Vibration and dynamic buckling control of imperfect hybrid FGM plates with temperature-dependent material properties subjected to thermo-electro-mechanical loading conditions

Composite Structures 88 (2009) 240–252

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Vibration and dynamic buckling control of imperfect hybrid FGM plates with temperature-dependent material properties subjected to thermo-electro-mechanical loading conditions M. Shariyat * Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Available online 7 April 2008 Keywords: Dynamic buckling Piezoelectric plate FGM Vibration control Initial imperfection Temperature-dependency

a b s t r a c t Dynamic buckling analysis of FGM plates has not been accomplished so far. In the present paper, vibration and dynamic buckling of FGM rectangular plates with surface-bonded or embedded piezoelectric sensors and actuators subjected to thermo-electro-mechanical loading conditions are investigated. A finite element formulation based on a higher-order shear deformation theory is developed. Both initial geometric imperfections of the plate and temperature-dependency of the material properties are taken into account. Dynamic buckling of plates already pre-stressed by other forms of loading conditions is assumed to occur under suddenly applied thermal or mechanical loads. A nine-node second order Lagrangian element, an efficient numerical algorithm for solving the resulted highly non-linear governing equations, and an instability criterion already proposed by the author are employed. A simple negative velocity feedback control is used to actively control the dynamic response of the plate. Results show that generally, initial geometric imperfections lead to an increased fundamental bending natural frequency and decreased buckling loads. Furthermore, buckling mitigation due to utilizing integrated piezoelectric sensors and actuators is mainly achieved in extremely high gain values. Therefore, the piezoelectricity effect on the buckling load is small in applicable voltages. It is also noticed that the temperature-dependency and initial geometric imperfections remarkably affect the buckling loads. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded structures are mainly developed to operate in environments with extremely high temperatures. Low thermal conductivity, low coefficient of thermal expansion and core ductility have enabled the FGM materials to withstand higher temperature gradients for a given heat flux. Extensive thermal stress studies performed by Noda [1] and Tanigawa [2] reveal that the weakness of the fiber reinforced laminated composite materials, such as delamination, huge residual stresses, and locally large plastic deformations, may be avoided or reduced in FGM materials. Since functionally graded structures are most commonly used in high temperature environments where significant changes in properties of the constituent materials are to be expected, it is essential to take into consideration this temperature-dependency for accurate prediction of the mechanical response. Reddy and Chin [3], have investigated this phenomenon and proposed a nonlinear equation to describe variations of the mechanical properties with temperature change. * Tel.: +98 9122727199; fax: +98 21 88674748. E-mail addresses: [email protected], [email protected] 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.04.003

Effects of utilizing piezoelectric layers on the free vibration of the FGM rectangular plates have been investigated by many authors. Yang et al. [4] studied large amplitude vibration of a pre-stressed FGM plate with two surface-mounted piezoelectric actuator layers under uniform temperature change, using Reddy’s higher-order shear deformation plate theory. Huang and Shen [5] studied the nonlinear vibration and dynamic response of FGM plates with surface-bonded piezoelectric layers and uniform temperature based on a higher-order shear deformation theory. Material properties of the substrate FGM layer were assumed to be temperature-dependent whereas the material properties of piezoelectric layers were assumed to be independent of the temperature. Active vibration control of the FGM plates may be accomplished through utilizing integrated piezoelectric sensor and actuator layers. Reddy and Cheng [6] gave 3D asymptotic solutions for FGM plates with a smart material layer to suppress the vibration amplitude. He et al. [7] presented a finite element formulation for shape and vibration control of the FGM plates using a constant velocity feedback control algorithm. Liew et al. [8–10] presented a finite element formulation for the shape and vibration control of FGM plates.

M. Shariyat / Composite Structures 88 (2009) 240–252

Numerous researches have been accomplished in the static thermal buckling analysis of simple FGM plates field. Stability of moderately thick FGM rectangular plates under thermal loads is analyzed by Lanhe [11]. Two types of thermal loading, uniform temperature rise and gradient through the thickness are considered. Morimoto [12] presented a thermal buckling analysis of FGM rectangular plates subjected to partial in-plane heating and uniform temperature rise through thickness. A three-dimensional thermomechanical buckling analysis is developed by Na and Kim [13] for FGM structures that are composed of ceramic, FGM, and metal layers. Abrate [14] studied the free vibration, static buckling, and static deformation of the FGM plates. It was shown that the natural frequencies, buckling loads and static deflections of functionally graded plates are always proportional to those of homogeneous isotropic plates. A thermal postbuckling analysis is presented by Shen [15] for imperfect FGM plates subjected to an in-plane parabolic temperature distribution or heat conduction. The material properties of the FGM layers are assumed to be temperature-dependent. Samsam Shariat and Eslami [16] proposed a closed-form solution for buckling of rectangular thick functionally graded plates under mechanical and thermal loads. Yang et al. [17] investigated the sensitivity of the post-buckling behavior of the FGM plates to initial geometrical imperfections. Na and Kim [13,18] used solid elements to calculate the buckling temperature of the FGM plates. Recently, Park and Kim [19] presented a thermal postbuckling and vibration analysis for FGM plates with temperature-dependent material properties using the finite element method. Lanhe et al. [20] investigated the dynamic stability of FGM plates subjected to aero-thermomechanical loads, using the first order shear deformation theory and the moving least squares differential quadrature method. Wu et al. [21] used the first order shear deformation theory and the fast converging finite double Chebyshev polynomials to study the post-buckling response of FGM plates subjected to thermal and mechanical loadings. Thermal buckling of the FGM hybrid plates was investigated by limited researchers. Liew et al. [22] examined postbuckling behavior of FGM rectangular plates with surface-bonded piezoelectric actuators subjected to combined action of uniform temperature change, in-plane forces, and constant applied actuator voltages. Reddy’s higher-order shear deformation plate theory was employed. Shen [23] presented a postbuckling analysis for a simply supported, imperfect FGM plate with piezoelectric actuators subjected to an electro-mechanical loading and a uniform temperature. The material properties of both FGM and piezoelectric layers were assumed to be temperature-dependent. A higher order shear deformation plate theory was used. Huang and Shen [5] studied the nonlinear vibration and dynamic response of FGM

