Bending of multiferroic laminated rectangular plates with imperfect interlaminar bonding

European Journal of Mechanics A/Solids 28 (2009) 720–727

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European Journal of Mechanics A/Solids www.elsevier.com/locate/ejmsol

Bending of multiferroic laminated rectangular plates with imperfect interlaminar bonding W.Q. Chen a,b,∗ , Y.Y. Zhou a , C.F. Lü c , H.J. Ding c a b c

Department of Engineering Mechanics, Yuquan Campus, Zhejiang University, Hangzhou 310027, PR China State Key Lab of CAD & CG, Zijingang Campus, Zhejiang University, Hangzhou 310058, PR China Department of Civil Engineering, Zijingang Campus, Zhejiang University, Hangzhou 310058, PR China

a r t i c l e

i n f o

Article history: Received 16 July 2008 Accepted 25 February 2009 Available online 3 March 2009 Keywords: Multiferroic materials Magnetoelectric coupling Imperfect interface State space method

a b s t r a c t The bending problem of a multiferroic rectangular plate with magnetoelectric coupling and imperfect interfaces is investigated via three-dimensional exact theory. A generalized spring layer model is proposed to characterize the imperfection of the bonding behavior at interfaces. In particular, the linear relation between the electric displacement and the jump of electric potential, the corresponding one for the magnetic field as well as linear relations among different physical fields are adopted. State space formulations are established, which, compared to the analysis for perfect laminates, only introduces a so-called interfacial transfer matrix. The present analysis can be readily used for the piezoelectric, piezomagnetic and elastic laminates by setting the proper material constants as zero. Numerical results are presented and discussed. © 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction Owing to the coupling effects between elastic, magnetic and electric fields, like piezoelectric materials, magneto-electro-elastic or multiferroic materials have significant application prospects in sensor technology, memory devices and smart structures (Spaldin and Fiebig, 2005; Eerenstein et al., 2006). Accordingly, mechanics problems of such smart/functional materials have attracted intensive research interests in recent years (Huang et al., 1998; Li, 2000; Tan and Tong, 2002; Chen et al., 2002; Wang and Mai, 2003; Wang and Ding, 2006). As regards three-dimensional structural analyses, the pioneering work belongs to Pan (2001), who obtained a threedimensional exact solution for bending of anisotropic magnetoelectro-elastic cross-ply laminated plates using the transfer matrix method. Subsequently, Pan and Heyliger (2002, 2003) investigated free vibration and cylindrical bending of laminated rectangular plates, and Wang et al. (2003) treated the bending of rectangular laminates using the state space method (SSM). Chen and Lee (2003) proposed an order-reduced state space method for inhomogeneous or functionally graded transversely isotropic magneto-electro-elastic rectangular plates. At the same time, Wang and Zhong (2003a) obtained an exact solution for a finitely long magneto-electro-elastic layered cylindrical shell bearing pressure and varying temperature.

*

Corresponding author at: Department of Engineering Mechanics, Yuquan Campus, Zhejiang University, Hangzhou 310027, PR China. Tel.: +86 571 87952284; fax: +86 571 87952165. E-mail address: [email protected] (W.Q. Chen). 0997-7538/$ – see front matter © 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.02.008

In the abovementioned analyses for layered multiferroic structures, interlaminar bonding was assumed perfect, that is, continuity conditions in electric, magnetic, and elastic fields at interfaces are satisfied. In practice, however, the practical manufacturing techniques or aging of adhesive layers may lead to the initiation and growth of microcracks or cavities, thus reducing the bonding strength and resulting in possible interfacial slip and debonding. The effect of such bonding imperfection on structural behavior has been intensively investigated (Cheng et al., 1996; Cheng et al., 2000; Librescu and Schmidt, 2001; Shu, 2001), most of which are based on various simplified plates and shell theories. However, exact three-dimensional analyses have shown that direct applications of the traditional high-order theories to the imperfect layered structures may result in significant errors (Chen and Lee, 2003, 2004; Chen et al., 2004). It is necessary to develop three-dimensional exact solutions for multiferroic layered structures for verifying the simplified results in the future. Wang and Zhong (2003b) presented an exact solution for an infinite angleply laminated cylindrical shell with piezoelectric surface layers. It was assumed that mechanical displacements may be discontinuous across the interfaces between the piezoelectric surface layers and the host shell, while all other quantities are continuous. Chang and Carman (2007) investigated the effect of bonding imperfection on the effective magnetoelectric (ME) voltage coefficient by a shear lag analysis. Wang and Pan (2007) presented exact solutions of simply-supported and multilayered piezoelectric plates with thermal effect as well as imperfect interfaces by adopting a special form of the transfer matrix.

