Composite Structures 174 (2017) 176–186

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Evaluation of effective material properties in magneto-electro-elasticcomposite materials

http://dx.doi.org/10.1016/j.compstruct.2017.03.1040263-8223/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: jan.sladek@savba.sk (J. Sladek).

Jan Sladek a,⇑, Vladimir Sladek a, Miroslav Repka a, Jozef Kasala b, Peter Bishay c

a Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakiab Faculty of Special Technology, University of Trencin, 91150 Trencin, SlovakiacCollege of Engineering and Computer Science, California State University, Northridge, CA, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 March 2017Revised 28 March 2017Accepted 29 March 2017Available online 12 April 2017

Keywords:Meshless local integral equationGradient theorySize effectMagnetoelectroelastic solidEffective material parameters

The meshless local integral equation method is developed to analyze general two-dimensional boundaryvalue problems in size-dependent magnetoelectroelastic solids. A consistent theory is developed for sizedependent magnetoelectroelasticity. The strain gradients are considered in the constitutive equations forelectric displacement and magnetic induction. The governing equations are derived with the correspond-ing boundary conditions using the variational principle. The local integral equations are subsequentlyderived and the meshless moving least square (MLS) numerical method is implemented to solve theseequations.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Modern smart structures made of piezoelectric and piezomag-netic materials offer certain potential performance advantagesover conventional ones due to their capability of convertingenergy from one type to another, among magnetic, electric, andmechanical [1,2]. It is well known that some composite materialscan provide superior properties compared to their virginmonolithic constituent materials [3]. The irregularity in the spatialarrangement of the fibers and their geometry can influence theestimation of the effective material properties of the unidirectionalcomposite. The experimental approaches are not convenientfor an optimal design of composites due to the expensive costof the measurements and the low efficiency. Therefore,mathematical and numerical models are frequently utilized toget homogenized material properties directly from those of theconstituents and from their microstructure. Then, these materialproperties are used for numerical analyses at macroscopic scale.However, sometimes such analyses are not accurate enough,especially if the size of the structure is comparable to the materiallength scale.

Due to their superior physical, electrical, optical, chemical andother properties, the nano/micro structures were expanded intomany areas such as nano-electromechanical devices [4], spaceand bio-engineering [5], actuators [6], and nanocomposites [7,8].The validity of using the classical continuum mechanics for micro/nano structures is inconclusive. Experimental as well as discreteatomistic methods such as molecular dynamics (MD) simulations[9,10] have been utilized in understanding and analysing the beha-viour of nano/micro structures. However, these methods are highlyexpensive and not applicable to real sized structures due to theextremely high computer hardware requirements. Fortunately,the continuum theory can be applied for these structures afterproper improvements or enhancements [11]. Recent experimentshave demonstrated that when the dimensions of the structureare of the same order of the material length scale, the stiffness ofthe structure increases. Size-dependent behaviour has beenobserved in many studies [12–15]. This phenomenon can beexplained by the presence of significant strain gradients whichappear because of the small size of the microstructural elements.Thus, the behaviour of the mesoscopic structural elements couldbe described by the continuum theory enriched by incorporationof strain gradients. Then, a consistent size-dependent continuummechanics theory can be a more efficient alternative to the atomis-tic models.

Magnetoelectroelastic (MEE) composites are made of piezoelec-tric and magnetostrictive phases coupled by a strain field. From

J. Sladek et al. / Composite Structures 174 (2017) 176–186 177

earlier investigation for macro-sized layered MEE, it is well-knownthat effective composite coefficients are higher than their con-stituents [16]. A similar enhancement of effective coefficients hasbeen observed for fiber composites. Kuo and Wang [17] optimizedthe effective magnetoelectric (ME) voltage coefficient of fibrouscomposites made of piezoelectric and piezomagnetic phases.They showed that at an optimal orientation of fibers, the effectivein-plane and out-of-plane ME coefficient can be enhancedmany-fold. Wang and Pan [18] investigated the influence of imper-fect matrix-fiber bond on the ME coefficient under longitudinalshear.

The influence of strain gradients in the gradient theory on theeffective material properties of MEE composites, if coated piezo-electric fiber is embedded into the piezomagnetic matrix, is inves-tigated in the present paper. This problem was analyzed beforeusing the classical elasticity theory. Kuo [19] and Kuo and Pan[20] analyzed fibrous composites with piezoelectric and piezomag-netic phases only under anti-plane shear deformation, whereasSladek et al. [21] analyzed the same composites under in-planedeformation. Ebrahimi and Dabbagh [22] also studied the flexuralwave propagation responses of smart FG MEE nanoplates via non-local strain gradient theory. If the thickness of the coating layer isvery small, the strain gradients should be included into the magne-toelectroelasticity mathematical model. The governing equationsfor magnetoelectroelastic solids in gradient elasticity with the cor-responding boundary conditions are derived from the variationalprinciple in the present paper.

Meshless methods are becoming very popular in numericalanalyses of engineering problems. A variety of meshless methodshas been proposed so far. They can be derived either from aweak- or strong-form. The weak formulation can be performedon the global domain or a set of local subdomains. Backgroundcells are required in the global weak formulation to performnumerical integration. However, no background cells are requiredin the local weak formulation. The moving least squares (MLS)approximation is often used for approximating the trial functions.It yields C1 continuity counterpart to conventional discretizationmethods, where discontinuity of secondary fields at the interfacesof elements occurs. Meshless methods have been also applied topiezoelectric problems [23,24]. Many meshless formulations canbe defined on the base of the meshless local Petrov-Galerkin(MLPG) method, since trial and test functions can be chosen fromdifferent functional spaces [25–27]. The test functions in the MLPGmethod can be arbitrary. The formulation with a Heaviside stepfunction as the test functions [28,29] leads to a simple form andit has been applied successfully to various boundary value prob-lems [30–34].

