Free vibrations of simply-supported piezoelectric adaptive plates: an exact sandwich formulation

Thin-Walled Structures 40 (2002) 573–593 www.elsevier.com/locate/tws

Free vibrations of simply-supported piezoelectric adaptive plates: an exact sandwich formulation A. Benjeddou ∗, J.-F. Deu¨, S. Letombe

1

Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Me´tiers, 2 rue Conte´, F-75003 Paris, France Received 29 June 2001; received in revised form 15 November 2001; accepted 20 November 2001

Abstract An exact two-dimensional analytical solution is proposed for the free-vibration analysis of simply-supported piezoelectric adaptive plates. It is based on an original sandwich formulation that considers layerwise first-order shear-deformation theory and quadratic non-uniform electric potential, with no assumptions on electric field and displacement components. Thus, the electric-charge conservation equation is exactly satisfied and the induced potential, hence the electromechanical coupling, is correctly represented. Also, two-dimensional electromechanical equations of motion and generalized piezoelectric constitutive equations, corresponding to introduced stress and electric displacement resultants, are derived and presented for the first time. The proposed approach was numerically validated through modal analysis of several hybrid plates with graphite-epoxy cross-ply substrates and embedded or surface-bonded piezoelectric layers. Compared to available uncoupled and coupled (exact) three-dimensional elasticity (Navier and state space) and finite-element (layerwise and mixed equivalent singlelayer/layerwise) solutions, the obtained results were the closest to the exact coupled threedimensional ones, making the present approach very reliable.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Analytical solutions; Vibration; Piezoelectric; Adaptive structures; Simply-supported; Plate; Sandwich theory; Quadratic electric potential

∗

1

Corresponding author. Tel.: +33-1-4027-2760; fax: +33-1-4027-2716. E-mail address: [email protected] (A. Benjeddou). Present address: LMT-Cachan, 61 avenue du Pre´sident Wilson, 94235 Cachan Cedex, France.

0263-8231/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 2 ) 0 0 0 1 3 - 7

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1. Introduction Modelling and analysis of multilayer piezoelectric beams and plates have reached a relative maturity as attested by the numerous papers listed in the latest assessments [1,2], reviews [3,4] and surveys [5,6]. Careful analysis of the corresponding literature indicates that approximate theories were often adopted. These mainly differ by the simplifying a priori assumptions concerning the piezoelectric effect representation, the directions of the electric field and/or displacement, and the through-thickness distributions of the mechanical displacements and electric potential. Hence, regarding these criteria, the various reported theories can be classified into uncoupled and coupled ones, depending on the presence or not of electric fundamental variables (i.e. depending on the consideration or not of the electric-charge equilibrium equation). Available models can also be classified into global or equivalent single-layer (ESL) models, and discrete-layer or layerwise ones, depending on the through-thickness variations of the mechanical and electric fields. Different models from previous classifications have been combined, leading in particular to uncoupled and coupled ESL theories. In either of the latter, the in-plane components of the electric field and/or displacement are often considered to be zero. The result is that the charge equilibrium equation is only approximated, leading to a small electromechanical coupling effect [5,7]. It was recently demonstrated, through an asymptotic analysis, that both assumptions cannot be considered simultaneously [8]. Theoretical and numerical comparisons for piezoelectric laminated plates have given favour to the assumption of zero in-plane electric displacement components [7]. However, a critical analysis of piezoelectric multilayer-beam solutions has shown that the latter assumption introduces a large error in estimating the electric field inside piezoelectric layers, whereas the assumption of zero in-plane electric-field components poses some difficulties in obtaining analytical solutions for the free-vibration analysis of smart beams [4]. For the purpose of verifying the accuracy of the results provided by the widely used approximate theories or computational numerical models, the need for exact, closed-form and other accurate analytical solutions has greatly increased during the last few years. In this regard, several three-dimensional analytical solutions have been proposed in the literature, for the free-vibration analysis of simply-supported piezoelectric laminated plates. Both coupled [9,10] and uncoupled [11,12] solutions have been proposed. Explicit representations of the fundamental variables have then been given, via their exponential expansions through the plate thickness [9,11]. However, using state space approaches [10,12], only implicit representations of the independent variables have been obtained. In contrast to the above three-dimensional solutions, analytical two-dimensional vibration solutions have been presented in the literature only as application examples of other theoretical [7,13,14] or numerical [15,16] analyses. Some of them [7,16] have retained only the transverse components of the electric field and/or displacement. Moreover, most of them [7,13,14,16] were based on mixed ESL theory for the mechanical behaviour, and a layerwise discretization for the electric potential. Hence, the third-order shear deformation theory (TSDT) was combined with a layer-

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575

wise linear or quadratic [13], linear [14], and linear uniform [16] electric-potential approximations in the piezoelectric layers of the laminated plate. However, in Ref. [7] a first-order shear deformation theory (FSDT) was associated with a layerwise quadratic electric potential in the piezoelectric laminae. Full linear layerwise (mechanical and electric) electromechanical and finite-element models have been proposed in Ref. [15], and applied to free-vibration analysis of simply-supported piezoelectric laminated plates. The above literature survey clearly shows that a two-dimensional analytical coupled full layerwise solution, that considers all components of the electric field and displacement, has not been proposed for free-vibration analysis of simply-supported piezoelectric adaptive sandwich plates. It is therefore the aim of the present work to fill this void. Layerwise FSDT kinematics, and quadratic non-uniform electric potential, are assumed in the proposed coupled electromechanical analytical solution. Mechanical displacement and electric potential Fourier-series amplitudes are retained as fundamental variables, so that an explicit representation of the piezoelectric effect is achieved, and the full electromechanical coupling is maintained. Mechanical and electric unknowns are reduced, by using the displacement and potential interface continuity conditions; the unknown electric constants are condensed, so that only nine mechanical ones remain in the reduced eigenvalue system to be solved in the vibration analyses. In the following sections, the exact solution is first presented. It includes the electromechanical basic equations, the two-dimensional mechanical and electric equations of motion for both short-circuited (SC) and open-circuited (OC) piezoelectric layer electrodes, and the resulting free-vibration problem to be solved. Its implementation is then verified through comparisons with available results from coupled and uncoupled three-dimensional elasticity solutions as well as layerwise and mixed ESL/layerwise finite-element solutions, as found in the literature.

2. Exact two-dimensional solution The piezoelectric adaptive sandwich plate to be studied is shown in Fig. 1. For simplicity, all layers are considered piezoelectric. They are assumed to be throughthickness polarized, but can have different thickness and material properties. Elastic

Fig. 1.

The piezoelectric adaptive sandwich plate: geometry and notations.

