Materials with Memory (C.I.M.E. Summer Schools, 74)

Dario Graffi ( E d.) Materials with Memory Lectures given at a Summer School of the Centro Internazionale Matematico E...

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Dario Graffi ( E d.)

Materials with Memory Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 2-11, 1977

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

ISBN 978-3-642-11095-5 e-ISBN: 978-3-642-11096-2 DOI:10.1007/978-3-642-11096-2 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli 1979 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.

I Ciclo

-

)

Bressanone d a l 2 a l l 1 1 1 giugno 1977

MATERIALS W I T H MEMORY

C o o r d i n a t o r e : Prof.

R. W.A.

BOUC

- GI

DAY

M.

FABRIZIO

G.

FICHERA

M.

RIBARIC

R.S.

RIVLIN

D.

GRAFFI

GEYMONAT : P e r i o d i c problems i n thermoviscoelasticity Pag. : The thermodinamics of m a t e r i a l s w i t h memory " : S u l l a nozione d i s t a t o n e l l a t e r momeccanica d e i c o n t i n u i l1 : A n a l y t i c problems o f h e r e d i t a r y phenomena : The b l a c k box approach and s y s t g ms w i t h memory " r The t h e o r y of c o n s t i t u t i v e equations l1

"

7

55 95 112

173 185

CEN TRO INTERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.)

P E R I O D I C PROBLEMS I N THERMOVISCOELASTICITY

Re BOUC e G. GEYMONAT

Corso tenuto a Bressanone d a l 2 a l l 1 1 1 giugno 1977

Periodic Problems i n Therrnoviscoelastici t y Two Seminars given a t the Centro Internazionale Maternatico Estivo, Bressanone, I t a l y , june 3-11, 1977.

R. Bouc Laboratoire de Mecanique e t dlAcoustique C.N.R:S. ,!?arseille G. Geymonat Pol i tecni co di Torino

We thank Professor D. GRAFFI f o r his kind invitation t o t h f s t a l k , the aim of which is t o give a survey of some recent work done a t Laboratoire de Mecanique e t dlAcous+ique of C.N.R.S. Marseille, p a r t l y i n collaboration with M. JEAN, 5. NAYROLES and M. RAOUS, specially during the second author's year of leave from the Politecnico di Torino.

Contents Introduction. 1. Background on the mechanics of continuous media with hidden variables.

2. A lineasization result. 3, Some exsrnples.

4. Some resugts Qn the nonlinear heat equation.

5. Duality and virtual work. 6. A viscoelastic constitutive equation with periodic coefficients. 7. The periodic bilateral problem f o r a Maxwell body. 8. The periodic unilateral Signorini problem.

.

B i b ?i o ~ r a p h y

Introduction Following the fundamental work of V. Volterra 1371, I381 a 1391 ,1401 , hereditary phenomena in mechanics have been deeply studied. A great part of the work that.has been done (see 118 1,135 1,136 I,. ) treats the case called, by Volterra himself, "the case of closed cycle" (see T. VOGEL 134 I) which corresponds to the case where the relaxation function in viscoelasticity is of the type G(t--c ). However, in 1907, HATT 1191 has discovered the phenomenon of creep in concrete whiohpresents stress- independent deformations which, in addition to thermal dilatation, includes shrinkage ; the material properties of concrete change indeed as a result of internal chemical reactions and the deformation problem coupled with complicate moisture diffusion through the material, as well as heat conduction. For these reasons, in a first approximation, concrete may be regarded as an aging viscoelastic material whose creep law can be written in a rate-type form, i.e. as a system of first-order differential equations, involving hidden strains, with time-dependent coefficients 6 More recently it appears that also for other materials, especially polymers in a temperature depending situation, the relaxation function is not of type G(t i ) but following a fundamental remark of Morland and Lee 127 1, the relaxation function can be written as G(5 E') where E = E ( 0 ) i s the reduced time (fee also PIPKIN 1311). From another poirit of view the extension of phenomenological laws based on spring and dashpot models to the temperature depending case has been proposed by many experimentalists (see e.g. 1 4 /) specially for metal s. In this paper we shall recall in 51 some results on the continuum mechanics of materials with hidden coordinates (indeed hidden strains) and some consequences of the Clausius-Duhem inequality on the constitutive equations due to Coleman-Gurtin / 12 1 and Bowen 1 10 .1

..

.

-

-

(r)For a very deep review of the bas; c facts on this subject see Z.P. BATANTI 3 I.

10

Because we are interested in the study of a phenomenon w i t h high temperature variations, we develop i n 9 2 , in the case of infinitesimal strains, a linearization of the equations obtained i n 51 only w i t h respect t o strains and hidden coordinates. We conclude t h i s analysis in 53, w i t h some remarks on the possibility of uncoupling the nonlinear heat equation, similar t o those developped by Crochet-Naghdi 1 13 If o r therrnorheologicall y simple solids. In 54 we recall very briefly how the nonlinear heat equation obtained i n t h i s way can be studied in the framework of nonlinear evolution equations a s developped i n the book of Lions 1 24 1 In 95 we s t a r t the study of the equation of motion (with temperature a s a data, i . e . a given function of time and space-variables),recalling some results on duality and virtual work principle. In 56 we consider a constitutive equation of axw well-type where the "stiffness" and "viscosityn matrix are temperature-dependent and thus are time dependent. More nrecirely the temperature i s T-periodic i n time and therefore the stiffness and the viscos i t y are also T-periodic. With t h i s constitutive equation we survey, i n 97 and 58 some results concerning existence, uniqueness, asymptotic s t a b i l i t y of a T-periodic stress-field for the dynamic and quasi-static periodic b i l a teral problem (1 6 1 ,I 9 1 ,I 17 1 ) and also f o r the quasi-static Signorini unilateral periodic problem (1 7 1 , 1 8 b. We refer t o the lectures of G . Fi chera i n the present session for the corresponding Cauchy-prcblems . The applicaiions of our results on the thermal fatigue of metals due t o cycle heating and cooling will be developped i n the thesis of M. Raous 132 By lack of time we cannot develop these f i r s t results ; we can only say t h a t the numerical experiments agree wf t h the ecperimentaT results of F.K.G. ODQVIST and N.G. OHLSONl 30 f"The virginal specimens behaved i n a normal way, whereas those already cracked apparently proved t o be stronger against the formation of new cracks!'.

.

1.

1- Background on t h e mechanics o f continuous medium. with hidden v a r i a b l e s .

1.1- The mechanical and thermal behaviour i n t h e time i n t e r v a l ? ' ? C R ,o f a nonpolar body occupying t h e reference configuration Q C Ft3 a t time toE C '? i s completely determined by a vector function p(X.t) (giving t h e p o s i t i o n a t time t of a material point which has t h e p o s i t i o n X i n t h e reference configuration Q ) and by a p o s i t i v e s c a l a r function B(X.t) (giving t h e a b s o l u t e temperature a t time t o f a material p o i n t which has th. p o s i t i o n X 9n t h e reference configuration S l ). As usual we d e f i n e F(X,t) = Grad p(X,t), t h e deformation g r a d i e n t t e n s o r and we s h a l l assume t h a t p(X,t) is always smoothly i n v e r t i b l e , i .e.

(1.1)

d e t F(X,t)

>

forall

0

tee

Using Lagrange's coordinates, t h e local form o f t h e laws o f balance o f l i n e a r momentum, o f moment of momentum and o f energy a r e the following (4 (Xw) ( s e e 116 1) : (1-2)

Div FS

+

pof

Po

(1-4)

E

=

0,

=

tr SE

.-

Div q

+par

where q,=p#)is t h e mass d e n s i t y i n t h e reference position. 5 i s t h e synmet r i c second Piola-Kirchoff ( o r Lagrangean) s t r e s s tensor. E = $F'F 3 ) is t h e Lagrangean s t r a i n tensor, E = aE is t h e Lagrangean s t r a i n r a t e , Po P = Po a t 2 is t h e i n e r t i a forceat f is t h e body f o r c e f i e l d p e r u n i t

.

(x) If A (xx) t r ( . )

.

3

.

-

is an m x n matrix, A' denotes t h e transposed matrix. = t r a c e of (.).

-

mass, E i s the internal energy of the body per unite mass, q i s the heat conduction vector, r i s the heat supply field per unit mass. Let u s also recall that the law of the conservation of the mass allows us t o compute the mass density a t the time t with the formula det F =

"0 P

The local Clausius-Duhem inequality

where q i s the specific entropy per unit mass can also be written, using (1.4), i n the form

where g = Grad 8 Defining the Helmoltz free energy per unit mass by

we can also write (1.6) i n the form (1.8)

- P,

~1a

-

p0q6

+tr(~.~j-fi 2

e

o

1.2- The characteristics of material composing the body are usually stated by additional equations, the so-called cansti'tutive equatjons, which give thes t r e s s , the internal energy, the entropy and the heat conduction in terms of the Lagrangean strain tensor and the temperature field. Obviously the constitutive equations depend on the properties of the material that we are modelling, and i n the following we construct a model for solid-like materials (e.9. metals, polymers, concrete,. . ) whose response depends t o a large extent on their past history (a qualitative explanation of t h i s fact can be given in terms of various microstructural rearrangements due t o dislocatians motions, longchain molecules, phase transformations,...).

.

13

We will account for such microscopic structural rearrangements by the introduction of additional state variables called internal or hidden coordinates and denoted collectively by E which in a certain average global sense represents the internal changes. As is pointed out by S. Nemat-Nasser (129 Ip. 110) : "The representation is macroscopic in the sense that there may exist multiple (in fact, probably infinitely many) microstates corresponding to the same values of these coordinates. However, inasmuch as these coordinates are characterized by certain constitutive relations involving various parameters, which are fixed by means of suitable macroscopic experiments, they signify the most phenomenologically dominant aspects o f the microstructural changes". On? can assume that the hidden coordinates are various tensorial qutntities that transform in a suitable way under a change of frame, here we shall assume for simplicity that F is a symmetric positive definite tensor invariant by orthogonal change of frame. '? 1.3- A thermodynamic process is a set of functions of X E 0 and t a ?

that satisfy (1.2), (1.3) and (1.4). In order to be frame indifferent, the lagrangean stress, the free energy, the entropy and the heat flux are defined as functions ofthe material point and of the actual values of the state variables E, E, 0 g, E (the thermodynamic state) :

.

I n order t o f i x t h e v a r i a t i o n o f t h e hidden coordinate 5 we s h a l l assume ("1 : For a l l X e R , there e x i s t s a function h o f t h a t alonq any process durinq the time i n t e r v a l B

E,

' 6 , 8 , g,

Moreover f o r a l l t o e and a l l E0 there e x i s t s a unique ( (X,t) f y i n q (1.13) f o r a l l t E e and 5&t,) = COW.

5 such

satis-

1.4- The c o n s t i t u t i v e equations o f materials t h a t we have i n mind are based on analogies t o spring-and-dashpot models ; indeed these simple models display q u a l i t a t i v e l y retarded-elastic, creep and r e l a x a t i o n phenomena t h a t are encountered i n polymers, concrete, metals

- (Thermoelasticity)

w

Example 1

5, = h

We take

K

.

0 and S = K( 0 )E

5

elasticity. If

eo

+ A(B

)

. We have the usual thermo-

i s the reference temperature i n

the reference configuration, without stress, we must

- K(8)

eo

write A ( O ) . =

E

i s the thermal d i l a t a t i o n tensor. , ,e = X( 8 ) ( 8 - go) i s the thermal d i l a t a t i o n and thus S = K( €r )(E t eo).

,~(8)(8-0,)

where x . ( 8 )

-

Fig.1 Examp'le 2- (Maxwell body) We have t h e r e l a t i o n s

E

(1.14) 5

1

S

=

K(e )

s

.=

v(e)i

[E

+

eo

-5)

from which i t f o l l o w s

(x) As has been pointed out by G. CAPRIZ and L.M. SAHA 111 ](see a l s o F. 1 33 1) the Clausius-Duhem i n e q u a l i t y implies t h a t eg t h e r depends on the other f i e l d s o r $ i s independent o f 4 SIDOROFF

.

i

One can a l s o consider N Maxwell elements i n p a r a l l e l . This model i s very i n t e r e s t i n g f o r concrete (see Z.P. BATANT 1 3 1 , where i t i s a l s o studied a possible dependence from the temperature and t h e humidity). Example 3- (Standard Sol i d )

s

Fig. 3 Example 4- (Jeffrey's element)

Fig. 4 W i t h respect t o the choice of a good model and t h e influence o f the thermodynamics we wish only quote S. NEPAT-NASSER (1 29 1 p. 110) : "In general, the selection of the hidden coordinates represents a s i g n i f i c a n t problem. An experimentalist can only monitor c e r t a i n *inputs" and measure certain "outputs". The material then represents a black box, whose internal s t r u c t u r e i s manifested through such input-output r e l a t i o n s . The optimal selection of suitable internal variables, minimum i n number, which provides maximum information f o r a given input-output setup, is an i n t e r e s t i n g nontrivial problem outside the realm of therr,odynamics.

Thennodynamics can only provides-ageneral framework within rvhich one must operate. The detailed selection of the parameters, however, must be guided by other considerations". 1.5- Ife shall now recall here some results essentially obtained by B.D. COLEMAN and M.E. GURTIN 1 12 1 and by R.M. BOWEN 1 10 1 on the thermodynamics with hidden variables. I t is clear t h a t i n order t o specify a process f o r the body a l l we need do is t o give the motion p(X,t), the temperature f i e l d 0 (X,t) and the value Fo(X) of the hidden variables a t some instant . tocz G , f o r then a l l the other quantities can be computed. Moreover from the conservation laws one can choose the f i e l d s f (X,t) and r(X,t) t o maintain the motion and the temperature. Theorem 1 (1 12 /

,I 10 1 ).

i ) The Clausius-Duhem inequality (1.8) is verified for a l l X C Q and a l l admissible thermodynamic process A ( i .e. a therand t 6 modynamic process which i s consistent with (1.9) t o (1.13))if and only i f the constitutive .equations (1.9), ...,(1.13) satisfy : (1.18)

$ and ;i a r e independent frbm

and g

CI

i i ) If we assume a priori that h , S and are independent from E, the Clausius-Duhem inequality (1.8) i s verified i f and only i f (1.18 bis)

h

and $

are independent from

a$ and S = (1.19bis) rl=-ae AX a$ j * A 9 (1.20 bis) t r po ac e

po

<

and g

a$ a E 0

0

1.6- The results obtaSned in theorem 1 imply some simplifications in the energy equation (1.4) ; indeed in the case i ) such equation can be written

pee

(1.4 bis)

+

a6

and in the case i i ) (1.4 t e r )

po8

0

=

-

ae

=

tr(s

-%*I;

A

aE

- oiv6+p0r

such equation can be written

+ tr

i

po 35

where

E

t r pO

=

- ~ l v + por A

g

.

1.7- As simple choice of the constitutive equations (1.9)-(1.13). suggested by the examples 12, 3,4 and compatible w i t h the Clausius-Duhem inequality (see th.1) we shall assume from now on the following :

where : BO, k and B1 are positive semi-definite tensors i n order t o ensure the validity of (1.20) ;moreover Bo, A2, A3 and A4 are tensorial quantities symmetric in the f i r s t 2 indices i n order t o ensure the validity of (1.3). and B1 i s symmetric according t o the Onsager principle. Let us also ranark that the expression of the entropy follows from (1.21) and (1.19).

2- A l i n e a r i z a t i o n r e s u l t .

2.1- We s h a l l now study what kind o f s i m p l i f i c a t i o n can be achieved i n the equations obtained i n 3 1 i n the hypothesis o f the i n f i n i t e s i m a l s t r a i n s ; hokever we shall made no assumptions on the v a r i a t i o n o f the temperature 0 (see M. J. CROCHET-P.M. NAGHDI 113 1 f o r analogous considerations i n the case o f thermorheological l y ,simple sol.ids). More precisely, l e t us w r i t e the equations o f 5 I i n a non-dimefisional set-up and l e t us define

. i.x,t 1-11 ax a2ui

sup

at

In the sequel we assume t h a t 6 i s small w i t h respect t o the unity. We shall w r i t e t h a t a function $I i s 0( 6") f o r n 3 0 i f there e x i s t s a constant C > 0 such t h a t 141 4 C 6" uniformly i n a11 the domain o f d e f i nition of 4 . To construct the linearized system we shall o n l y take the terms containing the lowest powers o f 6

.

2.2- From (2.1) we obtain

(2.2)

F = II

+

Grad

.

u

P f U

E =

f [ Grad u + (Grad u ) " ]

E =

$ [ Grad ;+ (Grad ;lx1 + 0( 6 ' )

(2-3)

Therefore i f we define N

(2.4)

E =

( Grad

u

+

(Grad u)'

+ 0( 2)

.

.5

=

$ [grad ;+ (Grad

.

d

4

then E = O ( 6 ) rJ

(2.3 b i s )

E = o(6)

a

E = E

+ 0(62)

Moreover remarking t h a t

1 =

det'F 'density a t time t i s given by (2-5)

p

=

po(l

and

rc.

E = E+0(62) 1

- Div u + 0(62)

- Div u + 0 ( ~ 3 ~ ) )=

po

+

we f i n d t h a t the mass

0(6 )

and so we can consider, i n a f i r s t approximation, t h a t the mass density i s time-i ndependevt, because po = O(1). 2.3-

I n order t o 1inearize the equation o f motion (1.2) we need some informa-

tions on the order of magnitude of the d i f f e r e n t terms t h a t appear i n (1.22). These informations are deduced from the following r e s t r i c t i o n s on the constit u t i ve equations t h a t wi 11 be b e t t e r d i scusszd on two examples

.

