Dynamic buckling of imperfect laminated plates with piezoelectric sensors and actuators subjected to thermo-electro-mechanical loadings, considering the temperature-dependency of the material properties

Composite Structures 88 (2009) 228–239

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Dynamic buckling of imperfect laminated plates with piezoelectric sensors and actuators subjected to thermo-electro-mechanical loadings, considering the temperature-dependency of the material properties M. Shariyat * Faculty of Mechanical Engineering, KNT University of technology, Pardis Street, Molla-Sadra Street, Vanak Square, Tehran, Iran

a r t i c l e

i n f o

Article history: Available online 4 April 2008 Keywords: Dynamic buckling Piezoelectric Temperature-dependency Active control Thermo-electro-mechanical loads Vibration

a b s t r a c t Dynamic buckling of piezolaminated plates under thermo-electro-mechanical loads has not been investigated so far. In the present paper, effects of the thermo-piezoelasticity on the dynamic buckling under suddenly applied thermal and mechanical loads are investigated for imperfect rectangular composite plates with surface-bonded or embedded piezoelectric sensors and actuators. A finite element formulation based on a higher-order shear deformation theory is developed. Both the initial geometric imperfections of the plate and the temperature-dependency of the material properties are taken into account. Complex dynamic loading combinations include in-plane mechanical loads, heating, and electrical actuations are considered. A nine-node second order Lagrangian element, an efficient numerical algorithm for solving the resulted highly nonlinear governing equations, and an instability criterion already proposed by the author are employed. A simple negative proportional feedback control is used to actively control the transient response of the plate. Results show that buckling mitigation due to utilizing integrated piezoelectric sensors and actuators is mainly achieved in extremely high gain values. It is also noticed that in many cases, effects of the control voltage on the results may be ignored compared to the temperaturedependency of the material properties and initial geometric imperfections effects. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Due to some considerations such as, economical fuel consumption and higher speed achievement, there is an increasing interest to decrease the thickness of the structural parts in aerospace, ground vehicles, hydrospace, and ship manufacturing industries. Smart and intelligent structures are developed to enhance the performance of the structural components (e.g. load carrying capacity, NVH, crash or buckling behaviors) and to compensate for the strength reduction due to using reduced thicknesses in such cases. In some cases, the load bearing substrates of these smart structures are made of composite materials. Using the benefits of direct and converse piezoelectric effects, surface bonded integrated piezoelectric sensor and actuator layers may be employed to adequately suppress the transient vibration or to control the deflection, shape and buckling of the structure. Active buckling control may be useful to effectively monitor and control the structure oscillations in cases where there is a small structural damping or there is a lack of other forms of damping. Behavior of the laminated piezoelectric plates has attracted the attention of many researches [1–4]. A review of some of the recent * 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.2008.03.044

developments in thermo-piezoelasticity with relevance to smart structures can be found in Yang [5]. Numerous authors have studied the piezoelectricity effects on the dynamic behavior of the composite structures. Influences of utilizing smart materials on the free vibration and natural frequencies of the laminated piezoelectric plates have been addressed by many authors [6–11]. Shi and Atluri [12] presented an analytical model to actively control the nonlinear transient vibrations of laminated piezoelectric rectangular plates with distributed sensors and actuators. Saravanos et al. [13] studied the active dynamic behavior control of some laminated piezoelectric plates using the three-dimensional theory of elasticity. Han and Lee [14] employed the layerwise theory and a closed loop control algorithm to actively control the dynamic behavior of composite plates with distributed piezoelectric actuators. Reddy [15] extended an equivalent single-layer plate model and presented Navier solutions for rectangular laminates with integrated sensors and actuators. Based on the classical laminated plate theory, Liu et al. [16] presented a finite element formulation to model the dynamic as well as static response of laminated composite plates containing integrated piezoelectric sensors and actuators. A piezolaminated quadrilateral plate/shell element was developed by Balamurugan and Narayanan [17] based on the first order shear deformation theory and the active vibration control of plates and shells was studied.

M. Shariyat / Composite Structures 88 (2009) 228–239

Thornburg and Chattopadhyay [18] presented a coupled theory that includes the nonlinear piezoelectric effects. Moita et al. [19] developed a finite element model based on the classical theory. Gao and Shen [20] adopted the first order shear deformation theory to present incremental finite element equations for analysis of geometrically nonlinear transient vibration response and control of plates with piezoelectric patches subjected to pulse loads. Kulkarni and Bajoria [21] presented a finite element formulation based on the higher order shear deformation theory to study the active vibration control performance of a piezolaminated curved beam with distributed sensors and actuators. Narayanan and Balamurugan [22] developed finite element models for laminated structures with distributed piezoelectric sensor and actuator layers based on the first order shear deformation theory. Constant-gain, negative velocity feedback, Lyapunov feedback as well as a linear quadratic regulator (LQR) approach have been used for active vibration control of structures subjected to impact, harmonic and random excitations. A nonlinear analysis of piezoelectric composite laminated beams and plates were presented by Varelis and Saravanos [23]. Simoes Moita et al. [24], derived a finite element formulation for active control of forced vibrations of thin plate/ shell laminated structures with integrated piezoelectric layers, based on the third order shear deformation theory. Recently, Pietrzakowski [25] studied the vibration of active rectangular plates using a velocity feedback control. The in-plane spatial variation of the potential is determined by solution of the coupled electromechanical governing equations with natural boundary conditions corresponding to both flexural and electric potential fields. Kirchhoff-type and Mindlin-type theories were employed. Qing et al. [26] proposed an efficient numerical Laplace inversion method for dynamic analysis of composite laminates with piezoelectric layers. Although influences of using active control on the dynamic response of the piezolaminated plates have been studied extensively by many authors, as was stated in the previous paragraph, limited researches have been performed in the dynamic buckling of piezolaminated plates field. Indeed, even the works accomplished in the dynamic buckling analysis of traditional rectangular composite plates are very limited [27–31]. Thompson and Loughlan [32] examined the active static buckling control using an analytical model and an experimental procedure. Oh et al. [33,34] presented a layerwise formulation for static buckling analysis of smart plates subjected to thermal and electrical effects. The Newton–Raphson iterative method was used to solve the nonlinear governing equations. The buckling temperature is determined through reducing the problem to an eigen value one. Shen presented static postbuckling analyses for imperfect plates with fully covered or embedded piezoelectric actuators subjected to thermal [35] and thermo-electro-mechanical [36] loads based on Reddy’s higher-order shear deformation plate theory. A mixed Galerkin-perturbation technique was used to determine the thermal buckling temperature and the postbuckling equilibrium paths. Matsunaga [37] studied the natural frequencies and the static buckling stresses of angleply laminated composite plates subjected to in-plane stresses using a higher-order theory. Varelis and Saravanos [38] studied the static buckling control of the smart beams and plates. Varelis et al. [39] introduced a finite element formulation for static linear buckling and postbuckling analysis of plates and beams loaded by the piezoactuators. The Newton–Rapshon iterative method was employed to solve the governing equations. Kapuria and Achary [40] developed a coupled zigzag theory for static buckling analysis of the hybrid piezoelectric plates. A global third order variation across the thickness with a layerwise linear variation is assumed for the in-plane displacement components. Buckling loads are derived using the approach of Noor and Burton [41]. Recently, Kundu et al. [42] investigated the nonlinear post buckling of piezoelectric

