Effective permittivity of a fiber-reinforced composite with transversely isotropic constituents

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Journal of Electrostatics xxx (2013) 1e10

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Journal of Electrostatics

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Effective permittivity of a fiber-reinforced composite withtransversely isotropic constituents

Eduardo López-López a,1, Federico J. Sabina a,*, Raúl Guinovart-Díaz b,Julián Bravo-Castillero b, Reinaldo Rodríguez-Ramos b

a Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726,Delegación Álvaro Obregón, 01000 México D.F., Mexicob Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado, Havana 4, CP-10400, Cuba

a r t i c l e i n f o

Article history:Received 16 December 2011Received in revised form5 June 2012Accepted 30 January 2013Available online xxx

Keywords:Effective permittivityAsymptotic homogenization methodDielectric anisotropyFiber-reinforced compositePotential theory

* Corresponding author. Tel.: þ52 5556223544; faxE-mail addresses: [email protected]

mym.iimas.unam.mx (F.J. Sabina), guino@[email protected] (J. Bravo-Castillero), reinaldo@Ramos).

1 Permanent address: Facultad de Ciencias Básicas,Universidad Autónoma de Tlaxcala, Calzada Apizaq90300 México.

0304-3886/$ e see front matter � 2013 Elsevier B.V.http://dx.doi.org/10.1016/j.elstat.2013.01.014

Please cite this article in press as: E. Lópezconstituents, Journal of Electrostatics (2013)

a b s t r a c t

Simple closed-form expressions for effective permittivity of fiber-reinforced composites with trans-versely isotropic constituents are found using asymptotic homogenization. Circular cylindrical fibers aredistributed in a square array. The analysis considers four orientations of constituents transverse sym-metry axis relative to fibers direction. Local problems defined on a periodic square unit cell are solved bymeans of complex potential theory using Weierstrassian and Natanzon’s functions, assuming that thecontrast of permittivities is small. Derived closed-form formulas are compared with finite element cal-culations and, in the isotropic case, with standard mixture rules and classical bounds, obtaining excellentresults even when contrast is large.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Composite materials have been studied theoretically for a longtime [12,38,39]. These studies have led to mixing rules relatingthe macroscopic dielectric properties of heterogeneous media tothose of their constituent phases and the internal structure of themixture [44].

Being able to characterize a heterogeneous material bymeans ofits effective properties is essential to researchers and industries infields as varied as automotive, construction, biomedical, sports,aerospace, remote sensing, etc. Besides, the rising popularity ofmicrowaves for drying [17,50], material processing [29] and sensing[31,34] make determination of effective properties of composites ahot topic. Several techniques and numerous studies have beendeveloped to predict the behavior of composite materials. Willis

: þ52 5556223564.x (E. López-López), [email protected] (R. Guinovart-Díaz),matcom.uh.cu (R. Rodríguez-

Ingeniería y Tecnología de lauito S/N, Apizaco, Tlaxcala,

All rights reserved.

-López, et al., Effective perm, http://dx.doi.org/10.1016/j.e

[52] classified them into four main categories: asymptotic, self-consistent, variational, and modeling methods. A summary ofmost important macroscopic mixture relations can be found inTinga [49].

Among these techniques, the effective medium theory (EMT) isconsidered as the most powerful approach to estimate the effectiveproperties for the composite systems, such as cosmic dusts, aero-sols, and porous media [30]. The simplest approximation is the oneused by Wiener, it takes simple arithmetic and harmonic averagesand provides an upper and a lower bound for the effectivepermittivity [9]. This approach does not make any assumption oninclusion shape, however, it is important. A more rigorous boundswere found by Hashin and Shtrikman [16] using variational prin-ciples. Although the original derivation was done for magneticsusceptibility, they can be applied to permittivity, conductivity andelastic problems. One of the oldest and most popular EMT is theMaxwell-Garnett (MG) mixing rule that was developed for opticalproperties of a medium [27]. The assumption that separation be-tween inclusions is large, limits the applicability of MG to dilutesystems with a low volume fraction. Corrections to the MG theoryhave been proposed before [2,3,13]. An important MG theoryimprovement came with another popular mixing rule, the Brug-geman approximation (B), that was developed for the effectiveelectrical properties of a composite system [7]. Another MG and B

ittivity of a fiber-reinforced composite with transversely isotropiclstat.2013.01.014

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Fig. 1. Possible ATS orientation: i) P3. Both ATS are parallel to Ox3-axis, ii) O2. Both ATSare orthogonal to Ox2-axis, iii) P1. Both ATS are parallel to Ox1-axis, iv) O3. (Not shown)with a suitable redefinition of parameter a2 it becomes a particular case of P1 case.

Fig. 2. Schematic view of periodic fiber-reinforced composite, the coordinate systemy1y2 and its periodic unit cell, it comprises a single fiber S2 and the matrix S1 thatsurrounds it. S ¼ S1 W S2, G ¼ vS2 and S1 X S2 ¼ B

E. López-López et al. / Journal of Electrostatics xxx (2013) 1e102

theory improvement was given by Looyenga [24] (L), where pre-dictions agree well with experiments [17,19,46,53]. Besides theabove mentioned techniques, there are many other methods forfinding effective dielectric behavior of composites [5,11,15,18,48],for instance.

