Homogenization of heterogeneous piezoelectric medium

Mechanks~Commmk,

~[',~MmOlll

mio~,Voi. 24, No. I.pp. 75--S4, 1997 Prinu~din the USA. All ri$1m~ e d 00936413/97 $1%00 + .00

Pn soo~lS(~JOOSl-X

HOMOGENIZATION OF HETEROGENEOUS PIEZOELECTRIC MEDIUM

Castillero and R. R. Ramos Institute Cybernetics, Mathematics and Physics (ICIMAF) Acoustic Group, Calle D No. 353 e/t 15 y 17, Vedado, Havana, Cuba J. A. O t e r o , J. B.

(Received 4 October 1995; accepted for print 11 August 1996)

Introduction

In [8] the variational method called F-convergence was used to obtain the effective moduli of a piezoelectric composite with fine periodic structure. In [5,9] these results were extended to investigate the dynamical behaviour and the method of Bloch expansions wa.~ used. The double scale expansion method was applied in [2] to compute macrobehaviour in thermopiezoelectric solids. In the present paper by means of the asymptotic averaging method [1,6] the procedure of constructing the formal asymptotic solution of a typical boundary value problem of the linear piezoelectricity theory for an heterogeneous and periodic medium is developed. The homogenized equations and the effective coefficients was derived for a one dimensional example. The results are a generalization of those thal appear in [7, Chapt.4].

Analysis

We consider a linear static piezoelectric boundary value problem in terms of the ff displacement vector and the ~ electric potential in the domain ~ C R3 with the boundary F= 0~ , cf.[4}: [C~./ktuk.~ + e,,,,j~o,,,],~ + X, = O, = O.

(l)

Dinilr3 = 0.

(2)

(e..tu,~.t - ~.~..~),,

u(Iro = 0;

~lr, = ~o;

aijnjlr , = S~'; 75

76

J.A. OTERO,J.B. CASTILLEROand R.R. RAMOS O'lS = Ctjklgkl

--

em~s Era,

D, = e,mlgml -Jr e , m E m.

ek~ = 1/2(uk.I + ut,k),

Em = - P . m .

(3)

(1)

i,j,k,l,m=1,2,3.

a_, ~_, /~, L3, )(, ¢o, Sro and 77 are, respectively, the stress tensor, the strain tensor, the electric field vector, the electric displacement vector, the body force vector, the electric potential on F~ , the mechanical load on F1 and the outward unit normal vector. F=F0 U F1, Fo N FI=0, F=F2 W Fa, F2 Cl ra= 0 . The moduli: C,k~ (elastic), e,,,,~ (piezoelectric) and e,m (dielectric) satisfy the usual symmetry conditions C . ~ l = C ~ ; , = C . ; k = C~ikl, e . , . = e . . , ,

~,,,, =

~,,,,.

(1), (2), (3) and (4) are, respectively, the static equations, typical boundary conditions. constitutive equations ahd the geometrical relations.

Homogenization

Let the material functions Ci.7~t, e,,,j, e,m be Y-periodic functions. As usual, Y is the so called periodic cell, say Y = (0, I/1) x (0, Y2) x (0, Y3). We set Cqkt= Cokt((), e , m = % , - ( 0 and qm=ei,,(0. Here ( = ((,,~2,~¢a) is the so called local co-ordinate or the fast co-ordinate in which case £-= (Xl, x~, x3) is referred to as the global or slow co-ordinate: (= ~. and a = -~ is a small parameter, which represents the ratio of the characteristic lengths of the periodic cell 1 for Y, and L for the body as a whole, see [7, p. 101]. When the asymptotic solution for the problem ( 1), (2), (with periodic material functions) is sought in the form ,,,(~)

~(e)

= ~,o(~, () + ~,4(~,0 + ~,,,~(~, () +

....

= ~°(e, E) + o~'(e, () + ~:~:(e, E) +

As shown in [2], the functions u i0 and tp° do not depend on ( linearity of this problem it results that u~(aT,0 = Nij,(() 0--~ + (1)i,(() &p° Ozp Oz.' .

~'(e,O

au o

= M.,(~)T~

" + P,(()

. P

...

