Applied Mathematical Modelling 36 (2012) 1132–1147

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Applied Mathematical Modelling

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

On the effects of coupling between in-plane and out-of-planevibrating modes of smart functionally graded circular/annular plates

Sh.H. Hashemi a, K. Khorshidi b, M. Es’haghi a, M. Fadaee a,⇑, M. Karimi a

a School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16848-13114, Iranb Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-8-8849, Iran

a r t i c l e i n f o

Article history:Received 30 December 2010Received in revised form 2 July 2011Accepted 12 July 2011Available online 23 July 2011

Keywords:PiezoelectricSandwich structuresFree vibrationMindlin plate theoryFGM

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.07.051

⇑ Corresponding author. Tel.: +98 2177 240 190;E-mail addresses: Eshaghi@mecheng.iust.ac.ir (M

a b s t r a c t

In recent years many articles concerned with the mechanics of functionally graded plateshave been published. The variation in material properties through the thickness of theplate introduces a coupling between in-plane and transverse displacements, the couplingis important in the vibration of functionally graded plates (FGPs), but none have producedan exact closed-form solution for the in-plane as well as transverse vibrations of smart cir-cular/annular FGPs. Therefore, this paper develops an exact closed-form solution for thefree vibration of piezoelectric coupled thick circular/annular FGPs subjected to differentboundary conditions on the basis of the Mindlin’s first-order shear deformation theory.Through the comparison of present results with those available, the accuracy of the presentmethod was verified. The effects of coupling between in-plane and transverse displace-ments on the frequency parameters are proved to be significant. It is concluded that thedeveloped model can describe vibrational behavior of smart FGM plates more realistic.Due to the inherent features of the present solution, all findings will be a useful benchmarkfor evaluating other analytical and numerical methods developed by researchers in thefuture.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

A functionally graded material (FGM) is a composite, consisting of two or more phases, which is fabricated such that itscomposition varies in some spatial direction. This design is intended to take advantage of certain desirable features of eachof constituent phases. For example, if the FGM is to be used to separate regions of high and low temperature, it may consistof pure ceramic, at the hotter end, because of the ceramic’s better resistance to the higher temperatures. In contrast, the coolerend may be pure metal because of its better mechanical and heat-transfer properties. In addition to high resistance to temper-ature gradients, FGMs have other advantages like reduction in residual and thermal stresses, high wear resistance, and anincrease in strength to weight ratio. They are of interest for a wide range of applications: thermal barrier coatings for turbineblades (electricity production), armor protection for military applications, fusion energy devices, biomedical materialsincluding bone and dental implants, etc. In nature, we can see examples of FGMs in bamboo, bones, and teeth. The metal–cera-mic composite plates are widely used in aircrafts, space vehicles, reactor vessels, automotive and other engineeringapplications.

. All rights reserved.

fax: +98 2177 24 0488.. Es’haghi), Fadaee@iust.ac.ir (M. Fadaee).

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1133

The Classical Plate Theory (CPT) is based on the assumption that with small deflections the normal of mid-plane remainsas normal to the deformed mid-plane after deformation of the plate. Due to this approximation, i.e. to neglect the influenceof shear deformation and rotary inertia, the free vibration frequencies obtained by the classical theory of plates are re-stricted to plates with small thickness in comparison with their lateral dimensions. Leissa [1] have presented the numericalvalues in this monograph. Itao and Crandall [2] have also presented the values for the first 701 modes of vibration of cir-cular plates with free edges. It is well known that CPT assumptions are satisfactory for low mode computation of thin platesand lead to inaccuracy in calculating higher modes. In fact, the CPT underestimates deflections and overestimates frequen-cies. In order to eliminate the above deficiency of the CPT, Deresiewicz and Mindlin [3] proposed the first-order shear defor-mation theory (FSDT), including the effects of shear deformation and rotary inertia for moderately thick plates. In theMindlin theory, the constant shear stress condition violates the statical condition of zero shear stress at the free surfaces.To compensate for the error, Mindlin introduces shear correction factors to modify the shear forces. In other words, sincethe transverse shear strain is assumed to be constant through the thickness of the plate, a shear correction coefficient isneeded in the FSDT to account for the prediction of uniform shear stress distribution. This coefficient depends on materialproperties, geometric dimensions and boundary conditions of the plate. Several papers were devoted to free vibration anal-ysis of moderately thick plates [4–7]. Some of researchers have considered the problem of thick plates using a variationalasymptotic approach. By using the method of power series expansion of displacement components, Matsunaga [8] derived aset of fundamental dynamic equations of a two-dimensional (2-D) higher-order theory for rectangular functionally graded(FG) plates, through Hamilton’s principle. Then, Natural frequencies and buckling stresses of FG plates are analyzed bytaking into account the effects of transverse shear and normal deformations and rotatory inertia.

Due to the widespread use of the piezoelectric materials in sensors and actuators, the study of embedded or surface-mounted piezoelectric materials has received considerable attention in recent years. Circular plates composed entirely orin part by piezoelectric layers introduce the electrostatic potential as an additional variable and increases the complexityof solution because of the coupling between the elastic and electric variables and the additional boundary conditions.Heyliger and Saravanos [9] have studied the exact free vibration behavior of laminated piezoelectric plates for rectangulargeometries. Exact free vibration of piezoelectric laminates in cylindrical bending has been considered by Heyliger andSaravanos [10]. Ding et al. [11] derived the axisymmetric state space formulation of piezoelectric laminated circular platesBased on three-dimensional elastic theory of piezoelectric materials. Wang et al. [12] and Liu et al. [13] analyzed the freevibration of a piezoelectric coupled thin circular plate, based on Kirchhoff plate model and moderately thick circular plate,based on Mindlin plate theory, respectively. The form of the electric potential field in the piezoelectric layer is assumedsuch that the Maxwell static electricity equation is approximately satisfied. Duan et al. [14] used the Mindlin plate theory(MPT) to investigate the free vibration analysis of piezoelectric coupled thin and thick annular plate. Liu et al. [15] pre-sented a finite element method to analyze the three-dimensional vibration of piezoelectric coupled circular and annularplates.

Recently considerable interest has also been focused on investigating the performance of FGM plates integrated withpiezoelectric actuators. The vibration analysis of a rectangular laminated elastic plate with embedded piezoelectric sensorsand actuators was done by Batra et al. [16,17]. Reddy and Cheng [18] investigated the bending of a functionally gradedrectangular plate with an attached piezoelectric actuator. The transfer matrix and asymptotic expansion techniques areemployed to obtain a three dimensional asymptotic solution. Liew et al. [19] presented a generic static and dynamic finiteelement formulation for modeling and control of piezoelectric shell laminates under coupled displacement, temperatureand electric potential fields. The base shell is of functionally graded material type which consists of combined ceramic–metal materials with different mixing ratios of the ceramic and metal constituents. Ootao and Tanigawa [20,21] theoret-ically investigated the simply supported FGM rectangular plate integrated with a piezoelectric plate subjected to transientthermal loading. Wang and Noda [22] analyzed a smart FGM composite structure composed of a layer of metal, a layer ofpiezoelectric and a FGM layer in between. Ebrahimi and Rastgoo presented an analytical solution for the free axisymmetricvibration of piezoelectric coupled thin circular [23] and thin annular [24] FGM plates. Also Ebrahimi et al. presented a the-oretical analysis of smart moderately thick shear deformable annular [25] and circular [26] functionally graded plate byusing Mindlin’s plate theory. Hosseini Hashemi et al. presented an analytical method to analyze vibration of thick plateswith an exact-closed solution. They investigated free vibration of piezoelectric coupled thick circular/annular functionallygraded plates (FGPs) subjected to different boundary conditions on the basis of the Reddy’s third-order shear deformationtheory (TSDT) [27], thick circular isotropic plate [28] and smart Reddy plate [29]. It should be noted that in the five latterstudies [23–27], the authors neglected the in-plane displacements on mid-plane. Therefore, in conjunction with our recentwork [27], we attempt to solve this problem, that is, to provide an exact closed-form solution for the in-plane as well astransverse vibrations of smart circular/annular FGPs. Very recently, Es’haghi et al. [30] have proposed an exact solution forvibration analysis of piezoelectric coupled FG Mindlin plates those have open circuit piezoelectric patches and have beenused as sensors.