In the present paper, the MLPG method, with the gradient the-ory, is applied to 2-D problems of magnetoelectroelasticity. Thecorresponding governing equations are satisfied in a weak formon small fictitious subdomains. Nodal points are randomly dis-tributed over the analyzed domain. Each node is surrounded bya small circle for simplicity, but without loss of shape generality.Numerical integration over circles can be easily carried out. Thelocal integral equations have a very simple nonsingular form.The MLS scheme is used for spatial approximations of the displace-ments, the electric and magnetic potentials [35]. Performing thespatial integrations, a system of linear algebraic equations forthe unknown nodal values is obtained. The proposed computa-tional method is applied to evaluation of effective material proper-ties of a piezomagnetic matrix with regularly distributedpiezoelectric fibers with a nano-sized coating layer. The influenceof the coating layer on the effective MEE coefficients isinvestigated.

2. Basic equations for electric- and magnetic-strain gradienttheory

The electric field-strain gradient theory for nano-dielectricsintroduced by Hu and Shen [36] is extended here for a nano-magnetoelectroelastic material. The strain gradients exist only inthe higher order stress, electric displacement and magnetic induc-tion fields. Then, the constitutive equations are given by

rij ¼ cijklckl � ekijEk � qkijHk;

sjkl ¼ �f ijklEi � hijklHi þ gjklmnigmni;

Dk ¼ ekijcij þ eklEl þ aklHl þ f klmnglmn;

Bk ¼ qkijcij þ aklEl þ lklHl þ hklmnglmn; ;

ð1Þ

where cijkl; ekij; qkij; ekl;akl;lkl and f ijkl; hijkl; gjklmni are the materialproperty tensors. Particularly, ekl;lkl; cijkl are the second-order per-mittivity, permeability and the fourth-order elasticity constant ten-sors, respectively. Symbol ekij denotes the piezoelectric coefficient,qkij is the piezomagnetic coefficient, akl is the electromagnetic coef-ficient. The electric field–strain gradient coupling coefficient tensorf ijkl represents the higher-order electromechanical coupling inducedby the strain gradient, and hijkl is the magnetic field–strain gradientcoupling coefficient tensor. The higher-order elastic parametersgjklmni represent the purely the strain gradient elasticity theory.Symbols sijk and Di denote the higher-order stress tensor and elec-tric displacement vector, respectively. A subscript preceded by acomma denotes differentiation with respect to the correspondingCartesian coordinate. The Eistein summation is employed forrepeated lowercase indices.

The strain tensor cij, the electric field vector Ej and the magneticfield vector Hl are related to the displacements ui, the electricpotential / and the magnetic potential w, respectively by

cij ¼12ðui;j þ uj;iÞ; Ej ¼ �/;j; Hj ¼ �w;j ð2Þ

The strain gradient tensor [37] is defined as

gijk ¼ cij;k ¼12ðui;jk þ uj;ikÞ ð3Þ

with exhibiting the symmetry cij ¼ cji, gijk ¼ gjik. The constitutiveequations (1) for orthotropicmaterial can bewritten in amatrix form:

r11

r33

r13

26664

37775 ¼

c11 c13 0

c13 c33 0

0 0 c44

26664

37775

c11

c33

2c13

26664

37775�

0 e31

0 e33

e15 0

26664

37775

E1

E3

" #

�0 q31

0 q33

q15 0

26664

37775

H1

H3

" #

or r ¼ Cc� LE�MH ð4Þ

D1

D3

� �¼ 0 0 e15

e31 e33 0

� � c11c332c13

24

35þ e11 0

0 e33

� �E1

E3

� �þ a11 0

0 a33

� �H1

H3

� �

þm2 0 0 e15 0 0 e15e31 e31 0 e33 e33 0

� �g111

g331

2g131

g113

g333

2g133

266666664

377777775

or D¼ LTcþ JEþAHþm2Fg; ð5Þ

178 J. Sladek et al. / Composite Structures 174 (2017) 176–186

B1

B3

� �¼ 0 0 q15

q31 q33 0

� � c11c332c13

264

375þ a11 0

0 a33

� �E1

E3

� �þ l11 0

0 l33

� �

� H1

H3

� �þm2 0 0 q15 0 0 q15

q31 q31 0 q33 q33 0

� �g111

g331

2g131

g113

g333

2g133

2666666664

3777777775

or B ¼ MTcþ AEþ IHþm2Og; ð6Þ

s111s331s131s113s333s133

2666666664

3777777775¼ �m2

0 e310 e33e15 00 e310 e33e15 0

2666666664

3777777775

E1

E3

� ��m2

0 q31

0 q33

q15 00 q31

0 q33

q15 0

2666666664

3777777775

H1

H3

� �

þ l2

c11 c13 0 0 0 0c13 c33 0 0 0 00 0 c44 0 0 00 0 0 c11 c13 00 0 0 c13 c33 00 0 0 0 0 c44

2666666664

3777777775

g111

g331

2g131

g113

g333

2g133

2666666664

3777777775

or s ¼ �m2FTE�m2OTHþ l2Ggð7Þ

where gradient theory parameters gjklmni are proportional to elasticparameters cklmn by the internal length material parameter l [38].Similarly, the electric field-strain gradient and magnetic field-strain gradient coupling coefficient f ijkl and hijkl are expressed bypiezoelectric and piezomagnetic coefficients, respectively, and ascaling parameter m.

