1Nonlinear thermal dynamic and vibration analysis of shear deformablepiezoelectric functionally graded material plates

Nguyen Dinh Duc1*, Pham Hong Cong1, Ngo Duc Tuan2,

Phuong Tran2, Pham Thi Ngoc An1

1VNU, University of Engineering and Technology, 144 Xuan Thuy - Cau Giay Hanoi - Vietnam2The University of Melbourne Parkville, VIC, 3010, Australia

Abstract: The nonlinear dynamic analysis and vibration of imperfect functionally gradedmaterial (FGM) plates with piezoelectric actuators (PFGM) on elastic foundations subjected to acombination of electrical, thermal, and damping loadings are investigated in this paper. One ofthe salient features of this work is the consideration of temperature on the piezoelectric layer, andthe material properties of the PFGM plates are assumed to be temperature-dependent. Thegoverning equations are established based on Reddys third-order shear deformation plate theoryand are solved by the stress function, the Galerkin method, and the Runge-Kutta method. In thenumerical results, the effects of geometrical parameters; material properties; imperfections;

elastic foundations; and electrical, thermal and damping loads on the vibration and nonlineardynamic response of the PFGM plates are discussed.

Keywords: Nonlinear thermal dynamic analysis and vibration; piezoelectric FGM (PFGM)plates; third-order shear deformation plate theory; electrical, thermal and damping loads.

1. IntroductionFunctionally graded materials (FGMs) are composite, microscopically

homogeneous materials with mechanical and thermal properties that vary smoothly andcontinuously from one surface to another. Typically, these materials are made from amixture of metal and ceramic, or a combination of different metals by gradually varyingthe volume fraction of the constituents. Due to their high heat resistance, FGMs have

many practical applications, such as use in reactor vessels, aircrafts, space vehicles,

defence industries, and in other engineering structures.

* Corresponding author: N.D. Duc; Tel.: +84 915 966 626; E-mail: ducnd@vnu.edu.vn

2Therefore, in recent years, many investigations have been carried out on thebuckling, dynamic and vibration response of FGM structures based on third-order sheardeformation plate theory. Talha and Singh (2010) investigated the static response andfree vibration analysis of FGM plates using higher-order shear deformation theory; andUngbhakorn and Wattanasakulpong (2013) presented the thermo-elastic vibrationanalysis of third-order shear deformable functionally graded plates with distributed patchmass in a thermal environment. Golmakani and Kadkhodayan (2011) investigated thenonlinear bending analysis of annular FGM plates using third-order shear deformationplate theory. Zhang (2013) successfully constructed a model of FGM rectangular platesbased on physical neutral surface and higher-order shear deformation theory. Belabed etal. (2014) presented an efficient and simple third-order shear and normal deformationtheory for functionally graded material (FGM) plates. Dozio (2014) investigated the exactfree vibration analysis of Levy FGM plates with third-order shear and normaldeformation theories. Swaminathan and Ragounadin (2004) used third-order refinedtheory to study the static analysis of anti-symmetric angle-ply composite and sandwichplates. Qian et al. (2004) investigated the static and dynamic deformations of thickfunctionally graded elastic plates by using higher-order shear and normal deformableplate theory as well as the meshless local Petrov-Galerkin method.

Piezoelectric materials have been extensively used in various smart structures asdistributed sensors and actuators for active structural control purposes. Many studies havebeen conducted on linear vibration analysis and vibration control of fibre reinforcedcomposite (FRC) laminated plates, with integrated vibration analysis of piezoelectricfunctionally graded materials (PFGM) sensors using an accurate method by Eshaghi etal. (2011). Liew et al. (2003) studied the post-buckling of piezoelectric FGM platessubject to thermo-electro-mechanical loading. Hui-Shen Shen also investigated the post-buckling of FGM plates with piezoelectric actuators (Shen, 2005) under thermo-electro-mechanical loadings, FGM plates with piezoelectric fibre reinforced composite actuators(Shen, 2009), and nonlinear vibration and dynamic response of FGM plates with

3piezoelectric fibre reinforced composite actuators (Xia and Shen, 2009). He et al. (2001)investigated the active control of FGM plates with integrated piezoelectric sensors andactuators. Chen et al. (2008) studied the stability of piezoelectric FGM rectangular platessubjected to non-uniformly distributed load, heat and voltage. Shariyat (2009)investigated vibration and dynamic buckling control of imperfect hybrid FGM plates withtemperature-dependent material properties subjected to thermo-electro-mechanical loadconditions. Duc et al. (2015) investigated the nonlinear dynamic analysis and vibration ofshear deformable piezoelectric FGM double curved shallow shells under damping-thermo-electro-mechanical loads.

Hybrid structures where a substrate made of FGMs is coupled with surface-bondedpiezoelectric actuators (PFGM) are usually exposed to high temperature environments.However, since this area is relatively new, published literature on the analysis of PFGMis limited and most research is focused on cases with mechanical stability problems.Therefore, the nonlinear static and dynamic response, and vibration of PFGM structuresneeds to be further investigated, especially the structures subjected to thermal loads.Moreover, there are no recent publications on the nonlinear dynamic and vibrationanalysis of PFGM plates where the temperature effects on the piezoelectric layer are

considered.This paper will focus on studying the nonlinear dynamic response and nonlinear

vibration of an imperfect shear deformable PFGM plate. The PFGM plates are assumedto be resting on elastic foundations and are subjected to the combined action of electrical,thermal and damping loads. Piezoelectric layers and material properties of FGM platesare assumed to be temperature-dependent. In the numerical results, the effects ofgeometrical parameters; material properties; imperfections; elastic foundations; andelectrical, thermal and damping loads on the vibration and nonlinear dynamic response ofthe PFGM plates are discussed.