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick circular/annular functionallygraded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped bound-ary conditions on the basis of the Mindlin’s first-order shear deformation theory (FSDT). First, results obtained by the presentsolution are compared with existing numerical data. Second, the effect of plate parameters such as thickness-radius ratios,power index, as well as coupling effects between in-plane and out-of-plane displacements and boundary conditions is compre-hensively investigated.

1134 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

2. Governing equations

2.1. Geometrical configuration

Consider a flat, moderately thick piezoelectric coupled FGM annular plate, including one host layer in the middle and twoidentical piezoelectric layers bonded perfectly to the upper and lower surfaces of the host layer, with outer radius r0, innerradius r1, host plate thickness 2h and piezoelectric layer thickness hp, as depicted in Fig. 1. Both top and bottom surfaces ofeach piezoelectric layer are fully covered by electrodes which are shortly connected. The thickness of electrodes is assumedto be extremely small compared to the plate thickness. Thus, in the following formulation, the mechanical effects of the elec-trodes are neglected. Both piezoelectric layers are polarized perpendicular to the mid-plane in the positive direction of thethickness. The plate geometry and dimensions are defined in an orthogonal cylindrical coordinate system (r,h,z) to extractmathematical formulations. The origin of the coordinate system is taken at the center of the plate in the middle plane. Forconvenience in the formulation, suffix ‘p’ and ‘h’ is used to denote each piezoelectric layer and the host structure,respectively.

2.2. Material properties

Functionally graded materials (FGMs) are composite materials, the mechanical properties of which vary continuously dueto gradually changing the volume fraction of the constituent materials, usually in the thickness direction. In this study, theproperties of the plate are assumed to vary through the thickness of the plate with a power-law distribution of the volumefractions of the two materials in between the two surfaces. In fact, the top surface (z = h) of the plate is metal-rich whereasthe bottom surface (z = �h) is ceramic-rich. Poisson’s ratio m is assumed to be constant and is taken as 0.3 throughout theanalysis. Young’s modulus and mass density are assumed to vary continuously through the plate thickness as

EðzÞ ¼ ðEm � EcÞVf ðzÞ þ Ec; ð1aÞqðzÞ ¼ ðqm � qcÞVf ðzÞ þ qc; ð1bÞ

in which the subscripts m and c represent the metallic and ceramic constituents, respectively, E and q are modulus ofelasticity and density, respectively and the volume fraction Vf may be given by

Vf ðzÞ ¼z

2hþ 1

2

� �g

; ð2Þ

where g is the power law index and takes only positive values. For g = 0 and g =1, the plate is fully metallic and ceramic,respectively; whereas the composition of metal and ceramic is linear for g = 1.

2.3. The FSDT assumptions

According to the FSDT, in which the in-plane displacements are expanded as linear functions of the thickness coordinateand the transverse deflection is constant through the plate thickness, the displacement field is used as follows:

Fig. 1. Sketch of a FGM annular plate surface mounted with two piezoelectric layers.

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1135

uðr; h; z; tÞ ¼ u0ðr; h; tÞ þ zwrðr; h; tÞ; ð3aÞvðr; h; z; tÞ ¼ v0ðr; h; tÞ þ zwhðr; h; tÞ; ð3bÞwðr; h; z; tÞ ¼ w0ðr; h; tÞ; ð3cÞ

where u0 and v0 denote the in-plane displacements on mid-plane in radial and circumferential direction, respectively; w0 isthe transverse displacement; wr and wh are the slope rotations in the r � z and h � z planes at z = 0, and t is the time. Forsimplicity, the notation w is used for w0 in the following derivation of the governing equations of the plate.

By neglecting the normal strain in the thickness direction ezz, the strains associated with the displacements in Eqs. (3a)–(3c) are given for small deformation as

err ¼@u@r; ehh ¼

urþ @v

r@h; ezz ¼ 0;

erh ¼@v@rþ @u

r@h� v

r; erz ¼

@u@zþ @w@r

; ehz ¼@v@zþ @w

r@h;

ð4a—fÞ

where @(�)/@r (� = u, v and w), for example, denotes the partial derivative with respect to r; err and ehh are the normal strainsand erh, erz and ehz are the shear strains.

Based on Hooke’s law, the stress–displacement relations are defined as

rhr ¼

EðzÞð1� m2Þ ðerr þ mehhÞ; rh

h ¼EðzÞð1� m2Þ ehh þ merrð Þ;

rhrh ¼

EðzÞ2ð1þ mÞ erh; rh

rz ¼ j2 EðzÞ2ð1þ mÞ erz; rh

hz ¼ j2 EðzÞ2ð1þ mÞ ehz;

ð5a—eÞ

where j2 is shear correction factor and in this study is taken as j2 = p2/12. A sinusoidal function for the electric potential isadopted as follows [14]:

/ ¼uðr; h; tÞ sin p z

hp� 1

a

� �� �; h 6 z 6 H;

uðr; h; tÞ sin �p zhpþ 1

a

� �� �; �H 6 z 6 �h;

8><>: ð6Þ

where a is a non-dimensional parameter as defined a = hp/h, H = h + hp and u(r,h, t) is the electric potential on the mid-surface of the piezoelectric layer. It is seen from Eq. (6) that the electric boundary condition of the short-circuit (/ = 0) atthe internal surfaces z = ±h and the external surfaces z = ±H of each piezoelectric layer is completely satisfied.

The constitutive relations in the piezoelectric layer are written as

rpr ¼ Cp

11er þ Cp12eh � �e31Ez; ð7aÞ

rph ¼ Cp

12er þ Cp11eh � �e31Ez; ð7bÞ

sprh ¼

12

Cp11 � Cp

12

� �erh; ð7cÞ

sprz ¼ j2Cp

55erz þ e15Er; ð7dÞsp

hz ¼ j2Cp55ehz þ e15Eh; ð7eÞ

where Cp11;C

p12 and �e31 are the reduced material constants of the piezoelectric medium for plane stress problems given by

Cp12 ¼ Cp

12 � Cp13

� �2=Cp

33

� �; Cp

11 ¼ Cp11 � Cp

13

� �2=Cp

33

� �; �e31 ¼ e31 � Cp

13e33=Cp33

� �; ð8a—cÞ

where Cp11; Cp

12; Cp13; Cp

33 and Cp55 are the moduli of elasticity under constant electric field; e31, e33 and e15 are the piezoelectric

constants; Er, Eh and Ez are the electric field intensities in the radial, tangential and transverse direction, respectively, given by