The techniques mentioned above have been used for a long timefor predicting the behavior of an isotropic composite with isotropicphases. However, this estimations are not enough when predictingeffective properties of anisotropic materials with anisotropic pha-ses. Levy and Stroud [23] used the MG theory to find the effectivedielectric behavior for a medium of anisotropic inclusionsembedded in an isotropic host. Sushko [47] used an approach,based upon the notion of macroscopic compact groups of particles,to derive dielectric mixing rules for macroscopically homogeneousand isotropic multicomponent mixtures of anisotropic inhomoge-neous dielectric particles. Sihvola [45] studied isotropic inclusionsin an anisotropic medium. Bianisotropic host media has beenconsidered as well [51] into the framework of MG and B theory.

As far as authors know, there are few previous works onmodeling different axis of transverse symmetry orientation ofconstituents. Kar-Gupta and Venkatesh [20] studied the electro-mechanical behavior, numerically, of a 1e3 composite by chang-ing the polarization direction of matrix. Sakthivel and Arockiarajan[42] proposed an analytical model based on parallel and seriestheory for 1e3e2 piezoelectric composite. A number of methodshave been applied to composites with different relative orientationof the material symmetry axis of each component [21,22,32,33,42],etc. The antiparallel case analyzed in Kar-Gupta and Venkatesh [21]can be considered as a particular case of the formulas developed inSabina et al. [41] and in Bravo-Castillero et al. [6]. In those papers,closed-form expressions for effective coefficients of a piezoelectricfiber-reinforced composite are found using the asymptotic ho-mogenization method (AHM). This mathematical technique is usedfor studying partial differential equations with rapidly oscillatingcoefficients [1,4,43]. Composites are subjected to fields which varyon a lengthscale, due to their microstructures. The AHM replacesthe rapidly oscillating coefficients of the differential equation withconstant effective coefficients that depend on the solution of theso-called local problems.

Within this paper, it is proposed an application of the AHM todetermine the effective dielectric behavior of a fiber reinforcedcomposite of circular cylinders distributed in a square periodicarrangement with material symmetry axis orthogonal to the fiberdirection. In Section 2 the problem at hand is established as well asrelevant definitions and notation, finally, some outstanding equa-tions for AHM are presented. The way local problems are solved ispresented in Section 3, aside from formulas for effective coefficientsfor each considered case. Numerical calculations and comparison ofobtained formulas can be found in Section 4. Empiric intervals ofvalidity are found and they are explicitly written in this section too.Finally, some concluding remarks are expressed in Section 5.

2. Statement of the problem

An anisotropic fiber-reinforced dielectric material, occupying aregion U in a tridimensional Euclidian space, is considered here.Parallel fibers, with circular cross-section, are aligned, withoutoverlapping, periodically into a square symmetry, their direction istaken as the Ox3-axis of a cartesian reference system. Fibers andmatrix have transversely isotropic dielectric properties. Within thispaper we study four possible orientations of the axis of transversesymmetry (ATS) of each component: i) P3. ATS of each componentcoincide with the Ox3-axis; ii) O2. The ATS of matrix is parallel tothe Ox1-axis; the ATS of fiber, to the Ox3-axis; iii) P1. Matrix andfiber ATS are parallel to the Ox1-axis; iv) O3. Matrix and fiber ATS

Please cite this article in press as: E. López-López, et al., Effective permconstituents, Journal of Electrostatics (2013), http://dx.doi.org/10.1016/j.e

are orthogonal to each other and with the fibers direction. With asuitable redefinition of parameter a2 (defined in Eq. (13)), this is aparticular case of P1 case (see Fig. 1).

Overall properties of the periodic media described above aresought by means of the asymptotic homogenization method (AHM)[10]. The periodic unit cell S is taken as a square in the y1y2-plane,where y ¼ x/ε is the local (fast) coordinate related to x, the macro-scopic (slow) variable by means of the small parameter ε. The pe-riodic unit cell consists of two phases: a single fiber (S1) and thematrix that surrounds it (S2), each of volume fraction V1 and V2,respectively. In addition, they are such that V1þ V2¼1, S1W S2¼ S,S1X S2¼B. The interface between fibers andmatrix is denoted byG(Fig. 2).

Here the electric displacement D(x) is linearly related to theelectric field E(x) through the second rank permittivity tensor, kij(x)by:

DiðxÞ ¼ kijðxÞEjðxÞ; (1)

where Einstein’s summation convention over repeated indices isassumed. Latin indices take values in {1, 2, 3}, while the Greekindices in {1, 2}. In what follows, unless otherwise indicated, thisnotation is assumed on repeated indices.

The permittivity of matrix and fibers is represented by kð1Þij andkð2Þij , respectively. They are assumed to be transversely isotropicsecond rank tensors.


E. López-López et al. / Journal of Electrostatics xxx (2013) 1e10 3

The permittivity tensor of the composite depends on itsmicrostructure, which is periodic in the y1 and y2 directions and itchanges rapidly. Within this paper, it is assumed that the period ineach direction equals one.