Let us consider f i r s t the following i n i t i a l value problem

where

B1, A5 are defined i n (1.24) and where we assume B1 = 0(1), A5 = O(1)

and 4 = 0( 6 " ) . n 0. We have existence and uniqueness o f the solution f o r a11 t € C and we can w r i t e

so t h a t we deduce

and by Gronwall Lemma i~(t)l'6C,6~" Vtte provided f bebounded. C1,C2,C3 are p o s i t i v e absolute constants. We ran prove now e a s i l y the foliowing Lemma.

Lemma 1-

Let h be given by (1.24) and l e t us consider the following i n i t i a l value problem

i=

(2.6) Let

%?be

h(X. E. 2 . 8 . 9.5 )

; E (to) =

C

.

to"

bounded and

then we have ~ ( t =)

4

?(t)+ O(6')

E(t)

(2.7) ~ ( t ) i s the unique solution of

N

5

-

Take y

we have y we have y

d 2 = h(X. E. E,

rrl

= 5

- E0

= 0( 6 ) = O(6')

e

.

u

4

rlt,)

9.5)

= 5,

- Blr~l + A4 E + A5gQl = O(6 ) , then . Putting now 5 - 6 = y and g = -BlA4(E-E) = O(6') 4

and 41 =

4

d

. Q.E.D.

Recalling (1.22) ,(2.3

bis) and (2.7) we can now w r i t e

Let us denote respectively by

.

= O(6 )

J .

where

Proof

- Eo

e0 and So the temperature f i e l d and the

second P i o la-Kirchof f stress tensor i n the reference configuration (where E = E = 0), we have

(2-8)

So(X)

=

from which we define

Po

C A2(X.eo)

A40(.90) S,(X)I

.

B2(X, 8 go) by

We shall also made the following assumptions

Having done the good hypothesis we f i n d t h a t

and so we can define

and we obtain

We can then take as linearized equation of motion the following (2.12)

fl

~iv[S+s,]+

with from (2.91,

(2.10)

pof

=

..

pou

In order to linearize the energy equation (1.4 bis) we remark first of all that (H3) implies 2.4-

and the hypothesis (HZ) implies

Mreover we find from (1.21) and rl =

a$ , --ae

Let us made the following final assumptions

r J u

then, recalling also (1.12),(1.23), we obtain the.fo1loninglinearization(inE,S ) of the energy equation (1.4 bis) :

2.5- S u m i n g up the previous considerations we have done a linearization, only ivith respect to the infinitesimal strain ,the hidden variables and the displacement u under the assumptions (Hl),(HZ), (H3). (H4), (H5). In this way we have obtained the system of equations rJ

(2.12)

(2.4)

Div t s + S o ] + pof

1E =

[Grad u

+

(Grad u)*

= poZ

1

f. F =

[Grad

6+

to be completed with suitable initial and boundary conditions.

(Grad b ) * l

3- Some examples. 3.1-

As a f i r s t example we shall take the case o f thermoelasticity

(example 1,

5

with

po A3(X, 0) = K(X, 0 ) (the s t i f f n e s s ) and

x

1) w i t h

So = 0. From (2.10) we have

i s the thermal d i l a t a t i o n tensor. The only assumption t o discuss i s (H4). I f the variations o f 6 are small near the reference

i.e. 18

- e0l

f ied. Moreover

= O(&)

,1 1

eo,

= O ( & ) , 141 -048) ,161=.0(&) t h e n ( H 4 ) i s s a t i r

,

A Tinearization o f the energy equation (2.15) gives then the classical equations o f the l i n e a r thermoelasticity. These t

equations are coupled by a term o f the type e0 K(eo) ~(8,) E i n the energy equation ; fortunately f o r most applications t h e coupling can be neglected (see the example of BOLEY-WIENER1 5 1 ) i s linear i n 0

.

. Note t h a t the heat equation

I n thecaseof great temperature variations, (H4) may also be v e r i f i e d ; i t suffices t h a t the product

K(X,e)

x(X.8)

(8 -go)

be small as i t

appears j n some metals (see M. RAOUS 1321). Furthermore i n t h i s case the aLAl c, E, which i s o f O(i3). i s negligible w i t h respect t o the term term.

a2r0

z

aA2 ae E

k

which i s o f O(1) and, i n .the same way, the term po8

negligible with respect t o the term p 0

a2n,

O,,2

;

is

The heat equation, which i s nonlinear i n 0 is indeed uncoupled from the motion equation.

3.2- As second example we shall consider the Maxwell model of example 2, 5 1. In this case from (1.14), (1.15), (1.22) and (1.24) we deduce that B1(X. 0.g)

=

P, v-'(x,e)

- poAq(X,O)

=

poA5(X,0)

POA2(X,e)

=

- p0A 1(X.0)

r;

p,A3(X81

K(X,e)

-

= K(X,O)eo with eo = - X ( X , ~ ) ( B go)

where we have So = $, .; 0 i n the reference configuration. We see that the only hypotheses t o be discussed ale (H4) and (HZ), which i n t h i s case are equivalent. Indeed the discussion can be done like i n the example o f thermoslasticity investigated in 3.1. In particular it appears that in the case o f great temperature variations the nonlinear heat equation can be u~icoupled from the motion equation. This fact has also been pointed out by CROCHET and NAGHDI 1 131

.

4- Some results on the nonlinear heat equation.

4.1- Taking into account the examples of the previous section we ssnall a t first study a nonlinear heat equation of the type (4.1)

pO

- ~ i v [k(X.6)

c (9)

Grad 9

1=

pOrI(X.,B~+ por2(X.t)

subjected t o the boundary conditions 9(X,t) (4.2)

=

go(X,t)

nX. k(X,9) Grad 9 =

on

ro (given

temperature)

gl(X,t)

on T1

imposed flux)

nX. k(X.9) Grad 9 + a( 9- g2(X,t))

= 0 on

T2 (radiation condition)

and the i n i t i a l condition

Or, i n the case where r2, go, gl, g2 are T-periodic fT > 0) i n time, the periodicity condition (4.3 bis)

rO,rl, r2

e(x, t ) =

e(X, t+T)

d(t. X I ;

are open subsets of the boundary an such #at

an =

T u F l tJF2.

4.2- The problems (4.1), (4.2), (4.3) and f4.1), (4,2), f4.3 bis] can be solved from the point of view of the nonlinear operator theory. Indeed we can apply theorems of 5 5 o f BARDOS-BREZIS [ 2 1. I f g(X,t) = ,O("), under very mild conditions of the type

(4.4)

the elements k 1X.n ) a r e bounded in rl and measurable i n X fE R 3%~

(x) According t o the trace theorems it is always possible t o make the change = 9 Bo where 8 , = g on and then aI = 0 on To. of variable

-

0

Z - kiSj(X,n)

i ,J

(4.6)

ti t j

rl(Xa rj)

is bounded i n

rl(X, rj)

= rg(X)n

Q

L

&a

i

c2 i

with a > 0

and measurable i n X o r e l s e

+ r4(X) with r3(X) 9 0 and

a(X) i n (4.2)3 is measurable and non-negative, i t is not d i f f i c u l t t o prove t h a t t h e operator (4-7)

- Div(

k!X, 9 ) Grad0 ) - pprl(X. e ) , with (4.2) is an "operator of t h e calculus o f v a r i a t i o n s n i n t h e sense of LIONS [ 24 ; chap.2,§23 and s o of type M (see e.g. Lions loc. c i t . ) . For a proof o f t h i s type of r e s u l t s s e e AMIEL-GEYMONAT I 1 1 and KENMOCHI 121 1 In t h e case of Cauchy problem additional deep r e s u l t s a r e obtained by LADYZENSKAJA-SOLONNIKOV-URALCEVA 1 22 1 .

.

5- Duality and v i r t u a l work.

n be

5.1- Let n = 2

a bounded connected open set i n

o r n = 3) w i t h boundary

an

closed subset o f

an

R

( i n practice one takes

s u f f i c i e n t l y smooth. Let

0. Let v = (vl,.

with (n-1)-measure,

.

anl

-.,vn)

be a be the u n i t

aa exterior t o n )n i s the set o f f i e l d s o f isplacements u = (ul, .,un) with ui E H1( ) , i = 1,. , n ; H1( n ) i s the usual Sobolev space : f o r t h e i r properties see 1 25 1. normal t o

..

..

I f u 4 H1(n )n then the p a c e you on 112

(an!"

; then

uN = I: yo ui vi €

you 6 H o f the trace o f the displacemeklon Let

W

where

a2Q

(a2a ) = 0 y), and

of

~ ' ( n ) ~ ; yo v

v €

=

an

i s a closed subset o f

.

an H

= 0 on

aSl

with

i s well-defined and

112

(an

af

) i s the normal component

and vN = 0 on

(n-1)-measure

ap}

>, 0 ( i f meas

then the condition v, = 0 must be dropped i n the d e f i n i t i o n l e t V be equiped w i t h the h i l b e r t i a n structure induced by

H I ( s-2 )n. Let E be the space of infinitesimal tensor s t r a i n fields, i.e. o f w i t h e.. E L 2 ( n ) and l e t % be symmetric matrices e = (eij)i ,j=l,. 1J the space o f tensor stresses fields, i.e. o f symmetric matrices with s €. L2(n ). The spaces E and S form a s = ('i j)i ,j=l,. .,n i,j dual system w i t h the separating b i l i n e a r form

.

which represents, from a mechanical point ~f view, the opposite o f t h e work of the stress

E

s

i n the deformation

may be i d e n t i f i e d t o S

we shall denote by

11. 11 the

The load space

L

, and

e

. From a mathematical

then (5.1)

corresponding norm i n

and the space

V

p o i n t o f view

represents t h e scalar product ;

E or

S

.

are i n d u a l i t y w i t h respect t o the

separating b i l i n e a r form a v , 4% which represents the work o f the strenath 4 under the displacement v ; i f I$ = (f,h), where f i s a regular volume

R and h a ~ e g u l a rsurface force' on 3n \ alQ having only a tangential component on a+l ( t h i s means t h a t L: hi(x) vi(x) = 0 for i a.e. x E a 2 n), then force distributed i n

I t is easy t o see t h a t t h i s formula is t r u e when f 1. 6 L 2 ( n ) and hi E L 2 ( aa\als2) but i t s validity can be extended t o a much more general situation, a t l e a s t when both boundaries of alR and a a r e regular i n an. 2 D will denote the synunetrfc gradient operator

.

1 I t is a l i n e a r continuous operator from H ( n )" i n t o E Thanks t o Korn's inequality and t o the f a c t t h a t meas ( a n ) > 0 D is a 1 one-to-one bicontinuous mapping from v onto DW and D l f is closed i n IE (see e.g. DUVAUT-LIONS 114 Ichap. 3).

Let t~ denote the transpose of (5.4)

< Dv, s >

D

, defined

= t v r t ~ s >

dj€

.

by

v

. ds

6 3;

I t i s easy t o see t h a t t~ i s linear. continour and onto ; formally t ~ =s + means (we use the followi:~gclassical notations : n IN IfJ Sijvivj* siT 3 z sijvj sNvi and sT = (siT))

S=l

-

and the methods of LIONS-MAGENES 125 1 render t h i s interpretation rigorous. For a more detailed analysis of the duality and the virtual work princ i p l e , see MOREAU 1 26 1 , NAYROLES 1 28 1

.

6- A viscoelastic constitutive equation w i t h periodic coefficients. Let T be a positive number. Let us assume that e(X,t) i s the unique T-periodic solution of the T-periodic boundary value problem associated with the non5inear heat equation (4.1) and l e t us consider the constitutive equation of Maxwell type as i n example 2 o f 91.4, i.e.

-

where eo = x(X,8)( 8- 8,) is the thermal dilatation field which corresponds t o a non-stressed state in the reference configuration R(S, = 0). For simplicity we put

rl

(6-3)

s = S (The total stress)

(6-4)

e = E

d

+

eo (The total strain)

and then

We assume :

AI-

K(X,t) and V(X,t) are s.metrica1 fourth order tensor, measurable and bounded on R X R and such that for almost a l l (X,t)

and there exist 0 < k 4 matrices (V . .) 1J

, 0 < r s7

such t h a t far a11 symmetrical

k z v?. ( 2 Kijlmo(.t) 'J ijm

vijvlm

-i,j

(k

Z v?. i j 1J

W2- The d i f f e r e n t i a l system

atdy + v-'(x,t)

(6.9)

i s uniformely ( i n X € A(X,

= 0

K(X,t) y

Q ) exponentially stable @

.

L e t A(X,~,T ) be the fondamental resolvant o f (6.9) such t h a t T ) = j = I d e n t i t y From A2 we have f o r almost a11 X

where

.

cl, c2 are some positive constants

Furthermore from (6.5).

(6.10) we can obtain and integral correspondance

between s

and e

o f the form (6.11)

F(X,t. r )

aG

where F(X,~,T ) = provided the i n i t i a l that :

Obvious!y,

$(x)

E,(X)

=

[: e(X.t)

- e(X. t ) 1 d~

( X S ~ S) T , G(X,t.r ) = K(X,t) A(X,t,r ) r e s u l t s from a *pastu s t r a i n h i s t o r y 3, such

A

0 (

) ~r

. Ue remark t h a t

we can also j f o m a l l ~ ) w r i t e the inverse form :

( x ) In fact, if V-'K (xx) w i t h aging .

i s s p e t r i c , then A 1

.+. 1 2

w i t h c.

I

.

1 and c2

.

;.

Remarks1) From (6.5). (6.11), (6.13) it appears t h a t t o a T-periodic s t r e s s s(X,t) corresponds a strain e(X,t) = a(X)t + p(X,t) which i s a sum of a secular term a(X)t and a periodic term p(X,t) (and conversely). 2) I t will appear clearly l a t e r on, t h a t the assumption A 2 can be weakened by the following one : A2 bis : The differential system (6.9) has no non-trivial T-periodic solution. However then the integral representation (6.11) does not necesseraly holds.

7- The periodic bilateral problem f o r a Maxwell body.

7.1- We consider in t h i s section some questions concerning existence. uniqueness and asymptotic s t a b i l i t y of a T-periodic solution of the equation of motion (2.12) i n the case of a17 the data are T-periodic in time and with the constitutive equation that we have studied i n 5 6. The data are :p(X)f(X,t) a density of forces i n fi , h(X.t) a surfacic 0 density of forces on a0 la,Q with only a tangential component on a2Q(see 55). e0(x.t)(*)the thermal dilatation f i e l d and two functions ul(X,t) and u2(X,t) which f i x the displacement u(X,t) on and a2Q i .e.

ap

.

A l l the data are assumed T-periodic in time. I t is convenient t o introd x e a smooth function uo(X.t) ( a t least(")uo(t) c H'(Q)") such t h a t uo = u1 on als2 and uoN = u2 on , and put in (2.12)

ap

I t is also convenient t o introduce

where D, e , s are defined in (5.31, (6.3), (6.4) and from (5.4) we can define so as a particular solution of

From (2.12). (6.5).

{w)

(7.2). (7.3), (7.4). (7.5). we have now t o solve

He assume here that the nonlinear heat equation is uncoupled from the motion motion equation.

(xx) I t i s always possible i f the boundary an i s smooth and i f the boundaries in aQ of alR and a2 n are regular.

K ( t ) and V ( t )

I n (7.6)

are l i n e a r continuous h i 1b e r t i a n operators i n E.

I n t h e f o l l o w i n g we s h a l l use the notations : Let

7.2-

h i 1bert. space. L e t T be a p o s i t i v e number. L$ (H) space o f (classes o f ) functions g defined a.e on g(t)

g(t+T),

=

H be a r e a l

denotes the h i 1b e r t

R , T-periodic,

with values i n H and such t h a t

r;r

For these functions we define

and we s h a l l use the decomposition

where

<

= 0 . L e t us also denote by J

(7.10)

V n)

(7.11)

E

= { v

oftheform

v(t)=$t+q(t)

withBEV.q~L;(v)]

{e

o f the f o k

e(t) = a t t p(t)

w i t h a CE,

=

w i t h scalar product (el,

V

and

e2)e =

( al. a2)E +

E ( = S) are defined i n Q 5.

(E))

( ~ ~ ( ~ 1~, ( dt~ 1 ) ~

35 7.3- The dynamical case. Recalling the remark 1 o f r

.

56

our problem i s the following :

4 ,

Find a e L2# (E) , E € E v e V s a t i s f y i n g (7.6). The two f o l l o w i n g lemma are basic f o r the proof o f t h e existence and uniqueness theorem. For d e t a i l s we r e f e r t o

1 9 1.

Under assumptions Al, A2 (cf.5 6), from (6.11) we can define a continuous l i n e a r mapping exists

Y-'

from

s = X ( e ) from

L) (E) onto ( e €

?

FI

to

L$ (E). Rreever, there

= OJ w i t h

Lemma 3Under - Al,

A2 and

:

K i s smooth enough and there e x i s t s y

>

0 such

t h a t f o r a l l syrranetrical (vi j)

(7.12)

ijam then for a l l

e €

e E

(7.13)

'isuch t h a t 6 E L)

< h f ) ,*(e)(t)

(E),

7 = 03

> dl&ylcl

we have

11eIlf

e

E Using v a r i a t i o n a l techniques i t can be proved Theorem 2a)

Exlztenc~

Assume, e , , s o E0 L system 17.6) possess a solution a €

*;

:

Ci

L> (V1) (XI

[x) Using r e s u l t s o f

1 25 1

be satisfied.Thm the (E) ,Al. 12.rJ and (7.12) & L2# (E), be E, v & V w i t h ~ ( t =) B t + q ( t ) ,

we can improve the r e g u l a r i t y i n t.

b)

Uniqueness-

a K 0 (and then.(7.12)is satisfied) the elements o, B ,qrr are If moreover = unique. Any displacement-solution is written v ( t ) = Bt + $(t) 9 C where C 6 V is arbitrary.

