creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally...

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European Journal of Mechanics A/Solids 30 (2011) 547e558

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European Journal of Mechanics A/Solids

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

Creep buckling and post-buckling analysis of the laminated piezoelectricviscoelastic functionally graded plates

Y.Q. Mao a,b,*, Y.M. Fu a,b, H.L. Dai a,b

aCollege of Mechanical and Vehicle Engineering, Hunan University, Hunan, Changsha 410082, PR ChinabKey Laboratory of Advanced Design and Simulation Techniques for Special Equipment, Ministry of Education, Hunan University, Changsha 410082, PR China

a r t i c l e i n f o

Article history:Received 16 November 2010Accepted 14 March 2011Available online 21 March 2011

Keywords:ViscoelasticFunctionally graded laminated platesTransverse shear deformationInitial deflectionCreep buckling and post-buckling

* Corresponding author. College of Mechanical andUniversity, Hunan, Changsha 410082, PR China.

0997-7538/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.euromechsol.2011.03.004

a b s t r a c t

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally gradedmaterial (FGM) plates are studied in this research. Considering the transverse shear deformation andgeometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelasticFGM plates with initial deflection is established. And then nonlinear creep governing equations of thelaminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived onthe basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finitedifference method and the Newmark scheme, the whole problem is solved by the iterative method. Innumerical examples, the effects of geometric nonlinearity, transverse shear deformation, the appliedelectric load, the volume fraction and the geometric parameters on the creep buckling and post-bucklingof laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

With the development of the scientific technology, the piezoe-lastic laminated plates and shells, mounted or imbedded withpiezoelectric layers as sensor or actuator, have been used in a widevariety of industries, such as the aerial industry, national defenseindustry, ocean industry, architecture etc. The research on dynamicproperty of these piezoelastic structures has received considerableattention in recent years. But these laminated piezoelastic struc-tures always suffer from the delamination damage due to thediscontinuity of the material property. So was born a new typeof materials known as functionally graded materials (FGMs),comprising of two or more materials varying continuously alonga certain direction, which has attracted much attention in manystructural members and used in a wide variety of industries for itsadvanced properties. In recent years, with the expanding applica-tions of FGMs, many works on buckling and post-buckling behav-iors of FGM plates and shell structures have been carried out (Wuet al., 2007; Ma and Wang, 2003; Lee and Kim, 2009; Yang et al.,2006), and the bending and vibration of FGM plate and shellwere investigated by researchers (Golmakani and Kadkhodayan,2010; Seyyed, 2010).

Vehicle Engineering, Hunan

son SAS. All rights reserved.

However, these FGM structures, often under severe conditions(for example high temperature), would exhibit obvious creepbuckling property within the serving time due to its instinctviscoelastic property. As for the thermal insulation structures madeof the FGM under a set of pressure loads, though the pressure maybe lower than the mechanical critical load, these structures wouldbe failure due to the creep post-buckling when the serving time islong enough. Therefore, the analysis on creep buckling and post-buckling of the viscoelastic FGM structures would be of greatvalue on engineering applications. Recently, researches on behaviorof viscoelastic laminated plate and beams have been done (Wei etal., 2006, 2007), but researches on viscoelastic FGM structureswere still limited. Most of the researches on viscoelastic FGMstructures are carried out on the basis of the elastic-viscoelasticcorrespondence principle. Paulino et al. (2001) identified that thecorrespondence principle can still be used to obtain the viscoelasticsolution for a class of FGMs exhibiting relaxation (or creep) func-tions with separable kernels in space and time. Jin (2006) hada further discussion on the application of the correspondenceprinciple on the viscoelastic functionally graded material analysis.According to the correspondence principle, Pan et al. (2009)obtained the viscoelastic crack tip field under applied strain fromthe linear elastic results and studied the viscoelastic fracture ofmultiple cracks in a functionally graded strip. Khazanovich (2008)demonstrated that the correspondence principle was valid fornon-homogeneous materials with separable relaxation moduli

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x

b

ax

y

xNxN

b

Metal layer

Piezoelectric layer

zy

FGM

a

FGM

3h

2h

1h

a

b

Fig. 1. Schematic diagram of the piezoelectric viscoelastic FGM plate.

Y.Q. Mao et al. / European Journal of Mechanics A/Solids 30 (2011) 547e558548

even if the time-dependences of the relaxation moduli in shear anddilatation were not necessarily time translation invariant. Butobviously the application of the correspondence principle inviscoelastic FGMs is limited when the problem is complicated,especially when the transverse shear deformation and geometricnonlinearity of the FGM structure are considered.

Multi-scale micromechanical model for predicting the visco-elastic property of FGMs is another feasible way for the viscoelasticanalysis of the FGM structures recently. Kamran and Anastasia(2009) introduced a micromechanical model for predictingeffective thermo-viscoelastic behaviors of a FGM. Anastatsia(2009) presented an integrated micromechanical-structuralframework to analyze coupled heat conduction and deformationsof FGMs having temperature and stress dependent viscoelasticconstituents. Assuming the extensional relaxation modulus asE ¼ E0expðby=hÞf ðtÞ, Jin and Paulino (2002) studied the visco-elastic functionally graded strip containing a crack subjected toin-plane loading. But the literature survey shows that the analysisof the creep buckling and post-buckling of the viscoelastic FGMplate has not been carried out due to the complication of the model,especially when the geometric nonlinearity and transverse sheardeformation are considered synthetically.

In this paper, based on the elastic piezoelectric theory andviscoelastic theory, the constitutive relation of the piezoelectricviscoelastic FGM plates is established by adopting the Boltzmannsuperposition principle, and the nonlinear creep buckling govern-ing equations of the laminated piezoelectric viscoelastic FGM plateswith initial deflection are derived on the basis of the Reddy’shigher-order shear deformation plate theory. The problem is solvedby using the finite difference method and the Newmark schemesynthetically. In numerical examples, the effects of geometricnonlinearity, transverse shear deformation, the applied electricload and the volume fraction on the creep buckling and post-buckling of the laminated piezoelectric viscoelastic FGM plateswith initial deflection are investigated in details.

2. Basic equations

A laminated structure with two orthotropic viscoelastic FGMplate core as shown in Fig. 1 is considered. Two piezoelectricmembranes are mounted on the top and bottom surfaces of thestructure. The thickness of the metal, FGM and piezoelectric layerare h1, h2 and h3, respectively. And the laminated plate is a in thelength and b in the width. The Cartesian coordinate system oxyz isset on themid-plane (z¼ 0). An in-plane load Nx ¼ �pHðH ¼ h1 þ2h2 þ 2h3Þ is exerted on the laminated piezoelectric viscoelasticFGM plates.

As depicted in Fig. 1, the bottom and top surfaces of the FGMcore are orthotropic metal and orthotropic ceramic, which hold thesame material property to the piezoelectric and metal layer,respectively. The region between these two surfaces comprisesmaterial with different mix ratios of the ceramic and metal. Thenpf , the effective material property of the FGM layer, can beexpressed as in the following form:

pf ¼ ð1� VðzÞÞpm þ VðzÞpc (1)

where, pm and pc are the properties of the metal and ceramic,respectively; V(z) is the volume fraction of the metal constituent ofthe FGM and is assumed to follow the power law function as

VðzÞ ¼ ðz=h2 � h1=2h2Þn ðh1=2� z� h1=2þh2ÞVðzÞ ¼ ð� z=h2 � h1=2h2Þn ð � h1=2� h2 � z��h1=2Þ

(2)

where, nð0 � n � NÞ is the volume fraction index and representsthe material variation profile through the thickness of the FGM

layer, and may be varied to obtain the optimum distribution of theconstituent materials. Then the effective elastic modulus E andshear modulus G of the functionally graded layer may be obtainedby following equations:

Ef ðzÞ ¼ ð1� VðzÞÞEm þ VðzÞEcrf ðzÞ ¼ ð1� VðzÞÞrm þ VðzÞrcnf ¼ const:

(3)

here, the subscript c and m refer to ceramic and metal materialrespectively. The Poisson ratio v is considered to be constant alongthe thickness of the FGM.