Consider a piezoelectric solid in domain Vwith boundary C. Thevariation of the electric Gibbs free energy in gradient theory ofmagnetoelectroelastic solids is given by

dU ¼ZVðrijdeij þ sijkdgijk þ Dkd/;k þ Hkdw;kÞdV : ð8Þ

Applying the Gauss divergence theorem, one gets

dU ¼ZVðrijdui;j þ sijkdui;jk þ Dkd/;k þ Hkdw;kÞdV

¼ �ZVðrij;jdui þ sijk;kdui;j þ Dk;kd/þ Hk;kdwÞdV

þZCðnjrijdui þ nksijkdui;j þ nkDkd/þ nkHkdwÞdC

¼ �ZV½ðrij;j � sijk;jkÞdui þ Dk;kd/þ Hk;kdw�dV

þZC½njðrij � sijk;kÞdui þ nksijkdui;j þ nkDkd/þ nkHkdw�dC

¼ �ZV½ðrij;j � sijk;jkÞdui þ Dk;kd/þ Hk;kdw�dV

þZC½tidui þ Ridsi þ Qd/þ Sdw�dC; ð9Þ

where the traction vector is defined as

ti ¼ njðrij � sijk;kÞ � @qi

@pþXc

kqiðxcÞkdðx� xcÞ ð10Þ

with qi :¼ nkpjsijk ð11Þand kqiðxcÞk :¼ qiðxc þ 0Þ � qiðxc � 0Þ is the jump at a corner on theoriented boundary contour C. Symbol pi is the Cartesian component

of the unit tangent vector on C, si :¼ @ui@n , Ri :¼ nknjsijk, Q :¼ nkDk,

S :¼ nkHk.In derivation of dU, the following identity is invoked:

nksijkdui;j ¼ qi@dui

@pþ Rid

@ui

@n

� �ð12Þ

The primary fields ðui; si; /; wÞ are energetically conjugatedwith ðti; Ri; Q ; SÞ. The work of the external generalized forcesð�ti; �Ri; �Q ; �SÞ is given by

dW ¼ZCt

�tiduidCþZCR

�RidsidCþZCQ

�Qd/dCþZCS

�SdwdC ð13Þ

From the principle of virtual work dU � dW ¼ 0, the followinggoverning equations are obtained from Eqs. (9) and (13),

rij;jðxÞ � sijk;jkðxÞ ¼ 0; Dk;kðxÞ ¼ 0; Hk;kðxÞ ¼ 0; ð14Þand two kinds of boundary conditions (b.c.) are:

Essential b:c: :

uiðxÞ ¼ �uiðxÞ onCu; Cu � C

siðxÞ ¼ �siðxÞ onCs; Cs � C

/ðxÞ ¼ �/ðxÞ onC/; C/ � C

wðxÞ ¼ �wðxÞ onCw; Cw � C

ð15Þ

Natural b:c: : tiðxÞ ¼ �tiðxÞ onCt ; Ct [ Cu ¼ C; Ct \ Cu ¼ £

RiðxÞ ¼ �RiðxÞ onCR; CR [ Cs ¼ C; CR [ Cs ¼ £

QðxÞ ¼ �QðxÞ onCQ ; CQ [ C/ ¼ C; CQ [ C/ ¼ £

SðxÞ ¼ �SðxÞ onCS; CS [ Cw ¼ C; CS [ Cw ¼ £

ð16Þ

3. The meshless local integral equation method

The MLPG method is the local weak-form with local fictitioussubdomains Xs. It is a small region constructed for each nodeinside the global domain [27,35]. The shape of local subdomainscould be of any geometry and size. In the present paper, the localsubdomains are circular for numerical simplicity. The local weak-form of the first governing equation (14) can be written asZXs

rij;jðxÞ � sijk;jkðxÞ� �

u�ikðxÞ dX ¼ 0 ð17Þ

where u�ikðxÞ is a test function.

Applying the Gauss divergence theorem to the weak-form (17)one obtainsZ@Xs

½rijðxÞ� sijk;kðxÞ�njðxÞu�ikðxÞdC�

ZXs

½rijðxÞ � sijk;kðxÞ�u�ik;jðxÞdX¼ 0;

ð18Þwhere @Xs is the boundary of the local subdomain. It consists ofthree parts @Xs ¼ Ls [ Cst [ Csu [35], where Ls is the local boundarythat is totally inside the global domain, Cst is the part of the localboundary which coincides with the global boundary with prescribedtractions, i.e., Cst ¼ @Xs \ Ct , and similarly Csu is the part of the localboundary with prescribed displacements, i.e., Csu ¼ @Xs \ Cu.

