A multilayered piezoelectric shell theory

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A multilayered piezoelectric shell theory

Claire Ossadzow-David, Maurice Touratier

To cite this version:

Claire Ossadzow-David, Maurice Touratier. A multilayered piezoelectric shell theory. Composites Science and Technology, Elsevier, 2004, 64 (13-14), pp.2121-2137. �10.1016/j.compscitech.2004.03.005�.

�hal-00019243�

A multilayered piezoelectric shell theory

Claire Ossadzow-David

^a,*

, Maurice Touratier

^b

aLMM – UMR CNRS 7607, UniversiteParis VI, Bo^ıte courrier No 0162, 4, Place Jussieu, 75252 Paris Cedex 05, France

bLMSP– UMR CNRS 8106 – ENSAM – ESEM 151, Bd de l’H^opital, 75013 Paris Cedex, France

Thispaperpresentsatwo-dimensionaltheoryfortheanalysisofpiezoelectricshells.Thetheoryisbasedonanhybridapproachin whichthecontinuityconditionsforbothmechanicalandelectricunknownsatlayerinterfacesaswellastheimposedconditionson theboundingsurfacesandattheinterfacesareindependentlysatisfied.Then, thepiezoelectricboundary-valueproblemisstated usingsuchkindofmechanicaldisplacementsandelectrostaticpotential,inconjunctionwiththecoupledpiezoelectricconstitutive law. The accuracy of the proposed theory is assessedthrough investigation of significant problems,for which an exact three- dimensionalsolutionisknown.

Keywords:A. Layered structures; Piezoelectric; C. Shell theory; Continuity; Constraints

1. Introduction

The development of the so-called ‘‘smart-structures’’, e.g., made of piezoelectric composites, require nowadays more and more precision in their design and sizing. The importance of efficient models has, so far, led to nu- merous theories.

The modeling of piezoelectric shells mostly concerns cases attached to specific geometries (cylindrical and spherical): Toupin [1] studied the static response of a radially polarized spherical piezoelectric shell; Adelman and Stavsky [2,3] examined cases involving hollow pie- zoelectric cylinders. Sun and Chen [4] and Karlash [5]

studied wave propagation in layered piezoelectric cyl- inders; Paul et al. [6,7] examined free vibration prob- lems. Siao et al. [8] proposed a semi-analytic model for layered piezoelectric cylinders taking into account a layerwise behavior of the composite.

Analytic solutions for laminated piezoelectric cylin- ders were proposed by Mitchell et al. [9], Xu and Noor [10], Heyliger [11] and Dumir et al. [12].

Drumheller and Kalnins [13] used classical shell the- ory for free vibrations of shells of revolution. Haskins

and Kalnins [14] proposed the development of electrical and mechanical quantities as expansions of the thickness variable. Tzou and Garde [15] used the Kirchhoff–Love hypothesis to derive the governing equations for thin shells, but did not take into account the charge equa- tion. It was done by Tzou and Zhong [16], this time with a shear-deformation theory. Other piezoelectric shell models and finite element approximations, based on single-layer models, were also developed by Tzou [17].

A Reissner–Mindlin shear-deformation shell finite element with surface bonded piezoelectric layers was developed by Lammering [18]. Koconis et al. [19] used a Ritz method for three-layered shells with embedded piezoelectric actuators.

Tzou and Yee [20] proposed a coupled theory where the piezoelectric shells are considered as a layerwise as- sembly of curvilinear solid piezoelectric triangular ele- ments. Heyliger et al. [21] developed a finite-element for laminated piezoelectric shells. Saravanos [22] used a coupled mixed theory for curvilinear composite piezo- electric laminates with the first-order shear deformation theory hypothesis and a layerwise approximation of the electrostatic potential, along with the corresponding fi- nite element for piezoelectric shells.

We propose here a new two-dimensional theory for the modeling of deep multilayered piezoelectric shells. It

*Corresponding author. Fax: +33-1-44-27-52-59.

E-mail address:david@lmm.jussieu.fr(C. Ossadzow-David).

extends our previous works on plates and shells [23,24]

to piezoelectricity by combining our previous equivalent single-layer approach for the displacement field, with quadratic variations of the electrostatic potential through the piezoelectric layers. Both quantities are automatically continued at layer interfaces. In addition, transverse shear stresses as well as the electric displace- ment are independently continued, using at the first stage uncoupled constitutive law for those two fields.

Refinements of the shear and membrane terms are taken into account, in the displacement field, by means of trigonometric functions. Moreover, we allow values for the electrostatic potential to be imposed either on the top and bottom surfaces of the structure, or at layer interfaces.

Finally, the piezoelectric boundary-value problem is constructed using the consistent coupled constitutive law, in conjunction with the above displacements and electrostatic potential fields. The proposed piezoelectric shell model is evaluated for significant problems, for which the exact three-dimensional solution is known [12].

2. The piezoelectric shell model 2.1. Geometric considerations for shells

We consider an undeformed laminated shell of con- stant thickness h, consisting of an arrangement of a fi- nite number N of piezoelectric layers (see Fig. 1). The space occupied by the shell will be denoted V . The boundary of the shell is the reunion of the upper surface S

h

, the lower surface S

₀

, and the edge faces A.