Moreover, due to the

HOMOGENIZATION OF HETEROGENEOUS MEDIA

77

This leads us to the assumption that in this case one can try to seek an asymptotic expansion of the solution in the following form

u,

=

(q)

-

~ aqtN}f2,...k,,(,~-')Vzk,...j,,,(:~) + ¢o,, ...k,,(,~)S.k,...k,,(-£')], q=O

¢ = ~ a'+[M~,{...k+(()V,,,+,J,,...,+,<,(£) + PI?!..,~,,,(OS,,h...,~,,(E)].

(5/

q=O

Here ;,~rtq), M (q), ~(q) and P(q) are the periodic local functions, independent of £ . These are Y-periodic functions and satisfying the following conditions:N! °) = +0 (the Kronecker deha), P<°)=l, M(°)= ¢__(°)=0 and N (q), M (q), ¢__(q), .P(q) are equal to zero for q < 0. \,\"e require from the periodic local functions that (N~")) = O,

(M(':')) = O,

(¢cq~) = O,

(p_c,,)) = O,

q > O.

/~l

And impose the no slip (or self adjoint) condition [[N~O)]I = 0,

[IMP")]] = 0,

[[¢__(")11= 0,

[[P~+)I] = 0

/:)

where ( f ) = ~ fy f d Y denotes the average of f over Y, IYI is the volume measure of "f . and [[u]] means the difference of values of the function "u" on opposite sides of Y . Consequently. /';,(£)= ( u , ( £ 0 ) is the averaged mechanical vector and 5"(£)= (v+(£. (-)} is the averaged electrical potential. We now substitute the expansions (5) into equations (1), (2) and we collect the terms multiplying c+q . After some manipulations, we obtain the ensuing boundary value pro-

}d,'ms:

aqrh(q) t O--~k,...k+V-,-,k,.-.~+J(~)

+

rtq) ,,ok,...~ S.~k,...k+~(~)] + X, = O,

q:O

~-'~ aq rt/q) (q) t ;j,,,,,~,+...k,, I/..,,.,,..,+,...,+,,+(~) - s ,,,,,+,...j,+S,,,,,+,...,+,,,(.~)]

= O.

(s)

q=0

" S k.,...<, k ( £ )]1,-o = o, Ea'+[N}.~2,...,~,,(Ovj,+,...,%,(e) + 0!+) ,+, ...k,,(~¢) q=0

~-"~aq [ M(q) . + p(+) ..k,...~+(()V.,~,...k,(x) ,,...~+(()S.k,..k+(~)]lr2

= ~o,

q=O :yo

~-" aqrh(q) I; (q) t O,,,;k,...k,,v,,.,nk,...k,,t x) + rm,jk,...kqS.mk,...k,,(E)]njlr, " ""

~0 = ,-,,

q=0

~" [t,.,,,,+, (+) ...,+, V, .,.,+,j,, ...,+, (.r.) - .s!~,+, ..~+S.mk,...~+(x)]n, lr3 -- O. q=0

(9)

78

J.A. OTERO, J.B. CASTII.I FRO and R.R. RAMOS

Where h_(q), t_(q), _r(q) and s_(q) are the constant tensorial functions or "constant functions". which vanish for q > 0. To find the functions V, and S , we seek for the solution of (8), (9) in the form of asymptotic series

I/~ =

oPw~ pl,

S = ~

p=O

aPy (p}

(10.)

p=O

Putting ( 1 0 ) i n (8), (9) we obtain the following sequence of recurrent boundary' vahw problems with constant coefficients: h(0) {p) ~_ _(o). {p} X,{p) = O, ijmnWn,m~ t m i l Y , m J "~ _ _

t(o), Ao).(v}+ y(p} = O, i . a , . ~{v} , . - ~,~v,,~i p=0,1,2

(p} + r miy~,m (°) • {P}~n -hi j°) n m w n,m ! JIt,

t(o) (v} _ s(o),,(v}/n, imlWm.I im~l,m !

(I 1 I

....