W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

In this paper, the state space method (SSM) (Fan and Ye, 1990; Chen and Ding, 2001; Ding and Chen, 2001; Ding et al., 2006) is adopted to analyze the bending of multiferroic layered rectangular plates with imperfect interfaces. To simulate weak bonding of magneto-electro-elastic materials, in addition to the general linear spring model for elastic field, similar linear relations (Fan and Sze, 2001) are adopted for electric potential and electric displacement, as well as magnetic potential and magnetic induction. To account for the coupling characteristic of the interfaces, the analogous linear relations are also assumed to exist between different fields. Numerical calculations are finally performed to investigate the effect of bonding imperfection on the static response of multiferroic composite laminates. 2. State space formulations The state equation for orthotropic magneto-electro-elastic materials has been derived by Wang et al. (2003). Here, we first present the state equation for general anisotropic magneto-electroelastic materials with thermal effect. As for elastic materials (Ding et al., 2006) and piezoelectric materials (Ding and Chen, 2001; Tarn, 2002a, 2002b), the following formulations are readily derived for magneto-electro- elastic materials,

     −1  −1  C B1 D11 C11 ∂ U U = + 11T T T ˜1 σ ∂z σ 1 1 D21 D11 L2 B     ∂2 0 0 +ρ 2 , − G ∂ t KU    −1  U − B˜ 1 T , σ 2 = C˜ 11 L2 CT12 C11

(1) (2)

σ1

where −1 C12 L2 , D11 = −L1 − C11

D21 = −LT2 C˜ 11 L2 , −1 C˜ 11 = C22 − CT12 C11 C12 ,

˜ 1 = B2 − CT C−1 B1 , B 12 11 ⎡ ⎢ ⎢ C11 = ⎢ ⎣ ⎡

c 33

c 34 c 44

c 35 c 45 c 55

(3) e 33 e 34 e 35

−ε33

sym. c 13

c 23

c 36

e 13

q33 q34 q35 −d33

⎤ ⎥ ⎥ ⎥, ⎦

−μ33 e 23

q13

⎧ ⎫ β3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ β4 ⎪ ⎬ , B1 = β5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − p3 ⎪ ⎭ −m3 ⎤ q23

e 24 q14 q24 ⎥ ⎢ c 14 c 24 c 46 e 14 ⎢ ⎥ C12 = ⎢ c 15 c 25 c 56 e 15 e 25 q15 q25 ⎥ , ⎣e −d13 −d23 ⎦ 31 e 32 e 36 −ε13 −ε23 q31 q32 q36 −d13 −d23 −μ13 −μ23 ⎤ ⎡

⎢ ⎢ ⎢ ⎢ C22 = ⎢ ⎢ ⎢ ⎣

c 11

sym.

c 12 c 22

⎫ ⎧ β1 ⎪ ⎪ ⎪ ⎪ ⎪ β2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ β6 ⎪ B2 = − p 1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − p2 ⎪ ⎪ ⎪ ⎪ ⎪ − m ⎪ ⎪ 1⎭ ⎩ −m2

c 16 c 26 c 66

e 11 e 12 e 16

e 21 e 22 e 26

−ε11 −ε12 −ε22

q11 q12 q16 −d11 −d12

−μ11

q21 q22 q26 −d12 −d22

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ −μ12 ⎦ −μ22

⎡

0 ⎢ ∂/∂ y 0 ⎢ L1 = ⎢ ∂/∂ x 0 ⎣ 0 0 0 0 ⎡ 0 ∂/∂ x 0 ⎢0 ⎢ 0 ⎢0 ⎢ L2 = ⎢ 0 ∂/∂ y ⎢ 0 ⎢0 ⎣0 0 0 0 ⎡ 1 0 0 ⎢0 1 0 ⎢ K = ⎢0 0 1 ⎣0 0 0 0 0 0

σ2 =

0

⎫ ⎧ σx ⎪ ⎪ ⎪ ⎪ ⎪ σy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ τxy ⎪ D

x ⎪ ⎪ ⎪ Dy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B ⎪ ⎪ x⎪ ⎭ ⎩

721

⎤

0 0 0 0 0 0⎥ ⎥ 0 0 0⎥, ⎦ 0 0 0 0 0 0 ⎤ 0 0 0 ∂/∂ y 0 0 ⎥ ⎥ ∂/∂ x 0 0 ⎥ ⎥ 0 ∂/∂ x 0 ⎥, ⎥ 0 ∂/∂ y 0 ⎥ 0 0 ∂/∂ x ⎦ 0 0 ∂/∂ y ⎧ ⎫ ⎤ 0 0 w⎪ ⎪ ⎪ ⎪ ⎪ 0 0⎥ ⎨u⎪ ⎬ ⎥ U= v , 0 0⎥, ⎪ ⎪ ⎦ ⎪ ⎪ 0 0 ⎪ ⎩φ⎪ ⎭ 0 0 ψ