If a Heaviside step function is chosen as the test function u�ikðxÞ

in each subdomain as

u�ikðxÞ ¼

dik at x 2 Xs

0 at x R Xs

�; ð19Þ

the local weak-form (18) is transformed into the following localboundary-domain integral equationsZLsþCsu

njðrij � sijk;kÞdCþ qiðx fstÞ � qiðxs

stÞ ¼ �ZCst

�tidC; ð20Þ

J. Sladek et al. / Composite Structures 174 (2017) 176–186 179

where definition of the traction vector (10) is utilized and x fst , xs

st

stand for the final and starting points on Cst .Similarly, the local weak-form of the second governing equation

(14) can be written asZXs

Dk;kðxÞv�ðxÞ dX ¼ 0; ð21Þ

where v�ðxÞ is a test function.Applying the Gauss divergence theorem to Eq. (21) and choos-

ing the Heaviside step function as the test function v�ðxÞ one canobtainZLsþCs/

QðxÞdC ¼ �ZCsQ

�QðxÞdC; ð22Þ

where

QðxÞ ¼DkðxÞnkðxÞ ¼ ekijui;jðxÞ� ekl/;lðxÞ�aklw;lðxÞþ f klmnul;mnðxÞ� �

nk:

ð23ÞThe local integral equation corresponding to the third governing

equation (14) has the formZLsþCsw

SðxÞdC ¼ �ZCsS

�SðxÞdC; ð24Þ

where magnetic flux is given by

SðxÞ ¼ BkðxÞnkðxÞ ¼ qkijui;jðxÞ�akl/;lðxÞ�lklw;lðxÞþhklmnul;mnðxÞ� �

nj:

ð25ÞIn the MLPG method the test and the trial functions can be from

different functional spaces. The MLS approximation is used forapproximation of trial functions. A number of nodes is spread overthe domain of influence. The approximated functions for primaryfields (mechanical displacements, electric and magnetic potentials)are given by [33]

uhðxÞ ¼ NTðxÞ � u ¼Xna¼1

NaðxÞua; /hðxÞ ¼Xna¼1

NaðxÞ/a;

whðxÞ ¼Xna¼1

NaðxÞwa ð26Þ

where the nodal values ua ¼ ðua1; u

a3ÞT , /a and wa are fictitious

parameters for displacements, electric and magnetic potentials,respectively, and NaðxÞ is the shape function related to the node a.The number of nodes n used for the approximation is determinedby the weight function waðxÞ. The forth-order spline-type weightfunction is applied in the present work

waðxÞ ¼ 1� 6 da

ra

2þ 8 da

ra

3� 3 da

ra

4; 0 6 da 6 ra

0; da P ra

8<: ð27Þ

where da ¼ kx� xak and ra is the size of the support domain. Thisweight function has the C1� continuity over the entire domain,and therefore the continuity conditions of tractions, electric chargeand magnetic flux are satisfied.

The traction vector tiðxÞ at a boundary point x 2 @Xs can beeasily approximated in terms of primary fields as

thðxÞ ¼ NðxÞCXna¼1

BaðxÞua þ NðxÞLXna¼1

PaðxÞ/a þ NðxÞMXna¼1

PaðxÞwa

�m2NðxÞLXna¼1

Pa1ðxÞ/a �m2NðxÞL

Xna¼1

Pa3ðxÞ/a

�m2NðxÞMXna¼1

Pa1ðxÞwa �m2NðxÞM

Xna¼1

Pa3ðxÞwa

þ l2NðxÞCXna¼1

Ba1ðxÞua þ l2NðxÞC

Xna¼1

Ba3ðxÞua; ð28Þ

where the matrices C; L, and M are defined in Eq. (4), the matrixNðxÞ is related to the normal vector n(x) on @Xs by

NðxÞ ¼ n1 0 n3

0 n3 n1

� �ð29Þ

and finally, the matrices Ba and Pa are represented by the gradi-ents of the shape functions as

BaðxÞ ¼Na

;1 0

0 Na;3

Na;3 Na

;1

2664

3775; B1aðxÞ ¼

Na;111 0

0 Na;311

Na;311 Na

;111

2664

3775;

B3aðxÞ ¼Na

;133 0

0 Na;333

Na;333 Na

;133

2664

3775;

PaðxÞ ¼Na

;1

Na;3

" #; P1aðxÞ ¼

Na;111

Na;311

" #; P3aðxÞ ¼

Na;133

Na;333

" #: ð30Þ

Similarly, the normal component of the electric displacementvector QðxÞ can be approximated by

QhðxÞ ¼ N1ðxÞLTXna¼1

BaðxÞua � N1ðxÞJXna¼1

PaðxÞ/a

� N1ðxÞAXna¼1

PaðxÞwa þm2N1ðxÞLTXna¼1

Ba4ðxÞua

þm2N1ðxÞLTXna¼1

Ba5ðxÞua; ð31Þ

where the matrices LT; J, and A are defined in Eq. (5) and

N1ðxÞ ¼ n1 n3½ �; Ba4ðxÞ ¼

Na;11 0

0 Na;31

Na;31 Na

;11

264

375 ; Ba

5ðxÞ ¼Na

;13 0

0 Na;33

Na;33 Na

;13

264

375 :

ð32Þ

The magnetic flux SðxÞ is approximated by

ShðxÞ ¼ N1ðxÞMTXna¼1

BaðxÞua � N1ðxÞAXna¼1

PaðxÞ/a

� N1ðxÞIXna¼1

PaðxÞwa þm2N1ðxÞMTXna¼1

Ba4ðxÞua

þm2N1ðxÞMTXna¼1

Ba5ðxÞua; ð33Þ

with the matrices MT and I being defined in Eq. (6).The essential boundary conditions are satisfied in the strong-

form at nodal points. One obtains the discretized form of theseboundary conditions from the approximation formula (26)

