Non-linear dynamic stability of piezoelectric functionally gradedcarbon nanotube-reinforced composite plates with initialgeometric imperfection

M. Rafiee a,b, X.Q. He a,n, K.M. Liew a

a Department of Civil and Architectural Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kongb Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Iran

a r t i c l e i n f o

Article history:Received 6 May 2013Received in revised form30 October 2013Accepted 30 October 2013Available online 20 November 2013

Keywords:Non-linear dynamic stabilityBucklingParametric resonancePiezoelectricNanotube reinforced compositeHarmonic balance method

a b s t r a c t

This paper deals with non-linear dynamic stability of initially imperfect piezoelectric functionally gradedcarbon nanotube reinforced composite (FG-CNTRC) plates under a combined thermal and electricalloadings and interaction of parametric and external resonance. The excitation, which derives fromharmonically varying actuators voltage, results in both external and parametric excitation. The governingequations of the piezoelectric CNTRC plates are derived based on first order shear deformation platetheory (FSDT) and von Kármán geometric non-linearity. The material properties of FG-CNTRC plate areassumed to be graded in the thickness direction. The single-walled carbon nanotubes (SWCNTs) areassumed aligned, straight and a uniform layout. The linear buckling and vibration behavior of perfect andimperfect plates are obtained in the first step. Then, Galerkin's method is employed to derive the non-linear governing equations of the problem with quadratic and cubic non-linearities associated with mid-plane stretching. Periodic solutions and their stability are determined by using the harmonic balancemethod with simply supported boundary conditions. The effect of the applied voltage, temperaturechange, plate geometry, imperfection, the volume fraction and distribution pattern of the SWCNTs on theparametric resonance, in particular the positions and sizes of the instability regions of the smart CNTRCplates as well as amplitude of steady state vibration are investigated through a detailed parametric study.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Molecular scale tubes of graphitic carbon namely called carbonnanotubes (CNTs) have outstanding properties having Young'smoduli as high as 1 TPa and tensile strength up to 63 GPa. CNTsare considered as one of the most promising reinforcementmaterials for high performance structural and multifunctionalcomposites with tremendous application potentials due to theirexceptional mechanical, thermal and electrical properties [1–3].

The dynamic instability of a structure subjected to periodic axialcompressive forces has been a very important topic in structuraldynamics and is of practical importance in different engineeringdisciplines. The periodic axial forces may cause parametric vibration,a phenomenon that is characterized by unbounded growth of a smalldisturbance. It may eventually cause damages. Nevertheless, researchworks in the area of smart composite plates are limited in number.Wu et al. [4] investigated the dynamic stability of thick functionallygraded material plates subjected to aero-thermo-mechanical loads,

using a novel numerical solution technique, the moving least squaresdifferential quadrature method. Pradyumna and Gupta ([5,6]presented a finite element solution to study the non-linear dynamicstability characteristics of piezoelectric composite plates and shellssubjected to periodic in-plane loads. An analysis on the non-lineardynamics of a simply supported functionally graded materials (FGMs)rectangular plate subjected to the transversal and in-plane excitationspresented in a thermal environment by Hao et al. [7]. Based onSchapery's 3-D constitutive relationship and von Kármán's platetheory, the non-linear dynamic response of viscoelastic symmetriccross-ply laminated plates with transverse matrix crack presented byFu and Lu [8]. The non-linear dynamic responses of a compositelaminated cantilever rectangular plate under the in-plane andmoment excitations are studied by Zhang et al. [9]. Balamuruganet al. [10] investigated the dynamic instability of anisotropiclaminated composite plates considering geometric non-linearity.The influence of the quadratic and cubic terms on non-lineardynamic characteristics of the angle-ply composite laminatedrectangular plate with parametric and external excitations isinvestigated by Sayed and Mousa [11]. The non-linear oscillationsand chaotic dynamics of a simply supported antisymmetric cross-plylaminated composite rectangular thin plate under parametric

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International Journal of Non-Linear Mechanics

0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijnonlinmec.2013.10.011

n Corresponding author. Tel.: þ852 34424760.E-mail address: bcxqhe@cityu.edu.hk (X.Q. He).

International Journal of Non-Linear Mechanics 59 (2014) 37–51

excitation investigated by Ye et al. [12]. Using finite element method,the non-linear instability behavior of plates subjected to periodic in-plane load has been studied by Ganapathi et al. [13]. Chattopadhyayand Radu [14] used a higher order shear deformation theory toinvestigate the instability associated with composite plates subject todynamic loads. Both transverse shear and rotary inertia effects aretaken into account in their study. Eshmatov [15] presented theanalyses of the non-linear vibrations and dynamic stability ofviscoelastic orthotropic plates based on the Kirchhoff–Love hypoth-esis and Reissner–Mindlin generalized theory with the incorporationof shear deformation and rotatory inertia in geometrically non-linearstatements.

The mechanical behavior of multiwalled carbon nanotube/epoxycomposites in both tension and compression studied by Schadleret al. [16]. Bekyarova et al. [17] reported an approach to the develop-ment of advanced structural composites based on engineered multi-scale carbon nanotube–carbon fiber reinforcement. Static analysis offunctionally graded carbon nanotube-reinforced composite plateembedded in piezoelectric layers by using theory of elasticityinvestigated by Alibeigloo [18]. Zhu et al. [19] presented the bendingand free vibration analyses of thin-to-moderately thick SWCNT-reinforced composite plates using the finite element method andthe first-order shear deformation plate theory. Hedayati and Sob-hani Aragh [20] investigated the influence of graded agglomeratedCNTs on vibration of CNT-reinforced annular sectorial plates restingon Pasternak foundation. In another study, Sobhani Aragh et al. [21]used Eshelby–Mori–Tanaka approach to study the vibrational beha-vior of continuously graded carbon nanotube-reinforced cylindricalpanels. Shen and his coworkers conducted several studies toinvestigate the non-linear static and dynamic behavior of function-ally graded carbon nanotube-reinforced composite plates (see [22–26]). For instance, Wang and Shen [25] studied the non-linearvibration of SWCNT-reinforced composite plates in a thermalenvironment. Shen and Zhang [26] investigated the thermal buck-ling of composite plates reinforced by SWCNTs (FG distribution).Rafiee et al. [27,28] conducted a comprehensive analytical study toinvestigate the non-linear vibration and dynamic response of simplysupported functionally graded material shells under combinedelectrical, thermal, mechanical and aerodynamic loading withpiezoelectric layers. Yas and his coworkers (see [29–32]) studiedthe free vibration, buckling and dynamic response of carbonnanotube-reinforced composite beams and/or panels. Moradi-Dastjerdi et al. [33] investigated the dynamic analysis of functionallygraded nanocomposite cylinders reinforced by carbon nanotube bya mesh-free method. Rafiee et al. [34–36] conducted severalresearch works to study the non-linear vibration of functionallygraded beam structures. Rafiee and Kalhori [34] and Shooshtari andRafiee [35] studied the non-linear free and forced vibration of afunctionally graded beam based on Euler–Bernoulli beam theoryand von Kármán geometric non-linearity under various boundaryconditions. Recently, Rafiee et al. [36] presented a perturbationbased multiple time scales analytical solution to investigate thenon-linear vibration of functionally graded CNTs-reinforced beamswith surface bonded piezoelectric layers under combined electrical,thermal and mechanical loading.

So far, no previous work regarding the non-linear parametricresonance of piezoelectric functionally graded carbon nanotube-reinforced composite plates has been conducted and reported inthe open literature.

The non-linear dynamic stability of piezoelectric functionallygraded SWCNT-reinforced composite (CNTRC) plates subjected to auniform temperature change and an applied voltage based on FSDTtheory and von Kármán type non-linear kinematics is investigated inthis study. The non-linear equation with quadratic and cubic non-linear terms is obtained via Galerkin procedure. Analytical solutionfor linear buckling, vibration and parametric resonance has been

presented. Harmonic balance method is then employed to deter-mine the non-linear dynamic stability of the plates with immovablesimply supported boundary conditions. A comprehensive parametricstudy has been conducted to investigate the effects of the geometricnon-linearities, applied voltage, piezoelectric layers, plate geometry,temperature rise, initial geometric imperfection, volume fraction ofcarbon nanotubes on the positions and sizes of the instabilityregions of the piezoelectric SWCNT-reinforced composite plates.

2. Problem formulation

Fig. 1 shows a CNTRC plate with the coordinate plane (x,y) oflength a, width b and thickness h with two surface-bondedpiezoelectric actuator layers of thickness hp. The total thicknessof the plate is H. The SWCNT reinforcement is either uniformlydistributed (referred to as UD) or functionally graded in thethickness direction, referred to as FG which includes FG-O andFG-X as shown in Fig. 1. Piezoelectric actuators are symmetricallyand perfectly bonded (neglecting the adhesive thickness) on thetop and bottom surface of the CNTRC host and are assumed tohave temperature-independent material properties.

2.1. Material properties of the CNTRC host and piezoelectric layers

The CNTRC material is made from a mixture of isotropic matrixand SWCNTs with a graded distribution in the thickness direction.The constituent materials are linear elastic throughout the defor-mation. The plate is initially stress free at T0 (in Kelvin) and issubjected to a uniform temperature variation ΔT¼T�T0 andapplied voltage V(t).

hHhp

hp

Fig. 1. Configurations of the carbon nanotube reinforced composite piezoelectricplates: (a) FGO CNTRC piezoelectric plate; (b) FGX CNTRC piezoelectric plate; and(c) UD CNTRC piezoelectric plate.

