Natural frequencies of laminated piezoelectric plates with internal electrodes

ZAMM · Z. Angew. Math. Mech. 86, No. 5, 410 – 420 (2006) / DOI 10.1002/zamm.200310254

Natural frequencies of laminated piezoelectric plateswith internal electrodes

C.W. Lim1, Zhen-Qiang Cheng2, and J. N. Reddy2,∗

1 Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong2 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

Received 13 May 2003, revised and accepted 13 November 2005Published online 28 February 2006

Key words piezoelectricity, vibration, asymptotic method, laminated plate, internal electrodeMSC (2000) 74F15, 74M45

Natural vibrations of laminated piezoelectric plates with internal electrodes are analyzed using the transfer matrix methodand the asymptotic expansion method. The steady-state equations of three-dimensional linear piezoelectricity reduce toa hierarchy of two-dimensional equations of the same homogeneous operator. The leading-order equations are easilysolvable, whereas the higher-order equations may contain secular terms and are not straightforward to find their solutions.The solvability condition is established to calculate higher-order frequency parameters. The present theoretical formulationis used to provide new results by calculating fundamental frequencies of a rectangular laminated plate with two surface-affixed piezoelectric actuators, a parallel bimorph, a four-layered multimorph and a functionally graded plate attached withan actuator.

c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Smart composite structures and micro-electro-mechanical systems (MEMS) have great potential applications in mechanical,civil and aerospace engineering. These electromechanical systems need accurate modeling of both strain and electric fields.However, most studies considered only one-way interaction between the mechanical field and the electric field. For example,in the analysis of smart structure with actuators, the electric field was calculated directly from applied electric voltage, whichis then introduced into constitutive relations as a prescribed eigenstrain. This one-way interaction is known as strain actuation.In fact, not only the piezoelectric actuation change the strain field during active control of the passive structure, but also theresulting new strain field in the substrate can in turn affect the electric field distribution. This is known as bi-way interactionin smart composites structures [1,2]. The bi-way interaction requires proper incorporation of an internal electrode in thesmart structures.

An internal electrode is present in many cases of a piezoelectric structural system. The electric voltage is applied to twoconducting electrodes that are attached on the two surfaces of an actuator. For a smart structure comprising of a substrate andactuators, one internal electrode is present for the case of a surface-affixed actuator (Figs. 1a and b) and two internal electrodesare present for the case of an embedded actuator (Fig. 1c). Another class of laminates containing internal electrodes is aparallel bimorph (Fig. 1d) with an internal electrode and a multimorph with interdigital electrodes (Fig. 1e). The upper andlower layers of the parallel bimorph shown in Fig. 1d are made of the same material and polarized in the same direction. Themultimorph (piezoelectric multilayered actuator) is composed of piezoelectric ceramic layers and internal electrode layerspiled up alternately, with each thin film internal electrode connected to every other to create two terminals. The adjacentceramic layers are made of the same material and may have either same or opposite poling directions. The adjacent layersin the multimorph shown in Fig. 1e are of opposite directions in polarization.

For a structural element with internal conducting electrodes, the transverse electric displacement must be different on thetwo sides of an electrode. In this case, the discontinuity in the transverse electric displacement must be accommodated. Thereare only a few papers [3–7] addressing this issue. Specifically, [3] was devoted to a simplified one-dimensional treatmentof a resonator and [4–7] were only concerned with static bending behavior of laminated piezoelectric plates. Therefore, itis desirable to develop a three-dimensional method for the vibration analysis of laminated piezoelectric plates with internalelectrodes.

Tarn and Wang [8] proposed using the technique of multiple time scales expansion to study dynamic response of alaminated elastic plate. The multiple time scales expansion was used in order to treat secular terms in their high-orderequations. In our viewpoint, a forcing vibration problem does not generate the secular terms whereas a free vibration

∗ Corresponding author, e-mail: [email protected]

c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ZAMM · Z. Angew. Math. Mech. 86, No. 5 (2006) 411

actuator

substrate

actuator

actuator

actuator

(a) (b) (c)

(d) (e)

Fig. 1 Profiles of smart structural elements: (a) substrate with a surface-affixed actuator; (b) substrate with two surface-affixed actuators; (c) laminate with an embedded actuator; (d) bimorph (parallel poling); (e) multimorph (alternate poling).

problem does. The multiple time scales expansion is so complicated that higher-order solutions are getting more and moreinvolved. In order to develop a simpler and more practical alternative, some efforts were made in the analysis of a functionallygraded plate [9] and a laminated piezoelectric plate without internal electrodes [10]. Instead of using the multiple time scalesexpansion for free vibration, we expand the frequency parameter to refine their multiple time scales expansion method in arather simpler way. The present work extends this simpler technique to the vibration analysis of a laminated piezoelectricplate with internal electrodes. The transfer matrix formulation is combined with the asymptotic expansion method to derivea hierarchy of two-dimensional equations from a three-dimensional framework.

