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published online 23 October 2013Journal of Intelligent Material Systems and StructuresKuzhichalil P Jayachandran, Jose M Guedes and Helder C Rodrigues

A generic homogenization model for magnetoelectric multiferroic  

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A generic homogenization model formagnetoelectric multiferroics

Kuzhichalil P Jayachandran, Jose M Guedes and Helder C Rodrigues

AbstractA generic homogenization modeling framework which incorporates crystallographic domain features is introduced andcomputationally implemented for magnetoelectric multiferroics of all symmetries. The homogenization, mathematicallyapplicable to heterogeneous media with contrasts in physical properties, replaces the heterogeneity of the multiferroicsby an equivalent effective medium with uniform physical characteristics. A statistically representative unit-cell is proposedto encompass all forms of multiferroics and their composites in bulk. The variational formulation of the coupledmagneto-electromechanical problem reveals the nature of interaction between mechanical, electrical, and magnetic fieldsof a multiferroic at a microscopic scale with high resolution. Furthermore, the mathematical homogenization theory ofthe multiferroic is implemented in finite element method by solving the coupled equilibrium electrical, magnetic, andmechanical fields. A ‘‘multiferroic finite element’’ is conceived for this purpose. The model is applied to a two-phase mul-tiferroic magnetoelectric composite to demonstrate its validity by characterizing the equivalent physical properties.

KeywordsMagnetoelectric effect, multiferroics, micromechanical modeling, microstructure, finite element method

Introduction

Materials simultaneously possessing two or more of theferroic order parameters—ferromagnetism, ferroelectri-city, ferroelasticity, and ferrotorroidicity—in the samephase (Schmid, 1994, 2008) are referred to as multifer-roics. Recently, the term multiferroics is meant predo-minantly for the coexistence of magnetism andferroelectricity and has shown great potential for infor-mation storage and sensor technologies (Eerenstein etal., 2006; Gajek et al., 2007; Martin et al., 2010). Thereare 31 point groups that allow a spontaneous magneti-zation Ms, and 31 that allow a spontaneous electricalpolarization Ps. 13 point groups intersect both thesesets permitting both spontaneous polarization Ps andspontaneous magnetization Ms in the same phase (Hill,2000; Schmid, 2008; Shuvalov and Belov, 1962). Thedegree of complexity of domain patterns and, for thatmatter, the degree of experimental difficulties (insynthesizing single-domain crystals) increase with thenumber of possible domain states. In single-phase mul-tiferroics such as bismuth ferrite (BiFO3), magnetic andferroelectric domains coexist closely in bulk and thinfilm forms (Chu et al., 2008; Ke et al., 2010; Lebeugleet al., 2008).

A key feature of ferroic materials is the formation ofdomains consequent to the appearance of a ferroic

order at the phase transition. The lesser the symmetryof the ferroic phase, the more the number of possibledomain states. For instance, the less symmetric mono-clinic multiferroic crystal with space group m and hav-ing Ps and Ms in (110)c possesses 12 and 24ferroelectric and ferromagnetic domain states, respec-tively. Yet the more symmetric orthorhombic 2-mmcrystals with Ps kMs k ½100�c have only 6 ferroelectricand 12 ferromagnetic domain states. (Here, the orienta-tions of Ps and Ms, i.e., ½hkl�c, are given relative to thecubic reference system.) However, the domain architec-ture differs in various multiferroics (Lebeugle et al.,2008; Meier et al., 2009). Domain orientation essen-tially is the manifestation of the orientation of underly-ing unit-cell. It should be emphasized here that the cellreferred to here is a true unit-cell whose definitioninvolves the magnetic structure of the multiferroic crys-tal which may be different from pure crystallographicunit-cell. Crystallographic orientation can be

IDMEC, Instituto Superior Tecnico, Technical University of Lisbon, Lisbon,

Portugal

Corresponding author:

Kuzhichalil P Jayachandran, IIRBS, Mahatma Gandhi University, Kottayam,

Kerala – 686560, India.

Email: kpjayachandran@gmail.com

characterized by a set of Euler angles (f, u,q)(Goldstein, 1978). Developments in the direction ofcontrolled manipulation of multiferroic domains couldopen opportunities for novel device architectures.

Theoretical treatment of equivalent or effectiveproperties of complex material systems would often besought as a means for material characterization whereexperiments would be difficult to be accomplished. Thecostly and delicate process of synthesis of crystals offairly large size and associated deleterious effects suchas depoling and twinning affects the accuracy of themeasurement of physical properties (Ahart et al., 2008).Recently, continuum models incorporating domain-scale information such as orientation and its distribu-tion have been used to describe the macroscopic fea-tures of coupled crystalline materials (Jayachandranet al., 2010; Li et al., 2010). An efficient and convenientway with affordable computational times is the use ofhomogenization technique. It has been demonstratedthat the figures of merit of multiferroic thin films,where the electrical, magnetic, and mechanical fieldsare coupled, can be tuned by the design of crystal mor-phology at the microstructure level (Martin et al.,2010).

Here, we have developed a generic homogenizationmethod theoretically for multiferroics irrespective of itscrystallographic symmetry. A two-scale asymptoticanalysis combined with a variational formulation of theunderlying mechanical, electrical, and magnetic fields iscarried out to unveil their interaction in the microscopicscale. Furthermore, the numerical solution of thecoupled magneto-electromechanical problem is soughtusing the finite element method (FEM) to eventuallycompute the homogenized (effective) properties of mul-tiferroics. FEM is a feasible computational approachand a natural choice of numerical approximation in thehomogenization procedure. Subsequently, an exampleis shown by applying the method to a two-phase multi-ferroic composite of a piezoelectric and a piezomagneticmaterial. The variational asymptotic homogenizationused in this study provides no direct simulation of mag-netic or electric domains. This framework instead mod-els the piezoelectric and magnetoelectric (ME) couplingin a continuum setting (where piezomagnetism istreated analogous to piezoelectricity), where the scalesinvolved are of the order of micrometers.