7.4- The quasi-static case. can be neglected In many interesting problem, the acceleration term i n (7.6)2 (see e.g. BOLEY-WIENER [ 5 1 and M. RAOUS 132 1 ) ; i t is the socalled "quasi-static hypothesis" We want t o show that the quasi-static periodic problem reduces t o the study of an ordinary differential equation i n E. For t h i s , l e t I = DV C E be the hilbert space nith the scalar product 1 of E. We put E = I 8 J where J = I . Note t h a t ker t~ = J Recalling the "Virtual Work Principle'' (5.1), (5.2). (5.3), (5.4)

.

.

we see t h a t the equations (7.6), i n the quasi-static case, become

= 0

(7.6 bis)

and are equivalent t o

(7 -6 t e r )

I

~ ( t )L I

E

, \(V

E I,

~ ( t ) + K(t) 5

=

G

v. ~ ( t (E )

-E

)

+

~ ( t eo )

- ;s

= 0

K ( ~ ) E + K(t) eo

5 I, and our problem i s t o find ~ ( t =) a t + p(t) w i t h a .5 I, p ( t ) t 5 ( t ) = a t + q ( t ) with q ( t ) G E (the same secular term as E in order t o have a ( t ) T-periodic). We refer t o 1 9 I f 17 1 f o r d e t a i l s concerning the proof of the existence and uniqueness theorem. We have f i r s t l y :

Lemma 4-

Let the assumption A1 be satisfied and l e t eo,-s&L; (5). Then for a.e. t 15. R, there exists an operator L(t) : E + I such that L(t) = L ( t + T ) , JI L(t)ll 4 l/k and

Moreover i f we put (7.15)

x(t) = ( j

we have a.e.

- K(t) L(t))K(t)

(j = identity i n E)

t 6 R

1) g ( t ) i s a T-periodic symmetric uniformely bounded operator from E to J 2) ker $ ( t ) = I 3) dx d J , e x, X ( t ) x > II II XU + l D ( t ) ~ ( t ) x l I ~ I -5 E

> L

Prom t h i s lemma, problem (7.6 t e r ) i s equivalent t o (7-6 quart.) (7.6quat-t.)

:Find

V(t) r(t)

~ ( t ) =a t + q ( t ) w i t h a B I , q ~ L ) ( E ) s u c h t h a t

i + x(t) 5

=

r ( t ) with

= K ( t ) L(t) (so

- K(t) eo) G L i (E)

which reduces t o (7.6 quint.)

: Find a E I ,

q E L i (E) such t h a t

In order t o solve t h i s l a s t problem the following l e m a i s basic. Lemma 5If there exists a T-periodic solution z

E

the3 (7.6 quint.) has a solution a -

#

L2 (J) t o the equation #

E I , q € L2 (E) given by

.

where c I is arbitrary. Conversely, i f a and q = q Isolutions of (7.6 quint.), then z = q", s a t i s f y (7.16) O

+ q,

are

We can give now the main theorem.

Theorem 3Let A 1 ,(7.12) be s a t i s f i e d and l e t eGso (E) Then (7.16) has one and only one solution z E L$ ( J ) such t h a t i E L$n- (J). Consequently problem (7.6 t e r ) has a solution (2 , and a l l the solutions A 4 are E = E + c 5 = 5 + c where c E I is arbitrary'. Moreover the corres= K(t) ( 2 2) + K(t) so-e, is unique 8 ponding stress

gL2 .

.

2)

-

-

.

We study now the asumptotic s t a b i l i t y of $ This r e s u l t is useful f o r the numerical investigation of the problem (see 1 17 1). Because t h e operators L(t) and x ( t ) a r e T-periodic l i n e a r and continuous the d i f f e r e n t i a l equation (7.6 quint.) is an ordinary l i n e a r , T-period i c d i f f e r e n t i a l equation i n E, and the Cauchy problem

Go

has one and only one solution, C I and h L$ (E). From t h i s , it can be proved the following asymptotic s t a b i l i t y theorem : Theorem 4- (Asymptotic s t a b i l i t y of $ ) Let A 1 ,(7-12) be s a t i s f i e d and ac be the Cauchy stress-solution correspondinq t o go and eC = ~ ~ ' 0 . Then we have

39 (7.20)

$&

-1 c uc.

K

so t h a t , from (7.12)(x) (7.21)

11 uC(t)ll

(

E

(x) I t can be proved t h a t

meaning.

uC,

-

gc

+

v

1

exp

-k

( 0C '

11 oC(0)ll

E

exists and

- 7K

-

ic k L

~

K

t a-e.

i (E)

]

U

=~

0

t b 0 0

and so uc(0) has a

8- The periodic unilateral Signorini problem. 8.1- We are dealing now with a signorini(*) periodic viscoelastic problem i n the quasi s t a t i c case. In a naive formulation we impose on a part a4R of the boundary an the complementary possibilities between the unknown displacement v = u - u o (u) and the unknown s t r e s s o = s so :

-

The f i r s t corresponding t o "no-contact and no-reaction" and the second one t o "contact w i t h only normal reaction", i .e. "contact without friction". The unknown s t r e s s o balancing the reaction on a 4 Q 8.2- Example . Let u s consider a plane medium with a crack (on a4n ). This medium i s submitted t o T-periodic forces or stresses of mechanical o r thermal nature. This problem, f o r a given configuration of the crack, bears unilateral constraints since the edge of the crack can part but cannot interpenet r a t e each other. The crack i s supposed t o l i e i n a plane of symmetry for the mechanical problem, which allows us t o formulate Fig. 5 conditions of contact mathematically identical t o those of contact without f r i c t i o n , as i n (8.1) (for deta5ls see

1

8

1).

(x) Ti?? Signorini problem has been f i r s t solved by G. FICHERA~5 [ i n the elast i c ca e. ( X X )~i tag, u0 = o on a4n

.

41

8.3- In order t o give a rigourous formulation of t h e Signorini problem we must f i r s t of a1 1 precise what vN 4 0 on -a@ means. Let a3n and a4n be open d i s j o i n t subset of an w i t h (n-1)-measure 2 0 such t h a t an = aln u a2h 0 3 3 IJ and we shall suppose t h a t on a2n U a3n a regular surface force, with only a tangential component on a2n a is given (the tangential force i s zero on a2R i n case of example of Fiq.5). , Let ~ ~ ' ~ ( a $ U a4n ) =($ & H ' / ~ ( $ Q) ; supp$sagLJa4n ;it i s a closed subspace of HI/'( an ) and s o i t i s a h i l b e r t space f o r t h e induced norm ;

ap,

1

B = HX1/2 (a4* ) = [ $ € L2(a4n) ; t h e r e e x i s t s * e ~ : ' ~ ( a ~ua4n n ) with

i t is a h i l b e r t space f c r the natural quotient topology. ; 4J E Ho1/2(a3Q c)a4n ) and q = $ r;l

a4n

3.

I f the boundary of a451 i n an i s regular a more manageable characterization of H:/' (a4n ) i s given i n 1 25 1. Let L be t h e usual positive cone i n L2(an ), i.e. L = { $ € L ~ ( ~)R ; f o r every f representative of $ , f ( x ) 2 0 a.e. x e an 1; then L2 (28 ) is an ordered topological vector space, moreover L2(an ) is a l a t t i c e , i.e. f o r every f i n i t e family[ml one has sup{$ @ n l c L 2 ( a n). Using tne t r a c e theorem and t h e order properties of H ( R ) established by LEWY-STAMPACCHIA 1 23 1 i t i s possible t o prove (see 1 20 1 ) t h a t t11/2(a~ ) i s an ordered topological vector space s u b l a t t i c e of L 2 ( aR ). The same r e s u l t s a r e valid f o r H ; ' ' ( ~ ~ u a4n), obviously. Let now d ii~'2(a4Q) then we define

-

-

,..., mnl

....,

supx ( 41 ,$ 2 ) = r e s t r i c t i o n to' a4n of sup($ 1 2) where JI i I a i = 1, 2 and . qi 6 Ho1/2 (a3n Ua4n)

,+

p=

$i*

I t i s e a s i l y seen t h a t does not depend on the choice of t h e representatives ql and q2 and a.lso t h a t t h i s d e f i n i t i o n coincides with i n L2(a4n). In the same way t h e positive the definition of 6 of H:'*(~~G) i s the r e s t r i c t i o n t o a4D of t h e positive cone cone --

-

42

H:"(a3Qua4n)

of

and we have

B = H : ' ~ ( ~ ~ Qi )s an ordered topological vector space sublattice of

-

L~ (a451) Let now L :

V-

~ : ' ~ ( a ~ nbe ) defined by

L v = restriction t o

a4n

of vM

Obviously L is linear and continuous ; moreover L i s onto as composition of the trace application u ~ ( y ~ = u ~ ) and~ the projection of a vector of IR" onto the linear subspace generated by v(x). We can now formulate rigorously the condition vN < 0 on a@ as

6 , the dual space of IB , is a hilbert space of d i s t r i h u t i ~ n sdefined in aR that can easily be characterized by using the r e s u l t s of LIONS-MAGENES125 The duality between IB and C will be denoted by a point and b g is the virtual worK of the normal contact force g f o r the virtual displacement b normal t o the boundary ; when g e L2 (a4Q) then

.

KO

Let C 6 be the polar cone t o can be written as :

5

.

then the condition uN ( 0 on a4n

The transpose t~ : 6 +.lL of L for the duality bc-tween B , 6 a n d v , IL i s linear continuous, one-to-one and has a closed image t~ 6 in IL Thanks t o t h i s f a c t K and have mutually polar images 5. L-'(KJcv = t~ KO c r h . Moreover thanks t o the i n j e c t i v i t y of D and the and closure of D V in IE , and also have mutual 1.y polar images t l o C = D LCS IE and CO = D) 1CS The situation i s sumiired up i n the fol lot~ingdiagram :

.

GO

-

-

-

-

.

KO

GO IG-

.

It is now easy to see that the unilateral constraints (8.1) can be written as the complementarity system :

(8.4)

L v E

-J

,

aN

E

-KO,

Lr.uN = 0

or in the two others equivalent ways :

An other formulation of (8.6) is

8.4- We give now a mathematical formulation of the periodic Signorini problem. Because we seek v(t) of the form v(t) = $t + q(t) with 8 e V, q 6 L$ (V), it is impossible to satisfy vN(X,t) $0 on aqQ for a1 1 t € I? However we are interested by the asymptotic character-of the periodic solution and so we shzll only ask that the unilateral conditions b2 satisfied for all t to. More precisely, let us consider the following linear continuous mapping

.

z

which can be deduced f r o m (6.11) provided Al, A2 be satisfied. The periodic Signorini problem for a Maxwell body can be stated as follows, recalling (8.7) :

44

(8.10) : Find ( t o , a, p, a ) E 1RxExL2 (E)xL2 (E) such that # # i ) a.e. i i ) a.e. VV

t 6 If?

o ( t ) + sO(t) = % ( a ) ( t ) + $(eo+

t h

[ .tO,+a,

-6 - C

,

p)(t)

~ ( t =) at + p(t) 6 - C and

- c(t),a(t)>>

0

With the method of convex analysis i t can be proved that this problem can be reduced t o (see 1 7 1) : (8.10 bis) : Find ( a i ) a.e.

t

ER

.

p, a) L E x L> (E) x L$ (E) such that a(t)

+

so(t) =

&O ( a ) ( t ) +gl(eO+ p ) ( t )

8.5- A mechanical comment. Connection between the formulations (8.10)

and (8.10 bis) can be more easily understood i f they are expressed in terms of the duality between IB, the space of the restriction on a451 o f the normal t o the boundary displacements, and 6 the space of associated normal forces. In these terms (8.10) becomes (8.11)

f o r a.e.

t > to : b ( t ) 6-5

,

g(t)

62,b(t).g(t)

where b(t) should be of the form b(t) = B t + q ( t ) w i t h B EIB q E L2 ( IB). I n the same terms (8.10 bis) becomes

= 0

and

#

(8.13)

f o r a.e.

t 6 R

q ( t ) E-L,

g(t)

,

q(t)-g(t) = 0

The cjnsti tutive law is implicltely determined as the general solution o f a boundary value problem on i2 , that we shall write :

where E 2 ( q )

depends only on

z,

i.e.

=

3f2(q).

Equation ( 8.12 ) implies, a t l e a s t formally, t h a t f o r a.e.

x

€a@

and furthermore, since

%(x)

B(x)

= 0

g i s non-positive,

i s zero (together w i t h a l l Hence we see t h a t the normal reaction g(x,t) the other reactions, thanks t o T si = 0) a t a.e. p o i n t x where the secular term B(x)

i s negative ; t h i s i s a mechanically obvious r e s u l t : f o r time t

large enough there w i l l be no contact a t t h i s point. Moreover i t i s easy t o understand why q may be choosen as a negative function. Indeed l e t (b,gj be a solution of (8.11) and (8.12) and l e t bo be a constant f i e l d belonging t o bo. then (b

+

5

B

such t h a t

= 0 and f o r a.e.

t EIR

q(t)

+

bO, g) i s another solution of (8.11) and (8.14)

boE-s as i s seen d i r e c t l y .

I f we choose, as i s possible

then (b + bo, g) i s a solution o f (8.11) and (8.14) and so we may ask q t o be a negative function a t l e a s t i f ( 8.:5 ) defines. an element o f IB , Hence i t appears natural ask f o r the existence of such a supremum which w i l l be ensured using the r e s u l t s o f

8.6-

17

/and

1 20 1 .

I n order t o solve problem (8.10 bis) o r equivalently problem (8.10) we

must introduce the following assumption on K and V A3- &,t)

i s o f the form V(X,t)

= v ( t ) Vo(X)

.

where

twv(t)

is a

T-periodic scalar function, continuously d i f f e r e n t i a b l e and w i t h p o s i t i v e values and there exists a positive constant

JJ K(X,tl)

- K(X,t2)J

kItl-

t such t h a t f o r a l l (X,

t

t ),

1' 2

t21

b!ith A 3 the periodic Signorini problem (8.10 b i s ) can be stated i n terms o f E ar,d a* = a/v

which s a t i s f y the system :

Assuming A t , A2 defined i n (8.81,

we denote by J(:

(8.9).

A;

and

It can be proved (see

the corresponding mappings

Lemma 6Assurni ng Al, A2 and A 3, -the 1inear mappings (8.8).

(8.9) and corresponding t o

K(t) vel(t)

.

1 7 1)

and

:Yo

and 9(; defined i n are continuous.

Furthermore

ii)' d p e ~ ; p E)~

;(p)

=

X;(*p)

and $;(P)

= 0

From t h i s lemma we can prove Theorem 5Assuming A l . A2 and A.3,

eo,-6,

s ,? & i L2 (T ;E) then : -0-0-#

The problem (8.10 bis) has a solution

Let ( ~l,-pl~ul) -

( a, p a ),

be a solution ; then (-9Lp2,-v2)

i s an another solu-

t i o n i f and only i f : a1 = a2

,

N

p1 =

4

al = u2

p2,

Furthermore we can choose pl

-

t h a t every solution

-

,

p2(t)

e-g

such t h a t

p2 must be w r i t t e n

sup

pp =-pI

=

and

(p ) = 0 t E 1

+, where

.

o

t h i s rneans

h E

-g

and c ti, al > = 0. I We refer t o

P

{

S~PE P )

17

land] 20

1

for d e f i n i t i o n and .properties o f the mapp.ir;g

and f o r d e t a i l s o f the proof.

Let us also point out the following mean asymptotic stability result Theorem 6Let (al, cl) and (02, s2-) be solutions of the Cauchy Signorini corresponding to the same %,-so but to the initial i$,-E;. Then for till a > 0 .tx+a

( x ) For Cauchy probleas see 114 land

1 7 1.

Bibliography

1

11 R. Amiel, G. Geymonat- Viscous Fluid Flow i n Chemically Reacting and Diffusing Systems.. Applications of Methods of Functional Analysis t o Problems in Mechanics, Lecture Notes in Mathematics, 503, Spri nger-Verl ag (1976).

.

1

21 C. Bardos, H. Brezis- Sur une classe de problemes d'Cvolution non lin6aires. 3. Diff. -Eq., 5 (1969), 345-394.

1

31 Z.P.

1

41 6. Belloni , 6. Bernasconi- Sforzi Deformazioni e lor0 legami .

I

51 B.A.

1

61

1

31 R. Bouc, 6. Geymonat, M. Jean, B. Nayroles- Cauchy and Periodfc Unilateral Problems f o r Aging Linear Viscoelastic Materials: Journal of Math. Anal. and Appl. 61 ( 1 P 7 7 \ , 7 33

/

81 R. Bouc, G. Geymonat, M. Jean, B. Nayroles- Hilbertian Unilateral Problems i n Viscoelasticity. I.U.T.A.M./I.M.U. :"Applications of Methods of Functional Analysis t o problems in tlechanics'l',Lecture Notes i n Mathematics, 503, Spri nger-Verl ag , 1976.