Er ¼ �@/@r¼ � @u

@rsin p z

hp� 1

a

� �� �; ð9aÞ

Eh ¼ �@/r@h¼ � @u

r@hsin p z

hp� 1

a

� �� �; ð9bÞ

Ez ¼ �@/@z¼ � p

hpu cos p z

hp� 1

a

� �� �: ð9cÞ

The corresponding electric displacements Dr, Dh and Dz are given by

Dr ¼ e15erz þ N11Er ; ð10aÞDh ¼ e15ehz þ N11Eh; ð10bÞDz ¼ �e31 er þ ehð Þ þ N33Ez; ð10cÞ

where N33 is the reduced dielectric constant, N11 and N33 are the dielectric constants and

N11 ¼ N11; N33 ¼ N33 þ e233=Cp

33: ð11a;bÞ

1136 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

2.4. Equations of motion

For free vibration, the kinetic energy T and the strain energy V of a piezoelectric coupled annular Mindlin plate isexpressed as

T ¼ 12

Z ro

ri

Z 2p

0

Z h

�hqðzÞ _u2 þ _v2 þ _w2� �

r dr dhdzþZ ro

ri

Z 2p

0

Z H

hqp _u2 þ _v2 þ _w2� �

r dr dhdz; ð12Þ

and

V ¼ 12

Z ro

ri

Z 2p

0

Z h

�hrh

r err þ rhhehh þ sh

rzerz þ shhzehz þ sh

rherh� �

r dr dhdzþZ ro

ri

Z 2p

0

�Z H

hrp

r err þ rphehh þ sp

rzerz þ sphzehz þ sp

rherh þ DrEr þ DhEh þ DzEz� �

r dr dhdz; ð13Þ

where qp is the density of the piezoelectric layer and dot-overscript convention represents the differentiation with respect tothe time variable t.

After applying Hamilton’s principle, five equations of motion for dynamic behavior of piezoelectric coupled annularMindlin plates can be found as follows:

du0 :@Nr

@rþ @Nrh

r@hþ Nr � Nh

r¼ I1€u0 þ I2

€wr; ð14aÞ

dv0 :@Nh

r@hþ @Nrh

@rþ 2

Nrh

r¼ I1 €v0 þ I2

€wh; ð14bÞ

dwr :@Mr

@rþ @Mrh

r@hþMr �Mh

r� Q r ¼ I2€u0 þ I3

€wr; ð14cÞ

dwh :@Mh

r@hþ @Mrh

@rþ 2

Mrh

r� Q h ¼ I2 €v0 þ I3

€wh; ð14dÞ

dw :@Q r

@rþ @Q h

r@hþ Q r

r¼ I1 €w; ð14eÞ

where the inertias Ii(i = 1, 2, 3) are defined by

ðI1; I2; I3Þ ¼Z h

�hqðzÞð1; z; z2Þdzþ

Z H

hqpð1; z; z2Þdzþ

Z �h

�Hqpð1; z; z2Þdz ð15Þ

and the expressions for the stress resultants Ni, Mi, Qi (i = r,h,rh) are

Ni ¼Z h

�hrh

i dzþZ H

hrp

i dzþZ �h

�Hrp

i dz i ¼ r; h ð16aÞ

Mi ¼Z h

�hrh

i zdzþZ H

hrp

i zdzþZ �h

�Hrp

i zdz i ¼ r; h ð16bÞ

Mrh ¼Z h

�hrh

rhzdzþZ H

hrp

rhzdzþZ �h

�Hrp

rhzdz; ð16cÞ

Qi ¼Z h

�hrh

izdzþZ H

hrp

izdzþZ �h

�Hrp

izdz; i ¼ r; h ð16dÞ

Based on the strain–displacement relations given in Eq. (4) and stress distribution given in Eqs. (5) and (7), the resultantbending moments, twisting moments, and the transverse shear forces in terms of u0, v0, wr, wh, u and w are obtained as follows:

Nr ¼ ðAþ FÞ @u0

@rþ ðAþ F � 2S1Þ

u0

rþ @v0

r@h

� �þ B

@wr

@rþ m

wr

rþ @wh

r@h

� �� �; ð17aÞ

Nh ¼ ðAþ F � 2S1Þ@u0

@rþ ðAþ FÞ u0

rþ @v0

r@h

� �þ B

wr

rþ @wh

r@hþ m

@wr

@r

� �; ð17bÞ

Nrh ¼ S1@u0

r@hþ @v0

@r� v0

r

� �þ S3

@wr

r@hþ @wh

@r� wh

r

� �; ð17cÞ

Mr ¼ B@u0

@rþ m

u0

rþ @v0

r@h

� �� �þ ðC þ DÞ @wr

@rþ ðC þ D� 2S2Þ

wr

rþ @wh

r@h

� �� 4

php�e31u; ð17dÞ

Mh ¼ Bu0

rþ @v0

r@hþ m

@u0

@r

� �þ ðC þ D� 2S2Þ

@wr

@rþ ðC þ DÞ wr

rþ @wh

r@h

� �� 4

php�e31u; ð17eÞ

Mrh ¼ S3@u0

r@hþ @v0

@r� v0

r

� �þ S2

@wr

r@hþ @wh

@r� wh

r

� �; ð17fÞ

Qr ¼ G@w@rþ wr

� �� R1

@u@r

; ð17gÞ

Qh ¼ G@wr@hþ wh

� �� R1

@ur@h

; ð17hÞ

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1137

where the unknown constants in the above equations are

ðA;B;CÞ ¼Z h

�h

EðzÞ1� m2 ð1; z; z

2Þdz; D ¼ 2Z hþhp

hC11z2dz; ð18a;bÞ

F ¼ 2hpC11; G ¼ 2j2hpC55 þZ h

�hj2 EðzÞ

2ð1þ mÞdz; R1 ¼4phpe15; ð18c—eÞ

ðS1; S2Þ ¼12ðA;CÞð1� mÞ þ ðF;DÞ 1� C12

C11

! !; S3 ¼

12

Bð1� mÞ: ð18f ; gÞ

Substituting Eq. (17) into Eq. (14) yields

ðAþ FÞ @2u0

@r2 þ@u0

r@r� u0

r2 þ@2v0

r@r@h� @v0

r2@h

!þ S1

@2u0

r2@h2 �@v0

r2@h� @2v0

r@r@h

!þ B

@2wr

@r2 þ@wr

r@r� wr

r2 þ@2wh

r@r@h� @wh

r2@h

!

þ S3@2wr

r2@h2 �@wh

r2@h� @2wh

r@r@h

!¼ I1€u0 þ I2

€wr; ð19aÞ

ðAþ FÞ @u0

r2@hþ @2v0

r2@h2 þ@2u0

r@r@h

!þ S1 �

@2u0

r@r@hþ @u0

r2@h� v0

r2 þ@v0

r@rþ @

2v0

@r2

!þ B

@wr

r2@hþ @2wh

r2@h2 þ@2wr

r@r@h

!

þ S3 �@2wr

r@r@hþ @wr

r2@h� wh

r2 þ@wh

r@rþ @

2wh

@r2

!¼ I1 €v0 þ I2

€wh; ð19bÞ

B@2u0

@r2 þ@u0

r@r� u0

r2 þ@2v0

r@r@h� @v0

r2@h

!þ S3

@2u0

r2@h2 �@v0

r2@h� @2v0

r@r@h

!þ ðC þ DÞ @2wr

@r2 þ@wr

r@r� wr

r2 þ@2wh

r@r@h� @wh

r2@h

!

