Non-linear Continuum Theories (C.I.M.E. Summer Schools, 36)

G. Grioli • C. Truesdell ( E ds.)

Non-linear Continuum Theories Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, May 31-June 9, 1965

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11032-0 e-ISBN: 978-3-642-11033-7 DOI:10.1007/978-3-642-11033-7 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Ed. Cremonese, Roma 1966 With kind permission of C.I.M.E.

Printed on acid-free paper

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CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. )

lo Ciclo

- Bressanone - dal 31 maggio

a1 9 giugno 1965

"NOX-LINEAR CONTINUUM THEORIESn

Prof, C, Truardell eoordiReteri: prof, et, orieli

B. COLEMAN-M. E . GURTIN : Thermodinamics and wave propagation in E l a s t i c and Viscoelastic media

Pag. 1

L.DE VITO : Sui fondamenti della Meccanica d i s i s t e m i Continui (11)

pag. 19

G. FICHERA : P r o b l e m i elastostatici con a1 contorno

pag. 107

ambigue condizioni

G. GRIOLI : Sistemi a trasformazioni r e v e r s i b i l i

109

W. NOLL : The foundations of Mechanics

11

159

R. A. TOUPIN : Elasticity and Electro-magnetic

11

203

CHAO-CHENG WANG : Subfluids

It

345

CEWTlZO I N T E R N A Z I O X A L E M A T E M A T I C O E S T I V O ('2. I. M. E. )

B. C O L E M A N M.E. GURTIN

T H E R M O D Y N A M I C S A N D W A V E P R O P A G A T I O N IN E L A S T I C -AND V I S C O E L A S T I C

MEDIA

Corso tenuto a Brecsanone dal 31 maggio a1 9 giugno 1965

THERMODYNAMICS AND WAVE PROPAGATION IN ELASTIC AND VISCOELASTIC MEDIA by

D. Coleman

Bernadrd

Mellon Institute, Pittsburgh Morton E . Gurtin Brown University Providence

In continuum physics the word wave is used with s e v e r a l distinct to s o m e it is

meanings. To s o m e a wave i s a sinusoidal disturbance, any

m e m b e r of a c e r t a i n c l a s s

and t o

of solutions t o a hyperbolic equation,

o t h e r s it is a propagating singuiar s u r f a c e .

(Christoffel,

Here we follow

Hugoniot, Hadamard, and Duhem and use the word in the

l a s t sense; thus, r e s p e c t to the

e t o be a s u r f a c e , moving with

we define a z

m a t e r i a l , a c r o s s which s o m e kinematical variable, such

a s the acceleration o r the velocity, s u f f e r s a jump discontinuity. In the present age of sonic booms and nuclear explosions , even the layman is f a m i l i a r

with "shock wavesn

.

To find, howe~rer, applications f o r a

general theory of

propagating singuiar surfaces, one does not have to

t u r n to the l a t e s t

accomplishments of physics ; it suffices t o think c a r e -

fully of the

motion of an object s t r u c k with a h a m m e r .

Here we review s o m e a s p e c t s of the c l a s s i c a l theory of propagation

in elastic m a t e r i a l s

and discuss recent extensions

wave of

the c l a s s i c a i theory to m a t e r i a l s with m e m o r y .

Let xi(Xj, t ) give r i a l point which occupies

*we

the spatial position a t time the position

use Cartesian t e n s o r notation.

X. in J

t

of the mate-

the r e f e r e n c e configuration.

- 4 -

-

B. Coleman

According t o (t)

=

the

M. E. Gurtin

Duhem-Hadamard c l a s s i f i c a t i o n s c h e m e , a s u r f a c e

is a

wave of o r d e r

if

N

the

N1 t h - o r d e r d e r i v a t i v e s

Z , but all l o w e r d e r i x.(X., t ) exhibit jump discontinuities a t 1 J v a t i v e s a r e continuous a c r o s s , I A s h o c k i s a wave of o r d e r 1;

of

.

that t)

is,

a -

xi(X., t ) i s continuous , but the velocity x . = xi(X., J at J a X . ( X t ) show jumps the d e f o r m a t i o n g r a d i e n t F . . = IJ ax; 1 k'

and

.

across 1

At

an

. acceleration

J

wave

second

derivatives, such

C)

0

a s the

acceleration

L

x . = q?;xi(x., t ) , a r e t h e f i r s t t o s u f f e r jumps; 1 J at hence , a n a c c e l e r a t i o n wave i s a s i n g u l a r s u r f a c e of o r d e r two. O u r i n t e r e s t h e r e is in waves

with

In the t e r m i n o l o g y

N

the s p e e d

the r a t e

Z

particles

along

instantaneously s i t u a t e d

t h e amplitude

of wave of

order

;

and Toupin [1960, 2 1 and

of T r u e s d e l l

T r u e s d e l l [1961, 2 1 , of advance of

22

of propagation i t s unit on

Z

V

normal

.