4Nomenclature

, ,a b h length, width, and thickness of the FGM layer

ah thickness of the piezoelectric layer,u v displacement components along the x, y directions, respectively

,x y rotations of normal to the mid-surface with respect to the x and y axes,

respectively

, a thermal expansion coefficients of the FGM layer and the piezoelectric layers,respectively

T temperature increment in the environment containing the plate1k Winkler foundation modulus

2k shear layer foundation stiffness of the Pasternak model.

2. Problem description and the governing equations

2.1. Problem description

Consider a piezoelectric FGM (PFGM) plate on elastic foundations. The PFGMplate includes the FGM layer, and the piezoelectric layers are perfectly bonded on its top

and bottom surfaces, as shown in Figure 1. A coordinate system , ,x y z is established inwhich the ,x y plane is on the middle surface of the PFGM plate, and z is the thicknessdirection.

5Figure 1. Geometry of the piezoelectric FGM (PFGM) plates on elastic foundations

The substrate FGM layer is made from a mixture of ceramic and metal; the mixing

ratio is varied continuously and smoothly in the z -direction. By applying the power lawdistribution, the volume fractions of the ceramic and metal are assumed as:

2( ) , ( ) 1 ( ),2

N

c m c

z hV z V z V zh (1)

where N is the volume fraction index ( 0 N ); and subscripts m and c stand forthe metal and ceramic constituents, respectively. The effective properties Pr

eff of the

FGM plate, such as the elastic modulus E , the mass density , and the thermal expansion

coefficient , are determined by the linear rule of mixture as:e ( ) ( ) ( ),ff c c m mPr z PrV z Pr V z (2)

in which Pr denotes a temperature-dependent material property. The effective propertiesof the FGM plate are obtained by substituting Eq. (1) into Eq. (2) as: ( , ), ( , ), ( , ) ( , ), ( , ), ( , )

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

m m m

N

cm cm cm

E z T z T z T E z T z T z T

z hE z T z T z Th

(3)

where

( , ) ( , ) ( , ), ( , ) ( , ) ( , ),( , ) ( , ) ( , ),

cm c m cm c m

cm c m

E z T E z T E z T z T z T z Tz T z T z T

(4)

6and Poissons ratio is assumed to be constant ( )z v const .A material property Pr can be expressed as a nonlinear function of temperature

(Duc, 2014; Touloukian, 1967) as: 1 2 30 1 1 2 31 ,Pr P P T PT PT PT (5)

in which 0T T T ; 0 300T K (room temperature), and 0 1 1 2, , ,P P P P and 3P arecoefficients characterising the constituent materials.

2.2. Governing equationsIn this study, Reddys third-order shear deformation plate theory is used to derive

the governing equations and determine the nonlinear vibration of the thick PFGM plates.The strain components across the plate thickness at a distance z from the mid-

plane are (Reddy, 2004):0 1 3

0 20 1 3 3 2

0 20 1 3

, ,

x x x x

xz xz xz

y y y yyz yz yz

xy xy xy xy

k kk

z k z k zk

k k

(6)

where2

0 12

0 1

0 1

2

23

31

3

12

1, ,

2

x

x x

yy y

xy xyyx

x

x

yy

xy

u w

x x xkv w ky y y

ku v w w

y x x y y x

w

x xkk c

yk

0 22

10 22

2

w w

w, , 3 ,

w w

w2

x x

xz xz

yz yzy y

yx

kx xc

kyy y

y x x y

(7)

where 21 4 / 3c h .

7The stress-strain relationships of the thick PFGM plates are defined by Hookeslaw, taking into account the piezoelectric and thermal effects, and are given by:

11 12 31

12 22 32

44 24

55 15

66

0 0 0 0 00 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0 0

x x

y y a

yz yz

zx zx

xy xy

Q Q eQ Q e

Q T eQ e

Q

,

x

y

z

EE

E

(8)

where

11 22 12 44 55 662 2, , ,1 1 2 1E vE EQ Q Q Q Q Qv v v

(9)

The piezoelectric stiffness 31 32 15 24, , ,e e e e can be expressed in terms of the dielectric

constants 31 32 15 24, , ,d d d d ; and the elastic stiffness ( 11,22,12,44,55,66)aijQ ij of thepiezoelectric layers can be expressed as:

31 31 11 32 12 32 31 12 32 22 24 24 44 15 15 55, , , .a a a a a ae d Q d Q e d Q d Q e d Q e d Q Note that for an actuator

layer, .aij ijQ QAs only the transverse electric field component ZE is dominant in the plate-type

piezoelectric material, it is assumed that:

0 0 / ,T T

x y z a aE E E V h (10)where aV is the voltage that is applied to the actuators in the thickness direction.