241

plates with surface-bonded piezoelectric layers in thermal environment. The temperature field was considered to vary in the thickness direction of the plate. Material properties of the substrate FGM layer were assumed to be temperature-dependent whereas material properties of the piezoelectric layers were assumed to be independent of the temperature. Higher-order shear deformation plate theory was used. Chen et al. [24] employed the element free Galerkin method and Mindlin plate theory to analyze buckling of piezoelectric FGM rectangular plates subjected to non-uniformly distributed loads, heat and voltage. Main novelties of the present work are incorporating the initial geometric imperfections, temperature-dependency, and dynamic thermal buckling analysis of the piezoelectric FGM plates. The above review reveals that no dynamic buckling analysis has been performed for the FGM hybrid plates so far. Furthermore, only few researches have been accomplished in thermal buckling of adaptive composite plates. These works have focused on the static thermal buckling. Dynamic thermal buckling analysis of the piezolaminated plates has not been performed yet. Temperature dependency of the material properties is also neglected in majority of the previous works. In the present paper, both vibration and buckling control of imperfect FGM plates with surface bonded or embedded integrated piezoelectric sensor and actuator layers are studied. A finite element formulation based on a high order shear deformation theory is used to investigate vibration and buckling control under thermo-electro-mechanical loads. Temperature-dependency of the material properties, initial thermal stresses and imperfections, and control voltage effects on the natural frequencies and buckling under thermo-electro-mechanical loads are also taken into consideration. A revised Budiansky’s criterion recently proposed by the author [25] is employed to detect the buckling load. In the results section, dynamic buckling is investigated for two complicated cases: dynamic thermal buckling of hybrid plates already subjected to electric potential and dynamic buckling of plates already subjected to thermal and electrical loads, under a suddenly applied mechanical compression. 2. The governing equations A hybrid plate constructed of an FGM substrate and one or two surface-bonded or embedded piezoelectric layers is considered. The FGM substrate is assumed to be made of a mixture of two constituent materials so that the top layer of the FGM plate is ceramicrich while its bottom layer is metal-rich. The coordinate system and the geometric parameters of the plate are shown in Fig. 1. In Fig. 1 hs and ha are thickness values of the sensor and the actuator layers, respectively.

Fig. 1. The coordinate system and the geometric parameters of the plate.

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M. Shariyat / Composite Structures 88 (2009) 240–252

In the present analyses, Reddy’s third order shear deformation description of the displacement field is adopted [26,27]: uðx; y; zÞ ¼ u0 ðx; yÞ þ zwx ðx; yÞ  c1 z3 ðw0;x þ wx Þ   vðx; y; zÞ ¼ v0 ðx; yÞ þ zwy ðx; yÞ  c1 z3 w0;y þ wy

ð1Þ

wðx; y; zÞ ¼ w0 ðx; yÞ where u0, v0, w0, wx, wy are the displacement and the rotation components of the reference plane of the plate, respectively. The symbol ‘‘,” stands for the partial derivative, and c1 ¼

4

ð2Þ

2

3h

u0 ¼ NU; v0 ¼ NV; w0 ¼ NW; wx ¼ NWx ; wy ¼ NWy

ð3Þ

where N is the shape function matrix. Therefore, Eq. (1) may be rewritten as

¼ @d

3 N;x 0 7 0 N;y 7 7 N;y N;x 7 7 7 0 0 5 0 0

3 c1 N;x 0 7 0 c1 N;y 7 7 c1 N;y c1 N;x 7 7 7 0 0 5 0 0

ð7Þ

Generally, Drichlet type boundary conditions (displacement) and Neumann type boundary conditions (stress) cannot be incorporated simultaneously [28], especially, in a finite element solution. Since simultaneous continuity of the displacement components and continuity of the contact shear and normal stresses at the mutual contact surfaces of the neighboring layers may not hold, it seems that in some circumstances, the layerwise theory may lead to results that are less accurate than the results of the third order shear-deformation theory [27]. If the nodal vectors of the displacement and the rotation components of the reference plane are denoted by U, V, W, Wx, Wy respectively, and similar shape function are chosen for them, one may write:

9 2 8 uðx; y; z; t Þ > N > > > > > > > > 6 > 0 > = 6 < vðx; y; z; t Þ > 6 dðx; y; z; tÞ ¼ wðx; y;z;t Þ ¼ 6 0 6 > 6 > > > > wx ðx; y; z; tÞ > > > 40 > > > > ; : wy ðx; y; z; tÞ 0

^e ¼ NðdðeÞ ÞdðeÞ ¼ ðN0 þ zN1 þ z2 N2 þ z3 N3 ÞdðeÞ 3 2 2  ;x ÞN;x 0 0 0 N;x 0 12 ðw0;x þ 2w 0 0 7 6 6  ;y ÞN;y 0 07 6 0 N;y 12 ðw0;y þ 2w 60 0 0 7 6 6 6  ;x ÞN;y þ w  ;y N;x 0 0 7 N0 ¼ 6 7 N1 ¼ 6 0 0 0 6 N;y N;x ðw0;x þ w 7 6 6 0 N5 4 0 0 N;y 40 0 0 0 0 0 N 0 0 0 N;x 3 2 2 0 0 0 0 0 0 0 c1 N;xx 7 60 0 0 6 0 0 7 6 6 0 0 c1 N;yy 7 6 6 7 6 6 0 0 N2 ¼ 6 0 0 0 7N3 ¼ 6 0 0 2c1 N;xy 7 6 6 N 0 3c N 0 0 3c 4 40 0 0 1 ;y 1 5 0 0 3c1 N;x 3c1 N 0 0 0 0

3 0 c1 z3 N;x ðz  c1 z3 ÞN 0 7 3 3 0 ðz  c1 z ÞN 7 N c1 z N;y 7 ðeÞ 7d 0 N 0 0 7 7 5 0 0 N 0 0

0

0

N

ðeÞ

T

dðeÞ ¼ ½ UðtÞ VðtÞ WðtÞ Wx ðtÞ Wy ðtÞ 

The tensoric coupled piezothermoelastic constitutive equations may be written as [29]: rij ¼ cijkl ðekl  e0kl Þ  kij  DT  ekij Ek ; Di ¼ eikl ðekl  e0kl Þ þ nik Ek þ pi DT ij

where r is the stress tensor, c is the tensor of the elastic coefficients at constant electric field and temperature, ekl is the strain tensor, e0kl is the initial strain tensor, kij is the temperature-stress coefficient tensor, ekij is the tensor of the piezoelectric coefficients at a constant temperature, Ek is the electric field vector, DT is the temperature rise from a stress free reference temperature, Di is the electric displacement tensor, nik is the dielectric coefficient tensor at constant elastic stress and temperature, and pi is the pyroelectric constant vector. It is assumed that thermal loads are not applied as thermal shocks. Otherwise, relaxation time constants must be included to account for the thermoelastic wave propagation [30]. For the present analysis, Eq. (8) may be rewritten in the following matrix form: r ¼ Q ð^e  a  DTÞ  eT E;

1 1  2 exx ¼ u;x þ ðw;x Þ2 ; eyy ¼ v;y þ w;y ; ezz ¼ 0 2 2 cxy ¼ u;y þ v;x þ w;x w;y ; cxz ¼ u;z þ w;x ; cyz ¼ v;z þ w;y

Q 11 ¼ Q 22 ¼

ð5Þ

It is easily verified that the strain-components of the imperfect plate may be related to the final displacement components and  through the following the initial transverse deformations ðwÞ equations:     ^e ¼ e  e0 ¼ e0 þ ze1 þ z2 e2 þ z3 e3 ; eT ¼ exx eyy cxy cyz cxz 8 9 8 9 > >  ;x w0;x u0;x þ 12 ðw0;x Þ2 þ w wx;x > > > > > > > > > > > >  2 > > > > > > > > > > >  ;y w0;y v0;y þ 12 w0;y þ w < wy;y > < = = 0 1 ¼ w þ w e ¼ u0;y þ v0;x þ w0;x w0;y þ w ; e   x;y y;x w þ w w ;x 0;y ;y 0;x > > > > > > > > > > > > 0 > > > > wy þ w0;y > > > > > > > > : > ; > : ; 0 wx þ w0;x   8 8 9 9 c1 wx;x þ w0;xx 0 > > > > > > > > > > > >   > > > > > > > > 0 c1 wy;y þ w0;yy > > > > < < = =   0 e2 ¼ ; e3 ¼ c1 wy;x þ wx;y þ 2w0;xy > > > >  > > > > > > > 3c1 wy þ w0;y > > > > > 0 > > > > > > > > : : ; ; 3c1 ðwx þ w0;x Þ 0 ð6Þ e, e0, and ^e are the strain, the initial strain, and the net strain vectors, respectively. Based on Eq. (6), one may write:

D ¼ e^e þ nE þ p  DT

ð9Þ

Q and a are the elasticity constants and the coefficients of thermal expansion matrices, respectively. As it will be explained later, these matrices are dependent on the z coordinate of the layer. The nonzero components of these matrices are:

ð4Þ Using von Karman type strain–displacement relations:

ð8Þ

ijkl

EðzÞ ; 1  m2 ðzÞ

Q 33 ¼ Q 44 ¼ Q 55 ¼

Q 12 ¼ Q 21 ¼

EðzÞ ; 2½1 þ mðzÞ

m:EðzÞ 1  m2 ðzÞ

ð10Þ

a1 ¼ a2 ¼ aðzÞ

where E(z) is the elasticity modulus. The electrical field E is calculated based on the gradient of the electric potential u: KT ¼ h ox

E ¼ Ku;

oy

ð11Þ

oz i

A nine-node quadratic Lagrangian element is used in the present research. The element is located on the reference plane and has six degrees of freedom in each node: u0, v0, w0, wx, wy displacement components and the electrical potential / per piezoelectric layer. Therefore, if the electric potential is assumed to vary through the thickness of the piezoelectric layers only, the electric field vector in each layer may be described as E ¼ Bu uðeÞ where 2

0

60 6 6 1 6 hs Bu ¼ 6 60 6 6 40 0

ð12Þ

0 0 0 0 0 1 ha

3 7 7 7 7 7; 7 7 7 5

uðeÞT ¼ h us

ua i

ð13Þ

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M. Shariyat / Composite Structures 88 (2009) 240–252

us and ua are the electrical potential vectors of the sensor and the actuator layers, respectively. The total potential energy of the plate is composed of various quantities: P ¼ PS þ PE þ PExt þ PI

ð14Þ

where, PS, PE, PExt, and PI are strain energy, electrical energy, external loads and electrical charges energies, and inertia forces energy, respectively. Therefore, neglecting the body forces, according to the principle of virtual displacement: dP ¼ dPS þ dPE þ dPExt þ dPI ¼ 0

ð15Þ

Z  T  d^eT r dV ¼ d^e ½Q ð^e  a  DTÞ  eT E dV V V " ! Z ( 3 3 X X ðeÞT ðeÞ T iþj i ^ ¼ dd Ni ½ð a DT Q Nj Þd  ^

dPS ¼

i¼0

ð23Þ

Ksdu dðeÞ

þ

Ksuu uðeÞ s

¼

GsT

ð24Þ

Eq. (24) gives the sensor voltage as sðeÞ

sðeÞ1 uðeÞ ðKud dðeÞ þ GsT Þ s ¼ Kuu

Subscripts/superscripts ‘s’ and ‘a’ denote the sensor and actuator layers, respectively. Therefore, Eq. (22) takes the following form: ðeÞ ðeÞ dðeÞ þ ðKdd  Kdu K1 Mdd € ¼ FT þ Fc þ Fs  Kdu K1 uu Kud Þd uu GT  Kdu ua

Since evaluating the general effect of the active control on the vibration and buckling of the FGM hybrid plates is of interest in the present approach, the following control law is implemented for the actuators:

#)

dA

_ ðeÞ uðeÞ a ¼ Gu s

i¼0 ðeÞT

¼ dd dPE ¼ 

ðeÞ

Kdd d

Z

ðeÞT

 dd

ðeÞT

FT þ dd

ðeÞ

Kdu u

ð16Þ

dET D dV Vp

ðeÞT

¼ du

Z Vp

BTu

e

3 X

! ~  DT dV Ni ~zi dðeÞ þ nBu uðeÞ þ p

i¼0

¼ duðeÞT Kud dðeÞ þ duðeÞT Kuu uðeÞ þ duðeÞT GT

ð17Þ

¼ ddðeÞT Fc  ddðeÞT

Z

@T f s dC þ duðeÞT

C1

Z

Bu q dC

C2

¼ ddðeÞT Fc  ddðeÞT Fs  duðeÞT Fq dPI ¼ ddðeÞT

Z

ð18Þ ð19Þ

where V, Vp and A are the volume, volume corresponding to the piezoelectric layer and area of the element, respectively, n is the number of layers, and k is the layer counter (including the piezoelectric layers) and

k¼1

XZ k¼1

zkþ1

zk zkþ1 zk

where G is the feedback gain and its appropriate value may be determined through successive solutions. It is assumed that the FGM substrate consists of two constituent materials so that its top layer (z = zt) is ceramic-rich whereas its bottom layer (z = zb) is metal-rich and the material composition varies continuously through the thickness. Temperature-dependency of an arbitrary material property (P) of the FGM substrate may be expressed as [31]:

^m ¼ Q k zm dz; a

~¼ zm dz; p

n Z X

XZ k¼1

zkþ1

k¼1 zk zkþ1 zk

Q k ak zm dz; ~ a¼

XZ k¼1

zkþ1

ak dz zk

pk dz ð20Þ

~ are calculated for the piezoelectric layers only. Fc, fs, ~zm , ~ a, and p and q are the external concentrated force, distributed force (traction), and charge vectors, respectively and W is the work done by the external loads. Since the temperature distribution is extracted from separate equations, increments of the thermal energies (entropy increase and heat flux incremental works) are neglected in Eq. (14). Substituting Eqs. (16)–(19) into Eq. (15), and noting that Eq. (15) must hold for any arbitrary dd(e) and du(e) vectors, the governing equations of the element will be: ( ðeÞ )

  ( €ðeÞ )  FT þ Fc þ Fs sKdd Kdu Mdd 0 d d ð21Þ þ ¼ ðeÞ ðeÞ Kud Kuu Fq  GT 0 0 € u u Because KTdu ¼ Kud . Separating the actuator and sensor components and noting that the external applied charge is zero for the sensor layer, the following equations may be extracted from Eq. (21):