A

N i s a vector

of

a wave 1 i s

relative t o the

convenient m e - a s u r e of s . defined by 1

Let formula*

.

the Piola

- Kirchboff

s t r e s s tensor

S . . be defined 1J

by t h e

Whenever a n index i s r e p e a t e d in a product of two t e r m s , s u m mation o v e r that index i s understood.

- 5 -

B. Coleman- M. E , Gurtin

SikFjk = T . . 1J

P

where

T . . i s the familiar s t r e s s t e n s o r of Cauchy

r

and

1J

sent

mass

density. F o r an elastic material,

influences

a r e ignored,

S is ij

i s the p r e -

when thermodynamic

a function

of

the deformation g r a -

dient : S . . = S. . ( F ) 1J 1J ki

(3) The c l a s s i c a l

Fresnel-Hadamard theorem

asserts

that for

any

s

and speed of propagation V i of an acceleration wave traveling in the direction n must obey k the propagation condition m a t e r i a l obeying

where the t e n s o r

Ericksen

[ 1953

(2) the amplitude

Qij(nk), called the acoustic

, 11 ,

condition

of

(4)

Even incluences

21 ,

[ 1961,

of compressible elastic m a t e r i a l s , have

elastic waves

order

with

and

allow

N

> 2

Qij(nk)

in elasticity S ij

is

*x

given

by

working in the theory of isotropic incompres-

sible hyperelastic m a t e r i a l s , and Truesdell t h e theory

tensor,

must given

theory one to depend

a l s o obey by

(5)

should not

working in

shown that

all -

the propagation

. include thermodynamic

only

on

Fkl

but

also

-

F o r the mos t general statement of the Fresnel-Hadamard Theo r e m , and for a detailed discussion of i t s consequences, s e e Trusdell 11961, 21

- 6 -

on a thermodynamic variable such cific ,

1'

F. ., 0 ,

and

1J

(6),

one

Onthe

then

waves

a

a r e taken

a

a s the t e m p e r a t u r e

t o be continuous

0

o r the spe-

a c r o s s t h e wave;

homothermal if , in

wave i s said to be

hand,

the wave of

M. E. Gurtin

addition to

has

other

theorem

-

q. In the thermodynamic theory of acceleration waves,

entropy

An acceleration to

B. Coleman

if,

in

place

i s called homentropic

of

.

(7)

,

It is the content

-

of the f i r s t

Duhem, that homothermal (homentropic) acceleration

in

elastic

in

(5)

sae Fdnh c n theorem

materials

is taken of

(4) , proveded that the derivative

obey

a t constant

Duhem

temperature

(entropy)

.

The

gives the physical circumstances in which

acceleration waves a r e homothermal o r homentropic : Every a c c e l e r a tion wave in an e l a s t i c m a t e r i a l obeying F o u r i e r ' s law of heat conduction ry

with positive-definite t h e r m a l conductivity i s homothermal: eveacceleration wave in

heat i s

Y

homentropic

an elastic material which

does not conduct

.

* The two t h e o r e m s of Duhem a r e explained and given proofs by Truesdell [ 1961 , 2 1

.

modern

- ,[

An elastic m a t e r i a l s t r a i n function function @ ;

of

i. e . ,

(3)

B. Coleman

-

M. E. Gurtin

i s said to be hyperelastic if the s t r e s s -

i s obtained by

differentiating a s t o r e d energy

if Sij(FkL ) =

It i s a f a m i l i a r a s s e r t i o n

6 (Fkz, 1.

a F 1~ ..

of c l a s s i c a l thermodynamics that e v e r y ela-

s t i c material i s hyperelastic in the sense that

s.1J.(Fu

(11)

where and

i s the E

the

A theorem m a t e r i a l the

It follows from

enternal

this

tensor

result

the classical equations (10) of an

elastic m a t e r i a l

tropic waves.