Assuming no source terms, the governing equations of thephenomenon in the domain U with boundary vU are given by(see [44], for instance)

� v

vxi

"kijðyÞ

vTðxÞvxj

#¼ 0 in U=G; (2a)

kTk ¼ 0 on G; (2b)

��kijðyÞ vTðxÞvxjni

�� ¼ 0 on G; (2c)

TðxÞ ¼ 40 on vU; (2d)

in the above equation, and in what follows, when there is no placefor a confusion, the superscripts that identify the relevant material,1 for matrix, 2 for fibers are dropped. T is the sought electric po-tential defined on Sg, g ¼ 1, 2. G denotes the lines separating thefibers from the matrix (interface) and n is the outward unit vectornormal to G. Equations (2b) and (2c) are conditions of continuity ofelectric potential and the flux across the interface G, respectively.On the boundary of the composite an electric potential 40 has beenprescribed. The double bar notation denotes the jump of thefunction f(y) across the interface,

kf ðyÞk ¼ f ð1ÞðyÞ � f ð2ÞðyÞ for y˛G:

Problem (2) is an elliptic PDE subject to interface continuity anddouble periodicity conditions, which has a unique solution [1]. It isworth noticing that this is a problem with rapidly varying periodiccoefficients very difficult, if not impossible, to solve numericallywhen ε is small. These kind of problems are commonly solved usingthe AHM. Then, it is assumed that electric potential can be writtenas an asymptotic expansion, that is:

TðxÞ ¼ T0ðx; yÞ þ εT1ðx; yÞ þ ε2T2ðx; yÞ þ. (3)

Following AHM, whose details can be found, for example, inCioranescu and Donato [10], it can be proved that the function T0 (x,y) depends only on x, it is actually the homogenized potential.Function T1 (x, y), is the approximation to O(ε); it is given by theexpression

T1ðx; yÞ ¼ pqðyÞvT0vxp

; (4)

where the functions pqðyÞ are the solution to the so-called localproblems pL; p ¼ 1;2;3, given below in (6). Further, by solvingthose problems, a formula for each non-vanishing effective co-efficients is obtained

kip ¼ZS

�kip þ kik

vpqðyÞvyk

�dy; (5)

where S is the unit periodic cell, depicted in Fig. 2. Each doublyperiodic functions pqðyÞ is the solution to the corresponding local


problem pL that satisfies the elliptic PDE (6a) and interface condi-tions (6b) and (6c). A null-average condition over the unit cell (6d)is required for uniqueness of solution:

kijðyÞv2 pqðyÞvyivyj

¼ 0 in S; (6a)

kpqðyÞk ¼ 0 on G; (6b)

��kijv pqðyÞvyj

ni

�� ¼ ��kipni�� on G; (6c)

hpqðyÞi ¼ZS

pqðyÞdy ¼ 0; (6d)

The local problems (6) are solved using the complex potentialtheory by means of Weierstrassian elliptic functions of periods 1and i.

Using Green’s theorem and Eq. (6b), effective coefficients in Eq.(5) become

k1p ¼ �k1p�� kk11k

ZG

pqð1Þdy2; (7a)

k2p ¼ �k2p�þ kk22k

ZG

pqð1Þdy1; (7b)

k3p ¼ �k3p�; (7c)

the first term in Eq. (7) is the arithmetic average between physicalproperties of constituents. Each non-vanishing effective coefficientdepends on the solution of one and only one local problem. Localproblem 3L, due to Eq. (6c) andmaterial symmetry, is homogeneous.Therefore, its unique solution is the trivial one. This problem is notstudied here, resonant solutions are beyond the scope of this article.Some relevant articles are [28,35,36], for instance.

2.1. Case P3. Both ATS of constituents are parallel to fiber direction

Formulation of local problems pL in P3 is stated here. Cumber-some notation is avoided denoting matrix and fibers permittivitiesby

kð1ÞðyÞ ¼ diag�k1; k1; k

01�; kð2ÞðyÞ ¼ diag

�k2; k2; k

02�; (8)

According to notation (8), the problem (6) for P3 case, is nowdefined as

Dpq ¼ 0 in Sg; g ¼ 1;2; (9a)

kpqk ¼ 0 in G; (9b)

��k v pq

vn

�� ¼ �kkkn[dp[ in G; (9c)

hpqi ¼ 0; (9d)

where the symbol D is the Laplacian operator, the Kronecker deltafunction, dij, is defined as 1 if i ¼ j and 0 otherwise.



The latter is a classical problem in literature. Its completemethod of solution and analysis can be found, as a particular case,for example in López-López et al. [25], Rodríguez-Ramos et al. [40].In those papers analytical formulas for effective coefficients of apiezoelectric composite material can be found.

2.2. Case O2. Matrix ATS is orthogonal to the fiber direction

From Eq. (6), the formulation of local problems for O2 is:

Dpqð1Þ ¼ a1

v2 pqð1Þ

vy21in S1; (10a)

Dpqð2Þ ¼ 0 in S2; (10b)

kpqk ¼ 0 on G; (10c)

��k v pq

vn

�� ¼ �kkkn[dp[ þ a1

1þ v pq

ð1Þ

vy1

!k1n[d1[ on G;

(10d)

hpqi ¼ 0; (10e)

where the anisotropic parameter related to the matrix is defined as

a1 ¼ 1� k01=k1: (11)

When a1 ¼ 0 Equation (9), for an isotropic medium, is recovered.