The force and moment resultants are expressed by:

8

3

/23

2/2

3/2

32

/2

3

3

2

10 0 0

1 1 11 1 1

10 0 0

1 1 11 1 1

0 0 0

1

x y x y x yx x x h

y y y x y x y x yh

z z z

h

h

a a a

a a a

a

a

v z v z vN M PEN M P v z v z v dzv

N M P

v z v z vE

v z v z v Tdzv

v z v z v

v z v z v

Ev

/2

3/2 31 32 31 32 31 32

332 31 32 31 32 31

3

3

231 32 31 32

0 0 0

1 1 11 1 1

0 0 0

1

a

a

h h

h a a a

a a a z

a a a

a a a a

a

a a a

T

dzd v d z d v d z d v dd v d z d v d z d v d E

v z v z v

v z v z v T

Ev d v d z d v d z

/2

3/2 31 32

332 31 32 31 32 31

0 0 0

a

h

h h a

a a a z

dzd v d

d v d z d v d z d v d E

(11)

where

2 2/2 /2

2 2/2 /2

/2 2

2/2

/23

/2

2 1 2 1

2 1

, , 1, ,2(1 )

a

a

h hx x xz xz xz xza

y y yz yz yz yzah h h

h hxz xza

yz yza h

h

xy xy xy xyh

Q R z zEE dz dzQ R z zv vzE dzzv

EN M P z z dz

(12)

Substituting Eqs. (6) into Eq. (11) leads to:

9

0 0 1 11 2 4 2 3 5

2 3 34 5 7 1 2 4

1 2 41 2 4 31 322

0 0 1 11 2 4 2 3 5

2 3 34 5 7

, , , ,1, ,

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

1

, , , ,1, ,

1 , ,

x y x y

x x x

x y

a a a a z z z a

a

y x y x

y y y

y x

E E E E E E vN M P

E E E v

E E E d v dv

E E E E E E vN M P

E E E v

1 2 4

1 2 41 2 4 32 312

0 1 31 2 4 2 3 5 4 5 7

0 21 3 3 5

01 3 3 5

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

11

, , , , , , , , ,

2(1 )1

, , , ,

2(1 )1

, , ,

2(1 )

a a a a z z z a

a

xy xy xy xy xy xy

x x xz xz

y y yz

E E E d v dv

N M P E E E E E E E E E

Q R E E E E

Q R E E E E

2

,yz

(13)

where

/22 3 4 6

1 2 3 4 5 7/2

/23

1 2 4/2

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

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

h

hh

h

E E E E E E E z z z z z z dz

E z z T z z z dz

/2/21 2 4 3 3

/2 /2

/2/23 3

1 2 4/2 /2

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

, , 1, , 1, ,

a

a

a

a

h hh

a a a a a a a

h h h

h hh

z z z a z a z

h h h

E z T z z z dz E z T z z z dz

E E E E E z z dz E E z z dz

(14)

The nonlinear equilibrium equations of a perfect PFGM plate based on Reddysthird-order shear deformation plate theory are (Reddy, 2004):

22 3

0 1 1 32 2 2

w,

xyx xNN uI J c I

x y t t x t (15a)

22 3

0 1 1 32 2 2

w,

xy y yN N vI J c Ix y t t y t

(15b)

10

2 2 2

2 2 2

2 22 2 2 2

1 1 2 0 02 2 2 2 2

4 4 3 321 6 1 32 2 2 2 2 2

w w w2

w w w w2 w 2

w w

y yx xx xy y

xy yx

Q RQ Rc N N N

x y x y x x y y

P PPc k k q I I

x x y y x y t t

u vc I c I

x t y t x t y t

33

4 2 2 ,yxJ

x t y t

(15c)

22 3

1 2 1 2 1 42 2 2

w,

xy xyx x xx x

M PM P uc Q c R J K c J

x y x y t t x t (15d)22 3

1 2 1 2 1 42 2 2

w,

xy y xy y yy y

M M P P vc Q c R J K c J

x y x y t t y t (15e)

in which q is an external pressure uniformly distributed on the surface of the plate, is

the damping coefficient, and

/2/2

, (i 0,1,2,3,4,6),h

ii

h

I z z dz

2

1 2 2 2 1 4 1 6 2 1, 2 , 3 ,i i iJ I c I K I c I c I c c (16)

From the constitutive relations (13), one can write:

0 0 1 1 3 32 4 1

1

1

21 1 31 32 32 31 32 31 31 32

0 1 32 4

1

1, , , , , 1,1

1,1 1 111 , ,

1 2 1 ,

x y x y y x x y x y

a a

a z a a a a

xy xy xy xy

N N N N E EE

v v

E v E d v d d v d v d v d d v d

N E EE

(17)

with the Airy stress function , ,f x y t defined as:2 2 2

2 2, , .x y xyf f fN N N

y x x y (18)Replacing Eq. (18) into Eqs. (15a) and (15b) leads to:

22 31 31

2 2 20 0

w,

xc Iu J

t I t I x t (19a)

11

22 31 31

2 2 20 0

w.

y c Iv Jt I t I y t

(19b)Substituting Eqs. (19a) and (19b) into Eqs. (15c) (15e) yields:

2 2 2

2 2 2

2 22 2 2 2

1 1 2 02 2 2 2 2

2 2 4 421 3

0 1 6 2 2 2 20

1 3 14 1

w w w2

w w w2 w

w w w2

y yx xx xy y

xy yx

Q RQ Rc N N N

x y x y x x y y

P PPc k k q I

x x y y x y t

c II c It I x t y t

J I cJ c

331 3 1

4 12 20 0

,yx J I cJ c

I x t I y t

(20a)

221

1 2 2 20

31 3 1

1 4 20

w,

xy xyx x xx x

M PM P Jc Q c R K

x y x y I t

c I Jc J

I x t

(20b)

221

1 2 2 20

31 3 1

1 4 20

w.

xy y xy y yy y

M M P P Jc Q c R K

x y x y I t

c I Jc J

I y t

(20c)

The introduction of Eqs. (7) and (17) into Eqs. (13), and then into Eqs. (20) gives:

2*11 12 13 1 0 0232 2 34 4

21 3 1 3 1 1 3 11 6 4 1 4 12 2 2 2 2 2

0 0 0

w ww w, w , 2

w w,

x y

yx

A A A P f P f q I It t

c I J I c J I cc I J c J c

I x t y t I x t I y t

22 3* * 1 3 1121 22 23 21 2 1 42 20 0

ww w ,x

x y

c I JJA A A A K c JI t I x t

22 3* * 1 3 1131 32 33 31 2 1 42 2

0 0

ww w ,

yx y

c I JJA A A A K c JI t I y t

(21)

in which the coefficients are given in Appendix A, and the imperfection function

12

*( , )w x y represents an initial small deviation of the plate surface.The geometrical compatibility equation for an imperfect PFGM plate may be

derived as:22 0 2 02 0 2 2 2 2 2 * 2 2 * 2 2 *

2 2 2 2 2 2 2 2

w w w w w w w w w2y xyxy x x y x y x y x y x y x y y x

(22)

Inserting Eqs. (17) into Eq. (22), the compatibility equation of the imperfectPFGM plate is:

22 2 2 2 2 * 2 2 * 2 2 *4

1 2 2 2 2 2 2w w w w w w w w w2 0.f E

x y x y x y x y x y y x

(23)

The nonlinear Eqs. (21) and (23), in terms of two dependent unknowns w and f,are used to investigate the nonlinear vibration and dynamic stability of the imperfectthick piezoelectric FGM plates using Reddys third-order shear deformation plate theory.

3. Solution procedure

Consider simply supported thick PFGM plates with all immovable edges, resting

on elastic foundations. The boundary conditions in the current study are:

0

0

w 0, 0,w = 0, 0, ,

y x x x

x y y y

u M N N at x av M N N at y b

(24)

where 0 0,x yN N are fictitious compressive edge loads at the immovable edges.

Based on the abovementioned conditions (Eq. (24)), the approximate solutions canbe represented by:

w , , W sin sin ,

, , cos sin ,

, , sin os ,

x x

y y

m x n yx y t t

a bm x n y

x y t ta b

m x n yx y t t c

a b

(25)

13

where ( ), ,x yW t are the time-dependent amplitudes; and ,m n are the numbers of half-waves in the axial and circumferential directions, respectively.

The initial imperfection *w is assumed in the following form:

* 0w , sin sin ,m x n yx y Wa b (26)

in which 0W is a known initial amplitude.

Introducing Eqs. (25) and (26) into the compatibility equation (Eq. (24)), andsolving the obtained equation for the unknown f , leads to:

2 21 2 0 01 1, , cos2 cos2 ,2 2x yf x y t A t x A t y N y N x (27)

with 2 21 11 0 2 02 22 ; 2 .32 32E EA W W W A W W W

Replacing Eqs. (25) (27) into Eqs. (21), and then applying the Galerkin method tothe resulting equations yields:

3 2 211 12 13 1 0 0 0 0 1 0 0222

2 0 2 22 2 2 2

2

16 W W2 ,

x y x y

yx

l W l l nW n N N W W nWW W W

m nq n Imn t t a t b t

(28a)

2 2

21 0 22 23 1 22 2

W( ) ,xx y

ml W W l lt a t

(28b)

2 2

31 0 32 33 1 22 2

W( ) ,yx y

nl W W l lt b t

(28c)

in which the detail of the coefficients is found in Appendix A.

The plate is subjected to uniform external pressure q (pascal) and simultaneouslyexposed to thermal environments. The condition expressing the immovability on the

edges, 0u (on 0,x a ) and v 0 (on 0,y b ), is satisfied in an average sense as:

14

0 0 0 0

0, 0.b a a bu vdxdy dxdy

x y (29)

From Eqs. (7) and (17), we obtain the following expressions:

22 2 22 1 4 1

2 2 21 1 1 1

1 131 32 32 312

1 1

2 2 22 1 4

2 2 21 1 1

1 12

1 1 ,1 1

1 12

x x

za a a

a a

y y

u f f E c E w wv

x E y x E x E x x x EE

v d v d v d v dE v E v

v f f E c E w wv

y E x y E y E y y

2

1

1

1 132 31 31 322

1 1

1 1 .1 1

za a a

a a

y EE

v d v d v d v dE v E v

(30)