Helmoltz

of Hadamard

acoustic

(Fkt ,771

2

1J

specific

specific

a a F..

,771 =

free

.

energy asserts

, then and

that

must

(5)

for

IL = a

E

-8

TJ

,

hyperelastic

be s y m m e t r i c :

first

(ll),

energy

theorem

that

the

of

Duhem, and

acoustic

tensor

must be s y m m e t r i c in homothermal and homeri-

B. Coleman

-

M . E . Gurtin 4

The recent y e a r s

have s e e n t h e development of general t h e o r i e s

of non-linear m a t e r i a l s with memorye

.

In these studies it i s a s s u m e d

that the present s t r e s s depends not only on the p r e s e n t value of the s t r a i n but

also

on the past history of the s t r a i n ,

and one a t t e m p t s

t o solve problems without specializing the functional expressing this deLet us

pendence.

.

val

a,

[o,

define a function

) a s follows;

dt) k1

over

the half-closed inter-

n< s < a,. This

function

tion

gradient

i s just

i s called

.

equivalent and -

to

the

define a

The past history

the r e s t r i c t i o n

F") (s) a g r e e s (r)kL fined f o r s=O.

the historv ur, t o t i m e of

F;

Obviously,

past history

all

of t h e p r e s e n t to b e

Following a

t i s given i s determined when F k0) can

the deforma-

s

\

0 but i s

left

unde-

a knowledge of the history

F(r)ke.

material

of

F ( ~ ) of the deformation gradient (r)kE t o the open interval (0, a) ; i. e . ,

F Ice ( ~ )(s) f o r

with

a knowledge

simple

,+,

t

value No11

material

.

For

F ( ~ ) is kc ,*, F k g ( t ) = :F : (0)

11958,

1] ,

we

for which the s t r e s s

such

a material

we

write

' s e e , for example, r h e works of Green and Rivlin [1957, 1 1 , No11 0958, 1,l C o l e m a n a n d N o l l 6960, 11[1961, 1 1 , and Wang [1965, 7 1, which a r e s u m m a r i z e d and extended in the exposition of Truesdell and No11 l.1965, 6 1.

- 9 -

Here

S..

that f o r

a general

t h e p r e s e n t value

gradient,

we w r i t e (14)

in

and

(15),

we s e e

history

1J

depends on 1J of the d e f o r m a t i o n S,.

the f o r m *

(t) S- l J. . (F(r)kl

1J

for short,

simple material

and t h e p a s t

S. = where,

M. E . Gurtin

a functional ; i. e . . a function whose a r g u m e n t i s a ( t ) , and whose value is a t e n s o r , Fkl S . . . When we wish

emphasize

both

-

is

-1J

function, to

B. Coleman

. .FkL

,

= F ( ~ )(0). On c o m p a r i n g ( 3 ) Fkl kl t h e e l a s t i c m a t e r i a l s c o n s i d e r e d in t h e p r e -

we put that

~ i o u ss e c t i o n a r e t h o s e s i m p l e m a t e r i a l s f o r which t h e dependence (t) Sij(F(r)w; Fkc ) on the p a s t h i s t o r y F (( r~) i)j i s negligible. H e r e we do not a s s u m e that influence of F ( ~ ) on S c a n be (r) ij neglected; we a s s u m e m e r e l y that t h i s influence i; compatible with t h e p r i n c i p l e of fading m e m o r y ; used

by

Coleman

and

t h e m a t i c a l t h e intuitive r y distant

Let

us

h here

a s m o o t h n e s s postulate 1 1 to

11 [1961,

that s t r a i n s

occurred

denote

by

by differentiating

value of deformation

(16).

idea

[ 196C,

with

render ma-

which o c c u r r e d in t h e ve-

p a s t should have a s m a l l e r influence on the p r e s e n t s t r e s s

than S t r a i n s which

obtained

No11

i.e.,

is no

in

S. .(F.(t . ) t h e f o u r t h - o r d e r t e n s o r - 1 ~ kt mn the s t r e s s with r e s p e c t to t h e p r e s e n t DF

gradient

summation

the r e c e n t p a s t .

