Non Linear Constitutive Relations and Homogenization

G.M. de La Penha, L.A. Medeiros (eds.) Contenporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations Worth-Holland Publishing Company (1978)

NON LINEAR CONSTITUTIVE RELATIONS AND HOMOGENIZATION

L. TARTAR University of Paris

-

Sud,

France

1. Introduction. - A few years ago there were two classes of methods to

handle non linear partial differential equations arising i n Continuum Mechanics o r Physics: the first one used a compactness argument and started with the w o r k of Leray, the second one a monotonicity or convexity argument initiated by Minty and Zarantonello and developped by BrGzis, Browder, Lions and others (cf. Lions Ell).

Although Leray's work was motivated

by Mechanics (cf. Leray [l])

the origins of the other method

were numerous but its applications to Mechanics were soon recognized and investigated (cf. Duvaut-Lions [l]).

It was

known, but not really emphasized, that these methods were inadequate for most

practical problems.

After a few years of darkness light came with the work of Ball [l]; adding ideas from homogenization gave rise to a new concept: compensated compactness (cf. Murat El], [l])

Tartar

and then t o the principles of the method presented here

(cf. Tartar [ a ] ) . Weak convergence plays an essential role and will be related to microscopic and macroscopic properties.

This

method seems perfectly adequate f o r non linear partial differential equations coming f r o m Mechanics and Physics (it was

NON LINEAR CONSTITUTIVE RELATIONS A N D HOMOGENIZATION

developed

473

with this purpose) but the approach is philosophic-

ally quite different from the classical one (except for Statistical Mechanics and Quantum Mechanics perhaps). Although the method could be used for stationary

or

evolutionary equations, it cannot yet be used for nonlinear hyperbolic systems: indeed one has first to know what a solution is, the concept o f entropy (which is not completely understood) being a n important restriction added to the equations.

We will avoid this problem here and accept all

solutions (in the sense of distributions) of the equations that we will consider. We will work on partial differential equations and not o n boundary value problems, but the method can o f course be

used to treat non linear boundary conditions.

2. Microscopic-macroscopic properties and weak convergence.

Partial differential equations are used as a mathematical model to describe the values o f physical quantities and

their adequacy is tested through prediction

o f the output of an experience knowing

the input.

Although there is no mathematical difficulty to work with functions there is a physical impossibility t o know everywhere its values (except for analytic cases); measurements o r identifications of some type will be used giving a finite number of parameters and from them an approximation of the function will be derived.

There is no a priori reason

that this approximate function satisfy any equation.

Of

course a large number of measurements may give a better know-

474

L.

TARTAR

ledge of the function but one is usually forced to work above some length (or time) scale in order to avoid perturbing the phenomenon under study (cf. Heisenberg principle of incertitude in Quantum Mechanics). The information one can obtain by measurements is called macroscopic, but the equations are usually valid for a physical quantity which is microscopic.

There is an implicit

belief that, if the phenomenon has been well analysed, all the relevant quantities appear in the equations and the macroscopic quantities will satisfy the same equations as the microscopic ones.

[It is my belief that some classical

equations have not been derived with sufficient care]. T h e numerical analysis approach is similar: to compute the solution of a n equation an approximation, depending o n a finite number of parameters, will be derived;

by increasing

the number of parameters it is hoped that this approximation converges(in

a weak o r strong sense) to the solution.

If the

macroscopic quantities were satisfying a different equation than the microscopic ones the limit of numerical approximations will usually satisfy the macroscopic equations (this fact will in this case depend upon the numerical method used).

Weak convergence seems to be the appropriate mathmatical tool to handle the above situation.

As,

o n bounded sets of

functions, the weak topology can usually be defined by a metric we will interpret the above analysis by saying that if two functions are nearby for the weak topology they are almost undiscernable through measurements.

NON LINEAR C O N S T I T U T I V E R E L A T I O N S A N D H O M O G E N I Z A T I O N

jt7

5

A s a n example c o n s i d e r a measurement o f t h e e l e c t r i c field

u

E(x)

i n a conductor.

is the electrostatic potential.

x0

w i l l c o n s i s t i n measuring

example

u(xo)

and

u(xo+h),

Suppose t h a t

u(x)

if

0.