2 State-space equations

Let the bottom plane of an undeformed plate of uniform thicknessh be located atx3 = 0 of a rectangular Cartesian coordinatesystem xi (i = 1, 2, 3). A comma followed by a subscript i denotes the partial derivative with respect to xi, and a repeatedindex implies summation over the range of the index with Latin indices ranging from 1 to 3 and Greek indices from 1 to 2.

In the absence of body force and electric charge density, the linear governing equations for steady-state deformations ofa piezoelectric plate and Gauss’ law of electrostatics are [11,12]

τij,j + ρω2ui = 0 , Di,i = 0 , (1)

where ρ is the mass density, ω an angular frequency, τij the stress tensor components, Di the electric displacement com-ponents, and uk the mechanical displacements. The time-harmonic factor exp(iωt) has been omitted and each physicalquantity refers to its spatial part. For a monoclinic piezoelectric material with reflectional symmetry in planes parallel to thesurfaces of the plate, the constitutive relations are

ταβ = cαβωρSωρ + cαβ33S33 − e3αβE3 ,

τα3 = 2cα3ω3Sω3 − eωα3Eω ,

τ33 = c33ωρSωρ + c3333S33 − e333E3 ,

Dα = 2eαω3Sω3 + εαωEω ,

D3 = e3ωρSωρ + e333S33 + ε33E3 .

(2)

Here c is the fourth-order elasticity tensor, e the third-order piezoelectric tensor and ε the second-order dielectric tensor.These material moduli exhibit the following symmetries

cijkl = cjikl = cklij , ekij = ekji , εik = εki. (3)

The material properties are functions of x3. For a laminated plate comprised of different homogeneous materials, the materialmoduli are piecewise constant functions of x3. The components of infinitesimal strain tensor Skl and electric field Ek arerelated to uk and the electric potential ϕ through the relations

Skl = 12(uk,l + ul,k) , Ek = −ϕ,k . (4)

www.zamm-journal.org c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

412 C.W. Lim et al.: Natural frequencies of laminated piezoelectric plates with internal electrodes

Eqs. (1), (2), and (4) can be rewritten in the following state-space equation

∂z

[FG

]= χ

[0 AB 0

][FG

]− χρΨ

[0 AB 0

][FG

], (5)

where Ψ = ω2, z = x3/χ, ∂z ≡ ∂/∂z, χ = h/a, a is a typical in-plane dimension. The thickness coordinate x3 has beenscaled to z extending from 0 to a. The state-space functions are

F =

u1

u2

τ33

D3

, G =

τ13

τ23

u3

ϕ

. (6)

In the case of a laminated plate in perfect bonding and without internal electrodes, F and G are continuous across eachinterlaminar interface. However, if there is an internal electrode, D3 is not continuous across it. The only nonzero elementsof the 4 × 4 constant matrices A and B are A33 = B11 = B22 = 1. The 4 × 4 operator matrices A and B contain thein-plane differential operator ∂α ≡ ∂/∂xα and depend on z only through the material moduli:

A =

[I −Jβ∂β

−JTβ ∂β Kβρ∂β∂ρ

], B =

[−Lβρ∂β∂ρ −Mβ∂β

−MTβ ∂β N

], (7)

where

I = (Iωα) =

[c1313 c1323

c1323 c2323

]−1

, N = (Nαω) =

[c3333 e333

e333 −ε33

]−1

,

[Jω1

β Jω2β

]=[δωβ Iωαeβα3

],[Mα1

β Mα2β

]=[cαβ33 e3αβ

]N,

K11βρ = K12

βρ = K21βρ = 0, K22

βρ = Jω2β eρω3 + εβρ, Lαω

βρ = cαβωρ −Mα1β c33ωρ −Mα2

β e3ωρ. (8)

δωβ is the Kronecker delta symbol. Superscripts are used to denote the location, i.e. row and column, of matrix elements,while the Greek subscripts are for the usual tensor notation. The in-plane stresses and in-plane electric displacements aregiven by

ταβ = Lαωβρ uω,ρ +Mα1

β τ33 +Mα2β D3, Dρ = Jα2

ρ τα3 −K22βρϕ,β . (9)

3 Asymptotic approach

The two bounding surfaces and r electroded interfaces of the plate are coated with very thin conducting electrodes to carryan alternating forcing electric potential. For simplicity, the thickness of each electrode is neglected and it is modeled asa mathematical surface with a specified electric potential. The distance between the ith internal electrode surface and thebottom most surface of the plate is z = (i)a. In particular, we set (0)a = 0 and (r+1)a = a for the position of the bottommost and top most surfaces. This physical model includes the important case of laminated plates with surface-affixed and/orembedded sensors and actuators and the case of bimorphs and multimorphs.