holding fixed the p a s t h i s t o r y :

over

k

and

I

in equations

(15) and

- 10 This tensor sponse t o s m a l l Equation

role

(16)

defines into

the thermodynamics of The theory under

a linear

functionals

in the theory

memory is

the moduli f o r

s t r a i n impulses superimposed

mapping functionals tral

1 , 2 1 gives

11964,

B. Coleman

of

instantaneous r e (t) Flee

on

at time

11965,

D~

1-41 and in

1964, 1, 2 1

memory

.

s u r f a c e s propagating in m a t e r i a l s with

not an empty subject. Among the m a t e r i a l s subsumed

the c l a s s of simple m a t e r i a l s

with

fading

memory a r e t h e

materials

of

the theory of l i n e a r viscoelasticity. In that theory, Sips

[1951,

,

Lee

11

and

Kanter

1 1 have exhibited explicit

solutions

others showing acceleration

acceleration that

gives r i s e t o Recently

general gation

11965.

materials

theorems given

o r greater

in

material

- Truesdell

traveling with

I,

1-4

in

Chu

we

to construct

waves. F u r t h e r , showing shock and

.

have been able

with

[1962,

equations

simple fluid with fading

nonlinear field equations

non-linear

Ericksen

exact solutions

a special

have the following extension and

elementary e x e r c i s e

1 1 has obtained

waves for

and

the dynamical

from

11966,

an

of

,

It

these solutions

is

2 1

11953,

showing shock waves. Pipkin

t.

mn o p e r a t o r plays a cen-

this

wave propagation

of singular

Gurtin

differential operator

;

m a t e r i a l s with

- M. E.

to

memory

extend

memory the c l a s s i c a l

to

propa-

the previous section. F o r example, we [1965,

4 ] of

the

Fresnel-Hadamard

t h e o r e m s . Consider a wave of o r d e r

the direction

fading memory

;

such

n a

k

in

a general

wave o b e p

2

simple

the propaga-

B. Coleman

tion condition

, and

(4)

the t e n s o r

of

Q. .(nk) , now called the instan-

i s given by the following remarkably

simple

(5) :

F o r the validity of this in front

>' E. I. Gurtin

1J

taneous acoustic t e n s o r , generalization

-

theorem

the past history of the m a t e r i a l just

of the wave may be a r b i t r a r y , subject only to c e r t a i n natural

.

t a m e n e s s hypotheses

Of course, we know that the s t r e s s in a general m a t e r i a l with

memory depends not only on the history of the deformation gradient but a l s o on the history of a thermodynamic variable. Hence, (14) should be replaced by e i t h e r

S.. = 1J

where the function tion

(20)

7

e(t) i s

5. . ( F(kt L) , J t ) -lJ

the history

) ,

of the temperature a d the func-

( t ) the history of the specific entropy

(s) = e ( t - s ) ,

(t) (s) = v ( t - s )

o < s < w .

* Propagation conditions of the type (4) a r e known for acceleration waves in s e v e r a l special m a t e r i a l s ; for elastic m a t e r i a l s (2. 18) was derived by Hadamard [1901, 1 1 and for the theory of linear viscoelasticity by H e r r e r a and Gurtin 6965, 5 1. We have recently seen a manuscript by Varley p965, 7 1 in which he a r r i v e s at (4) for acceleration waves in m a t e r i a l s of integral type.

- 12 -

-

B. Coleman

is a s s u m e d , it is a l s o

When (18)

c i f i c Helmholtz the histories

free (t) Fk2 ,

and

heat

that

sent

the

temperature

*

energy

M. E. Gurtin

r e a s o n a b l e t o a s s u m e that t h e s p e i s given

by

a functional

e

of

&t)

and

flux v e c t o r gradient

q, depends

=a

gm

not

only on the p r e -

I

but a l s o

axm

on

F ( ~ ) and k1

e(t)

When

(19)

is

assumed,

it

is r e a s o n a b l e

s p e c i f i c i n t e r n a l e n e r g y i s given

Starting from man

assumptions 1 ] has

[1964,

compatible with

by a functional

shown

only

if

and

relations,

fading

t h e s e functionals

s.( F k( tL) ,d t ) ) =

c

1J

D

F.. 1J

where

is t h e o p e r a t o r

with

memory

defined in

memory

'

FtL and n ( t ) ,.

of

$. .

that the functionals

t h e p r i n c i p l e of the

postulate that the

s o m e w h a t m o r e g e n e r a l than

of t h e r m o d y n a m i c s

g- through

to

-1J

these, and

$,

Cole,

- 1J

are

a n d the s e c o n d law

a r e d e t e r m i n e d by p-

p ( d t ) , e ( t ) ),

- kZ

(16). T h e s e

the classical

relations

gene-

r e l a t i o n s (10) and

- 13 and

(11) for e l a s t i c Even

brought

when the history

ma1 and homentropic

by

(17)

ction

tive

(4) s t i l l

holds f o r homother-

[ 1965,

-Sij F(:

I.