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5138

The effective material properties of the CNTRC plate can bepredicted by [22]

E11h ¼ η1 VCNECN11 þVh

mEmh ;

η2E22h

¼ VCN

ECN22þVh

m

Emh;

η3G12h

¼ VCN

GCN12

þVhm

Gmh

ð1Þ

where ECN11 , ECN22 and GCN

12 are Young’s moduli and shear modulus ofSWCNTs, respectively, and Emh and Gm

h represent the correspondingproperties of the isotropic matrix of the host. To account for thescale-dependent material properties, ηjðj¼ 1;2;3Þ, the CNT effi-ciency parameters, calculated by matching the effective propertiesof CNTRC obtained from the MD simulations with those from therule of mixture, are introduced. VCN and Vh

m refer to the volumefractions of the carbon nanotubes and the matrix, respectively.

The uniform and two functionally graded distributions of thecarbon nanotubes along the thickness direction are depicted inFig. 1 are assumed to be as following:

VCN ¼ Vn

CN ðUD� CNTRCÞVCN ¼ 2 1�2 zj j

h

� �Vn

CN ðFGO� CNTRCÞ

VCN ¼ 2 2 zj jh

� �Vn

CN ðFGX� CNTRCÞ

8>>>><>>>>:

ð2Þ

where

Vn

CN ¼ wCN

wCNþ ðρCN=ρmh Þ�ðρCN=ρmh ÞwCNð3Þ

and wCN is the mass fraction of the SWCNTs, ρCN and ρm are themass densities of the carbon nanotube and matrix, respectively.The thermal expansion coefficients in the longitudinal and trans-verse directions can be expressed as

α11h ¼ VCNαCN11 þVh

mαmh ð4aÞ

α22h ¼ ð1þνCN12 ÞVCNαCN22 þð1þνmh ÞVh

mαmh �ν12hα11h ð4bÞ

where αCN11 ; αCN22 and αmh are thermal expansion coefficients, and νCN12

and νmh are Poisson’s ratios, respectively, of the carbon nanotubeand matrix. Poisson’s ratio and mass density ρ can be calculated by

ν12h ¼ Vn

CNνCN12 þVh

mνmh ; ρh ¼ VCNρ

CNþVhmρ

mh ð5Þ

where νCN12 and νmh are Poisson’s ratios of carbon nanotube andmatrix, respectively. It is assumed that the material property ofnanotube and matrix is a function of temperature, so that theeffective material properties of CNTRCs, like Young’s modulus,shear modulus and thermal expansion coefficients, are functionsof temperature and position.

2.2. Displacement field model

The displacement field for the plate based on the first ordershear deformation plate theory (FSDT) is assumed as

uðx; y; zÞ ¼ uðx; yÞþzψxðx; yÞ

vðx; y; zÞ ¼ vðx; yÞþzψyðx; yÞ

wðx; y; zÞ ¼wðx; yÞ ð6Þ

where, u, v and w denote the displacements of a point along the (x,y, z) coordinates. u, v and w are corresponding displacements of apoint on the mid-plane. ψx and ψy are the rotations of normal tothe mid-plane about the y- and x-axis, respectively. The non-linearstrain–displacement relationships of uniform plate with initial

geometrical imperfection undergoing large deflections are

εx

εy

γxyγyzγxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

ε0xε0y

γ0xy

γ0yz

γ0xz

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;þz

ε1xε1y

γ1xy

γ1yz

γ1xz

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

¼

∂u∂xþ ∂w0

∂x∂w∂xþ1

2∂w∂x

� �2∂v∂yþ∂w0

∂y∂w∂yþ1

2∂w∂y

� �ð∂u∂yþ ∂v

∂xþ ∂w∂x

∂w∂yþ ∂w0

∂x∂w0∂y þ ∂w0

∂x∂w∂yþ ∂w

∂x∂w0∂y Þ

∂w∂xþ ∂w0

∂x þψx

∂w∂yþ∂w0

∂y þψy

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;þz

∂ψx∂x∂ψy

∂y∂ψx∂y þ

∂ψy

∂x

00

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

ð7Þ

where w0 is the initial geometrical imperfection.

2.3. Constitutive equations

Under the assumption that each layer possesses a plane ofelastic symmetry parallel to the x–y plane, the constitutiveequations for a layer can be written as

sxsysxysyzsxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

Q11 Q12 0 0 0Q12 Q22 0 0 00 0 Q66 0 00 0 0 Q44 00 0 0 0 Q55

26666664

37777775

εx

εy

γxyγyzγxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;�

α11

α22

000

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ΔT

0BBBBBB@

1CCCCCCA

�

0 0 e310 0 e320 0 00 e24 0e15 0 0

26666664

37777775

00Ez

8><>:

9>=>; ð8Þ

where Qij are the plane stress-reduced stiffnesses defined in termsof the engineering constants in the material axes of the layer as

Q11 ¼ Q11h ¼ E11h1� ν12hν21h

; Q12 ¼Q12h ¼ ν12hE22h1� ν12hν21h

; Q22 ¼ Q22h ¼ E22h1� ν12hν21h

;

Q44 ¼ Q44h ¼ G23h; Q55 ¼Q55h ¼ G13h; Q66 ¼ Q66h ¼ G12h;

ð9Þfor the CNTRC host, and

Q11 ¼Q11p ¼ Q22 ¼Q22p ¼ E11p1�ν2p

; Q12 ¼Q12h ¼ νpE11p1� ν2p

;

Q44 ¼Q55 ¼Q66 ¼ G12p ¼ E11p2ð1þνpÞ;

ð10Þ

for the piezoelectric layers. It should be noted that thermalexpansion coefficients of piezoelectric layers are α11p ¼ α22p; andand νp and ρp are Poisson’s ratio and mass density of piezoelectriclayers, respectively.

A linear distribution of the electric potential through thethickness direction is assumed here. For the panel type piezo-electric material, only thickness direction electric field Ez isdominant, and Ez is defined as Ez¼�ϒ,z, where ϒ is the potentialfield. The thickness of the piezoelectric layer is very thin and self-induced electric potential is much smaller than the appliedvoltage, and the voltage applied to the actuator in the thicknessonly, then the relationship between applied voltage V(t) andelectric field intensity within a piezoelectric actuator can bedescribed as [27,28,36]

Ez ¼VðtÞhp

: ð11Þ

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–51 39

2.4. Equations of motion

The governing equations of the first order theory will bederived using Hamilton’s principle:

Z T

0ðδUþδV�δKÞdt ¼ 0; ð12Þ

where δU is the virtual strain energy, δV is the virtual work doneby applied forces, and δK is the virtual kinetic energy.

By performing Hamilton’s principle, five equations of motionare obtained as [40]

∂Nx∂x þ

∂Nxy

∂y ¼ I0∂2u∂t2 þ I1

∂2ψx

∂t2

∂Nxy

∂x þ ∂Ny

∂y ¼ I0∂2v∂t2 þ I1

∂2ψy

∂t2

∂Qx∂x þ

∂Qy

∂y þ ∂∂x Nx

∂w∂xþ ∂w0

∂x

� �þNxy∂w∂yþ∂w0

∂y

� �� �þ ∂

∂y Nxy∂w∂xþ ∂w0

∂x

� �þNy∂w∂yþ ∂w0

∂y

� �� �þqðx; y; tÞ ¼ I0∂

2w∂t2

∂Mx∂x þ∂Mxy

∂y �Qx ¼ I2∂2ψx

∂t2 þ I1∂2u∂t2

∂Mxy

∂x þ∂My

∂y �Qy ¼ I2∂2ψy

∂t2 þ I1∂2v∂t2

ð13Þ

where q is the applied transverse external excitation. I0, I1, and I2are the mass moments of inertia and can be expressed as

I0I1I2

8><>:

9>=>;¼

Z H=2

�H=2

1z

z2

8><>:

9>=>;ρ dz: ð14Þ

The force, moment and transverse shear force resultants perunit length of the plate expressed in terms of the stress throughthe thickness are

Nx ¼Z h=2

�h=2shxx dzþ

Z h=2þhp

h=2spxx dzþ

Z �h=2

� h=2þhpð Þspxx dz ð15aÞ

Ny ¼Z h=2

�h=2shyy dzþ

Z h=2þhp

h=2spyy dzþ

Z �h=2

� h=2þhpð Þspyy dz ð15bÞ

Nxy ¼Z h=2

�h=2shxy dzþ

Z h=2þhp

h=2spxy dzþ

Z �h=2

�ðh=2þhpÞspxy dz ð15cÞ

Mx ¼Z h=2

�h=2zsxx dzþ

Z h=2þhp

h=2zspxx dzþ

Z �h=2

�ðh=2þhpÞzspxx dz ð16aÞ

My ¼Z h=2

�h=2zsyy dzþ

Z h=2þhp

h=2zspyy dzþ

Z �h=2

�ðh=2þhpÞzspyy dz ð16bÞ

Mxy ¼Z h=2

�h=2zsxy dzþ

Z h=2þhp

h=2zspxy dzþ

Z �h=2

� h=2þhpð Þzspxy dz ð16cÞ

Qx ¼ SZ h=2

�h=2shxz dzþ

Z h=2þhp

h=2spxz dzþ

Z �h=2

�ðh=2þhpÞspxz dz

!ð17aÞ

Qy ¼ SZ h=2

�h=2shyz dzþ

Z h=2þhp

h=2spyz dzþ

Z �h=2

� h=2þhpð Þspyz dz

!ð17bÞ

where S is the shear correction coefficient. Substituting Eqs. (7) and(8) into Eqs. (15) and (16) gives the expressions of force and

moment resultants as

Nx

Ny

Nxy

Mx

My

Mxy

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

A11 A12 A16 B11 B12 B16

A12 A22 A26 B12 B22 B26

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

B12 B22 B26 D12 D22 D26

B16 B26 B66 D16 D26 D66

26666666664

37777777775

ε0xε0y

γ0xy

ε1xε1y

γ1xy

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

�

NTx

NTy

NTxy

MTx

MTy

MTxy

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

�

NPx

NPy

NPxy

MPx

MPy

MPxy

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

; ð18aÞ

Qx

Qy

( )¼ S

A55 A45

A45 A44

" #γ0yz

γ0xz

( )ð18bÞ

The superscripts “T” and “P” in Eq. (18a) represent the thermal andelectric loads, respectively, and Aij;Bij and Dij i; j¼ 1;2;6ð Þ in Eq. (18)are the stretching stiffness, stretching–bending coupling stiffness andbending stiffness coefficients, respectively, which are defined as

ðAij;Bij;DijÞ ¼Z H=2

�H=2Qij 1; z; z2� �

dz ði; j¼ 1;2;6Þ; ð19aÞ

Aij ¼Z H=2

�H=2Qij dz ði; j¼ 4;5Þ: ð19bÞ

The stretching–bending coupling effect will not be appeared forsymmetric distribution of the carbon nanotubes along the thick-ness direction of the nanocomposite plates.