In the present analysis, the viscoelastic property is consideredonly for the FGM layer and the metal layer, which is assumed to bevarying along the thickness of FGM, and the ceramic relaxationcoefficient ac is considered to be zero, i.e. the viscoelastic propertyof the ceramic material is neglected. Then the relaxation coefficienta of viscoelastic FGM varies along the thickness as

aðzÞ ¼ ð1� VðzÞÞam (4)

here, am is the relaxation coefficient of the metal material. With thedecrease of the ceramic constituent and increase of the metalconstituent, the viscoelastic property of the structure would bemore obvious.

Reddy (1984) proposed a higher-order shear deformable theory.This theory takes into account the transverse shear deformation ofthe structure and meets the condition that the shear forces onboth of bottom and top surfaces equal zero. So based on theReddy’s higher-order shear theory, the displacement field in thepiezoelectric viscoelastic FGM plates with initial deflection isgiven by

u1ðx; y; z; tÞ ¼ uðx; y; zÞ þ g1ðzÞj1ðx; y; tÞ � g2ðzÞwx � g2ðzÞw0x

u2ðx; y; z; tÞ ¼ vðx; y; zÞ þ g1ðzÞj2ðx; y; tÞ � g2ðzÞwy � g2ðzÞw0y

u3ðx; y; z; tÞ ¼ wðx; y; tÞ þw0ðx; y; tÞ ð5Þ

Y.Q. Mao et al. / European Journal of Mechanics A/Solids 30 (2011) 547e558 549

here, u1;u2;u3 are the displacement components of a point at thetime t, and u,v,w are the displacement components of a point in themid-plane ðz ¼ 0Þ; w0 is the initial deflection of the laminatedplate; j1 and j2 denote the rotations of the transverse normalabout the y and x axes, respectively. The functions g1ðzÞ and g2ðzÞare given as

g1ðzÞ ¼ z� 4z3

3H2; g2ðzÞ ¼ 4z3

3H2 (6)

where H is the total thickness of the laminated plate.Based on the Von Karman plate theory, nonlinear strain

displacement relations of the plate with initial deflection can begiven as

ex ¼ ux þ 12ðwxÞ2þwxw0

x þ g1j1;x � g2wxx � g2w0xx

ey ¼ vy þ 12

�wy

�2þwyw0y þ g1j2;y � g2wyy � g2w0

yy

exy ¼ uy þ vx þwxwy þw0xwy þwxw0

y þ g1�j1;y þ j2;x

��2g2wxy � 2g2w

0xy

exz ¼ g1;zj1 � g2;zwx � g2;zw0x þwx þw0

x

eyz ¼ g1;zj2 � g2;zwy � g2;zw0y þwy þw0

y (7)

Considering the linear stress strain relation of the viscoelasticfunctionally graded layer and the metal layer, the following relationcan be obtained by applying the strain energy equivalent theoryand Boltzmann superposition theory:

sðtÞ[Q ðtÞ5eðtÞ¼ Q ð0ÞeðtÞ þZt

0

dQ ðt � sÞdðt � sÞ eðsÞds (8)

where, sðtÞ[ fsxðtÞ; syðtÞ; sxyðtÞ; sxzðtÞ; syzðtÞgT , eðtÞ[ fexðtÞ; eyðtÞ;exyðtÞ; exzðtÞ; eyzðtÞgT ; Q ð0Þ and Q ðtÞ are the initial elastic matrix andtime-dependent relaxation matrix of the viscoelastic FGM; }5} isthe Steltjes convolution sign defined as

l5k ¼Zt

�N

lðt � sÞdkðsÞ ¼ lðtÞkð0Þ þZt

0

lðt � sÞ _kðsÞds

¼ lð0ÞkðtÞ þZt

0

_lðt � sÞkðsÞds(9)

8>>>><>>>>:

sxsysxzsyzsxy

9>>>>=>>>>;

¼

266666664

cðcÞ11 cðcÞ12 0 0 0

cðcÞ22 0 0 0

cðcÞ44 0 0

cðcÞ55 0

cðcÞ66

377777775

8>>>><>>>>:

exeyexzeyzexy

9>>>>=>>>>;

�

266664

0 0 e0310 0 e0320 e24 0e15 0 00 0 0

3777758<:

ExEyEz

9=;

8<:

DxDyDz

9=; ¼

24 0 0 0 e15 0

0 0 e24 0 0e031 e032 0 0 0

358>>>><>>>>:

exeyexzeyzexy

9>>>>=>>>>;

þ24 g11 0 0

0 g22 00 0 g033

358<:

ExEyEz

9=;

(14)

As for the anisotropy viscoelastic FGM, the relaxation matrix inEq. (8)can be written as

Q ðtÞ ¼

266666664

cðf Þ11 cðf Þ12 0 0 0

cðf Þ21 cðf Þ22 0 0 0

0 0 cðf Þ44 0 0

0 0 0 cðf Þ55 0

0 0 0 0 cðf Þ66

377777775

(10)

where

cðf Þ11 ¼ Eðf Þ1 ðt; zÞ1� n12n21

cðf Þ12 ¼ Eðf Þ2 ðt; zÞn121� n12n21

cðf Þ22 ¼ Eðf Þ2 ðt; zÞ1� n12n21

cðf Þ66 ¼ Gðf Þ12ðt; zÞ cðf Þ44 ¼ Gðf Þ

13ðt; zÞ cðf Þ55 ¼ Gðf Þ23ðt; zÞ

(11)

here, the superscript f refers to the viscoelastic functionallygraded material. Eðf Þ1 ðt; zÞ; Eðf Þ2 ðt; zÞ, Gðf Þ

13 ðt; zÞ;Gðf Þ23 ðt; zÞ are the time-

dependent relaxation modulus varying along the thickness ofthe viscoelastic FGM plate. For a specified moment, thesevarying relaxation modulus of the structure can be obtained byEq. (3).

And as for the anisotropy metal material, the relaxation matrixin Eq. (8) can be written as

Q ðtÞ ¼

266666664

cðmÞ11 cðmÞ

12 0 0 0

cðmÞ21 cðmÞ

22 0 0 0

0 0 cðmÞ44 0 0

0 0 0 cðmÞ55 0

0 0 0 0 cðmÞ66

377777775

(12)

where

cðmÞ11 ¼ EðmÞ

1 ðt;zÞ1� n12n21

cðmÞ12 ¼ EðmÞ

2 ðt;zÞn121� n12n21

cðmÞ22 ¼ EðmÞ

2 ðt;zÞ1� n12n21

cðmÞ66 ¼ GðmÞ

12 ðt;zÞ cðmÞ44 ¼ GðmÞ

13 ðt;zÞ cðmÞ55 ¼ GðmÞ

23 ðt;zÞ(13)

here, the superscript m refers to the viscoelastic metal material.EðmÞ1 ðt;zÞ;EðmÞ

2 ðt;zÞ;GðmÞ12 ðt;zÞ;GðmÞ

13 ðt;zÞ;GðmÞ23 ðt;zÞ are the time-

dependent relaxation modulus varying along the thickness of themetal layer.

