and Gaussrsquo law ofelectrostatics be satisfied In the absence of body forces theequation for elastic equilibrium is given by120590119894119895119895 = 0 (2)In (2) and in the following the Einstein summation conven-tion is used The comma signifies partial differentiation withrespect to the coordinate following it The Gauss equation ofelectrostatics in the absence of free charges can be expressedas

119863119894119894 = 0 (3)The components of the strain tensor 120576119894119895 and the componentsof the electric field 119864119894 are related to displacement (119906119894) andelectric potential (120593) by the following equations

120576119894119895 = 12 (119906119894119895 + 119906119895119894) 119864119894 = minus120593119894(4)



Θ119894119869 = 120590119894119895 for 119869 = 1 2 3119863119894 for 119869 = 4 (5)


119885119870119897 = 120576119896119897 for119870 = 1 2 3minus120601119897 for119870 = 4 (6)


Σ119894119869119870119897 =


(7)


Θ119894119869 = Σ119894119869119870119897119885119870119897 (8)


(11) 997888rarr 1(22) 997888rarr 2(33) 997888rarr 3


(9)


[120590D] = [C eT

e minus120581][120576

minusE] (10)


Σ = [C eT

e minus120581] (11)


Σm

=

[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]

(12)


Σf

=

[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]

(13)




Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)A (14)







Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)AMT (17)


F = Σminus1 (18)



(19)


120590119894119895 = 1119881 int120590119894119895119889119881 = 1119881nelsum119899=1

120590(119899)119894119895 119881(119899)

120576119894119895 = 1119881 int 120576119894119895119889119881 = 1119881nelsum119899=1

120576(119899)119894119895 119881(119899)

119863119894 = 1119881 int119863119894119889119881 = 1119881nelsum119899=1

119863(119899)119894 119881(119899)

119864119894 = 1119881 int119864119894119889119881 = 1119881nelsum119899=1

119864(119899)119894 119881(119899)

(20)










120590111205902212059033120590231205901312059012119863111986321198633

=

[[[[[[[[[[[[[[[[[[[[[[

119862eff11 119862eff12 119862eff13 0 0 0 0 0 119890eff13

119862eff12 119862eff11 119862eff13 0 0 0 0 0 119890eff13

119862eff13 119862eff13 119862eff33 0 0 0 0 0 119890eff33

0 0 0 119862eff44 0 0 0 119890eff15 0

0 0 0 0 119862eff44 0 119890eff15 0 0

0 0 0 0 0 119862eff66 0 0 0


]]]]]]]]]]]]]]]]]]]]]]


(21)


119906119894 = 1205760119894119895119909119895120593 = minus1198640119894 119909119894

(22)


3 Types of RVE





y

z

x

(a)

y

z

x

(b)

y

z

x

(c)

y

z

x

(d)

y

z

x

(e)

y

z

x

(f)












119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810



119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23






6 Results




RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




Θ119894119869 = 120590119894119895 for 119869 = 1 2 3119863119894 for 119869 = 4 (5)


119885119870119897 = 120576119896119897 for119870 = 1 2 3minus120601119897 for119870 = 4 (6)


Σ119894119869119870119897 =


(7)


Θ119894119869 = Σ119894119869119870119897119885119870119897 (8)


(11) 997888rarr 1(22) 997888rarr 2(33) 997888rarr 3


(9)


[120590D] = [C eT

e minus120581][120576

minusE] (10)


Σ = [C eT

e minus120581] (11)


Σm

=

[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]

(12)


Σf

=

[[[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]]]

(13)




Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)A (14)







Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)AMT (17)


F = Σminus1 (18)



(19)


120590119894119895 = 1119881 int120590119894119895119889119881 = 1119881nelsum119899=1

120590(119899)119894119895 119881(119899)

120576119894119895 = 1119881 int 120576119894119895119889119881 = 1119881nelsum119899=1

120576(119899)119894119895 119881(119899)

119863119894 = 1119881 int119863119894119889119881 = 1119881nelsum119899=1

119863(119899)119894 119881(119899)

119864119894 = 1119881 int119864119894119889119881 = 1119881nelsum119899=1

119864(119899)119894 119881(119899)

(20)










120590111205902212059033120590231205901312059012119863111986321198633

=

[[[[[[[[[[[[[[[[[[[[[[

119862eff11 119862eff12 119862eff13 0 0 0 0 0 119890eff13

119862eff12 119862eff11 119862eff13 0 0 0 0 0 119890eff13

119862eff13 119862eff13 119862eff33 0 0 0 0 0 119890eff33

0 0 0 119862eff44 0 0 0 119890eff15 0

0 0 0 0 119862eff44 0 119890eff15 0 0

0 0 0 0 0 119862eff66 0 0 0


]]]]]]]]]]]]]]]]]]]]]]


(21)


119906119894 = 1205760119894119895119909119895120593 = minus1198640119894 119909119894

(22)


3 Types of RVE





y

z

x

(a)

y

z

x

(b)

y

z

x

(c)

y

z

x

(d)

y

z

x

(e)

y

z

x

(f)












119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810



119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23






6 Results




RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)A (14)







Σ = Σ119898 + V119891 (Σ119891 minus Σ119898)AMT (17)


F = Σminus1 (18)



(19)


120590119894119895 = 1119881 int120590119894119895119889119881 = 1119881nelsum119899=1

120590(119899)119894119895 119881(119899)

120576119894119895 = 1119881 int 120576119894119895119889119881 = 1119881nelsum119899=1

120576(119899)119894119895 119881(119899)

119863119894 = 1119881 int119863119894119889119881 = 1119881nelsum119899=1

119863(119899)119894 119881(119899)

119864119894 = 1119881 int119864119894119889119881 = 1119881nelsum119899=1

119864(119899)119894 119881(119899)