The state-space functions F and G and the frequency parameter Ψ are expanded in terms of the small parameter χ as[FG

]=

∞∑n=0

χ2n

[χf (n)

g(n)

], Ψ =

∞∑n=0

χ2nψ(n), (10)

and then substituted into eq. (5) to give the following recursion relations

∂zg(0) = 0, ∂zg(n+1) = Bf (n) −n∑

k=0

ρψ(k)Bf(n−k)

,

∂zf (0) = Ag(0), ∂zf (n+1) = Ag(n+1) −n∑

k=0

χ−2ρψ(k)Ag(n−k)

, (n ≥ 0). (11)

c© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.zamm-journal.org


Insofar as the free vibration problem is concerned in this paper, the condition of zero tractions on the two bounding surfacesand the condition of zero electric potentials on r + 2 electroded surfaces may be expressed by

g(n)α (0) = τ

(n)α3 (0) = 0, g(n)

α (a) = τ(n)α3 (a) = 0, (12)

f(n)3 (0) = τ

(n)33 (0) = 0, f

(n)3 (a) = τ

(n)33 (a) = 0, (13)

g(n)4 ((i)a) = ϕ(n)((i)a) = 0, (i = 0, · · · , r + 1). (14)

Because of the presence of internal electrodes, the transverse electric displacement is not continuous across each of theinternally electroded surfaces. The electric displacement jump across the jth internal electrode is denoted as

(j)∆D(n)3 ≡ D

(n)3 (xρ,

(j)a+) −D(n)3 (xρ,

(j)a−), (j = 1, · · ·, r). (15)

In such a case, the electric potential ϕ((j)a) at z = (j)a is given instead.By integrating the differential equations (11) with respect to z and using the given condition in eqs. (12)–(14), a solution

is obtained as

g(0) =

00

U(0)3

0

, g(n+1) =

00

U(n+1)3

0

+QBf (n) −

n∑k=0

ψ(k)

Qρf

(n−k)1

Qρf(n−k)2

00

,

f (0) =

U

(0)1 − zU

(0)3,1

U(0)2 − zU

(0)3,2

0D

(0)0

+

000

r∑j=1

(j)∆D(0)3 H(z − (j)a)

,

f (n+1) =

U

(n+1)1

U(n+1)2

0D

(n+1)0

+

000

r∑j=1

(j)∆D(n+1)3 H(z − (j)a)

+QAg(n+1)−

n∑k=0

χ−2ψ(k)

00

Qρg(n−k)3

0

, (n ≥ 0), (16)

where H(z − (j)a) is the Heaviside step function, and

U(n)i ≡ u

(n)i (xρ, 0), D

(n)0 ≡ D

(n)3 (xρ, 0+), Q(· · · ) ≡

∫ z

0(· · · )dz. (17)

Furthermore, f (n) can be alternatively written as

f (n) = X(n) + H(n) +

000

r∑j=1

(j)∆D(n)3 H(z − (j)a)

, (18)

where

X(n) =

U

(n)1 − zU

(n)3,1

U(n)2 − zU

(n)3,2

0D

(n)0

, H(0) = 0,



H(n+1) = QAQBf (n) −n∑

k=0

ψ(k)

QA

Qρf

(n−k)1

Qρf(n−k)2

00

+ χ−2

00

Qρg(n−k)3

0

, (n ≥ 0). (19)

The conditions (12)–(14) for zero tractions on the top surface of the plate and the zero electric potentials at the internalelectrodes and on the top surface of the plate may be expressed through (16) as

QBαLf(n)L =

n∑k=0

ψ(k)Qρf (n−k)α , (20)

QzBαLf(n)L,α =

n∑k=0

ψ(k)Qρ[zf (n−k)

α,α + χ−2g(n−k)3

], (21)

(i)QB4Lf(n)L = 0, (i = 1, · · · , r + 1), (22)

where an upper case subscript L takes the values from 1 to 4 and the usual summation convention applies to L, and

Q(· · · ) ≡∫ a

0(· · · )dz = (r+1)Q, (i)Q(· · · ) ≡

∫ (i)a

0(· · · )dz, (i = 1, · · · , r + 1). (23)

The following r equations can be derived from (22)

[Q− (i)Q]B4L(X(n)L +H

(n)L ) +

r∑j=1

Cij(j)∆D(n)

3 = 0, (i = 1, · · · , r), (24)

where eq. (18) has been substituted, and Cij is defined by

Cij =

[Q− (i)Q]B44, (i ≥ j)

