An Innovative Variational Method and Numerical Modeling of Multi-layered Piezoelectric Laminated Plates

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An Innovative Variational Method and Numerical Modeling of

Multi-layered Piezoelectric Laminated Plates

Li-Jeng Hunag*

Department of Civil Engineering

National Kaohsiung University of Applied Sciences *Associate Professor

Abstract

A state space approach based on Hellinger- Reissner variational principle for the analysis of simply supported multi-layered elastic and piezoelectric plates subjected to sinusoidal mechanical or electrical loadings is depicted. The approach is based on the state space formalism for the generalized plane deformation problem of piezoelectricity. Appropriate choosing the primary state variables, the system equations of boundary-value problem can be arranged into an initial-value problem with standard form of state equations and output equations in the linear control theory. The exact solutions can be obtained for various surface conditions via transfer matrix of the state equations. A [0/90/0] Gr/Ep composite and a [PZT-5/0/90/0/PZT-5] piezoelectric laminate will be taken as the examples for study the effect of span-to-thickness ratio and piezoelectric conditions on the piezoelastic behaviors in detail. Results show that this approach using only first approximation gives exact solutions and the piezoelastic analysis based on thin plate theory cannot be applied for those elastic and piezoelectric laminates with low span-to-thickness ratio.

Keywords : Piezoelectric plates, sinusoidal loading, state-space formalism, numerical modelling

1. Introduction

Piezoelectric materials have been widely applied to various technologies in aerospace and mechanical engineering, such as sonars, frequency control, acoustic interferometry, deformation and motions detecting, etc. (Tiersten, 1969; Mason, 1981). Intelligent or smart structures can be obtained by measuring strains and deformations by bonding or embedding a piezoelectric element in a structure and them give localized strain through which the deformation of structure can be controlled. However, effective sensing and control of a smart structure can be achieved only based on thorough and correct modeling the coupled piezoelastic behavior of the structures of piezoelectric materials.

Recently, developments and studies of mechanical behavior and technical applications of piezoelectric ceramics and polymers have been conducted by many researchers. The interest is placed on the interactions between mechanical fields and electric fields in various piezoelectric structures. Structural problem of piezoelectric plates arises in modeling flat piezoelectric sensors and actuators. Most analyses are based on the Kirchhoff hypothesis of deformation and cannot be applied for those structures with low span-to-thickness ratio. For thick piezoelectric plates an analysis based on full three-dimensional piezoelectricity is required. The analyses using potential functions, as conducted in Lehkniskii (1968, 1981), lead to systems of coupled higher-order

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differential equations and are complicated for solution (Bisegna and Maceri, 1996). Although there exist another approaches for three-dimensional analysis (Lee and Jiang, 1996) based on state space approach, the results and the calculations are not concise for practical application.

In this paper we employ a simple yet rigorous approach to study the simply supported multi-layered elastic and piezoelectric laminate subjected to sinusoidal mechanical or electrical loadings. The work is an extension and application of earlier study on the stress analysis and free-edge effects using state space approach (Wang et al., 2000; Tarn and Wang, 2001) and the bending of single-layer piezoelectric laminae (Huang and Tarn, 2001). Appropriate choosing the primary state variables, the system equations of boundary-value problem can be arranged into an initial-value problem with standard form of state equations and output equations in the linear control theory. The exact solutions can be obtained for various surface conditions via transfer matrix of the state equations, which is an 8×8 matrix for piezoelectric laminae and 6×6 for the special

case without piezoelectric effects. Thus, the present formulation can recover as a special case pure elasticity such that both the elasticity and piezoelasticity can be studied in the same way and it is possible to conduct a direct comparison between the two theories in detail, thereby depicting intricate electro-mechanic coupling effects. A [0/90/0] Graphic/Epoxy composite is employed to check the validation of the approach and the results are compared with those elasticity solutions obtained by Pagano.

A [PZT-5/0/90/0/PZT-5] multi-layered piezoelectric laminate will be taken as the example for study the effect of span-to-thickness ratio and piezoelectric conditions on the piezoelastic behaviors of the piezoelectric laminate in detail.