Since there are very few single-phase multiferroicmaterials (Hill, 2000; Khomskii, 2009), and most ofthem show very weak ME coupling at room tempera-ture, ME composites present an alternative for engi-neering ME coupling (Eerenstein et al., 2006). Most ofthe theoretical developments (Benveniste, 1995;Benveniste and Milton, 2003; Li and Dunn, 1998; Nanet al., 2001, 2005; Srinivasan et al., 2001) are based onthe strain-mediated two-phase multilayer composites ofmagnetostrictive and piezoelectric phases proposed byHarshe et al. (Avellaneda and Harshe, 1994; Harshe

et al., 1993). Experimental and theoretical develop-ments in these classes of materials are compiled inrecent reviews (Bichurin et al., 2010; Nan et al., 2008;Srinivasan, 2010). FEMs (Blackburn et al., 2008;Galopin et al., 2008) and micromechanical methods(Bravo-Castillero et al., 2008; Corcolle et al., 2008;Tang and Yu, 2008) have been used separately tomodel the heterogeneity of the ME composites.Nonetheless, most of the theoretical developments arelimited to two-phase ME composites which form onlya subset of the multiferroics that encompasses a verydiverse class of materials. The homogenization frame-work implemented in this article provides a uniqueplatform to treat linear interactions of all forms of mul-tiferroics, namely, single-phase multiferroic, single crys-tals, polycrystals, as well as multiferroic composites.The representative microstructure (Figure 1) is com-posed of domains in single crystals, while crystalliteswith subgranular domain structure make the polycrys-tal microstructure. For composites, the microstructureis formed by single or polycrystalline magnetic and fer-roelectric materials. The accompanying FEM imple-mentation can encompass all these features by properchoice of finite elements.

Constitutive laws

We consider a multiferroic body occupying a volume Oand that physical properties of the material change

Figure 1. A schematic picture of the microstructure(representative unit-cell) of a multiferroic material.Various colors assigned to the constituents indicate the manifold

orientations of polarization (or magnetization). The arbitrary-shaped

constituents of the microstructure would assume different functions

according to the nature of the material phase. For instance, it will be a

domain if the material is a single-phase multiferroic and would be grain if

the material is a polycrystalline multiferroic and so on.

2 Journal of Intelligent Material Systems and Structures 0(0)

periodically and the period is equal to the dimension ofan elementary cell Y . A linear but not necessarilyhomogeneous material is assumed here. The nonlinearME effect, resulted from the coupling interactionbetween magnetostriction (nonlinear magneto-mechanical effect) and piezoelectricity, is not consid-ered in this model. A hierarchical schematic picture isshown in Figure 2, showing the various coordinatesinvolved as well as the composition of the multiferroicmacrostructure and the microstructure. For smalldeformation, the linear constitutive laws of multifer-roics in the absence of heat flux are given by

sij = CEHijkl ekl � ekijEk � eM

kijHk ð1Þ

Di = eijkejk + keHij Ej +aijHj ð2Þ

Bi = eMijkejk +ajiEj +meE

ij Hj ð3Þ

and the field equations are given by

sij, j + bi = r€ui ð4Þ

Di, i = 0

Bi, i = 0

�ð5Þ

Here, s, e, u, b, r, E, D, H, and B are stress, strain, dis-placement, body force, mass density, electrical field vec-tor, electric displacement vector, magnetic field vector,and magnetic flux density, respectively. CEH, e, eM, keH,meE, and a are stiffness, strain to (electric, magnetic)field coupling constants (or piezoelectric and piezomag-netic coefficients), permittivity (dielectric), (magnetic)

permeability, and ME coupling constant, respectively.Equation (5) resulted from the quasi-static assumptionon the steady-state Maxwell’s equations for electromag-netic phenomena. In addition to the above equations,we have the strain–mechanical displacement relationsand the electrical (magnetic) field–electrical (magnetic)potential relations

eij =1

2(ui, j + uj, i) ð6Þ

Ei = � u, i

Hi = � c, i

�ð7Þ

where u and c are scalar potentials. A comma in thesubscript denotes spatial differentiation, and summa-tion is applied to repeated indices. (We have usedEinstein summation convention, that repeated indicesare implicitly summed over, throughout this text.)Obviously, the above system of equations is to be juxta-posed with the boundary and initial conditions for oneto seek the solution of the multiferroic constitutiveequations.

In homogenization theory, it is assumed that thematerial is locally formed by the spatial repetition ofvery small microstructure (representative volume ele-ment (RVE) or unit-cell) when compared with the over-all macroscopic dimensions. The homogenization,mathematically applicable to heterogeneous media withreasonable contrasts in physical properties, replaces theheterogeneity by an equivalent effective medium withuniform physical characteristics (Sanchez-Palencia,1980). The constitutive laws in equations (1) to (3),

Figure 2. A schematic picture showing the progression of different coordinate transformations involved in the homogenization ofmultiferroics.The macroscopic multiferroic Ω is constituted by juxtaposing a number of identical unit-cells of size Y in three-dimensional (3D) space. (Here, Ω and

Y whose coordinate systems are related through y=x=e are drawn in two-dimensions for brevity.) Also, the orientation of a multiferroic material is

essentially that of the underlying crystallographic orientation. Also shown how the orientation is characterized by Euler angles (f, u,q) in a typical

ABO3 perovskite crystal. (x0, y0, z0) is the rotated (by Euler angles (f, u,q)) crystallographic coordinate system (x, y, z). The rotation A would be

obtained by the coordinate transformations aij given in equation (33).

Jayachandran et al. 3

therefore, can be rewritten with the effective propertiesand average fields as

hsiji = eCEHijkl hekli � eekijhEki � eeM

kijhHki ð8Þ

hDii = eeijkhejki+ ekeHij hEji+ eaijhHji ð9Þ

hBii = eeMijkhejki+ eajihEji+ eme E

ij hHji ð10Þ

where hji denotes volume averages, namely,

((1= Vj j)Ð

VdV ), and the coefficients with tilde, namely,eCEH

ijkl , eeMijk , . . ., refer to the homogenized magneto-

electromechanical properties. The indicesi, j, k, l = 1, . . . , d, where d is the dimension of thespace (physically d = 2 or d = 3).

Homogenization theory of multiferroics

Magneto-electromechanical problem of multiferroics

Let O be bounded region of space R3 of coordinates

x (or xi) (Figure 2). For a homogeneous material, thephysical properties do not depend on x (the globalframe of reference). Nonetheless, if the material is het-erogeneous (not homogeneous), the magneto-electromechanical properties, CEH, e, eM, keH, meE, anda, effectively depend on x, such that CEH[CEH(x),e[e(x), eM[eM(x), keH[keH(x), meE[meE(x), anda[a(x). But for materials with a periodic structure,they are periodic functions of space variables. In cer-tain cases, the length of the period is very small withrespect to the other lengths appearing in the problem.In such cases, for instance, the present case of multifer-roics, the solution is approximately same as the corre-sponding solution for a homogenized material withconstant matrices, eCEH,ee,eeM, ~keH, emeE, and ~a.