Putting Eqs. (25) (27) into Eq. (30), then substituting the obtained result intoEq. (29), we have:

2 21

0 2 1 4 1 42 2

12 2 21 1

31 3222

11 1

,

1 18 1

x x y

a za

a a

N E c E v c E v Wv mn v

E Ev W d v d

v vv

(31a)

2 21

0 2 1 4 1 42 2

12 2 21 1

32 3122

11 1

,

1 18 1

y x y

a za

a a

N E c E v c E v Wv mn v

E Ev W d v d

v vv

(31b)

Substituting Eqs. (31a) and (31b) into Eqs. (28), the system of motion Eqs. (28) isrewritten as follows:

2 23 1

11 12 13 1 0 10 0

1 0 0 14 0 15 0 0

2222

17 0 2 0 2 22 2 2 2

1

2

16 W W2 ,

x y

x y 16

yx

m nl W l l nW n l W Wa b v

nWW W W l W W l W W +l W W W

m nl W W W q n Imn t t a t b t

(32a)

15

2 2

21 0 22 23 1 22 2

W( ) ,xx y

ml W W l lt a t

(32b)2 2

31 0 32 33 1 22 2

W( ) ,yx y

nl W W l lt b t

(32c)

in which the detail of the coefficients are found in Appendix A.

Taking the linear parts of Eqs. (32) and setting 0,q t the natural frequencies ofthe perfect plate is the smallest value of the three frequencies in the axial,

circumferential and radial directions, which can be determined by solving the followingdeterminant:

2 22 2 21

10 11 0 2 12 2 13 2

2 221 2 22 1 23

2 231 2 32 33 1

1

0.

m n m nl l n n l la b v a b

ml l la

nl l lb

(33)

Assume that the PFGM plate is acted on by a uniformly distributed transverse load

sinq Q t , where Q is the amplitude of the uniformly excited load, and is thefrequency of the load. Then the system of Eqs. (32) have the form:

2 23 1

11 12 13 1 0 10 0

1 0 0 14 0 15 0 0

2222

17 0 2 0 2 22 2 2 2

1

2

16 W W2 ,

x y

x y 16

yx

m nl W l l nW n l W Wa b v

nWW W W l W W l W W +l W W W

m nl W W W q n Imn t t a t b t

2 2

21 0 22 23 1 22 2

W( ) ,xx y

ml W W l lt a t

2 2

31 0 32 33 1 22 2

W( ) ,yx y

nl W W l lt b t

(34)

16

The nonlinear dynamic responses of the thick PFGM plates can be obtained bysolving Eq. (34), combined with the initial conditions by using the fourth-order Runge-Kutta method.

4. Numerical results and discussion

4.1. Comparison of results

For verification of the present method for determining the nonlinear dynamic

response of PFGM plates, the natural frequencies ( )L Hz of the Ti-6A1-4V/Al2O3 platewith symmetrically fully covered G-1195N piezoelectric layers are calculated andcompared in Table 1 with the theoretical results of Xia and Shen (2009). The materialproperties of the constituent materials of the FGM hybrid plates are:

3320.24 , 0.26, 3750 /c c c

E GPa kg m for aluminum oxide;3105.70 , 0.2981, 4429 /

m m mE GPa kg m for Ti6Al4V, and 63 ,aE GPa

3 1031 320.3, 7600 / , 2.54 10 /a a kg m d d m V

for the G-1195N piezoelectriclayer.

From Table 1, it can be seen that the present values are not significantly differentfrom the results of (Xia and Shen, 2009).Table 1. Comparisons of natural frequency ( )Hz for FGM plates with piezoelectric

actuators bonded on the top and bottom surfaces, 0.4 , 4 , 0.1 .a b m h mm ha mm ( , )m n N

0 0.5 1 5 15

(1,1)

Ref. (Xiaand Shen,

2009)

143.03 184.33 198.40 227.58 243.23

Presentstudy

145.00 186.01 199.10 230.11 249.30

(1,2) and (2,1)Ref. (Xiaand Shen,

2009)

358.68 460.97 494.58 568.50 607.57

17

Presentstudy

360.80 463.05 496.51 575.03 618.82

(2,2)

Ref. (Xiaand Shen,

2009)

563.35 727.27 778.29 898.58 962.85

Presentstudy

265.23 731.60 778.11 905.35 970.80

(1,3) and (3,1)

Ref. (Xiaand Shen,

2009)

717.26 922.58 991.77 1141.47 1223.24

Presentstudy

718.30 926.90 995.20 1150.00 1235.80

4.2. Nonlinear dynamic and vibration analysis of PFGM platesThe effective temperature-dependent material properties in Eq. (5) are listed in

Table 2. Poissons ratio is 0.3v . The initial conditions are:

(0) 0, (0) 0, (0) 0, (0) 0, (0) 0, (0) 0.yxx ydddWW

dt dt dt

G-1195N is selected for the piezoelectric layers and its material properties are:10 0 363 , 0.3, 1.2 10 1/ , 7600 / ,

a a a aE GPa C kg m

1031 32 2.54 10 / .d d m V

The thickness of each piezoelectric layer is 1 mm.Table 2. Material properties of the constituent materials of the considered PFGMplates

Material Property P0 P-1 P1 P2 P3

Si3N4(Ceramic)

