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Effective property of multiferroic fibrous composites with imperfect interfaces
View the table of contents for this issue, or go to the journal homepage for more 2013 Smart Mater. Struct. 22 105005
(http://iopscience.iop.org/0964-1726/22/10/105005)
Smart Mater. Struct. 22 (2013) 105005 (8pp) doi:10.1088/0964-1726/22/10/105005
Effective property of multiferroic fibrous
composites with imperfect interfaces
Hsin-Yi Kuo
Department of Civil Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan E-mail:[email protected]
Received 11 April 2013, in final form 6 August 2013 Published 29 August 2013
Online atstacks.iop.org/SMS/22/105005
Abstract
This paper studies the effective behavior of piezoelectric and piezomagnetic circular fibrous composites with imperfect interfaces under longitudinal shear with in-plane electromagnetic fields. Two kinds of imperfect contact are investigated: mechanically stiff and
dielectrically/magnetically highly conducting interfaces, and mechanically compliant and dielectrically/magnetically weakly conducting interfaces. For the former case, the potential field is continuous, while the normal component of the flux undergoes a discontinuity across the interface. For the latter case, the normal component of the flux is continuous, while there is a jump of potential field at such a contact. The classic work of Rayleigh (1892 Phil. Mag. 34 481–502) in a periodic conductive perfect composite is generalized to the current coupled magnetoelectroelastic composites with imperfect interfaces. It is shown that the expression of the effective property has exactly the same form as that in the ideal coupling composite. Finally, this method is used to study BaTiO3–CoFe2O4composites and provide insights into
enhancing the effective magnetoelectric voltage coefficient by properly choosing the interface. (Some figures may appear in colour only in the online journal)
1. Introduction
Magnetoelectricity (ME) in multiferroic composites, which is related to inducing an electric polarization by a magnetic field or conversely inducing a magnetization by an electric field, has been the topic of a number of theoretical and experimental investigations in recent years. The coupling between the electric and magnetic fields provides opportunities for technological applications in sensing, actuation, and data storage (Fiebig2005, Ramesh and Spaldin2007, Kumar et al
2009). A state of the art of recent development can be found in Eerenstein et al (2006), Nan et al (2008), and Bichurin et al(2010). The ME effect in the multiferroic composite is achieved through the product property: an applied magnetic field generates a strain in the ferromagnetic material, which in turn induces a strain in the ferroelectric material, resulting in a polarization. Each phase possesses either magnetostrictive or piezoelectric properties, and the product ME effect is a new property determined by the mechanical interaction between the two phases. Therefore, the interface is critical in achieving the giant magnetoelectricity.
In earlier investigation the interface between the ferroelectric and ferromagnetic constituents was primarily assumed to be perfect or ideal coupling (see, for instance, Harsh´e et al 1993, Nan 1994, Benveniste 1995, Li and Dunn 1998, Liu and Kuo 2012, Kuo and Bhattacharya
2013). However, measured ME coupling coefficients may be notably discrepant with the above theories for both the ME particulate composites and laminates. To explain the discrepancies, an interface coupling parameter that defines the degree to which the deformation of the piezoelectric layer follows that of the magnetostrictive layer was introduced by Bichurin et al (2003). Nan et al (2003) studied the influence of the interfacial bonding on the ME effect in the multiferroic PZT–Terfenol-D laminated composite by means of the Green’s function approach. Chang and Carman (2007) proposed a quasistatic theoretical model including shear lag and demagnetization effect for predicting the ME effects in an ME laminate. Wang and Pan (2007) used the complex variable approach together with the Mori–Tanaka mean field method to derive the effective moduli of multiferroic fibrous composites. All the above studies were primarily concerned
c
Smart Mater. Struct. 22 (2013) 105005 H-Y Kuo
with the soft interface at which different tangential strains or electromagnetic fields may occur. On the other hand, owing to the minimization of components, the surface elasticity theory, which describes the membrane-type stiff interface, has been recently developed to account for the effects of surfaces and interfaces at nanoscales (Benveniste and Miloh2001, Sharma et al2003, Chen et al2007). Pan et al (2009) generalized this idea to the field of the multiferroic fibrous nanocomposite with a size effect along its interface.