The interface between the ith and ði þ 1Þth layer is denoted by S

i

, the distance between S

₀

and S

_iþ1

; z

i

.

The reference surface coincides with the bottom sur- face of the shell S

₀

.

In this paper, the Einsteinian summation convention applies to repeated indices, where Latin indices range from 1 to 3 while Greek indices range from 1 to 2.

The Cartesian coordinate system of the shell will be denoted by ðx

1

; x

₂

; zÞ.

A point M out of the reference surface being given, let us denote P the point of the reference surface closest to M . Covariant base vectors ð ~ a

_i

Þ, ð ~ g

_i

Þ and contravariant base vectors ð ~ a

ⁱ

Þ, ð ~ g

ⁱ

Þ in the undeformed state of the shell are introduced such as:

~ a

a

¼ ~ P

;a

; ~ a

3

¼ ~ a

₁

^ ~ a

₂

~ a

₁

^~ a

₂

; ~ a

1

^ ~ a

2

~ a

3

> 0;

~ g

_i

¼ M ~

_;i

; ~ g

₁

^ ~ g

₂

~ g

₃

> 0; ~ a

^a

~ a

_b

¼ d

^a_b

;

~ a

³

¼ ~ a

₃

; ~ g

^a

~ g

_b

¼ g

^a_b

; ~ g

³

¼ ~ g

₃

:

ð1Þ

Differentiation with respect to x

_i

is denoted by ‘‘

;i

’’, [d

^b_a

] being the identity tensor.

It is recalled that M ~ ¼ P

*

þ z ~ a

3

: ð2Þ

The above equations ensure the following relations (see, for instance [25]):

~ g

_a

¼ l

^b_a

~ a

_b

; ~ g

₃

¼ ~ a

₃

; ~ g

^a

¼ l

^a1_b

~ a

^b

; ~ g

³

¼ ~ a

³

;

~ g

_a

¼ g

_ab

~ g

^b

; ~ g

^a

¼ g

^ab

~ g

_b

; ~ a

_a

¼ a

_ab

~ a

^b

; ~ a

^a

¼ a

^ab

~ a

_b

: ð3Þ The components of the shifter tensor are denoted by

l

^a_b

¼ d

^a_b

zb

^a_b

; ð4Þ

those of the curvature tensor by

b

ab

¼ ~ a

a;b

~ a

³

; ð5Þ

Nomenclature

V space occupied by the shell h total thickness of the shell R radius of curvature of the shell S

_h

top surface of the shell

S

0

bottom surface of the shell

S

_i

interface between the ith and ði þ 1Þth layer A lateral surface of the shell

ðx

₁

; x

₂

; zÞ Cartesian coordinate system of the shell z

i

distance between S

0

and S

i

z

_ið0Þ

distance between S

₀

and the midsurface of the ith layer

C

^ðiÞ_kl

components of the elastic stiffness tensor of the ith layer

s

_k

shear-strains

0

derivation with respect to the thickness co- ordinate z

u electrostatic potential

u

^1B

ðx

a

; tÞ electrostatic potential on S

₀

u

^Nþ1;B

ðx

a

; tÞ electrostatic potential on S

_h

u

^iB

ðx

a

; tÞ electrostatic potential on S

_i

u

^iT

ðx

a

; tÞ electrostatic potential on S

_iþ1

u

^iM

ðx

a

; tÞ electrostatic potential on the midsurface of the ith layer

E

_l

components of the electric field

D

k

components of the electric displacement e

^ðiÞ_kl

components of the rotated piezoelectric ten-

sor of the ith layer

e

^ðiÞ_kl

components of the dielectric tensor of the ith layer

q mass density d variational operator

differentiation with respect to time t

e

0

permittivity of vacuum (e

0

¼ 8:85 10

¹²

F/m)

and its mixed components by

b

^a_b

¼ ~ a

_3;b

~ a

^a

: ð6Þ

The surface metrics a

1

and a

2

are related to the a

_ab

co- efficients via

ða

a

Þ

²

¼ a

_aa

ð7Þ

(no summation on a index).

In the following, the curvilinear coordinates (or shell

coordinates) are assumed orthogonal, and are such that

the x

₁

- and x

₂

-curves are lines of curvature on the ref- erence surface z ¼ 0; z-curves are straight lines perpen- dicular to the surface z ¼ 0. R

₁

and R

₂

are the values of the principal radii of curvature of the reference surface.