229

laminated doubly curved shells based on the first order shear deformation theory and using the finite element method. A total Lagrangian approach associated to arc-length method is used to solve the equilibrium equations. The foregoing review reveals that only few researches have been developed for buckling analysis of adaptive composite plates. Besides, works that are presented in the mentioned field are focused only on the static thermal buckling. Furthermore, almost in all of the reviewed researches, effects of the initial geometric imperfections were ignored. Dynamic thermal buckling analysis of piezolaminated plates has not been accomplished yet. Temperature-dependency of the material properties is also neglected in majority of the previous works. The main motivation of present investigation is to study the dynamic buckling control of imperfect multilayered composite plates with surface-bonded or embedded integrated piezoelectric sensor and actuator layers. However, effects of various parameters on the dynamic behavior are also investigated to enable presenting the final dynamic buckling results and discussions in a more adequate manner. A finite element formulation based on a high order shear deformation theory is used to investigate the buckling control under thermo-electromechanical loads. Temperature-dependency of the material properties, initial thermal stresses and imperfections, and control voltage effects on the static and dynamic buckling behaviors under thermo-electro-mechanical loads are also considered. A modified Budiansky criterion recently proposed by the author is employed to detect the buckling load [43]. In the results section, dynamic buckling is investigated for two complicated cases: dynamic thermal buckling of hybrid plates already subjected to electrical potential field and dynamic buckling of a plate already subjected to thermal and electrical loads, under a suddenly applied mechanical compression. 2. The governing equations The coordinate system and the geometric parameters of the plate are shown in Fig. 1. The plate is considered to be composed of a multilayered composite substrate and one or two surfacebonded or embedded piezoelectric layers. In Fig. 1, hs and ha are thickness values of the sensor and the actuator layers, respectively. In the present analysis, Reddy’s third order shear deformation description of the displacement field is adopted [44]: 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 ðw0;y þ wy Þ

ð1Þ

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

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

230

M. Shariyat / Composite Structures 88 (2009) 228–239

c1 ¼

4

ð2Þ

2

3h

Generally, Drichlet type boundary conditions (displacement components) and Neumann type boundary conditions (stresses or loads) cannot be incorporated simultaneously [45], especially, in a finite element solution. Since simultaneous satisfaction of 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 [46]. If the nodal vectors of the displacement and the rotation components of the mid-surface are denoted by U, V, W, Wx, Wy respectively, and similar shape function are chosen for them, one may write: u0 ¼ NUðtÞ; v0 ¼ NVðtÞ; w0 ¼ NWðtÞ; wx ¼ NWx ðtÞ; wy ¼ NWy ðtÞ ð3Þ where N is the shape function matrix. Therefore, Eq. (1) may be rewritten as: 8 9 uðx; y; z; tÞ > > > > > > > > > > > < vðx; y; z; tÞ > = wðx; y; z; tÞ dðx; y; z; tÞ ¼ > > > > > > wx ðx; y; z; tÞ > > > > > > : ; wy ðx; y; z; tÞ 3 2 0 N 0 c1 z3 N;x ðz  c1 z3 ÞN 7 6 0 ðz  c1 z3 ÞN 7 6 0 N c1 z3 N;y 7 ðeÞ 6 7d ¼ @dðeÞ 6 ¼60 0 N 0 0 7 7 6 5 40 0 0 N 0 0

0

0

0

N

T

dðeÞ ¼ ½ UðtÞ VðtÞ WðtÞ Wx ðtÞ Wy ðtÞ  ð4Þ Using von Karman-type strain–displacement relations: 1 1 exx ¼ u;x þ ðw;x Þ2 eyy ¼ v;y þ ðw;y Þ2 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

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 (w) expressions:   T   ^e ¼ e  e0 ¼ e0 þ ze1 þ z2 e2 þ z3 e3 e ¼ exx eyy cxy cyz cxz 8 9  ;x Þðw0;x Þ u0;x þ 12 ðw0;x Þ2 þ ðw > > > > > > > > > > > >  ;y Þðw0;y Þ v0;y þ 12 ðw0;y Þ2 þ ðw < = e0 ¼ u0;y þ v0;x þ w0;x w0;y þ w  ;x w0;y þ w  ;y w0;x ; > > > > > > > > wy þ w0;y > > > > : ; wx þ w0;x 8 8 9 9 0 wx;x > > > > > > > > > > > > > > > > > > > > > > > > <0 < wy;y = = 1 2 e ¼ wx;y þ wy;x ; e ¼ 0 ; > > > > > > > > > > > 3c1 ðwy þ w0;y Þ > 0 > > > > > > > > > > > > : : ; ; 3c1 ðwx þ w0;x Þ 0 8 9 c1 ðwx;x þ w0;xx Þ > > > > > > > > > > > > < c1 ðwy;y þ w0;yy Þ = 3 e ¼ c1 ðwy;x þ wx;y þ 2w0;xy Þ > > > > > > >0 > > > > > : ; 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: ^e ¼ NðdðeÞ ÞdðeÞ ¼ ðN0 þ zN1 þ z2 N2 þ z3 N3 ÞdðeÞ 2 1  ;x ÞN;x ðw0;x þ 2w 0 N;x 0 2 6 1  ;y ÞN;y 0 N;y 2 ðw0;y þ 2w 60 6  ;x ÞN;y þ w  ;y N;x 0 N0 ¼ 6 6 N;y N;x ðw0;x þ w 6 0 0 N;y 40 0