[Q− (j)Q]B44, (i ≤ j). (25)

Note that B44 = N22 is only related to the material moduli depending on x3 and thus (Cij) is a constant matrix. Eqs. (24)

may be viewed as a set of r linear algebraic equations and thus solved for the electric displacement jumps (j)∆D(n)3 as

(j)∆D(n)3 = −

r∑k=1

C−1jk [Q− (k)Q]B4L(X(n)

L +H(n)L ), (j = 1, · · · , r). (26)

Substituting eq. (26) into eq. (18) and the resulting equation into eqs. (20), (21), and (22) at i = r + 1 yields

[R − ψ(0)m]X(n) =

0, (n = 0)

ψ(n)mX(0)

+ Z(n), (n ≥ 1), (27)

where

X(n) =[U

(n)1 U

(n)2 U

(n)3 D

(n)0

]T, (28)

Z(n) = −RH(n) +n−1∑k=1

ψ(k)mX(n−k)

+n−1∑k=0

ψ(k)Qρ

H(n−k)1

H(n−k)2

zH(n−k)α,α + χ−2QB3Lf

(n−k−1)L

0

, (29)

Rαω = Rαω = −QLαω

βρ +r∑

j=1

r∑k=1

C−1jk [Q− (j)Q]Mα2

β [Q− (k)Q]Mω2ρ

∂β∂ρ,



Rα3 =

QzLαω

βρ +r∑

j=1

r∑k=1


β [Q− (k)Q]zMω2ρ

∂β∂ω∂ρ,

Rα3 = −QMα1

β −r∑

j=1

r∑k=1


β [Q− (k)Q]N21

∂β ,

Rα4 = Rα4 = −QMα2

β −r∑

j=1

r∑k=1


β [Q− (k)Q]N22

∂β ,

R3ω = R3ω = −QzLαω

βρ +r∑

j=1

r∑k=1

C−1jk [Q− (j)Q]zMα2

β [Q− (k)Q]Mω2ρ

∂α∂β∂ρ,

R33 =

Qz2Lαω

βρ +r∑

j=1

r∑k=1


β [Q− (k)Q]zMω2ρ

∂α∂β∂ω∂ρ,

R33 = −QzMα1

β −r∑

j=1

r∑k=1


β [Q− (k)Q]N21

∂α∂β ,

R34 = R34 = −QzMα2

β −r∑

j=1

r∑k=1


β [Q− (k)Q]N22

∂α∂β ,

R4ω = R4ω =

QMω2

β −r∑

j=1

r∑k=1

C−1jk [Q− (j)Q]N22[Q− (k)Q]Mω2

β

∂β ,

R43 = −QzMω2

β −r∑

j=1

r∑k=1

C−1jk [Q− (j)Q]N22[Q− (k)Q]zMω2

β

∂β∂ω,

R43 = −QN21 +r∑

j=1

r∑k=1

C−1jk [Q− (j)Q]N22[Q− (k)Q]N21,

R44 = R44 = −QN22 +r∑

j=1

r∑k=1

C−1jk [Q− (j)Q]N22[Q− (k)Q]N22,

m11 = m22 = Qρ, mα3 = −m3α = −Qzρ∂α, m33 = χ−2Qρ−Qz2ρ∂β∂β ,

m12 = m21 = mi4 = m4i = m44 = 0. (30)

Eq. (27) is the key equation in the asymptotic theory, from which the frequency parameter and the modal vector (28) of eachorder remain to be solved with specified edge conditions.

4 Solvability and normalization conditions

It can be recognized that the operator matrix [R − ψ(0)m] on the left-hand side of eq. (27), when degenerated frompiezoelectricity to elasticity, is identical with that of the classical Kirchhoff theory for the vibrations of a thin plate madeof a monoclinic elastic material. The leading-order frequency parameter ψ(0) and its corresponding modal vector X(0)

can be determined by a standard method. However, a method for solving higher-order frequency parameters in terms ofeq. (27) (n ≥ 1) is not straightforward. The operator matrix [R − ψ(0)m] for the higher-order equation (n ≥ 1) is sameas that of the leading-order equation (n = 0), whereas terms (n ≥ 1) on the right-hand side of eq. (27) do not vanish. Thenonhomogeneous terms (n ≥ 1) in eq. (27) involve lower-order solutions that may generate secular terms in the higher-orderequations. To ensure a uniformly valid expansion of the asymptotic approach, it is necessary to study the solvability conditionunder which the higher-order equations possess solutions that are bounded and free from secular terms.