E

then

x

C

ul(:)

u(x)

-

h

0’

for

E(xo) the

and t h a t t h e

where 1.

u

is

Within t h e

w i l l be i d e n t i f i e d with which i s a l m o s t

duo du1 x - dx (F); dx compared

to

-

uo(x)

-

the difference

du1 x - z(z)

- b u t i t s mean dx

0

i t is quite f a r f r o m i n a weak t o p o l o g y

X

i n any s t r o n g

0

(if

cp

T ( ~ ) c o n v e r g e s weakly t o t h e mean v a l u e o f to

near

T h i s d i f f e r e n c e i s a l m o s t i m p o s s i b l e t o measure

topology b u t n e a r

goes

is

i s t o o small;

6

= uo(x) +

11

important v a r i a t i o n s

value i s

a t two p o i n t s n e a r

and t h u s d e r i v e f o r

uo ( x o ) - u o ( x o + h )

with

But t h e e x a c t has

E

i s periodic with period

u1

a c c u r a c y o f measurement

E(x)

where

A measurement of

i s s m a l l compared t o

C

exact p o t e n t i a l i s

and

(x)

u(x,)-u(x,+h) h

approximation

smooth and

u

- du d x

E(x) =

We h a v e

i s periodic

cp

when

G

0).

I t seems t h a t a l l measurements a r e done t h r o u g h a v e r a g e s of p h y s i c a l q u a n t i t i e s ;

from t h e s e measurements

i d e n t i f i c a t i o n o f p h y s i c a l parameters

can b e o b t a i n e d :

other

n o t i o n s of c o n v e r g e n c e a r e r e l a t e d t o t h i s f a c t as i n homogenization

(cf. Tartar

11 )

.

The r e a d e r w i l l h a v e remarked t h a t

t h e above

c o n s i d e r a t i o n s a r e more p h i l o s o p h i c a l t h a n m a t h e m a t i c a l : u s i n g a compactness r e s u l t weak c o n v e r g e n c e i n some s p a c e may i m p l y s t r o n g c o n v e r g e n c e i n an o t h e r ;

as t h e above a n a l y s i s

d e p e n d s on t h e s p a c e u s e d i t h a s no i n t r i n s i c v a l u e arid e a c h

47 6

L.

TARTAR

m a t h e m a t i c i a n w i l l want t o c h o o s e i t s p r e f e r r e d s p a c e .

But

e q u a t i o n s w i t h d i s c o n t i n u o u s c o e f f i c i e n t s and h o m o g e n i z a t i o n show t h a t t h e r e a r e w e l l d e f i n e d s p a c e s a s s o c i a t e d t o a g i v e n p a r t i a l d i f f e r e n t i a l e q u a t i o n coming f r o m Mechanics or Physics;

o u r a n a l y s i s r e l i e s on t h i s f a c t .

3 . A p p l i c a t i o n s t-.o. __. n o.____ nlinear e l a s t i c i t y . The p a r t i a l d i f f e r e n t i a l e q u a t i o n s g o v e r n i n g t h e motion of a n e l a s t i c body a r e ( c f . G u r t i n [ 11 )

(1) (2)

bo(x,t) = (det F ( x , t ) ) b ( x , t , r ( x , t ) )

(3)

Fij

(4)

S

(5)

det F 7 0

--

i n general

a ri

ax. J

= :(F)

A

The f u n c t i o n a l

S

(6)

satisfies

;(F)

(7)

F~ = F

G(QF) = Q$(F)

for a l l for a l l

F

F

and

Q E Orth'

Of c o u r s e t h e r e a r e u s u a l l y some b o u n d a r y c o n d i t i o n s . Remark 1: I f

one adds t h e s t r o n g e l l i p t i c i t y c o n d i t i o n on t h e

e l a s t i c i t y tensor

A(F)

= DS(F):

( a @ b ) * A ( F ) ( a0 b ) > 0

for

a 0 b

f 0,

t h e above s y s t e m becomes h y p e r b o l i c a n d , by a n a l o g y w i t h other

more or l e s s u n d e r s t o o d s i t u a t i o n s , o n l y p a r t i c u l a r s o l u t i o n s