For the general problem of a transversely isotropic multiferroic composite, Benveniste (1995) has shown that it can be decomposed into two independent problems, plane strain with transverse electromagnetic fields and anti-plane shear with in-plane electromagnetic field. The anti-plane shear deformation therefore serves as part of the contribution that is present in a three-dimensional situation. Further, this out-of-plane deformation mode has wide applications in screw dislocation, in which the slip vector is parallel to the dislocation line (Zheng et al2007), and Mode III crack problems, in which a shear stress acting parallel to the plane of the crack and parallel to the crack front (Spyropoulos et al
2003, Wang and Mai2004, Gao et al2004, Hao and Liu2006, Guo and Lu2010).
Motivated by these advances, and in a departure from previous work, this paper develops a new formulation to study the effective behavior of multiferroic fibrous composites with imperfect interfaces under longitudinal shear with in-plane electromagnetic fields. Both the fiber and matrix are assumed to be transversely isotropic. Two kinds of imperfect interface are considered: (i) mechanically stiff and electromagnetically highly conducting, which is a generalization of a membrane-type interface, and (ii) mechanically compliant and electromagnetically weakly conducting, which is a general extension of the shear lag model. For the former case, the potentials (displacement, electric potential and magnetic potential) are continuous across the interface, while the normal component of flux (stress, electric displacement, and magnetic flux) undergoes a discontinuity which is proportional to the local surface Laplacian of the potential field. For the latter case, the normal fluxes are continuous, while the potentials are discontinuous at such contact. The jumps in potential components are further assumed to be proportional to their respective interface flux components. These general imperfect contacts could model various types of interfacial damage such as debonding, sliding, cracking, or surface effects across the interface.
This paper is organized as follows. In section 2 the basic formulation is introduced for a composite medium made of piezoelectric and piezomagnetic phases arranged in a microstructure consisting of parallel cylinders in a matrix subjected to anti-plane shear deformation and in-plane electromagnetic fields. Following Kuo and Bhattacharya (2013), each field in each medium is expanded in a series in section 3. Two kinds of imperfect contact are studied: mechanically stiff and dielectrically/magnetically highly conducting interfaces, and mechanically compliant and dielectrically/magnetically weakly conducting interfaces. Expressions for effective properties are obtained in section4.
Numerical results are shown in section 5 using composites of BaTiO3and CoFe2O4. It is shown that the effective ME
effect can be substantially enhanced by properly choosing the interface, providing an opportunity for controlling the ME effect and other effective moduli of the composites.
2. Formulation
Consider a composite consisting of a periodic array of parallel and separated prismatic circular cylinders with radius a. Assume that the cylinders and the matrix are made of distinct phases: transversely isotropic piezoelectric or piezomagnetic materials. A Cartesian coordinate system is introduced with the xy-axes in the plane of the cross-section and the z-axis along the axes of the cylinders.
Now assume that the composite is subjected to anti-plane shear strains ¯εzx, ¯εzy, in-plane electric fields ¯Ex, ¯Ey and
magnetic fields ¯Hx, ¯Hy at infinity. Thus the composite is in
a state of generalized anti-plane shear deformation and can be described by Benveniste (1995)
ux=uy=0, uz=w(x, y),
ϕ = ϕ(x, y), ψ = ψ(x, y),
(2.1)
where ux, uy, and uz are the mechanical displacements along
the x-, y-, and z-axes, and ϕ and ψ are the electric and magnetic potentials, respectively.