Premultiplying [R−ψ(0)m]X(n) by X(0)T and integrating it over the domain of the plate surface Ω leads to, after lengthybut straightforward manipulation,∫

ΩX(0)T [R − ψ(0)m]X(n)dΩ =

∫Ω

X(n)T [R − ψ(0)m]X(0)dΩ −∫

ΓU (0)

α N (n)αβ nβdΓ

+∫

ΓU (n)

α N (0)αβ nβdΓ +

∫ΓU

(0)3,αM(n)

αβ nβdΓ −∫

ΓU

(n)3,αM(0)

αβnβdΓ

−∫

ΓU

(0)3 [M(n)

αβ,α + ψ(0)QzρU(n)β − ψ(0)Qz2ρU

(n)3,β ]nβdΓ

+∫

ΓU

(n)3 [M(0)

αβ,α + ψ(0)QzρU(0)β − ψ(0)Qz2ρU

(0)3,β ]nβdΓ, (31)

where Γ denotes the boundary of the domain Ω, nβ is an outward normal vector to the boundary, and

[N (n)αβ , M(n)

αβ ] = Q[1, z][Lαω

βρU(n)ω,ρ − zLαω

βρU(n)3,ωρ +Mα2

β D(n)0

]

+r∑

j=1

r∑k=1

C−1jk [Q− (j)Q][1, z]Mα2

β [Q− (k)Q][Mω2

ρ U (n)ω,ρ − zMω2

ρ U(n)3,ωρ −N22D

(n)0

]. (32)

Eq. (31) is identically satisfied for n = 0. The first integral on the right-hand side of eq. (31) vanishes due to eq. (27) atn = 0. If the remaining boundary integrals on the right-hand side of eq. (31) vanish, i.e. specifying on the plate edge Γ:

U(n)α = 0, or N

(n)αβ nβ = 0 ,

U(n)3,α = 0, or M

(n)αβ nβ = 0 ,

U(n)3 = 0, or [M (n)

αβ,α + ψ(0)QzρU(n)β − ψ(0)Qz2ρU

(n)3,β ]nβ = 0, (n ≥ 0) ,

(33)

then by substituting eq. (27) (n ≥ 1) into eq. (31) it follows that the higher-order frequency parameter is determined by

ψ(n) = −∫Ω X(0)TZ(n)dΩ∫

Ω X(0)TmX(0)

dΩ, (n ≥ 1). (34)

With this, eq. (27) is solvable for higher-order modal vectors.In addition that the elasticity counterpart of the operator matrix [R − ψ(0)m] in eq. (27) is the same as for the free

vibration problem using the two-dimensional classical plate theory, interestingly the edge conditions (33) for n = 0 are alsothe same as in the classical plate theory, except modified Kirchhoff’s effective shear force for a free edge. N (0)

αβ and M(0)αβ

defined by eq. (32) in the case of pure elasticity reduce to the conventional stress resultant and moment in the classical platetheory [12].

It is also interesting to note that no electric edge condition is required for eq. (34) to be valid. The conditions (33) atn = 0 are satisfied by the conventional edges such as simply supported edges and clamped edges. The edge conditions forhigher-order equations (n ≥ 1) can also be satisfied provided that the edge conditions for higher orders are approximatelygiven in the same way as given in the classical Kirchhoff plate theory. It should be noted that in general, specifying the edgeboundary conditions in the sense of the Kirchhoff plate theory only yields the accurate leading-order interior solution. Theleading-order solution does not account for the through-thickness distribution of the edge boundary conditions and cannotbe valid near to the edges. As far as the interior solution is concerned, it is generally expected, by virtue of Saint-Venant’sprinciple, that the boundary layer effect will not be felt away from the local disturbance.

Wu et al. [13] used the solvability condition in order to eliminate secular terms in their multiple time scales expansionfor elastic laminated shells. Note that minor formal errors, which do not affect their example results, appear in their work,where the dynamic terms involving the frequency parameter should be incorporated as they are into eqs. (31) and (33) ofthe present paper and the solvability condition should be given in the integral form.

It is often convenient to render a modal vector unique by assigning a given value either to one of the components of themodal vector or to the magnitude of the modal vector. This process is known as normalization. Clearly, the normalizationprocess is devoid of physical significance and should be regarded as a mere convenience. In this paper, the normalizationcondition is defined by∫

ΩXTXdΩ = 1 , where X =

∞∑n=0

χ2nX(n) . (35)



More detailed expressions can be found in [13].

