Pergamon Mechanics Research Communications, Vol. 27, No. 2, pp. 191-195. 2000

Copyright 0 2000 Elsevi~ Science Ltd Printed in the USA. All rights reserved

0093-641 ~ front matter

PII: 80093.6413(00)0~J61.1

M O D E L L I N G O F T H R E E - L A Y E R E D P I E Z O E L E C T R I C P L A T E S

Jean-Marie NIANGA Laburatoire d'6tude des Structures, Hautes Etudes Industrielles (H.E.I), 13 rue de Toul, 59046 Lille Cedex, France. E-mail: Jean-Marie.Nianga~hei.fr

(Received 10 October 1999: accepted for print 10 January 2000)

l a t r o d u e t i e n

Starting from the comparison of the three-layered plate models in elasticity with the theory of piezoelectric thin plates, a modelling of piezoelectric layered plates is presented in this paper. But beyond an elementmy formalism, an identification criterion between the simple and the three-layerod structures is formulated. And so, it becomes natural to set limits to the future applications of piezoelectric layered structures. A piezoelectric three-layered plate is a composite structure that consists of two thin circular piezoelectric thin plates, here and there stuck to a metallic thin structure of the same shape, but generally with higher lateral dimensions and clamped periphery.The outer faces of the two piezoelectric plates bear electrodes. In reality, these faces are compietely covered with metal and as a matter of fact, constitute elcgtmdes(seefigl).On the other hand,when subjected to a potential difference between its two faces, the three-layered plate sustains a bending deformation, except on its medium surface .We note too, that the th,-ee-layered plate is a primordial element in the conception of many acoustical wave detectors used in seismology or in petroleum industries. TI~ theory expressed in this paper refers to the works of Mangin and Attou [1,21 in electroelnsticity and tO that of Destuynder [3,41 on the three-layered plates in elasticity. This paper is structured as follows:In Section 2 , are presented the equations of piezoelectric three-layered plate in which three-layered plates with membrane-effects on the one hand and the bending equations of three-layered plates on the other hand are analysed. In Section 3 , models of piezoelectric structures with a single layer are deduced from the previous paragraph.

191

192 J.-M. NIANGA

Embedding support

Piezoelectric plates Metalfic part

FIG I: A piezoelectric three-layered plate

gauations of Piezoelectric Three-Layered Plates. and the position Let E. v, h and Z be the Young's modulus, the Poissun's coefficient, the thickness of the metallic part with respect to the medium surface of the plate;for any constant function C(x3) defined on the metallic part, we have:

fZ+hl2 fZ-h/2 C(X3 ) dx3 -- h C (x3)' dZ-h/2 , , , c(,,3) dx3 = h.Z (,,3) JZ-h/2

fZ Z+h/2 2 72 -h/2 x3 C(x3) dx3 = ( 12hZ2 +h3)"

If, on grounds of expediency, an asymmetrical disposition of the material with respect to its medium surface 6o can be assumed, the general model of thin plates in linear piezoelectricity expressed in the

piezoelectric problem(see [1,2]) , consists now of finding in the space (H~(6o))2xH2(6o) a

displacement field (~(o), u(o)~ and in the space H~(o)), a electric potential ~b (°) such that :

V ~ = (V~, V~) ' dHo(,O)),

E h + L ,_-7 {o- ,.) ,... (~,o,),... ~1+ ,,,,, ,,(~,o,),,,,,,/~l}

-L EhZ (_~o,o.~o) ~o(l-:)

_- j' (~ (,,,,,,,)+ ~(,,,.x,)) ~.

(P1)

PIEZOELECTRIC PLATES 193

VV, eH2o(o~) ZEh _ f.(~__#5 a. ~o).~ + f . L , x ; c , ~.. u~o, o..~

-Lj',÷~,x, c . , ,,(~(°') o.,v, = j ' ( (~ (.,.,~) + ~-(x,,x,))+ oo(~ + y.)) v,.