We shall consider O as a heterogeneous body obtainedby the translation of microstructure of edges y0

i of size Y,as shown in Figure 1. The material properties, CEH, e,eM, keH, meE, and a, are all Y-periodic functions, suchthat we restrict attention to the case of periodic mediahaving a very small spatial period e, that is

CEHeijkl (x)=CEHe

ijkl (x, y); eekij(x)= ee

kij(x, y)

eMkij

e(x)= eMkij

e(x, y); keHij

e(x)= keHij

e(x, y)

meEij

e(x)=meEij

e(x, y); aeij(x)=ae

ij(x, y)

8><>: ð11Þ

with y(= x=e) 2 Y . Here, e is a real positive parameterand is the characteristic length of the spatial period.The magneto-electromechanical property tensors satisfythe usual symmetry and positivity properties as detailedin Appendix 1. Dependence of properties on y meansthat they vary within a very small region with dimen-sions much smaller than those of macroscopic level. y isa magnified frame of reference with respect to x. Themicroscopic coordinate y is magnified by a factor 1=econsidering the fact that y= x=e and e\\1. In other

words, x is a scaled down coordinate system comparedto y by a factor of e. Thus, y can trace out more detailsand information of the trajectory of the evolution of afunction compared to x.

Let the surface G of the body O be subjected to pre-scribed surface traction tk and surface charge per unitarea s and normal magnetic flux density Bn. The energyfunctional (Qin and Yang, 2009) for a ME multiferroiccan be written as

G(ue,ue,ce)=

ðe

½12

CEHeijkl (x, y)e

eije

ekl � ee

ikl(x, y)Eei e

ekl

�eMeikl (x, y)H

ei ee

kl �1

2keH e

ij (x, y)Eei E

ej � ae

ij(x, y)Eei H

ej

� 1

2meE e

ij (x, y)H ei H e

j �de

�ðG

tkuekdG+

ðG

suedG+

ðG

BncedG

�ðe

1

2re(x, y)( _ue

k)2de�

ðe

bek(x, y)u

ekde ð12Þ

Here, Bn is the prescribed value of the normal compo-nent of magnetic flux density on the boundary,re([r(x, y)) is the density and b([b(x, y)) is the bodyforce. Here, surface traction tk , surface charge densitys, and normal magnetic flux density Bn are assumed tobe independent of the spatial scale e.

Homogenization of multiferroics

We describe in this section the formal asymptotic pro-cedure for solving equation (12), as e! 0. One seeks aformal solution to the problem in the form of a stan-dard two-scale asymptotic expansion up to the first-order variation terms

ue = u0(x, y)+ eu1(x, y)ue = u0(x, y)+ eu1(x, y)ce = c0(x, y)+ ec1(x, y)

9=;, y= x=e ð13Þ

where u1(x, y), u1(x, y), and c1(x, y) are functions to bedetermined, which are Y-periodic in y. It can be shownthat the functions u0(x, y), u0(x, y), and c0(x, y) areconstants in y (Sanchez-Palencia, 1980). This points outthat u0(x), u0(x), and c0(x) are the homogeneous partof the solution and are functions in x alone and variesin the ‘‘slow’’ scale x. The u1(x, y), u1(x, y), and c1(x, y)are the local variations describing the heterogeneouspart of the solution and are associated with y= x=e inthe ‘‘fast’’ changing scale. Also the field perturbations,u1(x, y), u1(x, y), and c1(x, y) (denoted by the trailingterms in asymptotic expansion equation (13)) must beY -periodic, and this condition plays the role of theboundary conditions. Based on this notion, the gradi-ents of various fields from equation (13) can be writtenas

4 Journal of Intelligent Material Systems and Structures 0(0)

∂uei

∂xj

=∂ui

0(x)

∂xj

+ e∂u1

i (x, y)

∂xj

+∂u1

i (x, y)

∂yj

∂ue

∂xj

=∂u0(x)

∂xj

+ e∂u1(x, y)

∂xj

+∂u1(x, y)

∂yj

∂ce

∂xj

=∂c0(x)

∂xj

+ e∂c1(x, y)

∂xj

+∂c1(x, y)

∂yj

8>>>>>>>><>>>>>>>>:ð14Þ

when y= x=e. We would get the variation dG by takingthe variation of the energy functional in equation (12),as shown in Appendix 2.

Collecting coefficients of du0i , du1

i , . . . of the varia-tional equation (41) in Appendix 2, considering thatthey are independent and also considering that

lime!0

ðe

f(x, x=e)de=1

Yj j

ðe

ðY

f(x, y)dyde ð15Þ

will lead to a set of macroscopic (equations involvingthe variables du0

i , du0, and dc0) and microscopic (equa-tions involving the variables du1

i , du1, and dc1) equa-tions. Due to linearity of the problem, and assumingseparation of variables, we denote

u1k(x, y)= xmn

k (x, y)emn(u0(x))+Fm

k (x, y)

3∂u0(x)

∂xm

+Gmk (x, y)

∂c0(x)

∂xm

u1(x, y)=hmn(x, y)emn(u0(x))+Rm(x, y)

3∂u0(x)

∂xm

+Cm(x, y)∂c0(x)

∂xm

c1(x, y)= lmn(x, y)emn(u0(x))+Ym(x, y)

3∂u0(x)

∂xm

+Qm(x, y)∂c0(x)

∂xm

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

ð16Þ

Here, x, R, and Q are characteristic material, electric,and magnetic displacements. F, G, h, C, l, and Y arecharacteristic coupled functions. The materialfunctions (electromechanical property tensors such asCEH, e, and eM) are assumed to be functions of classL‘(Y ), which means they belong to the Lebesgue classof integrals, in the sense that an integrable functionis summable as well. Moreover, the characteristic func-tions introduced in equation (16) are required to belongto subspaces of the form Hper(Y )= fv 2 H1(Y )gsuch that v takes equal values on opposite sides ofY and Hper(Y ,R

3)= fv=(vi)jvi 2 Hper(Y )g, whereH1 is a Sobolev space. Here, v is any arbitraryfunction.