Chose the piezoelectric layer as anisotropy material, and itsconstitutional relations can be written as

where, the superscript c refers to the piezoelectric ceramic mate-rial; Dx, Dy and Dz represent the electric displacement components;Ex, Ey, Ez denote the electric field components; CðcÞ

ij ; e0ij; gij are the


elastic modulus, educed piezoelectric constant and dielectricpermittivity constant, respectively, and

cðcÞ11 ¼ EðcÞ1 ðtÞ1� n12n21

cðcÞ12 ¼ EðcÞ2 ðtÞn121� n12n21

cðcÞ22 ¼ EðcÞ2 ðtÞ1� n12n21

cðcÞ66 ¼ GðcÞ12ðtÞ cðcÞ44 ¼ GðcÞ

13 ðtÞ cðcÞ55 ¼ GðcÞ23ðtÞ

e03i ¼ e3i �ci3e33c33

ði; j ¼ 1;2Þ; g033 ¼ e33e33c33

þ g33

(15)

The relations between electric field components and the electricpotential are

Ex ¼ �Jx Ey ¼ �Jy Ez ¼ �Jz (16)

For the brevity and convenience of expression, the followingVoigt notations are adopted

x/1; y/2; z/3; xz/4; yz/5; xy/6 (17)

The nonlinear equilibrium equations of the piezoelectric visco-elastic FGM plates with initial deflection can be written as

N1;xþN6;y ¼ 0N2;yþN6;x ¼ 0P1;xxþP2;yyþ2P6;xyþQ4þQ5;xþN1ðwxxþw0

xxÞþN2�wyyþw0

yy�þ

2N6�wxyþw0

xy�¼ 0

M1;xþM6;y�Q5 ¼ 0M2;yþM6;x�Q4 ¼ 0 ð18Þ

8<:

N1N2N6

9=; ¼

26642Aðf Þ

11 þ AðmÞ11 2Aðf Þ

12 þ AðmÞ12 0

2Aðf Þ21 þ AðmÞ

21 2Aðf Þ22 þ AðmÞ

22 0

0 0 2Aðf Þ66 þ AðmÞ

66

37755

8<:

uvy

uy þ vx þ

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

M1M2M6P1P2P6

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

¼

2666666666664

2Aðf Þ11 þ A

ðmÞ11 2A

ðf Þ12 þ A

ðmÞ12 0 2Dðf Þ

11 þ Dðm11

2Aðf Þ21 þ A

ðmÞ12 2A

ðf Þ22 þ A

ðmÞ22 0 2Dðf Þ

21 þ Dðm12

0 0 2Aðf Þ66 þ A

ðmÞ66 0

2Dðf Þ11 þ DðmÞ

11 2Dðf Þ12 þ DðmÞ

12 0 2~Aðf Þ11 þ ~A

ðmÞ11

2Dðf Þ21 þ DðmÞ


22 0 2~Aðf Þ21 þ ~A

ðmÞ12

0 0 2Dðf Þ66 þ DðmÞ

66 0

�Q4Q5

�¼

"2Fðf Þ44 þ FðmÞ

44 0

0 2Fðf Þ55 þ FðmÞ55

#5

�j2 þwy þw0

yj1 þwx þw0

x

�

where superscript m and f refer to metal and FGM, respectively, and

Aðf Þij ¼

Zh12 þh2

h12

cðf Þij dz; Aðf Þij ¼

Zh12 þh2

h12

cðf Þij ðg1ðzÞÞ2dz; ~Aðf Þij ¼

Zh12 þh2

h12

cðf Þij ðg2ð

Dðf Þij ¼

Zh12 þh2

h12

cðf Þij g1ðzÞg2ðzÞdz; Fðf Þ44 ¼Zh12 þh2

h12

cðf Þ44

�dg1dz

2

dz; Fðf Þ55 ¼Zh12 þh12

AðmÞij ¼

Zh12

�h12

cðmÞij dz; A

ðmÞij ¼

Zh12

�h12

cðmÞij ðg1ðzÞÞ2dz; ~A

ðmÞij ¼

Zh12

�h12

cðmÞij ðg2ð

DðmÞij ¼

Zh12

�h12

cðmÞij g1ðzÞg2ðzÞdz; FðmÞ

44 ¼Zh1

2

�h12

cðmÞ44

�dg1dz

2

dz; FðmÞ55 ¼

Z�

where, ðNi;Qi;Mi;PiÞ are the stress resultants of the piezoelectricviscoelastic FGM plates, and can be denoted as

Ni ¼ZH

2

�H2

sidz; Mi ¼ZH

2

�H2

sig1ðzÞdz; Pi ¼ZH

2

�H2

sig2ðzÞdz ði ¼ 1;2;6Þ

Qi ¼ZH

2

�H2

sig1;zðzÞdz ði ¼ 4;5Þ

ð19Þ

Now, the stress resultants ðNi;Qi;Mi;PiÞ are decomposed intotwo parts as following, one is that related to the piezoelectriclayer, and the other is related to the viscoelastic FGM and metallayer

Ni ¼ Ni þ NðpÞi ; Mi ¼ Mi þMðpÞ

i ; Pi ¼ Pi þ PðpÞi ;

Qi ¼ Qi þ Q ðpÞi (20)

here the superscript p refers to the piezoelectric layer.Substituting Eqs. (8) and (14) into Eq. (19), the stress resultants

ðNi;Qi;Mi; PiÞ and ðNðpÞi ;Q ðpÞ

i ;MðpÞi ; PðpÞi Þ can be written as follows.

The stress resultants ðNi;Qi;Mi; PiÞ related to the viscoelastic FGMand metal layers are

x þ 12w

2x þwxw0

xþ 1

2w2y þwyw0

ywxwy þw0

xwy þwxw0y

9=;

Þ 2Dðf Þ12 þ DðmÞ

12 0Þ 2Dðf Þ

22 þ DðmÞ22 0


66

2~Aðf Þ12 þ ~A

ðmÞ12 0

2~Aðf Þ22 þ ~A

ðmÞ22 0

0 2~Aðf Þ66 þ ~A

ðmÞ66

37777777777755

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

j1;xj2;y

j1;y þ j2;x�wxx �w0

xx�wyy �w0

yy�2wxy � 2w0

xy

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

(21)

zÞÞ2dz

h2

cðf Þ55

�dg1dz

2

dz

zÞÞ2dz

h12

h12

cðmÞ55

�dg1dz

2

dz


The stress resultants ðNðpÞi ;Q ðpÞ

i ;MðpÞi ; PðpÞi Þ related to the piezo-

electric layer with initial deflection are denoted as

8>><>>:

NðpÞ1

NðpÞ2

NðpÞ6

9>>=>>; ¼ 2

2664AðcÞ11 AðcÞ

12 0

AðcÞ21 AðcÞ

22 0

0 0 AðcÞ66

37758<:

ux þ 12w

2x þwxw0

x

vy þ 12w

2y þwyw0

yuy þ vx þwxwy þw0

xwy þwxw0y

9=;þ

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

2Zh1

2 þh2þh3

h12 þh2

ð�EzÞe031dz

2Zh1

2 þh2þh3

h12 þh2

ð�EzÞe032dz

0

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

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

MðpÞ1

MðpÞ2

MðpÞ6

PðpÞ1

PðpÞ2

PðpÞ6

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

¼ 2

2666666666664

AðcÞ11 A

ðcÞ12 0 DðcÞ

11 DðcÞ12 0

AðcÞ12 A

ðcÞ22 0 DðcÞ

12 DðcÞ22 0

0 0 AðcÞ66 0 0 DðcÞ

66

DðcÞ11 DðcÞ

12 0 ~AðcÞ11

~AðcÞ12 0

DðcÞ12 DðcÞ

22 0 ~AðcÞ12

~AðcÞ22 0

0 0 DðcÞ66 0 0 ~A

ðcÞ66

3777777777775

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

j1;xj2;y

j1;y þ j2;x�wxx �w0

xx�wyy �w0

yy�2wxy � 2w0

xy

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

þ

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

2Zh1

2 þh2þh3

h12 þh2

ð�EzÞg1ðzÞe031dz

2Zh1

2 þh2þh3

h12 þh2


0

2Zh1

2 þh2þh3

h12 þh2


2Zh1

2 þh2þh3

h12 þh2


0

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

(Q ðpÞ4

Q ðpÞ5

)¼ 2

"FðcÞ44 0

0 FðcÞ55

#�j2 þwy þw0

yj1 þwx þw0

x

�þ

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

2Zh1

2 þh2þh3

h12 þh2

��Ey��dg1

dz

2

e24dz

2Zh1

2 þh2þh3

h12 þh2

ð�ExÞ�dg1dz

2

e15dz

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

(22)

The first items on the right of the above three equations aremechanical stresses related to structure stiffness and the seconditems are mechanical stresses induced by the applied electric load.Stiffness coefficients in equations are

AðcÞij ¼

Zh12 þh2þh3

h12 þh2

cðcÞij dz; AðcÞij ¼

Zh12 þh2þh3

h12 þh2

cðcÞij ðg1ðzÞÞ2dz; ~AðcÞij ¼

Zh12 þh2þh3

h12 þh2

cðcÞij ðg2ðzÞÞ2dz

DðcÞij ¼

Zh12 þh2þh3

h12 þh2

cðcÞij g1ðzÞg2ðzÞdz; FðcÞ44 ¼Zh1

2 þh2þh3

h12 þh2

cðcÞ44

�dg1dz

2

dz; FðcÞ55 ¼Zh1

2 þh2þh3

h12 þh2

cðcÞ55

�dg1dz

2

dz

(23)

The main emphasis of this study is put on the effect of appliedelectric load on the creep buckling and post-buckling of thepiezoelectric viscoelastic FGM plates, so the direct piezoelectriceffect of the piezoelectric layer is neglected. If the electric fieldapplied on the plate is along the thickness of the plate, thereexists only the electric field component Ez. Assuming the electric

potential changes linearly across piezoelectric layer, the inducedelectric field Ez in the piezoelectric layer by the applied voltage Vecan be written as

Ez ¼ Ve

h3(24)

By substituting Eq. (24) into Eq. (22) we can obtain mechanicalforce induced by the applied voltage.


Substituting Eqs. (21) and (22) into Eq. (20), and thensubstituting Eq. (20) into Eq. (18), the nonlinear creep governingequations of the piezoelectric viscoelastic FGM plates with initialdeflection in the form of displacement functions can be obtained as

�Aðf Þ11 þ 1

2AðmÞ11

�5ðuxx þwxwxx þw0

xwxx þwxw0xxÞ þ

�Aðf Þ12 þ 1

2AðmÞ12

�5�

þ�Aðf Þ66 þ 1

2AðmÞ66

�5

�uyy þ vxy þwxywy þwxyw0

y þw0yywx þwxwyy þw

þAðcÞ11 ðuxx þwxwxx þw0

xwxx þwxw0xxÞ þ AðcÞ

12

�vxy þwywxy þw0

ywxy þ

þAðcÞ66

�uyy þ vxy þwxywy þwxyw0

y þw0yywx þwxwyy þw0

xwyy þw0xyw�

Aðf Þ12 þ 1

2AðmÞ12

�5�uxy þwxwxy þw0

xwxy þwxw0xy�þ �

Aðf Þ22 þ 1

2AðmÞ22

�5

þ�Aðf Þ66 þ 1

2AðmÞ66

�5

�uxy þ vxx þwxxwy þwxxw0

y þwxw0xy þwxwxy þw

þAðpÞ125

�uxy þwxwxy þw0

xwxy þwxw0xy�þ AðpÞ

225�vyy þwywyy þw0

yw

þAðpÞ665

�uxy þ vxx þwxxwy þwxxw0

y þwxw0xy þwxwxy þw0

xwxy þw0x�

Dðf Þ11 þ 1

2DðmÞ11

�5j1;xxx þ

�Dðf Þ12 þ 1

2DðmÞ12

�5j2;xyy þ

�~Aðf Þ11 þ 1

2~AðmÞ11

5ð�

5��wxxyy

�þ �Dðf Þ12 þ 1

2DðmÞ12

�5j1;xyy þ

�Dðf Þ22 þ 1

2DðmÞ22

�5j2;yyy þ

�~Aðf1

5��wxxyy

�þ �~Aðf Þ22 þ 1

2~AðmÞ22

5��wyyyy

�þ 2�Dðf Þ66 þ 1

2DðmÞ66

�5

�j1;xyy

5��2wxxyy

�þ �Fðf Þ44 þ 1

2FðmÞ44

�5

�j2;y þwyy

�þ�Fðf Þ55 þ 1

2FðmÞ55

�5

�j1;x

þ��

~Aðf Þ11 þ 1

2~AðmÞ11

5�ux þ 1

2w2x þwxw0

x�þ �

~Aðf Þ12 þ 1

2~AðmÞ12

5�vy þ 1

2w

þ��

~Aðf Þ12 þ 1

2~AðmÞ12

5�ux þ 1

2w2x þwxw0

x�þ �

~Aðf Þ22 þ 1

2~AðmÞ22

5�vy þ 1

2w

�wyy þw0

yy�þ 2

�~Aðf Þ66 þ 1

2~AðmÞ66

5�uy þ vx þwxwy þw0

xwy þwxw0y��

þDðcÞ12j2;xxy þ ~A

ðcÞ11ð�wxxxxÞ þ ~A

0ðpÞ12

��wxxyy�þ D0ðpÞ

12j1;xyy þ D0ðpÞ22j2;yyy

þ~AðcÞ22��wyyyy

�þ 2DðcÞ66

�j1;xyy þ j2;xxy

�þ 2~A

ðcÞ66��2wxxyy

�þ FðcÞ44

�j2;y

þ

0B@� pH þ 2

0B@ Zh1=2þh2þh3

h1=2þh2

ð�EzÞe031dzþZh1=2þh2þh3

h1=2þh2

ð�EzÞe032dz

1CA1CAðwx

�Aðf Þ11 þ 1

2AðmÞ11

�5j1;xx þ

�Aðf Þ12 þ 1

2AðmÞ12

�5j2;xy þ

�Dðf Þ11 þ 1

2DðmÞ11

�5ð�w

5��wxyy

�þ �Aðf Þ66 þ 1

2AðmÞ66

�5

�j1;hh þ j2;xy

�þ�~Aðf Þ66 þ 1

2~AðmÞ66

5��2

5ðj1 þwx þw0xÞ þ A

0ðcÞ11j1;xx þ A

0ðcÞ12j2;xy þ D0ðcÞ

11ð�wxxxÞ þ D0ðcÞ12

��wxy

þ~A0ðcÞ66��2wxxy

�� F 0ðcÞ55ðj1 þwx þw0xÞ ¼ 0�

Aðf Þ12 þ 1

2AðmÞ12

�5j1;xy þ

�Aðf Þ22 þ 1

2AðmÞ22

�5j2;yy þ

�Dðf Þ12 þ 1

2DðmÞ12

�5��w

5��wyyy


2AðmÞ66

�5�j1;xy þ j2;xx


2~AðmÞ66

5��2

5�j2 þwy þw0

y�þ A

ðcÞ12j1;xy þ A

ðcÞ22j2;yy þ DðcÞ

12

��wxxy�þ DðcÞ

22

��wyyy

þ~AðcÞ66��2wxxy

�� FðcÞ44

�j2 þwy þw0

y� ¼ 0

Considering a laminated piezoelectric FGM plates withfour simply and immovably supported edges, the boundary con-ditions are