þ S2@2wr

r2@h2 �@wh

r2@h� @2wh

r@r@h

!� G

@w@rþ wr

� �þ R2

@u@r¼ I2€u0 þ I3

€wr; ð19cÞ

B@u0

r2@hþ @2v0

r2@h2 þ@2u0

r@r@h

!þ S3 �

@2u0

r@r@hþ @u0

r2@h� v0

r2 þ@v0

r@rþ @

2v0

@r2

!þ ðC þ DÞ @wr

r2@hþ @2wh

r2@h2 þ@2wr

r@r@h

!

þ S2 �@2wr

r@r@hþ @wr

r2@h� wh

r2 þ@wh

r@rþ @

2wh

@r2

!� G

@wr@hþ wh

� �þ R2

@ur@h¼ I2 €v0 þ I3

€wh; ð19dÞ

G@wr

@rþ @wh

r@hþ wr

r

� �þ GDw� R1Du ¼ I1 €w; ð19eÞ

where the unknown constant R2 is defined as

R2 ¼4p

hp e15 � �e31ð Þ; ð20Þ

and the operator D is defined as

D ¼ @2

@r2 þ@

r@rþ @2

r2@h2 : ð21Þ

Note that all of the electrical variables primarily must satisfy Maxwell’s equation which requires that the divergence ofthe electric flux density vanishes at any point within the media. This condition can be satisfied approximately by enforcingthe integration of the electric flux divergence across the thickness of the piezoelectric layers to be zero for any r andh as

Z H

h

@ðrDrÞr@r

þ @Dh

r@hþ @Dz

@z

� �dz ¼ 0; ð22Þ

substituting Eqs. (10a)–(10c) into above equation and simplifying the result gives

@wr

@rþ @wh

r@hþ wr

r

� �þ R3Dw� R4Duþ R5u ¼ 0; ð23Þ

where R3, R4 and R5 are

R3 ¼e15

e15 þ �e31; R4 ¼

2N11

pðe15 þ �e31Þ; R5 ¼

2N33ph2

pðe15 þ �e31Þ: ð24a—cÞ

1138 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

2.5. Exact solutions for transverse displacement (w)

In order to solve six complex differential equations of motion, following steps must be taken so that Eqs. (19a)–(19e) and(23) become uncoupled:

1. Eq. (19a) is first differentiated with respect to r.2. Eq. (19a) is divided by r.3. Eq. (19b) is first differentiated with respect to h and then divided by r.4. An auxiliary function is defined as

W1 ¼@u0

@rþ 1

r@v0

@hþ u0

r; W2 ¼

@wr

@rþ 1

r@wh

@hþ wr

r: ð25a;bÞ

5. If three equations obtained from steps (1), (2) and (3) are added together, we will obtain

ðAþ FÞDW1 þ BDW2 ¼ I1€W1 þ I2

€W2: ð26Þ

6. Doing above five steps on Eqs. (19c) and (19d), respectively, yields

BDW1 þ ðC þ DÞDW2 � GDw� GW2 þ R2Du ¼ I2€W1 þ I3

€W2: ð27Þ

7. Eqs. (19e) and (23) must be rewritten by using Eq. (25) as

GW2 þ GDw� R1Du ¼ I1 €w; ð28ÞW2 þ R3Dw� R4Duþ R5u ¼ 0; ð29Þ

8. The next step in the analysis is to eliminate the parameters W1, W2 and u between Eqs. (26)–(29). After some mathemat-ical manipulation, the obtained equation is uncoupled from u0, v0, u, wr and wh.

Finally, an uncoupled eighth-order partial differential equation with constant coefficients is acquired in terms of w. Thesolution of w(r,h, t) for wave propagation in the circumferential direction can be written as

wðr; h; tÞ ¼ �wðrÞ cosðphÞ expðixtÞ; ð30Þ

where �wðrÞ is the amplitude of the z-direction displacement as a function of radial distance only; x is the natural frequencyof the plate; non-negative integer p represents the circumferential wave number of the corresponding mode shape; andi ¼

ffiffiffiffiffiffiffi�1p

. Considering Eq. (30), the eighth-order partial differential equation is obtained as follows:

P1DDDDwþ P2DDDwþ P3DDwþ P4Dwþ P5w ¼ 0; ð31Þ

in which the coefficients Pi(i = 1, . . . ,5) are given in Appendix A. Rewriting Eq. (31) in terms of �wðrÞ gives a differential equa-tion, namely

P1DDDD�wþ P2DDD�wþ P3DDwþ P4D �wþ P5 �w ¼ 0; ð32Þ

where the operator D is given by

D ¼ @2

@r2 þ@

r@r� p2

r2 ; ð33Þ

Eq. (32) can be transformed into the following form

ðD� x1ÞðD� x2ÞðD� x3ÞðD� x4Þ �w ¼ 0; ð34Þ

where x1, x2, x3 and x4 are the four roots of the following equation:

P1x4 þ P2x3 þ P3x2 þ P4xþ P5 ¼ 0; ð35Þ

The general solution to Eq. (32) may be expressed as

�w ¼ �w1 þ �w2 þ �w3 þ �w4; ð36Þ

in which �wiði ¼ 1;2;3;4Þ are obtained by four different kinds of Bessel’s equations as follows:

ðD� x1Þ �w1 ¼ 0; ðD� x2Þ �w2 ¼ 0; ðD� x3Þ �w3 ¼ 0; ðD� x4Þ �w4 ¼ 0: ð37a—dÞ

We can define parameters n1 = 3a + 2Y + 2b/W and n2 = 3a + 2Y � 2b/W where coefficients W, Y, a and b are given in Appen-dix B. The parameters n1 and n2 practically take real and negative values. Therefore, based on Ferrari’s formula [31], fourdistinct real roots of Eq. (35) are given by

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1139

x1; x2 ¼ �b

4aþW �

ffiffiffiffiffiffiffiffiffi�n1p

2; ð38a;bÞ

x3; x4 ¼ �b

4aþ�W �

ffiffiffiffiffiffiffiffiffi�n2p

2; ð38c;dÞ

in which the coefficients a and b are given in Appendix B. Thus, the solution of Eq. (32) can be given by

�wðrÞ ¼X4

i¼1

ciwi1ðp;virÞ þ ciþ4wi2ðp;virÞ� �

; ð39Þ

where

vi ¼ffiffiffiffiffiffiffijxij

p; ð40aÞ

wi1ðp;virÞ ¼JpðvirÞ; xi < 0;IpðvirÞ; xi > 0;

wi2ðp;virÞ ¼YpðvirÞ; xi < 0;KpðvirÞ; xi > 0;

i ¼ 1;2;3;4

ð40bÞ

ci(i = 1, . . . ,8) are unknown coefficients; Jp and Yp are the Bessel functions of the first and the second kind, respectively,whereas Ip and Kp are the modified Bessel functions of the first and the second kind, respectively.