Taking the spatial derivatives of the functions inequation (16) with respect to the microscopic coordi-nate y and substituting into the microscopic equationsresulted from equation (41) in Appendix 2 and makinguse of the periodic boundary conditions and the iden-tity in equation (15), we obtained the microscopic equa-tions as

1

Yj j

ðY

½CEHijkl (x, y)(dimdjn +

∂xmni

∂yj

)+ eikl(x, y)∂hmn

∂yi

+ eMikl(x, y)

∂lmn

∂yi

�3 ekl(du1(x, y))dY = 0

8du1 2 Hper(Y ,R3) ð17Þ

1

Yj j

ðY

½eikl(x, y)(dim +∂Rm

∂yi

)+CEHijkl (x, y)3

∂Fmi

∂yj

+ eMikl(x, y)

∂Ym

∂yi

�3 ekl(du1(x, y))dY = 0,

8du1 2 Hper(Y ,R3) ð18Þ

1

Yj j

ðY

½eMikl(x, y)(dim +

∂Qm

∂yi

)+CEHijkl (x, y)3

∂Gmi

∂yj

+ eikl(x, y)∂Cm

∂yi

�3 ekl(du1(x, y))dY = 0,

8du1 2 Hper(Y ,R3) ð19Þ

1

Yj j

ðY

½eikl(x, y)(dkmdln +∂xmn

k

∂yl

)� keHij (x, y)

∂hmn

∂yj

�aij(x, y)∂lmn

∂yj

�3 (∂du1(x, y)

∂yi

)dY = 0,

8du1 2 Hper(Y ) ð20Þ1Yj jÐY

½eikl(x, y)∂Fm

k

∂yl� keH

ij (x, y)(djm + ∂Rm

∂yj)� aij(x, y)

∂Ym

∂yj�

3 (∂du1(x, y)

∂yi

)dY = 0, 8du1 2 Hper(Y ) ð21Þ

1

Yj j

ðY

½eikl(x, y)∂Gm

k

∂yl

� keHij (x, y)

∂Cm

∂yj

� aij(x, y)

(djm +∂Qm

∂yj

)�3 (∂du1(x, y)

∂yi

)dY = 0, 8du1 2 Hper(Y )

ð22Þ1

Yj j

ðY

½eMikl(x, y)(dkmdln +

∂xmnk

∂yl

)� aij(x, y)∂hmn

∂yi

� meEij (x, y)

∂lmn

∂yj

�3 (∂dc1(x, y)

∂yi

)dY = 0, 8dc1 2 Hper(Y ) ð23Þ

1

Yj j

ðY

½eMikl(x, y)

∂Fmk

∂yl

� aij(x, y)3 (dim +∂Rm

∂yi

)

�meEij (x, y)

∂Ym

∂yj

�3 (∂dc1(x, y)

∂yi

)dY = 0,

8dc1 2 Hper(Y ) ð24Þ

1

Yj j

ðY

½eMikl(x, y)

∂Gmk

∂yl

� aij(x, y)∂Cm

∂yi

� �

�meEij (x, y)(dim +

∂Qm

∂yi

)�3 (∂dc1(x, y)

∂yi

)dY = 0,

8dc1 2 Hper(Y ) ð25Þ

Jayachandran et al. 5

where ekl(du1(x, y)) = (1=2)(∂du1

i (x, y)=∂yj + ∂du1j

(x, y)=∂yi). The solution of the microscopic equations(equations (17) to (25)) would yield the unknown peri-odic functions entering into equation (16). And this setof problems is called local problems. Substitution of thesolution of the local equations back into equation (16)and subsequently into equation (13) would unveil thecharacteristics of the local fields, namely, the displace-ments, magnetic, and electrical potentials of the multi-ferroic material.

After performing analogous mathematical opera-tions, as did for obtaining the microscopic equations(Equations (17) to (25)), one can arrive at the set ofmacroscopic equations. Having solved the local prob-lems, we can arrive at the homogenized (effective) mod-uli as

eCEHklmn =

1

Yj j

ðY

½CEHijpq(x, y) dimdjn +

∂xmni

∂yj

� �+ eipq(x, y)

∂hmn

∂yi

+ eMipq(x, y)

∂lmn

∂yi

3 dpkdql +∂xkl

p

∂yq

!dY ð26Þ

eeklm =1

Yj j

ðY

½eipq(x, y)(dim +∂Rm

∂yi

)+CEHijpq(x, y)

∂Fmi

∂yj

+ eMipq(x, y)3

∂Ym

∂yi

�(dpkdql +∂xkl

p

∂yq

)dY ð27Þ

eeMklm =

1

Yj j

ðY

½eMipq(x, y) dim +

∂Qm

∂yi

� �+CEH

ijkl (x, y)∂Gm

i

∂yj

+ eikl(x, y)3∂Cm

∂yi

� dpkdql +∂xkl

p

∂yq

!dY ð28Þ

ekeHnm =

1

Yj j

ðY

½epkl(x, y)∂Fm

k

∂yl

� keHpj (x, y)3 djm +

∂Rm

∂yj

� �

�apj(x, y)∂Ym

∂yj

�3 dnp +∂Rn

∂yp

� �dY ð29Þ

emeEij =

1

Yj j

ðY

heM

pkl(x, y)∂Gm

k

∂yl

� meEpj (x, y)3 djm +

∂Qm

∂yj

� �

�apj(x, y)∂Cm

∂yj

i3 ðdnp +

∂Qn

∂yp

)dY ð30Þ

eaij =1

Yj j

ðY

½epkl(x, y)∂Gm

k

∂yl

� keHpj (x, y)

∂Cm

∂yj

�apj(x, y)(djm +∂Qm

∂yj

)�3 (dnp +∂Rn

∂yp

)dY ð31Þ

Rigorously, the two-scale asymptotic analysis leadsdirectly to the strong formulation of the local problemsdepicted in equations (17) to (25). Consequently, thematerial functions that appear in those equations mustbe more regular (Glaka et al., 1992). Nevertheless, thepoint of departure is the variational (weak) formulation

of the problem shown in Appendix 2, which would leadto the local problems shown above. This would essen-tially map the underlying magneto-electromechanicalfields of a multiferroic material from a macroscopiccoordinates into a local coordinate system possessingmore resolutions.

Rotation of multiferroics

The expressions for homogenized material coefficientsthat appear in section ‘‘Homogenization of multifer-roics’’ could be used for finding out the effective prop-erties of polycrystalline and composite multiferroics.While dealing with both cases, we must take intoaccount the significant impact of the orientation effectof the underlying constituent domains (or crystals). Infact, multiferroic domains are volume elements with uni-form orientation of constituent crystallographic unit-cells (see Figure 2 in order to distinguish between therepresentative unit-cell or the microstructure used forsampling the macroscopic multiferroic material and thecrystallographic unit-cell).