2.3. Case P1. Matrix and fiber ATS are orthogonal to the fiberdirection

On the other hand, the local problems pL for P1 is as follows:

DpqðgÞ ¼ ag

v2 pqðgÞ

vy21in Sg; g ¼ 1;2; (12a)

kpqk ¼ 0 on G; (12b)

��kv pq

vn

�� ¼ �kkkn[dp[ þ��kav pq

vy1

��n1 þkkakn[d1[ on G; (12c)

hpqi ¼ 0; (12d)

here, the parameter of anisotropy relative to the fiber is defined as

a2 ¼ 1� k02=k2: (13)

Two new parameters arise in the above analysis, a1 and a2.Special attention should be paid when these parameters are zero:proposed cases, O2 and P1, are reduced to P3. This peculiaritysuggests that O2 and P1 can be analyzed as a perturbation of P3.Redefining the a2-parameter as: a2 ¼ 1� k2=k

02, O3 can be

analyzed exactly in the same way as P1.

3. Method of solution

Problems (10) and (12) are solved taking into account 258 re-ported materials, which correspond to the assumed crystal sym-metry [8]. By calculating the alpha parameter for each material, itwas found that 254 of them are such that jajh1. In addition 181 of


them satisfy jaj � 0:5. This means that nearly 71% of materials havetheir respective alpha parameter in the (�0.5, 0.5) interval. Let usassume, for themoment, that the a-parameters are small enough sothat, according with last observation of previous section, problems(10) and (12) can be considered as a perturbation of problem (9),i.e., it is assumed that the anisotropy is small.

3.1. Case P3

The function pqðgÞ0 is the solution to problem (9), where a

doubly-periodic harmonic function that satisfies the given jumpconditions (9b) and (9c) and the null average condition over theunit cell (9d) is sought. Many other papers deal with this problemusing the multipole expansion method like, for instance, Rayleigh[39], Perrins et al. [37]; by means of AHM, formulas and derivationscan be found in Rodríguez-Ramos et al. [40], López-López et al. [25],among others. Solution to problem (9) and explicit formulas foreffective coefficients can be found, for example, in López-Lópezet al. [26]. It is depicted here in a very short manner.

Due to the geometry of the problem, the right-hand-side ofjump conditions for local problem 3L, i.e. Eq. (9) with p ¼ 3, equalszero. Therefore the only solution is the trivial one. Let us considerthe case p ¼ 1 (case p ¼ 2 can be solved analogously), the sought

electric potential 1qðgÞ is proposed as

1qð1Þ ¼ Re

8><>:� pa1zþ

Xo

k¼1

N

akzðk�1ÞðzÞðk� 1Þ!

9>=>;;

1qð2Þ ¼ Re

8><>:X

o

k¼1

N

ckzk

9>=>;; (14)

here, the function z is the Weierstrass zeta function, a quasi-periodic complex-valued function on the variable z ¼ y1 þ iy2,with periods u1 ¼ 1 and u2 ¼ i; its k-th derivative zðkÞ is a doublyperiodic function with the same periods. The “o”, next to the sigmasymbol, means that summation should only be taken over oddindices. Coefficients ak and ck are unknown and real. It can beproved that coefficients ck depend on coefficients ak. This can bedone by substituting Eq. (14) in the interface conditions (9b) and(9c). This, also, yields the infinite linear systems

c�1k IþW

a ¼ V0;

c�1k I�2W

2a ¼ �V0; with

ck ¼ k1 � k2k1 þ k2

; (15)

for local problems 1L and 2L, respectively. Infinite order symmetricmatrices I, W ¼ [wkl] and 2W ¼ ½2wkl� are defined as

I ¼ diagð1;1;1;.Þ; (16a)

w11 ¼ �2w11 ¼ pR2; wkl ¼ 2wkl ¼ffiffiffik

p ffiffil

phklR

kþl; kþ l� 3;(16b)

column vector V0 ¼ (R, 0, 0,.)T has a unique non-zero component.The column vectors

a ¼ ffiffiffi

1p

R�1a1;ffiffiffi3

pR�3a3;

ffiffiffi5

pR�5a5;.

T;



2a ¼ ffiffiffi

1p

R�12a1;

ffiffiffi3

pR�3

2a3;ffiffiffi5

pR�5

2a5;.T

contain the sought coefficients. Matrix hkl is given by

h11 ¼�

p 1L;�p 2L;; hkl ¼

ðkþ l� 1Þ!k!l!

Skl; kþ l � 3; (17)

the lattice sums associated with Weierstrass’ functions are

Sk ¼Xm;n

0 1

bkmn

; (18)

the complex numbers bmn are defined as bmn ¼ mu1 þ nu2, withm;n˛Z=f0g. The symbol “0” next to the summation symbol in-dicates that the sum should not consider the term m ¼ n ¼ 0.

According to Eq. (7), now we are able to compute the effectivecoefficients. After some algebraic manipulation, using the double

periodicity of qðgÞ, Green’s Theorem and system (15), very simpleclosed-form expressions are achieved:

k11 ¼ k1ð1� 2pa1Þ; (19a)

k22 ¼ k1ð1þ 2p2a1Þ; (19b)

k33 ¼ k01V1 þ k02V2: (19c)

Latter formula is the arithmetic mean of permittivities of con-stituents. Coefficients a1 and 2a1 can be found using Eq. (15).

3.2. Case O2

The ATS of matrix lies in an orthogonal plane to the fiber axis. Letus consider only the p ¼ 1 case, in the interest of brevity. Formulasfor p ¼ 2 are obtained by following a similar procedure. Localproblem 3L, as mentioned before, becomes homogeneous.

When a1 ¼0, Eq. (10a) becomes the Laplace equation. Again, thepotential theory, in terms of Weierstrassian functions, can beemployed in a similar fashion as in previous section. Assuming thata1 is a small parameter, the following expansion is considered.