477

NON L I N E A R C O N S T I T U T I V E R E L A T I O N S AND I I O M O G E N I Z A T I O N

of this system are believed to be physical: some kind of

inequalities, called entropy conditions, are added;

at the

moment only a few examples are understood and the above system seems out of reach. Quite naturally one is

led to consider the stationary

equations, where functions only depend on stable

x.

stationary solutions will be observed;

Of course only but stability

involves the complete system so we prefer to forget about this point. A more curious point is that, when dealing with

stationary solutions, nobody

thinks of restricting the class

of solutions with entropy conditions as if they were automatically satisfied f o r stationnary discontinuities;

to be

sure o f this point one should know what these entropy inequalities are and this is not the case, but as for some simple hyperbolic systems the analog is false we have to be careful.

In order to avoid this question we add the follow-

ing Postulate: all discontinuous solutions (F or S, not of the stationary

r)

system of elasticity are accepted.

F r o m this and the philosophical approach of paragraph

2 we will derive a necessary condition on the function

A

S;

it is an interesting fact that this condition implies that stationary discontinuities (along a smooth surface) satisfy the entropy inequalities whatever they are (because they cannot rule out discontinuities in the linear case).

Of

course this does not prove (8).

If we do not want to accept (8) we may as well work

47 8

L.

TARTAR

d i r e c t l y w i t h t h e c o m p l e t e e v o l u t i o n problem.

Presumably we

have t o do s o f o r e l a s t i c f l u i d s . Our p h i l o s o p h i c a l a p p r o a c h t e l l s u s t h a t a weak l i m i t

(1) t o ( 5 ) i s a l s o a s o l u t i o n .

of stationnary solutions of

T o e x p r e s s t h a t we f i r s t h a v e t o p r e c i s e what weak t o p o l o g y

we u s e . I n l i n e a r e l a s t i c t h e o r y , u s i n g v a r i a t i o n a l methods and d i s c o n t i n u o u s c o e f f i c i e n t s , we know t h a t a n a t u r a l s p a c e is

~ ‘ ( n ) , sij

F.. E 1J

E L2(61);

by u s i n g m o r e s o p h i s t i c a t e d

r e s u l t s one c a n f i n d t h a t t h e s o l u t i o n s a t i s f i e s a n e s t i m a t e

E LP(n)

Fij,Sij

(2 s

p

f o r some

p 5:

a t the best

+a);

(when

c o e f f i c i e n t s a r e d i s c o n t i n u o u s ) we may e x p e c t a l l f u n c t i o n s

F . ., I J

S

t o b e bounded.

,

i J

We a r e l e d t o s a y t h a t ( a s s u m i n g ( 8 ) ) i f

i s a s e q u e n c e of weak

*

s o l u t i o n s of then

(r,F,S)

to

(rn,Fn,Sn)

( I ) t o ( 5 ) converging i n is a l s o a solution.

(r,F,S)

A

T h i s i s a n i m p l i c i t h y p o t h e s i s on t h e f u n c t i o n motivates

the

Definition 1

,.

-

~ ~ ( 6 2 )

which

S

i s an a d m i s s i b l e c o n s t i t u t i v e r e l a t i o n i f

S

i t s a t i s f i e s the preceding conditions. Let us n o t e f i r s t t h a t t h e only r e a l d i f f i c u l t y i s t o know i f

F

and

theorem o f B a l l [l], det F 2 0;

(5),

s t r o n g l y and

det Fn-det

F

L”

in

I n d e e d , by a

Ig(G)l +

+

det F

det G

As

r

n

4

r

0

w i l l imply

s t r o n g l y we have

bz(x,t)-bo(x,t)

in

L”

*

weak

a n a t u r a l growth c o n d i t i o n

s t a y s bounded and c a r e of

S = g(F).

a r e r e l a t e d by

S

>

+m

0

giving when

G

taking

b(x,t,rn)+b(x,t,r)

weak

*

taking care

479

N O N L I N E A R CONSTITUTIVE RELATIONS AND IIOMOGENIZATION

of

(2).

