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Nonlinear analysis of piezoelectric nanocomposite energy harvesting plates

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2014 Smart Mater. Struct. 23 065001

(http://iopscience.iop.org/0964-1726/23/6/065001)

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Smart Materials and Structures

Smart Mater. Struct. 23 (2014) 065001 (13pp) doi:10.1088/0964-1726/23/6/065001

Nonlinear analysis of piezoelectricnanocomposite energy harvesting plates

M Rafiee1,2, X Q He1 and K M Liew1

1 Department of Civil and Architectural Engineering, City University of Hong Kong, Tat Chee Avenue,Kowloon, Hong Kong2 Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Iran

E-mail: bcxqhe@cityu.edu.hk

Received 4 November 2013, revised 19 January 2014Accepted for publication 6 March 2014Published 10 April 2014

AbstractThis paper investigates the nonlinear analysis of energy harvesting from piezoelectricfunctionally graded carbon nanotube reinforced composite plates under combined thermal andmechanical loadings. The excitation, which derives from harmonically varying mechanicalin-plane loading, results in parametric excitation. The governing equations of the piezoelectricfunctionally graded carbon nanotube reinforced composite plates are derived based onclassical plate theory and von Karman geometric nonlinearity. The material properties of thenanocomposite plate are assumed to be graded in the thickness direction. The single-walledcarbon nanotubes (SWCNTs) are assumed to be aligned, straight and have a uniform layout.The linear buckling and vibration behavior of the nanocomposite plates is obtained in the firststep. Then, Galerkin’s method is employed to derive the nonlinear governing equations of theproblem with cubic nonlinearities associated with mid-plane stretching. Periodic solutions aredetermined by using the Poincare–Lindstedt perturbation scheme with movable simplysupported boundary conditions. The effects of temperature change, the volume fraction and thedistribution pattern of the SWCNTs on the parametric resonance, in particular the amplitudeof vibration and the average harvested power of the smart functionally graded carbon nanotubereinforced composite plates, are investigated through a detailed parametric study.

Keywords: nonlinear parametric resonance, buckling, vibration, piezoelectric, nanotubereinforced composite, energy harvesting

(Some figures may appear in colour only in the online journal)

1. Introduction

Energy harvesting from mechanical vibrations has becomeincreasingly important in recent years [1–3], mainly due tothe low power requirement of small electronic components,such as the wireless sensor networks used in monitoringapplications. Powering of small electronics using the vibra-tional energy available in their environment could reduce therequirement of external power sources, like batteries, thusavoiding the associated maintenance costs [1–3].

There are three basic vibration-to-electrical conversionmechanisms: electromagnetic, electrostatic and piezoelectric

transductions [3, 4]. Piezoelectric transduction has receivedthe greatest amount of attention when compared to the othertransduction systems due to its major advantages, such as largepower density, easy application (piezoelectric devices can befabricated on both the macro and the micro scale due to wellestablished production techniques), and the fact that the outputvoltage is obtained directly from the piezoelectric materialitself, which removes the requirement of an external voltageinput [1].

The energy conversion in piezoelectric materials is basedon variations in the dipolar moment when a strain is applied,

0964-1726/14/065001+13$33.00 1 c© 2014 IOP Publishing Ltd Printed in the UK

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

and therefore the formation of a potential difference which canbe used to power devices [3].

It is known that the use of single- or multi-walled car-bon nanotubes (SWCNTs or MWCNTs) as reinforcementsin composites significantly improves the mechanical prop-erties of those composites [5]. For instance, tensile tests ofmulti-walled carbon nanotube reinforced composites havedemonstrated that the addition of a small amount of thenano-scaled reinforcement results in remarkable increases inboth the elastic modulus and the breaking stress [6, 7]. Giventhe great engineering potential of composites reinforced withnanotubes under various loading conditions, their structuralbehavior has received extensive research attention in recentyears. Schadler et al [8] studied the mechanical behaviorof multi-walled carbon nanotube/epoxy composites in bothtension and compression. Bekyarova et al [9] reported anapproach to the development of advanced structural compos-ites based on engineered multiscale carbon nanotube–carbonfiber reinforcement. Nonlinear dynamic stability testing ofpiezoelectric functionally graded carbon nanotube reinforcedcomposite plates with initial geometric imperfections wasconducted by Rafiee et al [10]. Furthermore, the thermal bifur-cation buckling of carbon nanotube reinforced functionallygraded composite beams with piezoelectric layers was studiedby Rafiee et al [11]. Balamurugan and Narayanan [12–14]presented a series of studies to investigate active vibrationcontrol of multi-layer smart composite plates using the finiteelement method. In another study, Rafiee et al [15, 16] con-ducted a comprehensive analytical study to investigate thenonlinear vibration and dynamic response of simply supportedfunctionally graded material shells under combined electrical,thermal, mechanical and aerodynamic loading with piezoelec-tric layers. Rafiee et al [17–19] conducted several researchworks to study the nonlinear vibration of functionally gradedbeam structures. Rafiee and Kalhori [17] and Shooshtari andRafiee [18] studied the nonlinear free and forced vibration ofa functionally graded beam based on Euler–Bernoulli beamtheory and von Karman geometric nonlinearity under variousboundary conditions. Recently, Rafiee et al [19] presented aperturbation based multiple time scale analytical solution toinvestigate the nonlinear vibration of functionally graded CNT-reinforced beams with surface-bonded piezoelectric layersunder combined electrical, thermal and mechanical loading.

The parametric resonance of a structure subjected toperiodic axial compressive forces is a very important topicin structural dynamics and is of practical importance indifferent engineering disciplines. Periodic axial forces maycause parametric vibration, a phenomenon that is characterizedby unbounded growth of a small disturbance. It can be usedfor energy harvesting purposes; however, there are a limitednumber of relevant studies in the open literature. For instance,Daqaq et al [20] presented an analytical and experimentalinvestigation of energy harvesting via parametrically excitedcantilever beams. A global nonlinear distributed-parametermodel for a piezoelectric cantilever beam with a tip massenergy harvester under parametric excitation was developedby Abdelkefi et al [21]. The method of multiple scales wasused in their study to obtain analytical expressions for the tip

deflection, output voltage and harvested power near the firstprincipal parametric resonance. Zhu et al [22] investigatedthe design and analysis of a broadband piezoelectric energyharvester that used a simply supported piezoelectric beamcompressed by dynamic loading. A numerical model has beendeveloped by Jia et al [23] to explore the use of parametricresonance for MEMS applications.