Xna¼1

NaðfÞua ¼ �uðfÞ for f 2 Cu;Xna¼1

PaðfÞN1ðfÞua ¼ �sðfÞ for f 2 Cs

Xna¼1

NaðfÞ/a ¼ �/ðfÞ for f 2 C/;Xna¼1

NaðfÞwa ¼ �wðfÞ for f 2 Cw

ð34Þ

Substituting the MLS-approximation (28), (31), (33) into thelocal boundary-domain integral equations (20), (22) and (24), weobtain the system of algebraic equations for unknown nodalquantities

Fig. 1. Representative volume element (RVE) with piezomagnetic phase as matrixand piezoelectric phase as embedded fiber with a nano-sized coating layer ofthickness h.

180 J. Sladek et al. / Composite Structures 174 (2017) 176–186

Xn

a¼1

ZLsþCst

NðxÞCBaðxÞdCþZLsþCst

l2NðxÞCðBa1ðxÞþBa

3ðxÞÞdC� �

ua

þXna¼1

ZLsþCsQ

NðxÞLPaðxÞdC�ZLsþCsQ

m2NðxÞLðPa1ðxÞþPa

3ðxÞÞdC !

/a

þXna¼1

ZLsþCsS

NðxÞMPaðxÞdC�ZLsþCsS

NðxÞMðPa1ðxÞþPa

3ðxÞÞdC� �

wa

þ l2 Pðx fstÞC n1B

a4ðx f

stÞþn3Ba5ðx f

stÞh i

�PðxsstÞC n1B

a4ðxs

stÞþn3Ba5ðxs

st� �n o

ua

¼�ZCst

�tðxÞdC;

ð35Þ

Xn

a¼1

ZLsþCsQ

N1ðxÞLTBaðxÞdCþZLsþCsQ

m2N1ðxÞLTBa4ðxÞdC

þZLsþCsQ

m2N1ðxÞLTBa5ðxÞdC

!ua ��

Xna¼1

ZLsþCsQ

N1ðxÞJPaðxÞdC !

/a

�Xna¼1

ZLsþCsS

N1ðxÞAPaðxÞdC� �

wa ¼�ZCsQ

�QðxÞdC; ð36Þ

Xn

a¼1

ZLsþCsS

N1ðxÞMTBaðxÞdCþZLsþCsS

N1ðxÞMTBa4ðxÞdC

�

þZLsþCsS

N1ðxÞMTBa5ðxÞdC

�ua ��

Xna¼1

ZLsþCsQ

N1ðxÞAPaðxÞdC !

/a

�Xna¼1

ZLsþCsS

N1ðxÞIPaðxÞdC� �

wa ¼ �ZCsS

�SðxÞdC; ð37Þ

which are considered on the sub-domains adjacent to theinterior nodes as well as to the boundary nodes on Cst , CsQ andCsS and

PðxÞ ¼ p1 0 p3

0 p3 p1

� �: ð38Þ

Fig. 2. Boundary conditions appropriate for evaluation of ceff , ceff , eeff , qeff .

4. Coated circular piezoelectric fiber in piezomagnetic matrix

The proposed computational method is applied for evaluatingthe effective material parameters of a piezomagnetic matrix withregularly distributed piezoelectric fibers of a circular cross sec-tion. The fibers are coated by a general MEE material with thick-ness h. The thickness of the cladding layer is considered so smallthat it is necessary to apply the gradient theory to describe thesize effect. In idealized assessments of the effective elastic prop-erties of unidirectionally reinforced composites, it is assumed thatthe spatial arrangement of fibers is regular, which enables theapplication of effective elasticity estimates that are usually devel-oped by modelling the mechanics of a representative volume ele-ment (RVE). Then, it is sufficient to consider only one fiber in thesquare domain (a � a) as the RVE (Fig. 1). The polarization in ournumerical analyses is considered in the transverse direction, x3-axis.

The effective material coefficients of MEE solids are computedfrom the constitutive equations (4)–(6) rewritten for the averagevalues of the secondary fields and the average values of conjugatedfields. These fields are obtained from the numerical solution ofproperly selected boundary value problems in the RVE sample.Let us consider a rectangular RVE sample X ¼ f8x ¼ ðx1; x3Þ;x1 2 ½0; a�; x3 2 ½0; a�g. The volume average stress, electric dis-placement and magnetic induction are, respectively

hriji ¼ 1a2

ZXrijdX; hDii ¼ 1

a2

ZXDidX; hBii ¼ 1

a2

ZXBidX: ð39Þ

If a uniform strain along x1 and vanishing electric and magneticpotentials are considered as shown in Fig. 2, we get the followingaverage values of the secondary fields

hc11i ¼ �c11 ¼ const; hc33i ¼ 0; hc13i ¼ 0; hEii ¼ 0; hHii ¼ 0;hgijki ¼ 0 ð40Þ

Then, the effective material coefficients are given by

ceff11 ¼ hr11i�c11

; ceff13 ¼ hr33i�c11

; eeff31 ¼ hD3i�c11

; qeff31 ¼ hB3i

�c11: ð41Þ

The following effective material coefficients can also be com-puted as

ceff33 ¼ hr33i�c11

; ceff13 ¼ hr11i�c33

; eeff33 ¼ hD3i�c33

; qeff33 ¼ hB3i

�c33; ð42Þ

if we consider a uniform strain along x3 and vanishing electricand magnetic potentials