It should be noted that the second type Bessel functions become singular at r = 0. Therefore, transverse displacementfunction of circular plates is taken the following form

�wðrÞ ¼X4

i¼1

ciwi1ðp;virÞ; ð41Þ

2.6. Determination of electric potential in the piezoelectric layer

The solution of u(r,h, t) for wave propagation in the circumferential direction can be written as

uðr; h; tÞ ¼ �uðrÞ cosðphÞ expðixtÞ: ð42Þ

Using Eqs. (42) and (30) into Eqs. (26)–(29), eliminating W1 and W2 from these equations; and canceling the exp(ixt) term,give a relation between �uðrÞ and �wðrÞ, namely,

�u ¼ K2

K1DDD�wþ K3

K1DD�wþ K4

K1D�wþ K5

K1�w; ð43Þ

where the coefficients K1, K2, K3, K4 and K5 are given in Appendix C.From Eq. (37), the following equations can be written:

DDD�wi ¼ x3i �wi DD�wi ¼ x2

i �wi; D�wi ¼ xi �wi; i ¼ 1;2;3;4: ð44a—cÞ

The electric potential can be expressed by substituting Eq. (44) into Eq. (43) as follows:

�uðrÞ ¼X4

i¼1

Li �wiðrÞ; ð45Þ

where

Li ¼K2

K1x3

i þK3

K1x2

i þK4

K1xi þ

K5

K1; i ¼ 1;2;3;4: ð46Þ

2.7. Exact solutions for u0, v0, wr and wh

In order to determine the in-plane displacements on mid-plane u0, v0 and the slope rotations wr and wh, the followingforms are initially considered

uo ¼ a1@w1

@rþ a2

@w2

@rþ a3

@w3

@rþ a4

@w4

@rþ a5

@w5

r@hþ a6

@w6

r@h; ð47aÞ

v0 ¼ b1@w1

r@hþ b2

@w2

r@hþ b3

@w3

r@hþ b4

@w4

r@hþ b5

@w5

@rþ b6

@w6

@r; ð47bÞ

1140 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

wr ¼ a7@w1

@rþ a8

@w2

@rþ a9

@w3

@rþ a10

@w4

@rþ a11

@w5

r@hþ a12

@w6

r@h; ð47cÞ

wh ¼ b7@w1

r@hþ b8

@w2

r@hþ b9

@w3

r@hþ b10

@w4

r@hþ b11

@w5

@rþ b12

@w6

@r; ð47dÞ

where ai, bi(i = 1, . . . ,12) are unknown coefficients. The functions w5 and w6 are also unknown and must be determined. Theunknowns ai, bi, w5 and w6 can be obtained as follows by substituting Eq. (47) into Eq. (19),

ai ¼ bi ¼ðG� R2LiÞðBxi þ I2x2Þ

ðBxi þ I2x2Þ2 � ððAþ FÞxi þ I1x2ÞððC þ DÞxi þ ðI3x2 � GÞÞi ¼ 1; . . . ;4; ð48aÞ

aj ¼ bj ¼ðG� R2Lj�6ÞððAþ FÞxj�6 þ I1x2Þ

ððAþ FÞxj�6 þ I1x2ÞððC þ DÞxj�6 þ ðI3x2 � GÞÞ � ðBxj�6 þ I2x2Þ2j ¼ 7; . . . ;10 ð48bÞ

and the functions w5 and w6 takes the following forms:

wiðr; hÞ ¼ �wiðrÞ sin phð Þ; i ¼ 5;6; ð49Þ

where

�w5ðrÞ ¼ c9w51ðp;v5rÞ þ c10w52ðp;v5rÞ; ð50aÞ�w6ðrÞ ¼ c11w61ðp;v6rÞ þ c12w62ðp;v6rÞ ð50bÞ

and

x5 ¼�b2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

2 � 4b1b3

q2b1

; x6 ¼�b2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

2 � 4b1b3

q2b1

ð51a;bÞ

b1 ¼ S23 � S1S2; b2 ¼ 2S3I2x2 � S1ðI3x2 � GÞ � S2I1x2; b3 ¼ I2

2x4 � I1x2ðI3x2 � GÞ; ð51c—eÞ

vi ¼ffiffiffiffiffiffiffijxij

p; i ¼ 5;6; ð51fÞ

wj1ðp;vjrÞ ¼JpðvjrÞ; xj < 0;IpðvjrÞ; xj > 0;

(ð51gÞ

wj2ðp;vjrÞ ¼YpðvjrÞ; xj < 0;KpðvjrÞ; xj > 0;

(j ¼ 5;6 ð51hÞ

in which c9, c10, c11 and c12 are unknown coefficients that must be determined by applying the boundary conditions. The restof unknown coefficients are determined as follows:

ai ¼ �bi ¼ �S2xi þ ðI3x2 � GÞ

I2x2 þ S3xi; i ¼ 5;6; ð52aÞ

a11 ¼ a12 ¼ �b11 ¼ �b12 ¼ 1: ð52bÞ

For circular plates, �w5ðrÞ and �w6ðrÞ are defined as follows:

�w5ðrÞ ¼ c5w51ðp;v5rÞ; �w6ðrÞ ¼ c6w61ðp;v6rÞ: ð53a;bÞ

If the plate is insulated at the edge, the electrical flux conservation equation is given by

Z H

hDrðr; h; tÞdz ¼ 0: ð54Þ

Substituting Eq. (10a) into Eq. (54) yields the electric boundary condition

e15p wr þ@w@r

� �� 2N11

@u@r¼ 0: ð55Þ

2.8. Classical boundary conditions

Circular and annular plates may take any classical boundary conditions at their edges, including free, simply supportedand clamped.

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1141

The boundary conditions along the edges of the circular and annular plate are as follows:

– for a free edge

Mrh ¼ 0; Nrh ¼ 0; Mr ¼ 0; Nr ¼ 0; Q r ¼ 0; e15p wr þ@w@r

� �� 2N11

@u@r¼ 0 ð56a—eÞ

– for a soft simply supported edge

w ¼ 0; Mrh ¼ 0; Nrh ¼ 0; Mr ¼ 0; Nr ¼ 0; e15p wr þ@w@r

� �� 2N11

@u@r¼ 0; ð57a—eÞ

– for a hard simply supported edge

w ¼ 0; Mr ¼ 0; Nr ¼ 0; v0 ¼ 0; wh ¼ 0; e15p wr þ@w@r

� �� 2N11

@/@r¼ 0; ð58a—eÞ

– for a clamped edge

w ¼ 0; u0 ¼ 0; wr ¼ 0; v0 ¼ 0; wh ¼ 0; e15p wr þ@w@r

� �� 2N11

@u@r¼ 0: ð59a—dÞ

Natural frequencies of piezoelectric coupled circular and annular FG plates can be calculated by using above boundaryconditions.

3. Comparison studies

For convenience of notation, circular/annular plates are described by a symbolism defining the boundary conditions attheir edges, For example, C–S denotes an annular plate with clamped edge on the inner radius and simply supported (hardtype) on the outer radius. It should be noted that in this paper soft simply supported and hard simply supported boundaryconditions are denoted by Ss and S, respectively, and in all the computations the material properties of the piezoelectric layeris PZT4 and properties of five different FGM components for the host plate are listed in Table 1.

For verification of the present formulation, a comparison study of the results for annular plates without piezoelectric layerfor F–C boundary condition is made with the results from first-order shear deformation theory given by Tornabene et al. [32].The results are listed in Table 2 as a function of the gradient index (g) while the two modes ((p,n) = (0,0) and (0,1)) areconsidered for each value of g. The inner radius (r1), outer radius (r0) and the thickness of the annular plate are 0.5 m,2 m and 0.1 m, respectively. In this comparison the volume fraction Vf is defined as Vf(z) = (1/2 � z/h)g and Young’smodulus and mass density are assumed to vary continuously through the plate thickness as E(z) = (Ec � Em)Vf(z) + Em andq(z) = (qc � qm)Vf(z) + qm, respectively. It is evident from Table 2 that the present exact solution is in excellent agreementwith the results obtained on the basis of the Generalized Differential Quadrature (GDQ) method [32]. This is due to the factthat in the present paper and Ref. [32] the deflection equations taking into account a coupling of in-plane and transversemotions are used. Furthermore, this plate is thin and in thin plate, different methods used to obtain frequency, yield the sameresult.