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layers can be cross-ply composites, and are obtained through vanishing of their homogenized piezoelectric constants. ESL electromechanical fields are assumed for elastic laminated layers. Layerwise FSDT and quadratic approximations are assumed for the mechanical and electric behaviours, respectively. The plate edges are grounded (potential-free) and mechanically unloaded (stress-free), while upper and lower plate surfaces are mechanically unloaded and their corresponding surface electrodes are either SC or OC. All layers are considered mechanically and electrically perfectly bonded. Hence, mechanical displacements and transverse stresses, and electric potential and transverse displacements, have to be continuous through the plate interfaces. However, since the model is based on a displacement-potential formulation, only the latter variables are made continuous. 2.1. Electromechanical basic equations A Mindlin-type mechanical displacement field inside the kth layer can be written as: uak(x,y,z,t) ⫽ u0ak(x,y,t) ⫹ (z⫺z¯ k)bak(x,y,t) uzk(x,y,z,t) ⫽ w(x,y,t)

(1)

where x, y, z are the global reference coordinates and z¯k ⫽ (zk ⫹ zk ⫹ 1) / 2 for k ⫽ 1,2,3 are the transverse coordinates of the layer mid-planes; uak , a ⫽ x,y and ukz are the kth layer in-plane and transverse displacements; uk0a and w are the corresponding mid-plane displacements; bka is the bending rotation in the a–z plane. Since the plate is assumed to be simply-supported, the above displacement components have to satisfy the following Navier-type edge boundary conditions: sxxk ⫽ 0, uyk ⫽ uzk ⫽ 0 at x ⫽ 0, Lx syyk ⫽ 0, uxk ⫽ uzk ⫽ 0 at y ⫽ 0, Ly

(2)

where skaa is the normal stress to the plate at the a-fixed edge with length La. Considering a harmonic displacement solution for the vibration problem, midplane displacements and rotations of Eq. (1), that satisfy the edge boundary conditions of Eq. (2), can be written in the following Fourier series form: (u0xk,bxk)(x,y,t) ⫽ (Uk,Bxk)cos px sin qy exp(iwt) (u0yk,byk)(x,y,t) ⫽ (Vk,Byk)sin px cos qy exp(iwt)

(3)

w(x,y,t) ⫽ Wsin px sin qy exp(iwt) where p ⫽ mxπ / Lx, q ⫽ myπ / Ly, i2 ⫽ ⫺1; ma is the mode number in the a-direction and w is the circular frequency (rad/s). From the interface continuity conditions, the core in-plane displacement and rotation amplitudes in (3), can be expressed in terms of those of the faces:

A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593

冉

577

1 1 1 1 h1 1 h3 3 U2 ⫽ (U3 ⫹ U1) ⫹ (h1Bx1⫺h3Bx3); Bx2 ⫽ (U3⫺U1)⫺ B ⫹ Bx 2 4 h2 2 h2 x h2

冉

1 1 1 1 h1 1 h3 3 V2 ⫽ (V3 ⫹ V1) ⫹ (h1By1⫺h3By3); By2 ⫽ (V3⫺V1)⫺ B ⫹ By 2 4 h2 2 h2 y h2

冊

冊 (4)

Hence, the unknown mechanical constants reduce to: U1, V1, B1x , B1y , W, U3, V3, B3x, B3y

(5)

The kth layer linear strains are obtained from the usual strain–displacement relations: 1 skij ⫽ (uki,j ⫹ ukj,i) 2

(6)

where “,” indicates spatial partial differentiation. Using (1), the strains can be decomposed into membrane (eab), bending (␬ab) and transverse-shear (saz) contributions: sabk(x,y,z,t) ⫽ eabk(x,y,t) ⫹ (z⫺z¯ k)␬abk(x,y,t) 2sazk(x,y,z,t) ⫽ w,a(x,y,t) ⫹ bak(x,y,t)

(7a)

with 1 1 ekab ⫽ (uk0a,b ⫹ uk0b,a), ␬klm ⫽ (bkl,m ⫹ bkm,l) 2 2 These strains are then written using the condensed (engineering) notations as: Srk(x,y,z,t) ⫽ erk(x,y,t) ⫹ (z⫺z¯k)crk(x,y,t), r ⫽ 1,2,6 Ssk(x,y,z,t) ⫽ gsk(x,y,t), s ⫽ 4,5

(7b)

The generalized (membrane, bending and transverse-shear) components can then be written in terms of the independent mechanical variables of (5) by substituting (3) together with (4) into the above relations. This is handled with the symbolic software Maple. Corresponding results are not given here for brevity. The electric potential inside the kth piezoelectric layer is considered through-thickness quadratic, and written in the form: fk(x,y,z,t) ⫽ fk0(x,y,t) ⫹ (z⫺z¯k)fk1(x,y,t) ⫹ (z⫺z¯k)2fk2(x,y,t)

(8)

where the functions f (j ⫽ 0,1,2) are chosen from the plate grounded (SC) edge conditions: k j

fk ⫽ 0 at x ⫽ 0, Lx and y ⫽ 0, Ly

(9)

An electric potential solution that satisfies edge electric boundary conditions can be obtained by choosing its jth in-plane function in the form: fkj(x,y,t) ⫽ ⌽kj sin px sin qy exp(iwt)

(10)

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where ⌽kj (j ⫽ 0,1,2) are the kth layer unknown electric potential amplitudes. Their number can be reduced, by using the continuity conditions at the sandwich plate interfaces:

冉冊 冉冊 冉冊 冉冊

h1 h2 h1 2 1 h2 2 2 ⌽2 ⫹ ⌽2 ⌽01 ⫽ ⌽02⫺ ⌽11⫺ ⌽12⫺ 2 2 2 2 h3 h2 h3 2 3 h2 2 2 ⌽0 ⫽ ⌽0 ⫹ ⌽13 ⫹ ⌽12⫺ ⌽2 ⫹ ⌽2 2 2 2 2 3

2

(11)

With these relations, the electric unknown constants reduce to: ⌽20, ⌽11, ⌽21, ⌽31, ⌽12, ⌽22, ⌽32

(12)

Eqs. (11) and (12) are valid for OC electric boundary conditions on the upper and lower plate surface electrodes, which are h 2

⫽ 0 at z ⫽ ± D1,3 z

(13)

In fact, these conditions cannot be enforced for a displacement-potential based formulation while taking into account continuity of the electric transverse displacement. However, this is not the case for the SC electric boundary conditions, since the electric potential is a fundamental variable of the present model. They can be written as: f1,3 ⫽ 0 at z ⫽ ±

h 2

(14)

Enforcing these conditions and those of the continuity of electric potential, we obtain ⌽10 ⫽