ð28Þ

where T (K) is the temperature, and P0, P1, P1, P2, and P3 are some coefficients. The material properties of the piezoelectric layers are assumed to be linear functions of the temperature change [15]: P ¼ P 0 ð1 þ P 1 DTÞ

dðeÞ q@T @€ dðeÞ dV ¼ ddðeÞT Mdd €

V

n Z X

ð27Þ

P ¼ P 0 ðP1 =T þ 1 þ P1 T þ P2 T 2 þ P3 T 3 Þ

dPExt ¼ dW

ð25Þ

ð26Þ

j¼0

3 X NTi ~zi ÞeT Bu uðeÞ þð

~zm ¼

ð22Þ

a Kadu dðeÞ þ Kauu uðeÞ a ¼ Fq  GT

Z

A

^m ¼ Q

Mdd € dðeÞ þ Kdd dðeÞ þ Ksdu uðeÞ s ¼ FT þ Fc þ Fs

ð29Þ

where P is the material property and P0 is the material property at the ambient temperature. Temperature distribution may be determined using the classical heat transfer equation. The temperature is assumed to vary in the thickness direction only. Therefore, the heat transfer equation has the following form:

o oT k ¼ qC v T_ ð30Þ oz oz where the thermal conductivity (k), mass density (q), and the specific heat (Cv) are all functions of the temperature and the z-coordinate. Eq. (30) may be solved incorporating appropriate thermal boundary conditions. The material property at any point through the thickness of the FGM substrate is related through the following equation to the material properties of the constituent materials: PðrÞ ¼ P m ðvf Þm þ Pc ðvf Þc

ð31Þ

Pm and Pc are the temperature-dependent properties of the metal and the ceramic, respectively and each may be calculated from Eq. (28). (vf)m and (vf)c are the metal and the ceramic volume fractions, respectively. Using a simple power law, one may write:

n zt  z ðvf Þm ¼ 1  ðvf Þc ¼ ð32Þ zt  zb Consequently: PðrÞ ¼ ðPm  Pc Þ



zt  z zt  zb

n

þ Pc

ð33Þ

where n is the power-law exponent or the so-called volume fraction index.

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M. Shariyat / Composite Structures 88 (2009) 240–252

3. Results and discussion Results may be classified into two mains categories: vibration results and dynamic buckling results. Second order 9-node Lagrangian elements are employed to derive the results. 3.1. Vibration analysis results 3.1.1. Free vibration results For validation purposes, the dimensionless fundamental natural frequencies of simply supported square Al/Al2O3 FGM plates already studied by Matsunaga [32] are reexamined. Coefficients of Eq. (29) are given for various properties of the PZT-5A material in Table 1 [23,33]. The temperature coefficients (Eq. (28)) for stainless-steel SUS304, titanium alloy Ti–6Al–4V, alumina Al2O3, and silicon nitride Si3N4 used to derive the results of the present paper, are extracted from Reddy and Chin [3]. To save space, these data are not mentioned in the present article. The dimensionless fundamental natural frequency is defined as pffiffiffiffiffiffiffiffiffiffiffiffi ð34Þ X ¼ xh qc =Ec and given in Table 2 along with results reported by Matsunaga [32]. In Eq. (34), x is the fundamental natural frequency and the subscripts ‘c’ and ‘m’ denote the metal and the ceramic material properties, respectively. Results of Matsunaga [32] are calculated choosing a polynomial series description for the displacement components, similar to some of previous works of the author (Shariyat et al.) [34–37]. In this description, the parameters must be so carefully chosen that the shear strains of the outermost surfaces vanish.

Table 1 Material properties of the piezoelectric layers Parameter

P0

P1

E11 (GPa) E22 (GPa) G12, G13, G23 (GPa) m e13 (C/m2) e23 e42 e51 n11 (108 F/m) n22 n33 q (k/m3) a11, a11 (106/°C) k (W/mK) p11 (1012 C2/N m2) p22

61 61 24.2 0.3 7.2097.20912.322 12.322 1.53 1.53 1.5 7600 0.9 2.1 82.6 90.3

0.0005 0.0002 0.0002 0 0 0 0 0 0 0 0 0 0.0005 0 0 0

Table 2 A comparison among the fundamental natural frequencies of the square Al/Al2O3 FGM plates a/h

n

There exists a good agreement among the results. The differences appeared in the results are mainly due to the differences in the solution procedures. Matsunaga [32] used a semi-analytical solution. As a second example in the comparative study, free vibration of a simply supported square hybrid FGM plate with symmetrically fully covered G-1195N piezoelectric layers is considered. The top layer of the laminated plate is the piezoelectric actuator layer and the bottom layer is the piezoelectric sensing layer. The intermediate layer is an FGM plate that is fabricated from aluminum oxide and Ti–6Al–4V materials. Top layer of the substrate FGM plate is metal-rich. The material properties are adopted as mentioned in Refs. [5,7] as 3

Et ¼ Em ¼ 320:24 GPa; mm ¼ 0:26; qm ¼ 3750 kg=m ; Eb ¼ Ec ¼ 105:7 GPa 3

d31 ¼ d32 ¼ 254  10

Matsunaga [32]

Present

0 0.5 1 4 10

0.2121 0.1819 0.1640 0.1383 0.1306

0.2083 0.1762 0.1594 0.1312 0.1271

10

0 0.5 1 4 10

0.05777 0.04917 0.04426 0.03811 0.03642

0.05682 0.04876 0.04369 0.03792 0.03578

12

m=V

As before, subscripts ‘t’ and ‘b’ denote top and bottom surfaces, respectively and the subscript p denotes the piezoelectric layer. Furthermore: e ¼ dQ

ð35Þ

The side and the thickness of the substrate FGM plate are 400 and 5 mm, respectively and the thickness of each piezoelectric layer is 0.1 mm. The first five natural frequencies of the plate are calculated for various volume fraction indices and compared in Table 3 with results of FEM analysis of He et al. [7] based on the classical theory and the semi-analytical results of Huang and Shen [5]. The initial imperfection effect is studied simultaneously. In Table 3, W=h is the amplitude of the initial geometric imperfection to thickness ratio. It is assumed that the initial imperfection has the same form as the shape mode. Results show that the initial imperfection effect on the natural frequencies is noticeable especially for greater W=h ratios. Generally, initial geometric imperfections increase the bending stiffness and lead to increased natural frequencies.

Table 3 A comparison among results reported by different references for the first natural frequencies of simply supported square hybrid FGM plates Mode no.

Source

n=0

1

He et al. [7] Huang and Shen [5] Present W=h ¼ 0 Present W=h ¼ 0:1 Present W=h ¼ 0:5

144.25 143.25 142.83 144.56 154.68

185.45 184.73 183.24 185.50 199.04

198.92 198.78 198.06 200.57 215.43

230.4 229.47 228.61 231.84 248.30

2

He et al. [7] Huang and Shen [5] Present W=h ¼ 0 Present W=h ¼ 0:1 Present W=h ¼ 0:5

359.00 358.87 357.53 360.92 390.51

462.65 461.02 460.21 464.68 502.32

495.62 494.65 493.18 497.87 539.69

573.82 571.87 570.03 575.42 622.03

3

He et al. [7] Huang and Shen [5] Present W=h ¼ 0 Present W=h ¼ 0:1 Present W=h ¼ 0:5

359.00 358.87 357.53 360.92 390.51

462.47 461.02 460.21 464.68 502.32

495.62 494.65 493.18 497.87 539.69

573.82 571.87 570.03 575.42 622.03

4

He et al. [7] Huang and Shen [5] Present W=h ¼ 0 Present W=h ¼ 0:1 Present W=h ¼ 0:5

564.10 563.42 562.91 568.70 613.90

731.12 727.98 726.87 734.82 788.46

778.94 778.61 777.46 786.01 845.75

902.04 899.91 888.14 897.21 961.36

5

He et al. [7] Huang and Shen [5] Present W=h ¼ 0 Present W=h ¼ 0:1 Present W=h ¼ 0:5