1qðgÞ ¼ 1q

ðgÞ0 þ a1 1q

ðgÞ1 þ O

a21

; g ¼ 1;2; ja1j � 1: (20)

Upon substitution of Eq. (20) into Eq. (10) and after similar termsin powers of a1 are collected, a recurrent sequence of problems is

obtained for 1qðgÞ0 and 1q

ðgÞ1 :

To Oa01

:

D1qðgÞ0 ¼ 0 in Sg; (21a)

k1q0k ¼ 0 on G; (21b)

��k v1q0vn

�� ¼ �kkkn1 on G; (21c)

h1q0i ¼ 0: (21d)


To Oa11

:

D1qð1Þ1 ¼ v1q

ð1Þ0

vy21in S1; (22a)

D1qð2Þ1 ¼ 0 in S2; (22b)

k1q1k ¼ 0 on G; (22c)

��k v1q1vn

�� ¼ k1

1þ v1q

ð1Þ0

vy1

!n1 on G; (22d)

h1q1i ¼ 0: (22e)

Problem (21) has been studied in Eq. (9). Once the harmonic

function 1qðgÞ0 is known, the sought solution to problem (22), can be

seen to satisfy a biharmonic equation with the same double peri-odicity, since the right-hand-side of Eq. (22a) is harmonic. Againjump conditions (22c), (22d) and null average condition (22e) haveto be satisfied.

3.2.1. Solution to 1qðgÞ1

The function 1qðgÞ1 can be found by adapting Goursat’s method,

used in the context of plane elastic problems to solve the bihar-monic equation in terms of two harmonic function 4(z) and j(z) in

the form 1qð1Þ1 1 ¼ z4ðzÞ þ jðzÞ, where z is the complex conjugate of

z ¼ y1 þ iy2. To find a doubly periodic biharmonic function 1qðgÞ1 we

set it in the form of a series [40]

1qð1Þ1 ¼ Re

8><>:14a0zþ b0z

þX

o

k¼1

N"ak4

zzðkÞðzÞðk� 1Þ!þ

Qðk�1ÞðzÞðk� 1Þ!

!þ bk

zðk�1ÞðzÞðk� 1Þ!

#9>=>;;

ð23aÞ

1qð2Þ1 ¼ Re

8><>:X

o

l¼1

N

dlzl

9>=>; (23b)

with undetermined real coefficients bk, dk. QðzÞ is Natanzon’sfunction [14].

The Laurent expansion of 1qð1Þ1 about the origin is obtained as

follows:

1qð1Þ1 ðzÞ ¼ Re

8><>:X

o

l¼1

N 14

lalz

�l þ lAlzl þ A0

lzlþ blz

�l � Blzl�9>=>;;

(24)where



Al ¼Po

k¼1

Nkakhkl; A0

l ¼Po

k¼1

Nkakh

0kl; Bl ¼

Po

k¼1

Nkbkhkl;

hkl ¼ðkþ l� 1Þ!

k!l!Skl; h0kl ¼

ðkþ lÞ!k!l!

Tkþl for kþ l � 3;

h11 ¼(

p 1L

�p 2L; h011 ¼

(pþ 5S4=p 1L

p� 5S4=p 2L:

In addition to Sk as defined in Eq. (18), the lattice sums associ-ated with Natanzon’s function are

Tk ¼Xm;n

0 bmn

blþ1mn

: (25)

Following a similar procedure to that used in Sabina et al. [41] orin Bravo-Castillero et al. [6], an infinite system of algebraic equa-tions is reached to determine the coefficients of Laurent’s expan-sion (24):

c�1k IþW

a ¼ V0; (26a)

c�1k I�W

b ¼ V1 þ

�Uþ 1

4W0�a; (26b)

the symmetric infinite-order matrix W0 is defined as

W0 ¼ �w0

kl

� ¼

8><>:�pþ 5S4

p

�R2 k ¼ l ¼ 1

ffiffiffik

p ffiffil

ph0klR

kþl kþ l � 3

; (27)

the infinite order column vectors V1 and b, as

V1 ¼�14c�1k Rld1l;0;0;.

�T

; (28)

b ¼ ffiffiffi

1p

R�1b1ffiffiffi3

pR�3b3

ffiffiffi5

pR�5b5.

T; (29)

and the components of the infinite order tridiagonal matrixU ¼ [uk[], for [ ¼ 1, as

u11 ¼ 12

k1kkk ðck � 2Þ � 1

2c�1k

�c�1k ; (30a)

u13 ¼ffiffiffi3

p

2

12� k1kkk�c�1k ; (30b)

u1k ¼ 0 ck˛f5;7;9;.g; (30c)

while for [ � 3, as:

u[k ¼12

12c�1k þ 1

lk1kkk� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kðkþ 2Þq

dðl�2Þk þ12

1kk1kkk

1� c�1

k

�

� kdlk þ12

12� 1k� 2

k1kkk�c�1k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk� 2Þk

qdðlþ2Þk:

(30d)


It must be emphasized that Eq. (26a) is the same infinite alge-braic system for calculating the coefficients of the Laurent’s

expansion of solution to problem (21) 1qð1Þ0 of zeroth-order in a1.