= s °{p}, r' 3 -~- qO(~}

p=O,l,'2

(12)

....

where P

.X.,(p}

. (r-q} r(q) , (P-q} ) ~.~[Itllm~kl...kq~m,nkl...kq ) JV , n . l y k ] ..kq~lJ~tkl...kqpJ ' q=l P y{p} X-'r~(q) , (p-q} o(q) , {p-q} : / ~[%jmkl...kq~3,mkl...kq, - - °imkl..kq~,mkl...kqz " q--I P uOlP} ~-"FN(q) WlV-q} ¢~(q) , (P-q) II = -- /'~[ ijkl...kq j,kl...kq -~ ikl...kq~,kl..mkqJlFo* q:l P ~30{p} ~'-~rhar(q) , {P-q} p(q) ~ {P-q} 11 = -- 2_~[~r~jkl...kq~j,kl...kq + * kl...kqY,kl ...kqJ F2~ q=l P solp} w"fL(q) wlp-q} r(q) , {p-q} i n = -- ~l#[l~iynmkl.m.kq n,mkl...k q "~- raIjkl...kq~l,rakl...kqJ jIF1, q=l P qO{p} = _ ~--~f£(q) w{p-q} o(q) , {p-q} I_ , =

X-'-'FL(q)

t Urakl...kq

:,mk~.-kq

-- °ljk~.-kq~jk~...kqJn:W3'

p > 0

P > 0

P ~ 0

P > 0

P > 0

P ~

0

q=!

X , {°} ~o{o) --

X,,

~o,

y { o ) _ O,

u °{°} - O,

so{o)__ So,

qO{O>_- O.

79

HOMOGENIZATION OF HETER%ENFiOUS MEDIA

the The solution of the chain problems (1 l), (12) completes the procedure of constructing formal asymptotic solution of problem (l), (2). Finally, we only need to find the so called “periodic local functions” and the “constant functions”.

Constant

TO obtain boundary

Functions

and The Periodic

Local Functions

N(p+‘), M(q+‘) and h(q), t(q) (p(q+l), p(q+*) and _r(q), s(q)), the following value problems,

designated

P/qt’lq) ( P$(lQt’*q))are solved, where (.),] s g.

(cijmko+zN~~~,!..k.+, Cik,+zml c

%zmk,tl

Ni:k,...k,

+

Nh'+') mnkl...kp+,/l +

+

ek,+,:~"!~~?k,+,

ek,t,ik,t&;...kq

ek,+zmk,t,

N(P) mnk,...k,

mnkl...k,tl/l -

ck,tzk,tl

=

-

Mb+‘)

ck,tzm

nk,...kqt,fm

M%...k,

(q)

qtznh..J+tl = (+%,t2mlN::;/..k,,,,, kk,tdM

+

(q+l) nkl...k,+,/l

+

hi;;+2nk,...kq+,r

=

-l,O,

ek,tzmk,t, +

+

t$2nk,...klt,

q =

tk

)/I +

emikq+zM~%q+,,m

Nbl+') ek,+zml

non

I,...

Nr?!rk,...k,) -

%+&tl q =

Mii;...kp) 0,1,2...

(13)

80

J.A. OTERO, J.B. CASTILLERO and R.R. RAMOS

(“npt.‘.t~+2,7L - e,,,r~!~:,2!,k,+3,r)/, + (E’k ’ qtz P(‘+‘) k, k,+,

%mkqt2 ‘$:!.kqt,

-

p(qt’) fk qtz” k, . ..kqtl /n pm

L

‘kq+2kqtl

q-

k,...k

ek

ek,+zml

N$zn,

M$,

hj$,,

and

t

wmkptl

(15) (I

To obtain

)/t +

@k+l) mh . ..k.+,/l

t$),

=

-l,O,

we solve the periodic

1,

problem

P/l”).

Then

b\

(13). (14) we consider:

(Cimn

Ci~pqN~~,,lq t e,,,kf,f$p)l) = o

+

('=knt eiPqNij,),,iq - ~ip~~~lp),i= o

h!?) = rjmn t!?,,

(C..,,mn =

(eimn

Analogously

a$_,, PA’) and r$i,

P/j”‘.

by (15), (16) we consider:

Then

C

$

‘1JPq

+

N

p%,q

eipqN~~,,,q

siz) could be calculated

+

eP*J”:,!,p)

-

e,M~,flp).

by means

(17)

(18)

of the periodic

problem

HOMOGENIZATION OF HETEROGENEOUS MEDIA

81

p)~ ,o)

p.