vxy þwywxy þw0ywxy þwyw0

xy�

0xwyy þw0

xywy�

wyw0xy�

y� ¼ 0�vyy þwywyy þw0

ywyy þwyw0yy�

0xwxy þw0

xxwy�

yy þwyw0yy�

xwy� ¼ 0

wxxxxÞ þ�~Aðf Þ12 þ 1

2~AðmÞ12

Þ2 þ 1

2~AðmÞ12

þ j2;xxy

�þ 2

�~Aðf Þ66 þ 1

2~AðmÞ66

þwxx�

2y þwyw0

y

�ðwxx þw0

xxÞ

2y þwyw0

y

�

wxy þw0xy�þ DðcÞ

11j1;xxx

þ ~AðcÞ12��wxxyy

�þwyy

�þ FðcÞ55

�j1;x þwxx

�

x þw0xxÞ ¼ 0

xxxÞ þ�Dðf Þ12 þ 1

2DðmÞ12

�

wxxy��

Fðf Þ55 þ 12F

ðmÞ55

�

y�þ A

0ðcÞ66

�j1;yy þ j2;xy

�

xxy�þ �

Dðf Þ22 þ 1

2DðmÞ22

�

wxxy��

Fðf Þ44 þ 12F

ðmÞ44

��þ A

ðcÞ66

�j1;xy þ j2;xx

�

(25)


x ¼ 0; a : v ¼ w ¼ 0; j2 ¼ 0; My ¼ 0�Aðf Þ11 þ 1

2AðmÞ11

�5�ðuxx þwxwxx þw0

xwxx þwxw0xxÞ

þ�Aðf Þ11 þ 1

2AðmÞ11

5j1;xx �

�~Aðf Þ11 þ ~A

ðmÞ11

5ðwxxx �w0

xxxÞþAðcÞ

11 ðuxx þwxwxx þw0xwxx þwxw0

xxÞ ¼ 0

y ¼ 0; b : u ¼ w ¼ 0; j1 ¼ 0 ; Ny ¼ 0 ; My ¼ 0 (26)

3. Solution methodology

Eq. (25) is an integro-partial differential equation, and therelaxation modulus of the laminated piezoelectric viscoelastic FGMplate is both time and space dependent, so it is impossible to get ananalytical solution to this problem. In order to get the solutions tothe governing Eq. (25) and boundary condition Eq. (26), thedisplacement variables u; v;w;j1;j2 are separated both for spaceand for time domain. Theway of dealingwith the partial derivativeswith respect to the space and the convolution integral is introducedin details in the following.

The relaxation modulus in equations can be expanded as theProny-Dirichlet series,

e1ðzÞ ¼ E1ðzÞ=E ¼ e10ðzÞ þPki¼1

e1iðzÞexpð � a1iðzÞsÞ









(27)

where, k is the number of the terms to be expanded;e10ðzÞ; e1iðzÞ;a1iðzÞ;.;a23iðzÞ are the material parameters, all ofwhich are positive and varying along the thickness of the plate.

For the generality, the eðsÞ represents Aij;Aij;~Aij;Bij;.; ~Dij and f

refers to unknown variables u; v;w;j1;j2. And the viscoelastic FGMmaterial is assumed as standard linear solid in present model.Dividing time s equally into small time segments Ds, the followingrelations can be obtained with adoption of the Newton-Cotestrapezoidal formula.

e5fi;j ¼ eð0Þfi;jðtÞ þZt

o

_eðt � sÞfi;jðsÞds

¼ eð0Þfi;jðtÞ þZt

0

�a1e1e�a1tea1sfi;jðsÞds

¼ eð0Þfi;jðtÞ þ�� a1e1e�a1t

�264ZDt0

þZ2DtDt

þ.

ZNDtðN�1ÞDt

375

¼ eð0Þfi;jðtÞ þ�� a1e1e�a1t

�hDt2

�ea1Dt fi;jðDtÞ

þe0fi;jð0Þ þ ea12Dt fi;jð2DtÞ þ ea1Dt fi;jðDtÞ.

þea1NDt fi;jðNDtÞ þ ea1ðN�1ÞDt fi;jððN � 1ÞDtÞ�

(28)

According to the way dealing with the convolution in the above,Eq. (25) and the boundary conditions (26) can be converted to a set

of nonlinear algebraic equations by using the central differenceformulas and forward difference formulas, and the solutions tothese algebraic equations can be obtained by the iterative methods.In each iterative step J, the nonlinear items in equations and theboundary equations can be linearized in the following way:

ðx$yÞJ ¼ ðxÞJ$ðyÞJp (29)

where ðyÞJp is the average value of those obtained in previous twoiterations. For the initial iterative step, it can be determined byusing the quadratic extrapolation, i.e.

ðyÞJp ¼ AðyÞJ�1þBðyÞJ�2þCðyÞJ�3 (30)

and for different iterative steps, the coefficients A,B, and C can beexpressed as follows

J ¼ 1 : A ¼ 1; B ¼ 0; C ¼ 0J ¼ 2 : A ¼ 2; B ¼ �1; C ¼ 0J � 3 : A ¼ 3; B ¼ �3; C ¼ 1

(31)

Using the linearization method in Eq. (29), the governingequation (25), and boundary equation (26) can be transformed toa series of linear equation at each discrete point by applyingdifference quotient, and the solution to these linear equations canbe acquired. For each iteration step, the iterations lasts until thedifference of the present value and the former is smaller than0.05%. When the convergent solution in Jth step is held, theviscoelastic stiffness of the FGM plate related to period time can becalculated and the convolution in equation can be dealt withaccording to Eq. (28).

4. Numerical results

In this part, some numerical examples are carried out witha rectangular viscoelastic FGM plate, a common use component inengineering application. The maintenance of the components andavoidance of too large creeping post-buckling deflection are ofgreat importance for the sound application of some exact FGMdevices, such as devices in aviation industry and nuclear industry.In present model, two piezoelectric films mounted on the bottomand top surfaces of the viscoelastic FGM plate are applied to controlthe creeping speed of the structure, and the numerical examplesshow that a good effect can be obtained.

In section 4.1, we firstly obtained the instantaneous criticalcreep buckling load pE and endurance critical creep bucking loadpE for present model basing on linear theory. The Galerkin methodis mainly applied in this section for it is more convenient indealing with linear problem. Some comparisons are also carriedout to validate present model and the calculating method bydegenerating the present model. In section 4.2, some numericalexamples are carried out to investigate the post-buckling of thepiezoelectric FGM plate basing on nonlinear theory. The effects ofthe piezoelectric load and FGM volume fraction exponent n onpost-buckling behavior of the piezoelectric viscoelastic FGM plateare investigated and discussed, and some interesting conclusionshave been drawn which would be useful to the engineeringapplications.