Thermal and electrical force and moment resultants aredefined as

NTx

NTy

NTxy

MTx

MTy

MTxy

8>><>>:

9>>=>>;¼

Z h=2

h=2

Q11h Q12h 0Q12h Q22h 00 0 Q66h

264

375

α11hα22h0

8><>:

9>=>; 1; zð ÞΔT dz

þZ H=2

h=2

Q11p Q12p 0Q12p Q11p 00 0 Q66p

264

375

α11p

α11p0

8><>:

9>=>; 1; zð ÞΔT dz

þZ �h=2

�H=2

Q11p Q12p 0Q12p Q11p 00 0 Q66p

264

375

α11p

α11p

0

8><>:

9>=>; 1; zð ÞΔT dz;

ð20aÞ

NPx

NPy

NPxy

MPx

MPy

MPxy

8>><>>:

9>>=>>;¼

Z H=2

h=2

e31e320

8><>:

9>=>; 1; zð ÞV tð Þ

hpdz

þZ �h=2

�H=2

e31e320

8><>:

9>=>; 1; zð ÞV tð Þ

hpdz; ð20bÞ

When the upper and lower piezoelectric actuators are appliedby the same harmonic excitation voltage VðtÞ ¼ Vs0þVd0 cos ωet,which make the piezoelectric layers expand or compact, the forceper unit length of the plate can be expressed as

NPx ¼NP

y ¼NPs0þNP

d0 cos ωet ð21Þ

where NPs0 ¼ 2Vs0e31 and NP

d0 ¼ 2Vd0e31 which could be obtainedfrom Eq. (20a) and it is valid while the upper and lower

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5140

piezoelectric actuators are applied by the same constant static anddynamic component of excitation voltage of Vs0 and Vd0

respectively.For the case of symmetric distribution of the carbon nanotubes

along the thickness direction of the nanocomposite plates, I1 willbe zero and terms containing it will be omitted. With an accep-table accuracy, neglecting all the inertia terms on u and v in Eq.(13) since their influences are small compared to that of thetransverse inertia term, one may have

∂Nx∂x þ

∂Nxy

∂y ¼ 0;

∂Nxy

∂xþ∂Ny

∂y¼ 0;

∂Qx

∂xþ∂Qy

∂yþ ∂∂x

Nx∂w∂x

þ∂w0

∂x

� �þNxy

∂w∂y

þ∂w0

∂y

� �� �

þ ∂∂y

Nxy∂w∂x

þ∂w0

∂x

� �þNy

∂w∂y

þ∂w0

∂y

� �� �þq x; y; tð Þ

¼ I0∂2w∂t2

;∂Mx

∂xþ∂Mxy

∂y�Qx ¼ I2

∂2ψx

∂t2;∂Mxy

∂xþ∂My

∂y�Qy ¼ I2

∂2ψy

∂t2:

ð22ÞIntroducing Airy stress function Φ as

Nx ¼∂2Φ∂y2

; Ny ¼∂2Φ∂x2

; Nxy ¼ � ∂2Φ∂x∂y

ð23Þ

which satisfies first two equations of Eq. (22), reduces theequations of motion to

∂Qx∂x þ

∂Qy

∂y þ ∂∂x Nx

∂w∂xþ ∂w0

∂x

� �þNxy∂w∂yþ ∂w0

∂y

� �� �þ ∂

∂y Nxy∂w∂xþ ∂w0

∂x

� �þNy∂w∂yþ ∂w0

∂y

� �� �þq x; y; tð Þ ¼ I0∂

2w∂t2 ;

∂Mx∂x þ∂Mxy

∂y �Qx ¼ I2∂2ψx

∂t2 ;

∂Mxy

∂x þ ∂My

∂y �Qy ¼ I2∂2ψy

∂t2 :

ð24Þ

Along with compatibility equation of the form

∂2ε0x∂y2 þ

∂2ε0y∂x2 �

∂2γ0xy∂x∂y

¼ ∂2w∂x∂y

� �2� ∂2w

∂x2∂2w∂y2 þ2∂2w

∂x∂y∂2w0∂x∂y� ∂2w

∂x2∂2w0∂y2 � ∂2w

∂y2∂2w0∂x2 ;

ð25Þ

Eq. (18a) can be written in the alternative form

ε0

Mn

( )¼ An Bn

�ðBnÞT Dn

" #Nn

κ

( ); ð26Þ

where

An ¼ A�1; Bn ¼ �A�1B; Dn ¼D�BA�1B; Mn ¼M; Nn ¼N:

ð27Þand

fε0g ¼ε0xε0y0

8><>:

9>=>;; fκg ¼

ε1xε1y

γ1xy

8>><>>:

9>>=>>;: ð28Þ

Substituting Eqs. (26) and (18b) into Eqs. (24) and (25), givesthe equations of motion of symmetric multiscale composite plates(and identically, the compatibility equation) in terms of displace-ments and stress function.

S A45∂2w∂x∂yþ∂2w0

∂x∂y

� �þA45

∂ψy

∂x þA55∂2w∂x2 þ ∂2w0

∂x2

� �þA55

∂ψx∂x

h iS A44

∂2w∂y2 þ ∂2w0

∂y2

� �þA44

∂ψy

∂y þA45∂2w∂x∂yþ ∂2w0

∂x∂y

� �þA45

∂ψx∂y

h iþ ∂2Φ

∂y2∂2w∂x2 þ ∂2w0

∂x2

� �þ ∂2Φ

∂x2∂2w∂y2 þ ∂2w0

∂y2

� ��2∂2Φ

∂x∂y∂2w∂x∂yþ ∂2w0

∂x∂y

� �þq x; y; tð Þ ¼ I0∂

2w∂t2

ð29aÞ

Dn

11∂2ψx

∂x2þDn

12∂2ψy

∂x∂yþDn

66∂2ψx

∂y2þDn

66∂2ψy

∂x∂y�S A55

∂w∂x

� �þA55ψx

� ¼ I2

∂2ψx

∂t2

ð29bÞ

Dn

66∂2ψx

∂x∂yþDn

66∂2ψy

∂x2þDn

12∂2ψx

∂x∂yþDn

22∂2ψy

∂y2�S A44

∂w∂y

� �þA44ψy

� ¼ I2

∂2ψy

∂t2

ð29cÞ

An

22∂4Φ∂x4

�2An

26∂4Φ∂x3∂y

þð2An

12þAn

66Þ∂4Φ

∂x2∂y2�2An

16∂4Φ∂x∂y3

þAn

11∂4Φ∂y4

¼ ∂2w∂x∂y

� �2

�∂2w∂x2

∂2w∂y2

þ2∂2w∂x∂y

∂2w0

∂x∂y�∂2w

∂x2∂2w0

∂y2�∂2w

∂y2∂2w0

∂x2

ð29dÞ

3. Analytical solution approach

Boundary conditions for simply supported rectangular plates are

w¼ 0; Mx ¼ 0 at x¼ 0; a ð30aÞ

w¼ 0; My ¼ 0 at y¼ 0; b ð30bÞThe in-plane boundary conditions for immovable edges are

Z a

0ε0x �

∂w0

∂x∂w∂x

�12

∂w∂x

� �2)

y ¼ 0;b

dx¼ 0;

8<: ð31aÞ

Z a

0fNxygy ¼ 0;b dx¼ 0; ð31bÞ

Z b

0ε0y�

∂w0

∂y∂w∂y

�12

∂w∂y

� �2)

x ¼ 0;a

dy¼ 0;

8<: ð31cÞ

Z b

0fNxygx ¼ 0;α dy¼ 0: ð31dÞ

The conditions expressing the immovability conditions (31) arefulfilled on the average sense asZ b

0

Z a

0ε0x �

∂w0

∂x∂w∂x

�12

∂w∂x

� �!dx dy¼ 0;

ð32aÞ

Z b

0

Z a

0

∂2Φ∂x∂y

dx dy¼Z a

0

Z b

0

∂2Φ∂x∂y

dy dx¼ 0; ð32bÞ

Z a

0

Z b

0ε0y�

∂w0

∂y∂w∂y

�12

∂w∂y

� �2!dy dx¼ 0:

ð32cÞ

For the inplane condition of zero shear stresses at edges, thedeflection function is assumed as

w¼WðtÞYðx; yÞ ¼WðtÞ sin mπxa

� �sin

nπyb

� �ð33Þ

where m,n¼1,2,3,… are numbers of half waves in x and ydirections, respectively.