The physical quantities that appear in the homogeni-zation equations, namely, CEH

ijkl (x, y), eijk(x, y), eMijk(x, y),

keHij (x, y), meE

ij (x, y), and aij(x, y), are the magneto-electromechanical properties of the crystallite whichconstitutes the microscopic structure and can bedescribed in microscopic coordinates y as

CEHijkl (x, y)= aipajqakralsC

EH9pqrs(x, y, z) ð32aÞ

eijk(x, y)= aipajqakre0pqr(x, y, z) ð32bÞ

eMijk(x, y)= aipajqakre

M9pqr(x, yz) ð32cÞ

keHij (x, y)= aipajqkeH9

pq (x, y, z) ð32dÞ

meEij (x, y)= aipajqmeE9

pq (x, y, z) ð32eÞ

aij(x, y)= aipajqa0ij(x, y, z) ð32fÞ

where aij are the Euler transformation matrix compo-nents from crystallographic coordinate system (x, y, z)to the local microscopic coordinates y. Here, theprimed moduli are the ones expressed in crystallo-graphic coordinate system. Also, the ME response isdetermined along an arbitrary crystallographic direc-tion determined by the Euler angles (f, u,q) withrespect to the reference frame of the microstructure,that is, y. (Here, the microstructure refers to the RVEof the multiferroic single crystal used for the homogeni-zation.) Euler angles f, u, and q can completely specifythe orientation of the crystallographic coordinatesystem embedded in domains and thereby the orienta-tion relative to a fixed Cartesian coordinate system.Figure 2 explains how the Euler angles map the rota-tion of a typical perovskite.

6 Journal of Intelligent Material Systems and Structures 0(0)

The transformation matrix from the crystallographiccoordinate system (x, y, z) to the local coordinate systemy is given by (Goldstein, 1978)

aij =

cosq cosf� cos u sinf sinq cosq sinf+ cos u cosf sinq sinq sin u

� sinq cosf� cos u sinf cosq � sinq sinf+ cos u cosf cosq cosq sin u

sin u sinf � sin u cosf cos u

0@ 1Að33Þ

Before being used in the homogenization of poly-crystals and composites of multiferroics, the transfor-mation equations (32) and (33) could well be used forcharacterizing the transformed magneto-electromechanical moduli of rotated multiferroic singlecrystals. Here, we assume that the multiferroic singlecrystal to be monodomain. Nevertheless, it is possibleto model the rotation of real poly-domain single crys-tals using the methodology described in the followingsection but with the knowledge of the proper domainorientation distribution function. Symmetric reductionshould be applied according to the single-crystal sym-metry. Knowledge of the full set of magneto-electromechanical moduli along the spontaneous polar-ization/magnetization direction of a multiferroic singlecrystal is necessary to have a complete picture of theorientational effect. Nonetheless, due to the paucity ofthe full set of measured data of the multiferroics,single-crystal orientation analysis is not possible.

Polycrystalline multiferroics

Polycrystalline ceramics present ease in manufactureand in compositional modifications compared to singlecrystals. As-grown polycrystalline multiferroic materialis an aggregate of single crystalline grains (or crystal-lites) with randomly oriented (spontaneous) electric (ormagnetic) polarizations. Each grain in a polycrystallinematerial is assumed to be made of a single, pinned, che-mically homogeneous domain. External magnetic or

electrical loading can reorient either the polarizationaxis or the magnetization easy axis or both of most ofthe domains. Figure 3 shows the schematics of domainswitching in a multiferroic polycrystal due to electricalpoling. However, electric and magnetic states can bereversibly controlled by an electrical field alone, whichsimultaneously switches the electrical polarizationinside a ferroelectric domain and controls the equiva-lent magnetic domains (Lee et al., 2008). Hence, thematerial could be transformed from a randomlyoriented sample into a mostly oriented (textured) one.In a randomly oriented multiferroic polycrystal, theorientations of domains with reference to a fixed coor-dinate system (in this study, it is the microscopic coor-dinates y) would be different for each domain, that is,the distribution of Euler angles (f, u,q) subtended bythe domains would fall in a uniform distribution for anas-grown crystal (Figure 3(a)). Nonetheless, as theexternal load is applied, the domains get aligned andtheir orientation distribution slowly deviates from theuniform distribution (Figure 3(b)). One of the possibili-ties is that the resultant orientation distribution afterloading could be a Gaussian one with the mean pointstoward a preferred direction fully dictated by the crys-tallographic symmetry of the multiferroic (Lee et al.,2008; Schmid, 1994). The probability distribution func-tion (pdf) of Gaussian distribution is defined by

f (ajm,s)=1

(sffiffiffiffiffiffi2pp

)exp� (a� m)2

2s2

� �ð34Þ

where m and s are the parameters of the distribution,namely, the mean and the standard deviation, respec-tively. a stands for the Euler angles (f, u,q).

Here, the Gaussian distribution can encompass bothextremes of the polycrystal configuration: the perfectlytextured as well as the random sample. It can beexplained in the following way: as the standard devia-tion s! 0, the distribution becomes narrow and onecan approach perfect texture close to single crystals.While s! ‘, the distribution becomes flatter (close touniform distribution) and generates a randomlyoriented polycrystal with little net polarization/magne-tization, if not none. Euler angles generated accordingto the above equation (34) could be used in equations(32) and (33) to find out the transformed moduli ofeach domain/grain in the polycrystal. These trans-formed moduli could consequently be plugged intoequations (26) to (31) for the homogenization of thepolycrystals.

Numerical results and discussion

In order to solve the microscopic system of equationsgiven through equations (17) to (25) in section‘‘Homogenization of multiferroics,’’ we developed afinite element formulation. (Further details of finite

Figure 3. Possible switching of domains that reorient thepolarization vectors (arrows) from unpoled (a) to poled (b)configuration due to electrical field (E) loading in polycrystalmultiferroic material. We have not shown the equivalentmagnetic domains or the corresponding magnetization here.