2. Basic Equations of The Linear Theory of Piezoelectricity

The Hellinger-Reissner variational principle states that (Washizu 1975):

( )

(

)

( )

1 2 V σij ijε B σIJ dV S_TT u dSi i S_u u u T dSi− i i ∏ ∫∫∫= _ − _ −∫∫ −∫∫   (1)

where V,S_T,S_udenotes the volume, traction boundary and displacement boundary, respectively, of the domain,

σ ε

_ij

,

_ijare stress and strain components, T_i, are traction and displacement vector u_i

with T ,_i u_iare prescribed traction vector and displacement on S_T,S_u, respectively. It should be noticed that in this variational equation both stress field and displacement are independent.

Consider a square piezoelectric plate with span a and thickness h and is simply supported at the edge x₁ =0 and x₁ = a(Fig. 1). The piezoelectric material belongs to monoclinic system of

class 2 with the

x

₃ axis being the polarization direction. The constitutive equations (stress-strain-electrical field relationship) of the piezoelectric materials for the kth layer can be expressed as , T k i k Ck k e k

σ

=

ε

−

φ

(2a) ,i k k k k k D =e ε +κ φ (2b)

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where

{ }

σ Tk ={σ11,σ22,σ33,σ23,σ13,σ11}k is the stress tensor,

{ }

ε kT ={ε11, ε22,ε33,2ε23,2ε13,2ε11}k

is the strain tensor, φ_k is the electric potential, {D}Tk ={D1,D2,D3}k is the electric displacement

vector, and the elastic matrix, piezoelectric matrix and dielectric permittivity matrix is defined, respectively, as 11 12 13 16 12 22 23 26 13 23 33 36 44 45 45 55 16 26 36 66 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 k k c c c c c c c c c c c c C c c c c c c c c         =             , 31 32 33 14 24 15 25 36 0 0 0 0 0 0 0 0 0 0 T k k e e e e e e e e e         =             , 11 12 12 22 33 0 0 0 0 k k k k k k k κ     =      

in which cij are the 13 elastic constants measured at a constant electric field, kij are the permittivity constants measured at constant stress, e_ij the piezoelectric constants. The constitutive equations of a monoclinic anisotropic elastic material, having one plane of symmetry parallel to the

3

x plane, are a special case of (1) with e_ij =0. The comma denotes partial differentiation with respect to the suffix variables. The subscript k denotes the k-th layer, k runs from 1 to n.

The admissible displacement and stress field that make the functional in Eq.(1) stationary gives the following relationships: The strain-displacement relations are

, ,

1

2( )

ij ui j uj i

ε = + (3)

The equations of equilibrium in the absence of body force are

, 0

ij j

σ = (4)

The equations of electrostatics without free charge are

, 0

i i

D = (5)

Furthermore, the continuity conditions on the interfaces x3=zk,k=1,2,...,n−1 require

1 2 3 13 23 33 1 2 3 13 23 33 1

[ , , , ,u u u φ σ σ σ, , ]_k =[ , , , ,u u u φ σ σ σ, , ]_k₊ (6) The simply supported edge conditions u₃( , )0 x₃ =u a x₈( , )₃ = at 0 x₁=0 and x₁=a are

simulated by 0 0

,

)

(

,

)

(

₃ ₈ ₃ 3

x

=

u

a

x

=

u

, (7a) 11(0, )x3 11( , )a x3 0 σ =σ = . (7b)

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However, there exist various prescribed surface conditions on the upper and lower surfaces due to different mechanic and electric loading. First consider the case of mechanical loading at the top surface,

[σ ,σ ,σ ]_{xz yz zz} _x _h [0,0,q(x )],[σ ,σ ,σ ]₁ _{xz yz zz} _x ₀ [0,0,0]. 3

3= = = = (8)

with either both the top and bottom surface insulated, i.e., [ ₃] [ ₃] ₀ 0, 3 3 D _{x h =}₌ _{D x} ₌ = or grounded, i.e., [ ] [ ] ₀ 0, 3 3 x x h

φ ₌ = φ ₌ = or the combination such as [ ] [ ₃] ₀ 0.