The distance ds between two points Pðx

1

; x

₂

; 0Þ, P

⁰

ðx

1

þ dx

₁

; x

₂

þ dx

₂

; 0Þ of the reference surface S

₀

of the shell is given by

ðdsÞ

²

¼ a

²₁

ðdx

1

Þ

²

þ a

²₂

ðdx

2

Þ

²

; ð8Þ where a

1

and a

2

are the surface metrics

a

²_l

¼

oP oxl

oP

oxl

: ð9Þ

The distance dS between two points M ð x

₁

; x

₂

; z Þ, M

⁰

ð x

₁

þ dx

₁

; x

₂

þ dx

₂

; z þ dz Þ out of the reference sur- face is given by

ðdSÞ

²

¼ L

²₁

ðdx

1

Þ

²

þ L

²₂

ðdx

2

Þ

²

þ L

²₃

ðdzÞ

²

; ð10Þ where L

₁

, L

₂

and L

₃

are the so-called Lam e coefficients:

L

₁

¼ a

1

1

þ z R

₁

; L

₂

¼ a

2

1

þ z R

₂

; L

₃

¼ 1: ð11Þ

2.2. Kinematic assumptions

Geometric linear shells are considered, including an elastic-linear behaviour for laminates. The transverse normal stress is ignored and it is assumed that no tan- gential tractions are exerted on the upper and lower surfaces of the shell.

The components of the displacement field of any point M ðx

1

; x

₂

; zÞ of the volume occupied by the shell (V ), expressed for sake of commodity in the contra- variant basis ð ~ g

^a

;~ g

³

Þ, are assumed in the following form:

U

a

¼ u

a

þ zg

_a

þ f ðzÞw

_a

þ gðzÞc

⁰_a

þ

PN1

m¼1

u

_ðmÞa

ðz z

m

ÞHðz z

m

Þ;

U

₃

¼ w;

8<

:

ð12Þ

where (as suggested in [26] for f ðzÞ):

f ðzÞ ¼ h

p

sin

pz

h

; gðzÞ ¼ h

p

cos

pz

h

; ð13Þ

H being the

Heaviside step function, defined by:

H ð z z

_m

Þ ¼ 1 for z

P

z

_m

; 0 for z < z

_m

:

ð14Þ This step function has been previously used among others in Di Sciuva [27] and He [28].

Also, as in Touratier [26], the choice for f ð z Þ can be justified in a discrete-layer approach from the three-di- mensional works of Cheng [29] for thick plates.

In this displacement field, u

_a

are membrane dis- placements, c

⁰_a

are the components of the transverse shear strain vector at z ¼ 0, w is the transverse deflection of the shell.

O

x1

z i

z

zi+1

1 ^st layer N ^th layer

h

x2

Fig. 1. The multilayered piezoelectric shell.

The g

_a

and w

_a

are functions to be determined using the boundary conditions for the transverse shear stresses on the top and bottom surfaces of the shell. With the help of the u

_ðmÞ

a

, which represent the generalized ‘‘dis-

placements per layer’’ the continuity of the displace-

ments at layer interfaces are automatically satisfied from the Heaviside function. The generalized displacements per layer are then determined from the continuity con- ditions on the transverse shear stresses at the interfaces.

The transverse shear-stresses in each layer are given by the classic uncoupled constitutive law

r

^ðiÞ_6a^uncoupled

¼ C

_6a;6a^ðiÞ

s

_6a

; i ¼ 1;

. . .

; N 1; a ¼ 1; 2;

ð15Þ where the C

_6a^ðiÞ

are the corresponding stiffness compo- nent, and s

_6a

the shear strains.

The boundary conditions allow one to eliminate the g

_a

and w

_a

, from the following system:

g

_a

¼ w

_a

w

_ja

b

^m_a

u

_m

þ h

p

c

⁰_m

;

2w

_a

þ hb

^m_a

w

_m

2 h

p

b

^m_a

c

⁰_m

þ

X^N1

m¼1

d

^m_a

b

^m_a

z

_m

u

_ðmÞm

¼ 0 ð16Þ or

w

_a

¼ d

_a^b

c

⁰_b

þ

X^N1

m¼1

f

_ðmÞ^b

a

u

_ðmÞb

; ð17Þ

where

½d

_a^b

¼ ½hb

^b_a

2d

^b_a

¹

2 h

p

b

^b_a

;

f

_ðmÞ^b

a

h i

¼ hb

^b_a

2d

^b_a1

d

^m_a

b

^m_a

z

_m

;

ð18Þ

½d

_a^b

being the tensor of components d

_a^b

, given by the first relation of the above equation.

The transverse shear stresses can then be expressed as functions of u

_ðmÞ

a

, c

⁰_a

.

The continuity conditions for transverse shear stres- ses lead thus to a system of 2(N 1) equations with the 2(N 1) unknowns u

_ðmÞ

a

.

Those latter functions can then be expressed in terms of the c

⁰_a

,

u

_ðmÞa

¼ k

ðmÞ_a

c

⁰_a

ðno summation on aÞ; ð19Þ where the k

_ðmÞa

are given by the resolution of the previ- ous system.

The final form of the displacement field is given by:

U

_a

¼ l

^b_a

u

_b

zw

_ja

þ h

^b_a

c

⁰_b

; U

₃

¼ w;

ð20Þ where h

^b_a

are functions of the global thickness coordinate z,

h

^b_a

ðzÞ ¼ gðzÞd

^b_a

z h

p

b

^b_a

þ ½ f ðzÞ zd

_a^b

þ

X^N

m¼1

f

_ðmÞ^b

a

h

þ ð z z

_m

Þ H z ð z

_m

Þd

^b_ai

k

_ðmÞb

: ð21Þ All those formulae will be referred to as

Kinematic Field

(K.F.).