⎧ fz ⎪ ⎪ ⎪ ⎨ fx

G=

,

⎧ ⎫ σz ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ τ yz ⎪ ⎬

σ1 =

τ

xz ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Dz ⎪ ⎭

,

Bz

⎫ ⎪ ⎪ ⎪ ⎬

f

y ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − fe ⎪ ⎭ − fm

(4)

,

By

where φ , ψ , D i , B i and T are the electric potential, magnetic potential, electric displacement, magnetic induction and temperature difference relative to a referenced temperature at the stress free state, respectively, σi and τi j the normal (shear) stresses, u, v, w the displacement components in x-, y- and z-directions, respectively, c i j , εi j , e i j , q i j , di j , μi j , p i and mi respectively the elastic, dielectric, piezoelectric, piezomagnetic, electromagnetic, magnetic induction, pyroelectric and pyromagnetic constants, βi and ρ the thermal elastic modulus and density, respectively, f i (i = x, y , z) the body forces, and f e and f m the free charge and current density. Eq. (1) is known as the state equation, while Eq. (2) is called the output equation. For the static problem of orthotropic materials without body forces, free charge, current density and temperature effect, Eq. (1) is reduced to

  ∂ 0 A1 V, V = MV = ∂z A2 0

(5)

where V = [u , v , σz , D z , B z , τxz , τ yz , w , φ, ψ]T is the state vector, and A1 and A2 are expressed as follows

⎡

1 c 55

⎢ ⎢ ⎢ A1 = ⎢ ⎢ ⎣

0

− ∂∂x

1 c 44

− ∂∂y 0

15 − ce55

24 − ec44

15 − qc55

∂ ∂x ∂ ∂y

24 − qc44

0 2 2 k1 ∂∂x2 + k3 ∂∂y 2

⎢ ⎢ ⎣

∂

0 2 2 k2 ∂∂x2 + k4 ∂∂y 2 2

2

1 ∂ x2

2

2

− c 66 ∂∂y 2

−(γ2 + c 66 ) ∂ ∂x∂ y 2 −c 66 ∂∂x2

A2 = ⎢

2

2 − γ3 ∂∂y 2

− g11 ∂∂x

− g12 ∂∂x

− g13 ∂∂x

− g21 ∂∂y

− g22 ∂∂y

− g23 ∂∂y

α11 α

α12 α α22 α

α13 α α23 α α33 α

sym.

The induced variables are obtained as ⎧ ⎫ ⎡γ ∂ γ ∂ g g g 1 ∂x 2 ∂y 11 12 13 σx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σy ⎪ ⎪ ⎢ γ2 ∂∂x γ3 ∂∂y g21 g22 g23

⎪ ⎪ ⎪ ⎪ ⎪ ⎨ τxy ⎪ ⎬

⎢c ∂ ⎢ 66 ∂ y ⎢ Dx = ⎢ 0 ⎢ ⎪ ⎪ ⎪ Dy ⎪ ⎢ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ B 0 ⎪ ⎪ x ⎩ ⎭ By

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

k5 ∂∂x2 + k6 ∂∂y 2

sym.

⎡ −γ

⎤

∂ ∂x ∂ ∂y

0

0

0

0

0

0

0

0

0

c 66 ∂∂x

0

0

0

0

0

0

0

0

0

0

0

0

−k1 ∂∂x

0

0

0

0

0 −k3 ∂∂y

0

0

0

0

0

0

0

0

0

0

e 15 c 55 e 24 c 44 q15 c 55 q24 c 44

0

0

0

0 −k4 ∂∂y

−k2 ∂∂x

0

⎤ ⎥ ⎥ ⎥. ⎦ (6)

⎤

⎥ ⎥ ⎥ ∂ ⎥ −k2 ∂ x ⎥ V. ⎥ −k4 ∂∂y ⎥ ∂ ⎦ 0

0

−k5 ∂ x

−k6 ∂∂y

(7)

722

W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

Fig. 1. Geometry of a laminated rectangular plate.