3

3 D x

x h

φ ₌ = ₌ = In the case of

electrical loadings, such as a differential electric voltage is applied to top and bottom surface, then we can set ] [ , ₃] [ ( ),0],[ ,₁ ₃] ₀ [0,0 3 3 D _{x h} x _{D x} φ ₌ = ϕ φ ₌ = (9)

while both the top and bottom surfaces are stress free, i.e., [ , , ] [0, 0, 0], 3 xz yz _{zz x h} σ σ σ ₌ = [ , , ] ₀ [0,0,0] 3 xz yz _{zz x} σ σ σ ₌ =

3. State Space Formulation

In order to cast the equations of system into those in the state space formalism, the stress, strain and electric field variables can be separated into in-plane and transverse components, and the equations (1) to (5) can be decomposed into

1 2 3,3 1 ,3T P C L u C uP e σ = + + φ , (10) 33 2 33 3, 3 33 ,3 T P C L u c u e σ = + + φ , (11) 3( 3 ,3) 2T S C Lu u e L σ = + + φ, (12) 2( 3 ,3) P D =e Lu +u −κ Lφ, (13) 3 1 P 33 3,3 33 ,3 D =eL u e u+ −κ φ , (14) in which{σ_P}T ={σ σ σ₁₁, ₂₂, ₁₂}_k, { }σ_S T ={σ σ₃₁, ₂₃}_k,{D_P}T ={ ,D₁ D₂} {_k , { }u T ={ , }u u₁ ₂ _kand

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11 12 16 13 55 45 1 12 22 26 2 23 3 45 44 16 26 66 36 , , , k k k c c c c c c C c c c C c C c c c c c c           =_ _ =_ _ _{= } _          

[

]

15 14 11 12 1 31 32 36 2 25 24 12 22 , , k k k e e k k e e e e e e e κ k k     = =_ _ =_ _     1 1 2 2 2 1 0 0 , P L L ∂   ∂     =_ ∂ _ _{=  }_∂   ∂ ∂   

Choose{x}T ₌{u₁,u₂,u₃,_φ,_σ₁₃,_σ₂₃,_σ₃₃,D₃}_{as the state vector, and}_{ _} _{ _, _, _, _, _}

2 1 12 22 11 D D y T ₌ _σ _σ _σ _as

the output vector, the equilibrium equations, equation of electrostatics, etc. can be expressed as the state equations and output equations for the k-th layer, respectively,

3 { }_k _k{ }_k d x A x dx = (15) { }y _k =C x_k{ }_k (16)

in which the state matrix and output matrix is defined as

1 1 3 2 3 1 2 33 33 1 1 2 33 33 33 33 1 2 2 1 1 2 3 2 2 3 0 0 0 1 1 1 0 0 0 1 _ˆ 1 1 0 0 0 1 _{1 ˆ} 0 0 0 0 0 0 0 0 0 0 ( ) 0 0 T P P k T P P P P T T T T L C e L C C L e C L e c A L C L C L C L L L e C e L L e C _k κ α α α κ κ α α α α α κ − − − − − − −  − −    ₋      ₋ ₋    =   − − −      ₋     + −    % % % T 1 2 2 1 1 2 3 2 2 3 1 _{1 ˆ} 0 0 0 0 0 ( ) 0 0 P k T C L C C C e C e L e C _k α α κ − −     =   − +     % % with 2 1 33 33 33 c e k α ₌ ₊ − _, 1 1 1 1 2 2 33 33 1 1 33 33 2 1 1 2 1 1 1 ( ) T T T T C C C C c k e e e k C e e C α α − α − = − + − + % , 1 2 2 33 33 1T, C% =C +e k e− 1 1 2 33 33 2 33 33 1 ˆ T_.

C =e k C− −c k e− In the absence of the piezoelectric effect, 0 ij e = so that 33 c α = , 1 1 1 33 2 2 T C% =C −c C C− , 2 2

C% =C , Cˆ₂ =0and the system equations reduces to those for elastic materials.

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4. Solution

Introducing nondimensional variables, x= x h z₁/ , =x h₃/ and considering the first approximation by choosing one term in the trigonometric series:

3 33 3 ₃₃ 3 ( )cos ( )sin ( , ) ( , ) ( )sin ( , ) ( , ) ( )cos ( , ) ( , ) _{( )sin} ( )sin S S k k n x U z S n x W z u x z _S u x z _{n x} z x z _S x z n x S z S x z n x D x z _S _z S n x z S π π π φ σ π σ π π           _ _   _ _   __Φ _   _ _ =                         ∆     (17a)

and output variables, sin ( , ) ( , ) cos P P P P k k n x S x z _S D x z n x S π σ π =               __∆ _     (17b)

in which _S₌a_h_{is the span-to-thickness ratio. It is noticed that (17a) and (17b) satisfy the simply}

supported edge conditions (7a) and (7b). Then we have

* { } { } k k k d X A X dz = (18) { }Y _k =C_k∗{ }X _k (19) where T _k k U V W S S S X} { , , , , , , , }

{ = Φ ₁₃ ₂₃ ₃₃ ∆₃ is the state vector, and {Y}T ={S₁₁,S₂₂,S₁₂,∆₁,∆₂}is the output

vector, and both are functions of z; and A_k∗,C_k∗ are similar to A ,_k C_k in Eq. (15) and (16) except

in which LP,L is replaced with

            =                 − = 0 1 , 1 0 0 0 0 1 S n H S n P H π π .