This kinematic field has been developed in Ossadzow et al. [23,24].

2.3. The electrostatic potential

Within a standard variational procedure when for- mulating any piezoelectric boundary-value problem, it is easy to show that the piezoelectric constitutive law given by Eq. (35) introduces a strong coupling, including be- sides derivatives. This does not allow to exactly solve all the interfaces and boundary equations that could be used to reduce the number of unknowns (see an example in elasticity for thick plates in [30]).

Therefore, to approximate the electrostatic potential, we consider the purely electric state of the shell, as above in elasticity, in order to write interface conditions and boundary conditions at the top and bottom surfaces of the shell. Then, when formulating the piezoelectric boundary-value problem, we will include coupled elec- tromechanical constitutive laws. Eventually, throughout several examples, we will show that using the uncoupled law just when building the displacement and electro- static fields has not any significative influence on the distributions of stresses and electrostatic potential.

So, the electrostatic potential u is approximated un- der the following form:

uðx

1

; x

2

; z; tÞ ¼

X^N

i¼1

u

ⁱ

ðx

1

; x

2

; z; tÞv

ⁱ

ðzÞ; ð22Þ where the u

ⁱ

are the potentials ‘‘per layer’’, and v

ⁱ

the characteristic ith-layer function,

v

ⁱ

ðzÞ ¼ 1 if z 2 ½z

ⁱ

; z

^iþ1

; 0 if z 62 ½z

ⁱ

; z

^iþ1

:

ð23Þ Introducing, for each layer, the thickness coordinate n

ⁱ

(see Fig. 2), given by

1

6

n

ⁱ6

1; n

ⁱ

¼ 2ðz z

_ið0Þ

Þ

h

_i

; ð24Þ

where z

_ið0Þ

is the distance between S

₀

and the midsurface of the ith layer, the u

ⁱ

are taken as

u

ⁱ

ðx

j

; tÞ ¼

¹₂

n

ⁱ

ðn

ⁱ

1Þu

^iB

ðx

a

; tÞ þ ð1 ðn

ⁱ

Þ

²

Þu

^iM

ðx

a

; tÞ

þ

¹₂

n

ⁱ

ðn

ⁱ

þ 1Þu

^iT

ðx

a

; tÞ ði ¼ 1;

. . .

; NÞ; ð25Þ

u

^iB

ðx

a

; tÞ is the electrostatic potential on the bottom

surface S

_i

, u

^iT

ðx

a

; tÞ the electrostatic potential on the top

surface S

_iþ1

, and u

^iM

ðx

a

; tÞ the electrostatic potential on

the midsurface of the ith layer (see Fig. 2), so that

u

ⁱ

ðx

a

; z

_i

; tÞ ¼ u

^iB

ðx

a

; tÞ;

u

ⁱ

ðx

a

; z

_iþ1

; tÞ ¼ u

^iT

ðx

a

; tÞ;

u

ⁱ

x

_a

; z

_iþ1

z

_i

2 ; t

¼ u

^iM

ð x

_a

; t Þ ð i ¼ 1;

. . .

; N Þ : ð26Þ

Since

u

^iþ1;B

ð x

_a

; z

_i

; t Þ ¼ u

^iT

ð x

_a

; z

_i

; t Þ ð i ¼ 0;

. . .

; N Þ ð27Þ it is worth noting that the continuity of the electrostatic potential at layer interfaces is automatically satisfied, as it can be seen in Fig. 2.

Starting from the first layer, we choose to keep u

^iB

ðx

a

; tÞ and u

^iM

ðx

a

; tÞ as unknowns.

The u

ⁱ

can then be written as

u

ⁱ

ðx

j

; tÞ ¼

¹₂

n

ⁱ

ðn

ⁱ

1Þu

^iB

ðx

a

; tÞ þ ð1 ðn

ⁱ

Þ

²

Þu

^iM

ðx

a

; tÞ þ

¹₂

n

ⁱ

ðn

ⁱ

þ 1Þu

^iþ1;B

ðx

a

; tÞ ði ¼ 1;

. . .

; NÞ;

ð28Þ where u

^1B

ðx

a

; tÞ is the electrostatic potential on S

₀

, u

^Nþ1;B

ðx

a

; tÞ the electrostatic potential on S

_h

(see Fig. 2).

For future applications, we suppose that p values for the electrostatic potential are given, which means that p values for the u

^iB

ðx

a

; tÞ, are imposed, depending on the kind of electric boundary conditions. These values can be given on the top and bottom surfaces of the plate, or at layer interfaces.

The uncoupled piezoelectric constitutive law gives

D

^uncoupled₃

¼ e

33

E

₃

¼ e

33

u

_;3

: ð29Þ

The continuity of the purely electric D

₃

at layer inter- faces and the boundary conditions on top and bottom surfaces and at the p layer interfaces lead to a system of N þ p 1 equations which allow to eliminate part of the u

^iB

and u

^jM

, as it can be seen in the examples developed in Parts II and III.

The u

^iB

which are chosen to remain unknown will be denoted fu

^i1B

;

. . .

; u

^inB

g.

The u

^jM

which are chosen to remain unknown will be denoted fu

^j1M

;

. . .