In the above equations, the coefficients are determined by k1 = ε11 + e 215 /c 55 , 2 22 + e 24 /c 44 ,

k3 = ε

k5 = μ

2 11 + q 15 /c 55 ,

superscript (1) denotes the layer number, and (m, n) is the pair of half wave numbers in x- and y-directions. The solution in Eq. (9) satisfies the generalized simply supported boundary conditions (Pan, 2001; Chen and Lee, 2003). Because of the orthogonality of trigonometric functions, we will treat each term in the solution separately hereafter. Substitution of Eq. (9) into (5) yields

k2 = d11 + e 15 q15 /c 55 , k4 = d22 + e 24 q24 /c 44 , k6 = μ22 + q224 /c 44 ,

g 1i = (α1i c 13 + α2i e 31 + α3i q31 )/α , g 2i = (α1i c 23 + α2i e 32 + α3i q32 )/α ,

d

γ1 = c11 − c13 g11 − e 31 g12 − q31 g13 ,

dζ

⎡

γ3 = c22 − c23 g21 − e 32 g22 − q32 g23 ,   q33   c 33 e 33  α =  e 33 −ε33 −d33  ,   q33 −d33 −μ33

(8)

where αi j are the algebraic cofactors of α . The above equations are essentially the same as those in Wang et al. (2003), but arranged in a more concise way, with the symmetry of matrices A1 and A2 clearly shown in Eq. (6).

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ¯1 = ⎢ A ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

As clarified by Chen and Lee (2003), there are two kinds of boundary conditions, i.e. simple supports and rigid rolling supports (or in the generalized sense which accounts for electric and magnetic fields), for which exact solutions are obtainable. Here, the fully simply supported rectangular plate (see Fig. 1) is considered for example. Hence, assume that

(1) ⎪ ⎪ c 44 τ¯xz (ζ ) cos(mπ ξ ) sin(nπη) ⎪ ⎪ ⎪ ⎪ ⎪ ( 1 ) ⎪ ⎪ ⎪ ⎪ ¯ c 44 τ yz (ζ ) sin(mπ ξ ) cos(nπη) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯ (ζ ) sin(mπ ξ ) sin(nπη) ⎪ ⎪ hw ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ (1) (1) ¯ ⎪ ⎪ ⎪ h c 44 /ε33 φ(ζ ) sin(mπ ξ ) sin(nπη) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩  (1) (1) ¯ h c 44 /μ33 ψ(ζ ) sin(mπ ξ ) sin(nπη)

m=1 n=1 ⎪ ⎪ ⎪

(10)



(1 )

c 44 c 55

0 (1 )

c 44 c 44

15 −l1 − ce55 l1

15 − qc55 l1

(1 )



24 −l2 − ec44 l2

0



(1 )

c 44

ε33

(1 )

c 44

(1 ) ε33

0

⎢ ⎢ ⎢ ⎢ ⎢ A2 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(1 )

c 44



(1 )

c 44

μ(331)

0

k1 l21 +k3 l22

k l21 +k4 l22

2

(1 )

(1 ) (1 ) μ33 ε33

ε33

k5 l21 +k6 l22

γ1 l21 +c 66 l22

γ2 +c 66

c 44

c 44

(1)

(1)

 l1 l2

c 66 l21 +γ3 l22 (1)

c 44

g 11 l1

g 12

g 21 l2

g 22

(1) l 1



−

(1)

α11 c 44 α

μ(331) (1) ε33

c 44

(1) ε33

(1) l 2



c 44

(1) (1) α12 c 44 ε33 − α

−

(1) α22 ε33 α

sym.

⎤

μ33 ⎥ (1 )

24 − qc44 l2

sym.

3. Three-dimensional exact solutions

V=



¯1 A ¯, V 0

0 ¯2 A

¯ = [u¯ , v¯ , σ¯ z , D¯ z , B¯ z , τ¯xz , τ¯ yz , w ¯ ψ] ¯ T , and ¯ , φ, where V

γ2 = c12 − c13 g21 − e 31 g22 − q31 g23 ,

⎫ ⎧ h u¯ (ζ ) cos(mπ ξ ) sin(nπη) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h v¯ (ζ ) sin(mπ ξ ) cos(nπη) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( 1 ) ⎪ ⎪ ⎪ ⎪ ¯ c σ (ζ ) sin ( m π ξ ) sin ( n πη ) − ⎪ ⎪ 44 z ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ (1) (1) ¯ ⎪ ⎪ ⎪ − c 44 ε33 D z (ζ ) sin(mπ ξ ) sin(nπη) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ − c (1) μ(1) B¯ (ζ ) sin(mπ ξ ) sin(nπη) ⎪ ∞ ∞ ⎬ ⎨  44 33 z