0

6 60 6 N1 ¼ 6 60 6 40

N;x N;x

0

0

0

0

0

N;y

0

0

0

0 2 0 60 6 6 N2 ¼ 6 60 6 40

0 0

0 0

0

0

0

0

0

0

0

0

3c1 N;y

0

0 2 0 6 60 6 N3 ¼ 6 60 6 40

0 0

3c1 N;x c1 N;xx

3c1 N c1 N;x

0

c1 N;yy

0

0

2c1 N;xy

c1 N;y

0

0

0

0

0

0

0

2

0 0

0

N

3

0

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

0

3

0

3

7 07 7 07 7 7 N5 0

ð7Þ

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

The coupled piezothermoelastic constitutive equations may be written as [47]: r ¼ Q ð^e  a  DTÞ  eT E

D ¼ e^e þ nE þ p  DT

ð8Þ

where r is the stress tensor ðrT ¼ h rxx ryy sxy syz sxz iÞ, Q is the elasticity constants matrix, a is the vector of the thermal expansion coefficients, e is the matrix of the piezoelectric coefficients, E is the electric field vector, DT is the temperature rise from a stress free reference temperature, D is the electric displacement vector, n is the dielectric coefficient matrix, and p is the pyroelectric constants 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 [48]. The electrical field E is calculated based on the gradient of the electric potentialu: E ¼ Ku

KT ¼ h ox

oy

ð9Þ

oz i

A nine-node quadratic Lagrangian element is used in the present research. The elements are located on the reference plane of the plate and have six degrees of freedom in each node: five (u0, v0, w0, wx, wh) displacement components and an electrical potential / per piezoelectric layer. Therefore, if the electric potential is assumed to vary through the thickness (z-direction) of the piezoelectric layer only, the electric field vector in each layer will be: E ¼ Bu uðeÞ where 2

0

6 0 6 6 1 6 hs Bu ¼ 6 6 0 6 6 4 0 0

ð10Þ

0

3

0 7 7 7 0 7 7 0 7 7 7 0 5

uðeÞT ¼ h us

ua i

ð11Þ

1

ha

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

ð12Þ

M. Shariyat / Composite Structures 88 (2009) 228–239

where PS, PE, PExt, and PI are strain energy, electrical energy, external loads and electrical charges energies, and inertial forces energy, respectively. Therefore, in absence of the body forces, according to the principle of virtual displacement, one has: dP ¼ dPs þ dPE þ dPExt þ dPI ¼ 0 Z Z  T  d^eT r  dV ¼ d^e ½Q ð^e  a  DTÞ  eT E  dV dPs ¼ V V " " #! Z ( 3 3 X X ðeÞT ^ iþj Nj ÞdðeÞ  ^ ¼ dd NT ð ai DT Q

ð13Þ

i¼0

þð

3 X

j¼0

#)

NTi ~zi ÞeT Bu uðeÞ

Since evaluating the general effect of the active control on the vibration and buckling is of interest in the present approach, the following proportional control law is implemented for the actuators: ðeÞ uðeÞ a ¼ Gus

ð25Þ

where G is the feedback gain and its appropriate value may be determined through successive solutions. Generally, based on different possible choices of the electrical potential vector of the actuator layer, Eq. (24) gives the finite element governing equations of the plate in the following form:

i

A

231

M€ d þ CðdÞd_ þ KðdÞd ¼ F

dA

ð26Þ

i¼0

¼ ddðeÞT Kdd dðeÞ  ddðeÞT FT þ ddðeÞT Kdu uðeÞ Z dET D  dV dPE ¼  Vp

¼ duðeÞT

Z Vp

BTu ðe

3 X

ð14Þ

~  DTÞ  dV Ni ~zi dðeÞ þ nBu uðeÞ þ p

i¼0

¼ duðeÞT Kud dðeÞ þ duðeÞT Kuu uðeÞ þ duðeÞT GT Z Z @T f s :dC þ duðeÞT dPExt ¼ dW ¼ ddðeÞT Fc  ddðeÞT C1

ð15Þ Bu q:dC C2

¼ ddðeÞT Fc  ddðeÞT Fs  duðeÞT Fq Z dðeÞ q@T @€ dðeÞ  dV ¼ ddðeÞT Mdd € dPI ¼ ddðeÞT

ð16Þ ð17Þ

V

where V, Vp and A are the volume, volume of the piezoelectric layers and area of the element, respectively, n is the number of layers, and kis the layers counter (including the piezoelectric layers) and: n Z zkþ1 n Z zkþ1 X X ^m ¼ Q k zm  dz; ^ am ¼ Q k ak zm  dz; Q zk

k¼1

~ a¼

XZ k¼1

~zm ¼

zk

ak  dz

ð18Þ

zk

XZ k¼1

k¼1

zkþ1

zkþ1

zm  dz

~¼ p

XZ

zk

k¼1

zkþ1

zk

pk  dz

~ are calculated for the piezoelectric layers only. Fc, fs, and ~zm ; ~ a; and p 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. (12). Substituting Eqs. (14)–(17) into Eq. (13), and noting that Eq. (13) must hold for any arbitrary dd(e)and d u(e) vectors, the governing equations of the element will be: ( ðeÞ )

  ( €ðeÞ )  FT þ Fc þ Fs Kdd Kdu Mdd 0 d d ð19Þ þ ¼ Fq  GT Kud Kuu 0 0 € ðeÞ u uðeÞ 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. (19): dðeÞ þ Kdd dðeÞ þ Ksdu uðeÞ Mdd € s ¼ FT þ Fc þ Fs Kadu dðeÞ