717.80 717.65 716.08 732.16 782.68

925.45 922.83 920.11 940.47 1004.52

993.11 992.87 990.67 1012.60 1080.44

1148.12 1146.87 1144.28 1169.63 1246.27

X

5

3

mb ¼ 0:2981; qb ¼ 4429 kg=m ; Ep ¼ 63 GPa; mp ¼ 0:3; qp ¼ 7600 kg=m

n = 0.5

Present results for the imperfect plate are also included.

n=1

n=5

M. Shariyat / Composite Structures 88 (2009) 240–252

Having validated the present formulation, effect of the boundary conditions on the free vibration response of hybrid Al/Al2O3 FGM plates is evaluated. To this end, free vibration of rectangular FGM hybrid plates with 20  20 cm in-plane dimensions is investigated for various boundary conditions. The FGM substrate is composed of four plies each with 1 mm thickness. Only one piezoelectric PZT-5A layer with a thickness of 0.2 mm bonded on the top surface of the plate is employed, in this example. Five types of boundary conditions are considered for free vibration analysis: (a) All edges clamped (CCCC), (b) One edge clamped and three edges free (CFFF), (c) Two opposite edges clamped and two simply supported (CCSS [27]), (d) Two opposite edges simply supported and two free (SSFF [27]), (e) All edges simply supported (SSSS). Externally applied loads, including electrical charge and thermal and mechanical loads must set to zero in a free vibration analysis. Therefore, for each boundary condition, three cases: an FGM plate without piezoelectric layers (W), an FGM plate with one open circuit piezoelectric layer (OC), and an FGM plate with one closed circuit (short-cut circuit) piezoelectric layer (SC), are adopted for the free vibration analysis. Effect of the externally applied load may be determined through a forced vibration analysis. First five natural frequencies of the plates are given for n = 1 in Table 4 for various boundary conditions and different roles of the piezoelectric layer. Results of Table 4 reveal that the piezoelectric layer role in natural frequencies modification is ignorable in cases where no electrical potential field is applied. Furthermore, natural frequencies are lower when the piezoelectric layer is used in short-cut circuit.

245

Comparison of the natural frequency values appeared in Table 4 reveals that when looser boundary conditions are used, lower natural frequencies are resulted. Generally, looser boundary conditions lead to more flexibility, and subsequently in absence of local loadings or geometric variations, lead to greater displacements and stresses. Therefore, immature buckling and resonance are more likely to occur in plates with looser boundary conditions. To present influence of the boundary conditions in a more appropriate manner, results of Table 4 are depicted in Figs. 2–4. It is obvious that the piezoelectric layer has no effect on the mode shapes. Electrical actuation effect is more remarkable in adaptive control of transient rather than forced vibration. Numerous works in active control of FGM hybrid plates field may be found in literature. The active vibration control analysis introduced in this section is a simple one and is performed merely to enable some discussions in the next sections. A rectangular clamped hybrid FGM plate with two surface-bonded piezoelectric layers is adopted. One piezoelectric layer serves as a sensor and the other is used as an actuator and both are employed in a negative velocity feedback active control loop. The geometric parameters and material properties of the plate are similar to plates of the previous example. Appropriate gain value must be chosen based on successive analyses. Low gain values result in less damping and higher gains may lead the structure to instability after the initial damping. An impulsive force of amplitude 1 kN is applied on the mid-point of the clamped plate. After some computer runs, a gain value of 50 is chosen and the results are illustrated in Fig. 5. As it may be expected, the transient vibration vanishes after a short time. 3.1.2. Forced vibration analysis To investigate the piezoelectricity influence on the forced vibration response, transverse step and sinusoidal concentrated forces

Table 4 First five natural frequencies in (Hz) corresponding to different boundary conditions of the plate Boundary condition

Mode no.

W

OC

SC

CCCC

1 2 3 4 5

1375 2941 2942 4342 5749

1331 2845 2848 4201 5558

1328 2839 2844 4193 5548

CFFF

1 2 3 4 5

127 313 799 1006 1156

124 302 773 973 1117

123 300 769 971 1112

CCSS

1 2 3 4 5

1105 2081 2786 3711 4040

1067 2011 2695 3586 3899

1064 2008 2686 3577 3896

SSFF

1 2 3 4 5

380 606 1356 1477 1771

361 582 1309 1430 1708

359 578 1304 1424 1701

SSSS

1 2 3 4 5

767 1871 1872 2984 3885

731 1803 1804 2875 3753

729 1798 1802 2868 3744

W, OC, SC, denote without piezoelectric layer, with open circuit, and with short-cut circuit piezoelectric layer, respectively.

Fig. 2. Comparison of the natural frequencies of the FGM plate.

Fig. 3. Comparison of the natural frequencies of the FGM hybrid plate with an open circuit piezoelectric layer.

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M. Shariyat / Composite Structures 88 (2009) 240–252

Fig. 4. Comparison of the natural frequencies of the FGM hybrid plate with a shortcut circuit piezoelectric layer.

are applied individually on the plates mentioned in the latter examples. To this end, a clamped plate with one surface-bonded piezoelectric actuator layer is adopted and a force with an amplitude of 2 kN is exerted on the mid-point of the plate. The frequency of the applied load is 100 Hz for the harmonic load. The control voltage and the mechanical load are applied simultaneously. As explained before, electrical potential has minor effects on the natural frequencies of the hybrid plate. The effect increases with the applied electrical potential. In the present analysis, the contact surface of the piezoelectric layer and the FGM substrate is chosen as the reference plane for the electrical potential. Effect of the electrical actuation on the forced vibration response is depicted in Fig. 6, for the step load input. Since no structural damping is considered for the plate, the transient response does not diminish. The transient oscillations are performed with the fundamental natural frequency of the plate that is introduced in Table 4. Furthermore, a second transient response due to electrical potential application is also induced in the plate. Since only actuator layer is used, higher vibration modes are more remarkably invoked. As it may be noticed from Fig. 6, application of appropriate minus electrical potential may reduce the vibration amplitude. On the other hand, vibration amplitude may increase in case of non-appropriate electrical potential application.