In order to show the applicability, reliability, stability and effectiveness of the presented formulation for the piezoelectriccoupled annular plates, a comparison study of the natural frequencies of the present method with those of Duan et al. [14]using the analytical solution based on the improved plate theory (IPT) are listed in Table 3. In this table, the first three naturalfrequencies with two combinations of boundary conditions are considered for different thickness-radius ratio when h/r0 = 1/60 m and h/r0 = 1/20. The inner radius (r1) and outer radius (r0) of the annular plate are 0.1 m and 0.6 m, respectively. Thethickness ratio of the piezoelectric layer to the host plate is 1/10. It can be seen that when the gradient index (g) tends tozero, the FGM4 plate behaves like a homogeneous plate. As can be seen, there is an excellent agreement between the presentresults and available data.

4. Results and discussion

In this section, natural frequencies of the piezoelectric coupled annular plates are presented in tabular and graphicalforms for different plate and material of host plate. Also the effect of coupling effects between in-plane and out-of-plane dis-placements and boundary conditions on natural frequencies of the plate is comprehensively investigated.

4.1. Identification of the in-plane and out-of-plane modes

To show the effects of coupling between in-plane and transverse displacements, an interesting comparison study of thenatural frequencies of the present exact method (considering coupling effects) with those obtained based on the presentmethod without couplings between in-plane and bending effects are listed in Tables 4 and 5 for the piezoelectric coupled

Table 1Material properties.

PZT4 Piezoelectric

Elastic constants (GPa) Electric constants (C/m2) Dielectric constants (nC/Vm) Density (Kg/m3)

C13 C12 C55 C33 C11 e15 e31 e33 N33 N22 N11 q

73 71 26 115 132 10.5 �4.1 14.1 5.841 6.46 7.124 7500

Host plate

qc (kg/m3) Ec (Gpa) qm (kg/m3) Em (Gpa)

FGM1 3800 380 2700 70FGM2 8000 2800 2000 70FGM3 5700 168 2707 70FGM4 3800 380 7800 200FGM5 5700 200 2700 70

Table 2Comparison of frequencies x(Hz) of the piezoelectric coupled FGM3 annular plates under F–C boundary condition.

Method Mode number Gradient index

g = 0 g = 0.6 g = 1 g = 5 g = 20 g = 50 g = 100 g =1

Present (0,0) 70.1288 67.1309 66.7832 69.6396 68.8143 67.3364 66.5964 65.6876Tornabene et al. [32] 70.13 67.13 66.78 69.64 68.81 67.34 66.59 65.68FEM 70.3258 67.4564 66.8563 69.9247 68.8963 67.5267 66.7134 65.6996

Present (0,1) 296.993 284.618 283.124 294.457 290.959 284.921 281.898 278.185Tornabene et al. [32] 296.99 284.62 283.12 294.46 290.96 284.92 281.89 278.18FEM 298.235 286.025 283.954 295.326 291.003 285.381 282.056 278.986

1142 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

FGM1 circular/annular plates, respectively. In these tables, natural frequencies are compared with those of a 3-D finiteelement solution by ABAQUS 6.8, also two different thickness-radius ratios (h/r0 = 1/30 and h/r0 = 1/12) are examined, whichcorrespond to thin and moderately thick circular/annular plates, respectively, while three different boundary conditions areconsidered for each thickness of the host plate. The volume fraction exponent (g) is set to one; the material property of thepiezoelectric layer is PZT4 and hp/2his equal to 0.1.

As can be seen in Tables 4 and 5, a complete and accurate vibration spectrum of the piezoelectric coupled circular/annularfunctionally graded plates, including the modes due to bending, shears, and their coupling, are obtained by using the presentmethod. By comparing present results with those of a 3-D finite element solution by ABAQUS 6.8, it can be observed that thepresent method is in good agreement with FEM results. Let us neglect in-plane displacement on the mid-plane, all resultsproduced by this assumption are shown as uncoupled method (UM). It is obvious from Tables 4 and 5 that we miss allin-plane modes (bold frequencies) in latter method (UM), similar to other methods have been used in Refs. [23–27]. Alsoit is worth noting that the difference between the results obtained by the present method (considering coupling effects) withthose obtained by UM is considerable, this is attributed to the fact that the UM accounts only for the bending effect, the in-plane dynamics is neglected and it plays an important role in the frequency spectrum of FGM plates. Also, the number ofmissing modes increases for higher frequencies and thicker plates. It is shown herein that the commonly used uncoupledmethods (UM) do not predict accurate natural frequencies for FGM plates.

It should be noted that, for the finite element analysis, a three dimensional model was constructed with 3D-stress ele-ments (C3D20R) for the host plate and piezoelectric elements (C3D20RE) for the two piezoelectric sheets, which are 20-nodequadratic brick. The host plate was divided into a number of 3D-stress elements and the two identical piezoelectric layerswere also meshed with piezoelectric elements. The plate was divided into 14,350 elements in order to make sure that theresults are converged. To enforce the clamp boundary condition on the edge of the plate, a fully clamped (ENCASTRE) con-dition along a boundary can be considered, this constraint suppresses all displacements and rotations on the plate edge. Forsimulating simply supported boundary condition on the plate edge all translation in three directions and rotations aboutthree axes are constrained except the rotation about tangential axis and translation in the radial direction.

4.2. Effects of gradient index on the frequencies

Fig. 2 shows the behavior of the fundamental frequency x(Hz) as a function of power index for a circular FGM5 plateunder classical boundary conditions when r0 = 0.6 m, hp = 0.006 m and h = 0.03 m. It is obvious from Fig. 2 that regardlessof the boundary conditions at the plate edges, the power law index g has severe influences on the fundamental frequency,especially for 0 < g < 50. This is due to the fact that the fundamental frequency x is strongly dependent on E(z) and q(z) (Ref.[33]) and the power law index g has a highly significant influence on the Young’s modulus and density of Zirconia when0 < g < 50. It is noticeable that the fundamental frequency increases with increasing the Young’s modulus and decreases with

Table 3Comparison of frequencies x(Hz) of the piezoelectric coupled moderately thick FGM4 annular plates under different boundary conditions.

h Method g Boundary condition

C–Ss Ss–C

Mode number Mode number

(0,0) (1,0) (2,0) (0,0) (1,0) (2,0)

0.01 Present 10�3 290.372 311.739 397.419 349.406 381.734 503.07410�4 290.193 311.550 397.176 349.192 381.501 502.77710�5 290.164 311.532 397.132 349.191 381.499 502.76710�6 290.139 311.466 397.092 349.186 381.494 502.770

Duan et al. [14] 290.139 311.466 397.092 349.186 381.494 502.770

0.03 Present 10�3 806.717 861.242 1100.35 975.67 1044.21 1358.2810�4 806.227 860.719 1099.69 975.078 1043.58 1357.4610�5 806.177 860.666 1099.62 975.018 1043.52 1357.3810�6 805.961 860.392 1099.28 974.824 1043.26 1357.27