The imperfect shape can be of an arbitrary type, to describe thevarious possible imperfection modes, which take the form of theproducts of trigonometric functions and hyperbolic functions inthe x–y plane [46]

w0 ¼ η sech δ1xa�ψ1

� �h icos λ1π

xa�ψ1

� �h isech δ2

yb�ψ2

� �h icos λ2π

yb�ψ2

� �h ið34Þ

where η is a coefficient representing imperfection size, δ1 and δ2are the constants defining the localization degree of the imperfec-tion that is symmetric about x/a¼ψ1 and y/b¼ψ2, and λ1 and λ2 are

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–51 41

the half-wave numbers of the imperfection in x- and y-axis,respectively. This expression is capable of modeling a wide rangeof initial imperfection modes, including: (a) the sine type, whenδ1¼δ2¼0, λ1¼λ2¼1, ψ1¼ψ2¼0.5; (b) the localized type, whenδ1a0, δ2a0; and (c) the global type, when δ1¼δ2¼0, λ1a1 or

λ2a1. However, for sake of simplicity, only sine type imperfectionis investigated in this study. A list of the imperfection modes isgiven in Fig. 2 where Cases G1, G2, and G3 are global imperfectionmodes while Cases L1, L2, L3, and L4 are localized imper-fection modes.

δ1=0, λ1=1, ψ1=0.5δ2=0, λ2=1, ψ2=0.5

δ1=0, λ1=3, ψ1=0.5δ2=0, λ2=3, ψ2=0.5

δ1=15, λ1=2, ψ1=0.25δ2=0, λ2=1, ψ2=0.5

δ1=0, λ1=7, ψ1=0.5δ2=0, λ2=7, ψ2=0.5

δ1=15, λ1=2, ψ1=0.5δ2=0, λ2=3, ψ2=0.5

δ1=15, λ1=2, ψ1=0.5δ2=0, λ2=1, ψ2=0.5

δ1=15, λ1=2, ψ1=0.5δ2=0, λ2=7, ψ2=0.5

δ1=15, λ1=2, ψ1=0.5δ2=0, λ2=5, ψ2=0.5

Fig. 2. Configuration of different imperfection modes: (a) sine type; (b) Case G1; (c) Case G2; (d) Case G3; (e) Case L1; (f) Case L2; (g) Case L3; and (h) Case L4.

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5142

Substituting Eqs.(33) and (34) into Eq. (29d) and solving it, airystress function Φ, is obtained as

Φ¼ΦCþΦP : ð35ÞWhen the expansions of w and w0 are substituted in the right-

hand side of Eq. (29d), a partial differential equation for Airy stressfunction Φ is obtained, the particular solution can be determinedby incorporating Eq.(33) into Eq. (29d) and comparing coefficientsof the same harmonic components as

ΦP ¼ ϕ1 cos 2mπxa

� �þϕ2 cos 2

nπyb

� �þϕ3 cos 2

mπxa

þnπyb

� �þϕ4 cos 2

mπxa

�nπyb

� �ð36Þ

where ϕi (i¼1,2,3,4) have long expressions (Appendix A).The complementary solution ΦC will now be obtained such that

it satisfies inplane boundary conditions of Eqs.(31). It can beshown that ΦC is zero for movable inplane edges without inplaneloading [41].

The complementary solution of Airy stress function is assumed as

ΦC ¼ �12Nx0y2�1

2Ny0x2�Nxyxy ð37ÞUpon using Eq. (33) and enforcing the conditions of Eqs. (31),

unknown coefficients may be obtained as

Nx0 ¼1Ξ

π2W2mn tð Þ b2m2An

22�a2n2An

12

� ���2ηπ2Wmn tð Þ An

12a2n2þAn

22b2m2

� ��þNP

s0þNPd0 cos ωetþNT

x ð38aÞ

Ny0 ¼1Ξ

π2W2mn tð Þ a2n2An

11�b2m2An

12

� ���2ηπ2Wmn tð Þ An

12b2m2þa2n2An

11

� ��þNP

s0þNPd0 cos ωetþNT

y ð38bÞ

Nxy ¼ 0 ð38cÞwhere Ξ ¼ 8a2b2 An2

12�An

11An

22

� �: Eqs. (29b) and (29c) lead to a set

of equations with two unknown parameters:

Ln1ψxþLn2ψy ¼ Ln3w

Ln4ψxþLn5ψy ¼ Ln6w

(ð39Þ

Finding unknowns in terms of w gives

ψx ¼Ln2L

n

6�Ln5Ln

3

Ln2Ln

4�Ln1Ln

5w; ψy ¼

Ln3Ln

4�Ln1Ln

6

Ln2Ln

4�Ln1Ln

5w: ð40Þ

Lni are partial differential operators for CNTRC rectangularplates and are given as

Ln1 ¼Dn

11∂2∂x2þDn

66∂2∂y2�SA55� I2 ∂2

∂t2;

Ln2 ¼ ðDn

12þDn

66Þ ∂2∂x∂y;

Ln3 ¼ SA55∂∂x;

Ln4 ¼ ðDn

12þDn

66Þ ∂2∂x∂y;

Ln5 ¼Dn

66∂2∂x2þDn

22∂2∂y2�SA44� I2 ∂2

∂t2;

Ln6 ¼ SA44∂∂y:

ð41Þ

Then by substituting ψx and ψy into Eq. (40), it is obtained

S A44∂∂y

Ln3Ln

4�Ln1Ln

6

� �þA55∂∂x

Ln2Ln

6�Ln5Ln

3

� �� w

þ A44∂2

∂y2þA55

∂2

∂x2þ∂2Φ∂y2

∂2

∂x2

� �þ∂2Φ

∂x2∂2

∂y2

� ��

�2∂2Φ∂x∂y

∂2

∂x∂y

� �� I0

∂2

∂t2

�

Ln2Ln

4�Ln1Ln

5

� �wþw0ð Þ ¼ 0: ð42Þ

3.1. Non-linear parametric resonance analysis of perfect plates

Substituting Eqs. (33)–(35) together with Eq. (40) into Eq. (42),multiplying the resulting equation by sin mπx=a

� �sin nπy=b� �

, andintegrating over the domain, we obtain the following non-linearordinary-differential equation for free vibration of piezoelectricCNTRC plates:

J1 €WðtÞþðJ2þNx00ðNPs0þNP

d0 cos ωetþNTx Þ

þNy00ðNPs0þNP

d0 cos ωetþNTy ÞÞWðtÞ

þ J3W3ðtÞþ J4 €WðtÞW2ðtÞ ¼ 0 ð43Þ

where a super dot denotes differentiation with respect to time,and Ji (i¼1,2,3,4) are the constant coefficient obtained by applyingGalerkin’s method to Eq. (42) and

Nx00 ¼Z b

0

Z a

0sin

mπxa

� �sin

nπyb

� � ∂2

∂x2Ln2L

n

4�Ln1Ln

5

� �w dx dy ð44aÞ

Ny00 ¼Z b

0

Z a

0sin

mπxa

� �sin

nπyb

� � ∂2

∂y2Ln2L

n

4�Ln1Ln

5

� �w dx dy ð44bÞ

3.1.1. Linear buckling analysis of perfect smart nanocomposite platesBy neglecting the contribution of time-differentiated and non-

linear terms in Eq. (43), the critical buckling load can be deter-mined using

J2þNx00ðNPs0þNT

x ÞþNy00ðNPs0þNT

y Þ ¼ 0 ð45Þ

Eq. (45) may be used to determine the critical temperaturedifference and voltage as

ΔTcr ¼ �ðJ2þðNx00þNy00ÞNPs0Þ=(

Nx00

Z h=2

h=2ðQ11hα11hþQ12hα22hÞdzþ

Z H=2

h=2ðQ11pþQ12hÞα11p dz

!

þNy00

Z h=2

h=2ðQ21hα11hþQ22hα22hÞdz

þ2Z H=2

h=2ðQ21pþQ22hÞα11p dz

�)ð46Þ

Vs0cr ¼ �ðJ2þNx00NTx þNy00N

Ty Þ

2ðNx00þNy00Þe31ð47Þ

It should be noted that the critical voltage does not physicallyexist since the structure cannot undergo such a high value ofvoltage. It will be used only for calculating a dimensionlessparameter in non-linear dynamic stability analysis.

3.1.2. Linear vibration analysis of perfect smart nanocompositeplates

From Eq. (43) the fundamental frequencies of natural vibrationof the piezoelectric CNTRC plates can be determined by therelation [34–39]:

ωmnL ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2þNx00ðNP

s0þNTx ÞþNy00ðNP

s0þNTy Þ

J1

s: ð48Þ

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–51 43

3.1.3. Linear dynamic instability analysis of perfect smartnanocomposite plates

In order to perform a linear stability analysis, it is moreconvenient to rewrite Eq. (43) in the form:

IM €WþðΚ�2Β cos ωetÞW ¼ 0; ð49Þwhere

Κ ¼ J2þNx00ðNPs0þNT

x ÞþNy00ðNPs0þNT

y Þ;

Β¼ �12ðNx00þNy00ÞNP

d0;

IM ¼ J1: ð50ÞEq. (49) is in the form of a second order differential equation

with periodic coefficients of the Mathieu–Hill type. The regions ofunstable solutions are separated by periodic solutions havingperiod T and 2T with T ¼ 2π=ωe. The solutions with period 2T areof greater practical importance as the widths of these unstableregions are usually larger than those associated with solutionshaving period T. Using Bolotin’s first approximation, the periodicsolutions with period 2T can be sought in the form [45]

WðtÞ ¼ ∑1

j ¼ 1;3;5;:::ϑ1j sin

jωet2

� �þϑ2n cos

jωet2

� �; ð51Þ

where ϑ1j and ϑ2j are arbitrary constants. For the case of principalresonance ωe � 2ωmnL, the influence of higher harmonics in theexpansion of above equation can be ignored. Substituting Eq. (51)into Eq. (49) and equating the coefficients of sin ωet=2

� �and

cos ωet=2� �

terms, a set of linear homogeneous algebraic equa-tions in terms of ϑ11 and ϑ21 can be obtained. The conditions fornontrivial solutions are given by

Det� �1

4ω2e IM 0

0 Κþ Β�14ω

2e IM

24

35¼ 0: ð52Þ

Instead of solving the above non-linear geometric equations forΩ, the above equation can be rearranged to the standard form of ageneralized eigenvalue problem