¯ V¯ = ¯ =M V

 g 13

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤

μ(331)

(1) l 1

 g 23

c 44

μ(331) (1)



c 44

l2

(1) (1) α13 c 44 μ33 − α



−

(1) (1) α23 ε33 μ33 α (1) α μ − 33α 33

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

(11)

where l1 = mπ h/a and l2 = nπ h/b. For a typical layer, say the kth layer, the solution to Eq. (10) can be expressed as





¯ k (ζ − ζk−1 ) V¯ (k) (ζk−1 ) ¯ (k) (ζ ) = exp M V ,

(9)

where ξ = x/a, η = y /b, ζ = z/h are dimensionless coordinates, the over-bar represents the dimensionless unknown function, the

(12) (ζk−1  ζ  ζk , k = 1, 2, . . . , N ), k where ζ0 = 0, ζk = zk /h = j =1 h j /h, and hk is the thickness of the kth layer. Putting ζ = ζk in Eq. (12), yields

¯ = Qk V¯ , V 1 0 (k)

(k)

(13)

where the subscripts 1 and 0 represent the upper and lower sur¯ k (ζk − ζk−1 )] is the transfer faces of the kth layer, and Qk = exp[M matrix for that layer. Similarly, one has (k+1)

¯ V 1

(k+1)

= Qk+1 V¯ 0

.

(14)

W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

4. Interfacial transfer matrix In view of the imperfect interface model for elastic (Cheng et al., 2000), piezoelectric (Fan and Sze, 2001) and piezothermoelastic media (Wang and Pan, 2007), the following linear spring model is introduced,







(k) (k) σz(k+1) = σz(k) = K zz w (k+1) − w (k) + K ze φ (k+1) − φ (k)



(15)

(16)

1 0

0

0

0

R xx

0

0

0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

R yy 0 0 0 0 1

(k)

0 0 0 0 0 0

0 0 0 0 0 0

(k)

0 1

0

− R zm

0

0

− R em

(k)

0

0

(k)

(k)

0

0

0

− R ee

− R ze

(k)

− R zm − R em − R mm

0

0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ , 0⎥ ⎥ 0⎥ ⎥

(k)

− R ze

− R zz

1

j=N

Qj.

Suppose that the laminate is subjected to distributed normal pressures p (x, y ) and q(x, y ) on the bottom and top surfaces, respectively. The loads can be expanded in terms of Fourier series ∞  ∞ 

(1)

p (x, y ) = c 44

amn sin(mπ ξ ) sin(nπη),

m=1 n=1

(1)

q(x, y ) = c 44

∞  ∞ 

bmn sin(mπ ξ ) sin(nπη),

where



(1) 

1 1

[amn , bmn ] = 4/c 44





p (ξ, η), q(ξ, η) sin(mπ ξ ) sin(nπη) dξ dη.

0 0

For an arbitrary pair of (m, n), the mechanical boundary conditions at the top and bottom surfaces will be denoted as

σ¯ z (0) = amn ,

σ¯ z (1) = bmn ,

τ¯xz (1) = τ¯ yz (1) = τ¯xz (0) = τ¯ yz (0) = 0.

⎧ ⎫ ⎡ u¯ (0) ⎪ T 3, 1 ⎪ ⎪ ⎪ ⎪ ⎨ v¯ (0) ⎪ ⎬ ⎢ T 4, 1 ⎢ ¯ (0) = ⎢ T 5,1 w ⎪ ⎪ ⎣T ⎪ ⎪ ¯ ⎪ 6, 1 ⎩ φ(0) ⎪ ⎭ ¯ 0) ψ( T 7, 1

where R i j are the following dimensionless coefficients of the interface



(1)

(k)

(k)

(k)

(k)

(k) (1)





R ze = t 21

(1) (1)









(k)

(k)

(1)





R mm = t 33 μ33 / ht (k) , (k)

where t i j

(k) (1)

(k)

c 44 ε33 / ht (k) ,

R ee = t 22 ε33 / ht (k) ,

(k)

(k)

R zm = t 31 (k)





R zz = t 11 c 44 / ht (k) ,

(k)

R em = t 32





(1)

(1)





c 44 μ33 / ht (k) ,





(1) (1) ε33 μ33 / ht (k) ,

 (k) (k) (k)   K zz K ze K zm    (k) (k) (k)  t (k) =  K ze − K ee − K em  ,    K (k) − K (k) − K (k)  zm em mm

are the algebraic cofactors of t (k) .