þ

Kauu uðeÞ a

¼ Fq 

GaT

s Ksdu dðeÞ þ Ksuu uðeÞ s ¼ GT

ð20Þ ð21Þ ð22Þ

Eq. (22) gives the sensor voltage as: sðeÞ

sðeÞ1 uðeÞ ðKud dðeÞ þ GsT Þ s ¼ Kuu

ð23Þ

Subscripts/superscripts s and a denote the sensor and actuator layers, respectively. Therefore, Eq. (20) 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

ð24Þ

3. The proposed numerical solution and stability criterion Due to the nonlinear behavior of the plate, an incremental and iterative solution method based on adaptive relaxation [49] scheme as a corrector, is employed. The numerical time integration procedure and the instability point tracing of the present numerical solution algorithm are somewhat similar to those proposed in some of the previously published papers of the author [50–53]. Some authors, (e.g. Bisagni [54]) have compared results of the mentioned papers with the experimental data and have reported that results of the mentioned papers are more accurate than results reported by other references. Among various instability criteria proposed for the conservative systems so far [55], three criteria are commonly used: The total energy-phase plane or Lyapunov exponents approach [56], the equation of motion approach or Budiansky–Roth criterion [57–59], and the parametric instability [60,61]. Some authors has proposed criteria for dynamic buckling of imperfection sensitive nonconservative systems [62] and some others, studied the modal shifting and mode jumping phenomenona [63–65]. However, in the present investigation, the system is conservative. In the present research, time variations of the quantities are determined through using Newmark’s time integration method [66]. Since the plate is pre-loaded by electrical and mechanical/thermal loads, numerical solution is accomplished using the following steps. The proposed algorithm is somewhat similar to an algorithm recently published by the author [67]. 1. Pre-loads are divided to many increments. Subsequently, the pre-loading is accomplished through the same number of steps. 2. At the beginning of each loading step, values of the displacement, velocity, and acceleration components of the previous loading stage are utilized to construct the stiffness matrix. At the first stage, initial values are used as a first estimation. 3. The stiffness matrix is updated based on the values of the displacement components obtained at the end of the previous iteration. Since the pre-loading is performed in a static manner, components of the mass and the damping matrices are set equal to zero. Then, the modified iterative Newton method is applied to the incremental form of the static version of Eq. (26) to improve the solution accuracy and to obtain a convergent solution for the specified loading interval. Otherwise, numerical instability will occur. 4. When the static mechanical or thermal pre-loading is completed, the dynamic loading is started and accomplished during an incremental increasing. 5. Eq. (26) is solved by the implicit method [57]. As a common rule, the time step should be chosen less or equal to 1/(20f) where f is the natural frequency of the highest mode that contribute to the response. In the present research, time step is chosen as 105 (s). The resulted highly nonlinear equations are solved using the Newmark’s numerical integration method. Nodal displacement,

232

M. Shariyat / Composite Structures 88 (2009) 228–239

velocities, and accelerations of the end of each time step are determined using the modified Newton–Raphson iterative method until the convergence criterion is satisfied. In each iteration, the stiffness matrix may be updated. In some cases, over-relaxation or underrelaxation methods may be useful [49]. In the present research, a variable relaxation is used to guarantee the numerical stability. The following convergence criterion is chosen for the present research: kDHi k 6 0:001 kHi k

ð27Þ

ied by references [9,68] is reexamined. Four different situations are considered for the plate: without piezoelectric layers (referred to as ‘W’), with a single PZT-5A piezoelectric layer at the top surface (T) or at the mid-thickness (M), and with two PZT-5A piezoelectric layers bonded to the top and bottom surfaces (TB). The layers of the mentioned four groups of plates are of equal thickness and the material properties of the graphite–epoxy layers are [9]: E11 ¼ 181 ðGPaÞ; G23 ¼ 2:87 ðGPaÞ;

E22 ¼ 10:3 ðGPaÞ; m12 ¼ 0:28;

a11 ¼ 0:02  106 ð1=C  Þ;

G12 ¼ G13 ¼ 7:17 ðGPaÞ;

q ¼ 1580 ðkg=m3 Þ;

a22 ¼ 1:5  106 ð1=C  Þ

where Hi is a representative nodal value. material properties of the PZT-5A layers are as follows: 6. Values obtained at the end of each time interval are considered as initial values for the next time interval. 7. Trend of change of the maximum lateral displacement component versus loads (or vice versa) is determined. 8. Using an appropriate instability criterion, e.g. the one that is recently proposed by the author [43], the dynamic buckling load is determined. According to Budianky’s criterion, any abrupt change in the slope of the load–displacement curve implies a buckling occurrence. Due to the nonlinear behavior, the load–displacement curve of some structures does not show distinct buckling point. The modified Budiansky criterion previously proposed by the author [43], presents an instability criterion that is applicable to nonlinear systems and especially is more adequately applicable in a computerized solution procedure. In the present approach, variation of the load versus the maximum value of the transverse displacement component is considered. However, variation of the load with the in-plane shortening may be used instead. The instability behavior is more detectable for imperfect structures. For perfect plates, sometimes application of a transverse pulse to pass through the bifurcation point to the instable dynamic equilibrium path is necessary.