1

91

Bazsnt- Theory of Creep and Shrinkage i n Concrete Structures : A'Precis of Recent Developments. Mechanics Today, 2 , Pergamon Press, (1975). Editioni Scientifiche Universitarie. Tamburini Editore, Milano,(1975]

Boley, J.H. Uiener- Theory of thermal StressesJohn Wiley & Sons, (1960).

R. Bouc, 6. Geymonat, M. Jean, B. Nayroles- Solution periodique du probleme

quasi -statique d'un sol ide visco~lastiqdeZ coefficients pErio Jiques . 3. Mi5canique. 14 (1975). 609-637.

-

R. Bouc, G. Geymonat- Linearized Thermoviscoelasticity w i t h high temperat u r e Variation and related Periodic Problems. To appear.

f 101 R. M. Bowen- Thermochemistry of Reacting Materials. 3, of chemical Physics, 49, no 4 (1968).

-

1111 6. Capriz, L.M. Saha- Anelastic deformation as an internal parameter. Meccanica, 11 (1976), 36-41. 1121 B.D.

Coleman, M.E. Gurtin- Thermodynamics w i t h Internal State Variables. J. of chemical Physics, 2, n02, (1967).

.

1131 9.M. Crochet, P.M. Naghdi- On Thermo-Rheologically Simple'Solids In Symposium East Kilbride 1968, Springer-Verlag, Mien, New-York; (1970). 114i G. Duvaut, 3.L. Lions- Les 'ini5quations en Rscanique e t en Physique.

Dunod, Paris, (1972).

1 151

G. Fichera- Problemi e l a s t o s t a t i c e con vincole u n i l a t e r a l i il problema

1 161

P. Germain- Mecanique des milieux continus, I- Masson, Paris ,(f972).

1 171

6. Geymonat, M. Raous- Elements f i n i s en viscof2lastisit6 periodique. Seminaire du Laboratoire dlAnalyse Numgrique, Univ. Paris VI, mai 1976.

1 181

D. Graffi- Sulla t e o r i a del le o s c i l l a z i o n i e l a s t i c h e con e r e d i t a r i e t a . Nuovo Cim.. 5 (1928), 310-317.

de Signori ni con ambigue condizioni a1 contorno. Mem. Accad. Naz., Lincei, 8, (1964), 91-140.

1191 W.K.

Hatt- Notes on the e f f e c t of Time Element i n Loading Reinforced Concrete Beams. Proc. A.S.T.M. Z (1907), 421-433.

1201 M. Jean- Some complete L a t t i c e s of 186-195.

w'*~(Q). B e l l . U.M.I.,

g , A (1976),

[ 21 1 N. Kenmochi- Pseudo monotone operators and nonf i n e a r e l l i p t i c boundary value problems. 3. Math. Soc. Japan,

1221 O.A.

27,

(1975), 121-149.

Ladyzenskaja, -V.A. Solonni kov, N.N- Ouraltceva- Lineai- and quasil l n a a r Equations of Parabolic Type. Amer. Math. Soc., 13968).

,=,

1231 H , LewyS G. Stampacchia- On t h e r e g u l a r i t y of t h e safri.tian o f a varfat i c n a l inequality. Comm. Pure Appl . [.lath153-188, (19591. 1241 J.L.

Lions- Quelques methodes d e resolution d e s prob?Smes aux l i m i t e s nonlineaires. Dunod, Gauthier-Qillars, (1969).

1251 J.L. l i o n s , E. Magenes- Problemes aux limites non homog$nes e t appljra%ions, I e t I I . Dunod, P a r i s , (1968). 1251 9 . J , Moreau- On u n i l a t e r a l c o n s t r a i n t s , f r i c t i o n and p l a s t i c i t y .

In "New Variational Techniques i n Mathematical Physics", PI c i c l o C.I.M.E., 1973, 171-322, Ed. Cremonese, Rome (1974).

1271 E.W.

Morland , E.H. Lee- %tress Anal;ys-is f o r Linear Viscoelastic Materials with Temperature Variation. Trans. Soc. Rheology, -4, (1960), 233-263.

1281 B. Nayroles- Point de vue algebrique, convexitii e t integrandes convexes en mecanique des solides. In "New Variational Techniques i n Mathematical Physics",II Ciclo C.I.M.E. 1973, 323-404, Ed. Cremonese , Rome, ( 1974)

.

] 29 1 S . Nemat-Nasser- On Nonequi 1i brium Thermodynamics of Continua. Mechanics Today.

1301 F.K.G.

2, (1975),

94-158.

Odqvist, N.G. Ohlsort- Tfiermal Fatigue and Thermal Shock Investigat i o n s . In Symposium East Kilbride on Thermoinelasticity, Springer-Verlag, Wien, New-Yorb (1970).

1311 A. C. Pipkin- LecV~reson Viscoelastici t y Theory. Springer-Verlag , (1972). 1321 M. Raous- These (en preparation). Lab. Mecanique & Acoustique, C-N.R.S. Marseille. 1331 F. Sidoroff- The geometrical concept of intermediate confi uration and e l a s t i c - p l a s t i c f i n i t e s t r a i n . Arch. Mech.. g (7973) ,299-308. 1341 T. Vogel- Theorie des systemes Cvolutifs. Gauthier-Villars, Paris (1965).

/ 351 E.

Vol t e r r a - On e l a s t i c continua with hereditary c h a r a c t e r i s t i c s , J. Appl . Mech., (1951). 273-279.

18

1361 E. Vol t e r r a - A mathematical i n t e r p r e t a t i o n of some experiments on p l a s t i c s and rubberlike materials. Proc. 2nd Intern. Congr.Rheology, (1953), 73-78. 1371 V. Volterra- S u l l e equazioni integro-differenziali d e l l a t e o r i a d e l l 'elast i c i t5. R.C. Lincei (5), 18 (1909). 295-301. 1381 V. Volterra- Vibrazioni e l a s t i c h e nel caso d e l l a ereditti. R.C. Lincei (5), 21 (1912).

1 391

V. Vol t e r r a - L e ~ o n ss u r 12s fonctions de lignes. Gauthier-Vill ars , Paris , (1913).

1401 V. Vol t e r r a - Sur l a theorie mathematique des phiinomiines h6red-i t a i r e s . J. Mach. pures e t appl.. (1928). 249-298.

CENTRO IN TERNAZIONALE MATEMATICO E S T I V O (c.I.M.E.)

THE THERMODYNAMICS OF MATERIALS WITH MEMORY

W.A.

DAY

C o r s o tenuto a B r e s s a n o n e d a l 2 alllll giugno 1977

The Thermodynamics of Materials with Memory Eight lectures given at the Centro Internazionale Matematico Estivo, Bressanone, Italy, June 3-11, 1977.

W.A.

Day

(University of Oxford).

Contents Introduction.

I.

Thermodynamic restrictions on hereaztary response functionals

2,

Thermodynamic restrictions in mechanics.

3.

Thermodynamics based on an inequality.

.

Bibliography,

Introduction The mathematical study of thermodynamics for materials with memory began in 1964 with the publication of two important papers by Coleman'''.

In the previous year Coleman and Noll(*) had

proposed to interpret the second law as the proposition that in all processes which are compatible with the balance laws and the constitutive relations the Clausius-Duhem inequality

holds for each body 8 , where n is the unit outward normal to the boundary a ~ ,-q is the entropy density, r is the heat supply density, q is the heat flux vector and 9 is the absolute temperature.

Coleman explored the implications of that proposal for an

extensive, but not universal, class of materials with memory and thereby he initiated the subject of theee lectures. Although some of the resu-ltsI shall derive are the same as, or closely similar to, thosg of Coleman and of others who have followed up Coleman and Noll's proposal, the route by which I

(1) See [4] and 151. (2) See [ 2 ] .

shall arrive at the results will be quite different.

I advance

three reasons for going about things in another way. First,it is always an advantage to have available alternative approaches to a subject which will illuminate each.other and set up an exchange of ideas between them. Second, it must be remembered that Coleman and ~011's proposal is precisely that

- a proposal.

~t has proved to be most

fruitful but, at least in my view, there is no compelling reason for according it more than provisional status.

There is, there-

fore, much to be said for an alternative approach which starts not from the Clausius-Duhem inequality but from some agreed common ground and which attempts to move out from there into unfamiliar territory.

For tha.t reason tho first concrete situation

I shall examine will be mechanics, where it is easier for us all to agree on what are realistic thermodynamic assumptions,

I

suspect too that many of you who want to use the results are interested primarily in mechanical applications and naturally you will want those grounded on assumptions as free from controversy as can be.

Later I shall talk about genuine thermodynamics.

Third, there is a difficulty for Coleman and Nollls proposal in that it introduces entropy for materials with memory ab initio, as a concept which is well understood from the outset, whereas I think that entropy best enters as a derived concept.

I must make

it clear, though, that whether one finds a difficulty for their

proposal at this point depends upon one's presuppositions about the ingredients which can go legitimately into a physical theory and I shall not pursue the matter further here, (3) The plan of the lectures is as follows.

Part 1 (lectures

1-5) lays the abstract mathematical foundation for part 2 (lecture 6), which deals with thermodynamic restrictions in mechanics, and

for part 3 (lectures 7,8) which approaches genuine thermodynamics from the standpoint of a certain inequality Clausius-Duhem inequality.

- which is not the

In this approach entropy enters as a

derived concept and its values can be computed, at least in principle. Many of the results I shall present have appeared already in

my Springer Tract [ 8 ] ,

Since I am addressing an audience of

mathematicians I shall indulge a more austere style than that of the tract.

( 3

I have elaborated this objection in 1121.

1.

Thermodynamic restrictions on hereditary response functionals. In order to avoid repeating certain fundamental definitions,

arguments and constructions which are employed extensively in what follows I shall begin by expounding them in an abstract setting so as to dispose of them at the outset. Throughout R denotes the real numbers and V is a real Hilbert apace of elements a, p, y,

... equipped with the inner product (a,

@)

r, and the norm 1 l u l l = (a,a)

.

u

is a nonempty, open and arcwise

connected subset of V. A

map

:

U + R is smooth if there is a continuous S:U+V

such that for each a

E

U and.each c > 0 there is a neighbourhood

N of a such that N c U and

for all 3

E

U.

Here 5

3

Dcp is the derivative or gradient of ~ p .

A localised process of duration d 2

which is continuous and piecewise C

.

-

f > 0 is a map f: [O,df]+U f connects f (0) to f(df)

and it is said to be closed if f(0) = f(df).

The constant local-

ised process with value ri add duration t 2 0 is denoted by a (tli.e.

I £ f,g are localised processes and f(d f) = g(0) one can define a localised process f o g , of duration d +d f

n*

g*

which is the continu-

of f with g, by t

E

10,dfI

, t

E

fdf.dffdg1

r

(fog)(t) = s(t-df)

If f is a localised process and t

ei

'

[O,d f] the truncation of f at

t is the localised process f of duration t for which [O,tI

If a > 0 then fa is the localised process of duration a -'d

f which

is defined by

f

a

is a retardation of f if a < 1 and an acceleration of f if

The history of f up to t whose values are f (t-s) ft(S) =

*

E

[O.d ] is the map ft: [0,m) + U f

:

s

=

s E

[O.tl.

[t,m).

H denotes the collection of all histories of localised processes.

The constant history associated with an element a

6U

is a* i.e.

A

hereditary response functional is a map o

:

H

+ V such

that (i) for each localised process f, the map t

.r

u(f

t

is piecewise continuous on [O,d ] and (ii) the associated f equilibrium response function u*

: U -r

V, whose values are

,

o* fa) = o(a*)

a a U ,

is continuous. The following are three simple examples of hereditary response functionals. Let 5

Example 1.

Here

5.

IJ* =

Example 2.

:

U

-t

V 'be continuous and define

Such a o is elastic. More interestingly, let L(V,V) be the s ~ 3 c eof all

continuous linear maps from V into V, equipped with the norm sup(l IAul l

l lAl l let G r [O,W) G ( w) E

4

: 'aE

V, 1 la1 l = I),

L(V,V) be smooth and let it satisfy (i) there is

L(v,V) such that

and (ii) l6(t), ldt <

fl

-,

0

where the dot denotes the derivative, and let

Here u*(a)=G(=)a

,

Such a a is linear viscoelastic, G is the relaxation function, G(0) is the instantaneous modulus and G(m) is the equilibrium modulus. Example 3.

Let 5 : U x V + V be continuous and define

where ?(t-) is the backward time derivative of f at t. a*(n) = I(a.01

,

a

E

Here

U.

Such a a is of the rate type. If f is a localised process 1 set

df

w*(f,-

J

(u*(f(t)) ,zct, ,at.

0

I choose not to make V,U or a explicit yet but it may be t found helpful to think of f (t) and o(f ) as the strain and stress, respectively, at the time t.and of w as the work done.

If no, nl. n2 are maps which are defined on U,H,H x U, respectively, and which take values in a common codomain I say that =1*

extends n

0

if ni(a*) = no(a)

and that R extends 2

;r

1

,

a e U,

if

t n2(f ,f(t))

t = nl(f 1 .

t f

E

a.

With n

n2 are associated maps n*,n* which are defined on 1

n; (a) = nl (a*) ,

9(a)

a

2

n2 (a*.a)

,

U

by

n. t U.

~ h u sif nl extends n0 and rr2 extends n1 then no = n; = n;

.

The following is a standard result: 1.1 Supp3se that -

element a

0

5

:

U .r V is contiriuous and that for some fixed

e U and some fixed c c R,

J

f (5(f(t)).ht))dt

0

,

c

for every closed localised process f w i t h f(0) = ao. is a smooth map cp

:

U + R w&Dp

Then there

= 5.

The hereditary response functional

0

with thermodynamics if there is a fixed a

0

is said to be compatible F

u such that

w(f1 2 0 for every closed localised process f with f(0) = ao.

a is said to be elastic under retardation if, for every localised process f, lim w ( f a ) = w*(f). a*

Trivially an elastic hereditary response functional is elastic under retzrdation.

With little effort a linear visco-

elastic functional and a functional of the rate type can be shown to be so too.

If o is compatible with thermodynamics and 1.2. retardation there is a smooth map q

:

elastic under

U + R such that

1.2.1 a* = Dcp, and, if f is any localised process -

Proof. -

c U be as in the definition of compatibility with

Let .a

thermodynamics, let g be a closed localised process with g(o)=yy,0 and let a g,(o}

7

= .a

0.

Then ga too is a closed localised process with

and so ~(57,)

On letting a

2 0.

-+ o it follows that

for all such g and, in view of the definition of w*, of p with the properties 1.2.1

and 1.2.2

the existence

now follows.

The next three resu3ts are corollaries of 1.2. 1.3. -

If 0

is the elastic functional of example I and is compatible

with thermodynamics then

S = Dcp. Proof.

Immediate.

If a 1.4. -

is the linear viscoelastic functional of example 2

is compatible with thermodynamics then

G(W)

e L ( V , V ) is symmetric

and pfa)

Proof.

a

%(a.Gf*)a)

According to 1.2.1

and

s

a r U.

there is a smooth 9 such that

Hence G(W)

,

= ~'p(a)

is symmetric and the result follows. 1.5. If a is the functional of the rate type of example -

3 and is

compatible with thermodynamics then S(a,O) Proof.

a

Dvta)

.

a

E

U.

Immediate. Further progress is possible if an additional restriction is

imposed upon a.

g

is said to have fading memory if whenever f and

g are localised processes with f(d ) = g(o) then f lim w(f o g(o) o g) = w(f) t+w (t)

+

~(9).

Each of the examples 1,2,3 has fading memory in this sense but note that example 3 does not come Within the compass of Coleman's principle of fading memory 151 Convention.

.

From here on a is always assumed to be compatible

with thermodynamics, to be elastic under retardation and to have fading memory. The additional assumption of fading memory ensures that v, whose existence was established in 1.2, has the following properties. 1.6. -

L e t f be any localised process, let t

a [O,d ] and let e > 0. f

and, in particular, 1.6.2 w(f) 2 0 if f is any closed localised process. -

Moreover, whether f

closed or not,

for all sufficiently small a 1.6.4 -

7

0, and also

t

J

(a(fs),g(s))ds

2 g(f(t))

0

If -

8~

has a global minimum at f(o)

Proof.

Let g

1

E

U

- ip(f(o)).

then

be a localised process connecting a

0

to f(o), let

g be a localised process connecting f(d ) to a and let tl,t2>0. 2 f 0 The localised process

is closed and connects a

0

to .a

and so compatibility with thermo-

dynamics demands that

Letting t + m and appealing to the fading memory of a yields the 2 inequality

and letting t

1

+

and appealing to fading memory again gives

w

w(g,)

+ w(f) + w(g2) 2 0.

This last inequality must still hold if g and g are replaced by 1 2 their retardations i.e. ~ ( g ~ + , ~ ()f )+ ~ ( g ~ >* 0~, ) and taking the limit as a + o and using 1.2.2

which is 1.6.1

and from which 1.6.2

it follows that

follows,

Not only does f connect f(o) to f(d ) but so does each of f the retardations f

a

and, according to 1.6.1

w(fa) 2 cp(f (df))

- cp(f

(0)) =

and 1.2.2,

lim w(f a) , a+o

which implies 1.6.3. Finally, 1.6.4

f

follows by applying 1.6.1

to the truncation

and 1.6.5 is an immediate ccnsequence of 1.6.4, [o,tI Note that 1.6.2 says that w is nonnegative on any closed

localised process whereas compatibility with thermodynamics demands only that w be nonnegative on closed localised processes starting and ending at the element a r U. 0

is never less than the change in

9

will be generalised below as 1.7.2.