冉冊 冉冊

冉冊

h1 1 h1 2 1 2 h1 1 h3 3 h2 2 2 ⌽2, ⌽0 ⫽ ⌽1 ⫺ ⌽1⫺ ⌽2 ⌽⫺ 2 1 2 2 2 2

h3 h1 h3 h3 2 3 2 ⌽ ⫽ ⫺ ⌽31⫺ ⌽2, ⌽1 ⫽ ⫺ ⌽11⫺ ⌽31 2 2 h2 h2 3 0

(15)

These relations reduce further the unknown electric constants (12) for the SC case to only five parameters, which are: ⌽11, ⌽12, ⌽22, ⌽31, ⌽32

(16)

The kth layer electric field is obtained from the usual electric field-potential relation: Eki ⫽ ⫺fk,i

(17)

Taking into account (8), the previous expression can also be written as: Eak(x,y,z,t) ⫽ E0ak(x,y,t) ⫹ (z⫺z¯k)E1ak(x,y,t) ⫹ (z⫺z¯k)2E2ak(x,y,t) Ezk(x,y,z,t) ⫽ E0zk(x,y,t) ⫹ (z⫺z¯k)E1zk(x,y,t)

(18)

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where Ekja ⫽ ⫺fkj,a, Ek0z ⫽ ⫺fk1, Ek1z ⫽ ⫺2fk2 These generalized (zero, first and second order) electric fields can then be written in terms of the independent electric potential variables of Eqs. (12) or (16), depending on the electric boundary conditions considered on the upper and lower plate surface electrodes (i.e. OC or SC, respectively). This is handled using the symbolic software Maple, and corresponding details are omitted here for brevity. The linear strains (6) and electric field (7) are coupled through the converse and direct linear piezoelectric constitutive equations. They can be written down using the usual condensed (engineering) notations as: Tmk ⫽ CmnkSnk⫺elmkElk

(19)

Dik ⫽ einkSnk ⫹ 苸ilkElk

respectively, where Ckmn, ekin,苸kil (m,n ⫽ 1…6; i,l ⫽ 1,2,3 or x,y,z) are the elastic, piezoelectric and dielectric constants, respectively. Due to the zero normal-stress assumption of the FSDT, and decomposing the stresses and electric displacement into in-plane (p) and transverse (s) components, the above equations reduce respectively to:

冦冧 冤 (k)

T1

C11

T2

⫽ C12

∗

0

∗ 22

C

0

0

C66

C12

∗

0

T6

冥冦冧 冦 冧 (k)

∗

(k)

(k)

S1

e31

∗

S2

⫺ e32∗

S6

0

Ek3

or {Tp}(k) ⫽ [C∗p ](k){Sp}(k)⫺{ep∗}(k)Ek3

再冎 冋 T4

(k)

T5

⫽

C44 0

0

册再冎 冋 (k)

C55

S4

(k)

⫺

S5

0

册再冎

e24

e15 0

(20a)

(k)

E1

(k)

E2

or {Ts}(k) ⫽ [Cs](k){Ss}(k)⫺[es]T(k){Ekp} and

再冎 冋 D1

D2

(k)

⫽

0

册再冎 冋

e15

e24 0

(k)

S4 S5

(k)

⫹

(20b)

苸11 0

0

册再冎

苸22

(k)

E1

(k)

E2

or

with

{Dp}(k) ⫽ [es](k){Ss}(k) ⫹ [苸p](k){Ekp}

(21a)

k Dk3 ⫽ (e∗31Sk1 ⫹ e∗32Sk2) ⫹ 苸∗33Ek3orDk3 ⫽ 具ep∗典(k){Sp}(k) ⫹ 苸∗k 33 E3

(21b)

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A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593 k k k k ∗k k k k k ∗k k k k k C∗k ab ⫽ Cab⫺Ca3Cb3 / C33,e3a ⫽ e3a⫺e33Ca3 / C33,苸33 ⫽ 苸33⫺e33e33 / C33

These expressions suggest the following comments: 앫 In-plane stresses are coupled to the transverse electric field only (eq. (20a)), whereas the transverse-shear stresses are coupled to in-plane electric fields (eq. 20b). The latter vanish for uniform electric potential. Hence, in the absence of transverse shear strains, as often assumed for thin plates, transverse shear stresses vanish for uniform electric potential only. That is, even for thin plates, the latter stresses are still present if the in-plane electric fields are not zero (i.e. if the electric potential is not uniform). 앫 In-plane electric displacements are coupled to the transverse shear strains only (eq. (21a)). Hence, for thin plates, where the latter strains can be neglected, the former displacements remain non-zero due to in-plane electric field components. They vanish only if the electric potential is uniform. 앫 As discussed in Section 1, most available literature on either thin or thick piezoelectric adaptive plates neglect in-plane electric field and/or displacements. It means that only Eqs. (20a) and (21b) are retained as reduced converse and direct piezoelectric constitutive equations. This has important consequences. Neither the transverse shear stresses can be induced if the in-plane electric fields are neglected (eq. (20b)), nor can the transverse shear strains be measured if the in-plane electric displacements are not considered (eq. (21a)). The stress and electric displacement components (Eqs. (20) and (21)), satisfy the harmonic mechanical equation of motion and the electric charge equilibrium equation given, respectively, by: skij,j ⫽ ⫺rkw2uki, Dki,i ⫽ 0

(22)

where rk is the mass density of the kth layer material. The electric displacement conforms to the OC electric natural boundary conditions (eq. (13)), on the lower and upper plate surfaces, whereas the stresses are subject to the edge natural boundary conditions of (2) and the following additional ones at the plate surfaces: h s1,3 iz ⫽ 0 at z ⫽ ± . 2

(23)

2.2. Two-dimensional electromechanical equations of motion The three-dimensional piezoelectric problem defined by Eqs. (2), (6), (9), (13), (14), (17), (19), (22) and (23) can be written in variational form, using the virtualwork principle extended to piezoelectric media [5] for admissible virtual displacements and electric potentials, as:

冘冕 3

冕

[ (dsijksijk⫺dEikDik)d⍀k⫺w2 duikrkuikd⍀k] ⫽ 0

k⫽1

⍀

k

k

⍀

(24)

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581

The first integral term on the left-hand-side of the above equation represents the sandwich plate piezoelectric internal virtual work, and the second one is its inertial virtual work. Separating in-plane and transverse contributions in (24) leads to:

冘冕

冕

3

[ (dSrkTrk ⫹ dSskTsk)d⍀k⫺ (dEakDak ⫹ dE3kD3k)d⍀k

k⫽1

⍀

k

(25)

k

⫺w2

冕

⍀

(duakrkuak ⫹ duzkrkuzk)d⍀k] ⫽ 0

⍀k Substituting Eqs. (1), (7b) and (18), and integrating over the plate thickness, one obtains