Duan et al. [14] 805.961 860.392 1099.28 974.824 1043.26 1357.27

Table 4First eight Frequencies x(Hz) of the piezoelectric coupled FGM1 circular plates under C, S and F boundary conditions when hp/2h = 1/10, r0 = 0.6 m and g = 1.

h BC Method Mode number

(0,0) (1,0) (2,0) (0,1) (3,0) (1,1) (4,0) (2,1)

0.02 C Present 389.961 800.582 1293.13 1468.92 1860.07 2200.06 2493.35 2992.27FEM 391.27 802.37 1303.6 1476.1 1883.5 2212.7 2534.1 3024.7UM 414.669 850.313 1371.78 1557.83 1970.72 2329.69 2638.36 3163.87

S (0,0) (1,0) (2,0) (0,1) (1,1) (3,0) (1,2) (4,0)Present 189.964 530.486 966.719 1117.39 1353.81 1488.06 1791.54 2085FEM 193.31 532.63 974.73 1121.6 1361.9 1505.6 1797.6 2116.1UM 202.212 564.296 1027.52 1187.36 – 1580.23 1901.54 2211.98

F (2,0) (0,0) (3,0) (1,0) (4,0) (5,0) (2,1) (0,1)Present 205.634 344.767 472.699 773.598 820.571 1243.03 1310.15 1429.72FEM 204.95 350.77 473.12 776.92 823.71 1250.9 1318.19 1436UM 218.580 367.108 502.837 823.017 872.389 1320.65 1392.70 1519.92

0.05 C (0,0) (1,0) (2,0) (0,1) (3,0) (1,1) (1,2) (0,2)Present 912.369 1774.58 2722.15 3054.47 3732.88 3867.6 4323.72 4479.63FEM 922.75 1800.6 2787 3122.2 3848.6 3879.4 4437.4 4467.6UM 964.213 1866.33 2852.16 3197.58 3899.84 – 4508.22 –

S (0,0) (1,0) (1,1) (2,0) (0,1) (2,1) (3,0) (1,2)Present 463.529 1243.07 1353.07 2167.15 2470.75 2832.81 3188.77 3751.16FEM 473 1257.1 1362.8 2209 2511.3 2835.5 3272.2 3828.5UM 492.603 1316.87 – 2288.37 2606.45 – 3356.89 3942.94

F (2,0) (0,0) (3,0) (1,0) (4,0) (2,1) (5,0) (2,2)Present 495.975 820.980 1101.18 1747.71 1839.21 2717.75 2676.15 2839.01FEM 494.8 837.3 1104.7 1767.6 1853 2705.6 2724.9 2878.6UM 527.433 873.644 1168.81 1853.51 1948.20 2967.88 2829.13 –

Bold figures indicate in-plane modes.

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1143

increasing the density, generally. Consequently, for 0 < g < 0.44 & 2.2 < g < 50, the Young’s modulus of Zirconia has a largereffect on the frequency than the density of Zirconia. Hence, the Young’s modulus of Zirconia increases with increasing thepower law index g, leading to the increase of the fundamental frequency. On the contrary, the effect of the density becomesmore pronounced for 0.44 < g < 2.2. As a result, the density of Zirconia increases with increasing the power law index g, lead-ing to the decrease of the fundamental frequency. This behavior leads to create the local maximum and minimum values offundamental frequency at the points around g = 0.44 and g = 2.2, respectively. Most of the time when we use FGM plates weneed to alter the natural frequency by changing the power index (g), as can be seen, each curve have two relative extremawhich by choosing an appropriate value of g, the fundamental frequency of FGM plates attains its maximum/minimum valuethat it can be helpful for optimal design of FGM plates.

4.3. Effects of coupling terms on natural frequency

Fig. 3(a) and (b) shows the behavior of the natural frequencies x(Hz) as a function of thickness of host plate for aclamped FGM2 circular plate and S–S annular plate, respectively. It is evident from Eqs. (19a)–(19e) that the coupling

Table 5First eight frequencies x(Hz) of the piezoelectric coupled FGM1 annular plates under C–C, S–S and F–F boundary conditions when hp/2h = 1/10, r0 = 0.6 m,r1 = 0.1 m and g = 1.

h BC Method Mode number

(0,0) (1,0) (2,0) (3,0) (4,0) (0,1) (1,1) (5,0)

0.02 C–C Present 1165.87 1217.20 1440.01 1891.63 2497.88 3033.41 3119.76 3186.00FEM 1174 1275.6 1451.9 1914.6 2539.1 3104.5 3190.1 3250.9UM 1234.00 1287.93 1524.34 2003.05 2643.00 3195.50 3286.77 3367.25

S–S (0,0) (1,0) (2,0) (3,0) (4,0) (0,1) (1,1) (1,2)Present 599.581 691.984 1004.10 1493.62 2085.63 2148.82 2290.43 2592.79FEM 599.22 693.04 1011.2 1511.2 2117.5 2155.9 2301.8 2605.1UM 637.017 735.300 1066.94 1586.07 2212.63 2277.48 2427.42 –

F–F (2,0) (0,0) (3,0) (1,0) (4,0) (5,0) (2,1) (0,1)Present 199.136 327.126 471.828 750.797 820.482 1243.02 1280.41 1485.85FEM 199 333.39 472.32 755.3 823.45 1250.6 1291.1 1488.3UM 211.887 348.341 501.901 798.157 872.293 1320.64 1361.11 1580.55

0.05 C–C (0,0) (1,0) (2,0) (3,0) (4,0) (0,1) (0,2) (1,1)Present 2349.19 2446.47 2910.3 3765.0 4793.98 4813.48 5296.16 5322.45FEM 2402.8 2502.4 2983.7 3881.9 4798.0 4971.0 5330.0 5497.6UM 2441.06 2546.39 3039.14 3931.79 – 4997.60 5467.14 5656.11

S–S (0,0) (1,0) (2,0) (1,1) (2,1) (3,0) (0,1) (3,1)Present 1368.31 1574.41 2236.91 2591.61 2972.97 3198.31 3687.60 4222.08FEM 1378.8 1590.4 2278.1 2603.6 2975.0 3282.4 3724.0 4215.6UM 1442.36 1661.39 2360.16 - – 3366.61 – –

F–F (2,0) (0,0) (3,0) (1,0) (4,0) (2,1) (5,0) (2,2)Present 478.038 779.034 1098.75 1676.51 1838.97 2333.36 2676.13 2753.81FEM 478.35 795.46 1102.5 1699.5 1852.04 2337.3 2705 2804.8UM 508.126 829.546 1166.20 1773.92 1947.95 – 2829.11 2906.96

Bold figures indicate in-plane modes.

Fig. 2. Variation of the fundamental frequency x(Hz) for FGM5 circular plate under classical boundary conditions against the power index (g), whenr0 = 0.6 m, hp = 0.006 m and h = 0.03 m.

1144 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

Fig. 3. Investigation of coupling effects and thickness of host plate on the fundamental frequency x(Hz). (a) Clamped FGM2 circular plate for r0 = 0.6 m andg = 1; (b) S–S FGM2 annular plate, for ri = 0.1 m, r0 = 0.6 m and g = 1.