(20)










120590111205902212059033120590231205901312059012119863111986321198633

=

[[[[[[[[[[[[[[[[[[[[[[

119862eff11 119862eff12 119862eff13 0 0 0 0 0 119890eff13

119862eff12 119862eff11 119862eff13 0 0 0 0 0 119890eff13

119862eff13 119862eff13 119862eff33 0 0 0 0 0 119890eff33

0 0 0 119862eff44 0 0 0 119890eff15 0

0 0 0 0 119862eff44 0 119890eff15 0 0

0 0 0 0 0 119862eff66 0 0 0


]]]]]]]]]]]]]]]]]]]]]]


(21)


119906119894 = 1205760119894119895119909119895120593 = minus1198640119894 119909119894

(22)


3 Types of RVE





y

z

x

(a)

y

z

x

(b)

y

z

x

(c)

y

z

x

(d)

y

z

x

(e)

y

z

x

(f)












119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810



119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23






6 Results




RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of










120590111205902212059033120590231205901312059012119863111986321198633

=

[[[[[[[[[[[[[[[[[[[[[[

119862eff11 119862eff12 119862eff13 0 0 0 0 0 119890eff13

119862eff12 119862eff11 119862eff13 0 0 0 0 0 119890eff13

119862eff13 119862eff13 119862eff33 0 0 0 0 0 119890eff33

0 0 0 119862eff44 0 0 0 119890eff15 0

0 0 0 0 119862eff44 0 119890eff15 0 0

0 0 0 0 0 119862eff66 0 0 0


]]]]]]]]]]]]]]]]]]]]]]


(21)


119906119894 = 1205760119894119895119909119895120593 = minus1198640119894 119909119894

(22)


3 Types of RVE





y

z

x

(a)

y

z

x

(b)

y

z

x

(c)

y

z

x

(d)

y

z

x

(e)

y

z

x

(f)












119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810



119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23






6 Results




RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



y

z

x

(a)

y

z

x

(b)

y

z

x

(c)

y

z

x

(d)

y

z

x

(e)

y

z

x

(f)












119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810



119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23






6 Results




RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of





119862eff11 119862eff12

119906 = 0120601 = 0 119906 = ℎ1205760120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = 0120601 = 0119862eff11 = 1205901112057611

119862eff12 = 1205902212057611

119862eff13 119862eff33

119906 = 0120601 = 0 119906 = 0120601 = 0 V = 0120601 = 0 V = 0120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119862eff13 = 1205901112057633

119862eff33 = 1205903312057633

119862eff44

119906 = 0120601 = 0 119906 = 0120601 = 0119906 = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 0V = 0120601 = 0

119906 = 0V = 2ℎ1205760120601 = 0 119862eff

44 = 12059023(212057623)119862eff66

V = 0119908 = 0120601 = 0V = 0119908 = 0120601 = 0

119906 = 0119908 = 0120601 = 0119906 = 2ℎ1205760119908 = 0120601 = 0

119908 = 0120601 = 0 119908 = 0120601 = 0 119862eff66 = 12059012(212057612)

120581eff11 119906 = 0120601 = 0 119906 = 0120601 = 1198810 V = 0 V = 0 119908 = 0 119908 = 0 120581eff11 = 11986311198641

119890eff13 119890eff33 120581eff33 119906 = 0 119906 = 0 V = 0 V = 0 119908 = 0120601 = 0 119908 = 0120601 = 1198810



119906 = 0120601 = 0 119906 = ℎ12057611120601 = 0 V = 0120601 = 0 V = ℎ12057622120601 = 0 119908 = 0120601 = 0 119908 = ℎ1205760120601 = 0119884eff3 = 1205903312057633

]eff31 = minus1205761112057633119884eff2 ]eff23






6 Results




RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



RVE1RVE2RVE3RVE4

RVE5RVE6EMT

5

10

15

20

25

30

Ceff 11

(GPa

)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0123456789

10

Ceff 12

(GPa

)


(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

Ceff 13

(GPa

)


(c)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

120C

eff 33(G

Pa)

(d)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT

0

2

4

6

8

10

12

14

Ceff 44

(GPa

)


(e)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


2

4

6

8

10

12

Ceff 66

(GPa

)

(f)


12 (c) 119862eff13 (d) 119862eff

33 (e) 119862eff44 and (f) 119862eff

66


RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



RVE1RVE2RVE3RVE4

RVE5RVE6EMT

minus005

minus004

minus003

minus002

minus001

000

eeff 13

(Cm

2)


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


00

02

04

06

08

10

12

eeff 33

(Cm

2)

(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus014

minus012

minus010

minus008

minus006

minus004

minus002

000

eeff 15

(Cm

2)

(c)




11








13




RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


35

40

45

50

55

60

65120581eff 11

(times10

minus11

Fm

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


3

4

5

6

7

8

120581eff 33

(times10

minus11

Fm

)(b)















RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0005

minus0004

minus0003

minus0002

minus0001

0000

deff 31

(times10

minus9

CN

)

(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0000

0002

0004

0006

0008

0010

0012

deff 33

(times10

minus9

CN

)(b)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


minus0016

minus0012

minus0008

minus0004

0000

deff 15

(times10

minus9

CN

)

(c)





7 Discussions



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



RVE1RVE2RVE3RVE4

RVE5RVE6EMT


0

20

40

60

80

100

Yeff 3

(GPa

)

(a)

5

10

15

20

25

30

Yeff 2

(GPa

)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(b)



000

004

008

012

016

020

024

]eff 23

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


(a)

RVE1RVE2RVE3RVE4

RVE5RVE6EMT


022

024

026

028

030

]eff 31

(b)







8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




8 Conclusions







Acknowledgments


References






























































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of











































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of









CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


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