1.6.1

says that w(f)

introduced by f: that result 1.6.3

made arbitrarily close to the change in

says that w can be by the expedient of

retarding the localised process sufficiently,

Thus far the argument has established the existence of g :

u

-r

R with the satisfactory properties displayed in 1.2

Moreover, 1.2.1

1.6.

tine matter. on o that

ensures that the computation of

p

and

is a rou-

It is a major consequence of the assumptions made can be extended to

+

:H

+ R, in the sense of. 1.7.

However, the computation of $ is usually far from being routine if a has a genuine memory. 1.7 -

There is a map ) : H + R which extends q and which is such

that 1.7.1 -

if f is any localised process and 0

and

L.

-

0

and

proof. -

Let f be a localised process and let t

Localised process f d

1

t s t and ft = f ; 1

£1-

The truncation f sild t.

- tl 5 df,

5 t0 c

If f 1

5

[0,d 1. f

A

is said to be a continuation o f , f at t if it is a closed continuation if f (d

)=f(t).

£1

is an example of a closed continuation of f [o,tI is a closed continuation of f at t then, according

and consequently d,

where the right hand side depends upon the history ft only. Hence d, r =1 m(ftJ = sup(( f ), (s)d : f is a closed continuation 1 jt t which is the maximum recoverable work associated with f , is finite. t shows that m(f ) is nonnegative i.e. Choosing f = f 1 [o,tl

Moreover, it is easy to see that if ft is a constant history, a* say, an4 if f is a closed continuation of f at t then f is a 1 1 closed localised process and, because of 1.6.2,

which implies that m(n*)

i.e.

50

and hence that

c? work is recoverable from a constant history.

If #

:

H + R is defined by

the associated map $* : U

4

R has the values

$*(a) = t (a*) = cp(a)+m(a*)

p(a).

a E U.

which shows that 1.7.1.

)

does extend p and, together with 1.2.1,

proves

Also it is clear that the inequality 1.7-3 holds and so

it remains to verify 1.7.2. Let f,t ,t be as in the statement of 1.7, let g be any 0 1 closed continuation of f at t let t > 0 and let h be any local1' ised process which connects f(t ) to f(t ) . 1 0

is a closed continuation of f at t

0

m(f

to

)

2

-

dk

The localised process

and so

(cr(kS) ,<(s) > ds

Jt

.

0

and consequently to

w ( s o f(tl)

0

h) 2

(t) and taking the limit as t

J

- m(f

(u(fs),i(s~ ws

to

I

0 m

and appealing to the fact that u

has fading memory gives the inequality

If now h is replaced by its retardation h the limit as a + 0 and appealing to 1.2.2

and hence, as

a

it follows, on taking

that

On taking the infimum of the left hand side over all closed continuations q of f at t, it follows that

which, in view of the way I) has been defined, is 1.7.2.

.

Note that 1.7.2.i~ the Clausius-Planck inequality in this abstract setting.

1.7.3

says that among all histories ending with

a given value a c U the constant history a* minimises 1.7.1,

1.7.2,

1.7.3

(I. Each of

was derived by Coleman [ 4 ] but from different

hypotheses and by a different kind of argument. Note to? that not only does the proof of 1.7 establish the existence of 6 but it yields an explicit construction for it. The computation of m is usually a difficult task(4'.

However,

(1) For the conputation of maximum recoverable work in linear viscoelasticity see Breuer and Onat [ 3 ] and Day 171.

1.8. -

Lf 0

then m -

3

is elastic (example 1) or of the rate type (example 3)

0.

A simple example of a hereditary response functional which is not elastic nor of the rate type, which is compatible with

thermodynamics, which is elastic under retardation, which has fading memory and for which m and

can be computed explicitly is

$

provided by the linear viscoelastic Eunctional Example 4.

t a(£ ) = af(t)

- bJ

m

t exp(-s/r) f (s)ds,

t f

E

H,

0

where-a , b , ~E R and b , >~ 0 . Here a*(a) = (a-~b)a

a

v

E

U.

Observe that t(ft) 2 and

11

* = $*(f (t))

that wherever ?(t) exists t

l b ( f

f (a-rb) l l f(t)

)=(o(f

t

),i(t))

- bl(f(f)-:rcxp(-s/r)ft(s)dsll

2

0

-c
),

from which the Clausius-Planck inequality 1.7.2

follows on inte-

grating both sides over the interval [toeti] A further extension of $ to 3

I$

*

ext

is instantaneously elastic i,e. if

0

:

H x

u

4

R is possible if

admits an extension

Oext

:

t H x U + V such that (i) for each f

E

H the map

is continuous on U, and (ii) if f,g are localised processes with

f (d,)

= g(0) then

The elastic functional of example 1 is trivially instantaneously elastic with

and a little effort is enough to show that the linear viscoelastic functional of example 2 is instantaneously elastic, with

Functionals of the rate type (example 3) are not instantaneously elastic (unless they happen to be elastic). 1.9. ~f 0 is instantaneously elastic -

'ext:

H x U

.r

is smooth on U

Thus

t

R such that, for each f

and

qext satisfies

I$ admits an extension 5

H, the map

$ext (ft,f(t)) 2 $* ext (f('=)I.

1.9.4

and if -

t Let f

g (0) = f (t)

.

E

H.

then

0 < _ t o <_ tl (df

proof.

t

f

H and let g be a closed localised process with

Then the localised process ah

whose duration is t

+

a

-

-1d

f

g'

[o,t]

O

' a '

is a closed continuation of f at t

and so

Letting a +

and using the fact that

0

is instantaneously elastic

produces the inequality

which holds for every closed localised process g'with g(o)=f(t). Thus 1.1

implies the existence of a map

each ft c H the map

is smooth on U and

:

HxU

4

R such that for

If

text

is defined on HxU by

then 'ext

(ft,f(t)). = $(ft)

which means that $

ext

,

t f

E

H,

really is an extension of $,

has the property 1.9.1.

Moreover .pl ext

The remaining assertions follow with the

help of 1.7. The computation of

presents no difficulty.

For the linear

viscoelastic functional of example 4,

and

The remainder of part 1 is concerned with the general linear viscoelastic functional of example 2.

1,10.

If o

is compatible with thermodyn.amics then G ( O ) F L ( V , V ) &

symmeLric. Proof. -

According to 1.9.1,

and hence

is symmetric. It can be shown that ( 5 ) 1.11. If o is compatible with thermodynamics then, for every t 2 0, G(0)

- G(t)

is a positive semidefinite element of L(V,V).

-

and, consequently, so are ~ ( 0 )

- &(0).

G(W)

It is sometimes asserted that thermodynamics implies that the onsager relations hold in the sense that G(t) c L(V.V) symmetric for each t in 0

5

t

5 m.

is

Certainly the Onsager relat-

ions are not implied by compatibility with thermodynamics as understood here for (6)* 1.12. -

There is a relaxation function G

:

10,m) + L(V,V) such that

a is compatible with thermodynamics but G(t) is nonsymmetric for eacht&O
characterisation of the Onsager relations can be pro-

vided (7) in terms of the invariance of w under time-reversal.

f is any.localised process, its time-reversal is the localised process 7 which has the same duration as f and is defined by

-

f (t) = f (df-t)

1.13. -

Suppose that 0

E

U. &T

,

t

Q

[O,dfJ.

G(t) c L ( V , V ) is symmetric at

each t in O 5 t 5 - if and only if -

( 5 ) See Day [8]. (6) See Shu and Onat [6] and Day [ 8 ] . (7) See Day [a].

If

for every closed locblised process f with f (0) Sufficient conditions for compatibility with thermodynamics are (8)

1.14. -

Suppose that G

6

-

are smooth, that the Onsager relations

hold, and that, for each t 2 0 , -6(t) &z(t) definite elements of L(V,V),

-

-

- - -

are positive semi-

o is compatible with thermo-

-

(8) This result and its proof were suggested by the work of Volterra [I].

2.

Thermodynamic restrictions in mechanics

The main body of work is behind us,

In the lectures that

remain I shall sketch how the results of part 1 can be applied, first to mechanics and second to genuine thermodynamics. Consider a body B which is identified with the region of euclidean point space which it occupies in some reference configuration in which p

:

B + R is the mass density.

In all that

follows the spatial gradient v is always computed with respect to The translation space of euclidean point

that configuration.

space is an inner product space, with the inner product

and the

norm I I, and the application to nonlinear viscoelasticity usually involves choosing V = L (translation space, translation space), with the inner product T (a,B) = trace (si3 1, . the adjoint of p, and choosing where dT IS

u to be the open and

arcwise connected subset U = { ~ E V: det a r 0 ) . A

motion of B, of duration d > 0, is a map P

-

p

t

B x [O,d 1 P

4

euclidean point space

such that the velocity p and the deformation gradient vp exist,

det vp > 0, and, for each x e B, the map t and piecewise C' wise C

2

.

and the map t

4

h(x.t)

is continuous

vp(x,t) is continuous and piece-

The body is to be thought of as having been at rest in

the initial configuration x + p(x,O) at all times t < 0.

A pair (p,S) is a mechanical process for B if p is a motion of B and the map S : B x [0,8 1 4 V has a gradient vS. Here S is P the first Piola-Kirchhoff stress tensor, which is related to the symmetric stress tensor T by

The body force needed to sustain the mechanical process (p.S) is

- trace wherever

VS.

is defined.

2.1 If (p,S) is a mechanical process for B then -

where n is the unit outward normal to aB. A simple material has a constitutive relation of the form

S(x*t) t

r

t Sx( (VP),)

X E B .

t

E:

[Oldp]*

where ( v ) ~: (O,OD)4 U is the history of the deformation gradient

and the response functional is a map S

X

A

:

H + V such that

mechanical process (p,S) is said to be admissible if it

satisfies the constitutive relation. Note that 2.2.. -

To each motion p there corresponds a unique admissible

mechanical process (p,S)

.

2.3. If f is any localised process in U and 'if x -

there is a motion p,

of B.

with d

P

= d

E

B is arbitrary

and

f-

B is said to bk compatible with thermodynamics if there is a config'uration po

:

B

4

euclidean point space

such that whenever p is a motion of B with

for every x E B the corresponding admissible mechanical process satisfies

d

for every subbody C c B.

It now follows, with the help of 2.2 and 2.3, that

-

2.4.

L e t B be compatible with thermodynamics and let x

arbitrary.

i.e. -

Then for each closed localised process f

c B be

2 U,

wjth

Sx is compatible with thermodynamics in the sense of part 1. The following results can be read off from 1.7 and 1.9.

-

Suppose that B is compatible with thermodynamics, that

2.5.

x

E

B and thzt S

X

is a hereditary response functional which is

elastic under retardation and has fading memory.

-

map ,$I

:

H

4

Then there is a

R such that

and in any admissible mechanical process ;?,S)

whenever

whenever t

0

5 to 5 ti

E

[O,dp]

fd

P*

and -

.

Here $ is the response functional, at x X

E

B e for the

isothermal free energy density (per unit volume in the reference configuration).

2.6. -

If, in addition to the hypotheses of 2.5, SX is instan-

taneously elastic then

tx can be extended to a map Qx,ext'- H x U +

R

such that, in any admissible mechanical process (p,S), L

I

Applications to linear viscoelasticity usually involve a different choice of U, namely

T

u = { ~ c v a : = a ) ,

so that 0 c U (c.f. 1.13).

The response functional S

is

X

T

where id E V is the identity, sym a = +(a+& ) for each G~ :

[O,m)

+ L(V,V) is the relaxation function at x

E

B.

E

V, and

3.

Thermodynamics based on an inequality

The application of part 1 to thermodynamics involves taking V to be the direct sum V = L (translation space, translation space)

$

R,

for which the inner product is

T

( (a,a), ( B,b) ) = trace (a$

+ ab,

and taking U to be the open and arcwise connected subset U = {(a,a) c V : det a > 0, a > 0). A

(i)

motion and internal energy pair (p,e) for B consists of

a motion p , and (ii) a strictly positive tnap

.

[O*dP1+

R

such that, for each x E B, the map t + e(x,~j is continuous and L

piecewise C

.

The body is to be thought of as having been at rest

in the initial configuration x + p(x,O), energy x A

4

e(x,O),

with the constant internal

at all times t < 0.

quintuple (p,e,S,8 , q ) is a thermodynamic process for B if

(i) (p,e) is a motion and internal energy pair for B, (ii) ( p , S ) is a mechanical process for B, (iii) the map 8 : B x [O,dpl

4

R

is strictly positive and its gradient v 8 exists, and (iv) the map q

has a gradient vq.

:

B x [O,d ]

P

-r

translation space

Here 9 is the absolute temperature, q is the

heat flux vector and the heat supply needed to sustain the thermodynamic process (p,e,S, 8,q) is r = i

wherever

and

- trace

~p are defined.

T (S(vp) )

+

trace vq,

It is convenient to introduce the

generalised stress

so that

(G,;)1 = p1

(Z,

-T trace (~(vp) 1

- 29 i

and 3.1. -

(p,e,S, 9,q) is a thermodynamic process for B

and C c B

is any subbody,

For the sake of ease of exposition I confine my attention to a simple material which has constitutive relations of the form

for each x (79):

E

B and each t

are defined on (0.m)

Note that

-

Z(x,t) = L;(

T

f0.d 1, where the histories et and P X exactly as (vp)' is, and where X

t t (VP)~.~~), x E B, t

E

[o,dpl.

A thermodynamic process is said to be admissible if it satisfies the constitutive relations.

3.2. -

To each motion and internal energy pair (pee) there correa-

ponds a unique admissible thermodynamic process (p,e,S,8,a). It is necessary to make a technical assumption on the response functional to the effect that the second constitutive relation is always uniquely invertible in the form

In order to formulate the assumption precisely observe that if k is any localised process in U then, for each t

G

[O,dk] , k(t)

admits the unique decomposition k(t) = (kL(t) ,kR(t)) where kL(t)

E

L (translation space, translation space), det lc (t) L

: , 0,

and kR(t) E R The map 3

, kR(t) >

0.

is said to be invertibz if, given any localised process

X

k in U, there is a unique localised process y in d

g

u

such that

= d and k kL(t) = gL(t)

t kR(t) = ex(g )

,t

E

[o,dk1-

3.3. -

Suppose that 6 is invertible for every y s B. Let x c B b e Y arbitrary, let f be any localised process in U and let v

:

[O,df]

4

translation space

2 be any continuous and piecewise C m x . ible thernodynamic process (p,e,S, 8,q)

Then there is an admiss-

for B

such that

proof.

~ c t f and x be as stated and choose a motion p, of duration

df, and

2

strictly positive map

such that

With each y e B associate a localised process k

Y'

of duration d f'

by setting ky(t) ' (VP(Y,~),e(y,t) and define the localised process g

Y

invertibility of 0

Y-

g ~ and so

,

r

t E [O,dfl,

as in the definition of the

Then

~ t oy((w)Yt .gY8R )

= e(~,t)

.

Y

E

B.

t

E

IOldfl.

~f the map e

:

B x [O,df] + R

is defined by

it is easily verified that the admissible thermodynamic process (p,e,S,q,q) corresponding to the motion and internal energy pair (p,e) has the required properties.

.

B is said to be compatible with thermodynamics if there is a

map i

:

B

4

U such that,y

4

3

Y if x

5

(i(y)*) is independent of y r E and,

B is arbitrary and (p,e) is any motion and internal

pair for B with

GilcirG'

where the integrand is evaluated at x. Here i assigns to each x

E

B the element i(x) E U which can

be thought of as the initial state (i (x) ,i.,( x ) ) consisting of an L r\

initial deformation gradient i (x) and an initial internal energy

L

Note that each of the initial states has the same temper-

iR(x).

ature and so the initial temperature gradient vanishes. ~t follows, with

3.4. -

0

Y

the help of 3.1, 3 . 2 and 3 . 3 that

Suppose that B is compatible with thermodynamics and that

is invertible for each y

E

F.

T,et

x e B be arbitrary.

for each closed localised process f

Then

U with f(o) = i(x) and for

each continuous and piecewise cL map v

:

[O,df]

translation space

w i t h v(o) e 0 the inequality

holds. Choosing f = i(x) (t) implies the following hest conduction inequality: 3.5. If B and x -

are as in the statemen2 of 3 . 4 , v

:

[Oft] -r translation space

is continuous and piecewise

c2 4 v(0)

= 0

then

if

t 2 0

and

The heat conduction inequality does not necessarily imply that %(i for every s 3.6. -

E

[O,t]

(XI *,vs)-v(s)

5 o

.. However,

If there is a differentiable map A

q

:

translation space

4

translation space

such that %(i(x)*,v whenever B,x,t

v

0

and v

t

= $(~(t))

are as in the statsment of 3.5 then for every

translation space

E

sX(i(x)*,vO*)

-vO

< 0.

Also qx(i(x)*,O*)

= 0

and the heat conduction tensor

is positive semidefinite.

On the other hand, choosing v in 3.4 to vanish identically gives If B,x 3.7. -

i.e.

Ex

&$I

f are as in the statement of 3.4

then

is compatible with thermodynamics in the sense o f ~ 2 I s.

The following results can be read offf9) from 1.7 and 1.9. Here the partial derivatives of a map q

:

U + R are denoted by

D cp and D rp. L R

-

Suppose that B is compatible with thermodynamics, that 9

3.8.

is Y -

-

invertible for each y E B e that x e B and that 2X is a hereditary response functional which is elastic under retardation and hae fading memory.