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1145

between the in-plane and transverse components is due to the potential coupling and the kinetic coupling. In this section forfurther investigation of coupling effects on the frequency, the authors categorized the present method, uncoupled method(UM), as kinetic uncoupled method (KUM) and potential uncoupled method (PUM) according to the nature of the couplingeffects. In the uncoupled method (UM) the terms B and I2 are set equal to zero in Eqs. (19a)–(19e) and PUM and KUM areobtained by taking B = 0 and I2 = 0 in the mentioned equations, respectively.

As can be seen the KUM results are very close to those of the present results. This indicates that the kinetic couplinghas a smaller effect on the frequency parameters. Beside, by comparing the PUM and UM results, it can be inferred thatthe frequencies are significantly influenced by the potential coupling parameters, On the other hand, from Eqs. (15) and(18a), it can be concluded that the magnitude of the potential coupling parameter (i.e. B) and the kinetic couplingparameter (i.e. I2) are strongly dependent on (Ec � Em) and (qc � qm), respectively. Hence, the difference between Young‘smodulus of ceramic and metal in FGMs has a larger effect on the frequency parameters than the difference betweentheir mass density. Also, it can be figured out from Fig. 3(a) and (b) that the coupling effects for thicker plates are morepronounced.

5. Concluding remarks

In this paper, the free vibration of a three-layer piezoelectric laminated circular/annular FG plate based on the Mindlinplate theory is investigated for the case where the electrodes on the piezoelectric layers are shortly connected. The electricpotential distribution across thickness of piezoelectric layer is modeled by a sinusoidal function and Maxwell equation isenforced. Analytical solutions are presented and the exact closed-form characteristic equations, displacement field of theplate and the electric potential are derived for the first time. Comparison studies proved that the present method is in goodagreement with other methods reported in the literature for different boundary conditions of the plate. Parametric studieswere devoted to the effects of the coupling between in-plane and transverse displacements, the thickness-radius ratio, inner-outer radius ratio and material of host plate on the natural frequencies of the piezoelectric coupled circular/annular plate.The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be sig-nificant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic thanother models [23–27].

Appendix A

P1 ¼ B2GðR1 þ R2ðR3 � 1Þ � GR4Þ þ ðC þ DÞðAþ FÞGðGR4 � R1R3Þ; ðA:1Þ

P2 ¼ ðB2 � ðC þ DÞðAþ FÞÞG2R5 þ ðB2I1ðR1 � R2 � GR4Þ þ 2BGI2ðR1 þ R2ðR3 � 1Þ � GR4Þ þ Gð�ðAþ FÞI3ðR1R3 � GR4Þþ CI1ððAþ F þ GÞR4 � R1R3Þ þ DI1ððAþ F þ GÞR4 � R1R3ÞÞÞx2; ðA:2Þ

1146 Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147

P3 ¼ Gðð�B2I1 þ ðC þ DÞGI1 � 2BGI2ÞR5 þ AðGI3R5 þ I1ð�R1 þ GR4 þ ðC þ DÞR5ÞÞ þ FðGI3R5 þ I1ð�R1 þ GR4

þ ðC þ DÞR5ÞÞÞx2 � ð2BI1I2ðR1 � R2 � GR4Þ þ GðI22ðR1 þ R2ðR3 � 1Þ � GR4Þ þ I1ððC þ DÞI1R4 þ I3ðR4ðAþ F þ GÞ

� R1R3ÞÞÞÞx4; ðA:3Þ

P4 ¼ ðAþ FÞG2I1R5x2 � G �GI22R5 þ I1ð�2BI2 þ ðAþ F þ GÞI3ÞR5 þ I2

1 �R1 þ GR4 þ ðC þ DÞR5ð Þ� �

x4

þ I1 GI1I3R4 þ I22ðR1 � R2 � GR4Þ

� �x6; ðA:4Þ

P5 ¼ GI1R5x4 GI1 þ ðI22 � I1I3Þx2

� �: ðA:5Þ

Appendix B

a ¼ P1; ðB:1Þb ¼ P2; ðB:2Þc ¼ P3; ðB:3Þd ¼ P4; ðB:4Þe ¼ P5; ðB:5Þ

a ¼ �3b2

8a2 þca; ðB:6Þ

b ¼ b3

8a3 �bc

2a2 þda; ðB:7Þ

c ¼ � 3b4

256a4 þcb2

16a3 �bd4a2 þ

ea; ðB:8Þ

P ¼ � a2

12� c; ðB:9Þ

Q ¼ � a3

108þ ac

3� b2

8; ðB:10Þ

R ¼ �Q2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 2

4þ P3

27

!vuut ; ðB:11Þ

U ¼ffiffiffiR3p

; ðB:12Þ

Y ¼ �56aþ U � P

3U; ðB:13Þ

W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ 2Y

p: ðB:14Þ

Appendix C

K1 ¼1

ðR1 � GR4Þ3GR5 G2ððC þ DÞðAþ FÞR1 � B2R2ÞR2

5 þ G ðC þ DÞI1 þ ðAþ FÞI3ð ÞR1 � 2BI2R2ð ÞðGR4 � R1ÞR5x2�

þ I1I3R1 � I22R2

� �R1 � GR4ð Þ2x4

�; ðC:1Þ

K2 ¼ G �B2 þ ðC þ DÞðAþ FÞR3 þðR3 � 1ÞðB2R2 � ðC þ DÞðAþ FÞGR4Þ

R1 � GR4

!; ðC:2Þ

K3 ¼1

ðR1 � GR4Þ2G2 ðC þ DÞðAþ FÞR1 � B2R2

� �ðR3 � 1ÞR5

�

þðR1 � GR4Þ B2I1ðR1 � R2 � GR4Þ þ 2BGI2ðR1 þ R2ðR3 � 1Þ � GR4Þ þ G �ðAþ FÞI3ðR1R3 � GR4Þð�

þCI1ð�R1R3 þ ðAþ F þ GÞR4Þ þ DI1ð�R1R3 þ ðAþ F þ GÞR4ÞÞÞx2�; ðC:3Þ

Sh.H. Hashemi et al. / Applied Mathematical Modelling 36 (2012) 1132–1147 1147

K4 ¼1

ðR1 � GR4Þ3�G3 ðC þ DÞðAþ FÞR1 � B2R2

� �ðR3 � 1ÞR2

5 þ GðR1 � GR4ÞððAþ FÞI1ðR1 � GR4Þ2�

� �BR2 BI1 � 2GI2ðR3 � 1Þð Þ þ AR1 ðC þ DÞI1 � GI3ðR3 � 1Þð Þ � FGI3R1ðR3 � 1ÞðþCI1R1ðF þ G� GR3Þ þ DI1R1ðF þ G� GR3ÞÞR5Þx2 þ R1 � GR4ð Þ2 2BI1I2ðR1 � R2 � GR4Þð

þG I22ðR1 þ R2ðR3 � 1Þ � GR4Þ þ I1 ðC þ DÞI1R4 þ I3 �R1R3 þ ðAþ F þ GÞR4ð Þð Þ

� ��x4�; ðC:4Þ

K5 ¼1

ðR1 � GR4Þ3I1x2ðG2ððC þ DÞðAþ FÞR1 � B2R2ÞR2

5 þ GðR1 � GR4Þ

� I1ðR1 � GR4Þ2 � ðC þ DÞI1 þ ðAþ FÞI3ð ÞR1 � 2BI2R2ð ÞR5

� �x2

þ ðR1 � GR4Þ2 GI1I3R4 þ I22ðR1 � R2 � GR4Þ

� �x4Þ: ðC:5Þ

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