(1) a n d

('4)

remains;

Thus o n l y a

( 3 ) b e i n g l i n e a r p r e s e n t no d i f f i c u l t y .

Then

result

as f o r

of Murat-Tartar

(6)

note than

,

( c f . M u r a t [ 13

SnFnT--SFT

T a r t a r [ 11

by

).

I t i s n o t h a r d t o d e r i v e a n e c e s s a r y c o n d i t i o n for admissibility Theorem 1

-

(cf. n

If

T a r t a r [ 21 )

is admissible then i f

S

d e t Fi

> 0;

F2-F

F1,

F2

satisfy

5

- 1 8

1 -

T h e n we h a v e

(10)

5((1-e) F 1 + 8 F 2 ) = (1-8) g ( F 1 )

Remark 2 : from

As

F1 t o

( 9 ) i s the Rankine-Hugoniot F

2

A

wise

(a.e)

E

[O,ll.

condition f o r a

5

jump

this

n

s

d i s c o n t i n u i t i e s o c c u r o n l y on

l i m i t o f smooth s o l u t i o n s :

contradictory

for 8

i s a f f i n e a n d t h u s c a n b e o b t a i n e d as p o i n t -

S

a r e believed

€I;(F,)

a c r o s s a n h y p e r s u r f a c e of n o r m a l

s h o w s t h a t for a d m i s s i b l e lines where

+

entropy inequalities

t o hold f o r these solutions s o ( 8 ) i s not ( b u t may n o t b e p h y s i c a l ) .

I S the s t r o n g e l l i p t i c i t y condition holds o n l y o c c u r s for

F1

= F2;

then

presumably s o l u t i o n s of

e q u a t i o n s may b e s m o o t h i n t h i s c a s e

(F

and

S

(9)

the

HBlder

continuous), F o r h y p e r e l a s t i c m a t e r i a l s whose s t o r e d e n e r g y f u n c t i o n a l s a t i s f i e s t h e Legendre-Hadamard necessary

condition, the

c o n d i t i o n o f Theorem 1 i s s a t i s f i e d .

I t i s n o t known i f sufficient o r not.

this necessary condition i s

Some s u f f i c i e n t c o n d i t i o n s f o r actnisiibility

480

L. TARTAR

can be obtained but the main problem remains that it is hard to check on particular examples if they apply.

-

Example 1

F

for all

matrix

A sufficient condition for admissibility is that

(satisfying

MF

det F > 0)

such that

5

(We assume o f course that if

G

there exists an invertible

is bounded and

is continuous and I;(G)I

det G + 0).

thesis corresponds to the case

MF

+

+co

The monotonicity hypoI

I.

T o obtain a wider class of admissible conditions we will use the following important notion which is adapted to equations (1)(3). Definition -_

2

-

A functional

(rn,Fn,Sn) converging in

sequence

and such that then

is admissible if for any

cp(F,S)

Div Sn

cp(Fn,Sn) If

L"

is bounded in

converging weakly to depends only on

to the quasiconvexity o f

ep

F

(and

L" J,

++

weak

to

(r,F,S)

Fn = O r n )

implies

~

z q(F,S).

this notion is equivalent

(cf. Ball [l]).

The exact

structure of these admissible functions is not known but simple examples, generalizing Ball's polyconvex functions,

All quadratic admissible functionals are

can be obtained.

known (cf. Tartar 111 , [ 2 ] ) : if

F = X 8

Example 2

-

5

and

S5

they must satisfy

cp(F,S)

2

0

= 0.

A sufficient condition for admissibility is that

there is a family

(cp,)

a€A

of admissible functionals such

that (12)

s

= :(G)

is equivalent to cp ( G , s ) c Q

o

for all

a E A.

481

N O N L I N E A R C O N S T I T U T I V E R E L A T I O N S AND H O M O G E N I Z A T I O N

Then Example 1 is only a particular case where the functions VU

take the form

Example 3 (PU)u:A

-

.

A sufficient condition is that there is a family

of admissible functionals satisfying cpa(F-G,

(13)

T

c p ( G , S ) = MF(:(F)-S(G))'(F-G)

< 0 f o r all F,G and all

;(F)-:(G))

u E

A

and the maximality condition cpu(F-G,

(14)

;(F)-S)

s

0 for all

F and a implies S = : ( G ) .