3,4

replaced

by

For

Q; ; i s

ir.sl:antaneous acoustic thensor

(~ib))

of the wave,

to

condition

is

'J

, G'~)) ,

and

given the fun-

is

held fixed in the computation

homectropic acceleration

of the derivais

waves

D~ 7, (t' , pq

$.-1JF

, T ' ~ ) ) and the function Prl of the entropy at the wave, is held fixed in

ap-

the histo-

(16) and

(17).

These observations extend to m a t e r i a l s with m e m o r y the f i r s t of

Iluhem. The second theorem of Duhem

zation

[1965,

I:

3, 4

In

wave i s h o m e n t ~ o p i c . Here; aq: t e r i a l f o r which - 2 ,

a

-

of heat, e v e r y accele-

a non-conductor , evevy acceieration

by a definite condcctor,

we mean a ma-

computed

is always

using

(22),

definite t e n s e r . The

with

lations

q

*

,

po-

procf cf the theorem i s straightforward

uses

a generalization

i.e., 11

[ 1964,

a mate-

of t h e r e -

(24;. We should like

t h e following e x t e n s ~ o n tion

a

m

for a definite c c ~ d u c t o r, the proof for a non-conductcr, rial

theorem

a l s o has a d i r e c t gencrali-

a definite conductcr

ration wave i s homothermal ; in

sitive

ho-

the history of the t e m p e r a t u r e rip to the moment of

(16). F o r

plied ry

-lJ

,

0

arrival

5..

of a t h e r m o d y ~ a m i c variable

acceleration waves

waves the

with

Gurtin

materials

in, the propagation

mothermal

- X'I. E.

13. Coleman

(12) :

to

bring

i1965,

The relations

(24)

t o the a t t e ~ t i o n of 4 1 of the c l a s s i c a l imply thar,

and homentropic

s y m m e t r y condi-

even in

m e m o r y , the i n s t a n ~ a n e c u sa c o u s ~ i ct e n s o x f o r

experimenters

a m a t e r i a l with

both

waves a r e always s y m m e t r i c t e n s c r s in

homothermal ?he s e n s e

B.Coleman of equation

(12)

. This

ing t h e physical Truesdell materials,

[1961,

theorem

3,l

(24). F o r t u n a t e l y ,

working in the t h e o r y of e l a s t i c

h a s found s i t u a t i o n s in which

city can test the symmetry

M . E. Gurtin

a p p e a r s t o supply a method of t e s t -

a p p r o p r i a t e n e s s of the r e l a t i o n s 2 1[1964,

-

m e a s u r e m e n t s of wave velo-

of t h e a c o u s t i c t e n s o r . His a n a l y s e s

can

be applied with s m a l l modifications t o the theory d a c c e l e r a t i o n w a v e s e n t e r i n g a g e n e r a l m a t e r i a l with m e m o r y of t h e wave had a l w a y s been a t r e s t in

which

previous to a r r i v a l

a fixed configuration.

REFERENCES

,

,J. H a d a m a r d , Bull. Soc. Math. F r a n c e

R . Sips,

.J. P o l y m e r

J. L. E r i c k s e n ,

Sci.

2

,

5C-68.

285-293.

J. R a t i o n a l Mech. Anal.

2, 329-337.

E. H. L e e % L. K a n t e r , J. Appl. P h y s . 24, 1115-1122. u w

A. E. G r e e n 1

& R . S. Rivlin,

Arch. Rational

Mech. Anal.

1-21.

wJ

W. Noll,

Arch. Rational Mech

B. D. C o l e m a n br W. No11 C. T r u e s d e l l

. In :

ries

5,

. Anal.

z,

197-226

355-370.

& R . A. Toupin,

The classical

Fliiggets Encyclopedia of Physics,

B.D. Coleman

6

C. T r u e s d e l l ,

.