冘冕

冕

3

¯ 0zk ⫹ dE1zkD ¯ 1zk [ (derkNrk ⫹ dcrkMrk ⫹ dgskQsk)dA⫺ (dE0zkD

k⫽1

A

A

冕

¯ jak)dA⫺w2 [(du0akI0ku0ak ⫹ dwI0kw) ⫹ (du0akI1kbak ⫹ dEjakD

(26)

A

⫹ db I u

k k k a 1 0a

) ⫹ dbakI2kbak]dA] ⫽ 0

with

冕

zk+1

(Nkr,Mkr) ⫽

[1,(z⫺z¯ k)]Tkrdz, Qks ⫽

zk

冕

zk+1

⫽

zk

¯ kja ⫽ [1,(z⫺z¯k)]Dk3dz, D

冕

zk+1

zk

冕

Tksdz, Ikj ⫽

冕

zk+1

¯ k0z,D ¯ k1z) (z⫺z¯ k)jrkdz(D

zk

zk+1

(z⫺z¯k)jDkadz, j ⫽ 0,1,2

zk

where A is the plate middle-surface area. It is worth noting that, stemming from the piezoelectric constitutive equations (Eqs. (20) and (21)), the above stress and electric displacement resultants include both mechanical and electric contributions. Also, it may be noted that the inertial virtual work contains all translation, rotary and their coupling inertial contributions. The virtual generalized strains, electric fields and displacements should be expressed in terms of the independent mechanical (5) and electric (Eqs. (12) or (16)) variables, depending on the electric boundary conditions on the plate upper and lower surface electrodes, then substituted into (26). Next, integration by parts is used to lower the orders of the independent variables. Then, terms multiplying each independent variable are grouped together and simplifications made, so that the following mechanical equations of motion are obtained:

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A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593

1 1 1 2 2 dU 1: (N11 ⫹ N12),x ⫹ (N61 ⫹ N62),y⫺ (M1,x ⫹ M6,y ⫺Q52) ⫽ w 2Jx1 2 2 h2 1 1 1 2 2 dV 1: (N61 ⫹ N62),x ⫹ (N21 ⫹ N22),y⫺ (M6,x ⫹ M2,y ⫺Q42) ⫽ w 2Jy1 2 2 h2

冉 冉

h1 dBx1: M1 1⫺ M1 2 2h2 h1 dBy1: M6 1⫺ M6 2 2h2

冊 冉 冊 冉

冊 冉 冊 冉

冊 冊

h1 h1 h1 2 2 ⫹ M6 1⫺ M6 2 ⫺ Q5 1⫺ Q5 2 ⫹ (N1,x ⫹ N6,y ) ⫽ w2⌫x1 2h 2h 4 2 2 ,x ,y h1 h1 h1 2 2 ⫹ M2 1⫺ M2 2 ⫺ Q4 1⫺ Q4 2 ⫹ (N6,x ⫹ N2,y ) ⫽ w2⌫y1 2h 2h 4 2 2 ,x ,y

dW: (Q51 ⫹ Q52 ⫹ Q53),x ⫹ (Q41 ⫹ Q42 ⫹ Q43),y ⫽ w 2Jz

(27)

1 1 1 2 2 dU 3: (N13 ⫹ N12),x ⫹ (N63 ⫹ N62),y ⫹ (M1,x ⫹ M6,y ⫺Q52) ⫽ w 2Jx3 2 2 h2 1 1 1 2 2 dV 3: (N63 ⫹ N62),x ⫹ (N23 ⫹ N22),y ⫹ (M6,x ⫹ M2,y ⫺Q42) ⫽ w 2Jy3 2 2 h2

冉 冉

h3 dBx3: M1 3⫺ M1 2 2h2 h3 dBy3: M6 3⫺ M6 2 2h2

冊 冉 冊 冉

冊 冉 冊 冉

冊 冊

,x

h3 h3 h3 2 2 ⫹ M6 3⫺ M6 2 ⫺ Q5 3⫺ Q5 2 ⫺ (N1,x ⫹ N6,y ) ⫽ w 2⌫x3 2h2 2h2 4 ,y

,x

h3 h3 h3 2 2 ⫹ M2 3⫺ M2 2 ⫺ Q4 3⫺ Q4 2 ⫺ (N6,x ⫹ N2,y ) ⫽ w 2⌫y3 2h2 2h2 4 ,y

where

冉 冉

2 )U 1 ⫹ I1 1 ⫹ Jx1 ⫽ (I01 ⫹ I11

冊

h1 2 1 h3 2 3 2 B ⫹ I13 U 3⫺ I13 Bx I 2 11 x 2

冊

h3 h1 2 1 2 2 Jx3 ⫽ (I03 ⫹ I33 )U 3 ⫹ I1 3⫺ I33 2 Bx3 ⫹ I13 U 1 ⫹ I13 Bx 2 2 Jz ⫽ (I01 ⫹ I02 ⫹ I03)W

冉 冉

⌫x1 ⫽ I1 1 ⫹

冊 冋 冉冊 册 冊 冋 冉冊 册

h1 2 1 h1 2 2 1 h1 2 3 h1h3 2 3 I B ⫹ I13U ⫺ I11 U ⫹ I2 1 ⫹ I B 2 2 11 x 2 4 13 x

h3 h3 2 2 3 h3 2 1 h1h3 2 1 ⌫x3 ⫽ I1 3⫺ I33 2 U 3 ⫹ I2 3 ⫹ I B⫺ I U⫺ I B 2 2 33 x 2 13 4 13 x with I211 ⫽

I02 I12 I2 2 2 I0 2 I2 2 2 I0 2 I1 2 I22 ⫹ ⫺ ⫹ ⫺ 2,I33 ⫽ ⫹ 2,I13 ⫽ 4 h2 (h2) 4 (h2) 4 h2 (h2)2

The J and ⌫ y-components have expressions of the same form as their x counterparts, but with V and By instead of U and Bx variables. For open-circuited upper and lower plate surface electrodes, the electrostatic equations for the electric variables of (12) are:

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583

ˆ x,x ⫹ D ˆ y,y ⫽ 0 d⌽02: D 1 1 1 ˜ x,x ˜ y,y ¯ 0z ⫹D ⫹D ⫽0 d⌽11: D 2 ˜ x,x ⫹ D ˜ y,y ⫹ D ¯ 0z d⌽12: D ⫽0 3 3 ˜ x,x ˜ y,y ⫹D ⫹ d⌽13: D 1 1 ˘ x,x ˘ y,y ⫹D ⫹ d⌽21: D ˘ x,x ⫹ D ˘ y,y ⫹ d⌽22: D