4. Results and discussions To investigate effects of various parameters on the dynamic buckling of the piezolaminated plates, a sensitivity analysis, including the effects of the layer sequence, temperature-dependency of the material properties, boundary conditions, and loading type is accomplished. At the same time, some of the results are compared with results reported by other references, for validation purposes. Then, effects of the mentioned parameters on the static and dynamic buckling of the hybrid plates are determined. Second order 9-node Lagrangian elements are employed to derive the results. 4.1. Dynamic response investigation and validation 4.1.1. Transient analysis results For validation purposes, a square simply supported cross-ply laminated plate with eight graphite–epoxy layers previously stud-

E11 ¼ 61 ðGPaÞ;

m12 ¼ 0:35; 6



a11 ¼ a22 ¼ 1:5  10 ð1=C Þ;

q ¼ 7750 ðkg=m3 Þ; e13 ¼ e23 ¼ 171  1012 ðm=VÞ

Almost in all displacement-based formulations introduced so far, simply supported and free edge boundary conditions are incorporated approximately. An exact method, already proposed by the author is used to incorporate the boundary conditions [43]. The first natural frequencies are calculated for the mentioned four groups of the plates and compared in Table 1 with those obtained based on the three-dimensional (3-D) solutions of reference [68], the first order shear deformation theory of reference [69], and the high order theory of reference [9]. Based on a previous experience of the author [50–53,43,67,70], if the details of the employed numerical solution procedure remain the same, the stiffness of the shell or the plate structure generally reduces as a more accurate theory is used. Therefore as a rule of thumb, natural frequency of the structure is reduced when switching among classical, first order shear deformation, third order shear deformation, layerwise, and 3-D elasticity theories, respectively. Furthermore, the solution method may somewhat affect the results. However, there is a good agreement among results of Table 1. As a second example, effect of the initial temperature changes on the fundamental frequencies of the plate group denoted by ‘T’ is investigated. The initial temperature change is assumed to vary through the thickness direction only, so that: DTðzÞ ¼ DT Ave þ DT Diff  z=h

ð28Þ

where DTAve is the average temperature change of the top and bottom surfaces of the plate, and DTDiff is the difference between the temperatures of the top and bottom surfaces. Instead of DTAve, the dimensionless average temperatures change T Ave ¼ 103 DT Ave E22 qa2 x20 is used. T Ave is calculated based on the graphite–epoxy layer material properties and x0 is the fundamental frequency of the unstressed plate. The initially thermally stressed natural frequency to unstressed natural frequency ratios are evaluated for various values of DTAve, DTDiff, and h/aand are compared with results reported by references [9,68] in Table 2. There is a good agreement among the results. Since DTDiff/h denotes the temperature gradient, as the plate becomes thicker, the DTDiff effect diminishes and DTAve effect becomes more dominant. In Table 2, influence of the temperature-

Table 1 A Comparison among the fundamental frequencies (in rad/s) of the considered plates: without piezoelectric layers (referred to as ‘W’), with a single piezoelectric layer at the top surface (T) or at the mid-thickness (M), and with two piezoelectric layers located at the top and bottom surfaces (TB) Situation

W T M TB

a/h = 100

a/h = 10

Ref. [69]

Ref. [9]

Ref. [68]

Present

Ref. [69]

Ref. [9]

Ref. [68]

Present

340.86 300.64 285.16 283.93

340.79 292.80 284.78 269.12

333.02 290.38 285.26 268.86

337.82 291.67 286.03 271.23

3098.3 2703.7 2643.0 2516.7

3035.73 2656.37 2613.36 2397.06

2939.2 2554.7 2547.5 2357.7

2976.1 2627.8 2592.3 2381.2

233

M. Shariyat / Composite Structures 88 (2009) 228–239 Table 2 Comparison of thermally stressed to unstressed natural frequency ratios of the ‘T’ class plates h/a

DTDiff

0.01

0

10

100

Ref.

[9] [68] Present Present [9] [68] Present Present [9] [68] Present Present

T Ave 1

2

3

TID TD

0.7998 0.7965 0.7926 0.5782 0.8002 0.797 0.8014 0.5834 0.8029 0.8016 0.8041 0.5729

0.5995 0.593 0.5942 0.4161 0.5966 0.5935 0.5946 0.4132 0.5998 0.5981 0.6002 0.4190

0.3991 0.3894 0.3889 0.2683 0.3996 0.39 0.3913 0.2718 0.403 0.3946 0.4021 0.2734

TID TD

TID TD

0.1

100

[9] [68] Present TID Present TD

0.805 0.797 0.8104 0.5813

0.608 0.594 0.6071 0.4318

0.4102 0.3909 0.4116 0.2789

0.2

100

[9] [68] Present TID Present TD

0.8087 0.7981 0.8110 0.5907

0.6124 0.5959 0.6112 0.4412

0.4218 0.3935 0.4196 0.2862

Table 3 Material properties of the layers of the piezolaminated plates Parameter

Graphite/epoxy

PZT-5A

P0

P1

P0

P1

E11 (GPa) E22 G12 G13 G23 m12

150 9 7.1 7.1 2.5 0.3

0.0005 0.0002 0.0002 0.0002 0.0002 0

63 63 24.2 24.2 24.2 0.35

0.0005 0.0002 0.0002 0.0002 0.0002 0

e13 (C/m2) e23 e42 e51

0 0 0 0

0 0 0 0

7.2097.20912.322 12.322

0 0 0 0

n11(108F/m) n22 n33 q (k/m3)

0 0 0 1590

0 0 0 0

1.53 1.53 1.5 7600

0 0 0 0

a11 (106/C) a22

1.1 25.2

0.0005 0.0005

0.9 0.9

0.0005 0.0005

k11 (W/mK) k22

1.8 1.8

0 0

2.1 2.1

0 0

p11 (1012C2/Nm2) p22

0 0

0 0

82.6 90.3

0 0

(a) All edges clamped (CCCC). (b) One edge clamped and three edges free (CFFF). (c) Two opposite edges clamped and two simply supported (CCSS). (d) Two opposite edges clamped and two free (SSFF). (e) All edges simply supported (SSSS). Since externally applied loads, including electric charge, thermal loads, and mechanical loads must be set to zero in a free vibration analysis, for each boundary condition, three cases: composite plate without piezoelectric layers (W), hybrid plate with open circuit piezoelectric layer (OC), and hybrid plate with closed circuit (short-cut circuit) piezoelectric layer (SC), are chosen for the free vibration analysis. First five natural frequencies of the plate are given 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 externally applied control voltage exists. Furthermore, natural frequencies are lower when the piezoelectric layer is used in a short circuit. 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

Table 4 Natural frequencies in (Hz) corresponding to different boundary conditions

dependency of the material properties on the natural frequency is also studied. TID and TD abbreviations denote temperature independent material properties and temperature-dependent material properties assumptions, respectively. The material properties are assumed to be linear functions of the temperature change [71]: P ¼ P0 ð1 þ P 1 DTÞ

To evaluate the boundary condition influence on the natural frequencies, free vibration of a rectangular cross-ply graphite–epoxy plate with 20 (cm)  20 (cm) in-plane dimensions is investigated. The composite plate is assumed to be 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 considered, in the present example. Similar to the previous example, the material properties are assumed to be linear functions of the temperature change. Coefficients of Eq. (29) are given for various properties in Table 3. Five types of boundary conditions are adopted for the free vibration analysis:

Boundary condition

Mode no.