Then there is a map qx: H

4

R such that

and in any admissible thermodynamic process (p,e,S,B,q)

3.8.2

(6:x.t) - trace (s(x.t) (G(x,t)) T ))at 9(x,t)

C

rL1

J,

whenever 0

< to 5 t1 < dp * -and

t t rl,((V~)~~e~)5 q;;:(v~(x,t).e(x,t))

3.8.3 whenever

t

G

[O,dp].

Rere q is the response functional for the entropy density X at x 3.9. -

E

B (per unit volume in the reference configuration).

If, in addition to the hypotheses of 3.8,

eously elastic then

1 can be

extended to a mag

X

is instantan-

nxeeXt: H

x

1)

4

R

such that

(9) The negative sign which occurs in the definition of the e n t r c p y is conventional.

of course the partial derivatives which appear in 3.9.1

are

to be interpreted on the lines of 2.6.1.

Bibliography Not all the works listed are actually cited in the text. [I]

V. Volterra: Theory of functionals and of integral and integro-differential equations, Dover Publ. Inc., New York, 1959.

(21

B.D. Coleman and W. Noll: The thermodynamics of elastic materials with heat conduction-and viscosity. Arch.Rat.Mech.Ana1. 13, 167-178, 1963.

[3]

S. Breuer and E.T. Onat: On recoverable work in linear viscoelasticity. 2. Angew.Math.Phys. 15, 12-21, 1964.

[4]

B.D. Coleman: Thermodynamics of materials with memory. Arch.Rat.Mech.Ana1. 17, 1-46, 1964.

U

-

U

[5]

B.D.

Coleman: On thermodynamics, strain impulses and viscoelasticity. Arch.Rat.Mech.Ana1. 17, 230-254, 1964.

163

L.S. Shu and E.T. Onat: On anisotropic linear viscoelastic solids. Proceedings of the Fourth Symposium on Naval Structural Mechanics, Purdue University, April 1965. Reprinted in Mechanics and Chemistry of Solid Propellants, Pergamon, Oxford and New York, 1966.

[7]

W.A.

[El

W.A. Day: The Thermodynamics of Simple Materials with Fadirg Memory. Spri,nger Tracts in Natural philosophy, Volume 22, Springer-Verlag, Berlin, Heidelberg, New York, 1972.

[9]

M. E. Gurtin: qodern continuum thermodynamics. Today, 1, 168-213, 1972.

-

Day: Reversibility, recoverable work and free energy in,linear viscoelasticity. Quart.J.Mech.App1.Math. 23, 1-15, 1970. M

Mechanics

h)

[lo] B.D.

Coleman and D.R. Owen: A mathematical foundation for thermodynamics. Arch.Rat.Mech.Ana1, 54, 1-104, 1974.

-

[ill

W.A.

Day: Entropy and hidden variables in continuum thermodynamics. Arch.Rat.Mech.Ana1.

1121

W.A.

Day: An objection to using entropy as a primitive concept in continuum thermodynamics. Acta. Mechanica.

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

SULLA NOZIONE D I STAT0 NELLA TERMOMECCANICA D E I CONTINUI

M. FABRIZIO

Corso t e n u t o a Bressanone d a l 2 alllll giugno

1977

SULLA NOZIOE DI STAT0 ELLA TERFOMECCANICA

DEI COWINUI Mauro FABRIZI0 (Universitg di Salerno)

1ntrloduzi.one. LeproprietSdiuncorpocontinuovengono solitamentestudiatc da un punto di vista fenomenologico,cio2 una volta introdotto un numero di

grandezze fisiche (variabili) di tip0 macroscopico il cvrpo viene sollecitato da un process0 che fa variare alcune variabili in mani.era nota, quindi si calcola la risposta del materiale mediante il valore delle altre variabili. In tal modo l'dpn~tntomatehiaee. considerato si conporto come un sistema sollecitat0 da cntrate ed uscite. Qu:mdo si studiano materiali relativamente se:siplici si riesce a descriverli anche senza utilizzare la teoria dei sisteni, in quanto per questi materiali lo stato it facilmente esprimibile m e d i k e le variabili indipendenti, cioit ha un carattere esclusivamente cinematico. Ma quando il materiale presenta una maggiore complessit2, allora 2 necessario formalizzare concetti come quello di b;tato e di pkocedbo e cia b possibile impostando lo studio mediante appunto la teoria dei sistemi. Tale problema it stato affrontato prima da Onat [l] e poi da No11 [2] che ne ha dato una precisa e compiuta .formulazione. In quest0 lavoro dopo avere precisato il concetto di elemento materiale sulla base del lavoro di No11 [ Z ] , ho applicato tale nuova formulazione alla tennodinarnica dei continui awalendosi dei risultati di Coleman e Owen 131. Questi autori utilizzando una metodologia simile a quella di No11 sviluppano una tennodinarnica i cui principi sono formulati solo per cicli chiusi. Su questa base nella prima parte del lavoro danno una formulazione essenzialmente matematica della teoria,che viene nella seconda parte applicata a molti n i d ~ b e m p f i o i , di cui dimostriamo l'esistenza di una funzione entro-

pia, ritrovando i "classici" risultati di Coleman [4]. Inoltre sono riuscito ad estendere le-considerazionidi Coleman e Owen a materiali piO generali ed ho verificato l'esistenza oltre che di una funzione entropia, anche di una funzione energia. CiB mi S stato possibile perch2 ho sfruttato le tecniche e l'impostazione assiomatica del lavoro dl No11 che consente di descrivere una vasta classe di materiali, fra cui i materiali semielastici definiti da No11 [2] , i materiali con ''Fading memory" studiati da Coleman e Owen in [3], ma anche nuovi materiali con effetti non propriamente elastici. E' importante osservare che'per potere stabilire l'esistenza di una funzione energica, S necessario uscire dallo schema dei m a t W bmpLi.cL. E' per questo che ho considerato materiali per cui, pur valendo l'ipotesi dell'aziune Locale, cioS gli effetti del mot0 del continuo ad una distanza finita da X possono-essere trascurati nel calcolo dei funzionali costitutivi relativi a1 punto X, non risulta pera possibile ridurre tale dipendenza ai valori della storia del ghadiente di de~omzioneF o della t m p e m A u h c r 3 nel punto X, come awiene per i materiali semplici. Dimostro infine sotto ipotesi piuttosto ampie che tutti i funzionali costitutivi, ad eccezione del flusso di calore q, non variano per stati che differis~onosolo nei valori di _F, 3 circostanti il punt0 X. 1. Un corpo continuo B descritto matematicamente da un insieme 1(corpo), i cui elementi X, Y,j .. sono chiamti p u n t i m d W . In .9? B definita una classe @ di applicazioni 9 :28 + 8, dove 8S uno spazio euclideo tridimensionale di elementi x, y, (punti spaziali) . Lo spazio &ha la intuitiva interpretazione di sistema di riferimento, cio2 6 rappresenta lo spazio fisico gemtetrico in cui si trova inunerso .%?. Le applicazioni (p E Q, che assegnano ad ogni punto materiale X E .%? un Luogo nel riferimento 8 , le chiamerema tocaeizzaziuni. Inoltre supporremo che la classe @ verifichi le seyenti condizioni (Noll [ S j ) : i) Ciascuna cp E b iniettiva, il suo range q ( 9 ) B un sottoinsieme aperto di 8. ii) Per ogni +, cp E @ lVappl.icazioneX = $0(p-? : cp(.9?) + (b(.%?) e la sua inversa sono di classe C1 e Lipschitziane (*). Le chiameremo de~omazionie indicheremo con D la classe di tutte le deformazioni possibili per8. iii) Se cp E a e h E D allora h o E @.

.

...

("1 Tale condizione garantisce che le applicazioni X E D trasformino insiemi aperti e limitati in insiemi aperti e limitati.

Tali condizioni, molto naturali da un punto di vista fisico, conscntono di definire un atlante SUB. E' possibile pertanto determinare su 9 una unic;i struttura di variet2 differenziabile e quindi di spazio topologico (vedi Lang [6], cap. 2, 51). Ad ogni punto materiale X E a, la struttura di variets differenziabile su &? associa uno spazio tangente TX, mentre la struttura topologica associa a X una famiglia fX di intorni di X tali da costituire un filtro massi.male convergente a X. Lo spazio traslato di 8 lo rappresentiamo con E, il suo prodotto interno con ( ,-) Sia Lin (E) l'insieme delle applicazioni lineari da E ad E, mentre:

. .

dove AT B il trasporto di A. Lo spizio tangente TX consente di descrivere il materiale in un intorno "infinitesimo" di X e risulta uno spazio lsieare reale di dimensione tre. Ad ogni localizzazione (p E @ d i g in rfpossiamo associare il relativo gradiente VX q : TX + E che risulta una applicaziov,: K E Inv Lin(TX, E). Un mot0 di TX cC in E 2 una funzione M : [ 0 , dM] -t Inv Lin(E) . Fissata una localiz,zazionedi riferilaento !;r B possibile descrivere il mot0 in ~~.lazione a Kr ponendo:

La trasformazione lineare f: : E -+ E viene solitamente chiamata gnodit?nte di de~o&~aziane ed appartiene a in+(^). Infine indichiamo con 8(X) la t e m p u a t u h a a~huL1Lta relativa a1 p~mto materiale X. Se IX B un arbitrario intorno dipx, intenderemo per con,$igumzbne t m madinanica C' relativa a1 punto X la coppia: (1.1)

X ~ ( ~ )= (XE(~),

X8(~)) =

(?(X

- Y),

B(X - Y));

per

x-YE

cine la confiprazione 2 assegnata dal valore del gradiente dl cleionnazionc c dalla temperatura calcolati in un arbitrario intorno IX E .Fxdel punto X. Supponiamo pertanto che l'insieme di tutte le possibili configurazioni tennodinamicheX~sia un sottoinsieme r chiuso e semplicemente connesso di:

'cZ definite rispettivamente in I1 Diremo che due canfigurazioni X 1 IX) 2 per X7 I, sono equivalenti, cioS Cl 'c~, w esiste un .intornoIXc(IXn 'cZ ad IXcoincidono. E' possibile pertanto introdurre cui ie restrizioni nell'insieme r una relazione -che 6 transitiva, simetrica e riflessiva, risulta quindi una relazione di equivalenza. Indicheremo con Wlqinsiemequoziente di r rispetto a -, mentre C rappresenta il generic0 elemento dig. Pertanto % , che supporremo dotato di una opportuna topologia, S llinsiemedi tutteleclassi di equivalenza che chiameremo insieme delle conQ.igigwl~zzionit m monocinetichs.! Una funzione del tipo: 2

-

sar2 chiamato un process0 di durata $. La funzione P risulta definita negli estremi delllintervallo [O, dp] , dove i corrispondenti valori:

sono detti rispettivamente valore iniziale e finale del process@. Consideriamo quindi l'insieme dei processi P cmtinui in [0,dp] con derivata continua a tratti, cioS P esiste a mezzo di un numero finito di pu~ti, nell'intervallo [o, $1 e P m e t t e sempre limite destro e sinistro. Ad om1 P associamo la funzione P(a), che chiameremo pocenno te~umcivletico,definito nelllintervallo [o, $1 nel mu50 seguente:

(

1% P(t).

t

=

$

'+%

essendo a un elemento di in+(^). Processi di durata zero non sono esclusi zs verranno indicati P(ol (a). E' ora possibile caratterizzare in modo precis0 il concetto d4 eft.;:tbc di cahpo mediante il filtroFx, l'insizme 'Xdellc configurazioni tilmoc.-',::?tiche e quello dei processi termocinetici (ved- Noll 12 1, n. 7).

Definizione 1.1. Un & m m di cohpo h&ativo (Le puntv X t C O ~ ~ daLh t m n a ( Fx,V , n) , dove 2 d 6 i h h o madbhde teeativo (Le puntv X , %' CWinbiemed&e con&@uuzzioionitmnocinetiche e n l ' d i e m e di t.wXi i paoc a h i tehmocineaZci a c c a h i b X i (Le cotpo. Dalla definizione di processo utilizzata risulta che se Pl(a), P2(a) E n, f p1 = pi, lim Gl = a2 la funzione Tl(ai) * P2(a2) C O S ~definita:

5

'

"dp

dove

b ancora un processo appartenente a

.

n di durata dP1

+;dp2, che chiameremo can-

.tinuazione di Pl(al) con P2(a2) In seguito considereremo soltanto processi tennocinetici P(a) cheremo spesso semplicemente con P.

che indi-

2. Per pervenire ad una teoria precisa nell'ambito di una formulazione assiomatica, risulta necessario potere attribuire ad ogni elemento di corpo uno h t d o (fisic0)cheindividui ciascuna grandezza atta a caratterizzare la natura fisica dell'elemento stesso. Seguendo No11 (I2 ) , n. 7 ) , per stato intenderemo un ente (matematico) che nel modello costruito sia in grado di precisare tutte le informazioni relative all'elemento considerato. Quindi se indichiamo con il tenbohtlc degei b 6 a h z i di Ca~chydivisa per la d w W di madba p , con q il vettore @ u ~ b odi d o h e e con h il d o h c

L

~

fornito nella unita di tempo e di massa in a ( * ) ,stabiliamo la seguente definizione di dement0 rnatehiaee. Definizione 2.1. Un demento mcLtehicLee 2 individuate da un &emento di cotpa (YX, V , n) e d&'&ir-,ieme (c, i?,f , {, h) ddoe: XaLe che ciascun dement0 o i) i t Lo b p a h (topologico) degfi o-, (stato) 2 individudta d a U a copph o = (C, , or) , dove Ca E V Z la c o n 6 i g u ~ zione twmucinetica colmispondente &o n m o o e or t ckiamdo n m o h i d o m . ii) p u t u g n i P(a) E r[ u h t e una 6unziune p p da un ba.ttobieme non vuuto ed apehto 9 (P) E C a un bottoinbieme 9(P) E C. Mediante pp 2 pobdibdiee dehinine L'appeicazhne:

dedinita n&L1ia5ieme (c 0 n) = { (a, P(a) ) ; u E 9 (P) ) , ckiamata dunziane di f h t a v ~ i z i o n ee .tale che CA p(0,P) = ' iii) (T, 4, - 6) Z una t m n a di 6unziuionC cooi dedinita: A

A

Pertanto 1 , assegnano ad ogni stato o E Z il corrispondente valore del tensore degli sforzi 1, del flusso di calore 3 , mentre h, che risulta definita in c 0 n, assegna ad ogni coppia stato-processo il calore entrante nelI'unit2 di tempo. Questi oggetti dovranno infine verificare gli Assiomi I-VI specificati sotto. Dalla definizione della funzione $(a, P) si vede come tale funzione assepi il valore dello stato finale dell'elemento quando, partendo dallo stato a si esekaa il process0 P. Spesso useremo la notazione Po=$(a,P). Con nC, rC, LC,a inclicheremo rispettivamente gii insiemi:

(*) E' ben noto che sotto opportune condizioni di regolasits sui campi e sullo spazio degli stati, h risulta legato a 9_ dalla relazione: h = r - div q / 9 , essendo r il calore fornito per radiazione nell'unitl di tempo.

zC,a

=

{o

esiste un processo P

E ";

tale che

;(o,

P)

= a,

E

lim P

n e uno stato o

E 9(P)

= a)

ty'~

Assioma 1. Put a g n i P(a) E n a b b h o 9(P) = zpta. 1naLt.u h e l'l(al), P2(a,) E nc C U o B(P1) = 9(P2) a p p m e .9(P1) n d(P2) = 0. Infine consideriamo le funzioni:

Pertanto se indich'iamo con 3

=

(7 q,- K),

l'applicazione

1; chlameremo d u n z i a n d e di h i d p o b & ~ e scriveremo $(a; P) per intendere il valore della terna (f, 3, h) ottenuta dopo il processo P quando lo stato jniziale a .

+

Assioma 11. So Pl(al) , P2(a2) aeloAL7 :

Assioma 111. Due b

E

n,

o E

9 (PI) e : P

=

pi, lim

-

tdpl

b1 =

a2

M ol, o2 E ZC dono u g d b e e bo.fY.anta b e :

i P E n put c L .al, a2 E 9(P). Introduciamo ora una u n i d o ~ ~n aXt w l d e e sullo spazio degli stati in mniera formalmente simile a quella lseyita da No11 I21 . Le poche differenze sono dovute alle pr3priet5 della funzione S e a1 suo dominio. Tale definizione si awale del funzionale di risposta S(o; P) e dell'osservazione che Y = Sim(E) x E x R 5 m o spazio vettoriale di dimensione finita che possiede quindi una miformits naturale. put

Definizione 2.2. Ckiamehemo u n i 6 o h m i X natwraee hu Z Pa p i i i mzza u n i C,a 6

0

M

che "end& .LL &nz&ionaee

hu

wzi6omemcntc c o n t i n u o p a IwXL i

P E nC t a l i che 9(P)

=

zC,,.

d c d&o 6paZi.O z = u{zCta; C E V , a E L ~ ~ I + ( E )La ) h o r n . deLee ZopoLogic 6 t ~ECPa h d o * f e W e n U v e u n i b o m i Ckiamaemo t o p o t o g k ' i

.tii tlmbuLi.

Come facile corollario della Definizione 2.2 risulta (vedi No11 [2] Proposizione 12.1) :

?i uni6omemente c o n t i n u a .

Mentre per le proprietii degli spazi uniformi si ha (vedi Nagata (71 ,Teorema V I .13) Osservazione 2.2. Lo 6pazLo d e g e i b& C, con La hua natwLaee a%poLog.ia 2 uno hpazia di Hausdorff. E' opportuno osservare anche che la topologia naturale definita su z rende tale spazio uniforme (vedi Nagata [7], pag. 225) , inoltre gli insiemi Z C ,a

sono contemporaneamente aperti e chiusi. hono corn@& Assioma IV. Gfi hid E C ,a Se o E z indicheremo con no l'insieme: Ilo =

{p(o; P);

per tutti i P E

n

hinpetto &c

tale che

o E

uni60&nXi

naturcaei.