hc11i ¼ 0; hc33i ¼ �c33 ¼ const; hc13i ¼ 0; hEii ¼ 0; hHii ¼ 0;hgijki ¼ 0 ð43Þ

Similarly, one can select the average values of the secondaryfields

11 13 31 31

J. Sladek et al. / Composite Structures 174 (2017) 176–186 181

hc11i ¼ 0; hc33i ¼ 0; hc13i ¼ �c13 ¼ const; hEii ¼ 0; hHii ¼ 0;hgijki ¼ 0 ð44Þto compute effective material coefficients

ceff66 ¼ hr13i2�c13

; eeff15 ¼ hD1i2�c13

; qeff15 ¼ hB1i

2�c13: ð45Þ

If a uniform electric intensity field along x3 and vanishing dis-placements are considered, the average values of the secondaryfields are given as

hc11i ¼ 0; hc33i ¼ 0; hc13i ¼ 0; hEii ¼ �E3di3 ¼ const; hHii ¼ 0;hgijki ¼ 0 ð46Þ

Then, one can calculate the following effective materialcoefficients

eeff31 ¼ �hr11i�E3

; eeff33 ¼ �hr33i�E3

; eeff33 ¼ hD3i�E3

; aeff33 ¼ hB3i

�E3: ð47Þ

Similarly, a uniform electric intensity field along x1 and vanish-ing displacements are considered

hc11i ¼ 0; hc33i ¼ 0; hc13i ¼ 0; hEii ¼ �E1di1 ¼ const; hHii ¼ 0;hgijki ¼ 0 ð48Þ

for evaluating the effective material coefficients

eeff11 ¼ hD1i�E1

; eeff15 ¼ �hr13i�E1

; aeff11 ¼ hB1i

�E1: ð49Þ

If the following boundary conditions are considered

hc11i ¼ 0; hc33i ¼ 0; hc13i ¼ 0; hEii ¼ 0; hHii ¼ �H1di1 ¼ const;

hgijki ¼ 0 ð50Þthe effective material coefficients are computed as

qeff15 ¼ �hr13i

�H1; aeff

11 ¼ hD1i�H1

; leff11 ¼ hB1i

�H1: ð51Þ

Finally, if the average values of the secondary fields are consid-ered as

hc11i ¼ 0; hc33i ¼ 0; hc13i ¼ 0; hEii ¼ 0; hHii ¼ �H3di3 ¼ const;

hgijki ¼ 0 ð52Þthen, we can calculate the following effective material

coefficients

qeff31 ¼ �hr11i

�H3; qeff

33 ¼ �hr33i�H3

; aeff33 ¼ hD3i

�H3; leff

33 ¼ hB3i�H3

: ð53Þ

where the volume average values of the conjugated fields hr11i,hr33i, hr13i, hD3i, hB3i are calculated from Eq. (39), with the inte-grands being obtained from the solution of the considered boundaryvalue problem.

5. Numerical examples

In the numerical examples, the RVE has a piezoelectric fiberwith circular cross section in the square domain (a � a). The fiberis coated by the piezomagnetic material Terfenol-D. The matrixmaterial parameter is CoFe2O4 whose properties are given in[39]. The piezoelectric fiber is BaTiO3 with parameters

c11 ¼ 16:6� 1010 Nm�2; c13 ¼ 7:8� 1010 Nm�2;

c33 ¼ 16:2� 1010 Nm�2; c44 ¼ 4:3� 1010 Nm�2 ;

e15 ¼ 11:6 Cm�2; e31 ¼ �4:4 Cm�2; e33 ¼ 18:6 Cm�2 ;

e11 ¼ 11:2� 10�9 C2=Nm2; e33 ¼ 12:6� 10�9 C2=Nm2;

q15 ¼ 0:0 N=Am; q31 ¼ 0:0 N=Am; q33 ¼ 0:0 N=Am;

a11 ¼ 0:0 Ns=VC; a33 ¼ 0:0 Ns=VC;

l11 ¼ 5:0� 10�6 Ns2 C�2; l33 ¼ 10:0� 10�6 Ns2 C�2 :

The piezomagnetic material Terfenol-D has the following mate-rial coefficients:

c11 ¼ 0:854� 1010 Nm�2; c13 ¼ 0:391� 1010 Nm�2;

c33 ¼ 2:83� 1010 Nm�2 ; c44 ¼ 0:555� 1010 Nm�2 ;

e15 ¼ 0:0 Cm�2; e31 ¼ 0:0 Cm�2 ; e33 ¼ 0:0 Cm�2 ;

e11 ¼ 0:05� 10�9 C2=Nm2; e33 ¼ 0:05� 10�9 C2=Nm2 ;

q15 ¼ 155:5 N=Am; q31 ¼ �5:75 N=Am; q33 ¼ 270:1 N=Am;

a11 ¼ 0:0 Ns=VC; a33 ¼ 0:0 Ns=VC ;

l11 ¼ 8:6� 10�6 Ns2 C�2; l33 ¼ 2:3� 10�6 Ns2 C�2 :

The volume fraction of the fiber is introduced here asf ¼ pr2=a2, with r ¼ r0 þ h where r0 is the radius of the circularcross section of the fiber and h is the coating layer thickness. Thefollowing geometrical dimensions have been considered in thenumerical model: h ¼ 1 � 10�8 m and a ¼ 5 � 10�7m:

The coating layer thickness is kept constant for all the consid-ered volume fractions. Due to the symmetry of the problem withrespect to both Cartesian coordinates, only a quarter of the RVEis modelled. We have used 300 nodes for the MLS approximationof the physical quantities in the fiber and 2414 nodes in the matrix.The quarter of the coating layer is approximated by 90 nodes. Thelocal subdomains are assumed circular.