Thus, an expression is easily obtained for 1qðgÞ, enabling us to

calculate the effective coefficients (7), to the first-order in a1 as:

k11 ¼ k1ð1� 2pa1Þ þ a1

hk1

p1þ c�1

k

a1 � 1

� kkkA1

i;

(31a)

k22 ¼ k1ð1þ 2p2a1Þ þ a1

12k1Rþ 2N12a1R

�1 þ 2N3 2a3R�3

þ2k1p2b1R�1�Rp; (31b)

k33 ¼ k1V1 þ k02V2; (31c)

where

A1 ¼ � 14

1þ c�1

k

Rþ

�14c�1k � N1

�a1R

�1

� 3�14c�1k þ N3

�a3R

�3 þ1þ c�1

k

b1R

�1�Rp;

(32a)

N1 ¼ 12

k1kkk

1� 2c�1

k

� 12c�2k

�; (32b)

N3 ¼ 12

�12� k1kkk�c�1k ; (32c)

2N1 ¼ 12ðk1 þ k2Þ

�12c�1k þ k1

kkkck �12

�; (32d)

2N3 ¼ �32c�1k k1; (32e)

note that the pre-subscript “2” is explicitly used to denote quantitiesthat are obtained after solving the local problem 2L for the effectivecoefficient k22. For brevity, it is not given here but it can be solved bya procedure similar to the one already exposed. It should be notedtoo that (31c) is the arithmetic mean of the constituents properties.

3.3. Case P1

In the local problem pL, formulated in Eq. (12), two contrastparameters appear, a1 and a2 given by Eq. (11) and (13), respec-tively. We assume that each ag has the form

ag ¼ Lg1bþ Lg2b2 þ Lg3b

3;g ¼ 1;2: (33)

in terms of one small parameter b and known integers Lgk, g ¼ 1, 2,k ¼ 1, 2, 3.

To solve local problem (12) a new ansatz is proposed:

pqðgÞ ¼ pq

ðgÞ0 þ pq

ðgÞ1 bþ pq

ðgÞ2 b2 þ/;g ¼ 1;2; (34)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fiber volume fraction V2

κ 11/κ

(1)

11

χκ = 0.85

AHM predictionMG approximationB approximationL approximationHS upperHS lower

Fig. 3. Comparison, in the isotropic limit, of predictions of AHM, traditional mixingrules and HashineStrikman bounds. Here ck ¼ 0.85. For smaller absolute values of ck,MG separates from AHM at high V2 less than for high ck (not shown).


this assumption leads to a succession of recursive local problems:

To Ob0:

D1qðgÞ0 ¼ 0 inSg;g ¼ 1;2; (35a)

k1q0k ¼ 0 on G; (35b)

��k v1q0vn

�� ¼ �kkkn1 on G; (35c)

h1q0i ¼ 0: (35d)

To Ob1:

D1qðgÞ1 ¼ Lg1

v2 1qðgÞ0

vy21in Sg;g ¼ 1;2; (36a)

k1q1k ¼ 0 on G; (36b)

��k v1q1vn

�� ¼"��kLg1��þ

��kLg1v1qðgÞ0

vy1

��#n1 on G; (36c)

h1q0i ¼ 0: (36d)

which are immediately solved as before. Eq. (35) is like (21) and Eq.(36) is similar to (22). The inhomogeneous terms in Eq. (36) areslightly different from (21). Taken that into account effective coef-ficient formulas become

k11 ¼ k1ð1� 2pa1Þ ��kLg1

�� kLg1�� 1G02 þ kkk 1G12Rp�b;

(37a)

k22 ¼ k1ð1þ 2p2a1Þ þ kkkB11Rpb; (37b)

k33 ¼ k1V1 þ k2V2; (37c)

no summation over repeated g, where

1G02 ¼ �V2 þ1þ c�1

k

a1p; 1G12

¼ �14L111þ c�1

k

Rþ 1

2k2kkk L21Rþ

1þ c�1

k

b1R

�1

� 12

k1kkk L11

1� 2c�1

k

� 12c�1k

1þ c�1

k

L11

þ 3k2kkk L21

1þ c�1

k

�a1R

�1 � 12

�12� k1kkk�L11c

�1k

� 12L11c

�1k þ k2

kkk L211þ c�1

k

�3a3R

�3;

in Eq. (37b) the coefficient 2a1 is the first coefficient of Laurent’sexpansion of the solution to problem 2L, also


B11 ¼12

12þ12L11c

�1k � k2

kkk R�1þc�1

k

b1R

�1
� �
þ12

k1kkkþ

12

1þL11c

�1k

c�1k � k2

kkk1þc�1

k

�a1R

�1

þ12

�12L11�

k1kkk�c�1k �1

2L11c

�1k þ k2

kkk1þc�1

k

�3a3R

�3:

Eqs. (31) and (37) clearly exhibit the dependence of the effectivecoefficients upon: i) the size of the inclusion (fiber radius R), ii)physical properties of materials and contrast between them bymeans of ck, iii) geometry of the arrangement (harmonic andbiharmonic lattice sums) and iv) size of material perturbation(parameters a and b). It should be noted that when the anisotropicperturbation parameter a tends to zero, formulas in López-Lópezet al. [25] are recovered.