The constitutive laws of the kth phase for the non-vanishing fields can be recast in the compact form as
Σ(k)j =L(k)Z(k)j , j = x, y, (2.2) where for ease of terminology k = ‘m’(k = ‘i’) refers to the matrix (inclusion) phase,
Σ(k)j = σzj Dj Bj (k) , L(k)= C44 e15 q15 e15 −κ11 −λ11 q15 −λ11 −µ11 (k) , Z(k)j = εzj −Ej −Hj (k) . (2.3)
In equations (2.3), σzj and εzj are the shear stress and
strain; Dj and Ej are the electric displacement and electric
field vectors; Bj and Hj are the magnetic flux and magnetic
field vectors. The material constants C44, κ11, µ11, and λ11
are the elastic modulus, dielectric permittivity, magnetic permeability, and magnetoelectric coefficient, while e15 and
q15are the piezoelectric and piezomagnetic coefficients.
The shear strainsεzxandεzy, in-plane dielectric fields Ex
and Ey, and in-plane magnetic fields Hxand Hycan be derived 2
from the gradient of elastic displacement w, electric potential ϕ, and magnetic potential ψ as follows:
Zj=Φ,j= w,j ϕ,j ψ,j . (2.4)
Here the subscript j following a comma denotes the derivative with respect to x or y. In the absence of body force, electric charge density, and electric current density, the equilibrium equations are given by
L∇2Φ = 0, (2.5)
where ∇2 = ∂2/∂x2 + ∂2/∂y2 is the two-dimensional Laplace operator for the variables x and y. Since L is a nonsingular matrix, equation (2.5) can be decoupled into three independent Laplace equations,
∇2Φ = 0, (2.6)
in the interior of each phase. In other words, the three fields—displacement, electrostatic potential and magneto-static potential—are completely decoupled in the interior of each phase.
3. Circular cylinders with imperfect interfaces
Consider a situation where the composite is subjected to a macroscopically uniaxial loading along the positive x-axis
Φext= ¯Zxx, (3.1)
where ¯Zx =(¯εzx, − ¯Ex, − ¯Hx)t. Under the above generalized
anti-plane shear deformation, the potential field for the circular cylinder and its surrounding matrix can be expanded with respect to its center O in polar coordinates (r, θ) as (Kuo and Bhattacharya2013) Φ(m)(x) = a0+ ∞ X n=1 (anrn+bnr−n) cos nθ (3.2)
for the matrix, and
Φ(i)(x) = c0+ ∞
X
n=1
cnrncos nθ (3.3)
for the inclusion, where
an= Awn Aϕn Aψn , bn= Bwn Bϕn Bψn , cn= Cwn Cϕn Cψn . (3.4)
The coefficients an, bn, and cn are unknown constants to
be determined from the interface and boundary conditions. Note that the sine terms that would be present in a general expansion are missing since a uniaxial loading along the x-direction is imposed.
In order to treat the imperfect interface effect, we first resort to a more general three-phase composite of a similar distribution in which the inclusions possess a concentric elastic coating of thickness t and material parameter Lc=
diag(C44, −κ11, −µ11) (Torquato and Rintoul 1995, Hashin
2001, Miloh and Benveniste 1999). By passing to the limit where t → 0 and either L−1c →0 (mechanically stiff and dielectrically/magnetically highly conducting interface) or Lc → 0 (mechanically soft and dielectrically/magnetically
weakly conducting interface), we recover the distribution of interest in which the interfacial property is characterized by the parameters α and β given by
α = lim t→0 L−1c →0 (Lct) = αw 0 0 0 αϕ 0 0 0 αψ (3.5)
for the mechanically stiff and dielectrically/magnetically highly conducting case, and
β = lim t→0 Lc→0 (Lc/t) = βw 0 0 0 βϕ 0 0 0 βψ (3.6)
for the mechanically soft and dielectrically/magnetically weakly conducting case.