DetΚ� Β 00 Κþ Β

!�ω2

e

14 IM 0

0 14 IM

0@

1A

24

35¼ 0: ð53Þ

Expanding the determinant one may obtain

Β¼IMω2

e �4��� ���

4: ð54Þ

3.1.4. Non-linear dynamic instability analysis of perfect smartnanocomposite plates

For the case of non-linear dynamic instability, Eq. (43) take theform:

€Wþω2mnLð1�2β cos ωetÞWþβcW

3þβd €WW2 ¼ 0; ð55Þwhere β is a quantity that is called the excitation parameter anddefined as

β¼ �ðNx00þNy00ÞNPd0

2ðJ2þNx00ðNPs0þNT

x ÞþNy00ðNPs0þNT

y ÞÞ; ð56aÞ

βc ¼J3J1; ð56bÞ

βd ¼J4J1: ð56cÞ

Using Bolotin’s first approximation, considering the case ofprincipal resonance ωe � 2ω0, substituting Eq. (51) into Eq. (52)and equating the coefficients of the sin ωet=2

� �and cos ωet=2

� �

terms, a set of linear homogeneous algebraic equations in terms ofϑ11 and ϑ21 can be obtained. The conditions for nontrivial solutionsare given by

€Wþω2mnLð1�2β cos ωetÞWþβcW

3þβd€WW2 ¼ 0;

ω2mnLð1þ βÞ�1

4ω2e

h iϑ11þΞ1ðϑ11; ϑ21Þ ¼ 0; ð57aÞ

ω2mnLð1þ βÞ�1

4ω2e

h iϑ12þΞ2ðϑ11; ϑ21Þ ¼ 0; ð57bÞ

where Ξ1ðϑ11; ϑ21Þ and Ξ2ðϑ11; ϑ21Þ are defined as coefficients of theterms including sin ωet=2

� �and cos ωet=2

� �which obtained from

the first approximation of expansion in a Fourier series as

Ξ1ðϑ11; ϑ21Þ ¼316

Χ2ϑ11ð4βc�ω2eβdÞ ð58aÞ

Ξ2ðϑ11; ϑ21Þ ¼ 316 Χ

2ϑ21ð4βc�ω2eβdÞ ð58bÞ

where X is the amplitude of steady-state vibrations and is given by

Χ2 ¼ ϑ211þϑ221 ð59ÞBy substitution of Eqs. (58) into Eq. (57) a system of two

homogeneous linear equations with respect to ϑ11 and ϑ21 can beobtained. This system has solutions that differ from zero only inthe case where the determinant composed of the coefficientdisappears

Det1þ β� ω2

e4ω2

mnLþ 3a2

16ω2mnL

ð4βc�ω2eβdÞ 0

0 1� β� ω2e

4ω2mnL

þ 3a216ω2

mnLð4βc�ω2

eβdÞ

264

375¼ 0:

ð60ÞExpanding the determinant and solving the resulting equation

with respect to the amplitude, a, of the steady-state vibrations thefollowing equation is obtained:

Χ ¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16ω2

mnL

3ð4βc�ω2eβdÞ

�17 βþ ω2e

4ω2mnL

!vuut ð61Þ

It can be easily proved that in the �17 βþ ω2e=4ω

2mnL

� �� �term

of the above equation, only the positive one �1þ βþ ω2e=

��4ω2

mnLÞÞterm is the stable-solution, and the negative term is unstable-solution.

3.2. Non-linear parametric resonance analysis of imperfect plates

With a similar approach to Section 3.1, the non-linear govern-ing equation of motions for piezoelectric CNTRC plates with initialgeometric imperfection can be obtained as

K1€WðtÞþðK2þNix00ðNP

d0 cos ωetþNTx ÞþNiy00ðNP

d0 cos ωetþNTy ÞÞ

WðtÞþK3W2ðtÞþK4

€WðtÞWðtÞþK5W3ðtÞþK6

€WðtÞW2ðtÞ ¼ K7 cos ðωetÞ ð62Þwhere Ki (i¼1,2,3,…,6) are the constant coefficient obtained byapplying Galerkin’s method to Eq. (42) for imperfect plates and

Nix00 ¼Z b

0

Z a

0sin

mπxa

� �sin

nπyb

� � ∂2

∂x2ðLn2Ln4�Ln1L

n

5Þðwþw0Þdx dy

ð63aÞ

Niy00 ¼Z b

0

Z a

0sin

mπxa

� �sin

nπyb

� � ∂2

∂y2ðLn2Ln4�Ln1L

n

5Þðwþw0Þdx dy

ð63bÞAs it can be observed, Eq. (62) captures the quadratic and cubic

non-linearities associated with mid-plane stretching and thequadratic non-linearity is generated by predeformation. Also thispredeformation or initial geometric imperfection generates an

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5144

external excitation. The combination of inplane forces NPd0 cos ωet

and predeformation w0 creates a vertical component of the in-plane force which is NP

d0 ∂w0=∂x� �

cos ωet while the horizontalcomponent can be still considered as NP

d0 cos ωet [47].

3.2.1. Linear vibration analysis of imperfect smart nanocompositeplates

The fundamental frequencies of natural vibration of the imper-fect piezoelectric CNTRC plates can be determined from Eq. (62) bythe relation [34–39]:

ϖmnL ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2þNix00ðNP

s0þNTx ÞþNiy00ðNP

s0þNTy Þ

K1

s; ð64Þ

3.2.2. Linear dynamic instability analysis of imperfect smartnanocomposite plates

In order to perform a linear stability analysis, it is moreconvenient to rewrite Eq. (62) in the form:

IiM €WþðΚ i�2Βi cos ωetÞW ¼ 0; ð65Þwhere

Κ i ¼ K2þNix00ðNPs0þNT

x ÞþNiy00ðNPs0þNT

y Þ;

Βi ¼ �12ðNix00þNiy00ÞNP

d0;

IiM ¼ K1: ð66ÞAssuming the same assumption as Section 3.1.3 the following

relationship between the parameters in linear region for imperfectpiezoelectric CNTRC plates can be obtained

Βi ¼IiMω2

e �4Κ i

��� ���4

: ð67Þ

3.2.3. Non-linear dynamic instability analysis of imperfect smartnanocomposite plates

For the case of non-linear dynamic instability of imperfectpiezoelectric CNTRC plates, Eq. (62) can take the form:

€Wþϖ2mnL 1�2γ cos ωetð ÞWþγaW

2þγb €WWþγcW3þγd €WW2 ¼ γe cos ωet;

ð68Þwhere γ is a quantity that is called the excitation parameter anddefined as

γ ¼ �ðNix00þNiy00ÞNPd0

2ðK2þNix00ðNPs0þNT

x ÞþNiy00ðNPs0þNT

y ÞÞ; ð69aÞ

γa ¼K3

K1; γb ¼

K4

K1; γc ¼

K5

K1; γd ¼

K6

K1; γe ¼

K7

K1: ð69bÞ

3.2.3.1. Regions of resonance. The external excitation in Eq. (68) incombination with the non-linearities lead to primary (ωe �ϖmnL)and a number of secondary external resonances [38,39,47].Subharmonic resonances exist for ωe � 2ϖmnL and 3ϖmnL andsuperharmonic resonances exist for ωe �ϖmnL=2 and ωe �ϖmnL=3. Note that the subharmonic ωe � 2ϖmnL, primaryωe �ϖmnL and superharmonic ωe �ϖmnL=2 external resonancescoincide, respectively, with the principal (j¼1), second order(j¼2), and fourth order (j¼4) parametric resonances.

In the present study, it is desired to have flexibility in assigningthe relative importance of the parametric and external excitations.The amplitude of the external excitation is proportional to theamplitude of the initial geometric imperfection and ultimatelydominates for cases of (relatively) large imperfection [47]. Thus, asolution is sought that can capture the complete transition

between pure parametric and pure external resonant response.For this purpose, the method of harmonic balance is adopted [45].In the following, this method is used to determine periodicsolutions and their stability throughout the frequency rangeencompassing the one region of resonance.

3.2.3.2. Principal parametric resonance-subharmonic externalresonance response. In the principal parametric-subharmonicresonant region, the excitation frequency is approximately twicethe plate natural frequency. The harmonic balance solution musttherefore include a term with the frequency of the subharmonicresonance (ωe=2) as well as a term with the excitation frequency(ωe) [45,47]:

WðtÞ ¼ ζ1ðtÞ cosωe

2

� �þζ2ðtÞ sin

ωe

2

� �þζ3ðtÞ cos ðωeÞþζ4ðtÞ sin ðωeÞ

ð70Þ

Here ζ1ðtÞ; ζ2ðtÞ; ζ3ðtÞ and ζ4ðtÞ represent slowly varyingresponse amplitudes. The first two terms correspond to theprincipal parametric-subharmonic resonance, while the last twoterms capture (possible) primary external response. Differentiat-ing Eq. (70) twice, neglecting the second order derivatives of theamplitudes and multiplications of first order derivatives withother amplitudes [45] and substituting this result together withEq. (70) into Eq. (68) and equating coefficients of like harmonicsleads to the following state equations governing the slowlyvarying amplitudes leads to

_ζ1 ¼1

16ωeð16γað�ζ2ζ3þζ1ζ4Þþ12γcζ2ðζ21þζ22þ2ðζ23þζ24ÞÞ

�ð3γdζ32þ10γbζ1ζ4þζ2ð4�10γbζ3þ3γdðζ21þ6ðζ23þζ24ÞÞÞÞω2e

þ16ð1þ γÞζ2ϖ2mnLÞ ð71aÞ

_ζ2 ¼1

16ωeð�16γaðζ1ζ3þζ2ζ4Þ�12γcζ1ðζ21þζ22þ2ðζ23þζ24ÞÞ

þð3γdζ31þ10γbζ2ζ4þζ1ð4þ10γbζ3þ3γdðζ22þ6ðζ23þζ24ÞÞÞÞω2e

þ16ð�1þ γÞζ1ϖ2mnLÞ ð71bÞ

_ζ3 ¼1

8ωeð4γaζ1ζ2þ3γcζ4ð2ζ21þ2ζ22þζ23þζ24Þ

�ðγbζ1ζ2þζ4ð4þ3γdðζ21þζ22þζ23þζ24ÞÞÞω2e þ4ζ4ϖ

2mnLÞ ð71cÞ

_ζ4 ¼1

16ωeð8γeþ4γað�ζ21þζ22Þ�6γcζ3ð2ζ21þ2ζ22þζ23þζ24Þ

þðγbðζ21�ζ22Þþ2ζ3ð4þ3γdðζ21þζ22þζ23þζ24ÞÞÞω2e �8ζ3ϖ

2mnLÞ

ð71dÞ

The conditions _ζ1 ¼ _ζ2 ¼ _ζ3 ¼ _ζ4 ¼ 0 are satisfied by the singularpoints of Eqs. (71). and provide the steady amplitudes for periodicsolutions. The stability of each periodic solution is determined bylinearizing Eqs. (71) about the singular points and computing theeigenvalues of the resulting Jacobian matrix.