Now, it is needed to ensure a jump of secondary fields on inter-faces. It is well known that the C1� continuity is ensured over theentire domain in the MLS approximation. Therefore the continuityconditions of the tractions, electric displacement and magneticinduction are satisfied on interfaces too. However, this highly con-tinuous nature leads to difficulties when there is an imposed dis-continuity in the secondary fields. To simulate jumps oninterfaces Krongauz and Belytschko [40] introduced a jump shapefunction for 2-D problems. It is a trial function with a pre-imposeddiscontinuity in the gradient of the function on interfaces. It is verytedious to applied this approach for curvilinear interfaces. Cordesand Moran [41] solved also 2-D problems using Lagrangian multi-plier. The method requires a lot of computational effort for the dis-continuity of a complex geometrical shape.

It is much simpler to apply double nodes approach for the mate-rial discontinuity [42]. Two sets of collocation nodes are assignedon both the +side and the �side of the material interface at thesame location (Fig. 3). The MLS approximations are carried out sep-arately on particular sets of nodes related to the homogeneousdomains. Then, the support domains for the weights in theweighted MLS approximations must be truncated at the interfaceof the two media. Then, the high order continuity is kept withineach homogeneous part, but not across their interface. The localsubdomains considered around nodes should not cross the inter-face too.

On the interface CI , we must enforce the continuity for the dis-placements and potentials, as well as the equilibrium of tractions,electric charge and magnetic flux by collocating the following

equations at double nodes xd 2 @Xds \ CI ¼ Cd

I :

Interface between two media

Medium +Medium -

circular subdomain

semi-circular subdomain for node in +

Fig. 3. Double nodes approach for material discontinuities.

182 J. Sladek et al. / Composite Structures 174 (2017) 176–186

Xnþa¼1

NaðxdÞua ¼Xn�a¼1

NaðxdÞua; ð54Þ

Xnþa¼1

NaðxdÞ/a ¼Xn�a¼1

NaðxdÞ/a; ð55Þ

Fig. 4. Variation of effective elastic coefficients

Xnþa¼1

NaðxdÞwa ¼Xn�a¼1

NaðxdÞwa; ð56Þ

NðxdÞ CþðxdÞXnþa¼1

BaðxÞua � C�ðxdÞXn�a¼1

BaðxÞua þ LþðxdÞXnþa¼1

PaðxÞ/a

"

�L�ðxdÞXn�a¼1

PaðxÞ/a þMþðxdÞXnþa¼1

PaðxÞwa �M�ðxdÞXn�a¼1

PaðxÞwa

�m2LþðxdÞXnþa¼1

Pa1ðxÞ/a þm2L�ðxdÞ

Xn�a¼1

Pa1ðxÞ/a

�m2LþðxdÞXnþa¼1

Pa3ðxÞ/a þm2L�ðxdÞ

Xn�a¼1

Pa3ðxÞ/a

�m2MþðxdÞXnþa¼1

Pa1ðxÞwa þm2M�ðxdÞ

Xn�a¼1

Pa1ðxÞwa

�m2MþðxdÞXnþa¼1

Pa3ðxÞwa þm2M�ðxdÞ

Xn�a¼1

Pa3ðxÞwa

þ l2CþðxdÞXnþa¼1

Ba1ðxÞua � l2C�ðxdÞ

Xn�a¼1

Ba1ðxÞua

þl2CþðxdÞXnþa¼1

Ba3ðxÞua � l2C�ðxdÞ

Xn�a¼1

Ba3ðxÞua

#¼ 0: ð57Þ

with volume fraction of the coated fiber.

Fig. 5. Variation of effective piezoelectric coefficients with volume fraction of the coated fiber.

J. Sladek et al. / Composite Structures 174 (2017) 176–186 183

N1ðxÞ LþTðxdÞXnþa¼1

BaðxÞua � L�TðxdÞXn�a¼1

BaðxÞua þ JþðxdÞXnþa¼1

PaðxÞ/a

"

� J�ðxdÞXn�a¼1

PaðxÞ/a � AþðxdÞXnþa¼1

PaðxÞwa þ A�ðxdÞXn�a¼1

PaðxÞwa

þm2LþTðxdÞXnþa¼1

Ba4ðxÞua �m2L�TðxdÞ

Xn�a¼1

Ba4ðxÞuaþ

þm2LþTðxdÞXnþa¼1

Ba5ðxÞua �m2L�TðxdÞ

Xn�a¼1

Ba5ðxÞua

#¼ 0; ð58Þ

N1ðxdÞ MþTðxdÞXnþa¼1

BaðxÞua �M�TðxdÞXn�a¼1

BaðxÞua �AþðxdÞXnþa¼1

PaðxÞ/a

"

þA�ðxdÞXn�a¼1

PaðxÞ/a � IþðxdÞXnþa¼1

PaðxÞwa þ I�ðxdÞXn�a¼1

PaðxÞwa

þm2MþTxdXnþa¼1

Ba4ðxÞua �m2M�Txd

Xn�a¼1

Ba4ðxÞua

þm2MþTðxdÞXnþa¼1

Ba5ðxÞua �m2M�TðxdÞ

Xn�a¼1

Ba5ðxÞua

#¼ 0; ð59Þ

where nþ and n� are the numbers of nodes lying in the supportdomain in medium + and medium -, respectively. The normal vector

components in NðxdÞ is taken in the sense of outward normal to themedium +.