Effects of the applied electrical potential on the forced vibration response of the plate that is subjected to the harmonic force are illustrated in Fig. 7. The excitation frequency is equal to 100(Hz). Transient oscillations that are performed with the first natural frequency are combined with the forced vibration response of a plate that has no electrical excitation. Responses shown in Fig. 7 reveal that due to the employed control voltage, higher vibration modes are more magnified. Therefore, although piezoelectric layer may reduce the amplitude of the overall vibration of the plate in some cases, it induced noises that may results in harshness problems e.g. in passenger vehicles [38]. In the forgoing examples of the present section, only one piezoelectric layer is used. For this reason, contraction and expansion of the piezoelectric layer has led to bending excitation. This phenomenon is not desirable in buckling control. In many applications, two symmetric piezoelectric layers may be used. Therefore, contraction and expansion of the piezoelectric layers may induce an in-plane vibration. To study the nonlinear vibration and dynamic response of square hybrid FGM plates subjected to the combined action of transverse mechanical, electrical, and thermal loads, two types of hybrid FGM plate are considered. The first type is fully covered with PZT-5A piezoelectric actuators on the top surface (referred to as P/FGM), and the second has two piezoelectric layers symmetrically bonded to the top and bottom surfaces (referred to as P/ FGM/P). The substrate is a Si3N4/SUS304 FGM layer. The thickness of the substrate FGM layer is 1 mm and the thickness of each piezoelectric layer is 0.1 mm. The material properties of the piezoelectric layers are Ep ¼ 63 GPa; mp ¼ 0:26; ap ¼ 0:9  106 ðK1 Þ; kb ¼ 2:1 W=m K 3

qp ¼ 7600 kg=m ; d31 ¼ d32 ¼ 254  1012 m=V The temperature field is assumed to vary only in the thickness direction of the plate and may be determined by solving the steady-state heat conduction equation with appropriate thermal boundary conditions. The ambient temperature is assumed to be 300 K. The bottom surface of the FGM substrate is held at 300 K. The corresponding natural frequency results are listed in Table 5. In nonlinear structures, the natural frequencies are dependent on

Fig. 5. Active control of the transient vibrations of the FGM plate.

M. Shariyat / Composite Structures 88 (2009) 240–252

247

Fig. 6. Forced vibration response of the FGM hybrid plate under a concentrated step load.

Fig. 7. Forced vibration response of the FGM hybrid plate under a concentrated harmonic force.

the initial condition and loading. The dynamic load is assumed to be a suddenly applied uniform load fs = 2 MPa. Table 5 shows effects of the volume fraction index, control voltage and temperature field on the natural frequency parameter of the two types of the plates. The natural frequency parameter is defined as pffiffiffiffiffiffiffiffiffiffiffiffi ð36Þ X ¼ xða2 =hÞ qb =Eb where x is the fundamental natural frequency. Both TID and TD (temperature independent and temperature dependent FGM and piezoelectric materials properties) results are given in Table 5. It can be seen that the fundamental natural frequencies of these two plates decrease by increasing the temperature and the volume fraction index n. The plus voltage decreases whereas the minus voltage increases the fundamental natural frequency of the plate.

3.2. Dynamic buckling analysis Although numerous works may be found in literature in static thermal buckling analysis of hybrid FGM plates with piezoelectric actuators, no work is developed in the dynamic thermal buckling of the mentioned plates yet. Furthermore, the initial geometric imperfections role in buckling analysis is more remarkable than that of modal analysis. Generally, employing excessive electrical potential may lead to contraction of the plate and in turn to buckling, even in quasi-static electrical loading. Utilizing dynamic electrical passive control input (e.g. a suddenly applied voltage) may induce undesirable oscillations and subsequently may adversely affect the plate instability, especially when the structural damping is ignorable.

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Table 5 Natural frequency parameter of the two sets of hybrid FGM plates under various thermal and electrical loading conditions (Tb = 300 K) Stacking sequence

Tt (K)

Applied voltage

Volume fraction index 0

(P/FGM)

300

400

600

s (P/FGM/P)

300

400

600

0.5

2

4

TID

TD

TID

TD

TID

TD

TID

TD

Vt = 200 V Vt = 0 V Vt = 200 V Vt = 200 V Vt = 0 V Vt = 200 V Vt = 200 V Vt = 0 V Vt = 200 V

13.264 13.238 13.211 12.435 12.411 12.385 10.759 10.754 10.751

13.264 13.238 13.211 12.088 12.063 12.037 10.256 10.249 10.246

9.706 9.685 9.664 8.998 8.981 8.962 7.686 7.679 7.676

9.706 9.685 9.664 8.749 8.730 8.709 7.328 7.321 7.318

7.701 7.683 7.664 7.061 7.045 7.031 6.004 5.983 5.965

7.701 7.683 7.664 6.865 6.844 6.832 5.725 5.703 5.686

7.169 7.161 7.152 6.539 6.524 6.511 5.565 5.540 5.516

7.169 7.161 7.152 6.357 6.340 6.324 5.305 5.281 5.257

Vt = Vb = 200 V Vt = V b = 0 V Vt = Vb = 200 V Vt = Vb = 200 V Vt = V b = 0 V Vt = Vb = 200 V Vt = Vb = 200 V Vt = V b = 0 V Vt = Vb = 200 V

12.403 12.389 12.375 11.390 11.384 11.379 9.077 9.061 9.045

12.403 12.389 12.375 11.073 11.064 11.57 8.654 8.637 8.620

9.439 9.398 9.355 8.413 8.365 8.317 6.080 6.035 5.991

9.439 9.398 9.355 8.179 8.131 8.081 5.798 5.753 5.710

7.694 7.656 7.617 6.721 6.677 6.633 4.728 4.709 4.690

7.694 7.656 7.617 6.536 6.492 6.447 4.509 4.489 4.468

7.223 7.185 7.147 6.241 6.197 6.153 4.504 4.495 4.492

7.223 7.185 7.147 6.068 6.025 5.978 4.294 4.285 4.281

Time integration solution algorithm presented by Shariyat [39] is employed to solve the resulted highly nonlinear governing equations. 3.2.1. Dynamic thermal buckling As a first stage, a comparison made with thermal buckling results of simply supported (Si3N4/SUS304)S FGM square plates reported by Shen [15]. The side to thickness ratio is b/h = 20 and the thickness of the FGM layer is 10 mm. Table 6 presents the thermal buckling temperature rise DT(K) for perfect FGM rectangular plates with different volume fraction indices and aspect ratios subjected to a uniform temperature rise. It can be seen that, for the (Si3N4/SUS304)S plates, a fully metallic plate (n = 0) has the lowest buckling temperature rise and that the buckling temperature rise increases as the volume fraction index increases. It also can be seen that the buckling temperature decreases when the temperature dependency is taken into consideration. Up to about 16% decrease is noticed in the critical temperature when the temperature dependency of the materials is incorporated. Results shown in Table 6 reveal that the aspect ratio may considerably affect the buckling loads. As a second step, a parametric study is performed for simply supported (P/Si3N4/SUS304)S piezoelectric hybrid FGM plates that are subjected to gradual uniform temperature rise and constant electric field. Corresponding results are illustrated in Figs. 8 and 9. The thickness of the FGM substrate (that is containing of two FGM layers) is 10 mm and the thickness of each piezoelectric layer is 1 mm. Two control voltages Va = ± 200 V are used. In Fig. 8, ther-

mal buckling results of perfect and imperfect hybrid FGM plates with different side to thickness ratios are depicted. In each curve set, the upper curve is corresponding to Va = 200 V and the lower one is corresponding to Va = 200 V actuation voltage. Besides, W and W denote the maximum transverse deflection and the initial imperfection amplitude, respectively. Results show that minus control voltage may enhance the buckling behavior due to the induced contraction. However, it is noticed that the electrical actuation has minor influence on the buckling temperatures of the hybrid FGM plates. Furthermore, as it may be seen from results of Fig. 8, the initial geometric imperfections may remarkably reduce the buckling temperature. But as one proceeds in the postbuckling region, the slope of the imperfect plates grow with a greater rate so that at the end of the temperature-displacement curve, imperfect plate exhibits a higher stiffness recovery. Indeed, initial imperfection leads to a transverse displacement magnification and in turn, to an increased flexural stiffness in the postbuckling regime. From results appeared in Fig. 8, it may be deduced that in absence of the electrical actuation, the buckling temperature rise is approximately proportional to the square of the thickness to side ratio. In Fig. 9, effects of volume fraction index, aspect ratio, and temperature-dependency of the material properties on the buckling temperature are investigated. In each curve set, the upper curve is corresponding to the TID and the lower one is corresponding to the TD results. The dimensionless initial geometric imperfection amplitude is chosen as W=h ¼ 0:1. The results show that the critical buckling temperature rise decreases as the volume fraction