W

OC

SC

CCCC

1 2 3 4 5

1036 1671 2552 2947 2989

1038 1791 2500 3002 3241

1033 1780 2468 2974 3234

CFFF

1 2 3 4 5

141 187 525 880 938

133 195 580 838 923

132 189 569 833 904

CCSS

1 2 3 4 5

943 1284 2180 2509 2728

925 1354 2375 2450 2747

918 1341 2364 2413 2710

SSFF

1 2 3 4 5

394 437 718 1456 1577

374 435 788 1506 1568

370 425 771 1488 1545

SSSS

1 2 3 4 5

487 985 1642 1946 2010

504 1091 1610 2017 2225

500 1082 1590 1993 2217

ð29Þ

where P is the material property, P0 is the material property at the ambient temperature, and P1 values are given in Table 3 [71,72]. Results shown in Table 2 reveal that temperature-dependency can considerably affect the stiffness and subsequently, the natural frequency of the plate. Furthermore, temperature gradient effects across the thickness may be exaggerated when taking the material dependency into account.

W, OC, and SC denote without piezoelectric layer, with open circuit, and with shortcut circuit piezoelectric layer, respectively.

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M. Shariyat / Composite Structures 88 (2009) 228–239

conditions lead to more flexibility, and subsequently, in absence of local loading conditions, lead to greater displacements and stresses. Therefore, buckling and resonance are more likely to occur in plates with looser boundary conditions. 4.1.2. Forced vibration analysis To investigate the piezoelectricity influence on the forced vibration response, step and sinusoidal loads are applied individually. For this purpose, a CCCC plate, with one surface-bonded piezoelectric actuator layer is adopted and a force with amplitude of 1 (kN) is exerted on the mid-point of the plate. The frequency of the applied load is 100 (Hz) for the sinusoidal input. As explained before, electrical potential has minor effects on the natural frequencies of the piezolaminated plate. The effect increases with the applied electrical potential. In the present analysis, the contact surface of the piezoelectric layer with the composite substrate is chosen as a reference plane for the electrical potential. Effect of the actuator layer on the forced vibration response is depicted in Fig. 2, for the step input. Since no structural damping is considered for the plate, the transient response does not diminish with time. The transient oscillations are performed with the fundamental frequency of the plate which is given in Table 4. Furthermore, a second transient response due to the electrical potential application is also induced in the plate. Results of Fig. 2 imply that higher vibration modes are more remarkably invoked in plates with electrical actuation. As it may be noticed from Fig. 2, application of appropriate minus control voltage may reduce the vibration amplitude. On the other hand, application of nonappropriate electrical potential may adversely affect the vibration behavior. Effects of the electrical potential on the forced vibration response of the plate under the sinusoidal force are illustrated in Fig. 3. Transient oscillations with the fundamental natural frequency are combined with the forced vibration response of a plate with no electrical excitation. Responses shown in Fig. 3 reveal that higher vibration modes are more magnified in the presence of electrical actuation. Therefore, although using a piezoelectric layer may reduce the amplitude of the overall vibration of the plate in some cases, it induces noises that may result in harshness problems e.g. in passenger vehicles [73]. Present results show that in contrast to results reported by some researchers, passive control or static electrical inputs may not cause vibration damping.

Fig. 3. Forced vibration response of the piezoelaminated plate under the sinusoidal force.

A piezolaminated plate with properties listed in Table 3 is adopted to investigate the simultaneous effects of electrical potential application and temperature rise on the natural frequencies. The side of the square plate is 24 (mm) and the total thickness of the plate is 1.2 (mm). All orthotropic layers of the substrate are of equal thickness, whereas the thickness of piezoelectric layers is 0.1 (mm). A uniform temperature rise and a uniform electrical field are considered. Both TID and TD material properties are considered. In nonlinear structures, the initial conditions and the initial stresses may affect the natural frequencies of the structure. Results are derived for a suddenly applied uniform surface load p = 0.5 (MPa). For the sake of brevity, (0/90)2T antisymmetric cross-ply and (0/90)S symmetric cross-ply laminated plates with a double-thickness piezoelectric layer bonded to the top surface or embedded at the middle surface are referred to as (P/0/90/0/ 90), (P/0/90/90/0), (0/90/P/0/90) and (0/90/P/90/0), respectively, whereas hybrid plates with two piezoelectric layers bonded to the top and bottom surfaces are referred to as (P/0/90/0/90/P) and (P/0/90/90/0/P). Table 5 shows the dimensionless frequency pffiffiffiffiffiffiffiffiffiffiffiffi parameters X ¼ x0 ða2 =hÞ q=E22 (material properties are substituted according to the graphite/epoxy orthotropic layer) of the mentioned six hybrid laminated plates under three uniform temperature changes and six different electrical loading cases. Vt, Vm, and Vb represent the control voltages applied to the top, middle and bottom piezoelectric layers, respectively. Results of Table 5 imply that constant voltage inputs have minor effects on the first natural frequencies. Indeed, as results illustrated in Figs. 2 and 3 show, these inputs invoke the higher modes more remarkably. From results of Table 5, it may be easily deduced that errors resulted due to neglecting the temperature-dependency of the material properties are much remarkable than the piezoelectricity effects. 4.2. Dynamic buckling analysis

Fig. 2. Forced vibration response of the piezoelaminated plate under the step load.