5 Examples

A rectangular piezoelectric plate is considered. The following mixed boundary conditions are used to model simply supportedand grounded edges

u2 = u3 = τ11 = ϕ = 0 , at x1 = 0, a ,

u1 = u3 = τ22 = ϕ = 0 , at x2 = 0, b .(36)

The edges conditions (36) can be satisfied exactly by assuming

X(n) =

U

(n)1

U(n)2

U(n)3

D(n)0

=

U

(n)1 cos l1x1 sin l2x2

U(n)2 sin l1x1 cos l2x2

U(n)3 sin l1x1 sin l2x2

D(n)0 sin l1x1 sin l2x2

, (37)

where

l1 =m1π

a, l2 =

m2π

b, (38)

and a quantity with a superimposed hat denotes the amplitude of the corresponding physical quantity. A solution of eachorder may be obtained and numerical results may be computed to any desired degree of accuracy for the specific problem.The dimensionless frequency is defined by ω =

√ρ∗/c∗aω, where c∗ = 109N/m2 and ρ∗ = 103kg/m3. The frequency

solution of various orders is calculated from

Ψ = ω2 , or∞∑

n=0

χ2nψ(n) =

( ∞∑n=0

χ2nω(n)

)2

. (39)

The materials used in the calculation are lead zirconate titanate (PZT-4), fiber-reinforced composite (FRC) and bariumtitanate ceramic (BaTiO3) [14,15]. The material properties are shown in Table 1.

In the first example results obtained from the aforestated asymptotic approach are compared with those obtained from theexact solution [14]. A homogeneous FRC layer is perfectly bonded between two homogeneous PZT-4 piezoelectric layers.Thicknesses of the PZT-4 lamina and the FRC lamina equal 0.1h and 0.8h, respectively. The fundamental frequencies of thehybrid laminate (a/b = 0, a/h = 50, 4) without internal electrodes are calculated and given in Table 2, together with theexact values [14]. Because of slow convergence for the thick plate (a/h = 4), the repeated averaging technique for 5 partialsums (denoted as 5 terms-RA in Table 2) is used to accelerate the convergence of the asymptotic results. The technique canbe shown to be a special case of Euler’s transformation; more details may be found, for example, in the book [16]. Note thatthe small parameter expansion method may not be good for analysis of a thick plate.

It is seen that the present asymptotic solution converges very rapidly for the thin plate (a/h = 50). The second-orderapproximation has converged to the exact results for both flexural mode and in-plane stretching mode. However, this is notthe case for the thick plate (a/h = 4). The error in the 3rd-order approximation for the in-plane stretching mode is verysmall whereas it is significant for the flexural mode. Thus 13th-order result using 5 terms repeated averaging is calculatedand shows good agreement with the exact solution. Numerical convergence is reached at least to four significant digits forthe 13th-order results using 5 terms repeated averaging. In the following new examples, converged results to five significantdigits are reported.

The second example considers a two-ply elastic laminate with symmetrically affixed actuators on its top and bottomsurfaces, i.e. a four-ply (PZT-4/0˚FRC/90˚FRC/PZT-4) smart structural element (similar to Fig. 1b). Thicknesses of eachPZT-4 lamina and FRC lamina are 0.1h and 0.4h, respectively. Since the electric voltage is applied to two surfaces of anactuator, two internal conducting electrodes on the inner surfaces of the PZT-4 actuators are incorporated. Accordingly,four electrodes of vanishing thicknesses are present. The flexural-mode-predominant fundamental frequencies are shown inTable 3 for various aspect ratios and thickness-to-span ratios.

The third example is concerned with the free vibration of a rectangular parallel piezoelectric bimorph (Fig. 1d). Theupper and lower halves of the bimorph plate are made of the same material, have same thickness and are polarized in samex3 directions. The material of the bimorph is taken as BaTiO3. There is an internal electrode located at the interface. The



fundamental frequencies are given in Table 4. Incidentally, we also have calculated the frequencies of the bimorph when thetwo layers are polarized in opposite x3 direction. In the latter case, the piezoelectric coefficients are of opposite signs forthe upper and lower halves whereas all the other material properties are the same across the thickness. The results are thesame as in the formal case, i.e. the poling direction does not alter the natural frequencies of the bimorph.

The fourth example presents fundamental frequencies of a four-layered multimorph (Fig. 1e). Three internal electrodesare present. The material of the multimorph is made of BaTiO3. Comparing Table 5 with Table 4 for the same aspect ratioand thickness-to-span ratio, the fundamental frequency of the four-layered multimorph is slightly smaller than that of thebimorph.

The last example is devoted to a functionally graded plate attached with a piezoelectric actuator (Fig. 1a). The materialof the actuator is taken as BaTiO3. The functionally graded plate is made of a two-phase composite consisting of a matrixphase denoted by subscript 1 and a particulate phase denoted by the subscript 2. Denote V2 as the volume fraction of theparticulate phase. The composite is reinforced by spherical particles, randomly distributed in the plane of the plate. Thelocally effective mass density ρ is given by

ρ = ρ1(1 − V2) + ρ2V2. (40)

The locally effective bulk modulus K and shear modulus µ of the functionally graded material are estimated by the Mori-Tanaka method. The explicit formulas are (see, e.g. [6])

K −K1

K2 −K1=

V2

1 + (1 − V2) K2−K1

K1+43µ1

,µ− µ1

µ2 − µ1=

V2

1 + (1 − V2) µ2−µ1µ1+π1

, π1 =µ1(9K1 + 8µ1)6(K1 + 2µ1)

.