Jayachandran et al. 7

element formulation are given in Appendix 3.) A sim-ple, three-dimensional (3D) ‘‘multiferroic finite ele-ment’’ is conceived with the simplest nodalarrangement for a brick element employing the verticeswith 5 degrees of freedom (DOFs) (three spatial andone each for magnetic and electrical potentials). In thepresent context, 8-noded isoparametric elements with2 3 2 3 2 Gauss point integration are used to obtainsolutions. The use of the 2 3 2 3 2 Gauss point integra-tion has been proved to be optimal and adequate sinceit represents the best sampling point for the values ofderivatives such as the electrical/magnetic fields(Zienkiewicz, 1971). Substituting equation (42) inAppendix 3 into the microscopic equations (17) to (25)in section ‘‘Homogenization of multiferroics’’ andassembling the individual equations for each finite ele-ment, one can obtain the following global system ofequations for each load case mn or m

Kuu Kuu Kuc

Ktuu �Kuu �Kuc

Ktuc �Kt

uc �Kcc

24 35 xmn

hmn

lmn

8<:9=;=

Fmn1

Fmn2

Fmn3

8<:9=; ð35Þ

Kuu Kuu Kuc

Ktuu �Kuu �Kuc

Ktuc �Kt

uc �Kcc

24 35 Fm

Rm

Ym

8<:9=;=

fm1

fm2

fm3

8<:9=; ð36Þ

Kuu Kuu Kuc

Ktuu �Kuu �Kuc

Ktuc �Kt

uc �Kcc

24 35 Gm

Cm

Qm

8<:9=;=

fm1fm2fm3

8<:9=; ð37Þ

where the superscript t indicates transpose of the corre-sponding matrix. The globally assembled matricesKuu,Kuϕ,Kuc,Kϕϕ,Kϕc , and Kcc in equations (35) to(37) are the mechanical stiffness matrix, the piezoelec-tric stiffness matrix, the piezomagnetic stiffness matrix,the dielectric stiffness matrix, the ‘‘magnetoelectric stiff-ness’’ matrix, and the ‘‘diamagnetic stiffness’’ matrix,respectively, and Fmn

i , fmi , and fmi , where i= 1, 2, 3, arethe magneto-electromechanical load vectors.xmn, hmn, lmn,Fm,Rm, Ym, Gm,Cm, and Qm are theglobally assembled vectors of characteristic functionsof the multiferroic medium. All the load cases aresolved imposing periodic boundary conditions.

As the representative unit-cell is expected to capturethe response of the entire multiferroic system where the

macroscopic material is constructed by adding, con-tiguously to the representative unit-cell, a large numberof other identical unit-cells, particular care is taken toensure that the deformation across the boundaries ofthe representative unit-cell is compatible with the defor-mation of adjacent unit-cells. This requirement is appli-cable to the electrical and magnetic potentials as well.Hence, all the load cases are solved by enforcing peri-odic boundary conditions in the unit-cell for the displa-cements, electrical, and magnetic potentials.

The multiferroics offer the possibility of fast low-power electrical write operation and nondestructivemagnetic read operation. One of the order parameters,either electronic or magnetic, is, in general, a weakproperty resulting from a complex phase transforma-tion, orbital ordering, geometric frustration, and so onin materials (Martin et al., 2010). Nevertheless, compo-site multiferroics offer extraordinary coupling at roomtemperature and above. Here, we apply the methodol-ogy to a simple two-phase multiferroic ME compositedue to the rarity of single-phase single-crystal data(Eerenstein et al., 2006) to model an ideal multiferroic.Laminated two-phase composites of a magnetostrictive(or piezomagnetic) and electrostrictive (piezoelectric)material (Figure 4) introduce indirect ME couplingmediated through strain (Eerenstein et al., 2006; Ryu etal., 2001). The system used here is a composite ofcobalt ferrite (magnetostrictive) and barium titanate(electrostrictive). We have estimated the effectivemagneto-electromechanical properties of the laminateas a function of the volume fraction of the electrostric-tive (barium titanate, BaTiO3) medium, with the mag-netostrictive (cobalt ferrite, CoFe2O4) medium as thebase material.

The set of linear equations in equations (35) to (37)is coupled by the matrices Kuu,Kuc, and Kuc. Theseequations can be uncoupled into piezoelectric finite ele-ment equations and piezomagnetic finite element equa-tions by the following way. First, if we let thepiezoelectric stress tensor eijk = 0, then this wouldimply that Kuu = 0. Next, if we let the piezomagnetictensor eM

ijk = 0, the corresponding stiffness matrixKuc = 0. Moreover, the dielectric permittivity

keHij = 0) Kuu = 0

for the former case and magnetic permeability

meEij = 0) Kcc = 0

for the latter case. The ME coupling aij and the corre-sponding Kuc would obviously vanish in bothinstances. Since our model is general in the sense that itencompasses the ME multiferroic material as a whole,we apply the above notion to treat the magnetostrictiveand electrostrictive components of the two-phasecomposite.

Figure 4. Schematic diagram of a composite two-phasemultiferroic magnetoelectric laminate.

8 Journal of Intelligent Material Systems and Structures 0(0)

The material properties of the individual phases ofthe composite used for the simulation taken from thePan and Heyliger (2002) are listed in Table 1. Thehomogenized magneto-electromechanical propertiescalculated using the present model are shown in Figures5 to 9. The results compare well with the micromechani-cal calculation on laminated BaTiO32CoFe2O4 compo-site by Li and Dunn (1998). This is a proof of thevalidity of the present model. Experiments on the par-ticular geometry, that is, the 2-2 laminar ME compositeusing materials CoFe2O4 and BaTiO3 are scant in theliterature for direct comparison (Nan et al., 2008).Thus, a quantitative comparison is difficult due to lackof information regarding the ME coupling tensor per-taining to the CoFe2O42BaTiO3 laminate. However,geometries such as particulates, powder composites,and 1-3 fiber nanocomposites, heterostructure consist-ing of nanopillars of the ferro/ferrimagnetic phase(CoFe2O4) embedded in a ferroelectric matrix (BaTiO3)are reported (Zheng et al., 2004). Nevertheless, laminatecomposites are potentially useful because of the largecoupling between electrical and magnetic properties dueto large area of contact between phases. Yet the self-assembly of the nanocomposite occurs at volume frac-tions (0.35CoFe2O420.65BaTiO3) (Zheng et al., 2004)close to the volume fraction (’ 0.7BaTiO3) in ourmodel at which the laminate exhibits maximum longitu-dinal ME coupling (see Figure 9).