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The solution of the first-order matrix differential equations, (18) and (19), (Pease, 1965) is ) ) ( ) ( = _k − _k₋₁ _k

(

_k₋₁ k z T z z X

z

X , (20)

where )T_k(z is the 8×8 transfer matrix given by ) 1 ( 1) ( − ₋ = Ak∗ z−zk− k k z z

e

T . (21)

The interfacial continuity conditions, Eq. (6), are satisfied by letting 1 , , 2 , 1 ), ( ) ( 1 = = − + z X z k n X_k _k _k _k _L (22) There follows 1 , , 2 , 1 ), ( ) ( ) ( ₁ ₁ 1 = − − − = − + z T z z X z k n X_k _k _k _k _k _k _k _L (23)

Using Eq. (23) recursively yields

1 , , 2 , 1 , ), 0 ( ) ( ) (z =T z X z ₋₁≤z≤z k= n− X _k _k _k L (24) where         = = = − − − ₋ T z k n T k z T z T k k z z k k k ( ), 2,3,..., 1 ), ( ) ( 1 1 ) ( 1 1 (25)

Setting z=1 in Eq. (24) gives ) 0 ( ) 1 ( ) 1 ( T X X = , (26)

The solution of X(z) can be determined from Eq. (26) once the initial vector X(0) is given.

However, for a physical problem, not all the state variables at the bottom surfaces are prescribed and we should resort to the relationship between the top and bottom surface according to various surface conditions:

(1) Case 1: In the case of mechanic loadings at the top surface such as q(x)=q0sin(nπ x/S) with both the top and bottom surfaces electrically insulated, Eq. (26) can be partitioned into

            =       ) 0 ( ) 0 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( S U T T T T S U SS SU US UU (27)

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where _U ₌

[

_U_,_V_,_W_,_Φ

]

T _and _S

[

_S _S _S

]

T 3 33 23 13, , ,∆ = , then we have _S₍₁₎

[

₀_,₀_,_q _,₀

]

T 0 = ,

[

]

T S(0)= 0,0,0,0 and from (27) ) 1 ( ) 1 ( ) 0 ( T 1 S U = _SU− (28)

The primary states variables at any position z can be obtained as

) 0 ( ) ( ) (z T zU U = _UU , (29a) ) 0 ( ) ( ) (z T zU S = _SU . (29b)

(2) Case 2: When both the top and bottom surfaces are grounded, Eq. (26) can be cast into alternate form as ˆ ˆ ˆ₍₁₎ ₍₁₎ ₍₁₎ ˆ₍₀₎ ˆ ˆ ˆ₍₁₎ ₍₁₎ ₍₁₎ ˆ₍₀₎ UU US SU SS T T U U T T S S       =                  (30) where _U

[

_U _V _W

]

T 3 , , , ˆ ₌ _∆ _and _S_ˆ₌

[

_S _,_S _,_S _,_Φ

]

T 33 23 13 , then we have Sˆ(1)=

[

0,0,q0,0

]

T ,

[

]

T Sˆ(0)= 0,0,0,0 and from (30) ) 1 ( ˆ ) 1 ( ˆ ) 0 ( ˆ _T 1 _S U = _SU− , (31)

The primary states variables at any position z can be obtained as

) 0 ( ˆ ) ( ˆ ) ( ˆ _z _T _z_U U = _UU , (32a) ) 0 ( ˆ ) ( ˆ ) ( ˆ _z _T _z_U S = _SU (32b)

(3) Case 3: The case of mixed surface conditions such as the top surface being grounded while the

bottom surface being can be treated similarly.

(4) Case 4: If electrically loading exists such as D_Z(x,h)=∆₀sin(nπ x/S), we have Eq. (27) with

[

]

T

S(1)= 0,0,0,∆0 , S(0)=

[

0,0,0,0

]

Tand the solution is in the same form as Eq. (28) and (29).