Although numerous works may be found in literature in static thermal buckling analysis of composite laminated plates with piezoelectric actuators, no work is developed in the dynamic thermal buckling of the mentioned plates. Besides, although the initial geometric imperfections role is somewhat negligible in the modal analysis, their effects may be remarkable in the buckling analysis. Generally, employing excessive electrical potential may lead to contraction of the plate and in turn to buckling, even in quasi-static electrical loading [39]. Utilizing dynamic electrical passive control

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M. Shariyat / Composite Structures 88 (2009) 228–239 Table 5 pffiffiffiffiffiffiffiffiffiffiffiffi Non-dimensional frequency parameters X ¼ x0 ða2 =hÞ q=E22 of laminated plates with piezoelectric actuators under different control voltages and temperature gradients Stacking sequence

Electric potential (V)

Reference

Temperature rise DT = 0 (C)

(P/0/90/0/90)

Vt = 100

Vt = 0

Vt = 100

(P/0/90/90/0)

Vt = 100

Vt = 0

Vt = 100

(0/90/P/0/90)

Vt = 100

Vt = 0

Vt = 100

(0/90/P/90/0)

Vt = 100

Vt = 0

Vt = 100

(P/0/90/0/90/P)

Vt = 50

Vt = 0

Vt = 50

(P/0/90/90/0/P)

Vt = 50

Vt = 0

Vt = 50

[9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present [9] Present Present

TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD TID TD

inputs (e.g. a step input) may induce undesirable oscillations and subsequently may adversely affect the plate instability, especially when the structural damping is ignorable. 4.2.1. Dynamic thermal buckling As a first example, thermal postbuckling analysis is performed for symmetric cross-ply laminated graphite–epoxy plates with surface-bonded or embedded piezoelectric PZT-5A actuators under uniform temperature rise. Material properties are already given in Table 3. The total thickness of the plate is h = 1.2 (mm), the thickness of the piezoelectric layers is 0.1 (mm), and the graphite–epoxy orthotropic layers of the substrate are of equal thickness.

DT = 100 (C)

DT = 300 (C)

10.475 10.326 10.326 10.458 10.298 10.298 10.444 10.265 10.265

9.883 9.646 9.278 9.764 9.513 9.146 9.849 9.602 9.226

8.573 8.251 7.361 8.538 8.208 7.323 8.512 8.184 7.298

10.733 10.583 10.583 10.715 10.567 10.567 10.700 10.551 10.551

10.163 9.904 9.519 10.142 9.893 9.504 10.123 9.871 9.486

8.918 8.579 7.661 8.885 8.549 7.629 8.858 8.519 7.601

9.940 9.796 9.796 9.889 9.748 9.748 9.836 9.692 9.692

9.179 8.963 8.602 9.160 8.932 8.613 9.153 8.927 8.579

7.818 7.523 6.712 7.806 7.511 6.701 7.795 7.501 6.692

10.216 10.058 10.058 10.167 10.003 10.003 10.117 9.968 9.968

9.661 9.421 9.048 9.608 9.369 8.998 9.556 9.321 8.958

8.441 8.118 7.248 8.381 8.006 7.148 8.321 7.996 7.132

10.664 10.516 10.516 10.617 10.469 10.469 10.568 10.421 10.421

10.133 9.882 9.501 10.082 9.831 9.553 10.032 9.786 9.402

8.975 8.628 7.697 8.918 8.583 7.659 8.862 8.529 7.607

10.861 10.701 10.701 10.814 10.654 10.654 10.768 10.609 10.609

10.343 10.067 9.676 10.293 10.024 9.628 10.244 9.989 9.609

9.217 8.870 7.911 9.163 8.821 7.873 9.106 8.764 7.819

Figs. 4 and 5 show effects of the control voltage, the aspect ratio, and the stacking sequence on the static thermal buckling of the piezolaminated perfect and imperfect plates. In these figures, in each curve set, the upper, middle, and bottom curves are corresponding to Vt = Vb = 100 (V), Vt = Vb = 0 (V), and Vt = Vb = 100 (V) situations, respectively and W denotes the amplitude of the maximum transverse displacement component of the plate. In Fig. 4, the buckling and postbuckling behaviors of (P/(0/90)2)S perfect and imperfect ðW=h ¼ 0:1Þ square laminated plates with symmetrically fully covered piezoelectric layers and a (a/h = 50) width to thickness ratio, are compared.W is the amplitude of the initial transverse geometric imperfections. Fig. 5 presents a para-

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M. Shariyat / Composite Structures 88 (2009) 228–239

Fig. 4. Comparison of the buckling and postbuckling behaviors of perfect and imperfect (W0/h = 0.1)plates with a/h = 50 edge length to thickness ratio. In each curve set, the upper, middle, and bottom curves are corresponding to Vt = Vb = 100 (V), Vt = Vb = 0 (V), and Vt = Vb = 100 (V)situations, respectively.

Fig. 5. Results of the parametric study, for imperfect plates with (W0/h = 0.1) ratio. In each curve set, the upper, middle, and bottom curves are corresponding to Vt = Vb = 100 (V), Vt = Vb = 0 (V), and Vt = Vb = 100 (V)situations, respectively.

metric study results, for imperfect plates with a ðW=h ¼ 0:1Þ ratio. This study includes the width to thickness ratio, plate aspect ratio, and stacking sequence effects. Fig. 4 shows that the initial imperfection may considerably reduce the buckling temperature rise. Since the ratio of the initial imperfection amplitude to the transverse displacement amplitude ðW=WÞ diminishes in the postbuckling regime, postbuckling curves of the imperfect plates approached to those of the perfect plates as one proceeds in the postbuckling region. At the end of the postbuckling curves, indications of the stiffness recovery are

noticed which are more pronounced for the imperfect plates. Results also reveal that negative control voltages in contrast to the positive ones may enhance the buckling behavior of the plate. Results illustrated in Fig. 5 imply that the buckling temperature rises, in situations where no electrical actuation exists, are approximately proportional to the square of the thickness to edge length ratio as it is the case for the isotropic plates (whereas the natural frequency is approximately proportional to the mentioned ratio itself). Control voltage effect increases as the thickness to edge length ratio decreases. Since a greater couple is induced in the