To date, no previous work regarding the nonlinear energyharvesting analysis of piezoelectric carbon nanotube rein-forced composite plates has been conducted and reported inthe open literature.

A reduced-order model will be developed in this paperbased on Galerkin’s discretization of the governing equationswhich will derive from the extended Hamilton principleand Gauss law to investigate the output voltage and powerof a piezoelectric nanocomposite energy harvester underparametric resonance. A study of a piezoelectric carbonnanotube reinforced composite plate under periodic in-planeload and subjected to a uniform temperature rise will be carriedout within the context of classical plate theory, von Karmantype displacement–strain relationship. A Poincare–Lindstedtperturbation scheme method will be used to study the outputvoltage and harvested power of the piezoelectric functionallygraded carbon nanotube reinforced composite (FG-CNTRC)plates and an extensive parametric study will be carried out.

2. Problem formulation

Figure 1 shows a piezoelectric nanocomposite energy har-vesting plate with a coordinate plane (x, y) of length a,width b and thickness h with two surface-bonded piezoelectriclayers of thickness hp. The total thickness of the plate is H .The SWCNT reinforcement is either uniformly distributed(referred to as UD) or functionally graded in the thicknessdirection, referred to as FG, which includes FGO and FGX asshown in figure 1. Piezoelectric layers are symmetrically andperfectly bonded (neglecting the adhesive thickness) to the topand bottom surface of the CNTRC host and are assumed tohave temperature-independent material properties.

2.1. Material properties of the CNTRC host and piezoelectriclayers

The CNTRC material is made from a mixture of isotropicmatrix and SWCNTs with a graded distribution in the thicknessdirection. The constituent materials are linear elastic through-out the deformation. The plate is initially stress free at T0 (inkelvin) and is subjected to a uniform temperature variation1T = T –T0 and applied voltage V (t).

The effective material properties of the CNTRC plate canbe predicted by [10]

E11h = η1 VCN ECN11 + V h

m Emh

η2

E22h=

VCN

ECN22+

V hm

Emh

η3

G12h=

VCN

GCN12+

V hm

Gmh

(1)

where ECN11 , ECN

22 and GCN12 are the Young’s moduli and

shear modulus of the SWCNTs, respectively, and Emh and

2

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

UD-CNTRC

FGX-CNTRC

FGO-CNTRCz

z

y

H h

x

hp

hp

Piezoelectric layer

nanocomposite host

R

R

I(t) V(t)

V(t)

Cpiezo

(a)

(b)

(c)(e)

(d)

Figure 1. Configuration of the bimorph piezoelectric carbon nanotube reinforced composite plates: (a) FGO-CNTRC plate, (b)FGX-CNTRC plate, (c) UD-CNTRC plate, (d) material composition and other parameters through the thickness, (e) equivalent electricalcircuit of the harvesting system.

Gmh represent the corresponding properties of the isotropic

matrix of the host. To account for the scale-dependent materialproperties, η j ( j = 1, 2, 3), the CNT efficiency parameters,calculated by matching the effective properties of the CNTRCobtained from the MD simulations with those from the ruleof mixture, are introduced. VCN and V h

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

The uniform and two functionally graded distributions ofthe carbon nanotubes along the thickness direction are depictedin figure 1 and are assumed to be as follows:

VCN =

V ∗CN (UD-CNTRC)

2(

1−2 |z|

h

)V ∗CN (FGO-CNTRC)

2(

2 |z|h

)V ∗CN (FGX-CNTRC)

(2)

where

V ∗CN =wCN

wCN+ (ρCN/ρmh )− (ρCN/ρ

mh )wCN

(3)

and wCN is the mass fraction of the SWCNTs, ρCN andρm are the mass densities of the carbon nanotubes andmatrix, respectively. The thermal expansion coefficients in thelongitudinal and transverse directions can be expressed as

α11h = VCNαCN11 + V h

mαmh (4a)

α22h = (1+ νCN12 )VCNα

CN22 + (1+ ν

mh )V

hmα

mh − ν12hα11h (4b)

whereαCN11 ,αCN

22 andαmh are the thermal expansion coefficients,

and νCN12 and νm

h are the Poisson’s ratios, respectively, of the

carbon nanotubes and matrix. The Poisson’s ratio and massdensity ρ can be calculated by

ν12h = V ∗CNνCN12 + V h

mνmh ρh = VCNρ

CN+ V h

mρmh (5)

where νCN12 and νm

h are the Poisson’s ratios of the carbonnanotubes and matrix, respectively. It is assumed that thematerial properties of the nanotubes and the matrix arefunctions of the temperature, so that the effective materialproperties of the CNTRC, like the Young’s modulus, shearmodulus and thermal expansion coefficients, are functions oftemperature and position.

2.2. Displacement field model

The displacement field for the plate based on classical platetheory (CPT) is assumed as

u(x, y, z)= u(x, y)− z∂w(x, y)∂x

v(x, y, z)= v(x, y)− z∂w(x, y)∂y

w(x, y, z)=w(x, y)

(6)

where, u, v and w denote the displacements of a point alongthe (x , y, z) coordinates. u, v and w are the correspondingdisplacements of a point on the mid-plane. ψx and ψy arethe rotations of the normal to the mid-plane about the y-axisand x-axis, respectively. The nonlinear strain–displacementrelationships of a uniform plate undergoing large deflections

3

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

areεx

εy

γxy

=ε0

x

ε0y

γ 0xy

+ z

ε1

x

ε1y

γ 1xy

=

∂u∂x+

12

(∂w

∂x

)2

∂v

∂y+

12

(∂w

∂y

)∂u∂y+∂v

∂x+∂w

∂x∂w

∂y

+ z

−∂2w(x, y)∂x2

−∂2w(x, y)∂y2

−2∂2w(x, y)∂x∂y

. (7)