(5) Case 5: If electrically loading exists such as ϕ(x,h)=Ψ₀sin(nπ x/S), we have Eq. (30) with

[

]

T

S(1)= 0,0,0,Ψ0 , S(0)=

[

0,0,0,0

]

Tand the solution is in the same form as Eq. (31) and (32).

5. Examples and discussion

To check the validation of the state space approach for analysis of simply supported multi-layered laminate subjected to sinusoidal loadings a three-layered Graphite/Epoxy [0/90/0] composite as employed by Pagano (1969) is considered first. The present solutions agree well with the elasticity solutions obtained by Pagano as shown in Fig. 2. The phenomena that the solutions approach to the CPT solutions for larger aspect ratio are also shown. The state space approach developed can thus give the same exact elasticity solutions for composite laminates.

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In order to depict the idea of state space approach for the analysis of a simply supported multi-layered piezoelectric laminate subjected to sinusoidal loadings a [PZT-5/0/90/0/PZT-5] configuration is taken into account. This can be considered as to patch two PZT-5 ceramics onto a [0/90/0] Graphite/Epoxy laminate. The polarization direction is parallel to x axis and the top and ₃

the bottom surfaces are subjected to mechanical loadings or electrical loadings in various situations (Fig. 1). The material constants for PZT-5 are as follows:

. / 10 29 . 7 , / 10 13 . 8 , / 29 . 12 , / 63 . 15 , / 54 . 5 , / 10 58 . 22 , 2 / 9 10 05 . 21 55 44 , 2 / 9 10 09 . 75 23 13 12 , 2 / 9 10 87 . 110 33 , 2 / 9 10 35 . 120 22 11 9 33 9 22 11 2 24 15 2 33 2 32 31 2 9 66 m F k m F k k m C e e m C e m C e e m N c m N c c m N c c c m N c m N c c − − ₌ _× × = = = = = − = = × = × = = × = = = × = × = =

and the elastic constants for Graphite/Epoxy are

2 9 66 55 2 9 44 2 9 23 2 9 13 12 2 9 33 22 2 9 11 / 10 17 . 7 , / 10 87 . 2 , / 10 92 . 3 , / 10 36 . 4 , / 10 66 . 11 , / 10 44 . 183 m N c c m N c m N c m N c c m N c c m N c × = = × = × = × = = × = = × =

Figure 3 to 5 shows the through-thickness variation of the primary states, such as in-plane displacement, U, transverse displacement, W, in-plane stress, S₁₁, transverse shear stress, S , 13

electrical potential, Φ, transverse electric displacement, ∆ for a [PZT-5/0/90/0/PZT-5] laminate ₃ subjected to mechanical loading, q(x)=sin(πx/S) , i.e., q0 = n1, =1, with three kinds of electric conditions at the top and the bottom surface, respectively. All the primary states are dimensionless quantities and two span-to-thickness ratios, S=4 and S=10, are considered. All the three figures show that the mechanically loadings will induce electrical voltages and electric displacements as expected. However, the mechanical variables are not influenced drastically by different surface electric conditions. It is obviously seen that piezoelastic analysis based on the Kirchhoff hypothesis on the deformation cannot predict the correct piezoelastic interactions for a laminate with low span-to-thickness ratio as the same rule for elastic laminate without piezoelectric effects. Therefore in the analysis and design of sensors of deformations and/or motions, made of piezoelectric materials, exact three-dimensional theory should be employed, especially for those with low span-to-thickness ratio.

To investigate the effect of electrical loading on the mechanical states such as displacement and stresses for a piezoelectric laminate, we consider two kinds of electric loading. Figure 6 shows the through-thickness distribution of primary states for a [PZT-5/0/90/0/PZT-5] laminate subjected to ∆₃(x)=sin(πx/S) at the top surface with the bottom surface electrically insulated. In Figure 7 the results are for electrically loading with Φ(x)=sin(πx/S)while the bottom surface is grounded. Both the figures describe the influence and induction of displacements and stresses in a piezoelectric laminate due to electrical voltage or electrical displacements. It is noticed that the induced in-plane and transverse displacements and stresses are significantly different for S=4 and

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10 =

S . We should point out that those analysis of piezoelectric plates based on thin-plate theory can

not be applied to the piezoelectric laminate with low span-to-thickness ratio, especially for the prediction of efficiency of actuator made of piezoelectric materials for control of vibrations.