; u

^jmM

g.

It can be noted that

i

_n

þ j

_m

¼ 2N þ 1 ðN 1 þ pÞ ¼ N p þ 2: ð30Þ The electrostatic potential can then be written under the following form:

u ¼

Xⁱⁿ

i¼i1

Q

^iB

u

^iB

þ

X^jm

j¼j1

Q

^jM

u

^jM

; ð31Þ

where the Q

^iB

and Q

^jM

are polynomial functions of the global thickness coordinate z (see examples Part II), coming from the imposed boundary conditions for the potential, as described before.

After solving as explained hereafter in Section 2.5 any boundary-value problem, both all the mechanical un- knowns, and the unknowns electrostatic potential are the obtained. The only electric quantity that cannot be obtained without a post-processing correction is the fi- nal (coupled) electric displacement: we recall that the coupled piezoelectric constitutive law gives

D

^coupled₃

¼ e

33

E

₃

þ e

_3j

s

_j

: ð32Þ

It is very important to note that the e

33

E

₃

term are very small compared to the e

_3j

s

_j

terms; hence, if the final value of the electric displacement is not corrected in a post-processing phase, the e

_3j

s

_j

terms will prevail on the e

33

E

3

term. The e

3j

s

_j

terms are not continued at layer interfaces (see (29)). Thus, the final electric displacement requires such a correction. This correction leads to the following corrected value, which satisfies continuity at layer interfaces

D

~^coupled₃

¼ e

33

E

₃

þ e

_3j

s

_j

h

e

^ðkÞ_3j

s

^ðkÞ_j

ð Þv z

_k ^k

ð z Þ

e

^ðkþ1Þ_3j

s

^ðkþ1Þ_j

ð Þv z

_k ^kþ1

ð z Þ

i

; ð33Þ

ξⁱ

i ^th layer

ϕ ^{i B} ϕ ^{i+1, B}= ϕ ^{i T}

ϕ ^iM

hi

Fig. 2. Configuration of theith layer.

where the ð k Þ exponent characterizes the quantities re- lated to the kth layer and the ð k þ 1Þ exponent the quantities related to the ðk þ 1Þth layer.

For sake of commodity, we will denote by D

~^coupled

the final (coupled) electric displacement field (only its third component being of course corrected).

Therefore, at the kth layer interface, the continuity of the modified coupled electric displacement D

~^coupled₃

is satisfied (e

33

E

₃

) being continuous, from Eqs. (17)–(20), since we have:

lim

z!z_k0 z<zk

D

~^coupled₃

¼ lim

z!z_k0 z<zk

e

33

E

3

8>

<

>:

þ e

3j

s

_j

e

^ðkÞ_3j

s

^ðkÞ_j

ð Þ z

_k

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

0

9>

=

>;

¼ lim

z!zk

e

33

E

₃

f g

¼ lim

z!z_k0 z>zk

e

33

E

₃ 8>

<

>:

þ e

_3j

s

_j

e

^ðkþ1Þ_3j

s

^ðkþ1Þ_j

ð Þ z

_k

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

0

9>

=

>;

:

ð34Þ Hereafter, depending on the kind of electric boundary conditions, we will show how to build the potential field, since our method is new as it is possible to check, and different from a usual layerwise method, which does not exploit all the boundary conditions. It allows to reduce the size of any boundary-value problem.

2.4. The linear-piezoelectric constitutive law

The constitutive law of a piezoelectric material, adopted in our model, has been given by Tiersten [31], and is expressed by:

r

^coupled_i

¼ C

_ij

s

_j

e

_ki

E

_k

; D

^coupled_k

¼ e

kl

E

l

þ e

ki

s

i

;

ð35Þ where r

i

are the components of the stress tensor, C

ij

the elastic stiffness components, s

j

the components of small strain, e

_ki

the piezoelectric coefficients, E

_l

the compo- nents of the electric field, D

_k

the components of the electric displacement, and e

kl

are the dielectric constants.

The standard contracted notation, with i; j ¼ 1;

. . .

; 6 and k; l ¼ 1;

. . .

; 3, has been used in these equations. The poling direction is coincident with the z-axis.

Using results stated in Section 2.3, the electric-field is related to the electrostatic potential u through the re- lation

~ E ¼ grad

^!

u ¼

Xⁱⁿ

i¼i1

grad

^!

Q

^iB

u

^iB

X^jm

j¼j1

grad

^!

Q

^jM

u

^jM

:

ð36Þ For the materials used in this study, we assume that the nonzero components of the rotated piezoelectric

tensor e

_ki

, the elastic stiffness tensor C

_ij

, and the com- ponents of the dielectric tensor e

kl

are those of ortho- rhombic crystal. The nonzero elements of those two latter tensors will be taken as e

₃₁

, e

₃₂

, e

₃₃

, e

₂₄

, e

₁₅

and e

11

, e

22

, e

33

.