4. Numerical results and discussion

In order to assess the goodness of the above derived formulas,comparisons with other well-known analytical methods and finiteelement (FE) calculations is carried out in this section. The system(6) is solved using a commercial PDE solver based on FEMLAB ver.3.2. Lagrange-quadratic elements of second order are chosen as thebasis functions with triangular-shaped elements. In this study, non-uniform meshes were employed. About 900e1400 elements wereused, depending on the size of inclusion and the total number ofdegrees of freedom was around 1900e7600. With this, it is foundthat the final formulas are valid for a not so small anisotropic pa-rameters a1 and a2, for a given volume fraction V2 and materialcontrast ck. Let us first consider the range of materials available fora given ck. According to Choy et al. [8], there are 258 materials withphysical properties corresponding to those raised in this analysis.For every possible fiberematrix combination of those materials, ithas been calculated the ck corresponding parameter of the com-posite. Besides, for every reported material, it has been calculatedthe corresponding a parameter. Obviously, the ck parameter takesits values in the interval [�1, 1]. Most of the reported materials(98.4%) have the a parameter in the interval (�1, 1). In addition,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1

χκ= 0.35

κ 11/κ

(1)

11

Fiber volume fraction V2Fiber volume fraction VFiber volume fraction V

α1 = −0.5

α11 = −0.3 = −0.3 = −0.3

α1 = 0

α11 = 0.3

α1 = 0.5

AHMFE

Fig. 5. Behavior of the relative effective coefficient k11=kð1Þ11 in O2 case, for ck ¼ 0.35

and a1 ˛ [�0.5, 0.5]. Using these values the percentage difference between AHM andFE is less than 10%.


about 72% of them have the a parameter in the interval (�0.5, 0.5).Then the involved parameters are defined in:

ck˛½�1;1�; ag˛ð�0:5;0:5Þ; g ¼ 1;2; R˛½0;p=4� (38)

As ag tends to zero, the anisotropic cases O2 and P1 turn into anisotropic problem. It should be noticed also that, due to problemgeometry, we are dealing with a two dimensional situation.Therefore the inclusions can be considered as a circular inclusion ofradius R and predictions, in the isotropic limit, are compared withstandard mixture equations, such as MG [27], B [7] and L [24]. Fig. 3displays a plot of effective normalized permittivity k11=k

ð1Þ11 against

volume fraction V2 for a very large material contrast ck ¼ 0.85; acomparison between the predictions of standard mixing rules andthe derived formulas in the isotropic case, Eq. (19) with k1 ¼ k2, isshown. The MG approximation lies above our prediction, this, inturn, is located above the B and L approach. L being farther to AHMthan B. It should be noticed that our predictions, for a small volumefraction, agrees with the MG prediction, as it would be expectedfrom this approximation. Similar behavior is exhibited for smallervalues of ck and values as large as ck ¼ 0.95 including the negativesvalues, although not shown here. Our prediction can also becompared with classical bounds such as the well-known HashineStrikman (HS) bounds for conductivity problems [16]. Fig. 3shows, also, the plot of the corresponding HS bounds. For positivevalues of ck the AHM values are closer to the upper HS bound.Negative values of ck, not shown, exhibit an inverse behavior, i.e.the AHM formula is closer to the lower HS bound. The abovecomparison is given for a high contrast between phases. A similarbehavior is observed for smaller contrast values, not given here.

4.1. A comment about the solution of the infinite order algebraicsystem (26a)

System (26a) is solved by means of truncation to a finite orderand the Cramer’s rule. A fast convergence of successive truncationsis ensured because the system is regular (see Refs. [6,41]). Trun-cation order is a function of V2 and ck. Our numerical calculations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

1

2

3

4

5

6

7

8

κ 11/κ

(1)

11


χκ = −0.85

χκ = −0.35

χκ = 0

χκ = 0.85

AHMFE

Fig. 4. Comparison, for P3 case, of results obtained by calculating effective coefficientsusing AHM and FE of effective normalized coefficient k11=k

ð1Þ11 . It should be noticed that

the percentage difference between the two methods is smaller than 3%. Representativevalues of ck are shown.


show that this number can be fixed to be 7, a very small order forfast solution, to get a good accuracy in all the computations.

A measure of the accuracy of formulas (19), (31) and (37), in theanisotropic cases, is obtained by comparing them against FE cal-culations of the local problem that gave rise to those formulas. Fig. 4shows this comparison for the P3 case. It displays the plot ofdimensionless effective permittivity k=k1 against volume fractionV2 for ck ¼ �0.85, �0.55, 0, 0.85, obtained using AHM and FE. It isremarkable than in general the percentage difference between bothmethods is less than 3%. However around percolation value, i.e.when fibers get in contact, the difference is bigger. This differencecan be reduced by increasing the truncation order of the infinitesystem (26a). Plotted values of ck are taken from Choy et al. [8, pp.331e349], they correspond, for example, to a combination of ma-terials as the following:

ck Fiber Matrix

�0.85 PZT-6B Te�0.55 K3Li2Nb5O15 PZT-80 e e

0.85 Pb(Ti0.52Zr0.48)O3 KH14D0.60PO4

Having such a small percentage differences at such a high con-trasts (ck ¼ �0.85), indicates that the proposed formulas providesimilar results to those obtained via FE; however, significantlyfaster and easier to program. Calculations using FE were carried outin about 2 h, meanwhile those made using AHM take only a fewseconds using a standard PC.