!3(P))

che viene chiamato insieme degli stati a c c & 5 h i b ~da o.

noel

Assioma V. E b h t e uno b Z & o oo E c M e che 2 den60 i n Z, d o 2 la t o p o t o g k nalvcaee pecikatn ne&k Definizione ckiuswra d i noo, hecondo 2.2, 2 E. Se indichiamo con Eb.(X), fp(X) il gradiente di defomzione nel punto X relativo rispettivamente a110 stato o e a1 process0 termocinetico P, enuncinmo l'ultimo assioma che cornunque risulta inessenziale per molte delle ccnsiderazioni che seguono.

Assioma VI. P a ogni cop* a', a" di p m c(Li 0; = o;I hidub3 cCte p m ogni E > 0 e p a ogni intozno 0" di o" e b h t e un p m c e ~ b oPt E n di dulrata t < E M e che:

In p a h t i c o h e b e _Fg, (X) = FoI,(X), &om it pkocebbo Pr pu8 ebbme phedo cobabtante n e t v&ke d e t gmdien.2~Ept(X) ne.U'inte~v&o [O, tl . Poichb ci interessa formulare i principi della termodinamica su cicli chiusi ci limiteremo a considerare materiali per cui sia sempre possibile rigenerare lo stato di partenza. Condizione di invariabilitii. P a ogrci b&zto a E n t ' h i e m e nu E den60 2.

Tale condizione p d essere verificata per molti materiali trattati nella letteratura, come quelli elastici, semielastici, i fluidi perfetti e viscosi, materiali plastici e con variabili interne, ecc. Insieme alla ipotesi di invariabilitii supporremo anche: Condizione di limitatezza. Pm ogni ja,.P) E E 0 n, b e indickiamo C O I L : .ie ph~cebboPT(t) = P(t) (0 < t < T) di dwlata T < $, la dunzione S ( o , e m h m b L L c in [O, $1. [0, dp] +SPhislLeta

3. In questo numero ricorderemo orevemente le definizioni e i risultati principali della Parte I dell'impoltante lavoi.3 di B. D. Coleman e D. R. Owen 131 , che in seguito indicheremo brevemente [ C, 0 1 Un elemento materiale, cosi come b stato precisato nella Definizione 2.1 insieme con la condizione di invariabilitl, verifica la definizione di sistema, contenuta in [C, 0 1 (Def. 2.1). Inoltre lo spazio degli stati Z B di Hausclorff,1'applicazione pp :g(P) + 9(P) b continua, anzi iuniformemente continua rispetto alla topologia di E. Inoltre la condizione di invarinbilit; ci assicura che c m q u e sia o E E, no b denso in E. Mentre la composizione fra due processi b definita in (1.3), (1.4), (1.5). A questo punto b importante ricordare le definizioni di azione come enunciata in [C, 0) (Def. 2.2) e di azione con propriets di Clausius e con propriets di Conservazione ([ C, 0 1 , Def. 3.1, Def. 4.1).

.

Definizione 3.1. Una a z b n e p m un e t e m e n b m c L t U e 2 una 6unz.bne a: + R M e che: f i) Additiritii. SC P1(al), P2(a2) E n, P1 = P;,

z0n

Dimostrazione. Se on di Limitatezza:

-+

a, dove Con = Ca

=

,'P

risulta per la Condizione

rt

quindi per la continuits di

z(-,P) , h(-, P) abbiamo:

Pertanto E (P, .) B continua in a(~) per ogni P E ll. Inoltre se P,, P, En,conpf1 =pi, lim P1 =a2,ePT(t) =(P1fiP2)(t),contE[0, TI, abbiamo : t4~l A

E

L

'

=

['if

(a, PT)

. L(r)

+

h(0, Prll

Analoga dimostrazione vale per s(P, o l . Diamo quindi la definizione di potenziale e sopra-potenziale per una azione cosi come 6 formulata in ([ C, 0 1 , Definizione 3.2) : Definizione 3.3. Se a 8 una a z b n e , una dunziane A a v d o h i h d k d e t iiz tan p o t e n z X e (hopuz p o t e n z i d e ] p a a he: i ) AX dduminio 9 (A) d&a 6unzione A P denbo .in in; ii) b e ol, o2 E 9 ( A ) , &om pen ogni E > 0 c ' P un intorno 0 di a2 X ~ che:

S

Infine ricordiamo il teorema di [ C, 01 : Teorema 3.3. Se c'2 uno 6neR y u d e uvuz a z b n e ha .La pphophietZ di conhmvazbne ( Ceaub&], &0ha L'azbne ha un potenzia.Le 16opka-potenzia.Le) che E continuo (hemi-conCinuo 6 u p ~ m e n t e l . E' possibile cost formulare il primo e il second0 principio $ella termodinarnica nella forma ([ C, 0 1 , n. 5) (*) ; Phima P h i n c i p 2 . L'azione E(P, u) ha la proprieta di conservazione in oo. Secondo Phirzcip&. L7azione/h(P, o) ha la proprieta di Clausius inloo. Come conseguenza della prima e della seconda legge della termodinamica abbiamo per il Teorema 3.1: EhLte un inhieme EE Claoo di a w nei quati E(P, u) ha la pnophim di Conhmvazione e un imieme r.6 di b u n& 6 (P, o) ha pptr~p~P.#. di C L m i u . I n o h e c' E una 6unzLone colztinua E di n m o ckiamata dunziolzc e n m g h che 2 un poffenzia.Le p a trE, .ie dominio 9(E) 2 demo i n in e e e el, u 2 E E 9(E), &ow pwc ogni E > 0, c ' 2 un intophno 0 d i a2 a%& che:

7

p a ogni p m c e s ~ oP M e che Pol E 0 (*a) l . ~ d i n e ,bempptre p a /r Teolrema 3.1, e s h t e una dunzi.one di 6Z.uXo s, chinmata dunzionc entropia, che 2 un ~opm-po;tenzia.Lepwcla, eemicontinua nupehiomente neL dotninio 9( s ) demo i n in c en el, u2 E 9(s), &om peh ogru E > 0, c'E un in.tohn0 0 di o2 M e che:

pen ogtu pmcesno P M e che Pol E 0. Coleman e Owen nel loro lavoro, formulano anche un criterio per costruire iL potenziale o il sopra-potenziale una volta assegnata l'azione. I1 metodo 5 importante, ma non ne faremo cenno rimandando direttamente a ((C, 01, Teorema 3.3 e Teore~a4.4).

(*) La nostra formulazione differisce in parte da quella citata di Coleman e Owen, in quanto ora si specifica lo stato a nel quale deve valere il 0 Prino Principio. ("'3)

possono finizione. E

Inoltre ([C, 01, Teorema 4.6) dimostrano che tutti i potenziali per differire solo per una costante nel dominio oomune di de-

lim P1 =a2, e s e o f 9(P1), t"lpl

ii) Continuits.

Peh

ogni P E n, .Pa ~ u n z b n eap : 9 (P)

-+

R cobL d

c

~

E continua. Definizione 3.2. Sia .a E X c h m o 0 di oo Raee che pm ogni:

&om ao1

k3

una a z i o n e . Se p m ogni

E

> 0

c l E un &-

dioidnw che a ha la p~~ophiea% di Conservazione i n uo (di Clausius in

Inoltre 2 possibile sintetizzare alcuni risultati di [C, 01 cbl teorema:

Teorema 3.1. Se e.b.ibte uno &ahto oo n d q u d e uvllr az.ione !a .Pa p m p h i & i d i Clausius (Conservazione), atbm L 1 d i e m e d q L i b W XO(*) neL q u d e m e pt3j3hi&i B ~ P I L i d i r a t aB d m o C. 7n p a ; t t i c o h e n oo c zO. Indichiamo con & il gradiente di velocits, cio? !, = F ~ - l , con B = 2 = (1/~a) grad 8, allora risulta: Teorema 3.2. P a un &enrevL;to matehiaee, ne.R1.iporJatcui di v f i d . i z 3 d&e Condizioni di Invariabilit2 e di Limitatezza, Le dunzioni E , s de&inite nu I: 0 n n d modo deguente: t P ,1

=

0

,

1

L(t)

+

c(o, PT)l dr

t +

%(a, P,)

'

E(?)1

dt

3(7)

hono a z b n i .

(*) Per una p r e c i s a def i n i z i o n e d i ZO vedi

I C,

.

O] formula ( 3 -24)

~

:

4. In questo numero verificheremo che, assegnato il punto materiale X e comunque presi due strati a', a" E 9 ( E ) che differiscono solo nella coniiyrazione attuale per i valori circostanti il punto X, risulta:

Un discorso analogo vale per s(o). Fonnuliamb pertanto il seguente teorema:

Teorema 4.1. Petr ogni &emento (LFX,IV, n, E, S, 8) i l d o d n i o g(E) deLLa dunzhne enmgia E e qu&o 53 (s) d W dunzhne entrtoph s E .taec. chc? be a' E 9 (E) , o p p a e o ' E 9 ( s ) , LLUQJUX hiape&Xvam&e u i a p p W e n e ancke ogpL

* a;

u" peh cLL~

6

I:

cot(XI = CL7,,(X).

Ino&e:

'Jimostrazione. Dato o1 E 9 (E) , considerimo un generic0 o" = (Call, or), con Cu,,tX) = Cul(X) Per lVAssiomaVI, per ogni E < 0, esiste I I ~ \ intorno 0" di a" e un processo Pt di durata t < E/Q dove Q = sup Ih(o; PT) 1, tale che O
.

i) opt, 0') E 0" ii) PtCX) a CoI (XI mentre l'azione: (4.11

t ~(p,, a') Ijgf(ol;PT)

-L(T) d

6'

~ + fi(ol;PT) d~

verifica la disuguaglianza (4.3) . Per la condizione ii) risulta in particolare che P(X) = -FF-' - = 0 , si ha cost: (4-2)

=

0, quindi I, =

16(01;PT)I d r s t Q < ~

o' E

Per la Definizione 3.3 di [ C , 01 , a" 5 E-approssimabile da a ' , ma poi&$ 9 (E) , allora;per il Teorema 4-4 di [ C, 0 1 , a" E O (E) Quindi

.

Da cui per la limitazione (4.2) e l'arbitrarietg di E, abbiamo:

La dimostrazione relativa alla fimzione s(o) 2 naturalmente del tutto simile.

111 E. T. Gnat: The nofin 04 and i & impficcatians & inhenmodcj&cn &-$kc b o u , Proc. Iutam Symposium, 1966, pp. 292 - 314. (21 W. Noll: A new mathematicae theohg Anal. 48 (1972), pp. 1 50.

05

06 bhp.&2matcmki.b, Arch. Rational Mech.

-

t31

B. D. Coleman - D. R. Owen: A mathematical 6oundcLtion doh inhehmody&cb, Arch. Rational Mech. 54, (1974), pp. 1 - 104.

[-I1 B , D. Coleman: Thmodgnanticd od mat&

Lclith memohy; Arch. Rational Mech.

Anal., 17 (1964), pp. 1.- 46. Noll: Matehiaeec~unidohm b.impLe bodieb uLith inhomogeneitiu, Arch. Rational Mech. Anal. , 27 (1967) , pp. 1 32.

[S] W.

161

-

S. Lang: Intmductian to di~$ehenzi.abLemani6oLd4, New York Sydney, Interscience, 1962.

- London -

Nagata: Modehn genend Zopohgy, North Holland Publishing Company, Amsterdam, 1968.

[71 J.

CEN TRO INTERNAZIONAL E MATEMATICO ESTIVO (c.I.M.E)

ANALYTIC PROBLEMS OF HEREDITARY PHENOMENA

G.

FICHERA

Conso tenuto a Bressannone daf 2 allrIl giugno 1977

ANALYTIC

PROBLEMS

HEREDITARY

OF

PHENOMENA

Gaetano F i c h e r a U n i v e r s i t # t d i Roma

1. The V o l t e r r a i n t e ~ r o - d i f f e r e n t i a l e q u a t i o n s i n t h e mathe

-

m a t i c a l t h e o r y of e l a s t i c i t y . I n t h e c l a s s i c a l t h e o r y of e l a s t i c i t y when one t a k e s i n t o a c c o u n t t h e h e r e d i t y of t h e body l ~ n d e rc o n s i d e r a t i o ~ 1 , i . e . t h e t o r y of -

e-

t h e d e f o r m a t i o n s u n d e r which t h e body was s u b j e c t e d i n

t h e p a s t , one i s f a c e d w i t h t h e a n a l y t i c problems a r i s i n g from t h e i n v e s t i g a t i o n of a c l a s s of i n t e g r o - d i f f e r e n t i a l e q u a t i o n s . T h i s k i n d of e q u a t i o n s f i r s t a p p e a r e d , i n c o n n e c t i o n w i t h e l a s t i c (4)

p r o b l e m s , i n t h e work of L. Boltzmann f 1 1 , 1 2 1 . About t h i r t y y e a r s l a t e r V.Volterra s u p p l i e d t h e f i r s t approach t o an a n a l y t i c theo r y f o r t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s of h e r e d i t a r y l i n e a r e l a s t i c i t y 133, C43, E51,161, h71. L e t A be a t h r e e d i m e n s i o n a l bounded domain of t h e c a r t e s i a n s p a c e X'

of t h e r e a l v a r i a b l e s x ,

orthogonal c a r t e s i a n coordinates

, x,, M s . L e t x be t h e p o i n t w i t h x, , x, , x* . Suppose t h a t A is t h e

r e f e r e n c e c o n f i g u r a t i o n of a n e l a s t i c body i n t h e fzame-work of

t 4 ) Numbers i n b r a c k e t s r e f e r t o t h e b i b l i o g r a p h y a t t h e e n d of tbese lectures.

t h e l i n e a r t h e o r y of e l a s t i c i t y . Denote by 1 , 2 , 3 ) t h e e l a s t i c i t i e s of A

j k ( x ) ( i , a, j , k

Q

=

, s a t i s f y i n g t h e w e l l known symmetry

conditions

C o n s i d e r i n g t h e e l a s t ' i c h e r e d i t y of A

, t h e c o n s t i t u t i v e equa-

t i o n s a r e w r i t t e n a s follows:

where t is a r e a l v a r i a b l e ( t i m e ) ,

6.

.

i s t h e s t r e s s t e n s o r and

E., is t h e s t r a i n t e n s o r which, i n t h e l i n e a r i z e d t h e o r y lk

l a t e d t o t h e displacement v e c t o r

The f u n c t i o n s q.

ck jk

u S{U,

,Uz.U3

3

,

is re-

a s follows:

c a n be d e n o t e d , f o l l o w i n g V o l t e r r a ,

(x,t,s)

the heredity coefficients. From t h e e q u i l i b r t u m e q u a t i o n s 6;C,L = f;cx,+),

where f

5

If,, fa, f,f

is t h e body-f o r c e a c t i n g upon

A , one g e t s

,

r e c a l l i n g (1.11, (1.2), (1.3) t

i

(1.4)

j

x

u

'

-0

Yirjr/r ( x , ~ , T ) u . Ik (

~ 1 d~r

7

f t

where 4

Y ; R j k ( ~ , t =, ~5 ) { 9 ; k j L ( x . t . s )

+

'f'.Lkj(~.t.t)\.

I t is t a c i t l y assumed t h a t t h e f u n c t i o n s

qLjks a t i s f y

suitable

hypotheses such t h a t t h e i n t e g r a l s extended t o i n f i n i t e i n t e r v a l s be c o n v e r g e n t and t h e d e r i v a t i o n u n d e r t h e i n t e g r a l s i g n

(l)

, with

We assume t h r o u g h o u t t h e s e l e c t u r e s t h e summation c o n v e n t i o n ,

i . e . a summation must be u n d e r s t o o d when a n i n d e x is repeated twice. (')

By /k w e mean t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o t h e -s-p a c e '3= x k . By /LA we mean -. 9xk>xr

v ariable -

r e s p e c t t o t h e s p a c e v a r i a b l e s , be f e a s i b l e . Follouring V o l t e r r a we assume t h a t t h e " p a s t h i s t o r y W o fthe body

is known, i . e .

t h a t the functions 0

-OD

a r e known f o r x

€

A

.

and f o r any t (')

Set

Fi ( x , t )

= f;

(x.t)-

-

J;

(x,t).

Equations (1.4) a r e now w r i t t e n ri

I f we suppose t h a t t h e body i s clamped a l o n g i t s boundary, we must a s s o c i a t e t o (1.5) t h e boundary c o n d i t i o n s (1.6)

u; ( x , t ) = o

for

(x,t

1

t '3A x

Rr.

Hence t h e a n a l y t i c a l problem t o be i n v e s t i g a t e d c o n s i s t s i n m l v i n g f o r any t

0

t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s (1.5) w i t h t h e

boundary c o n d i t i o n s (1.6)

;

A s an example 1e.t u s c o n s i d e r t h e s i m p l e s t c a s e of a one-dimen=

s i o n a l problem. I n t h i s c a s e (1.5)

becomes

and we have t h e f o l l o w i n g boundary c o n d i t i o n s (1.8)

u ( o , t ) = u ( . r , t )= o

t ro.