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 asσx

σy

σxy

=Q11 Q12 0

Q12 Q22 00 0 Q66

εx

εy

γxy

−α11

α22

0

1T

−

[0 0 e310 0 e320 0 0

]{ 00Ez

}Dx

Dy

Dz

= 0 0 0

0 0 0e31 e32 0

{ εxεyγxy

}−

α11

α22

0

1T

+

[ε11 0 00 ε22 00 0 ε33

]{ 00Ez

}(8)

where {σ } is the stress vector, {ε} is the strain–displacementvector, {E} is the electrical field vector, {D} is the electricaldisplacement vector, [Q] is the elastic stiffness matrix, {α} isthe thermal expansion coefficient vector, [e] is the piezoelectricmatrix and [ε] is the permittivity matrix. Qi j are the planestress-reduced stiffnesses defined in terms of the engineeringconstants in the material axes of the layer as

Q11 = Q11h =E11h

1− ν12hν21h

Q12 = Q12h =ν12h E22h

1− ν12hν21h

Q22 = Q22h =E22h

1− ν12hν21hQ44 = Q44h =G23h Q55 = Q55h =G13h

Q66 = Q66h =G12h

(9)

for the CNTRC host, and

Q11 = Q11p = Q22 = Q22p =E11p

1− ν2p

Q12 = Q12h =νp E11p

1− ν2p

Q44 = Q55 = Q66 =G12p =E11p

2(1+ νp

)(10)

for the piezoelectric layers. It should be noted that the thermalexpansion coefficients of the piezoelectric layers are α11p =

α22p, and νp and ρp are the Poisson’s ratio and mass densityof the piezoelectric layers, respectively.

A linear distribution of the electric potential throughthe thickness direction is assumed here. For the plate typepiezoelectric material, only the thickness direction electricfield Ez is dominant, and Ez is defined as Ez =−ϒ,z , whereϒis the potential field. The thickness of the piezoelectric layersis very small, so the relationship between the applied voltageV (t) and the electric field intensity within a piezoelectricactuator can be described as [15, 16, 19]

Ez =V (t)X1hp

(11)

and X1 is either 1 for bimorph with piezoelectric layers inparallel or 2 for bimorph with piezoelectric layers in series.

2.4. Equations of motion

By using Hamilton’s principle, three equations of motion areobtained as [26]

∂Nx

∂x+∂Nxy

∂y= I0

∂2u∂t2 − I1

∂2

∂t2

(∂w

∂x

)∂Nxy

∂x+∂Ny

∂y= I0

∂2v

∂t2 − I1∂2

∂t2

(∂w

∂y

)∂2 Mx

∂x2 + 2∂2 Mxy

∂y∂x+∂2 My

∂y2 +∂

∂x

(Nx∂w

∂x+ Nxy

∂w

∂y

)+

∂

∂y

(Nxy

∂w

∂x+ Ny

∂w

∂y

)+ q (x, y, t)= I0

∂2w

∂t2

+ 2ς I0∂w

∂t− I2

∂2

∂t2

(∂2w

∂x2 +∂2w

∂y2

)+ I1

∂2

∂t2

(∂w

∂x+∂w

∂y

)

(12)

where q is the applied transverse external excitation. I0, I1 andI2 are the mass moments of inertia and can be expressed as{I0

I1I2

}=

∫ H/2

−H/2

{ 1zz2

}ρ dz. (13)

The force, moment and transverse shear force resultantsper unit length of the plate expressed in terms of the stressthrough the thickness are

Nx =

∫ h/2

−h/2σ h

xx dz+∫ h/2+hp

h/2σ

pxx dz

+

∫−h/2

−(h/2+hp)σ

pxx dz (14a)

Ny =

∫ h/2

−h/2σ h

yy dz+∫ h/2+hp

h/2σ

pyy dz

+

∫−h/2

−(h/2+hp)σ

pyy dz (14b)

4

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

Nxy =

∫ h/2

−h/2σ h

xy dz+∫ h/2+hp

h/2σ

pxy dz

+

∫−h/2

−(h/2+hp)σ

pxy dz (14c)

Mx =

∫ h/2

−h/2zσxx dz+

∫ h/2+hp

h/2zσ p

xx dz

+

∫−h/2

−(h/2+hp)zσ p

xx dz (15a)

My =

∫ h/2

−h/2zσyy dz+

∫ h/2+hp

h/2zσ p

yy dz

+

∫−h/2

−(h/2+hp)zσ p

yy dz (15b)

Mxy =

∫ h/2

−h/2zσxy dz+

∫ h/2+hp

h/2zσ p

xy dz

+

∫−h/2

−(h/2+hp)zσ p

xy dz. (15c)

Substituting equations (7) and (8) into equations (14)and (15) gives the expressions of force and moment resultantsas

Nx

Ny

Nxy

Mx

My

Mxy

=

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

ε0x

ε0y

γ 0xy

ε1x

ε1y

γ 1xy

−

N Tx

N Ty

N Txy

MTx

MTy

MTxy

−

N Px

N Py

N Pxy

MPx

MPy

MPxy

−

N Mx

N My

0000

. (16)

The superscripts ‘T’, ‘P’ and ‘M’ in equation (16) repre-sent the thermal, electric and mechanical loads, respectively,and Ai j , Bi j and Di j (i, j = 1, 2, 6) in equation (16) are thestretching stiffness, stretching–bending coupling stiffness andbending stiffness coefficients, respectively, which are definedas [27]

(Ai j , Bi j , Di j )=

∫ H/2

−H/2Qi j

(1, z, z2

)dz

(i, j = 1, 2, 6). (17)

The stretching–bending coupling effect will not appearfor a symmetric distribution of the carbon nanotubes along thethickness direction of the nanocomposite plates.

The thermal and electrical force and moment resultantsare defined as

N Tx MT

x

N Ty MT

y

N Txy MT

xy

=∫ h/2

h/2

Q11h Q12h 0Q12h Q22h 0

0 0 Q66h

α11h

α22h

0

× (1, z)1T dz+

∫ H/2

h/2

Q11p Q12p 0Q12p Q11p 0

0 0 Q66p

×

α11p

α11p

0

(1, z)1T dz

+

∫−h/2

−H/2

Q11p Q12p 0Q12p Q11p 0

0 0 Q66p

×

α11p

α11p

0

(1, z)1T dz (18a)

N P

x MPx

N Py MP

y

N Pxy MP

xy

=∫ H/2

h/2

e31

e32

0

(1, z)V (t)X1hp

dz

+

∫−h/2

−H/2

e31

e32

0

(1, z)V (t)X1hp

dz. (18b)

The in-plane load N M(t) is taken as

N Mx = (Psx + Pdx cosωet) N M

xcr (19a)

N My =

(Psy + Pdy cosωet

)N M

ycr (19b)

where N Mxs0 and N M

xd0 are constant static and dynamic compo-nents of the in-plane excitation in the x direction, respectively.N M

ys0 and N Myd0 have similar definitions.