Now consider that the interface is mechanically stiff and dielectrically/magnetically highly conducting. It has been shown that in this case, with α given by (3.5), the potential Φ is continuous across the interface∂V, while there is a jump in the normal component of the current (Miloh and Benveniste
1999, Pan et al2009). Specifically, one has Σ(m)j nj|∂V−Σ(i)j nj|∂V =α∇s2Φ(i)|∂V,
Φ(m)|∂V =Φ(i)|∂V, (3.7) where ∇s2= 1
r2 ∂ 2
∂θ2 is the surface Laplace operator, n is the unit
outward normal to the interface ∂V: r = a, and the repeated index j denotes the summation over the components x and y. The case where α = 0 corresponds to a perfect interface, whereas α−1=0 describes an isoexpansion and equipotential interface.
Using the orthogonality properties of trigonometric functions, the interface conditions (3.7) provide
an=a−2nTnbn,
cn=a−2n(Tn+I)bn, n ≥1,
(3.8) and a0=c0, where an, bn, cnare defined in (3.4), I is the 3 × 3
identity tensor, and
Tn=(L(m)−L(i)+Λn)−1(L(m)+L(i)−Λn), (3.9)
Λn=a−1nα. (3.10)
When α = 0, the results reduce to the perfectly bonded case (Kuo and Bhattacharya2013).
Next, consider the interface is mechanically soft and dielectrically/magnetically weakly conducting with interfacial imperfection matrix β given by (3.6). It can be shown that in
Smart Mater. Struct. 22 (2013) 105005 H-Y Kuo
Figure 1. The predicted ME voltage coefficients for composites of BaTiO3fibers in a CoFe2O4matrix: (a) mechanically stiff imperfect
interfaces characterized byαw=α0waC(i)44, αϕ=0, αψ=0 and (b) mechanically soft imperfect interfaces characterized by βw=βw
0a/C44(i), βϕ=0, βψ=0. Hereα w 0 andβ
w
0 are dimensionless parameters. In both (a) and (b), the solid line ‘–’ is based on the
presented solution for a hexagonal array. The dashed line ‘- -’ is from Pan et al (2009) for (a) and is from Wang and Pan (2007) for (b).
this case the potential Φ has a jump on the interface boundary ∂V, which is proportional to the normal component Σjnj of
the current, which is continuous across the interface (Miloh and Benveniste1999, Wang and Pan2007),
Σ(m)j nj|∂V =Σ(i)j nj|∂V,
Φ(m)|∂V−Φ(i)|∂V =βΣ(i)j nj|∂V.
(3.11)
The case where β = 0 corresponds to a perfectly bonded interface, whereas β−1=0 describes a completely debonded and electric charge-free (insulating) interface.
Analogous to (3.8), the interface conditions (3.11) give constraints
an=a−2nTnbn,
cn=a−2n[(I − Πn)Tn+I + Πn]bn, n ≥1,
(3.12) and a0=c0, where an, bn, cnare defined in (3.4), and
Tn=(L(m)−L(i)+Λn)−1(L(m)+L(i)+Λn), (3.13)
Λn=a−1nL(i)βL(m), Πn=a−1nβL(m). (3.14)
Again, when β = 0, the equation recovers the previous results of the ideal coupling contact.
Finally, imposing the periodicity conditions yields a generalized Rayleigh identity
an+ ∞ X m=1 m + n −1 n ! Sm+nbm= ¯Zxδn,1, (3.15)
whereδn,1is the Kronecker delta, and the quantities
Sm=
X
l6=O
Re(Xl+iYl)−m (3.16)
are the lattice sums characterizing the geometry of the periodic structure, and Xl+iYlis the center of the lth cylinder
when measured at the central point O. The index l runs over all cylinders’ centers underlying the periodic array except the central one. A list of non-zero normalized lattice sums for square and hexagonal arrays can be found in Berman and Greengard (1994).
Equations (3.15) and (3.8)1 or (3.12)1 constitute an
infinite set of linear algebraic equations. Upon appropriate truncations of the expansion terms at some finite order m = M, the expansion coefficients an, bn, and cncan be determined.