E (Pa) 348.43e9 0 -3.70e-4 2.160e-7 -8.946e-11 (kg/m3) 2370 0 0 0 0

1( )K 5.8723e-6 0 9.095e-4 0 0

SUS304

(Metal)

E (Pa) 201.04e9 0 3.079e-4 -6.534e-7 0 (kg/m3) 8166 0 0 0 0

1( )K 12.330e-6 0 8.086e-4 0 0

18

Table 3. Effect of power law index N on natural frequencies 1s of the PFGM platewith 1 2/ 1, / 20, 0, 0a b a h k k

,m n N0 0.5 1 5 10

1,1 440514 439507 435869 413582 412027 1,2 and 2,1 446552 440572 439795 414670 413110

2,2 442701 441634 436049 415756 414191 1,3 and 3,1 453151 451834 446206 416478 414909 2,3 and 3,2 457034 455623 437809 429853 428237

Table 3 shows the effects of the group numbers of ( , )m n and the volumecoefficient ratio N on natural frequencies. We can see that the volume coefficient ratio

and the natural frequencies have a reverse relationship: one is increased (decreased) whilethe other is decreased (increased). The values of ( , )m n are raised, while the naturalfrequencies are declined. Table 3 also shows that the lowest natural frequency

corresponds to mode , 1,1m n .

19

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0100.010.020.030.040.050.060.070.080.090.1

t(s)

W(m)

Q=600 kN/m2, Va=200 V, =1500 rad/s, =1000(m,n)=(1,1), b/a=1, b/h=20, T=300 K, k1=0, k2=0

N=0N=1N=5

Figure 2. Effect of power law index N on the nonlinear dynamic response of the PFGMplate.

Figure 2 shows the effect of the power law index 0, 1, 5N N on the nonlineardynamic response of the PFGM plate. As we can see in Figure 2, the vibration diagramsof the FGM plates in all three cases gradually decay. It is also noticed that the case of

5N gives the highest maximum amplitude, while the case of 0N yields the smallestamong the three values. After a very short vibration time, the obtained displacements are

constant for the three cases with 0, 1, 5N N , and the absolute value of the PFGMplate amplitude is reduced when the power law index N is decreased. This is reasonable

because when N is decreased, the ceramic volume fraction is increased; however, the

elastic module of ceramic is higher than metal ( c mE E ).

20

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.010.02

0.030.040.05

0.060.07

0.080.090.1

t(s)

W(m)

(m,n)=(1,1), b/h=20, T=300 K, k1=0, k2=0, N=1, Q=600 kN/m2, Va=200 V, =1500 rad/s, =1000

b/a=2b/a=1b/a=0.5

Figure 3. Effect of ratio /b a on the nonlinear dynamic response of the PFGM plate.

Figures 3 and 4 illustrate the effects of geometrical factors of the PFGM plateratios b/a and b/h on the nonlinear dynamic response, respectively. As we can see inFigure 3, after a very short vibration time, all displacements of the three cases with

/ 0.5,1,2b a reach the steady state, where their constant amplitudes are dependent onthe /b a ratio. Higher ratio /b a gives larger dynamic displacements. It also could beseen from Figure 4 that the absolute value of the PFGM plate amplitude increases when

increasing the ratio /b h .

21

0 0.002 0.004 0.006 0.008 0.010

0.02

0.04

0.06

0.08

0.1

0.12

0.14

t(s)

W(m)

(m,n)=(1,1), b/a=1, T=300 K, k1=0, k2=0, N=1, Q=600 kN/m2, Va=200 V, =1500 rad/s, =1000

b/h=20b/h=30b/h=40

Figure 4. Effect of ratio /b h on the nonlinear dynamic response of the PFGM plate.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0100.010.020.030.040.050.060.070.080.09

t(s)

W(m)

(m,n)=(1,1), b/a=1, b/h=20, T=300 K,=1500 rad/s, =1000,k2=0, N=1, Q=600 kN/m2, Va=200 V

k1=0k1=1 GPa/mk1=2 GPa/m

Figure 5. Effect of the linear Winkler foundation on the nonlinear dynamic response ofthe PFGM plate.

22

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.02

0.04

0.06

0.08

0.1

t(s)

W(m)

(m,n)=(1,1), b/a=1, b/h=20, T=300 K,=1500 rad/s, =1000,

k1=0, N=1, Q=600 kN/m2, Va=200 V

k2=0 k2=0.002 GPa.m k2=0.005 GPa.m

Figure 7. Effect of the Pasternak foundation on the nonlinear dynamic response of thePFGM plate.

Figures 6 and 7 describe the effects of coefficients of the Winkler 1k and Pasternakfoundations 2k , respectively, on the nonlinear dynamic response of the PFGM platewith 1, / 1, / 20, =1000, T=300 , 200

aN b a b h K V V . For the results

illustrated in Figure 6, the Pasternak stiffness value ( 2k ) is kept constant, while theWinkler value ( 1k ) is varied 1 0, 1, 2 /k GPa m . As we can see in this figure, theincrease in 1k value leads to a decrease in vibration amplitude. Similar results are

obtained for varying 2k and are presented in Figure 7. Figures 6 and 7 confirm the

important influences of elastic foundations on the dynamic response of PFGM plates. In

addition, the shear layer stiffness 2k of the Pasternak foundation model has a more

pronounced influence than the modulus 1k of the Winkler model on the nonlinear

dynamic response of the PFGM plate.

