Homogenization and thermopiezoelectricity

~t 2

are

a2u °

h -- C

lJkl

~Xj

k aX

c9 Ah 820 h 8--6 = l j -a x - ax i

8 uk 8x ~x i

r

o - 32~ CgX c~x

h eli

J

1

+

82o

h + gklJ

h ~lJ

aX J OXk

~0 h a--X- -

J + bi

'

(15)

,

j

2 0

h glkJ

(14)

Y

i

+

k h

~0 h

-0

.

Iax

J

i

Hence we conclude that u°(x,t), ~°(x,t) and are, respectively, displacement

field,

the electric

potential

and

the entropy of

(homogenized) theormopiezoeletric body while 8h(x,t)

the

the effective

is its temperature field.

The following relation holds true

°h(x't

We

observe

=

that

s°(x'y't

-

in general

Ic

8u ° 8u I k + ay, ~ ) +

s o depends

not

~k (

only

8~ ° + a~i

on

(x,t)

)

but

(16)

also

on

the

microscopic variable y.

The

boundary

ones.

Also

conditions

the

initial

for

the

homogenized

conditions

body

for u ° and o

are

still

remain

the

homogeneous

unchanged,

cf.(11).

A. GALKA,

320

Surprisingly,

initial

the

J.J.

TELEGA

condition

and R. W O J N A R

Oh c h a n g e s

for

and

is

given

in

the

next

section. Let

us pass

linearity

now t o

of the

ul(x,y,t)

= X~

k

the

formulae

problem

mn)

(y)e

for

studied

mn

the

effective

material

moduli.

Due t o

the

we h a v e

[ u°(x,t)

¢¢m)

] +

k

o~O(x,t)

(y)

OX

+ Fk(Y)Oh(x,t)

,

m

(17) 1

~ (x,y,t)

where

the f u n c t i o n

Let

us

are

functions

assume

materials.

H

per

X

that of

[ u°(x,t) ] + R(m)Cy)

(y)e

(mn)

~(m)

, _

the

class

,

F,

periodic Lm(Y).

~(mn) ,

Ox

R(m)

material

+ Q ( y ) 8 h ix, t)

a~°°(x't)

mn

and

m

Q are

functions

c

Such a case

covers

equal

on o p p o s i t e

'

Y-periodic.

ijkl

layered

(y),

gilk(Y)

'

etc.

'

thermopiezoelectric

We set

(Y) = {v e H I ( y ) Iv t a k e s

Hper ( y , ~ 3 )

The

~(mn)

=

unknown

following

values

= {v = (vl)lv I • Hper(Y),

i = 1,2,3}

periodic

into

local

functions

entering

sides

o f Y} ,

(18)

.

Eqs.(17)

(19)

are

solutions

to

the

problems.

P r o b l e m p1

loc

Find

(mn)

~ H

(y, N3)

and

~(mn)

e H

per

per

(mn) y

[Cijmn(Y)

+ C

lJkl

(y)eY

zl

-

(Y)

such that

@~(mn)

) + gk (y) lJ

- -

~Yk

]e ~ ( v ) d y = 0 ' J V v • H

J" [gl~,.,_ ( y ) y

+ gikl(Y)eY

1

(¢mn))

_ ~lk(y

)

Be( mn ) BY k

(y,~3)

,

] OW d y = 0 , ¥ w ~ H (Y) BY 1 per

.

per

HOMOGENIZATION

AND THERMOPIEZOELECTRICITY

321

Problem pZ loc Find

@(m)

e H

-

per

(Y,R 3)

and

R (m) e H

per

(Y)

such that

8R (m ) Yf [gmlj (y) + gklj (y) - - a yk + C! Jkl (Y)e~l (-@(m)) ]e lyj (v)dy = 0 , V v e H

f [elm(Y) y

+ e

8R (m) - ay - k -

ik

glkl

e y (@(m))] kl

aw

-

~y---~

dy = 0

(Y,R 3)

per

V w e H

'

,

(Y) . per

Problem p3

loc

Find

F e H

per

-

(y,~3)

and

Q e H

per

(Y)

such

that

f [~i (y) - c (y)e y (r) (y) 3Q ]e y (v)dy = 0 y J iJkl kl - - gklJ 3y---RlJ '

V ve

f [Ai(y ) + glkl(Y)e:l(F ) _ el OQ ] O w dy = 0 , y kay k 3y I

VweH

where

e y (v) = I

~j

2

(

av

av

t - - + ayj

j _ _ @Yl

Rigorously,

the

two-scale

formulation

of

the

involved

local

in them have

the point

of departure

Then the local problems

)

analysis

problems.

to be more

leads

directly

Consequently,

regular,

may be the weak

at

the

least

(Y)

per

(variational)

to

the

material

of class

,

strong

functions

CI(Y).

However,

form of the system

(9).

are as above.

one can determine

the homogenized

They are given by

~ (kl) h

CiJkl

--{Y,~3),

per

.

asymptotic

Having solved the local problems moduli.