4. Effective moduli
Now we turn to obtain the effective moduli of the composite from the solution of (3.15). The major distinction from previous studies is that the inclusions have interfacial imperfections. The effective material properties are defined in terms of average fields,
hΣji ≡L∗hZji, (4.1)
where the angular brackets denote the average over the representative volume element V (or unit cell in the periodic case), i.e., hΣji = 1 V Z V ΣjdV, hZji = 1 V Z V ZjdV, (4.2)
and L∗denotes the effective magnetoelectroelastic parameters of the composite.
When (3.1) is prescribed, statistical homogeneity in the fields Zxsimply implies
hZxi = ¯Zx. (4.3)
For a mechanically stiff and dielectrically/magnetically highly conducting interface, the average flux, hΣxi, is now
given by Miloh and Benveniste (1999) hΣxi = 1 V Z Vm Σ(m)x dV + Z Vi Σ(i)x dV + Z ∂V(Σ (m) j −Σ(i)j )njxds , (4.4)
which contains an additional integral involving the normal flux jump across the interface∂V. Substituting (3.7)1, (4.2)2,
(4.3), and the constitutive relation (2.2) into the above
Figure 2. The contour plots of the ME voltage coefficientα∗
E,11versus BaTiO3volume fraction and different interface parameters for a
composite of BaTiO3fibers in a CoFe2O4matrix. The interface imperfections include (a) mechanically stiff interface (αw=αw0
aC(i)44, αϕ =0, αψ=0), (b) electrically highly conducting interface (αw=0, αϕ= −αϕ
0aκ11(i), αψ=0), (c) magnetically highly conducting
interface (αw=0, αϕ=0, αψ= −αψ
0aµ(i)11), (d) mechanically compliant interface (β w=βw
0a/C(i)44, βϕ =0, βψ=0), (e) dielectrically
weakly conducting interface (βw=0, βϕ= −βϕ
0a/κ11(i), βψ=0), and (f) magnetically weakly conducting interface
(βw=0, βϕ=0, βψ= −βψ
0a/µ(i)11). Hereα w
0, α0ϕ, αψ0, β w
0, β0ϕ, andβ0ψare dimensionless parameters.
equation yields hΣxi =L(m) ¯ Zx− 1 V(L (m))−1 (L(m)−L(i)) × Z ∂VxΦ (i) ,rds − α Z ∂V1sΦ (i)xds. (4.5)
Here the subscript r following a comma denotes the derivative with respect to the r variable. Using multipole expansions of the potential fields in the inclusion (3.3) and recalling the relation (3.8)2, one obtains
hΣxi =L(m)( ¯Zx−2a−2fb1), (4.6)
where f is the volume fraction of the inclusion defined as f =πa2/V for square arrays and is f =√2π
3a
2/V for hexagonal
arrays.
On the other hand, for a mechanically compliant and dielectrically/magnetically weakly conducting interface, the average intensity, hZxi, is now given by Benveniste and Miloh
(1986) hZxi = 1 V Z Vm Φ(m),x dV + Z Vi Φ(i),x dV + Z ∂V(Φ (m)−Φ(i))n xds , (4.7)
Smart Mater. Struct. 22 (2013) 105005 H-Y Kuo
which contains an additional integral involving the potential jump across the interface ∂V. Substituting (3.11)2, (4.2)2,
(4.3), and the constitutive laws (2.2) into (4.2)1yields
hΣxi =L(m) ¯ Zx− 1 V β−1L(i) Z ∂VΦ (i) ,rnxds + (I − (L(m))−1L(i)) Z ∂VxΦ (i) ,r ds . (4.8) Making use of (3.3) and (3.12)2, one obtains again (4.6).