23

0 0.002 0.004 0.006 0.008 0.0100.010.020.030.040.050.060.070.080.09

t(s)

W(m)

(m,n)=(1,1), b/a=1, b/h=20, k1=0, k2=0, N=1T=300 K, Va=200 V, =1500 rad/s, =1000

Q=500 kN/m2Q=1500 kN/m2Q=3000 kN/m2

Figure 8. Effect of excitation force Q on the nonlinear dynamic responses of the PFGMplate.

Figure 8 considers the effect of harmonic uniform load with amplitudes2 2500 / , 1500 /Q kN m Q kN m and 23000 /Q kN m on the nonlinear dynamic

response of the PFGM plate. From the figure, it is seen that the absolute value of the

PFGM plate amplitude is considerably increased when excitation force amplitude Qincreases.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t(s)

W(m)

(m,n)=(1,1), b/a=1, b/h=20, k1=0, k2=0, N=1,Q=600 kN/m2, Va=200 V, =1500 rad/s, =1000

T=300 KT=600 KT=1200 K

Figure 9. Effect of temperature increment on the nonlinear dynamic response of the

24

PFGM plate.

Figure 9 considers the influence of temperature increment

300,600,1200T T K on the nonlinear response of the PFGM plate with1, / 1, / 20, 1000, 200

aN b a b h V V . It can be seen that the absolute value ofthe PFGM plate amplitude increases when increasing the value of temperature increment.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.02

0.04

0.06

0.08

0.1

t(s)

W(m)

(m,n)=(1,1), b/a=1, b/h=20, k1=0, k2=0, N=1,Q=600 kN/m2, T=300 K, Va=200 V, =1500 rad/s

=500=1000=2000

Figure 10. Effect of the viscous damping coefficient on the nonlinear dynamicresponse of the PFGM plate.

Figure 10 gives the effect of damping load with the viscous damping coefficient

500,1000,2000 on the nonlinear dynamic response of the PFGM plate. From thisfigure, we can see that the damping coefficient (DC) has a clear influence on the dynamicresponse of the PFGM plate, in which the higher DC causes smaller vibration amplitude.

The influence of initial imperfection with amplitude 0W on the nonlinear response

of the PFGM plate is presented in Figure 11. In the beginning, the imperfect plate inducesa larger vibration amplitude compared with the perfect one. After a short time, their

vibration amplitude values coincide with 3 30 0,10 ,5.10W .

25

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.02

0.04

0.06

0.08

0.1

0.12

0.14

t(s)

W(m

)

(m,n)=(1,1), b/a=1, b/h=20, k1=0, k2=0, T=300 K,N=1, Q=600 kN/m2, Va=200 V, =1500 rad/s, =1000

W0=0W0=10-3 mW0=5.10-3 m

Figure 11. Effect of imperfection 0W on the nonlinear dynamic response of the PFGM

plate.

Figure 12 shows the effect of applied voltage ( 0, 200a

V V and 500 V ) on thenonlinear dynamic response of the PFGM plate. Obviously, the applied voltage has aweak effect on the nonlinear dynamic response of the plate, and an increase of appliedvoltage leads to an increase of the absolute value of the PFGM plate amplitude. This isunderstandable because the piezoelectric layer is very thin and it is like a machine thattransforms electrical energy directly into mechanical energy, and vice versa.

26

0 0.002 0.004 0.006 0.008 0.010

0.02

0.04

0.06

0.08

0.1

0.12

0.14

t(s)

W(m)

(m,n)=(1,1), b/a=1, b/h=20, k1=0, k2=0, N=1, W0=0,T=300 K, Q=600 kN/m2, =1500 rad/s, =1000

Va=0Va=200 VVa=500 V

Figure 12. Effect of applied voltage on the nonlinear dynamic response of the PFGMplate.

5. Concluding remarksThis paper presents an analytical investigation into the nonlinear dynamic analysis

and vibration of imperfect piezoelectric functionally graded (PFGM) plates on elasticfoundations. The key outcomes of this work are:

- A system of partial differential equations was constructed to describe thenonlinear dynamic responses of composite PFGM plates, which incorporatevarious factors such as: electrical, thermal, and damping loading.

- The influences of temperature on the piezoelectric layer and the materialproperties of the PFGM plates were taken into account. They are nonlinearly

dependent on temperature.- A series of parametric studies were conducted to reveal the decaying nonlinear

dynamic responses of PFGM plates and their dependence on various factorsincluding: geometrical parameters; material properties; imperfections; elastic

27

foundations; and electrical, thermal and damping loads.

Funding. This work was supported by the Grant in Mechanics of the NationalFoundation for Science and Technology Development of Vietnam - NAFOSTED. Theauthors are grateful for this support.