For the case of a symmetric distribution of the carbonnanotubes along the thickness direction of the nanocompositeplates, I1 will be zero and terms containing it will be omitted.With an acceptable accuracy, neglecting all the inertia termson u and v in equation (12) since their influences are smallcompared to that of the transverse inertia term, one may obtain

∂Nx

∂x+∂Nxy

∂y= 0

∂Nxy

∂x+∂Ny

∂y= 0

∂2 Mx

∂x2 + 2∂2 Mxy

∂y∂x+∂2 My

∂y2 +∂

∂x

(Nx∂w

∂x+ Nxy

∂w

∂y

)+

∂

∂y

(Nxy

∂w

∂x+ Ny

∂w

∂y

)+ q (x, y, t)

= I0∂2w

∂t2 + 2ς I0∂w

∂t− I2

∂2

∂t2

(∂2w

∂x2 +∂2w

∂y2

).

(20)

Introducing the Airy stress function 8 as

Nx =∂28

∂y2 Ny =∂28

∂x2 Nxy =−∂28

∂x∂y(21)

5

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

which satisfies the first two equations of equation (20), reducesthe equations of motion to

∂2 Mx

∂x2 + 2∂2 Mxy

∂y∂x+∂2 My

∂y2 +∂

∂x

(Nx∂w

∂x+ Nxy

∂w

∂y

)+

∂

∂y

(Nxy

∂w

∂x+ Ny

∂w

∂y

)+ q (x, y, t)

= I0∂2w

∂t2 + 2ς I0∂w

∂t− I2

∂2

∂t2

(∂2w

∂x2 +∂2w

∂y2

). (22)

Along with a compatibility equation of the form

∂2ε0x

∂y2 +∂2ε0

y

∂x2 −∂2γ 0

xy

∂x∂y=

(∂2w

∂x∂y

)2

−∂2w

∂x2∂2w

∂y2 (23)

equation (16) can be written in the alternative form

{ε0

M

}=

[A∗ B∗

−(B∗)T D∗

]{Nκ

}(24)

where

A∗ =A−1 B∗ =−A−1B D∗ =D−BA−1B(25)

and

{ε0} =

ε0

x

ε0y

γ 0xy

{κ} =

ε1

x

ε1y

γ 1xy

. (26)

Substitution of equations (24) and (26) into equations(22)–(23) gives the equations of motion of symmetric CNT-reinforced composite plates (and identically, the compatibilityequation) in terms of displacements and the stress function

∂2

∂x2

(D11

∂2w

∂x2 + D12∂2w

∂y2

)+ 4D66

∂4w

∂x2∂y2

+∂2

∂y2

(D12

∂2w

∂x2 + D22∂2w

∂y2

)−∂28

∂y2

(∂2w

∂x2

)−∂28

∂x2

(∂2w

∂y2

)+ 2

∂28

∂x∂y

(∂2w

∂x∂y

)− q (x, y, t)

+ I0∂2w

∂t2 + 2ς I0∂w

∂t− I2

∂2

∂t2

(∂2w

∂x2 +∂2w

∂y2

)= 0 (27a)

A∗22∂48

∂x4 + (2A∗12+ A∗66)∂48

∂x2∂y2 + A∗11∂48

∂y4

=

(∂2w

∂x∂y

)2

−∂2w

∂x2∂2w

∂y2 . (27b)

3. Analytical solution approach

The boundary conditions for simply supported rectangularplates when four edges of the plate are simply supported and

freely movable (FM) are

w= Nxy = Mx = 0 Nx = Nx0 at x = 0, a (28a)

w= Nxy = My = 0 Ny = Ny0 at y = 0, b. (28b)

The in-plane boundary conditions for the movable edgesare ∫ a

0{Ny}y=0,b dx = 0 (29a)∫ a

0{Nxy}y=0,b dx = 0 (29b)∫ b

0{Nx }x=0,a dy = 0 (29c)∫ b

0{Nxy}x=0,a dy = 0. (29d)

These boundary conditions are exactly satisfied by thefollowing assumptions:

w=W (t)Y (x, y)=W (t) sin(mπx

a

)sin(nπy

b

)(30)

where m, n = 1, 2, 3, . . . are the numbers of half waves in thex and y directions, respectively.

By substituting equation (30) into equation (27b) andsolving it, the Airy stress function 8 is obtained as

8=8C+8P. (31)

When the expansion of w is substituted in the right-hand side of equation (27b), a partial differential equationfor the Airy stress function 8 is obtained; the particularsolution can be determined by incorporating equation (30)into equation (27b) and comparing coefficients of the sameharmonic components as

8P =W 2(t)2 (x, y)

=W 2(t)(φ1 cos 2

(mπxa

)+φ2 cos 2

(nπyb

))(32)

where φi (i = 1, 2) have the following expressions:

φ1 =a2n2

32b2m2 A∗22(33a)

φ2 =b2m2

32a2n2 A∗11. (33b)

The complementary solution 8C will now be obtainedsuch that it satisfies the in-plane boundary conditions ofequation (29). The complementary solution of the Airy stressfunction is assumed as

8C =−12 Nx0 y2

−12 Ny0x2

− Nxy xy. (34)

6

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

Upon using equation (30) and enforcing the conditions ofequations (28), the unknown coefficients may be obtained forthe freely movable (FM) edges as

Nx0 = − (Psx + Pdx cosωet) N Mxcr− N T

x − N Px (35a)

Ny0 = −(Psy + Pdy cosωet

)N M

ycr− N Ty − N P

y (35b)

Nxy = 0. (35c)

3.1. Nonlinear parametric resonance analysis of perfect plates

A second equation can be found in terms of the statesw and V .The total charge displaced qc(t) can be written as the integralof the electrical displacement over the surface area of theelectrodes

qc(t)=∫

As

Dz dAs. (36)

By differentiating qc(t) with respect to the time variablet , we get the measured electric current I (t) in the form

I (t)= qc(t)