(23)

Incorporating Eqs. (22) and (23) into the global transfer relation in Eq. (20), we obtain, T 3, 2 T 4, 2 T 5, 2 T 6, 2 T 7, 2

T 3, 8 T 4, 8 T 5, 8 T 6, 8 T 7, 8

T 3, 9 T 4, 9 T 5, 9 T 6, 9 T 7, 9

(17)

(k)

(22)

¯ z (0) = D¯ z (1) = B¯ z (0) = B¯ z (1) = 0. D

(k)

R ii = c 44 / K ii h (i = x, y ),

(21)

m=1 n=1

In addition, the electric and magnetic boundary conditions should be considered, which have many combinations. For brevity, the following conditions are considered

⎤

1 0 0⎥ ⎥ ⎥ 0 1 0⎦

(k)

0 0

0

(k)

(k)

Repeating the above manipulation for all interfaces, one derives

5. Bending analysis

where the interfacial transfer matrix Pk is defined as

⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ Pk = ⎢ 0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎣0

(19)

become unit, leading to T =

(k)

(k)

(k)

imperfections. In the case of a perfectly bonded laminate, all P j

(k) = Pk V¯ 1 ,

⎡

= Qk+1 Pk Qk V¯ 0 .

(1) = TV¯ 0 , (20) 1 where T = j = N Q j P j −1 , with P0 being an identity matrix of order 10 × 10, is the global transfer matrix for a laminate with interfacial

ficients of the interface, K ee and K mm are the electric permittivity and magnetic permeability coefficients, respectively, and the coefficients with different subscripts are the coupling ones between different fields. This linear model can be reduced to the imperfect interfacial model described in literature (Cheng et al., 2000; Fan and Sze, 2001) by making relevant coefficients zero. It shall be further pointed out that, the interfacial model given in Eq. (15) is not for the usual adhesive layer that is considered as elastic; rather, it is for a magneto-electro-elasto thin layer that is artificially introduced to explore its effect on the global response of the laminate, as will be shown in Section 6. The elastic adhesive bond is a particular case of the general model as mentioned above. In terms of Eq. (9), Eq. (15) can be written in the following matrix form (k+1)

(k+1)

¯ V 1

(N )

at the interface z = zk , i.e. the kth interface, between the kth and (k) (k + 1)th layers, where K ii (i = x, y , z) are the elastic stiffness coef-

¯ V 0

Eliminating the state vector at the upper surface of the kth layer and that the lower surface of the (k + 1)th layer from Eqs. (13), (14) and (16), one obtains

¯ V 1

 (k)  + K zm ψ (k+1) − ψ (k) ,  (k)  (k+1) τxz(k+1) = τxz(k) = K xx u − u (k) ,  (k+1) (k) (k)  τ yz = τ yz = K y y v (k+1) − v (k) ,   (k+1) (k) (k)  (k)  Dz = D z = K ez w (k+1) − w (k) − K ee φ (k+1) − φ (k)  (k)  − K em ψ (k+1) − ψ (k) ,   (k+1) (k) (k)  (k)  Bz = B z = K zm w (k+1) − w (k) − K em φ (k+1) − φ (k)  (k)  − K mm ψ (k+1) − ψ (k) ,

(k)

723

(18)

⎤

⎧

⎫

T 3,10 −1 ⎪ bmn − T 33 amn ⎪ ⎪ ⎪ T 4,10 ⎥ ⎪ ⎨ − T 43 amn ⎪ ⎬ ⎥ , T 5,10 ⎥ − T 53 amn ⎪ ⎦ ⎪ ⎪ ⎪ T 6,10 − T a ⎪ ⎪ 63 mn ⎩ ⎭ T 7,10 − T 73 amn (24)

where T i , j are the elements of the matrix T on the ith row and the jth column. It is seen that the final set of equations to be solved in the SSM for the present problem involves only five algebraic equations, hence assuring the superiority of the SSM over the conventional stress or displacement methods. Once the unknown variables at the lower surface of the plate is solved from Eq. (24), the state vector at any arbitrary point can then be determined by

  ¯ (ζ ) = exp M ¯ k (ζ − ζk−1 ) Pk−1 V



1 

 ¯ (1) Q j P j −1 V 0

j =k−1

(ζk−1  ζ  ζk ; k = 1, 2, . . . , N ). The induced variables then can be calculated based on Eq. (7).