3 ¯ 0z D ⫽0 1 ¯ 1z 2D ⫽0 2 ¯ 1z 2D ⫽0

3 3 3 ˘ x,x ˘ y,y ¯ 1z d⌽23: D ⫹D ⫹ 2D ⫽0

(28)

where h 1 ˆa ⫽ D ˜ a ⫽ 2(D ¯ 10a ⫹ D ¯ 20a ⫹ D ¯ 30a, D ¯ 30a)⫺D ¯ 21a, ¯ ⫺D D 2 0a

冉冊

2 ˘ a ⫽ ⫺ h2 (D ¯ 10a ⫹ D ¯ 30a) ⫺ D ¯ 22a, D 2

˜ 1a ⫽ D

冉冊 冉冊

2 h1 1 ˘ 1a ⫽ h1 D ¯ 11a, D ¯ 10a⫺D ¯ 12a ¯ 0a⫺D D 2 2

2 h 3 ˘ 3a ⫽ h3 D ˜ 3a ⫽ ⫺ 3D ¯ 31a, D ¯ 30a⫺D ¯ 32a ¯ 0a⫺D D 2 2

However, for the short-circuited case, the electrostatic equations corresponding to the independent electric variables of (16) reduce to: h1 2 12 12 1 ˜ x,x ˜ y,y ¯ 0z ¯ )⫽0 ⫹D ⫹ (D ⫺ D d⌽11 : D h2 0z h3 2 32 32 3 ˜ x,x ˜ y,y ¯ 0z ¯ )⫽0 d⌽13 : D ⫹D ⫹ (D ⫺ D h2 0z 1 1 1 ˘ x,x ˘ y,y ¯ 1z ⫹D ⫹ 2D ⫽0 d⌽21 : D 2 2 2 ˘ x,x ˘ y,y ¯ 1z ⫹D ⫹ 2D ⫽0 d⌽22 : D 3 3 3 3 ˘ ˘ ¯ 1z ⫽ 0 d⌽2 : Dx,x ⫹ Dy,y ⫹ 2D

(29)

with h1 1 h1 2 ˜ 12 ¯2 ¯1 ¯ ¯ D a ⫽ ⫺ (D0a ⫹ D0a)⫺(D1a⫺ D1a) 2 h2 ˜ 32 D a ⫽

h3 3 h 2 ˘ ka ⫽ (hk)2D ¯ 20a)⫺(D ¯ 31a⫺ 3D ¯ ),D ¯ k0a⫺D ¯ k2a. ¯ ⫹D (D 2 0a h2 1a 2

Mechanical (27) and electric (Eqs. (28) or (29)) equations of motion are coupled through the generalized (or integral) piezoelectric constitutive equations. These are

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obtained by evaluating the stress and electric displacement resultants introduced in (26). Hence, from Eqs. (20a) and (20b) and the stress resultants, the generalized membrane/bending and transverse-shear converse piezoelectric constitutive equations are:

再 冎 冋 册再冎 冋 册再 冎 再 冎 冋 册再冎 冋 册再冎 N

(k)

M

Q4

A B

⫽

(k)

Q5

(k)

C44 0

0

(k)

⫺

c

B D

⫽ hk

e

(k)

F G

G H

g4

E0z

⫺

(k)

(30a)

E1z

0 Py

(k)

g5

C55

(k)

Ex

(k)

Px 0

(k)

(30b)

Ey

with 具N典(k) ⫽ 具N1 N2 N6典(k), 具M典(k) ⫽ 具M1 M2 M6典(k) 具e典(k) ⫽ 具e1 e2 e6典(k), 具c典(k) ⫽ 具c1 c2 c6典(k),具Ea典(k) ⫽ 具E0a E1a E2a典(k) and [A](k) ⫽ hk[Cp∗](k), [B](k) ⫽

(hk)2 ∗ (k) (hk)3 ∗ (k) [Cp ] , [D](k) ⫽ [C ] 2 12 p

{F}(k) ⫽ hk{e∗p }(k), {G}(k) ⫽

(hk)2 ∗ (k) (hk)3 ∗ (k) {ep } , {H}(k) ⫽ {e } 2 12 p

具Px典(k) ⫽ 具hk

(hk)2 (hk)3 k (hk)2 (hk)3 k 典e15, 具Py典(k) ⫽ 具hk 典e 2 12 2 12 24

Similarly, substituting Eqs. (21a) and (21b) into the expressions for the electric displacement resultants, the generalized transverse and in-plane direct piezoelectric constitutive equations are, respectively:

再 冎 冋 册再冎 冋 册再 冎 再 冎 冋 册再冎 冋 册再冎 ¯ 0z D ¯ 1z D

(k)

¯ jx D ¯ jy D

(k)

FT GT

⫽

⫽

GT HT

(k)

e

(k)

c

(hk)j+1 0 e15 j ⫹ 1 e24 0

(k)

R00 R01

⫹

g4

(k)

R01 R11

(k)

g5

⫺

Rx 0

0

E0z

(k)

(31a)

E1z

(k)

Ry

Ex

Ey

(k)

(31b)

with k R00k ⫽ hk苸∗k 33 ,R01 ⫽

具Ra典(k) ⫽

冓

(hk)2 ∗k (hk)3 ∗k 苸33 ,R11k ⫽ 苸 , 2 12 33

冔

(hk)j+1 (hk)j+2 (hk)j+3 k 苸 j ⫹ 1 j ⫹ 2 j ⫹ 3 aa

Once generalized strains and electric fields are expressed (symbolically using Maple software) in terms of the independent mechanical (5) and electric (Eqs. (12) or (16)) variables, the above stress and electric displacement resultants can be written (also using the symbolic software Maple) in terms of the same variables.

A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593

585

2.3. Free-vibration problem Previously obtained stress and electric displacement resultants are written in terms of the independent mechanical and electric variables, then substituted into the mechanical (27) or electric (Eqs. (28) or (29)) equations of motion, depending on the electric boundary conditions considered for the upper and lower plate surface electrodes. Separating the mechanical and electric unknown constants, the resulting eigenvalue system to be solved for the free-vibration problem can be written as:

冉冋

KUU KU⌽ K⌽U K⌽⌽

册 冋 ⫺w2

册冊再 冎 再 冎

MUU 0

U

0

⌽

0

⫽

0

0

(32)

where [K], [M] are the stiffness and mass matrices, and {U}, {⌽} are the mechanical-displacement and electric-potential unknown vectors, respectively. The above system can be condensed so that the following reduced standard eigenvalue problem is solved to obtain the piezoelectric adaptive sandwich plate modal characteristics: ([K]⫺w2[M]){U} ⫽ {0}

(33)

with [K] ⫽ [KUU]⫺[KU⌽][K⌽⌽]⫺1[K⌽U], [M] ⫽ [MUU] The electric unknown constants are then computed a posteriori via the following relation: {⌽} ⫽ ⫺[K⌽⌽]⫺1[K⌽U]{U}

(34)

Once the mechanical and electric unknown constants are obtained, mechanical displacements, strains, in-plane stresses, electric potential and in-plane displacements can be calculated a posteriori. However, transverse stress and electric displacement components are better calculated from the three-dimensional divergence (equilibrium) relations.