Using the approximation of Nayfeh and Mook [38], closed formexpressions for the periodic solutions are presently derived. Thesteady state forms of Eq. (71c,d) are linearized and provide theamplitudes for linear response:

ζ3 ¼γe

ðϖ2mnL�ω2

e Þ; ð72aÞ

ζ4 � 0: ð72bÞ

From the steady forms of Eq. (71a,b), note that the trivialsolution, ζ1 ¼ ζ2 ¼ 0, is always a solution.

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–51 45

With χ2 ¼ ζ21þζ22 denoting the amplitude of the remaining(resonant) part of the solution, the steady forms of Eq. (71a,b)provide two additional solutions:

χ ¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�8γaζ3�12γcζ

23þð2þ5γbζ3þ9γdζ

23Þω2

e þ8ð�17 γÞϖ2mnLÞ

12γc�3γdω2e

s

ð73Þ

Refers to the same discussion for Eq. (61), only �1þ γð Þ8ϖ2mnL

term is the stable-solution, and the negative term is unstable-solution.

4. Numerical results and discussions

The study, here, has been focused mainly on the variouscharacteristic features of the phenomenon concerning the non-linear dynamic stability behavior of plates. Extensive numericaldata are provided to investigate the non-linear dynamic bucklinganalysis of CNTRC plates under electrical and thermal loadings.Material properties and effective thickness of SWCNTs used foranalysis are properly chosen in the present paper by MD simula-tions. Unless otherwise stated, η1 ¼ 0:137, η2 ¼ 1:022 and η3 ¼0:715 for the case of Vn

CN ¼ 0:12, and η1 ¼ 0:142, η2 ¼ 1:626 andη3 ¼ 1:138 for the case of Vn

CN ¼ 0:17, and η1 ¼ 0:141, η2 ¼ 1:585and η3 ¼ 1:109 for the case of Vn

CN ¼ 0:28 [23,36]. Here tempera-ture dependent (TD) PmPV material is considered. The parametersused are ρm¼1150 kg/m3, νm¼0.34, αm¼45(1þ0.0005ΔT)�10�6/K and Em¼(3.52�0.0034T) GPa, in which T¼T0þΔT and T0¼300 K (room temperature). The temperature independent (TID)material properties of CNTRC host are αm¼45�10�6/K andEm¼2.5 GPa at T¼300 K. The (10,10) SWCNTs are selected asreinforcements. Temperature dependent material properties ofSWCNTs are listed in Table 1. The temperature independentmaterial properties of piezoelectric material are E11p¼63.0 GPa,ρp¼7600 kg/m3, α11p ¼ α22p¼0.9�10�6/K, υp¼0.3 and e31¼e32¼17.6 C/m2.

4.1. Validation study

As a part of the validation of the present method, a comparisonof the buckling temperature (Tcr ¼ΔTcrþT0) of CNTRC squareplates under uniform temperature rise with different volumefraction of CNTs (hp/h¼0, Vs0¼0) with HSDT results by Shen andZhang [26] is listed in Table 2. A very good agreement is observed.

Comparison of the linear natural frequencies ωmnL ¼ωmnLa2ffiffiffiffiffiffiffiffiffiffiffiffiffiI0=E22

p=H for an unstressed UD-CNTRC square plate with differ-

ent volume fraction of CNTs is made and shown in Table 3.(a/h¼10, hp/h¼0, V0¼0). A good agreement between the resultscan be observed. In another validation study, an unstressed eight-layer (P/901/01/901/01/01/901/0/901/P) symmetric cross-ply plateunder immovable simply supported boundary condition is con-sidered. This example is chosen due to some comparative resultsavailable in the literature [42–44]. All the layers have the samethickness and all properties of the laminated plate are taken fromRef. [42] and are given below: (E11¼181.0 GPa, E22¼10.3 GPa, G12¼G13¼7.17 GPa, G23¼2.87

GPa, υ12¼0.28, ρ¼1580 kg/m3, a¼b¼1 m)The fundamental frequencies ωmnL (rad/s) for this plate is calcu-

lated and compared in Table 4 with three-dimensional (3-D)solutions of Xu et al. [42] and FSDPT solutions of Benjeddou et al.[43] and HSDT solution of Huang and Shen [44].

Table 5 compares the linear frequency parameter ω¼ωmnL

a2=π2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiI0=D11

presults of simply supported, moderately thick

(a/h¼10) stainless steel plates with sine type imperfection forseveral vibration modes. Close correlation is achieved. All material

Table 1Temperature-dependent material properties for (10,10) SWCNT (L¼9.26 nm,R¼0.68 nm, h¼0.067 nm, νcnt12 ¼ 0:175, ρCN¼1400 kg/m3) [36].

Temperature (K) ECN11 (TPa) ECN22 (TPa) GCN12 (TPa) αCN11 (�10�6/K) αCN22 (�10�6/K)

300 5.6466 7.08 1.9445 3.4584 5.1682500 5.5308 6.9348 1.9643 4.5361 5.0189700 5.4744 6.8641 1.9644 4.6677 4.8943

Table 2Comparison of the buckling temperature (Tcr ¼ΔTcrþT0) of CNTRC square platesunder uniform temperature rise with different volume fraction of CNTs (hp/h¼0,Vs0¼0).

Vn

CN Source a/h

10 20

UD FGX UD FGX

0.12 HSDT [26] 398.48 419.65 340.74 355.85Present-FSDT 399.67 420.14 341.05 355.96Error (%) 0.30 0.12 0.09 0.03

0.17 HSDT [26] 412.36 439.88 344.54 362.79Present-FSDT 413.88 442.83 344.87 363.26Error (%) 0.37 0.67 0.09 0.13

0.28 HSDT [26] 403.63 430.26 345.88 366.21Present-FSDT 404.89 440.68 346.31 369.23Error (%) 0.31 2.42 0.13 0.82

Table 3Comparison of the linear natural frequencies ωmnL ¼ωmnLa2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiI0=E22

p=H for an

unstressed UD-CNTRC square plate with different volume fraction of CNTs(a/h¼10, hp/h¼0, Vs0¼0).

Vn

CN Source (m, n)

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

0.12 HSDT [25] 12.27 16.81 29.44Present-FSDT 12.31 16.85 28.83Error (%) 0.29 0.25 2.07

0.17 HSDT [25] 15.39 21.43 37.44Present-FSDT 15.43 21.48 36.78Error (%) 0.30 0.22 1.75

0.28 HSDT [25] 17.30 23.21 40.75Present-FSDT 17.34 23.22 39.67Error (%) 0.24 0.07 2.64

Table 4Comparison of the linear fundamental frequency ωmnL (rad/s) for an unstressedeight-layer (PZT-5A/901/01/901/01/01/901/0/901/PZT-5A) symmetric cross-ply plate(E11¼181.0 GPa, E22¼10.3 GPa, G12¼G13¼7.17 GPa, G23¼2.87 GPa, υ12¼0.28,ρ¼1580 kg/m3, a¼b¼1 m, hp/h¼0, Vs0¼0).

Source a/H

100 10 5

3D [42] 268.86 2357.7 36482D –FSDT [43] 283.93 2516.7 3953.2HSDT [44] 269.12 2397.06 3769.61Present 265.99 2461.89 4074.42

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5146

properties of the laminated plate are taken from Ref. [46] and aregiven below:

E¼ 207:7877� 109

�ð1þ3:079� 10�4T�6:534

�10�7T2Þ GPa; ν¼ 0:3177; ρ¼ 8166 kg=m3

4.2. Linear and non-linear dynamics results

Numerical examples in terms of tabular data and figures areprovided to investigate the non-linear parametric resonancebehavior of CNTRC plates under electrical and thermal loadings,in which the width-to-thickness ratios (a/h) of plates is set to 10,and the thickness is taken to be 2.0 mm. Three types of squareplates, UD-CNTRC, FG-O and FG-X CNTRC, are considered withimmovable simply supported boundary condition.

4.2.1. Buckling resultsThe influence of piezoelectric to CNTRC thickness ratio (hp/h), and

applied voltage (Vs0) on the buckling temperature (Tcr ¼ΔTcrþT0) ofFGX-CNTRC square plates under uniform temperature rise(Vn

CN ¼ 0:12) is investigated and listed in Table 6. The thickness ofthe CNTRC host kept constant while the thickness of piezoelectriclayers varies. It can be seen that the piezoelectric layers can drasticallyincrease the critical temperature of the CNTRC plates while theinfluence of applied constant voltage is less evident. As it could bepredicted, the influence of the applied constant voltage on the thinplates is more prominent than the thicker one. According to Table 6,an increase in the critical temperature is more prominent in thick tomoderately thick plates as compared with thin plates.