(25)

724

W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

Table 1 Bending of an imperfect laminate ( B / F / B / F ). R

ζ

uˆ

0.0

0 .0 0.4 1.0 0 .0 0.4 1.0 0 .0 0.4 1.0

−1.13879 −0.265571 0.988274 −1.23391 −0.216318 1.06825 −1.32162 −0.172865 1.14491

0.3

0.6

vˆ

ˆ w

−1.70818 −0.398357 1.48241 −1.85087 −0.324476 1.60237 −1.98243 −0.259297 1.71736

τˆxz

−4.86760 −5.707617 −4.96115 −6.43039 −6.64019 −6.52389 −7.91162 −8.12265 −8.00506

6. Numerical example For numerical calculation, the multiferroic rectangular plate composed of BaTiO3 and CoFe2 O4 is considered. Material properties of the two constituents can be found in the paper of Pan (2001), and they are abbreviated as B and F in the following text for brevity. In all examples, the aspect ratios are taken as a/h = 6 and b/h = 4, and the plate is subjected to a sinusoidal load q(x, y ) = q0 sin(π ξ ) sin(πη) on the top surface. The following dimensionless quantities are employed

ˆ)= uˆ ( vˆ , w  ˆi = D

hq0

Di

B c 44

q0

B ε33



φˆ =

B u ( v , w )c 44

,

B B ε33 φ c 44

hq0

σˆ i (τˆi j ) =

,

 Bˆ i =

Bi

B c 44

q0

B μ33



,

ψˆ =

q0

,

,

B B ψ c 44 μ33

hq0

σi (τi j )

(26)

,

where the superscript B denotes BaTiO3 . Note that, all the results afterward refer to the amplitude, viz. uˆ represents the displacement in x direction at ξ = 0, η = 0.5, and φˆ corresponds to the value at ξ = η = 0.5, and so on. In calculation, each layer is assumed to have the same thickness, and the plate is arranged from the bottom to the top layer, such as B/F/B/F presenting a four-ply laminate with the bottom layer made from BaTiO3 and the top layer CoFe2 O4 . Preliminary numerical calculations show that the results for perfect laminate by the present formulations agree well with those in literature (Pan, 2001; Wang et al., 2003; Chen and Lee, 2003), hence verifying the formula and program in this paper. It should be noted that, the results for piezomagnetic laminates can be obtained by setting e i j = 0 in the present formulations, that for piezoelectric laminates by setting q i j = 0, and that for purely elastic laminates by setting e i j = 0 and qi j = 0 simultaneously. Therefore, further calculations for imperfect elastic laminate are performed, with the results identical to those given by Chen et al. (2003), further validating the present work. The following results are for multiferroic laminates with the (k) coupling coefficients of interfaces all vanishing, viz. R i j = 0 (k)

(k)

B B (i = j ). We further assume that c 44 /[ K xx h] = c 44 /[ K y y h] = (k) (k) (k) B B ˆ ε33 /[ K ee h] = μ33 /[ K mm h] = R k , and K zz → ∞. Table 1 lists the

numerical results for several physical quantities of B/F/B/F laminates with various interface coefficients Rˆ 1 = Rˆ 2 = Rˆ 3 = R. As mentioned earlier, the results for R = 0, i.e. the perfect B/F/B/F laminate, is identical to those calculated by other methods (Pan, 2001; Wang et al., 2003; Chen and Lee, 2003). It is seen that, with the increasing of R, the deflection of the rectangular plate increases, showing that the rigidity of plate decreases due to the bonding imperfection. Since the plate is quite thick, the transverse ˆ changes explicitly through the thickness direction displacement w rather than a constant as assumed in the classical plate theory or lower-order shear deformation theories. Furthermore, the electric potential difference and, in particular, the magnetic potential

0

τˆ yz 0

−0.822672

−1.23401

0 0 −0.824462 0 0 −0.826094 0

0 0 −1.23669 0 0 −1.23914 0

φˆ −0.214088 −0.545812 −0.632225 −0.217222 −0.537675 −0.621643 −0.219913 −0.530047 −0.612063