311,

'L

Mech. Anal.

B. -T. Chu, X a r c h 1962, A R P A r e p o r t

239-249.

$,

from

225-793.

Y

,?:

W. ~ 0 1 1 , R e v . Mod. P h y s .

Arch. Rational

field theo-

263-296. t h e Division

of E n g i n e e r ~ n g , B r o w n U n i v e r s i t y . B. D.

Coleman, Arch. Rational

B. D. C o l e m a n , C. T r u e s d e l l , Haifa,

Arch. Rational

&9Y, 1, 1-19

C o l e m a n , ck M. E.

. A%,

B. D.

,l7;

23 '-254

1962, pp. 187-199.

R a t i o n a l Mech. Anal.

nal

NIech. Anal.

Internat. Sympos. Second-Order Effects,

B. D. C o l e m a n , & M . E . ~ u r t i n ,

B. D.

Mech. Anal. ,I> 1-46.

.

Gurtin.

Arch. Rational

M e c h . A-

M.E. Gurtin ,

Arch. Rational

Mech.

4

Coleman

Anal. 19.5

I. H e r r e r a , R . , A r c h .

1965

4

B. D. C o l e m a n 19 5

.vJ

5

.

I.H e r r e r a ,

h

M. E. G u r t i n , A r c h . R a t i o n a l M e c h . Anal.

R . b M. E. G u r t i n , Q u a r t . Appl. Math.

?2

360-364. 6

C. T r u e s d e l l b, W. Noll, mechanics,

The

n o n - l i n e a r field t h e o r i e s

In : F 1 6 g g e t s E n c y c l o p e d i a of P h y s i c s ,

212 ,

in p r e s s .

1966

Arch. Rational

Mech. Anal.

Ll, L8,

7

E. V a r l e y ,

8

C. -C. Wang,

1

A. C. P i p k i n , Q u a r t . Appl. Math. , in p r e s s .

A r c h . R a t i o n a l Mech. Anal.

of

3, 215-225. 117-126.

C E N T R O INTERNAZIONALE MATEMATICO E S T I V O

(C.J. M. El )

..L. D E VITO

SUI FONDAMENTI D E L L A MECCANICA D E I S I S T E M I CONTINUI (11)

C o r s o tenuto a Bressanone da13 lmaggio a1 9 giugno 1965

SUI FONDAMENTI DELLA

MECCANICA DEI

SISTEMI CONTINUI (11) di Luciano De Vito Universitb- Roma In un recente Mazzaroli ordine di fa

- in

lo scrivente, ( I ) 81 6

e idee,

uno

lavoro, in collaborazione

dei

proposto dal Suoi c o r s i

tra

il

cominciato

dott,Innocente

a sviluppare un

Prof. Gaetano F i c h e r a giB alcuni alllIstituto

Nazionale

di

anni

Alta Matems-

tica -relative all~introduzione di un punto di vista "globaleIt nella teor i a della

"Meccanica dei s i s t e m i c o n t i n ~ i ' ~ in . luogo del punto d i

vista

"puntualen che pih

vista

"globaleU,

rappresentate benst

da

s p e s s o viene adottato

volume e di s u p e r f i c i e non aaranno pih

integrali

(in s e n s o ordinario) di funzioni continue,

da funzioni di

a110

studio

continuiu , p e r il

insieme affatto a r b i t r a r i e

delle

B

caso dei c o r p i

.

numerabil-

alltimpostazione

della I1Statica d e i s i s t - m i

indefinitamente e s t e s i . VerrA o r a qui

limitati (includendovi anche il c a s o dei c o r -

Strumento essenziale p e r tale studio B l a tore

purchb

pervenuti

equazioni cardinali

esposto il c a s o d e i corpi pi "fluidit1 )

Secondo tale punto di

l e f o r z e di

mente additive. Nel sucitato lavoro s i ed

.

nozione

di

"vet-

dotato di divergenza in s e n s o debole (o in senso integrale) " ,

che r i e n t r a , come c a s o particolarissimo, nella pih generale nozione di

k - m i s u r a dotata di

differenziale

in

s e n s o debole1$ introdotta

L. De Vito e I. Mazzar01i:~Suifondamenti della Meccanica s t e m i coritinuiw , Memorie Acc. Naz. Lincei , v. VII , 1965

.