Putting together (4.1) and (4.6) and noting that the coefficients b1depend linearly on the applied field ¯Zx, a set
of equations is then obtained for the effective property L∗. By applying different loading combinations between ¯εzx, ¯Ex
and ¯Hx, all the components of L∗ can be determined. Note
that, although the inclusions now have imperfect interfaces, equation (4.6) has exactly the same form as that in the perfect case (Kuo and Bhattacharya2013), but here the coefficients b1incorporate the effect of the imperfect contact.
5. Results and discussion
The above framework is applied below to the BaTiO3–CoFe2O4 (BTO–CFO) multiferroic composite,
which has been studied by other researchers. The hexagonal array, and both BTO fibers in a CFO matrix and CFO fibers in a BTO matrix, are considered. The independent material constants of BTO are C44=43 × 109N m−2, e15=
11.6 C m−2, κ11 = 11.2 × 10−9 C2 N−1 m−2, µ11 =
5 × 10−6 N s2 C−2; while those of CFO are C44 =
45.3 × 109 N m−2, q15 =550 N A−1 m−1, κ11 =0.08 ×
10−9C2N−1m−2, µ11=590 × 10−6N s2C−2(Wang and
Pan 2007). Here the xy-plane is the isotropic plane and the unique axis is along the z-direction. Note that in both materials ME coefficients are zero, i.e.λ11=0. A material property of
particular interest is the ME voltage coefficient α∗ E,11=λ ∗ 11/κ ∗ 11, (5.1)
where λ∗11(κ11∗) is the effective ME coefficient (dielectric permittivity) of the composite. It relates to the overall electric field that is generated in the composite when it is subjected to a magnetic field and is the figure of merit for magnetic field sensors.
To check the correctness of the formulation, the ME voltage coefficients for a composite of BTO fibers in a CFO matrix is studied first. Figure 1 shows how the ME voltage coefficient depends on the BTO volume fraction and different mechanical interfacial imperfections. The order of truncation is M = 4. Figure 1(a) is for mechanically stiff imperfect interfaces characterized byαw=αw
0aC(i)44andαϕ =0, αψ=0
while figure1(b) is for mechanically soft imperfect interfaces characterized byβw=βw
0a/C44(i)andβϕ =0, βψ =0. Here
αw
0 andβ0w are dimensionless parameters. The curves vary
nonlinearly with volume fraction, and they stop around f = π/(2√3) when the inclusions begin to touch each other for hexagonal arrays. The ME voltage coefficient decreases as αw
0 (β0w) increases. For comparisons, figure1 also plots the
effective moduli with those predicted by Pan et al (2009) (for
the mechanically stiff case) and Wang and Pan (2007) (for the mechanically soft case) who used the complex variable approach and the Mori–Tanaka method. In the Mori–Tanaka method, there is no upper limit on the volume fractions. Still, the overall magnitudes and trends agree well between the present periodic and their Mori–Tanaka method.
Figure 2 shows the ME voltage coefficient for a composite of BTO fibers in a CFO matrix as a function of inclusion volume fraction for different interfacial imperfec-tions: (a) mechanically stiff interface (αw=αw
0aC44(i), αϕ =
0, αψ = 0), (b) electrically highly conducting interface (αw=0, αϕ = −αϕ 0aκ (i) 11, αψ =0), (c) magnetically highly conducting interface (αw = 0, αϕ = 0, αψ = −αψ 0aµ (i) 11),
(d) mechanically soft interface (βw = βw
0a/C (i) 44, βϕ =
0, βψ =0), (e) dielectrically weakly conducting interface (βw=0, βϕ = −βϕ
0a/κ11(i), βψ =0), (f) magnetically weakly
conducting interface (βw = 0, βϕ =0, βψ = −βψ
0a/µ (i) 11).