4.2.2. Vibration resultsPresented in Table 7 is the effect of piezoelectric to CNTRC

thickness ratio (hp/h), temperature rise (ΔT) and applied voltage(V0) on the dimensionless linear fundamental natural frequenciesωmnL ¼ ωmnLa2

ffiffiffiffiffiffiffiffiffiffiffiffiffiI0=E22

p=H of FGX-CNTRC square plates (Vn

CN ¼ 0:12).As the same as prior sections, to investigate the effect of piezo-electric to CNTRC thickness ratio (hp/h), the thickness of the CNTRC

host kept constant while the thickness of piezoelectric layersvaries. It can be seen that the piezoelectric layers can significantlychange the linear fundamental natural frequencies of the CNTRCplates. However, the significance of this fact can be seen mostly inthick to moderately thick plates rather than the thin plates. Thenegative voltage (which is corresponding to the voltage applied inopposite polarization direction) increases the plate dimensionlesslinear fundamental natural frequencies. Nevertheless, it has asmall influence on the linear fundamental natural frequencies ofthe plates. Temperature rise (ΔT) decreases the dimensionlesslinear fundamental natural frequencies of the CNTRC plates whichis consistent with the fact that axial compressive stress state isgenerated in the plate by the temperature rise. As it could bepredicted, the influence of the temperature rise on the thin platesis more significant than the thicker one. It should be noted that ata sufficiently large compressive thermal load, the natural fre-quency approaches to zero, indicating that the plate bucklingoccurs.

4.2.3. Linear and non-linear parametric resonance resultsEach unstable region is bounded by two lines which originate

from a common point from the Ω-axis (see Fig. 3). The two curvesappear at first glance to be straight lines but are in fact, two very

Table 6Effects of piezoelectric to CNTRC thickness ratio (hp/h) and applied voltage (Vs0) onthe buckling temperature (Tcr ¼ΔTcrþT0) of FGX-CNTRC square plates underuniform temperature rise (Vn

CN ¼ 0:12)

a/h hp/h Vs0

0 �250 �500

10 0 420.137 – –

1/20 650.92 653.19 655.461/10 856.071 858.33 860.60

20 0 355.956 – –

1/20 414.669 416.94 419.211/10 471.221 473.48 475.75

50 0 311.956 – –

1/20 320.131 322.40 324.681/10 329.328 331.59 333.85

Table 7Effects of piezoelectric to CNTRC thickness ratio (hp/h) and applied voltage (Vs0) onthe linear natural frequencies ωmnL ¼ωmnLa2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiI0=E22

p=H of FGX-CNTRC square plates

with different volume fraction of CNTs (Vn

CN ¼ 0:12).

a/h hp/h Vs0

0 �250 �500

ΔT ¼ 0 K ΔT ¼ 10 K ΔT ¼ 0 K ΔT ¼ 10 K ΔT ¼ 0 K ΔT ¼ 10 K

10 0 13.25 12.7091 – – – –

1/20 17.5695 17.4245 17.6256 17.48 17.68 17.541/10 18.7006 18.653 18.74 18.69 18.78 18.73

20 0 18.1877 16.5391 – – – –

1/20 20.3099 19.525 20.51 19.74 20.71 19.941/10 21.056 20.5647 21.19 20.71 21.33 20.85

50 0 21.0536 8.44054 – – – –

1/20 21.3443 15.2069 22.52 16.83 23.63 18.321/10 21.879 17.86 22.71 18.88 23.51 19.85

Stable Region

Stable Region

Dim

ensi

onle

ss lo

adin

g B^

Dimensionless frequency parameter Ω

Transition curve

Unstable Region

Fig. 3. Transition curves for the linear Mathieu equation separating stable regionfrom unstable one.

Table 5Comparison of linear frequency parameters ω¼ωmnLa2=π2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiI0=D11

pfor simply

supported laminated square plates with sine type imperfection (a/h¼10, η¼0.2H).

Mode (m,n) Kitipornchai et al. [46] Present Error (%)

(1,1) 1.9379 1.9494 0.59(1,2) 4.6088 4.79116 3.96(2,1) 4.6088 4.79116 3.96(2,2) 7.0676 7.0698 0.03

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–51 47

slight “outward” curving plots. For the sake of tabular presenta-tion, each unstable region is defined by its point of origin from theΩ-axis with B ðor BiÞ ¼ 0. The angle subtended, ϴ, is also intro-duced to give a measure of the size of the unstable region. It iscalculated based on the arctangent of the right-angled triangle,ℜ1ℜ2ℜ3, as shown in Fig. 4. This angle gives an accurate measureof the slope of the boundary of the unstable region as calculationsdone with the smaller similar triangle, ℜ1ℜ

n

2ℜn

3 (Fig. 4), are within0.1% difference. Also dimensionless frequency parameterΩ¼ ωe=ða2

ffiffiffiffiffiffiffiffiffiffiffiffiffiρm=E0

p=hÞ is introduced to facilitate the demonstration

of results. Hence, Βand Βi are divided by 1020 to avoid presentinglarge numbers.

Effect of initial geometric imperfection of sine type on theunstable regions for the transverse mode of piezoelectric FGX-CNTRC plates is demonstrated in Table 8 (a/h¼10, hp/h¼1/10,h¼2 mm, Vs0 ¼ 0 V, Vd0 ¼ 500 V, T¼300 K, η¼ 0:2H). From Table 8it can be observed that initial geometric imperfection has asignificant influence on the fundamental natural frequency andthe sizes of unstable regions as well as the amplitude of vibrationresponse for the transverse mode of piezoelectric FGX-CNTRCplates. It can be seen that the dimensionless frequency parameteras well as fundamental natural frequency increased as a result ofpredeformation. The pre-deformation also increases slightly thewidth of the instability region (higher ϴ) and the amplitude ofsteady state vibration response decreases with introduction ofimperfection.

Table 9 together with Figs. 5 through 7 illustrate the effect ofdifferent distributions of carbon nanotubes and CNTs volume fractionVn

CN on the unstable regions for the transverse mode of perfectpiezoelectric CNTRC plates in linear dynamic stability analysis underdifferent thermal environments (a/h¼10, hp/h¼1/10, h¼2 mm,Vs0 ¼ 0 V, Vd0 ¼ 500 V). It can be seen that the increase of the CNTvolume fraction Vn

CN yields an increase of stable region as the angle ϴis higher. Containing more CNTs leads to stiffer and strongerstructure than a lesser one. It is noticeable that the stable region of

FGO CNTRC plates are smaller than that of UD-CNTRC plates whilethose of FGX CNTRC plates are larger though these three types ofbeams have the same mass fraction of the CNTs. This is because the

0.00

0.05

0.10

ℜ∗3

ℜ3ℜ2

ℜ∗2

Stable RegionStable Region

Dim

ensi

onle

ss lo

adin

g

Dimensionless frequency parameter Ω

Unstable Region

Θ

ℜ1

Fig. 4. A magnified view of transition curves for the linear dynamic stabilityequation separating stable region from unstable one.

Table 8Effects of initial geometric imperfection of sine type on the unstable regions for thetransverse mode of piezoelectric FGX-CNTRC plates (a/h¼10, hp/h¼1/10, h¼2 mm,Vs0 ¼ 0, Vd0 ¼ 500 V, T¼300 K, η¼ 0:2H).

Initialstatus

Instability parameter

Ω ϴ (deg) Amplitude(at Ω¼40)

Fundamentalnatural frequency

Perfect 37.4012 50.0426 0.3620 18.7006Imperfect 39.5349 48.4718 0.1251 19.7675

Table 9Effects of different distributions of carbon nanotubes and CNTs volume fraction Vn

CN

on the unstable regions for the transverse mode of perfect piezoelectric CNTRCplates in linear dynamic stability analysis under different thermal environment(a/h¼10, hp/h¼1/10, h¼2 mm, Vs0 ¼ 0, Vd0 ¼ 500 V)

T Vn

CN Distribution

UD FGO FGX

Ω ϴ (deg) Ω ϴ (deg) Ω ϴ (deg)

300 0.12 35.6187 53.1171 33.5072 56.4366 37.4012 50.04260.17 37.8542 45.0687 35.0431 49.4961 40.144 41.02450.28 40.6113 38.7739 36.881 44.4024 43.539 33.247

350 0.12 34.8511 56.1212 32.6158 59.4465 36.8653 52.92690.17 36.5111 48.9181 33.5487 53.4967 39.1689 44.52460.28 38.5592 43.0939 34.7682 49.0326 42.1588 36.815

400 0.12 33.8493 59.3281 a a 36.1543 56.0130.17 34.7663 53.2112 31.6049 57.902 37.9037 48.43140.28 35.8528 48.2221 31.9734 54.3973 40.363 41.0059

a Buckled.

30 33 36 39 42 450

2

4

6

8

Dimensionless frequency parameter (Ω)

UD-CNTRC FGO-CNTRC FGX-CNTRC

Dim

ensi

onle

ss a

mpl

itude

of l

oadi

ng

Fig. 5. Effect of different distribution of carbon nanotubes (UD, FGO and FGXCNTRC) on the unstable regions for the transverse mode of perfect piezoelectricCNTRC plates (a/h¼10, h¼2 mm, Vn

CN ¼ 0:12, T¼300 K, hp/h¼10, Vs0 ¼ 0,Vd0 ¼ 500 V).

30 33 36 39 42 450

2

4

6

8

V*CN=0.12

V*CN=0.17

V*CN=0.28

Dimensionless frequency parameter(Ω)

Dim

ensi

onle

ss a

mpl

itude

of l

oadi

ng

Fig. 6. Effect of different volume fraction of carbon nanotubes Vn

CN on the unstableregions for the transverse mode of perfect piezoelectric FGX-CNTRC plates (a/h¼10,h¼2 mm, T¼300 K, hp/h¼10, Vs0 ¼ 0, Vd0 ¼ 500 V).