Table 6 A comparison between the present buckling temperature rise results and those reported by Shen [15] for the FGM plates (T0 = 300 K) b/a

n=0

n = 0.5

n=1

n=2

Shen [15]

Present

Shen [15]

Present

Shen [15]

Present

Shen [15]

Present

TID 1 1.5 2

362.1022 193.0915 131.6516

359.3103 190.1729 129.8263

485.9651 259.1497 176.6851

482.1851 256.7952 174.2604

551.6215 294.1621 200.5562

547.9892 291.2417 198.3019

617.3345 329.2048 224.4478

613.2541 326.7460 222.7991

TD 1 1.5 2

302.5820 173.4869 121.9668

299.2819 171.4623 120.3381

391.9424 225.7363 159.6335

389.0836 222.4168 157.8351

436.5354 252.3783 178.9512

433.2913 249.8662 177.1792

477.3847 277.9374 197.7615

474.2283 276.1844 195.9885

M. Shariyat / Composite Structures 88 (2009) 240–252

249

Fig. 8. Thermal buckling results of perfect and imperfect hybrid FGM plates with different side to thickness ratios. In each curve set, the upper curve is corresponding to Va = 200 V and the lower one is corresponding to Va = 200 V actuation voltage.

Fig. 9. Effects of volume fraction index, aspect ratio, and temperature-dependency of the material properties on the buckling temperature of the perfect and the imperfect (W=h ¼ 0:1) FGM hybrid plates. In each curve set, the upper curve is corresponding to the TID and the lower one is corresponding to the TD results.

index n increases, and increases as the aspect ratio a/b of the plates becomes larger. It is also noticed that temperature- dependency of the material properties may considerably affect the results. For load-displacement curves with gradual slope variations, a modified version of Budiansky’s criterion recently introduced by the author [25] is used to determine the buckling load. Based on the mentioned criterion, the dynamic buckling temperature of the FGM hybrid plates is detected. Furthermore, influence of employing an active negative velocity feedback control to suppress the thermally-induced vibrations and mitigating the buckling phenomenon are investigated. In this regard, the square plates with a/ h = 40 ratio used in the foregoing static thermal analysis are reexamined. The only difference is that the top piezoelectric layer is used as a sensor and the bottom piezoelectric layer is employed as an actuator. Results of the perfect and imperfect ðW=h ¼ 0:1Þ

plates with or without active control (G = 104) under suddenly temperature rise are derived for both temperature-dependent and temperature-independent material properties assumptions. Table 7 summarizes the dimensionless thermal buckling loads which are obtained by dividing the dynamic buckling temperature rises by those obtained from the static thermal buckling analysis (with no applied voltages). From results of Table 7, it may be concluded that the active control effect is minor whereas temperaturedependency of the material properties may considerably affect the buckling temperature. Furthermore, enhancements in forced vibration response and subsequently, buckling behavior requires employing extremely high gains [40]. Generally, passive control is not useful in cases where there is a lack in the structural damping unless the oscillations induced due to the suddenly applied voltages are artificially damped.

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M. Shariyat / Composite Structures 88 (2009) 240–252

Table 7 Dimensionless dynamic thermal buckling results a/b

b/t

No control voltage

With active control

TID

TD

TID

TD

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

1

30 50

0.736803 0.706208

0.719944 0.691469

0.615837 0.598974

0.582115 0.557882

0.765176 0.736904

0.742461 0.710553

0.632093 0.609475

0.60544 0.583933

1.5

30 50

0.724888 0.691971

0.702476 0.682079

0.602609 0.56737

0.560406 0.556064

0.76447 0.734279

0.734888 0.709039

0.630983 0.598065

0.601098 0.57434

3.2.2. Dynamic buckling under an in-plane dynamic load To study the thermo-piezoelectric effects on the postbuckling behavior of FGM hybrid plates under uniaxial compression, several numerical examples were solved for perfect and imperfect, (P/ FGM)S and (FGM/P)S plates with (Si3N4/SUS304) substrate. The thickness of each of the FGM layers is 1 mm whereas the thickness of each piezoelectric layer is 1 mm, so that the total thickness of the plate is 2.2 mm. The dimensionless amplitude of the initial geometric imperfection of the plates is chosen equal to ðW=h ¼ 0:1Þ. As a first step, results of Ref. [15] are reexamined for FGM hybrid plates under static thermo-electro-mechanical loads. These results as well as the present results are given in Table 8. Table 8 gives the buckling load for uniaxially compressed square (P/FGM)S and (FGM/P)S plates (in kN) under uniform temperature rise and three sets of electrical loading conditions. Side to thickness ratio of the plates is a/h = 20. From the results, it may be noticed that the buckling load of the (P/FGM)S plate is lower than that of the (FGM/P)S plate. Buckling load increases as the volume fraction index n increases. It is obvious that the metallic plate has a lower stiffness than the ceramic plate. It can also be seen that the temperature-

dependency of the material properties reduces the buckling load. Although minus voltages may enhance the buckling behavior, it may be readily noticed that the control voltage has a very small effect on the buckling load of the hybrid plates. This is because the piezoelectric layer is much thinner than the FGM substrate. Although very high voltages may be utilized to influence the buckling response of a hybrid FGM plate, they are not recommended for applied cases. Fig. 10 illustrates the postbuckling load-deflection curves of perfect and imperfect square (P/FGM)S hybrid plates undergoing thermo-electro-mechanical loads (T 0 ¼ 300 K; W=h ¼ 0:1) considering the temperature-dependency of the materials. In each curve set, the upper, middle, and lower curves are corresponding to Va = 500 V, Va = 0 V, and Va = 500 V, respectively. Two different uniform temperature rises, three various electrical actuations and two different side to thickness ratios are considered. Fig. 10 shows that the temperature-dependency of the material may decrease the stiffness of the plate in the postbuckling region. Fig. 11 shows the effect of the volume fraction index on the load-deflection curves of the mentioned perfect and imperfect (P/

Table 8 A comparison between the static buckling loads (in kN) for axially compressed square (P/FGM)S and (FGM/P)S plates under uniform temperature rise and three sets of electrical loading conditions (a/h = 20, T0 = 300 K) Subject

(P/FGM)S TID

DT (K)

0

100

200

(P/FGM)S TD

100

200

(FGM/P)S TID

0

100

200

(FGM/P)S TD

100

200

Vt = Vb (V) or Vm (V)

n = 0.2

n=1

n=5

Shen [15]

Present

Shen [15]

Present

Shen [15]