4.1. Creep buckling of the laminated piezoelectric viscoelasticfunctionally graded plates

For the linear case, the creep buckling governing equations ofthe piezoelectric viscoelastic FGM plates with initial deflection canbe obtained by degeneration of Eq. (25).

�Aðf Þ11 þ 1

2AðmÞ11

�5uxx þ

�Aðf Þ12 þ 1

2AðmÞ12

�5vxy þ

�Aðf Þ66 þ 1

2AðmÞ66

�5

�uyy þ vxy

�þ AðcÞ11uxx þ AðcÞ

12vxy þ AðcÞ66

�uyy þ vxy

� ¼ 0�Aðf Þ12 þ 1

2AðmÞ12

�5uxy þ

�Aðf Þ22 þ 1

2AðmÞ22

�5vyy þ

�Aðf Þ66 þ 1

2AðmÞ66

�5

�uxy þ vxx

�þ AðpÞ125uxy þ AðpÞ

225vyy þ AðpÞ665

�uxy þ vxx

� ¼ 0�Dðf Þ11 þ 1

2DðmÞ11

�5j1;xxx þ

�Dðf Þ12 þ 1

2DðmÞ12

�5j2;xyy þ

�~Aðf Þ11 þ 1

2~AðmÞ11

5ð�wxxxxÞ þ

�~Aðf Þ12 þ 1

2~AðmÞ12

5��wxxyy


2DðmÞ12

�5j1;xyy þ

�Dðf Þ22 þ 1

2DðmÞ22

�5j2;yyy þ

�~Aðf Þ12 þ 1

2~AðmÞ12

5��wxxyy

�þ �~Aðf Þ22 þ 1

2~AðmÞ22

5��wyyyy

�þ 2�Dðf Þ66 þ 1

2DðmÞ66

�5

�j1;xyy þ j2;xxy

�þ 2

�~Aðf Þ66 þ 1

2~AðmÞ66

5��2wxxyy

�þ �Fðf Þ44 þ 1

2FðmÞ44

�5

�j2;y þwyy

�þ�Fðf Þ55 þ 1

2FðmÞ55

�5

�j1;x þwxx

�þ��

~Aðf Þ11 þ 1

2~AðmÞ11

5u;x þ

�~Aðf Þ12 þ 1

2~AðmÞ12

5vyðwxx þw0

xxÞ þ��

~Aðf Þ12 þ 1

2~AðmÞ12

5ux þ

�~Aðf Þ22 þ 1

2~AðmÞ22

5vy

�wyy þw0

yy�þ 2

�~Aðf Þ66 þ 1

2~AðmÞ66

5�uy þ vx

��wxy þw0

xy�þ DðcÞ

11j1;xxx

þDðcÞ12j2;xxy þ ~A

ðcÞ11ð�wxxxxÞ þ ~A

0ðpÞ12

��wxxyy�þ D0ðpÞ

12j1;xyy þ D0ðpÞ22j2;yyy þ ~A

ðcÞ12��wxxyy

�þ~AðcÞ22��wyyyy

�þ 2DðcÞ66

�j1;xyy þ j2;xxy

�þ 2~A

ðcÞ66��2wxxyy

�þ FðcÞ44

�j2;y þwyy

�þ FðcÞ55

�j1;x þwxx

�

þ

0B@� pH þ 2

0B@ Zh1=2þh2þh3

h1=2þh2

ð�EzÞe031dzþZh1=2þh2þh3

h1=2þh2

ð�EzÞe032dz

1CA1CAðwxx þw0

xxÞ ¼ 0

�Aðf Þ11 þ 1

2AðmÞ11

�5j1;xx þ

�Aðf Þ12 þ 1

2AðmÞ12

�5j2;xy þ

�Dðf Þ11 þ 1

2DðmÞ11

�5ð�wxxxÞ þ

�Dðf Þ12 þ 1

2DðmÞ12

�5��wxyy


2AðmÞ66

�5

�j1;hh þ j2;xy


2~AðmÞ66

5��2wxxy

�� Fðf Þ55 þ 1

2FðmÞ55

�

5ðj1 þwx þw0xÞ þ A

0ðcÞ11j1;xx þ A

0ðcÞ12j2;xy þ D0ðcÞ

11ð�wxxxÞ þ D0ðcÞ12

��wxyy�þ A

0ðcÞ66

�j1;yy þ j2;xy

�þ~A

0ðcÞ66��2wxxy

�� F 0ðcÞ55ðj1 þwx þw0xÞ ¼ 0�

Aðf Þ12 þ 1

2AðmÞ12

�5j1;xy þ

�Aðf Þ22 þ 1

2AðmÞ22

�5j2;yy þ

�Dðf Þ12 þ 1

2DðmÞ12

�5��wxxy


2DðmÞ22

�5��wyyy


2AðmÞ66

�5�j1;xy þ j2;xx


2~AðmÞ66

5��2wxxy

�� Fðf Þ55 þ 1

2FðmÞ55

�5�j2 þwy þw0

y�þ A

ðcÞ12j1;xy þ A

ðcÞ22j2;yy þ DðcÞ

12

��wxxy�þ DðcÞ

22

��wyyy�þ A

ðcÞ66

�j1;xy þ j2;xx

�þ~A

ðcÞ66��2wxxy

�� FðcÞ55

�j2 þwy þw0

y� ¼ 0

(32)


The displacement functions of the plate with simply andimmovably supported boundary conditions are chosen as

u ¼ u sinðkpxÞcosðlpyÞv ¼ v cosðkpxÞsinðlpyÞw ¼ w cosðkpxÞcosðlpyÞj1 ¼ 41sinðkpxÞcosðlpyÞj2 ¼ 42sinðkpxÞcosðlpyÞ

(33)

The initial deflection of the plate is set as

w0 ¼ w0cosðkpxÞcosðlphÞ (34)

where, k and l are the different modes in length andwidth directionof the FGM plate; w0 is the central initial deflection of the plate.Substituting Eqs. (33) and (34) into Eq. (32), application of theGalerkin method gives the following equations:

266664m11 m12 m13 m14 m15m21 m22 m23 m24 m25m31 m32 m33 m34 m35m41 m42 m43 m44 m45m51 m52 m53 m54 m55

3777755

8>>>><>>>>:

uvw4142

9>>>>=>>>>;

¼

8>>>><>>>>:

00

Kw0p2

00

9>>>>=>>>>;

(35)

where,K ¼ pH � 2ðR h1=2þh2þh3

h1=2þh2ð�EzÞe031dzþ

R h1=2þh2þh3

h1=2þh2ð�EzÞe032dzÞ,mij is

the material parameters relatedto the viscoelastic relaxation modulus, the mode value k, l and

the elastic modulus of the piezoelectric material. Substituting theinstantaneous elastic modulus E(0) and endurance elastic modulusEðNÞ in conjunction with the material parameters of the piezo-electric material into Eq. (35), five equations would be obtained.Considering the determinant of the above equations to be zerogives following equations

Det½m� ¼ 0 (36)

The instantaneous critical load pH and endurance critical load pEof the piezoelectric viscoelastic FGM plates can be obtained fordifferent modes by solving the above equations when k and l are setdifferent values.

To ensure the validity and accuracy of the present method, thepresent laminated model is degenerated to one single isotropicviscoelastic plate when no piezoelectric voltage is applied.Comparison of the results by the present model with that in Pengand Fu (2003) is presented in Fig. 2. In the Peng and Fu (2003),the relations between buckling load and time in Fig. 2 is obtained

0 70 140 210 280 350 420

54

60

66

72

78

84

90

Ref.[17]Present

mm.N/daol

1-

time/h

Fig. 2. Critical load of the viscoelastic plate.