This is also a particular case of Example 2 but it will he more convenient to handle homogenization. Remark 3: It is not known if hyperelastic material having a polyconvex stored energy functional (cf. B a l l [l]) have an admissible constitutive relation.

4. Homogenization. Homogeneous materials are very

often heterogeneous at

a microscopic level (we stay o f course far above the molecular level).

If the different components are small enough compared

to the experience scale the material will behave like a homogeneous material. as

(Of course this is the same approach

in paragraph 2). T o avoid technicalities we will work with a material

having a periodic structure of size

6

and assume that there

are no exterior forces; we have functions satisfying

(15)

Div Sc = 0

( r E , F E, s e )

L. TARTAR

482

a Tie

F6. . =

ax. J

IJ

Sc

,(:

=

Fe)

> 0

det F' G

,;(: If when (r,F,S)

QF ) = US(%, F')

g o e s to

E

we expect that

for

Q F Orth'.

(re ,FG , S ' )

0

(r,F,S)

converge weakly to

will satisfy the equations (Of c o u r s e

corresponding to some homogeneous material. does not go to

0

but if

almost undiscernable f r o m

(rE ,FE ,S')

is small enough

E

e

is

-

(r,F,S)).

We will obtain the homogenized constitutive relation S

1

by saying that

(21)

-

S = S(F)

(re ,FE, S @ )

if there exists a sequence

m

satisfying (15)(16)(17) converging i n L

-S

It is a belief that such an

weak

exists: a priori

-S

*

to (r,F,S)

may be

multivalued. Properties (18)(20) f o r

5;

corresponding properties of automatic by its definition. obtain information on

5:

-S

follow easily from

admissibility of

5

is

The only kind of question is to

i f for every

x,

...

S(x,F)

is of

the type of example 1, 2 , 3 ( o r in any other interesting class) what can be said on same family

To.

-S.

F o r the class of Example 3 , if the

is used f o r all

inequality ( 1 3 ) is true for

4 . ,

S;

x,

then the same

on the contrary Example 1

does not seem to b e a good setting for homogenization.

NON LINEAR CONSTITUTIVE RELATIONS AND HOMOGENIZATION

483

5. Comments. Non linear partial differential equations o f Continuum Mechanics or Physics should be stable under some kind of weak convergence, the natural spaces being pointed out by the case o f discontinuous coefficients (heterogeneous materials) and

homogenizati on. The main open problem is related to the so c a l l e d entropy inequalities and consists in asking which are the physical discontinuous solutions of the equations.

If one accepts all weak solutions o f the equations, which we have done here, w e are led to a simple necessary condition and some implicit sufficient conditions which are difficult to check.

An important problem is to derive a

necessary and sufficient condition or at least to give a simple way to check a sufficient condition. The same analysis can be done with boundary conditions and of course writing all this f o r manifolds with or without boundary is left as a good exercise for specialists in translations. Existence theorems are now reduced to construct approximations with a good a priori estimate, the admissibility property dealing with the passage to limit. The philosophical approach used (which is more or less known in Statistical and Quantum Mechanics) is i n opposition

with the classical use of strong topology, implicit function theorem and other local results (in n o r m topology).

484

L. TARTAR

Bibliography

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-

J.M. [l]

Convexity conditions and existence theorems

in nonlinear elasticity,

63, Duvaut, G.

337-403

-

-

Arch. Rational Mech. Anal.

(1977)

Lions, J.L.

[l]

-

Inequalities in Mechanics and

in Physics, Paris, Dunod 1972 (in French); Springer

1974.

-

Gurtin, M.E. [l]

O n the n o n linear theory of elasticity

in this volume. Leray, J. [l]

-

Etude de diverses Qquations int6grales n o n

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(1933). Lions, J . L .

-

[l]

Quelques d t h o d e s de r6solutions des

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Paris, Dunod-

Gauthier Villars, 1969. Murat, F. [ 13

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Cornpacite par compensation, t o appear in

Annali d i Pisa. Tartar, L . [ll

-

HomogenGisation dans les Gquations aux

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[a]

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