(41)

It is assumed that the volume fraction of the ceramic phase is of the power law type Vc = [(x3 −hE)/(h−hE)]n, wherehE is the thickness of the attached actuator. Note that the bottom surface of the functionally graded plate is metal-rich andthe top surface is ceramic-rich. The constituent materials of the plate are taken to be nickel-based alloy, Monel (70Ni-30Cu),and zirconia with their material properties [6]

Km = 227.24 × 109 N/m2, µm = 65.55 × 109 N/m2, ρm = 8800 kg/m3

Table 1 Material properties.

PZT-4 0FRC BaTiO3

c1111 (109N/m2) 139.02 134.86 150

c2222 (109N/m2) 139.02 14.352 150

c3333 (109N/m2) 115.45 14.352 146

c1122 (109N/m2) 77.846 5.1563 66

c1133 (109N/m2) 74.328 5.1563 66

c2233 (109N/m2) 74.328 7.1329 66

c2323 (109N/m2) 25.6 3.606 44

c3131 (109N/m2) 25.6 5.654 44

c1212 (109N/m2) 30.6 5.654 43

e311 (C/m2) −5.2 0 −4.35

e322 (C/m2) −5.2 0 −4.35

e333 (C/m2) 15.08 0 17.5

e223 (C/m2) 12.72 0 11.4

e113 (C/m2) 12.72 0 11.4

ε11/ε∗0 1475 3.5 1115

ε22/ε∗0 1475 3 1115

ε33/ε∗0 1300 3 1260

ρ (kg/m3) 1 1 5700∗ ε0 = 8.854185 × 10−12F/m

Table 2 Frequency comparison of the present results with the three-dimensional exact solution [14] of a hybrid laminate (PZT-4/0FRC/PZT-4)in cylindrical bending (a/b = 0).

Flexural mode In-plane stretching mode

a/h = 50Leading-order 19.788 1123.6

1st-order 19.704 1123.6

2nd-order 19.705 1123.6

3-D exact [14] 19.705 1123.7

a/h = 4Leading-order 241.27 1123.7

1st-order 97.163 1123.6

2nd-order 220.35 1119.7

3rd-order 112.52 1120.1

13th-order (5 terms-RA) 162.48 1116.0

3-D exact [14] 162.45 1116.0



Table 3 Flexural-mode-predominant fundamental frequen-cies of a two-layered elastic substrate bonded with two actua-tors (PZT-4/90FRC/0FRC/PZT-4).

a/b = 0 a/b = 0.5 a/b = 1a/h = 4 134.89 156.97 220.32

a/h = 10 68.649 81.627 123.27

a/h = 20 36.022 43.096 66.635

a/h = 50 14.620 17.527 27.319

a/h = 100 7.3254 8.7846 13.709

Table 4 Fundamental frequencies of flexural vibration of aparallel bimorph (BaTiO3).

a/b = 0 a/b = 0.5 a/b = 1a/h = 4 3.0203 3.7100 5.6325

a/h = 10 1.3022 1.6254 2.5741

a/h = 20 0.65902 0.82505 1.3179

a/h = 50 0.26452 0.33146 0.53084

a/h = 100 0.13232 0.16583 0.26569

Table 5 Fundamental frequencies of flexural vibration of a4-layered multimorph (BaTiO3).

a/b = 0 a/b = 0.5 a/b = 1a/h = 4 2.9911 3.6734 5.5728

a/h = 10 1.2907 1.6110 2.5511

a/h = 20 0.65327 0.81789 1.3065

a/h = 50 0.26223 0.32859 0.52626

a/h = 100 0.13118 0.16440 0.26340

Table 6 Flexural-mode-predominant fundamental frequen-cies of a functionally graded ceramic-metal plate with an at-tached actuator (BaTiO3) (a/h = 10, hE/h = 0.1).

Exponent a/b = 0 a/b = 0.5 a/b = 1n = 0.2 1.7379 2.1647 3.4236

n = 0.5 1.6003 1.9933 3.1529

n = 1 1.4930 1.8597 2.9417

n = 2 1.4069 1.7526 2.7726

n = 5 1.3414 1.6709 2.6432

Kc = 125.83 × 109 N/m2, µc = 58.077 × 109 N/m2, ρc = 3000 kg/m3, (41)

where the subscripts m and c stand for the metal and ceramic. In the numerical results Monel is chosen to serve as a matrixphase. The flexural-mode-predominant fundamental frequencies are presented in Table 6.