= e31∂

∂t

(A∗11

(∂28P

∂y2

)+ A∗12

(∂28P

∂x2 +∂28P

∂y2

)+ A∗22

(∂28P

∂x2

))−

e31 H2

∂

∂t

(∂2w

∂x2 +∂2w

∂y2

)+ ε33

V (t)X1hp

=−V (t)

R. (37)

Note that e32 = e31.By substituting equations (30)–(31) into equations (27a)

and (37), multiplying the resulting equation by sin(mπxa ) sin

( nπyb ) and integrating over the domain we obtain the following

nonlinear ordinary differential equation for the dynamics ofpiezoelectric CNTRC plates:

J0W + J1W + J2W + J3W 3+ JvV W

+ JdW cos (ωet)= 0 (38a)

Z0V + Z1V + Z2W + Z3W W = 0 (38b)

where

Z0 =

∫ a

0

∫ b

0

Y ε33

X1hpdy dx Z1 =

∫ a

0

∫ b

0

YR

dy dx

Z2 =−e31 H

2

∫ a

0

∫ b

0

(∂2Y∂x2 +

∂2Y∂y2

)Y dy dx

Z3 = 2e31

∫ a

0

∫ b

0

(A∗11

(∂22

∂y2

)+ A∗12

(∂22

∂x2 +∂22

∂y2

)+ A∗22

(∂22

∂x2

))Y dy dx

(39)

J0 =

∫ a

0

∫ b

0

(I0Y − I2

(∂2Y∂x2 +

∂2Y∂y2

))Y dy dx

J1 =

∫ a

0

∫ b

0

(∂2

∂x2

(D11

∂2Y∂x2 + D12

∂2Y∂y2

)+ 4D66

∂4Y∂x2∂y2 +

∂2

∂y2

×

(D12

∂2Y∂x2 + D22

∂2Y∂y2

))Y dy dx

+

∫ a

0

∫ b

0

((Psx N M

xcr+ N Tx )

×

(∂2Y∂x2

)+ (Psy N M

ycr+ N Ty )

(∂2Y∂y2

))Y dy dx

J2 =

∫ a

0

∫ b

02ς I0Y 2 dy dx

J3 =−

∫ a

0

∫ b

0

(∂28P

∂y2

(∂2Y∂x2

)+∂28P

∂x2

(∂2Y∂y2

)− 2

∂28P

∂x∂y

(∂2Y∂x∂y

))Y dy dx

Jd =−

∫ a

0

∫ b

0

{Pdx N M

xcr

(∂2Y∂x2

)+ Pdy N M

ycr

(∂2Y∂y2

)}Y dy dx

Jv =−2

X1

∫ a

0

∫ b

0

(e31

(∂2Y∂x2

)+ e32

(∂2Y∂y2

))Y dy dx

(40)

where a superscript dot denotes differentiation with respect totime, and Ji (i = 1, 2, 3, d, v) and Zi (i = 0, 1, 2, 3) are theconstant coefficients obtained by applying Galerkin’s method.

3.1.1. Linear buckling analysis of smart nanocomposite plates.By neglecting the contribution of time-differentiated andnonlinear terms in equation (38a), the critical buckling loadcan be determined using∫ a

0

∫ b

0

(∂2

∂x2

(D11

∂2Y∂x2 + D12

∂2Y∂y2

)+ 4D66

∂4Y∂x2∂y2

+∂2

∂y2

(D12

∂2Y∂x2 + D22

∂2Y∂y2

))Y dy dx

+

∫ a

0

∫ b

0

((Psx N M

xcr+ N Tx

)(∂2Y∂x2

)+

(Psy N M

ycr+ N Ty

)(∂2Y∂y2

))Y dy dx = 0. (41)

Equation (41) may be used to determine the criticaltemperature difference and in-plane mechanical load as

1T =−{∫ a

0

∫ b

0

(∂2

∂x2

(D11

∂2Y∂x2 + D12

∂2Y∂y2

)+ 4D66

∂4Y∂x2∂y2 +

∂2

∂y2

×

(D12

∂2Y∂x2 + D22

∂2Y∂y2

))Y dy dx

7

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

+

∫ a

0

∫ b

0

(Psx N M

xcr

(∂2Y∂x2

)+ Psy N M

ycr

(∂2Y∂y2

))Y dy dx

} /{∫ a

0

∫ b

0

(∫ h/2

h/2(Q11hα11h+ Q12hα22h) dz

+ 2∫ H/2

h/2

(Q11p+ Q12h

)α11p dz

)(∂2Y∂x2

)Y dy dx

+

∫ a

0

∫ b

0

(∫ h/2

h/2(Q21hα11h+ Q22hα22h) dz

+ 2∫ H/2

h/2

(Q21p+ Q22h

)α11p dz

)(∂2Y∂y2

)Y dy dx

}.

(42)

The critical axial buckling load (Psy = 0) corresponds to

N Mxcr =

{∫ a

0

∫ b

0

(∂2

∂x2

(D11

∂2Y∂x2 + D12

∂2Y∂y2

)+ 4D66

∂4Y∂x2∂y2 +

∂2

∂y2

(D12

∂2Y∂x2 + D22

∂2Y∂y2

))Y dy dx

+

∫ a

0

∫ b

0

(N T

x

(∂2Y∂x2

)+ N T

y

(∂2Y∂y2

))Y dy dx

}/∫ a

0

∫ b

0

(∂2Y∂x2

)Y dy dx . (43)

The corresponding expression may be derived whenPsx = 0 for the transverse buckling load,

N Mycr =

{∫ a

0

∫ b

0

(∂2

∂x2

(D11

∂2Y∂x2 + D12

∂2Y∂y2

)+ 4D66

∂4Y∂x2∂y2 +

∂2

∂y2

(D12

∂2Y∂x2 + D22

∂2Y∂y2

))Y dy dx

+

∫ a

0

∫ b

0

(N T

x

(∂2Y∂x2

)+ N T

y

(∂2Y∂y2

))Y dy dx

}/∫ a

0

∫ b

0

(∂2Y∂y2

)Y dy dx . (44)

Alternatively, the load Psy N Mycr may be assumed to be

equal to Psx N Mxcr until buckling takes place, with

Ps N Mcr =

{∫ a

0

∫ b

0

(∂2

∂x2

(D11

∂2Y∂x2 + D12

∂2Y∂y2

)+ 4D66

∂4Y∂x2∂y2 +

∂2

∂y2

(D12

∂2Y∂x2 + D22

∂2Y∂y2

))Y dy dx

+

∫ a

0

∫ b

0

(N T

x

(∂2Y∂x2

)+ N T

y

(∂2Y∂y2

))Y dy dx

}/∫ a

0

∫ b

0

(∂2Y∂x2 +

∂2Y∂y2

)Y dy dx (45)

where N Mcr is the critical mechanical load and Ps is the static

factor of the critical mechanical load. In this study, we considerthe case of equation (45) for all the mechanical loadingconditions unless otherwise specified.