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5148

form of distribution of reinforcements can affect the stiffness of theplates and it is thus expected that the desired stiffness can beachieved by adjusting the distribution of CNTs along the thicknessdirection of plates. It is concluded that reinforcements distributedclose to top and bottom are more efficient than those distributednearby the mid-plane for increasing the stiffness of plates. Alsotemperature rise leads to smaller stable regions for the piezoelectricCNTRC plates.

The effect of initial geometric imperfection on the unstableregions for the transverse mode of piezoelectric FGX-CNTRC platesin the linear dynamic stability analysis is shown in Fig. 8 (a/h¼10,hp/h¼1/10, h¼2 mm, Vs0 ¼ 0 V, Vd0 ¼ 500 V, T¼300 K, η¼ 0:2H).It can be observed that the predeformation causes the instabilityregion to shift to the right in response to the increased platenatural frequency. The predeformation also increases slightly thewidth of the instability region.

Fig. 9 shows the stable and unstable solution amplitude of steady-state vibrations for the transverse mode of a perfect piezoelectric FGX-CNTRC plates (a/h¼10, h¼2mm, Vn

CN ¼ 0:12, T¼300 K, hp/h¼10,Vs0 ¼ 0 V, Vd0 ¼ 500 V). The stable solution is corresponding to

Χ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16ω2

mnL=3ð4βc�ω2eβdÞ

� � �1þ βþ ω2e=4ω

2mnL

� �� �rand unstable

solution is corresponding to

Χ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16ω2

mnL=3ð4βc�ω2eβdÞ

� � �1� βþ ω2e=4ω

2mnL

� �� �r.

Fig. 10 depicts the effect of different volume fraction of carbonnanotubes Vn

CN on the amplitude of steady-state vibrations for thetransverse mode of perfect piezoelectric FGX-CNTRC plates (a/h¼10,h¼2 mm, T¼300 K, hp/h¼10, Vs0 ¼ 0 V, Vd0 ¼ 500 V). It can be seenthat the increase of the CNT volume fraction Vn

CN yields an increaseof stable amplitude of deflection. As we discussed earlier, this is dueto the fact that containing more CNTs leads to a stiffer and strongerstructure than a lesser one.

Frequency–amplitude relationship for different distributions ofcarbon nanotubes (UD, FGO and FGX-CNTRCs) of piezoelectricCNTRC plates are depicted in Fig. 11. It can be observed that theamplitude of FGX CNTRC plate is smaller than that of UD-CNTRCone while those of FGO CNTRC plates are larger though these threetypes of plates have the same mass fraction of the CNT. Thisdimensionless amplitude of steady-state vibrations of the CNTRCplates are subject to the similar influence from distributions ofcarbon nanotubes (UD, FGO and FGX-CNTRCs) observed in theprevious discussion, however the variations of stable amplitude ofsteady-state vibrations exhibit opposite trends relative to thechanges of origin of point Ω.

30 33 36 39 42 450

2

4

6

8T = 300 KT = 350 K T = 400 K

Dimensionless frequency parameter (

Dim

ensi

onle

ss a

mpl

itude

of l

oadi

ng

Ω)

Fig. 7. Effects of temperature difference on the unstable regions for the transversemode of perfect piezoelectric CNTRC plates (a/h¼10, h¼2 mm, Vn

CN ¼ 0:12,hp/h¼10, Vs0 ¼ 0, Vd0 ¼ 500 V).

30 33 36 39 42 450

2

4

6

8 Imperfect Perfect

Dimensionless frequency parameter (

Dim

ensi

onle

ss a

mpl

itude

of l

oadi

ng

Ω)

Fig. 8. Effects of initial geometric imperfection on the unstable regions for thetransverse mode of piezoelectric FGX-CNTRC plates in linear dynamic stabilityanalysis (a/h¼10, hp/h¼1/10, h¼2 mm, Vs0 ¼ 0, Vd0 ¼ 500 V, T¼300 K, η¼ 0:2H).

37.0 37.5 38.0 38.5 39.0

-0.15

0.00

0.15

0.30

Dim

ensi

onle

ss v

ibra

tion

ampl

itude

Stable Unstable

Dimensionless frequency parameter (Ω)

Fig. 9. Stable and unstable solution amplitude of steady-state vibrations for thetransverse mode of a perfect piezoelectric FGX-CNTRC plates (a/h¼10, h¼2 mm,Vn

CN ¼ 0:12, T¼300 K, hp/h¼10, Vs0 ¼ 0, Vd0 ¼ 500 V).

0 10 20 30 40 50 60 70 80-2

-1

0

1

2

Dim

ensi

onle

ss v

ibra

tion

ampl

itude

(X/H

)

V*CN=0.12

V*CN=0.17

V*CN=0.28

Dimensionless frequency parameter (Ω)

Fig. 10. Effect of different volume fraction of carbon nanotubes Vn

CN on theamplitude of steady-state vibrations for the transverse mode of perfect piezo-electric FGX-CNTRC plates (a/h¼10, h¼2 mm, T¼300 K, hp/h¼10, Vs0 ¼ 0,Vd0 ¼ 500 V).

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–51 49

The effect of temperature rise on the amplitude of steady-statevibrations for the transverse mode of perfect piezoelectric CNTRCplates (a/h¼10, h¼2 mm, Vn

CN ¼ 0:12, hp/h¼10, Vs0 ¼ 0 V, Vd0 ¼500 V) is illustrated in Fig. 12. It can be seen that the temperature

rise yields a decrease of stable amplitude of deflection in earlystage after resonance and thereafter leads to an increase in stableamplitude of deflection.

In Fig. 13 the (resonant) principal parametric-subharmonicamplitude, X/H, is plotted versus dimensionless excitation fre-quency parameter, for values of predeformation of 0.0 and 0.2H (a/h¼10, h¼2 mm, Vn

CN ¼ 0:12, T¼300 K, hp/h¼10, Vs0 ¼ 0 V,Vd0 ¼ 500 V). It can be observed that predeformation yields adecrease of stable amplitude of deflection.

5. Conclusions

Non-linear parametric instability of initially imperfect piezo-electric carbon nanotube-reinforced composite plates under per-iodic in-plane load and subjected to a harmonic applied voltageand uniform temperature rise has been carried out within thecontext of FSDT theory, von Kármán type displacement–strainrelationship. Galerkin procedure is used to obtain a second ordernon-linear ordinary equation with cubic non-linear terms. Har-monic balance method is used to study the non-linear dynamicstability of the smart piezoelectric CNTRC plates. An extensiveparametric study has been carried out. Numerical results showthat the distributions of reinforcement, the volume fraction ofSWCNTs, initial geometric imperfection and temperature rise havea significant effect on the size of stability regions and amplitude ofsteady state oscillations. Plate predeformation introduces firstorder (linear) coupling of in-plane and lateral response. Thiscoupling results in externally excited lateral response for prede-formed plates subjected to in-plane loading. Moreover, predefor-mation leads to a small increase in the fundamental naturalfrequency of plate and generates a quadratic non-linearity describ-ing mid-plane stretching.

Acknowledgments

The work described in this paper was fully supported by aresearch grant from the City University of Hong Kong (Project no.7003034). The authors are grateful for the financial support.

Appendix A

ϕ1 ¼a2n2

32b2m2An

22

WðtÞðWðtÞþ2HηÞ ðA:1Þ

ϕ2 ¼b2m2

32a2n2An

11WðtÞðWðtÞþ2HηÞ ðA:2Þ

ϕ3 ¼m2n2π4ηWðtÞ

8a2b2=ðAn

22b4m4�2An

26ab3m3n

þ2An

12a2b2m2n2þAn

66a2b2m2n2

�2An

16a3bmn3þAn

11a4n4Þ ðA:3Þ

ϕ4 ¼m2n2π4ηWðtÞ

8a2b2=ðAn

22b4m4

þ2An

26ab3m3nþ2An

12a2b2m2n2þAn

66a2b2m2n2

þ2An

16a3bmn3þAn

11a4n4Þ ðA:4Þ

References

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0 10 20 30 40 50 60 70 80-2

-1

0

1

2

Dim

ensi

onle

ss v

ibra

tion

ampl

itude

(X/H

)

T = 400 KT = 350 KT = 300 K

Dimensionless frequency parameter (Ω)

Fig. 12. Effects of temperature difference on the amplitude of steady-state vibra-tions for the transverse mode of perfect piezoelectric CNTRC plates (a/h¼10,h¼2 mm, Vn

CN ¼ 0:12, hp/h¼10, Vs0 ¼ 0, Vd0 ¼ 500 V).

0 10 20 30 40 50 60 70 80-2

-1

0

1

2

Dim

ensi

onle

ss v

ibra

tion

ampl

itude

Perfect Imperfect

Dimensionless frequency parameter (Ω)

Fig. 13. Effect of initial geometric imperfection on the amplitude of steady-statevibrations for the transverse mode of piezoelectric CNTRC plates (a/h¼10,h¼2 mm, Vn

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0 10 20 30 40 50 60 70 80-2

-1

0

1

2

Dim

ensi

onle

ss v

ibra

tion

ampl

itude

(X/H

)

Dimensionless frequency parameter (Ω)

FGO-CNTRC UD-CNTRC FGX-CNTRC

Fig. 11. Effect of different distribution of carbon nanotubes (UD, FGO and FGXCNTRC) on the amplitude of steady-state vibrations for the transverse mode ofperfect piezoelectric CNTRC plates (a/h¼10, h¼2 mm, Vn

CN ¼ 0:12, T¼300 K,hp/h¼10, Vs0 ¼ 0, Vd0 ¼ 500 V).

M. Rafiee et al. / International Journal of Non-Linear Mechanics 59 (2014) 37–5150

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