6. Concluding Remarks

Exact analysis of a simply supported multi-layered elastic and/or piezoelectric laminate subjected to sinusoidal mechanical or electrical loadings via state space approach based on the Hellinger-Reissner variational principle has been successfully conducted. The basic equations are first grouped and rearranged into a set of standard first-order matrix differential equations. Approximate solutions are obtained by the use of the first term of the trigonometric functions that satisfy the boundary conditions a priori. A [0/90/0] three-layer Graphite/Epoxy laminate is employed first to check the validation of this approach and is shown to agree well with the elasticity solutions obtained by Pagano. Further, a [PZT-5/0/90/0/PZT-5] is taken as typical example for piezoelectric laminate and various surface conditions are studied in detail. It is noticed that as span-to-thickness ratio of the simply supported laminate less than 10 to 20, classical thin plate theory can not be employed for piezoelectric and elastic plates and the present state space approach gives a satisfactory exact solutions.

References

[1] Bisegna, P. and F. Maceri, “An Exact Three-Dimensional Solution for Simply Supported Rectangular Piezoelectric Plates,” Transaction of ASME, Journal of Applied Mechanics, Vol.

63, 1996, pp. 628-638.

[2] Huang, L. J. and J. Q. Tarn, “A State Space Approach for Simply Supported Piezoelectric Laminate,” The 25th Conference on Theoretical and Applied Mechanics, Taichung, Taiwan, R.O.C., 15-16 December. 2001.

[3] Lee, J. S. and Jiang, L. Z., “Exact Electroelastic Analysis of Piezoelectric Laminae Via State Space Approach,” International Journal of Solids and Structures, Vol. 33, No. 7,1996, pp. 977-990.

[4] Lekhnitskii, S. G., Anisotropic Plates, 2nd edition. Gordon and Breach, New York, 1968. [5] Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body. Mir, Moscow., 1981.

[6] Mason, W. P., “Piezoelectricity, its History and Applications,” Journal of Acoustic Society

America, Vol. 70 , 1981, pp. 1561-1566.

[7] Pagano, N. J., “Exact Solutions for Composite Laminates in Cylindrical Bending,” Journal of

Composite Materials, Vol. 3, 1969, pp. 398-411.

[8] Pease, M. C., Methods of Matrix Algebra. Academic Press, New York, 1965.

[9] Tarn, J. Q. and Y. M. Wang, “Laminated Composite Tubes Under Extension, Torsion, Bending, Shearing and Pressuring: A State Space Approach, to be appeared in International Journal of Solids and Structures., 2001.

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[10] Tarn, J. Q. and L. J. Huang, “Saint-Venant End Effects in Multilayed Piezoelectric Laminates,”

International Journal of Solids and Structures, Vol. 39, No. 19, 2002, pp. 4979-4998.

[11] Tiersten, H. F., Linear Piezoelectric Plate Vibrations, Plenum Press, New York, 1969.

[12] Wang, Y. M., J. Q. Tarn, and C. K. Hsu, “State Space Approach for Stress Decay in Laminates”,

International Journal of Solids and Structures, Vol. 37, 2000, pp. 3535-3553.

[13] Washizu, K., Variational Methods in Elasticity and Plasticity, Pergamon Press, Ltd, 1975.

Figure 1 A schematic of a simply supported piezoelectric laminate

Figure 2 Through-thickness variation of U,S₁₁,S₃₃,S₁₃ for a simply-supported three-layer laminate subjected to sinusoidal mechanical load (‘— ’: S=4; ‘---’: S=10).

a h x1 x3 Polarization direction q(x1) Insulated Ψ(x1) Grounded

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Figure 3 Through-thickness variation of primary states for a simply-supported [PZT-5/0/90/0/PZT-5] piezoelectric laminate subjected to sinusoidal mechanical load (case 1). (‘— ’: S=4; ‘---’:

10 =

S ).

10 =

(13)

10 =

S ).

Figure 6 Through-thickness variation of primary states for a simply-supported [PZT-5/0/90/0/PZT-5] piezoelectric laminate subjected to sinusoidal electric displacement at the top surface (case 4). (‘— ’: S=4; ‘---’: S=10).

(14)

Figure 7 Through-thickness variation of primary states for a simply-supported [PZT-5/0/90/0/PZT-5] piezoelectric laminate subjected to sinusoidal electric potential at the top surface (case 5). (‘— ’:

4 =