3. Numerical validation and analysis The above-mentioned exact two-dimensional (2D) solution has been implemented in the Matlab software environment. To check its validity, it is now assessed through comparisons with: (i) (ii) (iii) (iv)

3D coupled [9] and uncoupled [11] Navier-type solutions 3D uncoupled state space method (SSM) [12] 3D coupled full layerwise finite element method (FEM) [15] 2D uncoupled mixed higher-order shear deformation theory (HSDT)ESL/layerwise FEM [16].

The properties of all materials used in this analysis are summarized in the Appen-

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dix A. Those of elastic laminated faces or core are homogenized for the corresponding lamination scheme. The procedure assumes a unique electric potential distribution when applied to conductive layers. That is, for simplicity, an ESL model is assumed in the latter case. 3.1. Comparison with 3D uncoupled Navier-type solution An adaptive rectangular simply-supported plate of dimensions Lx ⫽ 冑2L, Ly ⫽ L, with L ⫽ 30 cm and h ⫽ 0.404 cm, is considered. It is composed of a 10-layer cross-ply symmetric [0/90/0/90/0]s laminated core sandwiched between two 0.002 cm thick PZT-G1195 actuator and sensor layers. Each core lamina is made of a T300/976 Graphite-Epoxy (Gr/Ep), and is 0.04 cm thick. The plate circular frequencies wmxmy, mx,my ⫽ 1,2… are normalized using the following formula [11]: ⍀mxmy ⫽

wmxmy π ( )2 Lx

(35)

冪

D11 r(2)h2

where D11 is the flexural rigidity of the homogenized laminated core. The first six normalized circular frequencies are calculated by the present 2D solution and compared, in Table 1, to a classical laminated plate theory (CLPT) and a 3D uncoupled theory of Batra and Liang [11]. Table 1 also shows corresponding frequencies for a 10-times thicker piezoelectric laminated plate (h ⫽ 4.04 cm). Notice that the above normalization (35) is ambiguous for the 3D reference solution [11]. Besides, the CLPT solution is, in fact, not valid for the thick case. It is considered here for comparison with Ref. [11] only. From Table 1, it is clear that the present solution is satisfactory for the thin-plate (L / h ⫽ 74.3) case only. This can be due to the data uncertainty regarding some material properties and the normalization procedure (35). In fact, all material dielecTable 1 First six normalized circular frequencies of a simply-supported rectangular 12-layers hybrid plate L/h

Solution method

74.3

3D uncoupled [11] 2D, CLPT [11] Errora (%) 2D, present Errora (%) 3D uncoupled [11] 2D, CLPT [11] Errora (%) 2D, present Errora (%)

7.43

a

⍀11 1.944 2.002 3.0 2.035 4.7 1.664 1.975 18.7 1.822 9.5

⍀21 4.456 4.585 2.9 4.609 3.4 3.276 4.570 39.5 3.641 11.1

100(2D–3D uncoupled [11])/3D uncoupled [11].

⍀12 6.152 6.408 4.2 6.463 5.1 4.021 6.284 56.3 4.730 17.6

⍀22 7.736 8.007 3.5 8.111 4.8 4.992 7.900 58.3 5.800 16.2

⍀31 9.128 9.454 3.6 9.408 3.1 5.443 9.444 73.5 6.194 13.8

⍀32 11.528 11.938 3.6 12.016 4.2 6.657 11.858 78.1 7.730 16.1

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587

tric properties were not given in Ref. [11], and the C66 constant of the PZT-G1195 was incorrect. The former has been given in full and the latter has been corrected from Refs. [14,15]. Also, the mass density given in Ref. [14] was different from that used here [11] (see Appendix A). It is worth mentioning that in Ref. [11], the piezoelectric layers were modelled as membranes, i.e. with neither bending nor transverse shear effects. Moreover, the charge equations were approximated and the electric potential was considered linear. Hence, the electromechanical coupling was only partial. In fact, the reference solution [11] is not exact as claimed but approximate. This explains the relatively high discrepancies between the present exact sandwich solution and the 3D reference one [11] for the thick (L / h ⫽ 7.43) plate. The high errors of the CLPT solution are meaningless, since it is not valid here. 3.2. Comparison with 3D uncoupled state space solution Four elastic and hybrid plate configurations are now considered [12]: (a) an 8layer Gr/Ep cross-ply laminate [90/0/90/0/0/90/0/90]; (b) a PZT-5A layer at the top surface of an 8-layer Gr/Ep cross-ply laminate [90/0/90/0/0/90/0/90/p]; (c) a PZT5A layer sandwiched between two 4-layer Gr/Ep cross-ply laminated faces [90/0/90/0/p/0/90/0/90]; (d) two PZT-5A layers at the top and bottom surfaces of a Gr/Ep symmetric cross-ply substrate [p/90/0/90/0/0/90/0/90/p]. The plates are square of side length L ⫽ 1 [1], and have variable side-to-thickness ratio (L / h). All the layers have the same thickness, and their other material properties may be seen in Ref. [12] and are reproduced in the Appendix A. The fundamental frequencies (mx ⫽ my ⫽ 1) of the four configurations are calculated from the present 2D exact solution, and compared with those yielded by the 3D state space uncoupled solution proposed by Xu, Noor and Tang [12]. All results are summarized in Table 2 for L / h ⫽ 100, 10 and 5. Table 2 shows that the present solution is satisfactory up to the layerwise FSDT validity limit, i.e. L / h ⫽ 10. For lower thickness ratios, the results indicate that it is better to consider higher-order theories to get more accuracy. It is worth noticing Table 2 Fundamental circular frequencies (rad/s) of four configurations of simply-supported square hybrid plates L/h

10

5

Configuration 3D 2D Errora uncoupled present (%) [12]

3D 2D Errora uncoupled present (%) [12]

3D 2D Errora uncoupled present (%) [12]

(a) (b) (c) (d)

2939.2 2554.7 2547.5 2357.7

4576.8 3964.4 4049.1 3648.0

a

100

333.02 290.38 285.26 268.86 100(2D–3D)/3D.