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M. Shariyat / Composite Structures 88 (2009) 228–239 Table 6 Nondimensional dynamic thermal buckling results a/b

b/t

No control voltage TID

1 1.5

30 50 30 50

With active control TD

TID

TD

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

0.7297 0.6994 0.7179 0.6853

0.7031 0.6749 0.6858 0.6656

0.6099 0.5932 0.5968 0.5619

0.5666 0.5426 0.5451 0.5408

0.7578 0.7298 0.7571 0.7272

0.7254 0.6938 0.7179 0.6923

0.6260 0.6036 0.6249 0.5923

0.5897 0.5684 0.5854 0.5589

(P/(0/90)2)S plate whose piezoelectric layers are surface-bonded, the buckling temperature rise of the mentioned plate is somewhat higher than the (0/P/90/0/90)S plate that has two embedded piezoelectric layers. From Figs. 4 and 5 it may be readily deduced that applying positive control voltages, may lead to an additional expansion and subsequently, to a decreased buckling temperature rise. For load–displacement curves with gradual slope variation, a modified version of Budiansky criterion is recently introduced by the author [43] to determine the buckling load. Based on the mentioned criterion, the dynamic thermal buckling of the piezoelectric plates and influence of employing an active proportional closed loop control to suppress the thermally-induced vibrations and mitigating the buckling phenomenon are investigated in the present research. In this regard, the (P/(0/90)2)S square plates with a/h = 50 ratio used in the foregoing static thermal analysis are employed in a dynamic buckling analysis. 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 = 103) under suddenly temperature rise are derived for both temperature-dependent and temperatureindependent material properties assumptions. Table 6 summarizes the nondimensional thermal buckling loads which are obtained by dividing the dynamic buckling temperature rises by those obtained from the static thermal buckling analyses (with no applied voltages and neglecting the temperature-dependency of the material prop-

erties). From results of Table 6, it may be concluded that the active control effect is minor whereas the temperature-dependency of the material properties may considerably affect the buckling temperature. Further enhancements in forced vibration response and subsequently, buckling behavior requires employing extremely high gains [24]. Generally, passive control is not useful in cases where there is a lack in the structural damping unless the oscillations induced due to the voltage application are artificially damped. 4.2.2. Dynamic buckling under an in-plane load Dynamic buckling of the same piezoelectric laminated (P/(0/ 90)2)S plates is investigated under thermo-electro-mechanical loads as a more general case. For this purpose, as a base, static buckling of the (P/(0/90)2)S perfect and imperfect ðW=h ¼ 0:1Þ square piezolaminated plates withDT = 200 (°C) is investigated for two width-to-length ratios and three applied voltages. Corresponding postbuckling load–deflection curves are shown in Fig. 6. In this figure, load–deflection behavior of plates already under thermal and voltage inputs is investigated by suddenly loading the plates by an in-plane compression force ‘F’ which is applied in X-direction. Time history of the mentioned mechanical load is defines as a step function. In each curve set, the upper, middle, and lower curves are corresponding to Vt = Vb = 100(V), Vt = Vb = 0(V), and Vt = Vb = 100 (V) situations, respectively. As before, it is apparent that the minus control voltage increases the buckling load. Furthermore, it may be found that the control voltage has less effect

Fig. 6. Static buckling under thermo-electro-mechanical loading conditions for perfect and imperfect (W0/h = 0.1) piezolaminated plates. In each curve set, the upper, middle, and bottom curves are corresponding to Vt = Vb = 100 (V), Vt = Vb = 0 (V), and Vt = Vb = 100 (V) situations, respectively.

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M. Shariyat / Composite Structures 88 (2009) 228–239

Table 7 Nondimensional mechanical dynamic buckling loads of plates initially stressed by thermo-electrical loads a/b

b/t

No control voltage TID

1 1.5

30 50 30 50

With active control TD

TID

TD

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

Perfect

Imperfect

0.7963 0.7637 0.7836 0.7485

0.7677 0.7373 0.749 0.7272

0.6673 0.6492 0.6533 0.6156

0.6206 0.5947 0.5974 0.5928

0.8266 0.7965 0.8259 0.7937

0.7917 0.7576 0.7837 0.756

0.6846 0.6604 0.6834 0.6482

0.6455 0.6226 0.6409 0.6123

on postbuckling behavior of plates with lower width to thickness ratios. Results of the (P/(0/90)2)S plates (are not shown) are somewhat similar to those presented in Fig. 6 (within a 3% deviation is noticed). The offsets of the displacement–load curves of the imperfect plates are due to the presence of the initial thermal strains. A dynamic buckling analysis includes the thermo-electromechanical loads and temperature-dependency of the (P/(0/ 90)2)S square plates with a/h = 50 ratio, is performed and the relevant results are given in Table 7. As it may be noticed, due to presence of the initial temperature rise, the dynamic buckling ratios are greater than those appeared in Table 6. Furthermore, in contrast to the control voltage role, the temeperature-dependency role on the dynamic buckling results is significant. 5. Conclusions In the present paper, dynamic buckling of laminated plates with piezoelectric sensors and actuators under thermo-electro-mechanical loads are investigated. The main novelties of the present paper are: 1. Performing a dynamic buckling analysis for piezolaminated plates under thermo-electro-mechanical loading condition, for the first time. 2. Incorporating the influence of the temperature-dependency of the material properties in the modal and buckling analyses. 3. Considering the initial thermal stresses and control voltages effects in the buckling analyses. 4. Employing a revised version of Budiansky criterion recently proposed by the author to determine the buckling loads. 5. Incorporating the initial geometric imperfections effects. 6. Exactly applying the kinematic, force and moment boundary conditions. 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 dynamic response and the 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. References [1] Lee CK. Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. Part I: governing equations and reciprocal relationships. J Acoust Soc Am 1990;87(3):1144–58. [2] Pai PF, Nayfeh AH, Oh K, Mook DT. A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors. Int J Solids Struct 1993;30(12):1603–30. [3] Yu YY. On the ordinary, generalized, and pseudo-variational equations of motion in nonlinear elasticity, piezoelectricity, and classical plate theory. Am Soc Mech Eng J Appl Mech 1995;62:471–8.

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