H

c~ ( k l )

m

ljkl

lJmn

c3y n

gmlJ

Oy m

(effective)

322

A. GALKA,

J.J.

TELEGA

and R. W O J N A R

a~( k ) h

aR {k)

m ----

gkil

+


C

-

+

llmn @Yn

-

>

gmlJ @Ym

aF

h

k

~IJ

= <~|J

- ClJkl

aQ --

Oy I

gkiJ

>

Oy k

(20) aO

h =


+

Ki i

K

lJ

h

=

<•

>

Oy k

'

a_ # _k(m)

-

Elm

j

-;k

im

glkl

+

aR(m) - -

•

Oy I

ik

>

Oy k

aF

~h

=


i

Here 8- = ( O j)

® •

find

-

f [K y

i

e

aQ

H

(Y,~)

+

glkl

k

Ik ay k

is a solution

ay I

to the following

problem

such that

per

a® av (y) k ] k dy = 0 , Oy I Oy~

(y) - K Ik

Moreover

-

II

V ve

H

per

(Y, [R3 )

(21

we have

h

1



= IYI yI s ° ( x , y , t ) d y _

=

_

~:J

#heh

U0 i,l

(x,t)

+

(x,t)

-

~h

a~O(x,t)

l

ax i

(22 aF k

Chan~e

in the

Surprising the

Initial

phenomenon

temperature

prove

that,

Condition

of

the

in general,

~

~Q

>

for the Temperature

is observed effective oh(x,O)

+

in what solid,

~ T(x),

concerns cf.also

x e H.

(23

of the H o m o K e n i z e d

the

initial

[2,4].

Namely,

Body

condition we

shall

for now

H O M O G E N I Z A T I O N AND T H E R M O P I E Z O E L E C T R I C I T Y

323

For e > 0 the initial entropy is expressed by

c

sC(x'O) = ~IJ

(x)

OU i #c AC(x) a-~-- + (x)T(x) - I J

aF ax

(24) i

The d e r i v a t i o n of the initial c o n d i t i o n for e h relies on [ 3 ,

Lemma

1.

Let f e Lm(Y)

be a Y-periodic

where ~ is a bounded domain.

Consequently,

(x)

=

f(

ex )' x • ~,

Then

l I f(x)dy ) = ]-Y]-y

fc

function and fe

p. 21].

weak - " in Lm(fl) .

,

if g e L2(fl) then feg converges to g weakly in L2(~).

Hence we conclude that

au

lim se(x'O) = <~lJ > c ~0

On the other hand Eq.(22)

-

i

(x)

-a x

+ <#>T(x)



-

OF(x) Ox

j

(25)

l

implies

h = ~ i ]

aU (x) i ~X J

+

~

(

)

_

~heh'x'O"

Ah OF(x) I ~X

(26)

i

Because

lim se(x,O) = e ~0

,

(27)

therefore au T(x)

eh(x,O)

+ (<~l]

> - ~ih )

J

8h(x,O)

~ T(x),

i - (<~ > - kh ) OF i i ax j

= ,e h

Thus

~

in

general.

i

(28)

324

A. GALKA, J.J. TELEGA and R. WOJNAR

The change the r.h.s,

in the thermal

boundary condition

of (25) and (26) have to be equal.

of the homogenized body is well defined. is determined

by

(28).

h are different Yij

from

More

precisely,

<8>,

and

results

the homogenized

its initial temperature quantities

respectively.

responsible for the change in the thermal boundary condition, initia]

conditions

(II) I,

observed for the homogeneous

(11) 3

and

(11) 4

that

In such a way the initial entropy

Consequently,

<~i j>,

from the fact

are

initial conditions,

Just

~h

this

k~ and i fact is

provided that the

inhomogeneous.

No

change

is

because T=O, U=O and F=O imply

eh(x,O)=O.

References

1.

H. A t t o u c h , Variational Convergence for Functions and Operators, Pitman, London (1984) 2. S. B r a h i m - O t s m a n e , G.A. F r a n c f o r t and F. M u r a t , Homogenization in thermoelastieity, in: Random M e d i a a n d C o m p o s i t e s , e d . b y R.V. Kohn a n d G.W. M i l t o n , p. 13, SIAM, P h i l a d e l p h i a (1988) 3. B. D a c o r o g n a , Direct Methods in the Calculus of Variations, Springer Verlag, Berlin (1989) 4. G.A. F r a n c f o r t , Two v a r i a t i o n a l problems in thermoelasticity, Ph.D. Thesis, Stanford University, April ( 1 9 8 2 ) 5. G.A. Maugin, Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam (1988) 6. W. Nowacki, Electromagnetic Effects in Deformable Solids, (in Polish), PWN, Warszawa (1989) 7. J.F. Nye, Physical Properties of Crystal, The Clarendon Press, Oxford

8. 9.

i0.

(1957) E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer Verlag, Berlin (1980) 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, C.A. Maugin, On the linear piezoelectricity of composite materials, Math. Meth. in the Appl. Sci., 14, p.403, (1991)