340.86 300.64 285.16 283.93

2.4 3.6 ⫺0.04 5.6

3098.3 2703.7 2643.0 2516.7

5.4 5.8 3.7 6.7

5055.4 4337.0 4441.2 3953.2

10.5 9.4 9.7 8.4

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A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593

that configuration (c) is not practical for thickness-polarized piezoelectric materials. In fact, in this case, the induced bending moments are the lowest compared to configurations (b) and (d). Hence, the influence of the piezoelectric effect is meaningful only for the latter two configurations. This can explain the higher relative difference between both solutions, compared to configurations (a) and (c), and also the fact that both solutions are closer to each other for the latter two configurations. The present solution considers full electromechanical coupling and exact charge-equation representation with quadratic non-uniform electric potential. However, the reference solution [12] considered only linear uniform electric potential, and did not satisfy the electrostatic equation. Hence, the electromechanical coupling was only partial, as discussed earlier in the literature survey. 3.3. Comparison with full layerwise and mixed HSDT-ESL/layerwise FEM solutions Attention is now directed to the free vibration of a simply-supported five-ply square hybrid laminated plate. The latter consists of two PZT-4 layers surface-bonded to a three-ply Gr/Ep symmetric laminate [0/90/0]. Material properties may be seen in Ref. [15] and are also given in the Appendix. For comparison purposes, all materials have identical mass density r ⫽ 1 kg / m3. Two side-to-thickness ratios (L / h ⫽ 4, 50) and two electric boundary conditions (SC or OC) are considered. Each piezoelectric layer is 0.1 h thick, and all Gr/Ep layers have the same thickness. Fundamental (mx ⫽ my ⫽ 1) frequency parameters, l ⫽ wL2r1 / 2 / (2πh), [103 Hz (kg/m)1/2] are calculated (for both thickness ratios and both electric boundary conditions) with the present coupled 2D exact sandwich solution, and compared in Table 3 with the results obtained from: (i) a 3D exact coupled Navier-type solution [9]; (ii) a coupled linear full layerwise FEM solution with variable or constant transverse deflection [15]; (iii) uncoupled mixed HSDT-ESL/layerwise FEM solutions using nine-node quadrilateral finite elements with 11, 9 or 5 degrees of freedom per node [16], referred to as Q9-HSDT 11p, Q9-HSDT 9p, Q9-FSDT 5p, respectively. With regard to (iii), the first element has cubic in-plane and quadratic transverse displacements, whereas the second has cubic in-plane and constant transverse displacements. The three elements assume linear electric potential in the piezoelectric layers. It is clear from Table 3 that the present solution agrees very well with the 3D layerwise exact one [9], in particular for the thin-plate case (L / h ⫽ 50). In contrast with finite element results [15,16], both 3D and 2D exact solutions indicate that, for the thin plate, differences between SC and OC frequencies tend to vanish. Also, in contrast with the full layerwise FEM results [15], mixed HSDT/layerwise results [16] agree better with analytical solutions for the OC electric boundary conditions. Higher discrepancies appear for the SC condition, in particular with the Q9-FSDT 5p element for the thin-plate case, for which an error of 16.12% is observed. This may be due to the zero in-plane electric field and displacement assumptions inherent in Ref. [16], which are not imposed in [9,15] and herein. Higher errors for the thinplate case can also be due to possible shear locking of the FSDT element. Interest-

3D exact [9] FEM layerwise-w variable [15] FEM layerwise-w constant [15] Q9-HSDT 11p Q9-HSDT 9p Q9-FSDT 5p 2D exact (present)

Analysis method

Reference 6.41 5.38

1.85 ⫺0.32 ⫺0.24 0.05

259.173

250.497 245.161 245.349 246.068

142.817 144.531 147.489 148.283

151.222

145.377 151.964

% Difference L /h ⫽ 4

245.942 261.703

L / h ⫽ 50

OC

236.833 230.461 229.878 206.304 246.067

⫺1.76 ⫺0.58 1.45 1.97

245.941 239.628

L / h ⫽ 50

4.02

Reference 4.53

% Difference

139.683 138.513 142.068 148.246

145.323

⫺3.70 ⫺6.29 ⫺6.53 ⫺16.12 0.05

145.339 146.269

L/h ⫽ 4

SC

Reference ⫺2.57

% Difference

Table 3 Fundamental frequency parameters, l ⫽ wL2r1 / 2 / (2πh) [103 Hz (kg/m)1/2], of a square hybrid sandwich plate

⫺3.89 ⫺4.70 ⫺2.25 2.00

⫺0.01

Reference 0.64

% Difference

A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593 589

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A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593

ingly, the constant-deflection assumption in the full layerwise results [15] seems to give more satisfactory results than the variable-deflection one. This is also true for the Q9-HSDT 9p and Q9-FSDT 5p elements with respect to the OC condition. The latter element is also curiously more accurate than the former for the same electric condition, but for the thin plate only. To confirm the closeness of the SC and OC analytical natural frequencies, and to check the accuracy of the present sandwich solution for higher modes, the results of Table 3 are extended to the first five in-plane modes. For these, the computed results are compared in Table 4 with those of Ref. [16], and those given by a coupled layerwise 3D mixed SSM theory developed by the authors. The SSM solution generalizes the solution for piezoelectric laminated plates presented in Ref. [21] for single-layer piezoelectric plates. Table 4 confirms that the difference between OC and SC frequencies is much higher in FEM results than in the exact solutions. This Table 4 First five frequency parameters, lmxmy ⫽ wmxmyL2r1 / 2 / (2ph) [103 Hz (kg/m)1/2], of a simply-supported square hybrid sandwich plate L/h

Electric BC Solution method

50

SC

OC

4

SC

OC

a

3D coupled SSM 2D, present Errora (%) 2D Q-9HSDT 9p [16] Errora (%) 2D Q-9HSDT 11p [16] Errora (%) 3D coupled SSM 2D, present Errora (%) 2D Q-9HSDT 9p [16] Errora (%) 2D Q-9HSDT 11p [16] Errora (%) 3D coupled SSM 2D, present Errora (%) 2D Q-9HSDT 9p [16] Errora (%) 2D Q-9HSDT 11p [16] Errora (%) 3D coupled SSM 2D, present Errora (%) 2D Q-9HSDT 9p [16] Errora (%) 2D Q-9HSDT 11p [16] Errora (%)

l11

l12

l21

245.936 559.402 691.727 246.067 559.615 693.601 0.05 0.04 0.27 229.878 519.231 660.945 ⫺6.53 ⫺7.18 ⫺4.45 230.461 520.384 662.915 ⫺6.29 ⫺6.97 ⫺4.16 245.937 559.406 691.731 246.068 559.621 693.606 0.05 0.04 0.27 245.161 558.784 691.732 ⫺0.32 ⫺0.11 0 250.497 583.185 695.697 1.85 4.25 0.57 145.338 258.266 268.002 148.246 261.212 275.112 2.0 1.14 2.65 138.513 205.859 205.887 ⫺4.70 -20.29 -23.18 139.683 183.212 198.998 ⫺3.89 ⴚ29.06 ⴚ25.75 145.376 258.384 268.109 148.283 261.335 275.205 2.0 1.14 2.65 144.531 205.859 205.887 ⫺0.58 ⴚ20.33 ⴚ23.21 142.817 183.491 199.299 ⫺1.76 ⴚ28.99 ⴚ25.66