Appendix A

2 2

11 1 1 3 2 3 1 5 2 2

2 4 4 4 2 2 21 4

7 1 22 4 2 2 4 2 21

1 2 2 2 21

31 32 32 312 2 2 2 2

1 w ww 3 3

2(1 )w w w w w2 + w

1

w w w w

1 1a z

a a

a a

A E c E c E c Ex y

c EE k kx x y y E x y

E d v d d v dv x y v x y

12 1 1 3 2 3 1 5

321 2 4 4

5 1 72 31 1

322 4 4

1 5 1 72 2 21 1

13 1 1 3 2

,

1 3 32(1 )

1

1 1,

1 1 1 1

1 32(1 )

xx

x

x

y

A E c E c E c Ex

c E E EE c EE E x

E E Ec E c E

v E E x y

A E c E c

3 1 5

321 2 4 4

5 1 72 31 1

322 4 4

1 5 1 72 2 21 1

2 2 2 2 2 2

2 2 2 2

3

1

1 1,

1 1 1 1

w w w, 2 ,

y

y

y

E c Ey

c E E EE c EE E y

v E E v Ec E c E

E E x y

f f fP x fy x x y x y x y

28

2 3 31 1 4 2 4

21 1 7 52 2 31 1

1 1 3 2 3 1 5

2 2 2 222 2 4 4

22 3 1 5 1 72 2 21 1 1

1 1

w ww

11 w3 3 ,

2(1 )1 1 2

1 2(1 )1 3

2(1 )

x xx

c c E E EA c E EE E x y x

E c E c E c Ex

E E E EA E c E c Ex y E E E

E c

3 2 3 1 5

22 222 2 4 4

23 3 1 5 1 71 1 1

3 ,

1 2 ,2 1

x

yy

E c E c E

E E E EA E c E c EE E E x y

2 3 31 4 2 4

31 1 7 1 52 3 21 1

1 1 3 2 3 1 5

22 222 2 4 4

32 3 1 5 1 71 1 1

2 2

33 2 2 2

w ww

11 w3 3 ,

2(1 )1 2 ,

2 1

1 11 2(1 )

xx

y yy

c E E EA c E c EE E y x y

E c E c E c Ey

E E E EA E c E c Ev E E E x y

Ay x

2 222 2 4 4

3 1 5 1 71 1 1

1 1 3 2 3 1 5

2

1 3 3 ,2(1 ) y

E E E EE c E c EE E E

E c E c E c E

2 * 2 * 2 2 **1 1 1 3 2 3 1 5 2 2 2 22 2 * 2 2 *

2 2

1 w w w, 3 3 ( + )

2(1 )w w2 ,

fP w f E c E c E c Ex y y x

f fx y x y x y

**21 1 1 3 2 3 1 51 w3 3 ,2(1 )A E c E c E c E x

**21 1 1 3 2 3 1 51 w3 3 ,2(1 )A E c E c E c E y

29

22 22 21 4

11 7 121

2 2 2 21

2

2 21

31 32 32 312

+1

1

,

1

a

a

za a

a

c m n El E ka b E

m n m nka b v a b

E m nd v d d v dv a b

12 1 1 3 2 3 1 53 2

1 2 4 45 1 72

1 1

2 452 2

11 2

41 72

1

1 3 32(1 )

1

11 1

11 1

ml E c E c E c Ea

c m E E EE c Ea E E

E EEv E m n

ca bE

c EE

13 1 1 3 2 3 1 5

3 21 2 4 4

5 1 721 1

2 452 2

11 2

41 72

1

1 3 32(1 )

1

11 1

11 1

nl E c E c E c Eb

c n E E EE c Eb E E

v E EEE m n

ca bv E

c EE

2 20 1 1 3 2 3 1 51 3 32(1 )m n

n E c E c E c Ea b

4 4

11 ,16

E m nn

a b

30

2 222 3

2 0 1 60

,

I m nn I c I

I a b

2 2 21 1 4 2 4

21 1 7 521 1

1 1 3 2 3 1 5

2 2 2 222 2 4 4

22 3 1 5 1 721 1 1

+1

1 3 3 ,2(1 )

1 1 21 2(1 )

c m m n c E E El c E Ea a b E E

m E c E c E c Ea

m n E E E El E c E c Ea b E E E

1 1 3 2 3 1 5

2 2 222 2 4 4

23 3 1 5 1 71 1 1

1 3 3 ,2(1 )

1 2 ,2 1

E c E c E c E

mn E E E El E c E c Eab E E E

2 221 4 2 4

31 1 7 1 521 1

1 1 3 2 3 1 5

2 2 222 2 4 4

32 3 1 5 1 71 1 1

2

33 2

1

1 3 3 ,2(1 )

1 2 ,2 1

1 11 2(1

c n E E E m nl c E c Eb E E a b

n E c E c E c Eb

mn E E E El E c E c Ev ab E E E

nlb

22 2 4

3 1 521 1

22 41 7

1

1 1 3 2 3 1 5

2

)

1 3 3 ,2(1 )

E E EE c EE Em

a Ec E

E

E c E c E c E

2 2 2 21110 31 32 32 312 ,1 1 az a aa a

E m n m nl d v d d v dv a b v a b

31

2 2

2 1 4

14 2 2

4,

1

m nE c E va b ml

amn v

2 2

2 1 4

15 2 2

4,

1

m nE c E va b nl

bmn v

4 2 2 41 4

16 2 2

4 2,

1c E v

lmn v

4 2 2 4

117 1 2 2 ,8 1

E m m n nl n va a b bv

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