(~)

d~(x)

(eni 3 -~Lv,13pq__pn/q -~- e p u P ~ / p ) / 2 = 0 (Etn--~lpqTpn/qT

r(O) ,,,ij =(e,, 0 + ~

pO) nip +

p. ¢(~) \ ' ~ u p q pn/q/

"mJ"

= < , , . + - , , 'p(l) -/, •

(19)

niP]It =

tp

.

_

_ ~(x) ~

(20)

Equations (17), (19) represent respectively, a system of equations for finding ,NO),,~nand M ~ ), (I)~ and p O) taking cognizance of (6), (7). These problems are strong formulations of the local problems. In such a case the periodic functions have to be of class Cl(Y). Thi~ assumption may be significantly weakened provided that one passes to weak or variational formulations, cf[2,3]. Indeed, in the case of laminate composite with axis of symmetry in the direction normal to the layers, the periodic local functions N {q}, M (q), ¢__{q),P(q) and the material functions C u .... e. . . . e,, will only depend on one variable. For this kind of media we proved analytically, after considerable manipulations, that t{°}= E{°), see [9]. Consequently, the first problem (p=0) of the recurrent sequence of boundary value problems (11). (12) represents a typical boundary value problem of linear piezoelectricity for an homogeneous medium and has the form Chij . . . . w(O). , : + e mUY,m3 h . (o) + X i = O, e h w(O) imt

w!°)lro = O,

y(O)lr: = ~(o),

h

(o)

m.h -- f,mY,ml : O,

D~n, lr~ =

a ~ n , lr, = S o ,

O.

(21) ('22)

where a,hj = Cu,~mw h (o) ~ h (o) .... (x) + emuy,m (X), eijra l"U3.m~, ) -

C~

ran

= t,!0)

"'zjrn.n~

h

(o)

_(o)

em.13 ~ trot 2 ~ rrat 3,

h

s(o)

f't'm. ~ "lrn~

and it is called the averaged zeroth-order problem[l, p. 22]. The effective coefficients: ch.,,, (elastic), e,, uh (piezoelectric) and e h (dielectric) are given by (18) and (20). The above general expressions are the same that appear in [2,5,8,9]. The procedure of constructing the formal asymptotic solution of linear static piezoelectric equations with typical boundary conditions for an heterogeneous and periodic mediurn is developed by means of the Asymptotic Averaging Method. The original boundary value' problem with variable coefficients is transformed in a recurrent sequence of boundary

82

J.A. OTERO, J.B. CASTILLERO and R.R. RAMOS

value problems with constant, coefficients. Actually, this asymptotic analysis leads to the solution of two recurrent sequence of problems. The first of these problems (problem B(p), p=0,1 .... ) consists in the solution to multiple boundary value problems (I1), (12). For solving the problem B(p) it is necessary to solve the problems B(r), r=0,1 ..... p-l. The solution to each one of these problems permits to find the functions w{p/ aim ~/0'~. Then by (10) it is possible to determine the averaged functions V~ and S. The solution it) the original problem (1), (2) is achieved on the basis of V, and S by (5), however, local periodical functions __N(q), M'(q),~(q),P (ql are included in these formulas, whose search leads to the solution to the second recurrent sequence of problems which is made up of pJq+~.ql (eqs. (13), (14)), p(q+ , H 1,q) (eqs. (15), (16)). For a fixed value of q, eqs. (13), ( 1.5 ) represent respectively, a system for finding N (q+l) and M / q + l ) , ( I )(q+l) and p(q+l) taking into account (6), (7). After that, by using (14), (16) the constant tensorial functions h_(q),t(q),r(q)and ,¢(q) are obtained.