ψˆ 0.0250287 0.0261254 0.00168551 0.0196094 0.0214944 0.00651301 0.0185561 0.021073 0.0070068

difference between the bottom and top surfaces vary obviously with the interface coefficients. This phenomenon is interesting and may be used in feedback control of smart structures. The throughˆ τˆxz and ψˆ ) are thickness distributions of some field variables (u, given and compared in Fig. 2, which again indicates clearly the effect of bonding imperfection. Now we assume that Rˆ 1 = Rˆ 2 = 2 Rˆ 3 = R, namely, the third interface differs from the others. Since the effect of bonding imperfection on mechanical behavior of elastic materials was presented in detail in Chen and Lee (2003, 2004) and Chen et al. (2004) as well as is partly shown Fig. 2, only the effect on the electric and magnetic fields is discussed in the following. The throughthickness distributions of the dimensionless electric and magnetic potential amplitudes are displayed in Fig. 3. It is shown that, for the present B/F/B/F laminate, the weakening of interfaces leads to the redistributions in the electric and magnetic fields. Especially, the magnetic potential difference between the top and bottom surfaces decreases dramatically with R. This indicates that, to get an ideal ME coupling coefficient of the multiferroic laminate, the bonding between layers should be carefully prepared. Compared in Fig. 4 are the distributions of the electric and magnetic potential amplitudes for two different bonding conditions. The curves of R = 0.3(I) are for Rˆ 1 = Rˆ 2 = 2 Rˆ 3 = R with R = 0.3 as in the previous example, and those of R = 0.3(II) are for Rˆ 1 = 2 Rˆ 2 = Rˆ 3 = R = 0.3. It is seen that different bonding conditions may lead to quite different distributions of field variables in the plate. This may be utilized for a practical purpose for the adjustment of plate behavior. 7. Conclusions In this paper, an exact three-dimensional solution for orthotropic magneto-electro-elastic laminates with interlaminar bonding imperfections is presented. The state space method is adopted, which has great advantages in numerical efficiency and programming uniformity. The effect of the electro-magneto properties of interfaces on the electric and magnetic fields in the laminate is investigated, showing that the magnetic potential changes distinctly for a B/F/B/F laminate. It should be noted that, since the coupling coefficients of the electro-magneto-elastic interface are very small, their effect on the elastic field is insignificant. The present generalized linear spring model in Eq. (15) is a direct extension of the models for elastic and piezoelectric media. The reasons for such generalization lie in: (1) The present model can be easily reduced to the model for the elastic or piezoelectric media; (2) The linear model obviously facilitates the theoretical analysis, and it contains few parameters which may be much easier to be measured through experiments (Hashin, 1990); (3) Due to the popularity of the linear spring model in analysis of elastic structures and the correspondence principle that can be established between linear magneto-electro-elastic and

W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

725

(a) uˆ

(b) τˆxz

(c) ψˆ Fig. 2. Comparison of through-thickness distributions of field variables.

elastic theories (Karapetian et al., 2002), the present generalized linear model is theoretically reasonable. However, since what causes the weakening of interface may differ significantly from case to case, micromechanics models and experiments are required to determine the interface coefficients for practical problems.

In a practical laminate, the imperfect bond may take place within a certain region of the interface. In this case, it is in general impossible to get the exact solution. Nevertheless, one can evaluate the global weakness of bond using an appropriate micromechanics approach (such as the simple average method, the self-consistent method, the Mori–Tanaka method, the differential scheme, etc.), to

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W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

(a) φˆ

(b) ψˆ Fig. 3. Distributions of dimensionless electric and magnetic potentials along the thickness in multiferroic plate for Rˆ l = Rˆ 2 = 2 Rˆ 3 = R.

(a) φˆ

(b) ψˆ Fig. 4. Comparison of distributions of dimensionless electric and magnetic potentials along the thickness in multiferroic plate between two bonding conditions.

W.Q. Chen et al. / European Journal of Mechanics A/Solids 28 (2009) 720–727

(k)

get the effective compliance coefficients R i j of the interface. Thus, one may approximately get the change of field variables due to local bonding imperfection just as the examples considered in this paper. Acknowledgements

727

Hashin, Z., 1990. Thermoelastic properties of fiber composites with imperfect interface. Mechanics of Materials 8, 333–348. Huang, J.H., Chiu, Y.H., Liu, H.K., 1998. Magneto-electro-elastic Eshelby tensors for a piezoelectric-piezomagnetic composite reinforced by ellipsoidal inclusions. Journal of Applied Physics 83, 5364–5370. Karapetian, E., Kachanov, M., Sevostianov, I., 2002. The principle of correspondence between elastic and piezoelectric problems. Archive of Applied Mechanics 72, 564–587.

The work was supported by the National Natural Science Foundation of China (Nos. 10725210, 10832009, and 10702061), the National Basic Research Program of China (No. 2009CB623204), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060335107) and the Program for New Century Excellent Talents in University (No. NCET-05-05010).

Li, J.Y., 2000. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science 38, 1993–2011.

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