100(2D–3D coupled SSM)/3D coupled SSM.

l22 965.179 967.141 0.20 905.298 ⫺6.20 908.459 ⫺5.88 965.191 967.481 0.24 959.648 ⫺0.57 980.361 1.57 347.859 353.957 1.75 260.607 -25.08 263.631 ⴚ24.21 348.058 354.151 1.75 268.155 ⴚ22.96 267.975 ⴚ23.01

l13 1090.983 1091.458 0.04 1019.758 ⫺6.53 1022.091 ⫺6.32 1091.003 1091.481 0.04 1092.775 0.16 1145.406 4.99 387.554 391.924 1.13 264.698 -31.70 266.305 ⴚ31.29 387.796 392.191 1.13 270.956 ⴚ30.13 270.125 ⴚ30.34

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was also observed by Saravanos et al. [15], without giving any reason for this difference. Here, the authors suggest that investigation should focus on the effects of the discretization aspect, inherent in the FEM, and of the zero normal-stress edge conditions, which are explicitly enforced in the exact solutions but ignored in the displacement-based FEM. Table 4 also shows that, for the thin-plate case, the results of the present formulation outperform those given by the mixed HSDTESL/layerwise FEM. Curiously, for the OC thin plate, the constant-deflection-based FEM results are better than those of the variable-deflection-based ones. For the thickplate case, the present approach provides all the computed frequencies within a maximum error of 2.65%, whereas the mixed FEM solutions [16] provide accurate results only for the first frequency.

4. Conclusion An original sandwich formulation for the free-vibration analysis of simply-supported piezoelectric adaptive laminated plates has been presented. It can be considered as an exact two-dimensional solution in the sense that no assumptions are made other than the through-thickness layerwise first-order shear deformation kinematics and quadratic electric potential. In particular, all components of the electric field and displacement field were retained. The electrostatic-charge equation is exactly satisfied, and the induced potential as well as the electromechanical coupling are properly represented. Two-dimensional mechanical and electric equations of motion, and generalized piezoelectric constitutive equations for the induced stress and electric displacement resultants, were also derived for the first time. The proposed solution has been numerically validated through comparisons with available uncoupled and coupled 3D Navier-type and state space solutions, as well as 3D coupled layerwise and 2D uncoupled mixed HSDT-ESL/layerwise finite element solutions. The results of the present solution were the closest to the 3D coupled exact ones. Hence, this approach is considered reliable as a basis for developing other numerical and approximate two-dimensional formulations. Currently, the approach is being extended to in-plane polarized piezoelectric materials, so that their transverse shear response (known as the shear actuation mechanism [17-20]) can be exploited.

Appendix A. Material properties T300/976 graphite-epoxy [11] E11 ⫽ 150 GPa, E22 ⫽ E33 ⫽ 9 GPa, n12 ⫽ n23 ⫽ n13 ⫽ 0.3, G12 ⫽ G13 ⫽ 7.1 GPa, G23 ⫽ 2.5 GPa, r ⫽ 1600 kg m⫺3 苸11 /苸0 ⫽ 3.5, 苸22 /苸0 ⫽ 苸33 /苸0 ⫽ 3.0, 苸0 ⫽ 8.85 × 10⫺12 F / m [15].

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A. Benjeddou et al. / Thin-Walled Structures 40 (2002) 573–593

Graphite-epoxy [1,12,14] E11 ⫽ 181GPa, E22 ⫽ E33 ⫽ 10.3GPa, n12 ⫽ n13 ⫽ 0.28, n23 ⫽ 0.33, G12 ⫽ G13 ⫽ 7.17 GPa, G23 ⫽ 2.87 GPa, r ⫽ 1580 kg m⫺3 苸11 ⫽ 苸22 ⫽ 苸33 ⫽ 1.53 × 10⫺8 F / m Graphite-epoxy [3,15] E11 ⫽ 132.38 GPa, E22 ⫽ E33 ⫽ 10.76 GPa, n12 ⫽ n13 ⫽ 0.24, n23 ⫽ 0.49, G12 ⫽ G13 ⫽ 5.65 GPa, G23 ⫽ 3.61 GPa, r ⫽ 1578 kg m⫺3 苸11 /苸0 ⫽ 3.5, 苸22 /苸0 ⫽ 苸33 /苸0 ⫽ 3.0, 苸0 ⫽ 8.85 × 10⫺12 F / m. PZT-G1195 [11,13,14]

[C] ⫽

冤

148 76.2 74.2 0

0

0

76.2 148 74.2 0

0

0

74.2 74.2 131 0

0

0 0

0

0

0

25.4 0

0

0

0

0

25.4 0

0

0

0

0

0

冤

460 0

[苸] ⫽ 0 0

0

460 0 0

235

冥

冥 冤 冥

35.9

GPa,[e]T ⫽

0

0

⫺2.1

0

0

⫺2.1

0

0

9.5

0

9.2 0

9.2 0

0

0

0

0

Cm⫺2

苸0, 苸0 ⫽ 8.85 × 10⫺12 C / N m2,

r ⫽ 7500, [11], 7600 [14] kg m⫺3. PZT-5A [1,12] E11 ⫽ E22 ⫽ 61.0 GPa, E33 ⫽ 53.2 GPa, n12 ⫽ 0.35, n13 ⫽ n23 ⫽ 0.38, G12 ⫽ 22.6 GPa, G13 ⫽ G23 ⫽ 21.1 GPa, r ⫽ 7750 kg m⫺3 e31 ⫽ e32 ⫽ 7.209 C / m2, e33 ⫽ 15.118 C / m2, e24 ⫽ e15 ⫽ 12.322 C / m2, 苸11 ⫽ 苸22 ⫽ 1.53 × 10⫺8 F / m, 苸33 ⫽ 1.5 × 10⫺8 F / m. PZT-4 [3,4,15] E11 ⫽ E22 ⫽ 81.3 GPa, E33 ⫽ 64.5 GPa, n12 ⫽ 0.33, n13 ⫽ n23 ⫽ 0.43, G12 ⫽ 30.6 GPa, G13 ⫽ G23 ⫽ 25.6 GPa, r ⫽ 7600 kg m⫺3 e31 ⫽ e32 ⫽ ⫺5.2 C / m2, e33 ⫽ 15.08 C / m2, e24 ⫽ e15 ⫽ 12.72 C / m2, 苸11 / 苸0 ⫽ 苸22 / 苸0 ⫽ 1475, 苸33 / 苸0 ⫽ 1300, 苸0 ⫽ 8.85 × 10⫺12 F / m.

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