Taking, as usually for plates, into account a zero value of the transverse normal stress, the constitutive law ‘‘per layer’’ can be written as follows:

r

^coupled_i

¼ C

_ij^2D

s

_j

e

^2D_ki

E

_k

; D

^coupled_k

¼ e

^2D_kl

E

_l

þ e

^2D_kj

s

_j

; ð37Þ where

C

_ij^2D

¼ C

_ij

C

_i3

C

_3j

C

33

; e

^2D_ki

¼ e

_ki

C

_i3

e

_k3

C

33

;

e

^2D_kl

¼ e

kl

þ e

k3

e

l3

C

₃₃

: ð38Þ

2.5. The two-dimensional boundary-value problem

The equations of motion and the natural boundary conditions are derived via Hamilton’s principle:

Z t 0

Z

V

rⁱds_idV

þ Z

V

D_idu_;idV þ Z

V

~f_vdU~dV

þ Z

A

~f_sdU~dAþ Z

S0

ð phl_hþp0ÞdSþ Z

S

WdudV

dt

¼0:

ð39Þ

A superposed dot is used for differentiation with respect to time t; q is the mass density, and d the variational operator; f

_vⁱ

are components of body forces; f

_sⁱ

the prescribed components of the load on the undeformed lateral surface of the shell, p

0

and p

_h

the prescribed components of traction on the surfaces S

0

and S

_h

. W is the density of electric forces. l

_h

is the value of the de- terminant of the shifter tensor at z ¼ h.

Performing numerical integration through the thick- ness of the shell, the following equations of motion are deduced from Eq. (39), (K.F.), and (31):

N

_ð1Þ^ak

jk

þN

_ð2Þ^ab

b

j

¼ I

1€

u

a

I

2

w

€ja

þI

₃^a€

c

⁰_a

; N

_ð3Þ

þ M

_ð1Þ^am

jam

þ M

_ð2Þ^bm

bm

j

¼ I

2€

u

a aj

I

4

w

€jaa

þ I

₅^a€

c

⁰_{a aj}

þ I

1

w;

€

N

_ð5Þ^ak

k j

h

þN

_ð6Þ^ab

b j

þ N

_ð4Þ^a i

¼ I

₃^a€

u

a

I

₅^a

w

€ja

þI

₆^a€

c

⁰_a

;

N^iB

þ

M^iB_ja^a

¼ 0; i 2 f i

1

;. . . ;i

_n

g;

N^jM

þ

M^jM_ja ^a

¼ 0; j 2 f j

1

;

. . .

;j

_m

g ða ¼ 1; 2Þ:

ð40Þ

We introduce

D

ab

¼ 1 d

^a_b

: ð41Þ

We will denote by l the determinant of the shifter

tensor.

In Eq. (40):

the generalized stresses are given by:

N

_ð1Þ^ak

¼

Z h

0

C

_kj

s

_j

e

_3a

E

₃

l

^m_k

l

^a_m

l dz;

N

_ð2Þ^ab

¼

Z h

0

C

_6;6

s

₆

D

kb

l

^m_k

l

^a_m

ldz;

N

_ð3Þ

¼

Z h

0

C

_aj

s

_j

e

3a

E

₃

l

^m_a

b

_ma

þ C

_6;6

s

₆

D

ab

l

^m_a

b

_mb

h

_a

ldz;

N

_ð4Þ^a

¼

Z h

0

C

_6k;6k

s

_6k

f e

_k;6k

E

_k

g½l

^m_k

h

^a_m;3

: þ b

^m_k

h

^a_m

ldz;

N

_ð5Þ^ak

¼

Z h

0

C

kj

s

_j

e

3a

E

3

l

^m_k

h

^a_m

ldz;

N

_ð6Þ^ab

¼

Z h

0

C

_6;6

s

₆

D

kb

l

^m_k

h

^a_m

ldz;

M

_ð1Þ^am

¼

Z h

0

C

_aj

s

_j

e

_3a

E

₃

l

^m_a

þC

6;6

s

₆

D

ab

l

^m_a

lzdz;

M

_ð2Þ^bm

¼

Z h

0

C

6;6

s

6

D

ab

l

^m_a

zldz;

N^iB

¼

Z h

0

E

₃

e

33

Q

^iB0

ldz; i 2 f i

₁

;

. . .

;i

_n

g

; N^jM

¼

Z h 0

E

₃

e

33

Q

^jM0

l dz; j 2 f j

₁

;

. . .;

j

_m

g;

M^iBa

¼

Z h

0

E

a

e

aa

f þe

ka

s

a;a

gQ

^iB

ldz; i 2 f i

1

;. . . ;i

_n

g;

M^iMa

¼

Z h

0

E

_a

e

aa

f þ e

_ka

s

_a;a

g Q

^iM

ldz; j 2 f j

₁

;. . . ; j

_m

g

;

ð42Þ the generalized external mechanical forces by:

F

_v¹

a

¼

Z h 0

f

_va

l dz; F

_v²

a

¼

Z h 0

f

_va

zl dz;

F

_v³_a

¼

Z h

0

f

_va

h

a

l dz; F

_v³

¼

Z h

0

f

_v3

l dz; P ¼ p

h

þ p

0;

ð43Þ the generalized external electrostatic forces by:

W

^iB

¼

Z h

0

WQ

^iB

l dz; i 2 f i

₁

;

. . .