dei s i -

L. De Vito da G. F i c h e r a (2)

. Alla

considerazione

-

"Staticatl p e r corpi limitati ne quindi quello

premettere

delle equazioni cardinali della

second0 llattuale

lo studio dei vettori

sono, appunto, dedicati

i primi

vista

-

convie-

a divergenza debole e a gradiente debole. A ta-

(strettamente connesso) delle funzioni

l e studio

punto di

.

paragrafi

1 . Notazioni In tutto

quel

c3

che segue, ci

nel quale penseremo introdotto un

euclideo

il

punto

denoteremo siano

2

,x

xi=

del

in

,x , con 1%; (avendo

0

si

di

Con

il

la

3)

detto

a1

%'

con

a

(a,& )

la

gli

intervalli ( s u p e r i o r -

fissato

s i s t e m a cartesiano

di

x

, oppure,

I

su1 piano ove gli

delltaggiunzione di un mult'i-

llintervallo

a co lo,

9 p e r ogni

NE

q,CQ')

che, in

ove pensiamo

Allora tinuo r u tutto

FD'

(11) e , p e r ogni talchb di

(9),

pub F,l

imbolo

q,(D'),\,&

forza di

d Q i ~, p)e r

ogni VC%(D>~ ha:

= u d u . d ~ =- {Dl u, Avv dx

{ D l ~ xy Da

in

il

limite:

d r =-IDl a c ~ w dx r t 5' (Y)

r4

funzionale lineare

1k7,jJ

e continuo sulla varieta

di a v e r introdotto la norma :

11

riesce :

essere

,&?D')

che e s i s t e finito

prolungato

(che seguiteremo

in un funzionale l i n e a r e e cona denotare con il medesimo

) tale che :

11 D' I\AacDIJ*

q,(D'),

risulta

FD, Lm3, wet2

the 6 inclusa nella

.

XVI

"Cu 9

/LI~V(()-J)~~) una m i s u r a assolutamente continua con densitj. ,

/

,/e k91(0) esistono infiniti vettori K F & t ~ ) c h e hanno s u D-)JD eriesce: ddbol; /U

p e r divergenza

(24)9,1t

; 4 < 3 ' < ~ . 23, ,I(?< (ove s i convenga che, p e r q, ficato e

specificato

71

sono

Segue

dal

per /v

ie su

genza

D

esiste

le

migliori possibili

.

s i ha

un

IAl=[wD7" D{dr su

con

migliorare

N

) ;

2

ha

il s igni-

per

.

ye

5D f! d r =

A ~ D 'he) {B

i(

l e limitazioni

per

per

perfettamente

del t e o r . XIV

.

ha

Allora, p e r i i

p e r divergenza d ~ b o -

.

la misura e

.

aUdx= l i t U 1 ~ & y D )

Allora, il vettore

D-&D

un ragionamento

strazione

49;u)e

?I

vettore

r a r e l e limitazioni p e r di

,j

t e o r . XVI, in b a s e alla seguente osservazione. Posto -

la m i s u r a p ( ~ ) =

debole

; P 2 ( i ( < t @j+ , 3I-c9r b i . < t

di(?&+a

, =a&

*q 9 t e o r e m a precedente. L e limitazioni

nel

C=/?-(WDJ~~+ dx

t e o r . XIV

:4

Sia r('~~*~~‘$i;ilec I ~ r ha per d i r r r -

/1L . L1impossibilitB di rniglio-

q'

segue da quella e

/r'

analog0

in relazione ad un

- nota -

alla (19),

r+

fatto nella dimo-

L. De Vito

8. T r a c c i a s u l contorno p e r funzioni dotate di g r a d i e n t e debole

su I>-$D. Sussiste

teorema :

.Se M ~ & ' ( ~ha) g r a d i e n t e debole s u I)-PD r a p p r e s e n -

XVII tat0

il seguente

dalla m i s u r a

r-

e s i s t e una ed una s o l a funzione

ode delle s e e u e n t i ~ r o ~ r i e :t a

1) r i e s c e :

ove i l i m i t i devono p e n s a r s i fztti prescindendo da un i n s i e m e di m i s u r a nulla 2)

per

O V ~ Ke 3 ) s i ha :

(27)

) di

'I.'x'1i4G3D)

+

un conveniente

ed

ove

in

i n s i e m e di m i s u r a nulla ;

/1@ljd_l~4D~o

a c o n s i d e r a r e il

caso

in

, riesce :

r

h\r.

u4(y)

cui

piano. P o s s i a m o quindi s u p p o r r e che

sia

sia con-

- 40 tenuta nel

piano

1O -

tervallo

~

1

di

tale

(, x l + ~ < p 'kitkl;t24 J tenga llintervallo 1='

come v e r s o r e

Y(X)

e

s i0a

~

piano

&;+a

~ ix Ji

dalla chiusura

(definito dalle limitazioni anche che

delltin-

~''2 .