3.1.2. Linear vibration analysis of smart nanocomposite plates.From equation (38a) the fundamental frequencies of naturalvibration of the piezoelectric CNTRC plates can be determinedby the relation [17–25]

ωmnL =

√J1

J0. (46)

3.1.3. Nonlinear dynamic instability analysis of smart nanocom-posite plates. Equations (38a) can be rewritten in the follow-ing form [24]:

W +ω2mnLW + εK1W + εK2W 3

+ εKdW cosωet+ εKvW V = 0 (47)

V + S0V + S1W + S2W W = 0 (48)

where

ε= (a/h)2 K1 =J2

J0 (a/h)2

K2 =J3

J0 (a/h)2Kd =

Jd

J0 (a/h)2

Kv =Jv

J0 (a/h)2, S0 =

Z1

Z0

S1 =Z2

Z0S2 =

Z3

Z0.

(49)

To perform the method of Poincare–Lindstedt, one assumesthat

τ −ϕ

2=ωe

2t (50)

W (τ ; ε) =W0 (τ )+ εW1 (τ )+ ε2W2 (τ ) . . . . (51)

For the case of the principal parametric resonance, it issupposed that the frequency of excitation ωe and the linearfrequency of the system are close together as ωe ≈ 2ωL.Therefore, a detuning parameter σ is used to show the nearnessof ωe to 2ωL as

ωe = 2ωL+ εσ. (52)

By substituting equations (50)–(51) into equations (47)–(48) and equating the coefficients of ε0 and ε1 to zero, oneobtains

d2

dτ 2 W0+W0 = 0 (53)

ddτ

V +S0

ωmnLV =−S1

ddτ

W0− S2W0d

dτW0 (54)

and

d2

dτ 2 W1+W1 =−1

ω2mnL

(KvV + Kd cos (2τ −ϕ))W0

− σd2

dτ 2 W0−K1

ωmnL

ddτ

W0−K2

ω2mnL

W 30 . (55)

With this approach it turns out to be convenient to writethe solution of equation (53) as

W0 (τ )= A0 sin (τ ) (56)

8

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

where A0 is an unknown. Substitution of equation (56) intoequation (54) leads to

ddτ

V +S0

ωmnLV =−A0 (S1+ A0S2 sin (τ )) cos(τ ). (57)

The steady-state solution of equation (57) can be obtainedas

V (τ ) =12

A0ωmnL

(−

2S1 (S0 cos τ +ωmnL sin τ)S2

0 +ω2mnL

+A0S2 (2ωmnL cos 2τ − S0 sin 2τ)

S20 + 4ω2

mnL

). (58)

Substitution of equations (56) and (58) into equation (55)and elimination of the terms with sin τ and cos τ on theright-hand side which are resonant leads to

K 2d =

1411

{ω2

mnL

(−4

(S2

0 + 4ω2mnL

) ((S2

0 +ω2mnL

)S1

)+ A2

0S0S2

(S2

0 +ω2mnL

)Kv

)2

+

((S2

0 + 4ω2mnL

) (S2

0 +ω2mnL

) (4σω2

mnL− 3A20 K2

)+ 2A2

0S2ω2mnL

(S2

0 +ω2mnL

)Kv

)2}

(59)

where 11 =(S2

0 +ω2mnL

)2 (S20 + 4ω2

mnL)2. By solving equa-

tion (59) for A0, we obtain the amplitude vibration of thezeroth-order approximate solution

A20 =

112

(L1± 2

√L2

)(60)

where

L1 = 12σK2ω2mnL

(S2

0 + 4ω2mnL

)+ 4KvS2ω

2mnL

(K1S0− 2σω2

mnL)

L2 = 64ω4mnL

(3σK2

(S2

0 + 4ω2mnL

)+ KvS2

(K1S0− 2σω2

mnL))2

− 16(S2

0 + 4ω2mnL

)×(9K 2

2 S20 + (−6K2+ KvS2)

2 ω2mnL

)×(−K 2

d + 4ω2mnL

(K 2

1 + σ2ω2

mnL))

12 = 9K 22 S2

0 + (KvS2− 6K2)2 ω2

mnL.

(61)

The remaining terms on the right-hand side of equa-tion (55) are

<1+<21 sin 2τ +<22 cos 2τ +<311 sin 3τ

+ <312 sin (3τ +ϕ)+<32 cos 3τ (62)

where

<1 =A2

02

(KvS1

S20 +ω

2mnL

)

<21 =A2

0S0(S2

0 KvS1+ 4KvS1ω2mnL

)2ωmnL

(S4

0 + 5S20ω

2mnL+ 4ω4

mnL)

<22 =A2

02

(−

KvS1

S20 +ω

2mnL

)

<311 =A3

04

(K2

ω2mnL−

2KvS2

S20 + 4ω2

mnL

)

<312 =−A0

2

(Kd

ω2mnL

)

<32 =−A30

(KvS0S2

4ωmnL(S2

0 + 4ω2mnL

)) .