6 Conclusions

The three-dimensional free vibration equations of a laminated piezoelectric plate containing internal electrodes have beenreduced to a hierarchy of two-dimensional plate equations using the asymptotic expansion method. By expressing trans-verse electric displacement jumps across internal electrodes in terms of basic unknowns, the resulting governing differentialequations for piezoelectric laminates with internal electrodes have shown the same form as those for piezoelectric laminateswithout internal electrodes except different coefficients. Excellent agreement has been achieved between the present numer-ical results and available exact solution. New results have been presented for fundamental frequencies of a laminated platewith surface-affixed actuators, a bimorph, a multimorph, and a functionally graded plate with an actuator. The bi-way inter-action between strain and electric fields has been accounted for. The new fundamental frequencies may serve for benchmarkresults.

For modal analysis of very thin actuators attached to thin plates by using some two-dimensional shear deformation platetheories, the electric degree of freedom can be eliminated as a function of the mechanical degrees of freedom through astatic frequency-independent relationship. By carefully examining eq. (27), we feel that it is possible to condense the electricdegree of freedom in the matrix equation in our three-dimensional asymptotic analysis.

It is possible to extend the present method to calculations of static frequencies of slightly curved or deeply curved piezo-electric panels. This would be similar to solving the static bending problem of laminated piezoelectric circular cylindricalshells in [17].

Acknowledgements The research reported herein was carried out under the sponsorship of the Grant DAAD19-01-1-0483 from ArmyResearch Office and from City University of Hong Kong [Project No. 7001875 (BC)].



References

[1] J.A. Mitchell and J. N. Reddy, A refined hybrid plate theory for composite laminates with piezoelectric laminae, Int. J. SolidsStruct. 32, 2345–2367 (1995).

[2] H. Gu, A. Chattopadhyay, J. Li, and X. Zhou, A higher order temperature theory for coupled thermo-piezoelectric-mechanicalmodeling of smart composites, Int. J. Solids Struct. 37, 6479–6497 (2000).

[3] H. Nowotny, E. Benes, and M. Schmid, Layered piezoelectric resonators with an arbitrary number of electrodes (general one-dimensional treatment), J. Acoust. Soc. Am. 90, 1238–1245 (1991).

[4] Z. Q. Cheng and R. C. Batra, Three-dimensional asymptotic analysis of multiple-electroded piezoelectric laminates, AIAA J. 38,317–324 (2000).

[5] J. N. Reddy and Z. Q. Cheng, Deformations of piezothermoelastic laminates with internal electrodes, ZAMM 81, 347–359 (2001).[6] J. N. Reddy and Z. Q. Cheng, Three-dimensional solutions of smart functionally graded plates, J. Appl. Mech. 68, 234–241 (2001).[7] C.W. Lim, L. H. He, and A. K. Soh, Three-dimensional electromechanical responses of a parallel piezoelectric bimorph, Int. J.

Solids Struct. 38, 2833–2849 (2001).[8] J. Q. Tarn and Y. M. Wang, An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated plates, Int. J.

Solids Struct. 31, 231–246 (1994).[9] J. N. Reddy and Z. Q. Cheng, Frequency of functionally graded plates with three-dimensional asymptotic approach, J. Eng. Mech.

129, 896–900 (2003).[10] Z. Q. Cheng and J. N. Reddy, An asymptotic theory for vibrations of inhomogeneous/laminated piezoelectric plates, IEEE Trans.

Ultrson., Ferroelect., Freq. Contr. 50, 1563–1569 (2003).[11] H. F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum Press, New York, 1969).[12] J. N. Reddy, Mechanics of Laminated Composite Plates: Theory and Analysis (CRC Press, Boca Raton, Florida, 1997).[13] C. P. Wu, J. Q. Tarn, and S. M. Chi, An asymptotic theory for dynamic response of doubly curved laminated shells, Int. J. Solids

Struct. 33, 3813–3841 (1996).[14] P. Heyliger and S. Brooks, Free vibration of piezoelectric laminates in cylindrical bending, Int. J. Solids Struct. 32, 2945–2960

(1995).[15] D.A. Berlincourt, D. R. Curran, and H. Jaffe, Piezoelectric and piezomagnetic materials and their function in transducers, in

Physical Acoustics. Vol.I (part A). edited by W. P. Mason, pp. 169–270 (Academic Press, New York, 1964).[16] G. Dahlquist and A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, New Jersey, 1974).[17] Z. Q. Cheng and J. N. Reddy, Asymptotic theory for laminated piezoelectric circular cylindrical shells, AIAA J. 40, 553–558

(2002).

Documents

Natural frequencies of laminated piezoelectric plates with internal electrodes