Table 1Dimensionless instantaneous critical load PH for different buckling modes of thepiezoelectric viscoelastic FGM plate.

k ¼ 1 k ¼ 2 k ¼ 3 k ¼ 4

l ¼ 1 n ¼ 0.1 0.885372 0.379273 0.7400385 1.089n ¼ 1 0.783679 0.327319 0.6302487 0.876313n ¼ 2 0.6933753 0.313848 0.6086246 0.764706n ¼ 3 0.548711 0.299832 0.5057313 0.656565

l ¼ 2 n ¼ 0.1 3.96905 1.64996 3.21049 4.3842744n ¼ 1 3.02462 1.28398 2.129761 4.0107354n ¼ 2 2.74122 1.066065 1.312402 3.204324n ¼ 3 2.63495 1.0144709 1.109216 2.983806

l ¼ 3 n ¼ 0.1 8.3263 3.63212 5.169374 7.58578n ¼ 1 6.27223 2.64407 3.78424 4.0199478n ¼ 2 5.67509 2.20945 3.33421 3.8290256n ¼ 3 5.46041 2.06466 3.022449 3.754893

l ¼ 4 n ¼ 0.1 13.3021 5.83073 9.152478 14.294177n ¼ 1 9.83152 4.2041 6.122669 11.0468244n ¼ 2 8.87095 3.68702 5.47138 9.92527n ¼ 3 8.54939 3.62041 5.029979 9.1527955

Table 2Dimensionless instantaneous critical load PH and endurance critical load PE of thepiezoelectric viscoelastic FGM plate.

l1 ¼ 0.1 l1 ¼ 0.05 l1 ¼ 0.03 l1 ¼ 0.02

n ¼ 0.1 PE 5.41246 0.875268 0.1286471 0.0599563PH 5.52317 0.885372 0.1357033 0.0650049

n ¼ 1 PH 3.68353 0.719634 0.0908108 0.0346867PH 3.75895 0.783679 0.0939635 0.0358261

n ¼ 2 PE 1.11776 0.668021 0.092961 0.0339754PH 1.25491 0.693753 0.093586 0.035962

n ¼ 3 PE 1.10131 0.511627 0.032244 0.0200501PH 1.295439 0.548711 0.03597 0.0296501


through Laplace transformation and Bellman numerical inversion.Well agreement between these two results can be observed in Fig. 2where the instantaneous critical load is pH ¼ 85:52N=mm, and theendurance load is pE ¼ 57:18N=mm. As similar to the discussion inPeng and Fu (2003), it can be observed in the figure, when p � pH ,the buckling of the plate takes place immediately; when p� pE , thedestabilization of the plate will not happen; when pE � p � pH ,the delayed destabilization of the plate can be observed, which isthe typical property of the creep buckling of the viscoelasticstructure.

Without special indication, the FGM layer is set as the combi-nation of the anisotropy metal and ceramic material in thefollowing buckling and post-buckling analysis of the laminatedpiezoelectric viscoelastic FGM plates. The material parameters ofthe metal are set as (Zheng and Fu, 2005):

EðmÞ1 ¼

�9:25þ0:75e�aðmÞ

1 s�GPa; EðmÞ

2 ¼�5:32þ0:68e�aðmÞ

2 s�GPa

GðmÞ23 ¼ GðmÞ

13 ¼�0:05þ0:12e�aðmÞ

3 s�GPa;

GðmÞ12 ¼

�0:68þ0:12e�aðmÞ

4 s�GPa

nðmÞ12 ¼ 0:3;rðmÞ ¼ 7800:0 Kg=m3

aðmÞ1 ¼ 0:022; aðmÞ

2 ¼ 0:024; aðmÞ3 ¼ 0:026; aðmÞ

4 ¼ 0:028

The piezoelectric layer is chosen as PZT-5A, of which therelaxation coefficient aðcÞ is set to be zero, i.e. the viscoelasticproperty of the ceramic material is not considered in presentanalysis, and the material parameters are Fu and Wang, 2006:

EðcÞ1 ¼ EðcÞ2 ¼ 61:0 GPa; EðcÞ3 ¼ 53:2 GPa;

GðcÞ12 ¼ 22:6 GPa; GðcÞ

13 ¼ GðcÞ23 ¼ 21:1 GPa

e31 ¼ e32 ¼ 7:209 C=m2; e33 ¼ 15:118 C=m2;

e24 ¼ e15 ¼ 12:72 C=m2

n12 ¼ 0:35; r ¼ 7750:0 Kg=m3; g11 ¼ g22 ¼ 1:53� 10�8F=m;

g33 ¼ 1:5� 10�8 F=m

The critical buckling loads for different modes of the piezo-electric viscoelastic FGM are firstly investigated. The dimension-less instantaneous critical load PHðpH=EðcÞ1 Þ for different bucklingmodes of FGM plate are listed in Table 1. It can be seen theinstantaneous critical load PH increases as the volume fractionexponent of the viscoelastic FGM plate increases, and the instan-taneous critical load PH is higher along y direction for a highermode.

When keep the thickness of the piezoelectric and metal layerconstant, the effect of the thickness and volume fraction exponent nof FGM layer on the dimensionless instantaneous critical loadPHðpH=EðcÞ1 Þ and endurance critical load PEðpE=EðcÞ1 Þ are listed inTable 2. In the table, l1 is the ratio of the FGM layer thickness to thespan of the plateðh2=aÞ ; n is volume fraction exponent of theviscoelastic FGM plate. The geometric parameters are set ash1 ¼ 0:02 mh3 ¼ 0:005 m, a ¼ b ¼ 1 m. It can be found fromthe table the instantaneous (endurance) critical load decreasesas the volume fraction exponent value n increases and the increaseof the thickness ðl1Þ of the plate causes the increase of theinstantaneous (endurance) critical load of the structure.

4.2. Creep post-buckling of the laminated piezoelectric viscoelasticfunctionally graded plates

In the following, the discussion of the creep post-buckling of thepiezoelectric viscoelastic FGM plates is carried out in details whenthe same material parameters with that in section 4.1 are taken.With no special indication in the following numerical examples,the geometric parameters of the laminated plate are set ash1 ¼ 0:02 m, h2 ¼ 0:05 m, h3 ¼ 0:005 m, a ¼ b ¼ 1 m. Definethe total thickness of the laminated piezoelectric viscoelastic FGMplates as H ¼ h1 þ 2h2 þ 2h3. In the presented results, Wðw=HÞrepresents the non-dimensional central deflection of the plate,W0ðw0=HÞ represents the non-dimensional initial deflection of theplate and s represents the time step which is set as s ¼ 6 h in thefollowing numerical examples.