; i

_n

g;

W

^jM

¼

Z h

0

WQ

^jM

l dz; j 2 f j

1

;

. . .

; j

_m

g;

ð44Þ

and the inertia terms by:

I

1

¼

Z h

0

ql dz; I

2

¼

Z h

0

qzl dz;

I

₃^ðaÞ

¼

Z h

0

qh

a

ð z Þl dz; I

₄

¼

Z h

0

qz

²

l dz;

I

₅^ðaÞ

¼

Z h

0

qzh

a

ðzÞl dz; I

₆^ðaÞ

¼

Z h

0

qzh

²_a

ðzÞl dz:

ð45Þ

The boundary conditions for the shell leading to a

‘‘regular problem’’ are:

N

_ð1Þ^a

n

a

h

þ N

_ð2Þ^ab

n

b

i

¼ F

_v¹_a

or du

a

¼ 0;

M

_ð1Þ^a

;a

h

þ M

_ð2Þ^ab

;b

i

¼ F

_v²

a

or dw ¼ 0;

N

_ð4Þ^ab

n

b

¼ F

_v³_a

or dc

⁰_a

¼ 0;

M

_ð1Þ^a

n

a

h

þ M

_ð2Þ^ab

n

b

i

¼ F

v₃

or dw

;a

¼ 0;

M^iBa

n

_a

¼ W

^iB

or du

^iB

¼ 0; i 2

n

i

₁

;

. . .

; i

_no

; M^jMa

n

_a

¼ W

^jM

or du

^jM

¼ 0; j 2 f j

₁

;

. . .

; j

_m

g :

ð46Þ The equations of motion are deduced from Eq. (40) including the constitutive law given by Eqs. (35)–(38), (K.F.), Eqs. (31) and (42).

3. Validation of the piezoelectric shell model

In order to assess the accuracy of the present theory, we have considered problems for which a three-dimen- sional solution exists:

•

first, the free vibrations of five-layered piezoelectric plates;

•

second, a cylindrical orthotropic panel under pressure.

We recall that the electric-field is related to the elec- trostatic potential u through the relation

~ E ¼ grad

^!

u; ð47Þ

which gives here:

E

₁

¼

ou

ðR þ zÞox

₁

; E

₂

¼

ou

ðR þ zÞox

2

; E

₃

¼

ou

oz

: ð48Þ The piezoelectric and dielectric coefficients being very small compared to the elastic constants, we introduce non-dimensional quantities:

C

_ij^2D

¼ C

^2D_ij

C

₀

; e

^2D_ki

¼ E

0

e

^2D_ki

C

₀

; e

^2D_kl

¼ E

²₀

e

^2D_kl

C

₀

; ð49Þ where

C

₀

¼ C

¹₁₁

ð50Þ

and

E

₀

¼ 10

¹⁰

V=m: ð51Þ

Then the generalized displacements and potentials are written as follows:

u

_a

; w; c

_a

; u

^iB

; u

^jM

¼ C

₀

u

_a

; C

₀

w; C

₀

c

⁰_a

; C

0

u

^iB

E

₀

; C

0

u

^jM

E

₀

: ð52Þ

3.1. Free vibrations of five-layered piezoelectric plates

For this example, we consider a square, simply sup- ported five-layered piezoelectric plate, in closed circuit (u ¼ 0 at the top and bottom surfaces of the plates).

The simple support conditions for a square plate of

length

a

are simulated by:

u

₁

ðx

1

; 0; z; tÞ ¼ u

₁

ðx

1

; a; z; tÞ ¼ 0;

u

₂

ð0; x

₂

; z; t Þ ¼ u

₂

ð a; x

₂

; z; t Þ ¼ 0;

wðx

1

0; z; tÞ ¼ wðx

1

; a; z; tÞ ¼ 0;

wð0; x

₂

; z; tÞ ¼ wða; x

₂

; z; tÞ ¼ 0;

c

⁰₁

ð x

₁

; 0; z; t Þ ¼ c

⁰₁

ð x

₁

; a; z; t Þ ¼ 0;

c

⁰₂

ð0; x

2

; z; tÞ ¼ c

⁰₂

ða; x

2

; z; tÞ ¼ 0:

ð53Þ

The height of the plate is fixed at h ¼ 0:01 m.

The external layers are made of PZT4, the three in- ternal ones being a symmetric elastic Epoxy 0/90/0 cross- ply (see Table 1 for all the piezoelectric properties of those materials, e

0

being the permittivity of vacuum).

The same unit mass density will be used for all materials.

The piezoelectric coefficients of the three internal layers are identical; as for the dielectric coefficients, the three elastic layers have the same e

33

. The e

11

and e

22

are very close. We therefore consider that those three layers behave, on the electric point of view, as a single layer. The five-layered plate will thus be modelled as a three-layered elastic one, its core being the elastic–epoxy cross-ply.