It is seen from constitutive equations that strain-gradient piezo-electric model can be reduced to the classical piezoelectric model ifthe internal length material parameter, l, and scaling parameter, m,appearing in Eqs. (5)–(7) vanish. The classical piezoelectric modelhas been analyzed in an earlier paper by the authors [21]. To assessthe size effect, we can introduce the strain-gradient size-factor, q,as

l2 ¼ q � l20; m2 ¼ q �m20; ð60Þ

where m0 ¼ 2� 10�8 m and l0 ¼ 2� 10�9 m are fixed parametersselected for the present model.

The numerical results for effective elastic material parametersare presented in Fig. 4 for fiber volume fraction values in intervalfrom 0.05 to 0.3. All effective elastic coefficients decrease withgrowing fiber volume fraction. One can also observe that the effec-tive elastic coefficients increase with increasing strain gradientparameter q, but saturate when q reaches 4. However, thisenhancement is not significant due to the small volume fractionof the cladding layer on the whole composite content.

The variation of the effective piezoelectric coefficients with thevolume fraction of the coated fiber is presented in Fig. 5. The coef-ficients e15 and e33 show an increase with growing fiber volumefraction. It is interesting that the strain gradient produces an oppo-

Fig. 6. Variation of effective dielectric permittivities with volume fraction of the coated fiber.

184 J. Sladek et al. / Composite Structures 174 (2017) 176–186

site effect for piezoelectric coefficient e31 as compared to the otherones. The positive value is changing to negative with an increase inq. The magnitude of e31 is increasing with q. The magnitudes of e15and e33 can be tuned by different values of the strain gradientparameter q.

Fig. 7. Variation of effective piezomagnetic coeffic

The effective dielectric permittivities, shown in Fig. 6, areincreasing as fiber volume fraction increases. One can also observethat for fixed volume fraction, the effective dielectric permittivityis nearly independent of the strain gradient.

ients with volume fraction of the coated fiber.

Fig. 8. Variation of effective magnetoelectric coefficients with volume fraction of the coated fiber.

Fig. 9. Variation of effective magnetic permeabilities with volume fraction of the coated fiber.

J. Sladek et al. / Composite Structures 174 (2017) 176–186 185

Variation of the effective piezomagnetic coefficients with thevolume fraction of coated fiber is shown in Fig. 7. The effectivepiezomagnetic coefficients are reduced as the volume fraction ofthe fiber increases. The effect of strain gradients is insignificantfor all effective piezomagnetic coefficients.

Variation of the effective magnetoelectric coefficients with thevolume fraction of the coated fiber is shown in Fig. 8. The magni-tude of these coefficients highly increases as the fiber volume frac-tion increases. One can also observe a strong influence of the straingradient on the effective magnetoelectric coefficients. The effectivemagnetoelectric coefficients are enhanced as compared to the clas-sical models, particularly when q is large.

Finally, Fig. 9 shows the effect of fiber volume fraction on theeffective magnetic permeabilities. Both effective magnetic perme-abilities decrease with increasing volume fraction of the coatedpiezoelectric fiber. The influence of strain gradients on the effectivemagnetic permeabilities is also very small.

6. Conclusions

A new formulation based on the meshless local Petrov-Galerkin(MLPG) method is developed for the evaluation of the effectivematerial properties in MEE composite materials described bythe gradient theory. The strain-gradients are considered in the con-stitutive equations. The composite is made of a piezomagnetic

matrix with embedded piezoelectric fibers with nano-sized coatinglayer. The governing equations with the corresponding boundaryconditions are derived using the variational principle. The localintegral equations for the solution of 2D boundary value problemsof magnetoelectroelasticity are derived in the gradient theory.

It follows from the numerical analyses that the effective elastic,piezoelectric, magnetoelectric coefficients and magnetic perme-abilities are influenced significantly by the strain-gradient. Whilethe effective piezomagnetic coefficients and dielectric permeabili-ties are influenced very weakly. The strongest influence of the clad-ding layer is observed in the effective magnetoelectric (ME)coefficients. The influence of the coating layer on the effectiveME coefficient is dominant although the ME coefficients for allthree composite constituents (fiber, matrix, coating) are zero. Thus,our MLPG modeling results could open an opportunity for enhanc-ing the ME coefficients in coated fiber composites for possiblepromising future applications.

The strain-gradient theory should be employed if the dimen-sions of the analyzed problem are of the same order of magnitudeas the internal material length. Numerical results illustrate thatsize-effect phenomenon must be considered in such cases. TheMLPG method has been successfully applied to nano-MEE compos-ites. The present local integral equation method requires no funda-mental solutions and all integrands in the present formulation areregular. Therefore, the method is promising for numerical analysis

186 J. Sladek et al. / Composite Structures 174 (2017) 176–186

of general multi-field problems with more complex constitutiveequations even in the gradient theory.

Acknowledgement

The authors acknowledge the support by the Slovak Science andTechnology Assistance Agency registered under number APVV-14-0216 and VEGA-1/0145/16.

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