Here α0w, α0ϕ, α0ψ, β0w, β0ϕ, andβ0ψ are dimensionless param-eters. It is observed that except for the magnetically highly conducting interface (figure 2(c)), all the coupling constants are reduced as compared to that for a perfect case. The ME voltage constant in figure 2(c) is substantially enhanced as α0ψincreases. Whenαψ0 changes from 0 to 10, the maximum value of 0.3240 V cm−1Oe−1at f = 0.35 is ten times higher than 0.0306 V cm−1 Oe−1(f = 0.35), which is the optimal value of the perfectly bonded case. Further, the optimal value of the BTO volume fraction, at which the maximum ME voltage coefficients occurs, basically remains the same asαψ0 increases. Because αψ = −αψ
0aκ11(i) is size dependent, this
provides an excellent chance for enhancing the ME effect in nanocomposites using the size-dependent feature.
Now turn to the composite of CFO fibers in a BTO matrix. Figure 3 shows the ME voltage coefficient as a function of inclusion volume fraction for different interfacial imperfections, and is the counterpart of figure 2. Similarly, in most cases ((a), (c)–(f)) the ME coupling coefficient is reduced as compared to that for a perfect contact. However, for the composite with electrically highly conducting interface (figure 3(b)), the ME effect is substantially enhanced asαϕ0 increases. When α0ϕ changes from 0 to 100, the maximum value of 0.0842 V cm−1Oe−1at f = 0.90 is around 7.39 times higher than 0.0114 V cm−1Oe−1(f = 0.9069), which is the optimal value of the perfectly bonded contact. In addition, the optimal value of the CFO volume fraction, at which the maximum ME voltage coefficients occurs, decreases as αϕ0 increases. This also provides an alternative way to enhance the magnetoelectricity.
Note that the numerical results above show that, for a BTO–CFO composite, the mechanically compliant and dielectrically/magnetically weakly conducting interface all causes a decrease in the ME coupling.
6. Conclusions
A framework based on Rayleigh’s formalism is developed for predicting the field distributions and effective properties
Figure 3. The contour plots of the ME voltage coefficientαE∗,11versus CoFe2O4volume fraction and different interface parameters for a
composite of CoFe2O4fibers in a BaTiO3matrix. The interface imperfections include (a) mechanically stiff interface (αw=αw0
aC(i)44, αϕ =0, αψ=0), (b) electrically highly conducting interface (αw=0, αϕ= −αϕ
0aκ11(i), αψ=0), (c) magnetically highly conducting
interface (αw=0, αϕ=0, αψ= −αψ 0aµ
(i)
11), (d) mechanically compliant interface (β w=βw
0a/C (i)
44, βϕ =0, βψ=0), (e) dielectrically
weakly conducting interface (βw=0, βϕ= −βϕ 0a/κ
(i)
11, βψ=0), and (f) magnetically weakly conducting interface
(βw=0, βϕ=0, βψ= −βψ 0a/µ (i) 11). Hereα w 0, α ϕ 0, α ψ 0, β w 0, β ϕ 0, andβ ψ
0 are dimensionless parameters.
of the multiferroic composite consisting of regular arrays of circular cylinders with imperfect interfaces under general-ized anti-plane shear deformation. Both mechanically stiff and dielectrically/magnetically highly conducting interfaces, and mechanically compliant and dielectrically/magnetically weakly conducting interfaces, are considered. Expressions for the elastic, electric, and magnetic potentials for the cylinders and the matrix are derived, and used to compute the macroscopic behavior. It is shown that the effective properties solely depend on one set of particular constants b1, and
the formula of the effective property has exactly the same form as that in the perfectly bonded interface, although now the inclusions are with interfacial imperfection. Finally, as a practical example, the ME effects in BaTiO3–CoFe2O4
composites are presented and discussed. This example shows the important difference between two kinds of imperfect contact. The present theoretical framework provides a general guideline and an alternative way for enhancing the magnetoelectricity.
Acknowledgments
The financial support from the National Science Council under grant NSC 100-2628-E-009-022-MY2 is gratefully acknowledged. The author is also grateful to anonymous reviewers for their helpful comments and suggestions.
Smart Mater. Struct. 22 (2013) 105005 H-Y Kuo
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