Another important aspect to be noted in this study isthat neither the piezoelectric nor the magnetostrictivephase exhibits the ME effect manifested by the ME cou-pling eaij. Nonetheless, the composite phase derives thisproperty and the nonvanishing components, namely,ea11 and ea33, exhibit contrast in their dependence on thepiezoelectric component fraction as shown in Figures 8and 9. ea11 shows a maximum (absolute value) ataround vf = 0:5 of BaTiO3 (Figure 8). However, thelongitudinal component of ME coefficient ea33 showsthe maximum (absolute value) at around vf = 0:8 ofBaTiO3 (Figure 9). Moreover, from the simulationresults, we observe that they display a symmetry suchthat ea11[ea22. Only two (i.e. ea11([ea22) and ea33) of thenine components of eaij are found to be independentand nonvanishing in BaTiO32CoFe2O4 composite.

Conclusion

A rigorous theoretical model, in a generic setting, isdeveloped for the homogenization of multiferroics fortreating single-phase multiferroic single crystals as wellas polycrystals and for multiferroic composites. Themicrostructural features such as orientation and textureare incorporated fully into the modeling framework. Amultiferroic microstructure (or representative unit-cell)is conceived in the local scale. This unit-cell couldencompass the features of single crystals possessing

Table 1. Material properties of BaTiO3 and CoFe2O4 used forsimulation (Pan and Heyliger, 2002).

Physical property BaTiO3 CoFe2O4

C11 166 286C12 77 173C13 78 170.5C33 162 269.5C44 43 45.3C66 44.5 56.5e31 24.4 0e33 18.6 0e15 11.6 0keH

11 11.2 0.08keH

33 12.6 0.093eM

31 0 580.3eM

33 0 699.7eM

15 0 550meE

11 5 2590meE

33 10 157

Here, Cmn in units of 109 N/m2, ejn in C/m2, keHij in 10�9 C2/Nm2, eM

jn in

N/A m, and meEij in 10�6 Ns2/C2.

Figure 5. The homogenized elastic stiffnesses eCEHmn of the

laminated multiferroic magnetoelectric compositeBaTiO32CoFe2O4 versus the volume fraction vf of BaTiO3: (a)the homogenized eCEH

11 ,eCEH

12 , and eCEH13 and (b) eCEH

33 ,eCEH

44 , and eCEH66 .

Figure 6. The homogenized piezoelectric stress coefficientseeim of the laminated multiferroic magnetoelectric compositeBaTiO32CoFe2O4 versus the volume fraction vf of BaTiO3.

Jayachandran et al. 9

domain structure, polycrystals possessing crystalliteswith subgranular domain structure, and compositesformed by single or polycrystalline magnetic and ferro-electric materials. The homogenization expressions forentire magneto-electromechanical properties arederived assuming the macroscopic body is build by thecontiguous juxtaposition of the representative unit-cellin 3D space. The proposed modeling approach cantreat multiferroics irrespective of their crystallographicsymmetry, multiferroic polycrystals, and multiphaseME composites of any geometry and configuration,sandwich structures, and so on.

The variational formulation of the magneto-electromechanical problem enables us to derive a set of

local problems which essentially unravel the relation-ships among the underlying microscopic fields.Homogenization is implemented in FEM by solvingthese local coupled equilibrium electrical, magnetic,and mechanical fields to enable computation of theeffective properties. A ‘‘multiferroic finite element’’ isconceived with 5 DOFs (three for the spatial coordi-nates and one each for the electrical and magneticpotential) per node.

The transformation of the multiferroic’s physicalproperties due to the crystal rotation is characterizedthrough Euler angles (f, u,q). The correspondencebetween various scales and frames of referencesinvolved in this study is demonstrated (see Figure 2).The incorporation of the aggregate texture (orientationdistribution) of a multiferroic polycrystal into the pres-ent model is shown with a simple example of aGaussian distribution function. As an example case,the present model is applied to a two-phase multiferroicME composite to evaluate its homogenized physicalproperties. The general multiferroic FEM is used forthe solution by ‘‘turning off’’ (uncouple) the magneticDOF to treat the electrostrictive phase, and the electri-cal DOF is ‘‘turned off’’ to treat the magnetostrictivephase. The evolution of the homogenized physicalproperties with the varying volume fraction of the elec-trostrictive phase compares well with other models.The nonvanishing components of ME coupling eaij ten-sor is obtained for the product phase (i.e., magnetoelec-tric composite phase) although none of the constituentphases possess them individually. Hence, the modelcaptures the coupling between magnetic and ferroelec-tric subsystems in this case mediated via strain.

The size-effect and the nonlinear behavior of the‘‘responses’’ (i.e. the magnetization M in an electrical

Figure 7. The homogenized piezomagnetic coefficients eeMim of

the laminated multiferroic magnetoelectric compositeBaTiO32CoFe2O4 versus the volume fraction vf of thepiezoelectric BaTiO3.

Figure 8. The homogenized magnetoelectric coefficient ea11 ofthe laminated multiferroic magnetoelectric compositeBaTiO32CoFe2O4 versus the volume fraction vf of thepiezoelectric BaTiO3.

Figure 9. The homogenized magnetoelectric coefficient ea33 ofthe laminated multiferroic magnetoelectric compositeBaTiO32CoFe2O4 versus the volume fraction vf of piezoelectricBaTiO3.

10 Journal of Intelligent Material Systems and Structures 0(0)

field E and the inverse effect of polarization P gener-ated by the application of the magnetic field H) are notaccounted in the present model. Nevertheless, the size-effect scales down to crystallographic cell size (fewAngstroms) in ferroic materials that warrant an atomis-tically enriched continuum model to address the issue.A plausible approach to address both these deficienciesmight be to develop the energy functional with non-linear terms which contain force constants to be evalu-ated from atomistic potentials. A modeling frameworkin this direction that ensures the accuracy of the ato-mistic models and the robustness and economy of thecontinuum mechanics are in progress.Yet the homoge-nization model in the present work covers all the formsof multiferroics (and its composites) and captures theirlinear equivalent (effective) behaviour and offer com-putational efficiency besides unveiling the nature of theunderlying microscopic field characteristics.

Acknowledgements

We would like to acknowledge G. Srinivasan and K.P.Surendran for insightful discussions. K.P.J. acknowledges theaward of Ciencia 2007 from Fundacxao para a Ciencia e aTecnologia (FCT), Portugal.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research project is partially funded by FCT (Fundacaopara a Ciencia e a Tecnologia, Portugal) through projectPTDC/EME-PME/120630/2010.