Example

Let C(~), e({) and e({) be infinitely differentiable 1-periodic functions of~t E R satisfying the inequalities 0 < Cl <_ C({) _< C2, 0 < el _< e({) _< e~, 0 < (1 --< t'(~) ~ t 2 and X(x) be an infinitely differentiable function determined on a segment [0,1]. Here Cl, (7~, el. e ~, el. e2 are constants. We obtain the averaged equations and effective coefficients of the following problem using the above results

d,,

~[==o = O,

du

(c(~)~

+ ~(~)

~

)1~=, =.-';°.

du

_

~(~)~]

= 0

,Pl==o =

:o,

d~

(e(~)~ - ~(~)~)1~=, = o

(23)

(2.t)

where (--~, _ x c~ is a small parameter (a = ¼, n an integer). Here the expansions (5) take the form ,×)

" = E '~"( ,'v,,(~)v~ " ,o (~) + l,,,(~).S'(-)(~.)), ~l=O

= ~ o'~(M,,(()V(")(x)+ P,,(()S(~)(x)). n=O

(~sb

HOMOGI~rlZATION OF HETEROGENEOUS MEDIA

83

"['he periodic local functions are 1-periodic and satisfy that .No(() = 1, Po(() = 1, ~o({) = O. M0(() = 0 for ( 6 (0, 1), N.(O)=M.(O)=O, ¢.(O)=P,,(O)=O and (N.) = O, (M.) = O, (0,,) = O, (P.) = 0 for n/.O, with

(f(x, ())=fg f(x, ()d(. By substituting (25) into the equation (23) and the boundary conditions we have that

~

.

d"+~V(x)

[h. ~

d"+~S(x)

+~. d~.+-------71+X(x)=O,

nmO

f

a.[td~+2V(x) dx "+2

d"+2S(z) s. dx.+----------T]= 0

(7($)

nmO

•

¢

~o~"[N.(()~+

d"S(z)

.(()~]l==o =0, d n S( X )

+ Po(()

ll=:o

= ~o

nmO

_.. d"+~V(z)

cl tun

~xx--~

+rn

d"+'S(x)

11::,

~

= s °,

n----0

f a.[td"+lV(z) d"+lS(x)l, dx,~+l + s,, "~x~-~ JI~=L = 0 .

(27)

nmO

It can be proved that the constant functions h., r . , t. and s~ are equal to zero for n>0. Therefore, the averaged problem of infinite order of accuracy in a has the simplest form

chdlV(x) + e~d2S(x) + X(x) = O, dz2 dz2 e hd~V(z) ehd~S(z) - 0 dz2 dx2 Vl:=o = O,

+ eh dSd--@)l==, = S° ' (Ch dV(Z) d-~

(eh -dV~x - z)

-

Siz= 0 :

?a~}x~)l==,

dx

(28)

~0 =

o.

The homogenized (effective) moduli, n=O, are given by

ho = ( ~ ) / A tO = ro =

(C<-T-~)/A

- Ch, =

So = ( C--7--$--~le~)/A -

e~, ~h

where

(C(~

(~.~

(30)

84

J.A. OTERO, J.B. CASTILLERO and R.R. RAMOS

References

N.S. Bakhalov and G.P. Panasenko, Homogenization: Averaging Process in Periodic Media, Kluwer, Dordrecht (1989) .

A. Galka, J.J. Telega and R. Wojnar, Homogenization and therrnopiezoelectricity, Mech. Res. Comm, 19,315 (1992)

.

A. Galka, J.J. Telega and R. Wojna,r, Thermodiffusion in heterogeneous elastic solids and homogenization. Prace IPPT (IFTR Reports) 14 (1993) G.A. Maugin, Continuum Mechanics of Electromagnetic Solids, North-Holland. Amsterdam (1988)

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G.A. Maugin and N. Turbe, Homogenization of piezoelectric composites via Block expansions, Int. J. Appl. Electmag. in Mat, 2, 135 (1991)

.

O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam (1992)

7.

B.E. Pobedria, Mechanics of Composite Materials, Moscow University Press. Moscow (1984) (in Russian)

8.

.J.J. Telega, Piezoelectricity and homogenization. Application to biomechanics, in: Continuum Models and Discrete Systems, ed. by G.A.Maugin, vol.2,220 Longman, Essex (1991)

. N. Turbe, G.A.Maugin, On the linear piezoelectricity of composite materials, Math. Meth. in the Appl. Sci., 14,403, (1991)