D

con-

~;;%(~kl'tO) . Assurneremo v e r s o r e normale interno . Con l e con

;

per-)"r

notazioni introdotte in

costituita

. Supporremo

) 0

L. De Vito

il

-

1 , s i t r a t t a allora

di mostrare

che

esiste

g4(1;;) tale che : L((T$)M*($)JdjL=o,L 4(f,2+j=u*($d.j

una 14X(2;)sM X ( X ' + $ i * ~ ~

C->oS. P. ~111i.i prescindendo da un conveniente insie-

fatti m e di m i s u r a

nulla

per

/U ; e a %; ( n e l piano 1 '= 0

positiva e negativa di

4 10

rispetto

la:

vallo riesce :

del

I,i

. Indicate con

delltasse reale

con

X'

~(9. E)

l a derivata parziale di

senso

p e r ogni interval-

contenuto

in

Io2;

e per

ogni inter-

contenuto, con la s u a chiusura, in ( Q;O ),

2 $ ! ~ ~ ~ ~ ~ ;. ~ l ) ~ ~

4,ui(~-~;%i~:)=$ Post0

le variazioni

a:

, con 4i(f

s*P' ~@,?oo)

dalla p a r t i c o l a r e s c e l t a del v e r s o r e

propriamente regolare, & di verific?

immediata. L a funzione masi traccia di

k* ,

K

P e r l a funzione

su

d i cui si & provata l l e s i s t e n z a e ltunicitA, chia-

C)r)

(20)

t r a c c i a s u s s i s t o n o anche i seguenti t e o r e m i .

- d A ( ~ha) p e r

XVIII. Se K E

&

,

&a s u a t r a c c i a s u

ove

/('

2

@p

.

gradiente debole

appartiene a

gp2(~)

su

D-&D

la a i s u r a

e, indicata con 4 4k

, riesce :

una costante dipendente s o l o da

D.

( 2 0 ) ~ a l nozione e d i t r a c c i a 6 s t a t a data da G. F i c h e r a nel 1949(cfr. G. Ficher a , llSulllesistenza e s u l calcolo delle soluzioni dei problemi a1 contorno rel.ativi alltequilibrio di un coppo elastico", Ann. Sc. Norm. Sup. P i s a , 1950, Pubblicazione dell1INAC n. 248, 1949) e r i p r e s a , pih t a r d i , da Stampacchia ( c f r . G. Stampacchia, llProblemi a1 contorno p e r equazioni di tip0 ellettico a derivate p a r z i a l i e questioni di calcolo delle variazioni connesse", Ann, di Mat. pura e appl., 1952)

L. De Vito

L a ( 3 1 ) si deduce immediatamente applicando il gia pi0 esistenziale e di dualith di

G. F i c h e r a alltequazione

ePGV(b,qD) e ltincognita e 4 E$~'~(D)o&'(QJ)) spazio L3Y~))nr(J3j)) e normalizzato mediante la norma )I 11 s

ove il t e r m i n e

60

citato princi-

noto

lkqo)

I1 flgq,, +I XIX . L( E Irz_p€Y(D-T)D),e Se /Cc QE. &?(D) , la t r a c c i a u*

ha p e r gradiente debole su 6

D- .;'D

assolutamente continua con densith

di

U

su

~

~

appartiene a

e riesce :

&(I) u+eaA.ll&

(33kl\+4

=

rpl~',?~

f

n

~

limitazioni

4

non

con

il t e r m i n e densith

essere

f.

noto ,&

&T[D)

@ ;

una

migliorate.

Se o r a si tiene p r e s e n t e

g"('3D )

dette., come segue

perb,

m i s u r a assolutamente continua

Itincognita

spazio $(b)(~&f'(&~)nOrmalioeato

tenente a

~

.I