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Appendix 1

Symmetry and positive definiteness

The elastic moduli CEH eijkl , dielectric permittivity keH e

ij ,and magnetic permeability meE e

ij tensors are symmetricand positive definite with

CEHeijkl =CEHe

klij =CEHeijlk =CEHe

jikl , and9gc.0

: CEHeijkl jijjkl � gcjijjij, 8jij = jji 2 E

3s

keH eij = keH e

ji , and 9gk.0

: keH eji jijj � gkjiji, 8ji = jj 2 R

3

meE eij =meE e

ji , and 9gm.0

: meE eij jijj � gmjiji, 8ji = jj 2 R

3,

8>>>>>>>><>>>>>>>>:ð38Þ

where E3s is the space of symmetric 3 3 3 matrices. The

third-order piezoelectric and piezomagnetic tensors sat-isfy the symmetries

eeijk = ee

ikj; eMeijk = eMe

ikj ð39Þ

while the magnetoelectric tensor aeij has nine indepen-

dent components (Fiebig, 2005).

Appendix 2

Variation of G

Substituting for strain and electrical and magnetic fieldsfrom equations (6) and (7), the energy functional givenin equation (12) becomes

G(ue,ue,ce)=1

2

ðe

1

4(ue

i, j + uej, i)(u

ek, l + ue

l, k)

CEHeijkl (x, y)de+

1

2

ðe

ue, i(u

ek, l + ue

l, k)

eeikl(x, y)de+

1

2

ðe

ce, i(u

ek, l + ue

l, k)eMeikl (x, y)de

� 1

2

ðe

ue, iu

e, jk

eH eij (x, y)de�

ðe

ue, ic

e, j

3 aeij(x, y)de� 1

2

ðe

ce, ic

e, jm

eE eij (x, y)de

�ðG

tkuekdG+

ðG

suedG

+

ðG

cBncedG� 1

2

ðe

re(x, y)( _uek)

2de�ðe

bek(x, y)u

ekde

ð40Þ

Substituting the field gradients of equation (14)back into the functional G given in equation (40), tak-ing the variation and passing the limit e! 0, we wouldget

12 Journal of Intelligent Material Systems and Structures 0(0)

lime!0fdG(ue,ue,ce)g= lim

e!0fðe

½14(u0

i, j(x)+ u0j, i(x)

+∂u1

i (x, y)

∂yj

+∂u1

j (x, y)

∂yi

)

CEHijkl (x, y)+

1

2(u0

, i(x)+∂u1(x, y)

∂yi

)eikl(x, y)

+1

2(c0

, i(x)+∂c1(x, y)

∂yi

)eMikl(x, y)�

(du0k, l(x)+ du0

l, k(x)+∂du1

k(x, y)

∂yl

+∂du1

l (x, y)

∂yk

)de

�ðe

(u0, j(x)+

∂u1(x, y)

∂yj

)

3 (du0, j(x)+

∂du1(x, y)

∂yi

)keHij (x, y)de

�ðe

(c0, j(x)+

∂c1(x, y)

∂yj

)

3 (dc0, j(x)+

∂dc1(x, y)

∂yi

)meHij (x, y)de

�ðe

½(c0, j(x)+

∂c1(x, y)

∂yj

)

3 (du0, j(x)+

∂du1(x, y)

∂yi

)+ (u0, j(x)

+∂u1(x, y)

∂yj

)(dc0, j(x)+

∂dc1(x, y)

∂yi

)�

3 aij(x, y)de�ðG

tidu0i (x)dG

+

ðG

sdu0(x)dG+

ðG

Bndc0(x)dG

� 1

2

ðe

r(x, y) _u0k(x)d _u0

k(x)de�ðe

bk(x, y)uekdeg= 0 ð41Þ

where d denotes the variational derivative. Here, theperiodic boundary conditions assumed would accountfor the absence of magneto-electromechanical perturba-tions (corresponding to u1

i ,u1, and c1) in surface inte-

grands in the above equation. In other words, thesephysical quantities are nonlocal in the sense that theyare independent of the local variable.

Appendix 3

Numerical implementation of homogenization

We have developed a three-dimensional (3D) model forour numerical implementation. Solution of the localproblems (equations (17) to (25)) given in section‘‘Homogenization of multiferroics’’ is sought in order toevaluate the homogenized magneto-electromechanical

coefficients given in through equations (26) to (31). Thenine microscopic equations (equations (17) to (25))should be solved for as much number of the unknowns,namely, the characteristic functions xmn

i ,hmn, lmn,Rm,Fm

i ,Ym,Qm,Cm, andGm

k . Subsequently, the spatial deri-vatives of the characteristic functions would be utilizedin the evaluation of the homogenized coefficients. Finiteelement formulation for the magneto-electromechanicalproblem is developed along the lines of the standardpiezoelectric formulation of Allik and Hughes (1970).In the finite element method (FEM), the body to becomputed is subdivided into small but discrete elements,the finite elements. The characteristic functions aredetermined at the nodes of these elements. To generatethe magneto-electromechanical matrix relations fora finite element, the elemental characteristic functionsfxge, fhge, . . . are expressed in terms of a

(= 1, 2, . . . , n where n = number of nodes of a finiteelement) nodal values via interpolation functions fb

fxge = ½fx�f xagfhge = ½fh�f hagflge = ½fl�flagfRge = ½fR�fRagfFge = ½fF�fFagfYge = ½fY�fYagfQge = ½fQ�fQagfCge = ½fC�fCagfGge = ½fG�fhag

9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;

ð42Þ

where xa, ha, la, . . . are the corresponding nodal val-ues of the characteristic functions xa, ha, la, . . .,respectively. All other magnetic, electrical, and mechan-ical quantities can be similarly interpolated with appro-priate interpolation functions. The problem is reducedto the standard variational finite element form after theusual approximation of FEM and can be expressedconcisely as

Ku= f ð43Þ

where K is the global stiffness matrix, u is the vector ofunknown functions, and f is the load vector (see section‘‘Numerical results and discussion’’). The full set of lin-ear equations resulted from the global assembly isshown in equations (35) to (37). Also, the completematrix involving the globally assembled matricesKuu,Kuϕ,Kuc,Kϕϕ,Kϕc, andKcc in equations (35) to(37) is indefinite due to the negative diagonal terms inthe matrices Kϕϕ andKcc. Hence, we have used a sky-line solver for the solution of the finite elementequations.

Jayachandran et al. 13