Fig. 3 presents the effect of the geometric nonlinearity on thecreep post-buckling of the piezoelectric viscoelastic FGM plates. Inthe figure, real line refers to the linear result while the dashed linerefers to the nonlinear results. The initial deflection of the plate is

50 100 150 200

0.1

0.2

0.3

0.4

0.5

0.6

W

t

p=0.35pH

p=0.5pH

p=0.25pH

30 60 90 120 1500.0

0.3

0.6

0.9

1.2

1.5

W

τ

p=1.5pH

p=1.3pH

p=0.92pH

p=1.1pH

p=0.93pH

( )Ep p< ( )Ep p≥

Fig. 3. Effect of the nonlinearity on the creep post-buckling of the piezoelectric viscoelastic FGM plate.


set as W0 ¼ 0:1. The lowest instantaneous critical load andendurance critical load of the structure for this case can be calcu-lated as pH ¼ 7:83679� 108ðN=mÞ and pE ¼ 7:19634� 108ðN=mÞ,respectively, and pE=pH ¼ 0:9183. From the figure, it can be foundwhen p < PE and p is comparatively small, the result for the linearcase is nearly the same with that for the nonlinear case. As the in-plane pressure p increases, the deflection increases and the differ-ence between the linear result and nonlinear result becomes great.When p � pE , the laminated viscoelastic functionally graded platewould exhibit the instantaneous (endurance) instability when thelinearity is considered and the deflection increases to be infinite asthe time prolongs. But for the nonlinear case, the deflection of thestructure would go to be a finite value as the time elapses. So it canbe concluded the linear theory is not suitable for the creep post-buckling analysis of the viscoelastic structure when the in-planepressure p is comparatively large, especially when the in-planepressure is greater than the endurance critical load of the struc-ture. Therefore, the application of the nonlinear theory in theanalysis of the post-buckling of the viscoelastic structure isnecessary.

Fig. 4 gives the effect of the transverse shear deformation on thecreep post-buckling of the laminated piezoelectric viscoelasticfunctionally graded plate. In the figure, the geometric parametersare set as h1 ¼ 0:02 m; h3 ¼ 0:005 m; a ¼ b ¼ 1 m, W0 ¼ 0:1

20 40 60 80 1000.4

0.6

0.8

1.0

1.2

1.4

h2/a=0.050

h2/a=0.045

W

t

h2/a=0.042

Fig. 4. Effect of the transverse shear deformation on the creep post-buckling of thepiezoelectric viscoelastic FGM plate.

and the in-plane pressure is p ¼ 1:15pH . In the figure, the real lineindicates the result when transverse shear deformation is consid-ered and the dashed line indicates that when transverse sheardeformation is not considered. The figure shows the post-bucklingdeflection increases when the structure transverse shear defor-mation is considered and it differs increasingly great from the casewhen the transverse shear deformation is not considered as thetime prolongs. It also can be found from the figure that as thethickness-span ratio decreases (the thickness increases), theinfluence of the transverse shear deformation on the creep post-buckling of the viscoelastic FGM plate becomes apparent, whichpresents the necessity of the consideration of the transverse sheardeformation in the post-buckling analysis of the FGM structurewhen the thickness is comparatively large.

Fig. 5 shows the effect of the volume fraction exponent value non the creep post-buckling of the laminated piezoelectric visco-elastic functionally graded plate. The geometric parameters are setas h1 ¼ 0:02 m, h2 ¼ 0:05 m, h3 ¼ 0:005 m, W0 ¼ 0:1 and thein-plane pressure is p ¼ 0:85pH . It can be found from the figure thecentral dimensionless deflection of the structure under the in-plane pressure goes to be a finite value as the time increases. Anapparent influence of the volume fraction exponent value n on thecreep post-buckling of the laminated viscoelastic functionallygraded plate can be observed. And the creep post-buckling

20 40 60 80 100

0.35

0.40

0.45

0.50

0.55

0.60

n=2

n=1

n=0.5

n=0.1

W

t

n=0.1

Fig. 5. Effect of the volume fraction exponent value n on the creep post-buckling of thepiezoelectric viscoelastic FGM plate.

50 100 150 2001.0

1.1

1.2

1.3

W0=0.2W0=0.1

w+w

0

t

W0=0.01

Fig. 8. Effect of the initial imperfection on the creep post-buckling of the piezoelectricviscoelastic FGM plate.

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

λ1=0.065

λ1=0.065λ1=0.06

λ1=0.055

λ1=0.050

W

t

λ1=0.045

Fig. 6. Effect of the thickness-span ratio on the creep post-buckling of the viscoelasticFGM plate.


deflection of the structure decreases obviously as the n increasesand it turns to be a finite value (the stable state) more quickly. Sothe optimization of the volume fraction exponent value n of thefunctionally graded structure is necessary to avoid the failure of thestructure caused by too large creep post-buckling deflection.

The influence of the thickness-span ratio on the creep post-buckling of the laminated piezoelectric viscoelastic functionallygraded plate is presented in Fig. 6 when the material and geometricparameters are set the same as that in Fig. 5. It can be seen from thefigure the thickness-span ratio influences the laminated visco-elastic functionally graded plate greatly. With the increase of thethickness of the structure, the post-buckling deflection of thestructure decreases, and the deflection turns to be a finite value (thestable state) more quickly.

Fig. 7 discussed the effect of the applied electric load on thecreep post-buckling of the piezoelectric viscoelastic FGM plate. Thegeometric parameters are set as h1 ¼ 0:02 m, h2 ¼ 0:055 m,h3 ¼ 0:005 m, W0 ¼ 0:01 and n ¼ 1. The in-plane pressure is setas p ¼ 0:8PH . The applied positive electric load decreases thestructure deflection and the central deflection of the plate goes tobe a finite valuemore quickly. As the applied electric load increases,this influence becomes more apparent. When a negative electric

20 40 60 800.4

0.5

0.6

0.7

0.8

0.9

1.0

ve=11x106Vve=5x106V

ve=−5x106Vve=0V

Wc

τ

ve=−11x106V

Fig. 7. Effect of the applied electric load on the creep post-buckling of the piezoelectricviscoelastic FGM plate.

load is applied, a phenomenon contrary to the positive electric loadcan be observed. So it is a feasible way to control the creep post-buckling of the structure and make it more effectively work byapplying the positive electric load.

Fig. 8 gives the effect of initial structural deflection on the creeppost-buckling of the laminated piezoelectric viscoelastic function-ally graded plate. The volume fraction exponent value n is set asn ¼ 2. The lowest instantaneous critical load can be calculated aspH ¼ 6:93753� 108N, and the in-plane pressure is chosen asp ¼ 1:5pH . It can be seen from the figure that with the initialdeflection of the structure W0 increases, the deflection of thelaminated piezoelectric viscoelastic functionally graded plate Wdecreases, but the total deflection of the laminated piezoelectricviscoelastic functionally graded plates W þW0 will increase as theW0 increases, which confirms the research on elastic plate theory inChia (1980).

5. Conclusion

Based on high order transverse shear deformation andgeometric nonlinear theory, the buckling and post-buckling of thelaminated piezoelectric viscoelastic FGM plates with initialdeflection under the in-plane pressure are investigated by adoptingthe Boltzmann linear superposition principle. Effects of the struc-tural parameters on the creep post-buckling of the laminatedpiezoelectric viscoelastic FGM plate are discussed in details andsome useful conclusions have been drawn. It is observed thatgeometric linearity theory overestimates the creep post-bucklingdeformation of the structure, and for the geometric nonlinearitycase, the post-buckling deflection of the viscoelastic FGM platesunder in-plane pressure turns to be a finite value (a stable state) asthe time prolongs. Thin plate theory undervalues the post-bucklingdeflection of the viscoelastic FGM plates, and the greater of thethickness-span ratio, the more eminent of the influence of thetransverse shear deformation on the structure’s post-bucklingdeflection. The numerical results indicate that the volumefraction exponent value n of the FGM obviously affects the creeppost-buckling of the structure. It is also observed that the appliedpositive electric load decreases the structure deflection and thecentral deflection of the plate goes to be a finite value more quickly.So it is a feasible way to control the creep post-buckling of thestructure and make it more effective to work by applying thepositive electric load.


Acknowledgement

The work described in this paper is supported by the NationalNatural Science Foundation of China under Grant No. 10872066.

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