(63)

The next step is to solve equation (55) for W1 retainingonly the particular solution to obtain

W1 =<1−13<21 sin 2τ − 1

3<22 cos 2τ − 18<311 sin 3τ

−18<312 sin 3τ − 1

8<32 cos 3τ. (64)

The substitution for W0 and W1 from equations (56)and (64) into equation (51) and replacing A by its polar formand T0 by τ gives

W (τ )= A0 sin τ +<1−13<21 sin 2τ − 1

3<22 cos 2τ

−18<311 sin 3τ − 1

8<312 sin 3τ − 18<32 cos 3τ. (65)

The power generated is then represented by

J =V 2 (τ )

R(66)

and the average harvested power can be given by

Jave =1

2π

∫ 2π

0

V 2 (τ )

Rdτ. (67)

4. Numerical results and discussion

This study has been focused mainly on the various char-acteristic features of the phenomenon concerning the non-linear dynamic behavior of piezoelectric energy harvestingplates. Extensive numerical data are provided to investigatethe nonlinear analysis of CNTRC energy scavenger platesunder thermal loadings. The material properties and effectivethickness of the SWCNTs used for analysis are properlychosen in this paper by MD simulations. Unless otherwisestated, η1 = 0.137, η2 = 1.022 and η3 = 0.715 for the caseof V ∗CN = 0.12; η1 = 0.142, η2 = 1.626 and η3 = 1.138 forthe case of V ∗CN = 0.17; and η1 = 0.141, η2 = 1.585 andη3 = 1.109 for the case of V ∗CN = 0.28 [19]. Here, temperature-dependent (TD) PmPV material is considered. The parame-ters used are ρm

= 1150 kg m−3, νm= 0.34, αm

= 45(1+

9

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

Table 1. The temperature-dependent material properties for the (10, 10) SWCNTs (L = 9.26 nm, R = 0.68 nm, h = 0.067 nm, vcnt12 = 0.175,

pCN= 1400 kg m−3) (from [19]).

Temperature (K) ECN11 (TPa) ECN

22 (TPa) GCN12 (TPa) αCN

11 (×10−6 K−1) αCN12 (×10−6 K−1)

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 2. Comparison of the linear natural frequencies

ωmnL =ωmnLa2

H

√ρ0E0

for an unstressed UD-CNTRC square plate

(a/h = 100, hp/h = 0, T = 300 K, V ∗CN = 0.17).

SourceDistribution HSDT [31] Present-CPT Error (%)

UD 21.6989 21.8114 0.51

0.00051T )× 10−6 K−1 and Em= (3.52− 0.0034 T) GPa,

in which T = T0+1T and T0 = 300 K (room temperature).The temperature-independent (TID) material properties of theCNTRC host are αm

= 45× 10−6 K−1 and Em= 2.5 GPa

at T = 300 K. (10, 10) SWCNTs are selected as reinforce-ments. The temperature-dependent material properties of theSWCNTs are listed in table 1. The temperature-independentmaterial properties of the piezoelectric material are E11p =

63.0 GPa, ρp = 7600 kg m−3, α11p = α22p = 0.9× 10−6 K−1,υp = 0.3 and e31 = e32 = 17.6 C m−2. Other parameters areconsidered as ζz = 0.02, R = 25× 106, Ps = 0, Pd = 0.1,N M= 1.652× 106 N m−1 and ϒ = 2 unless otherwise speci-

fied.

4.1. Validation study

As part of the validation of the present method in terms ofnatural frequency, a comparison of the linear natural frequen-cies ωmnL =

ωmnLa2

H

√ρ0E0

for an unstressed UD-CNTRC squareplate is made and is shown in table 2 (a/h = 100, hp/h = 0,T = 300 K, V ∗CN = 0.17). Very good agreement between theresults can be observed.

In another validation study, an unstressed eight-layer(P/90◦/0◦/90◦/0◦/0◦/90◦/0/90◦/P) symmetric cross-ply plateunder simply supported boundary conditions is considered.This example is chosen as some comparative results areavailable in the literature [28–30]. All the layers have the samethickness and all the properties of the laminated plate are takenfrom [28] and are given below:

E11 = 181.0 GPa, E22 = 10.3 GPa, G12 = G13 =

7.17 GPa, G23 = 2.87 GPa, υ12 = 0.28, ρ = 1580 kg m−3,a = b= 1 m, a/h = 100, hp/h = 1/20.

The fundamental frequencyωmnL (rad s−1) for this plate iscalculated and compared in table 3 with the three-dimensional(3D) solution of Xu et al [28], the FSDPT solution ofBenjeddou et al [29] and the HSDT solution of Huang andShen [30].

Table 3. Comparison of the linear fundamental frequencyωmnL (rad s−1) for an unstressed eight-layer(PZT-5A/90◦/0◦/90◦/0◦/0◦/90◦/0/90◦/PZT-5A) symmetriccross-ply plate (E11 = 181.0 GPa, E22 = 10.3 GPa,G12 =G13 = 7.17 GPa, G23 = 2.87 GPa,ν12 = 0.28,ρ = 1580 kg m−3, a = b= 1 m, hp/h = 1/20).

3D [28] 2D-FSDT [29] HSDT [30]

Source 268.86 283.93 269.12Present 265.99 265.99 265.99Error (%) 1.079 6.745 1.177

Table 4. Effects of piezoelectric to CNTRC thickness ratio (hp/h),different distribution patterns of SWCNTs and volume fraction ofSWCNTs on the buckling temperature (Tcr =1Tcr+ T0) ofCNTRC square plates under uniform temperature rise (a/h = 50).

V ∗CN

Distribution hp/h 0.12 0.17 0.28

FGO-CNTRC 0 304.826 305.207 306.0021/10 321.840 317.703 315.4561/5 345.262 334.972 328.556

UD-CNTRC 0 308.463 308.96 310.0831/10 324.804 320.727 318.4871/5 347.325 337.008 330.15

FGX-CNTRC 0 312.782 313.824 316.731/10 329.729 326.267 326.1341/5 353.086 343.484 339.185

4.2. Buckling, vibration and harvested power results

The influence of the piezoelectric to CNTRC thickness ratio(hp/h), different distribution patterns of the SWCNTs and thevolume fraction of the SWCNTs on the buckling temperature(Tcr = 1Tcr + T0) of CNTRC square plates under uniformtemperature rise (a/h = 50) is investigated and is listed intable 4. The thickness of the CNTRC host is kept constantwhile the thickness of the piezoelectric layers is varied. It canbe seen that the piezoelectric layers can drastically increasethe critical temperature of the CNTRC plates. Also, it canbe found that the buckling temperature of a CNTRC plate isincreased by an increase in the nanotube volume fraction, anda plate with an intermediate nanotube volume fraction willnot have an intermediate buckling load. In contrast, for thecase of piezoelectric CNTRC plates, the buckling temperaturedecreases with increase in nanotube volume fraction. Thedifference between the thermal expansion coefficient of theCNTRC host and that of the piezoelectric layers is the reason.