Present

500 0 500 500 0 500 500 0 500

131.6953 131.1302 130.5651 109.9468 109.3817 108.8166 88.1983 87.6332 87.0681

130.6016 130.0432 129.4847 109.1171 108.5591 108.0011 87.65478 87.09757 86.54038

157.9360 157.3708 156.8056 137.6636 137.0984 136.5332 117.3912 116.8260 116.2608

156.5404 155.9816 155.4228 136.5007 135.9414 135.3827 116.4692 115.9109 115.3527

171.3940 170.8288 170.2635 153.4537 152.8885 152.3232 135.5134 134.9482 134.3829

169.8481 169.2892 168.7302 152.1088 151.5501 150.9911 134.3749 133.8163 133.2576

500 0 500 500 0 500

106.5790 106.0187 105.4583 79.0828 78.5270 77.9712

105.7918 105.2386 104.6854 78.67025 78.12272 77.57524

133.5544 132.9939 132.4334 107.4735 106.9176 106.3616

132.4388 131.8849 131.3310 106.6749 106.1261 105.5771

149.0257 148.4651 147.9045 125.3583 124.8023 124.2462

147.7311 147.1769 146.6227 124.3399 123.7906 123.2411

500 0 500 500 0 500 500 0 500

159.3017 158.7366 158.1714 137.5532 136.9881 136.4229 115.8047 115.2396 114.6745

157.8908 157.3320 156.7732 136.3909 135.8323 135.2737 114.9022 114.3441 113.7859

193.1219 192.5568 191.9915 172.8495 172.2843 171.7191 152.5771 152.0119 151.4467

191.3378 190.7788 190.2197 171.2875 170.7286 170.1696 151.2421 150.6833 150.1246

212.0199 211.4547 210.8894 194.0796 193.5144 192.9491 176.1393 175.5741 175.0088

210.0319 209.4727 208.9135 192.2851 191.7260 191.1669 174.5410 173.9820 173.4229

500 0 500 500 0 500

133.8241 133.2637 132.7033 105.6667 105.1109 104.5552

132.7054 132.1515 131.5977 104.8911 104.3424 103.7938

168.1422 167.5817 167.0212 141.3861 140.8302 140.2742

166.6324 166.0782 165.5239 140.1793 139.6298 139.0802

188.8832 188.3226 187.7620 164.5320 163.9759 163.4198

187.1452 186.5907 186.0362 163.0625 162.5126 161.9627

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Fig. 10. Postbuckling load-deflection curves of perfect and imperfect square (P/FGM)S hybrid plates undergoing thermo-electro-mechanical loads (T 0 ¼ 300 K, W=h ¼ 0:1) considering the temperature-dependency of the materials. In each curve set, the upper, middle, and lower curves are corresponding to Va = 500 V, Va = 0 V, and Va = 500 V, respectively.

Fig. 11. Effect of the volume fraction index on the load-deflection curves of the perfect and the imperfect square (P/FGM)S hybrid plates (T 0 ¼ 300 K; W=h ¼ 0:1) considering the temperature-dependency of the materials. In each curve set, the upper, middle, and lower curves are corresponding to Va = 500 V, Va = 0 V, and Va = 500 V, respectively.

Table 9 Dimensionless mechanical dynamic buckling loads of plates initially stressed by thermo-electrical loads a/b

b/t

No control voltage TID

With active control TD

TID

TD

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

1

30 50

0.8163 0.7837

0.7877 0.7573

0.6873 0.6692

0.6406 0.6147

0.8466 0.8165

0.8117 0.7776

0.7046 0.6804

0.6655 0.6426

1.5

30 50

0.8036 0.7685

0.769 0.7472

0.6733 0.6356

0.6174 0.6128

0.8459 0.8137

0.8037 0.776

0.7034 0.6682

0.6609 0.6323

FGM)S hybrid plates (a=b ¼ 1, T 0 ¼ 300 K, W=h ¼ 0:1) considering the temperature-dependency of the materials. As before, in each curve set, the upper, middle, and lower curves are corresponding

to Va = 500 V, Va = 0 V, and Va = 500 V, respectively. It is seen that the mechanical buckling loads of plates that are already under thermo-electrical loads, decrease as the volume fraction index de-

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M. Shariyat / Composite Structures 88 (2009) 240–252

creases. As it is expected, the initial thermal displacements are greater for lower volume fraction indices. A dynamic buckling analysis includes the thermo-electromechanical loads and temperature-dependency of the (P/FGM)S square plates with a/h = 40 ratio, is performed and the relevant results are given in Table 9 for T 0 ¼ 300 K; W=h ¼ 0:1, and Va = 500 V. The dynamic compressive load is suddenly applied in the x-direction. As it may be noticed, due to presence of the initial temperature rise, the dynamic buckling ratios are greater than those appeared in Table 7. Furthermore, in contrast to the control voltage role, the effect of the temperature-dependency of the material properties on the dynamic buckling results is significant. 4. Conclusions In the present paper, forced and free vibrations and dynamic buckling of FGM plates with piezoelectric sensors and actuators subjected to thermo-electro-mechanical loads are investigated. Influences of both the temperature-dependency of the material properties and the initial geometric imperfections are considered in the modal and buckling analyses. Results obtained reveal that the natural frequencies and the thermal and mechanical buckling loads are slightly higher when a minus control voltage is used. This effect is more pronounced for thinner plates. However, for thick plates, this influence is not appreciable in comparison with other significant parameters. Among various parameters that may influence the forced vibration and buckling results, temperature rise, width-to-thickness ratio, initial geometric imperfections, temperature-dependency of the material properties, aspect ratio, layer sequence, and load combination have significant effects. Relative enhancements in buckling behavior are noticed due to the piezoelectricity-induced damping caused by utilizing an adaptive feedback control. However, buckling mitigation due to utilizing integrated piezoelectric sensors and actuators is mainly achieved in extremely high gain values. References [1] Noda N. Thermal stresses in materials with temperature-dependent properties. Appl Mech Rev 1991;44:83–97. [2] Tanigawa Y. Some basic thermoelastic problems for non-homogeneous structural materials. Appl Mech Rev 1995;48:287–300. [3] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 1998;21:593–626. [4] Yang J, Kitipornchai S, Liew KM. Large amplitude vibration of thermo-electromechanically stressed FGM laminated plates. Comput Meth Appl Mech Eng 2003;192:3861–85. [5] Huang XL, Shen HS. Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. J Sound Vib 2006;289:25–53. [6] Reddy JN, Cheng ZQ. Three-dimensional solutions of smart functionally graded plates. Trans ASME, J Appl Mech 2001;68:234–41. [7] He XQ, Ng TY, Sivashanker S, Liew KM. Active control of FGM plates with integrated piezoelectric sensors and actuators. Int J Solids Struct 2001;38:1641–55. [8] Liew KM, He XQ, Ng TY, Sivashanker S. Active control of FGM plates subjected to a temperature gradient: modelling via finite element method based on FSDT. Int J Numer Meth Eng 2001;52:1253–71. [9] Liew KM, He XQ, Ng TY, Kitipornchai S. Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Comput Mech 2003;31:350–8. [10] Liew KM, Sivashanker S, He XQ, Ng TY. The modelling and design of smart structures using functionally graded materials and piezoelectric sensor/ actuator patches. Smart Mater Struct 2003;12:647–55.

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