P P Z - h / 2 ~ . , ' (0)'~ e el j.j, ~,,(A,4, )q~+jjz÷h,2~,, (A2¢(°))q,=j ". <W>q~ds

I~mwk 1 If the medium surface of the material coincides with that of its metallic part, then Z vanhtha and mnaequently in the previous equations and in the case of weak anisotropy, the bending model of three- layered plates uncouples from that of membrane effects as follows :

a) Mode l of Three-laFered Plates with MemOrane Effects

We have the problem:

VV=(VI, V2)E(H~(o))2,find ~(o) such thai:

L f i .c . ,~ (~,o,) ,~ (~)+ LL c . , ~ (~,o,),. (~) (,,,~

Eh vy,,,, (~ (°)) y,~, (V)} + L ~ {(i- ~), ~, (~,o,), o, (~)+

:I(r:-~.)~

b) Bending Eauations of Three-Lowred Plates

We have the problem:

VV; ~Ho2(O),find U~ °) such that:

j'J__lw2..2C ' ..~ ,,(°) 0 ~ + f . Eh 3 ~ 3 " ~ " 3 ~3 J. 12('~_'v)2 Au~ °)AV3

+I. f.' ~ c.~ o,,u~ °~ o.~,

: ~® (T~+3 (x,,x2) +'1"3- (x,,x2) + 0~ (T: (x,, x2)+ T: (x,,x2)) ) V,

(r3)

194 J.-M. NIANGA

For each of these models, we can add the following electric contribution :

V~O ~{9 ' ~ H'(co), ~, ~ } *(°) ~H'(co)such t -~n = 0 on y , find that:

/o,--_, ,o,, J-1 •11{ A2• ) ~t/ "l- L ~/2 "1 '

It is interesting to observe that, from the previous equations, we can easily deduce the corresponding equations of piezoelectric thin plates with a single layer as follows :

Equ#tions of Thin Elastic Plates.

a) Model of Thin Plates with Membrane Effects

Problem (P2) reduces to:

VV=(V,,V2) ~(Ho(c~) ) , 2,find ~(o) in (HIo(~)) 2 such that:

b) Bending Eauations Off Thin Plates

Problem (P3) reduces to :

t V EHo2(go),find u~ °) in H2o((o) such that:

C~s 0,o u~O) V3 = f. {(T3+ + T3_) + 0~ (T: + T:)} i/3

The electric aspect is expressed as follows:

~'q~ ~ ¢ ~Hl(co), ~,, ~ = 0 sur~, , find 1 ~b (°) ~H(~) such that:

(P4)

Remark 2 The local formulation of the bending model of three-layered plates shows that u~ °) verifies:

PIEZOELECTRIC PLATES 195

8 - h 3 E h 3 - ~ c*' (°*' u~°')~ ~20- v) A, ~o)

=(~ (x,,,O + ~(,,,,,~} + ~° (~(,,,x,) + ~(,,,x~))) 4 ) =o }

on Z i8 - hS Eh3 Au~ °) = 0 l"~-~ c'~°r#Or6 u~°)n~n~+ ~ 12(1- V ~)

(rs)

And if the bending model of thin plates in its local formulation is recapitulated (see [1,2] ),

2 u~O) _ ÷ ~ C ~ a ~ = T;(x,x2) + T; (x,x2) + ~= (T: (xtx2) + T~(x,x2)) on ¢0.

{ C*~'~Or6u~°) on r nanp = 0

u3=O

(e6)

is obtained. So, it can be observed that the structures with a single layer bend as the three-layered plates under the following condition:

/ 8 - h s C "~ EhS 8~e ~ - 2 C ~ / a ~ u~°) ]2 ~ x2~:v 2) =o

Consequently, the three-layered plates can be considered as a composite material which integrates in its formulation the physical characteristics of its components.

Conclul~ In this particular analysis, it follows that the three-dimensional solution of the model of three-layered piezoelectric plates, can be entirely described by the solution of a problem set on the medium surface, On the other hand, the three-layered plates models introduced here, naturally take the shear stress into

account, insofar as they play an essential role in the delaminatm" n of these structures. The interest of this analysis is that it doesn't require a priori assumptions .Modestly, this paper gives a new pmnflbility in the study and analysis of parameters that ~ c i p a t e in the conception of piezoelectric multilayered plates such as acoustical wave detectors.

[1] MlmglLG.A. and Attou.D. An asymptotic theot T of piezoelectric plates .Q.JL Mech., Appi. Math., Vol 43, Pt 3, p.347-362 (1990).

[2] Maugin.G.A. and Attan.D. Une thb~rie asymptotique des plaques minces pi~,o61eetriques, C.R. Acad. Sol.Paris. t.304, s~rie II n°15, p.865-868 (1987).

[3] Destuynder. Ph., Th~se d'6tat, Uuiversit~ Pet Marie Curie, Paris (1980)

[4] De~ynderAPh., Une thc~,orie asymptotique des plaques minces en 61asticit~ lim~aire Collection 1LM.A.2, Masson, Paris, (1988).