From Eqs. (22) and (25), the approximation of the electrostatic potential becomes thus:

uðx

j

; tÞ ¼ 1 2 n

¹

ðn

¹

1Þu

^1B

ðx

a

; tÞ þ ð1 ðn

¹

Þ

²

Þu

^1M

ðx

a

; tÞ þ 1

2 n

¹

ðn

¹

þ 1Þu

^2B

ðx

a

; tÞ

v

¹

þ 1

2 n

²

ðn

²

1Þu

^2B

ðx

a

; tÞ þ ð1 ðn

²

Þ

²

Þu

^2M

ðx

a

; tÞ þ 1

2 n

²

ðn

²

þ 1Þu

^3B

ðx

a

; tÞ

v

²

þ 1

2 n

³

ðn

³

1Þu

^3B

ðx

a

; tÞ þ ð1 ðn

³

Þ

²

Þu

^3M

ðx

a

; tÞ þ 1

2 n

³

ðn

³

þ 1Þu

^4B

ð x

_a

; t Þ

v

³

: ð54Þ

The plate being in closed circuit,

u

^1B

ð x

_a

; t Þ ¼ u

^4B

ð x

_a

; t Þ ¼ 0: ð55Þ Moreover, the symmetry allows to impose:

u

^2B

ð x

_a

; t Þ ¼ u

^3B

ð x

_a

; t Þ; u

^1M

ð x

_a

; t Þ ¼ u

^3M

ð x

_a

; t Þ: ð56Þ This results in

u x

j

;t

¼ ð1

ðn

¹

Þ

²

Þu

^1M

ðx

a

; tÞ þ 1

2 n

¹

ðn

¹

þ 1Þu

^2B

ðx

a

;tÞ

v

¹

þ 1

2 n

²

ðn

²

1Þu

^2B

ðx

a

; tÞ þ ð1 ðn

²

Þ

²

Þu

^2M

ðx

a

; tÞ þ 1

2 n

²

ðn

²

þ 1Þu

^2B

ðx

a

; tÞ

v

²

þ 1

2 n

³

ðn

³

1Þu

^2B

ðx

a

; tÞ þ ð1 ðn

³

Þ

²

Þu

^1M

ðx

a

; tÞ

v

³

: ð57Þ Due to the symmetries, the continuity of the uncoupled electric displacement D

₃

at layer interfaces is written at the first interface:

e

¹₃₃

u

¹_;3

ð x

a

; z

1;

t Þ ¼ e

²₃₃

u

²_;3

ð x

a

; z

1;

t Þ; ð58Þ where e

ⁱ₃₃

is the dielectric coefficient corresponding to the ith layer.

This leads to an equation which allows to express u

^2B

in terms of u

^1M

and u

^2M

u

^2B

ð x

_a

; t Þ ¼ k

^2B;1M

u

^1M

ð x

_a

; t Þ þ k

^2B;2M

u

^2M

ð x

_a

; t Þ; ð59Þ where k

^2B;1M

and k

^2B;2M

are the coefficients given by the resolution of the system (58).

The electrostatic potential can then be written under the following form:

u x

_j

; t

¼ Q

^1M

ð z Þu

^1M

ð x

_a

; t Þ þ Q

^2M

ð z Þu

^2M

ð x

_a

; t Þ ; ð60Þ where

Q

^1M

ð Þ ¼ ð1 z

ðn

¹

Þ

²

Þ þ 1

2 n

¹

ðn

¹

þ 1Þk

^2B;1M

v

¹

þ 1

2 n

²

ðn

²

1Þk

^2B;1M

þ 1

2 n

²

ðn

²

þ 1Þk

^2B;1M

v

²

þ ð1

ðn

³

Þ

²

Þ þ 1

2 n

³

ðn

³

1Þk

^2B;1M

v

³

;

Q

^2M

ðzÞ ¼ 1 2 n

¹

ðn

¹

þ 1Þk

^2B;2M

v

¹

þ 1

2 n

²

ðn

²

1Þk

^2B;2M

þ ð1 ðn

²

Þ

²

Þ þ 1

2 n

²

ðn

²

þ 1Þk

^2B;2M

v

²

þ 1 2 n

³

ðn

³

1Þk

^2B;2M

v

³

: ð61Þ The mechanical generalized displacements remaining unknowns are u

_a

, w, c

⁰_a

. The electrical generalized un- knowns are u

^1M

; u

^2M

.

The solution of Eqs. (40)–(46), including Eqs. (20), (37), (60), in terms of the generalized unknowns given by Eq. (52), is searched under the form:

Table 1

Elastic, piezoelectric and dielectric properties of PZT4, PVDF and Epoxy

Moduli PZT4 PVDF Graphite–Epoxy

C1111(GPa) 139 238.24 134.86

C2222(GPa) 139 23.6 14.352

C3333(GPa) 115 10.64 14.352

C1122(GPa) 77.8 3.98 5.1563

C1133(GPa) 74.3 2.19 5.1563

C2233(GPa) 74.3 1.92 7.1329

C2323(GPa) 25.6 2.15 3.606

C1313(GPa) 25.6 4.4 5.654

C1212(GPa) 30.6 6.43 5.654

e31(C/m²) )5.2 )0.13 0

e32(C/m²) )5.2 )0.145 0

e33(C/m²) 15.1 )0.276 0

e24(C/m²) 12.7 )0.009 0

e11=e0 1475 12.5 3.5

e22=e0 1475 11.98 3.0

e33=e0 1300 11.98 3.0