Hence, in terms of material properties, the interval where ourformulas (19) provide a value of the effective permittivity with anerror less than 3% is

337

k1 � k2 � 373

k1; (39)

provided that the radius of the fibers is not very close to thepercolation value. In such a case, it is only necessary to increase thesize of the truncated system (15). Our model is indeed valid for awide range of materials.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.5

0.6

0.7

0.8

0.9

1

1.1

1.2χκ= 0.35

κ 22/κ

(1)

11


α1 = −0.5

α1 = −0.3

α1 = 0

α1 = 0.3

α1 = 0.5

AHMFE

Fig. 6. Relative effective coefficient k22=kð1Þ11 for the O2 case and ck ¼ 0.35. All calcu-

lations in FE match with a1 ¼ 0 case in AHM.


A similar analysis about the O2 case reveals that the AHM pre-dictions differs from FE by less than 10% when the parameters aretaken as follows

0 � R � p4; �1

2� a � 1

2; �1

5� ck �

720

;

that is, when fiber and matrix are such that

0 � V2 � 12; �1

2k1 � k01 � 3

2k1;

1327

k1 � k2 � 32k1: (40)

Fig. 5 shows the behavior of the relative effective coefficient

k11=kð1Þ11 , as a function of V2, when ck ¼ 0.35, which is the maximum

value for which the percentage error difference is less than 10%,

0 0.2 0.4 0.6 0.8

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1α1=0.5 α2=0.5 χκ=0.2

Nor

mal

ized

effe

ctiv

e C

oeffi

cien

ts


κ22/ κ(1)11

κ11/ κ(1)11

AHMFE

0 0.2 0.4 0.6 0.80

1

2

3κ22

0 0.2 0.4 0.6 0.80

5

10κ11

Perc

enta

ge d

iffer

ence

bet

wee

n AH

M a

nd F

E


Fig. 7. Comparison between behavior of relative effective coefficients k11=kð1Þ11 and

k22=kð1Þ11 computed using AHM and FE in P1 case. The given parameters are the largest

for which the percentage difference between AHM and FE, for relative effective coef-ficient k11=k

ð1Þ11 , is less than 10%.


according to (40). Again, the error difference near the critical valueR¼p/4 can be reduced by increasing the truncation order in system(26a) if desired.

Fig. 6 shows relative effective coefficient k22=kð1Þ11 for the O2 case,

for five representative values of parameter a1, with ck ¼ 0.35, thesame value considered for coefficient k11. As in previous cases, itshould be noted that the greater the volume fraction the higher thepercentage error difference, which could be reduced by increasingthe truncation order in the corresponding infinite system. Thecomparison of AHM and FE calculations for the effective coefficientk22 shows better results for a1 ¼ 0 than otherwise, because theperturbation is considered only in the y1-direction. This is consis-tent with the fact that for FE calculations, for every value of a1,match the a1 ¼ 0 case of the formula (31b).

The same analysis was carried out in P1 case. It was found thattaking parameters such that

ck˛½�0:2;0:2�; a1;a2˛½�0:5;0:5�;

percentage difference between two methods is less than 10%. Thisdifference decreases to 5% restricting the anisotropy of each con-stituent to a1, a2 ˛ [�0.3, 0.3]. Fig. 7 shows a calculation of relativeeffective coefficients k11=k1 and k22=k1 for the P1 case, with ck. Inthis plot, the anisotropy parameters take values a1 ¼ a2 ¼ 0.5 andcontrast between material parameter has the value ck ¼ 0.2, whichis the largest possible in order to have a percentage error differenceless than 10%.

5. Conclusions

The quality and quantity of new applications of composite ma-terials depends largely on the ability to predict the effective prop-erties of the composite. This is not an easy work because effectiveproperties depend on the size, organization, orientation, etc. of theinclusions. Within this paper, using the AHM, explicit formulas hasbeen found for effective dielectric properties of a periodicallydistributed fiber-reinforced dielectric composite with anisotropicconstituents and four different orientations of the ATS. Effectiveproperties show explicit dependence on fiber volume fraction (V2),contrast between material properties (ck), anisotropy of constitu-ents ag and the geometry of the problem (S4). Nearly 71% of re-ported material in Choy et al. [8] fall within the ranges of reliabilitydetermined by the heuristic analysis carried out here.

In the isotropic limit, the predictions with AHM, which areexact, show that well-know classical mixtures rules are approxi-mately correct and the departure from exact values is given also.The AHM values lie within the HashineStrikman bounds. Foranisotropic materials, a comparison of calculations obtained bymeans of the proposed method and calculations made by the finiteelement method was carried out. This comparison helps to deter-mine heuristically a threshold of validity for the involved param-eters. Numerical experiments suggest that the analytical formulas,which were obtained under a small anisotropy perturbationassumption, can be used for values of ag as large as �0.5 � ag � 0.5having an accuracy of at most 10% in extreme values. In addition,this formulas are simple, mnemonic and much easier and faster toprogram than FE.

Other periodic arrangements, like hexagonal and paral-lelogrammic, can be handled by the method introduced in thispaper. It is worth noting that because of the mathematical simi-larity, results obtained in this paper can be easily applicable to theeffective thermal conductivities, magnetic permeability and massdiffusivity, as well as anti-plane shear problems. The analyticalformulas may be useful as a benchmark for numerical and exper-imental methods.



Acknowledgements

EL gratefully thanks the support of a fellowship from CONACYTand the support provided by the Instituto Tecnológico Superior deSan Martín Texmelucan. The funding of CONACYT project no.129658 is recognized. Thanks are due to Ana Pérez Arteaga andRamiro Chávez Tovar for computational help.

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Effective permittivity of a fiber-reinforced composite with transversely isotropic constituents

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