10

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

Figure 2. Effect of the different distribution patterns of SWCNTs onthe amplitude of steady-state vibrations for the transverse mode ofpiezoelectric CNTRC plates (a/h = 30, h = 2 mm, V ∗CN = 0.12,T = 300 K, hp/h = 5).

Table 5. Effects of piezoelectric to CNTRC thickness ratio (hp/h),different distribution patterns of SWCNTs and volume fraction ofSWCNTs on the dimensionless fundamental natural frequency

(ωmnL =ωmnLa2

H

√ρ0E0) of CNTRC square plates under uniform

temperature rise (a/h = 50, T = 300 K).

V ∗CN

Distribution hp/h 0.12 0.17 0.28

FGO-CNTRC 0 13.3763 16.1436 19.71111/10 18.8805 19.7872 21.14521/5 21.825 22.3256 23.097

UD-CNTRC 0 18.0598 21.8061 27.07721/10 20.5123 22.0434 24.50361/5 22.7471 23.6406 25.14

FGX-CNTRC 0 21.7682 26.3039 32.90821/10 22.0282 24.103 27.49581/5 23.6361 24.8948 27.0561

It is noticeable that the buckling temperatures of FGO-CNTRCplates are lower than those of UD-CNTRC plates, whileFGX-CNTRC plates have higher buckling temperatures thanthe other two types of distribution, when all three have thesame mass fraction of CNTs. This is because the form ofthe distribution of reinforcements can affect the stiffness ofthe plate and it is thus expected that the desired stiffnesscan be achieved by adjusting the distribution of CNTs alongthe thickness direction of the plates. It is concluded thatreinforcements distributed close to the top and bottom aremore efficient than those distributed near the mid-plane forincreasing the stiffness of plates.

Table 5 lists the effects of piezoelectric to CNTRC thick-ness ratio (hp/h), different distribution patterns and volumefraction of SWCNTs on the dimensionless fundamental natu-ral frequency (ωmnL =

ωmnLa2

H

√ρ0E0

) of CNTRC square platesunder uniform temperature rise (a/h = 50, T = 300 K). Itcan be observed that the fundamental natural frequency of

Figure 3. Effect of the different distribution patterns of SWCNTs onthe average harvested power of piezoelectric CNTRC plates(a/h = 30, h = 2 mm, V ∗CN = 0.12, T = 300 K, hp/h = 5).

the piezoelectric CNTRC plates is increased with increase innanotube volume fraction or piezoelectric to CNTRC thicknessratio (hp/h) or from FGO distribution to UD and FGX. Asmentioned above, the distribution pattern and volume fractionof SWCNTs play an important role in changing the vibrationcharacteristics of nanocomposite plates.

The influence of the different distribution patterns andvolume fraction of SWCNTs on the amplitude of steady-statevibrations for the transverse mode of piezoelectric CNTRCplates and the average harvested power is shown in figures 2to 5. It can be seen that the distribution patterns and volumefraction of SWCNTs affect the behavior of the system byshifting the back bone curves to the left or right side. Aninteresting finding is that by approaching from a negativevalue of the detuning parameter to a positive one, there isan interchange between the curves of different distributionpatterns starting at σ = 0, such that for a negative value of thedetuning parameter σ , FGO-CNTRC has the greatest vibrationamplitude and harvested power while FGX-CNTRC has thegreatest vibration amplitude and harvested power for positivevalues of the detuning parameter σ . By approaching from theright side to the left side, this phenomenon cannot be seenfor the different distribution patterns. On the other hand, inthe investigation of SWCNT volume fraction, this scenariomoves to the right side and this interchange can be seen for theright branch of the back bone curves and their correspondingharvested power curves (see figures 4 and 5).

The effect of temperature rise on the amplitude of steady-state vibrations and the average harvested power of piezoelec-tric UD-CNTRC plates is depicted in figures 6–7 (a/h = 30,h = 2 mm, V ∗CN = 0.12, hp/h = 5). Approaching from anegative value of the detuning parameter to a positive one,an interchange between the curves of different temperatureconditions can be observed at σ = 0. Similarly to the case offigure 2, for a negative value of the detuning parameter σ ,a higher temperature, i.e. 320 K, gives the greatest vibrationamplitude and harvested power while lower temperature risegives the greatest values for positive values of the detuning

11

Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

Figure 4. Effect of different volume fractions of SWCNTs on theamplitude of steady-state vibrations for the transverse mode ofpiezoelectric UD-CNTRC plates (a/h = 30, h = 2 mm, T = 300 K,hp/h = 5).

Figure 5. Effect of different volume fractions of SWCNTs on theaverage harvested power of piezoelectric UD-CNTRC plates(a/h = 30, h = 2 mm, V ∗CN = 0.12, T = 300 K, hp/h = 5).

parameterσ . By approaching from the right side to the left side,this phenomenon cannot be seen for the different temperaturerises.

5. Conclusions

A reduced-order model was developed based on Galerkin dis-cretization of the governing equations that were derived fromthe extended Hamilton principle and Gauss law to investigatethe output voltage and power of a piezoelectric nanocompositeenergy harvester under parametric resonance. A study ofa piezoelectric carbon nanotube reinforced composite plateunder periodic in-plane load and subjected to a uniform tem-perature rise has been carried out within the context of classicalplate theory, von Karman type displacement–strain relation-ship. A Poincare–Lindstedt perturbation scheme method wasused to study the output voltage and harvested power of thepiezoelectric CNTRC plates. An extensive parametric studyhas been carried out. The numerical results show that the

Figure 6. Effect of the temperature rise on the amplitude ofsteady-state vibrations for the transverse mode of piezoelectricUD-CNTRC plates (a/h = 30, h = 2 mm, V ∗CN = 0.12, hp/h = 5).

Figure 7. Effect of the temperature rise on the average harvestedpower of piezoelectric UD-CNTRC plates (a/h = 30, h = 2 mm,V ∗CN = 0.12, hp/h = 5).

distribution of the reinforcements, the volume fraction ofSWCNTs and the temperature rise have a significant effect onthe generated power and amplitude of steady-state oscillationsof nanocomposite energy scavengers.

Acknowledgments

This work was fully supported by a research grant (ProjectNo. CityU 114010) from the Research Grants Council ofthe Hong Kong Special Administrative Region, China. Theauthors are grateful for this financial support.

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Smart Mater. Struct. 23 (2014) 065001 M Rafiee et al

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