(4)Actually t o assume t h a t t h e body can be compared t o a good family g i r l of t h e p a s t y e a r s , such t h a t %wrything of her past is knowd'is a hit too much. En f ortunately the probl em becomes more complicat_ed i f this hyphothesis is not assumed (see Section 7) . V o l t e r r a a s u m e s f l x , t ) 5 0 , which,of c o u r s e , f r o m t h e p o i n t of view of a n a l y s i s is e q u i v a l e n t t o t h e hypothesis i n the t e x t .

We assume t h a t acx).o a(x)

e 4co, 11; y + ( x , t , t )

belongs t o

io,+=)

x CO,

+-)j

Set A(x)=

;

~ ( x , t )

lX3, a(?)

a

and moreover t h a t t h e f u n c t i o n

( 0 6 X s 1)

and ?v,(x,t,o)

belongs t o

,

R e c a l l i n g ( 1 . 8 ) we have uix,t,=

e 4 {[o,.I I x

a

v ( * , t )=

belong t o

c'[ COA3

CO,+-)

f

'3u

a x [ Q ( x ) - ] a. x

/ ~ ( x , ~ ) - - ( r , t ) d ~

where

The problem (I-. 71,(1.8) is e q u i v a l e n t t o t h e f o l l o w i n g one:

We have a x

where

J

I

/

I

~ ( x , j ) v ( ~ , i ) =d J G ( < , J ) n ( F , ' ) d T

.

0

a

*

S i n c e t h e f u n c t i o n G,(*,J)def i n e d i n t h e s q u a r e [o,il jump when t h e p o i n t

( ~ ~ c7r o) s s e s

x

C0,il

has a

t h e d i a g o n a l x = y d e r i v a t i o n under

t h e i n t e g r a l s i g n i s n o t p e r m i t t e d f o r computing u x x ( * , = ) . Since a'(x)

au

ox +

9 2 u a(x) -;:=

ax

Y

From ( 1 . 9 ) w e have 1

u(x.t

1 + /tdr j ' H

( r . . i , t , ~ ) v ,( y~) d y

where

Equation (1.10)

is a n i n t e g r a l

A n a l y s i s i s known a s t h e

e q u a t i o n , which i n

Volterra-Fredholm

classical

i n t e g r a l equation

( s e e f 8 1 , E91 1 . T h i s e q u a t i o n has one and only one

continuous

s o l u t i o n . Hence problem ( 1 . 7 ) , (1.8) h a s one and only one s o l u t i o n belonging t o

e2t0,11.

The a n a l y s i s of t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s (1.5)

, when

t h e dimension of t h e s p a c e is g r e a t e r t h a n one, is much more complicated. V o l t e r r a , a s an i n t r o d u c t i o n t o t h e g e n e r a l system (1.5), proposes t h e p r e l i m i n a r y s t u d y of t h e s c a l a r i n t e g r o - d i f f e r e n t i a l e q u a t i o n

w i t h t h e boundary c o n d i t i o n (1.12)

r*ix,t)=o

for (x,t)6aAxR7.

Equation (1.11) i s nowadays knou7n a s

the

Volterra integro-

d i f f e r e n t i a l equation. I t i s i n t e r e s t i n g t o p u t i n t o evidence t h e t e c h n i c a l d i f f i c u l t i e s which one has t o face,when one t r i e s t o apply t o (1.11)(1.12) t h e same procedure used f o r s o l v i n g t h e problem (1.7) I f we s e t v (x,+ ) = A 2 u

we have, because of (1.12),

(5)

A2 d e n o t e s t h e Laplace d i f f e r e n t i a l o p e r a t o r A,:

'3= ' 3 -q+3 ~ :

+

3:

ax:

.

,(1.8).

where G ( x , F ) i s t h e Green f u n c t i o n f o r t h e D i r i c h l e t problem f o r t h e Laplace o p e r a t o r i n t h e domain A

, which is .assumed t o s a t i s f y

s u i t a b l e r e g u l a r i t y hypotheses. I n t h i s c a s e f o r e x p r e s s i n g t h e second s p a t i a l d e r i v a t i v e s of u we need t o u s e s i n g u l a r i n t e g r a l s i n t h e Cauchy s e n s e ( s e e [ l o ] ,

Hence problem (1.11)

, (1.12) l e a d s t o t h e f o l l o w i n g

integral

equation

The c l a s s i c a l procedures f o r handling t h e Volterra-Fredholm e q u a t i o n s do n o t apply i n t h i s c a s e , because of t h e presence of t h e Cauchy s i n g u l a r i n t e g r a l s . I t would be of some i n t e r e s t

to

develop a t h e o r y of Volterra-Fredholm i n t e g r a l e q u a t i o n s , e x t e n d i n g t h e c l a s s i c a l t h e o r y , when Cauchy s i n g u l a r i n t e g r a l s

occur

w i t h r e s p e c t t o t h e space v a r i a b l e s . T h i s , i n o u r o p i n i o n , s h o u l d

2

be an i n t e r e s t i n g s u b j e c t f o r a good T h e s i s , e s p e c i a l l y i f '2A p e r m i t t e d t o have s i n g u l a r i t i e s . However i n t h e s e l e c t u r e s

we

s h a l l u s e a d i f f e r e n t approach. Volterra

r 71

proved f o r t h e problem (1.11)

theorem under q u i t e g e n e r a l hypotheses on e v e r y compact s e t ]

.

Pi

, (1.12) a

uniqueness

(t,z) [boundedness i n

We s h a l l o b t a i n t h e V o l t e r r a r e s u l t

as

a

p a r t i c u l a r c a s e of t h e t h e o r y we a r e going t o develop i n t h e next lectures.

2. The a b s t r a c t V o l t e r r a i n t e g r a l e q u a t i o n . L e t 5 be a complex Banach s p a c e , where t h e norm i s denoted '1

11

.

by

L e t u ( t ) be a f u n c t i o n of t h e rea.1 v a r i a b l e t with values

in S

(5)

.

Suppose u ( t ) is d e f i n e d i n t h e i n t e r v a l 1 of t h e r e a l axis

and is c o n t i n u o u s i n 1 , i . e . f o r e v e r y t o €I

W e shall write that u(t) r

e"(1,s ) .

L e t u s now suppose t h a t I : [ d > f . l i s bounded and c l o s e d ( i . e . c o m p a c t ) and l e t u s c o n s i d e r a decomposition of C d , p l i n t o p a r t i a l i n tervals

Lt

k ,tk+,3 d r

t,

f

tZ _L

. - - . 5- t,, 2

t m +=, ( 3 .

By u s i n g t h e e l e m e n t a r y arguments f o r d e f i n i n g t h e i n t e g r a l of a r e a l v a l u e d c o n t i n u o u s f u n c t i o n e x t e n d e d t o C d , p l , one p r o v e s that

, if

u ( t ) E C o ( I , ~ ) , t h ef o l l o w i n g l i m i t e x i s t s i n t h e s t r o n g

convergence of S :

n

$+*

where

Zlr

k.4

i s a r b i t r a r i l y chosen i n [tyl,ik+,fand

By d e f i n i t i o n

The i n t e g r a l i s a d d i t i v e , i .e . i f a

The i n t e g r a l is l i n e a r , i .e. if

_c

y

_z (3,

u f t ) = a, U, it) t &+u2(t)( 4,

$4

complex c o n s t a n t s , u , ( t ) , u l ( * ) c o n t i n u o u s on [ d , p J )

Moreover

When we c o n s i d e r an i n t e r v a l 1 of t h e r e a l a x i s w i t h o u t any f u r t h e r s p e c i f i c a t i o n . we mean t h a t I may be bounded o r unbounded, c l o s e d o r open.

(5)

Although t h e r e s u l t we a r e going t o c n n s i d e r i n t h e f i r s t p a r t of t h e s e l e c t u r e s hold i n t h e more g e n e r a l c o n t e x t of Lebesgue i n t e g r a t i o n t h e o r y , we s h a l l r e s t r i c t o u r s e l v e s t o c o n s i d e r i n t e g r a l s of g e n e r a l l y continuous f u n c t i o n s which can be e a s i l y i n t r o duced and have enough g e n e r a l i t y t o cover most of t h e applications t o Mechanics. I f I is an i n t e r v a l of t h e r e a l a x i s (bounded o r not),the function

it 1, d e f i n e d i n I and w i t h v a l u e s i n S ,i s g e n e r a l l y continuous in I -

i f any bounded i n t e r v a l

i n t o i n t e r v a l s J, ,--.,1 , t e r v a l Jk

c o n t a i n e d i n I can be decomposed

such t h a t

( C = 4 , ...,m ), considering

k(t)

i s continuous i n any i n -

1% a s an open i n t e r v a l . I t i s

e v i d e n t t h a t sums of g e n e r a l l y continuous f u n c t i o n s i n J

are

g e n e r a l l y continuous f u n c t i o n s . I f a ( + ) i s c o n t i n u o u s i n t h e open and bounded i n t e r v a l (

) , we s a y t h a t

4

u ( t ) is summable i n

(d,rJ)

(d,p) whenever t h e func-

is bounded.

I f u ( t ) is summable i n P

1u c t , d t d

p

(4, )

t h e n we d e f i n e

= Aim jY;t)dt. &--)O

d+E

I t is v e r y e a s y t o prove t h e e x i s t e n c e and t h e f i n i t e n e s s of the l i m i t . If

I i s bounded and

u ( t ) i s g e n e r a l l y continuous i n

t h a t u ( t ) is summable i n 1 i f u ( t ) open i n t e r v a l c o n t a i n e d i n

I

,we s a y

is summable i n every bounded

I where a c t )

i s continuous. I f we

decompose 1 i n t o a f i n i t e s e t of such i n t e r v a l s : J, , ... , 5,

,

we put

The i n t e g r a l on t h e l e f t hand s i d e does n o t depend on t h e part i c u l a r decomposition of I i n t o t h e open i n t e r v a l s J,,... , J, I f I i s unbounded and u(t) is g e n e r a l l y continuous i n I ,we say

t h a t u ( t ) i s summable i n I i f 1) u ( t ) i s summable i n e v e r y bounded i n t e r v a l J c o n t a i n e d i n I ;

2 ) a p o s i t i v e c o n s t a n t L e x i s t s s u c h t h a t f o r e v e r y bounded J c I

jl I u ( t , l l d t

5

L

.

J

I f o n l y c o n d i t i o n 1) i s s a t i s f i e d , t h e f u n c t i o n

u (f)

is

said

l o c a l l y summable i n 1. If

I

( t ) is summable i n t h e unbounded i n t e r v a l

u

n o t e by and by

, p t h e e x t r e m a o f I ( o n e o r b o t h o f them b e i n g i n f i n i t e )

ot G

a n d i f we de-

t h e e x t r e m a o f t h e bounded i n t e r v a l J c I

,b

, we s e t

The e x i s t e n c e a n d t h e f i n i t e n e s s o f t h e l i m i t c a n be p r o v e d v e y easily. The i n t e g r a l o f a g e n e r a l l y c o n t i n u o u s f u n c t i o n ~ ( t j e x t e n d e dt o an i n t e r v a l

I

f i n i t e o r n o t , e n j o y s a l l t h e u s u a l p r o p e r t i e s of

t h e i n t e g r a l s , f o r i n s t a n c e t h e p r o p e r t i e s e x p r e s s e d by (2.1).(2.2). (2.3). L e t u s now d e n o t e by A a n i n t e r v a l o f t h e r e a l a x i s and by L' a n open s e t o f t h e Banach s p a c e d e f i n e d on

A x A x

V

.

5

with values i n

L e t f ( t , r , v ) be a f u n c t i o n

S

a n d s a t i s f y i n g t h e f o l l m-r

ing conditions:

1) a r e a l v a l u e d n o n - n e g a t i v e f u n c t i o n L summable i n A

,

(T)

s u c h t h a t f o r a n y p a i r u ,v

exists, locally

o f v e c t o r s of

V

and f o r a n y t t A

2) f o r a n y v ( s ) 6 e G ( i , V ) [ i - e . c o n t i n u o u s i n 1 a n d w i t h v a l u e s

, w h e r e I is a n y c o m p a c t s u b i n t e r v a l o f A

i n V] fixed

t

€

A

t h e f u n c t i o n of f :

u o u s a n d ) summable i n 3 ) f i x e d t, i n A

,

1 ; the function &

,

and f o r e v e r y

[ t , ~v,( t ) J is ( g e n e r a l l y c o n t i n -

belongs t o

e

.

where 1 i s any compact s u b i n t e r v a l of A

O ( I , S ) ,

L e t W ( t ) be a g i v e n f u n c t i o n b e l o n g i n g t o ~ ' ( A , v ) . ws~h a l l c o n s i d e r t h e following a b s t r a c t Volterra non-linear

i n t e g r a l equations

The f o l l o w i n g e x i s t e n c e and u n i q u e n e s s theorem h o l d s . 2 . I . Under t h e s t a t e d h y p o t h e s e s f o r f ( t , r , w )

EXISTENCE: a compact s u b i n t e r v a l

t,

I

and

of A

*(t) we

have

e x i s t s containing

s u c h t h a t t h e r e e x i s t s v ( t ) e0(1,v) ~ which is a s o l u t i o n of (2.4)

f o r every t c l . UNIQUENESS: If

(2.5)

U C ~ E)

,t

eacr,v)

,

~ c t =,

q ( t ,+ j - f [ t , r , u ( r ) ] d t

(~cI),

0

t h e n uCt) 5 w ( t ) . L e t I be a compact s u b i n t e r v a l of A c o n t a i n i n g t, and b a p o s i t i v e r e a l number s u c h t h a t i ) t h e s e t of 6? x R X

is contained i n

d e f i n e d by t h e c o n d i t i o n s

A x A xV ;

ii) f o r every t 6

I

I t is e v i d e n t t h a t by t a k i n g t h e l e n g t h of

I

and b s m a l l enough

c o n d i t i o n s i ) ,i i ) a r e s a t i s f i e d . Let

2 be t h e s e t o f t h e f u n c t i o n s u ( t ) o f e0(I,S)d e f i n e d by t h e

condition

Iluit)-9(t)il 6 b

Yte.1.

Let us consider f o r every u ( t ) ~ e o ( I, v ) and f o r t € 1 t h e mapping

T u = 'P(t) + \ L f ' [ t , ~ , u ( ~ ) ] d i .

(2.6)

t.

I f we s e t z v = T u , and s u p p o s e

~

€ , 2we h a v e

t

I I ~ ( t ) - 4 ( t )Ll l )I/ f [ t , ~ . ~ ( i - ) ] d l - / ~ $ [ t , Z , q ( l ) (1] d ~ t.

to

1,

+

[+,T,QCTI]~T to

4

1/ t.

1

t ~ C ~ I I I u ~ ~ j - ~ ~ ~ ~ I l d ~

Hence (2.7)

L e t u s now c o n s i d e r t h e f u n c t i o n

p(t)

E

~-'(I,R+ )

.t

= e x p ~ L) ( ~ ) d ' t

t L t c ,

+ " ,

exp 2 J t 0 ~ ( = ) d ~ t ? t o

=

eO(I,S )

and c o n s i d e r

IJIu(t)jll : m a x ( p ( f ) l l u ( f ) l l ] .

(2.8)

I

Since

L i s , w i t h r e s p e c t t o t h e norm

eo( I , 5 ) , For

t

a s a Banach s p a c e through t h e norm

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

u(t) E

a s a complete m e t r i c s p a c e .

eocr,s), v ( t ) ce3(1,s)

/l

p ( t ) ~ I T u =- ~p (~t )~ ~ ~f t.

Hence

(2.8), a c l o s e d s u b s e t of

we have t

[ t , ~ , u ( t ) ] a -J r It

t . i . ~ ~ ~ ) ~ c ~ r ] l

t o

+

l l l ~ ~ - 5~ ~ I\\~ u l- w\ \ \ \ . I f T is r e s t r i c t e d t o 2 we have t h a t T i s a c o n t r a c t i o n mappi n g of C i n t o i t s e l f . By t h e c o n t r a c t i o n p r i n c i p l e of T e x i s t s i n

: v =T v

, i . e . a s o l u t i o n of

I f u s a t i s f i e s c o n d i t i o n (2.5)

hence

I*

we have

r v.

-

(6)

See, f o r i n s t a n c e ,

1127

,

p.394-399.

(6)

(2.4)

a f i x e d point

The t h e o r e m , which h a s b e e n p r o v e d , is a t h e o r e m " i n t h e s m a l l " s i n c e i n g e n e r a l t h e i n t e r v a l I where t h e s o l u t i o n e x i s t s w i l l n o t coincide with A

. However

i f a p a r t i c u l a r h y p o t h e s i s is made, we

have a t h e o r e m " i n t h e l a r g e " .

P r e c i s e l y t h e f o l l o w i n g theorem

holds. 2 - 1 1 . I f i n a l l t h e h y p o t h e s e s c o n c e r n i n g $ ( t , ~ , vwe ) may assume V

5

S , t h e n t h e s o l u t i o n ~ ( tof) ( 2 . 4 ) is d e f i n e d i n 4he wholeofA.

In fact i f V

I

5 ,we

of A c o n t a i n i n g

ii)

,

c a n c h o o s e a r b i t r a r i l y t h e compact subinterval

t o , w i t h o u t i m p o s i n g t h e two c o n d i t i o n s i) and

c o n s i d e r e d i n t h e p r o o f of t h e o r . 2 . 1 .

T u is g i v e n by ( 2 . 6 ) , w i l l b e l o n g t o ea(?,S).Thus ( 2 . 7 ) h o l d s a s s u m i n g 2 = t h e f u n c t i o n w ( t ) = T u , where

F o r a n y m(t) t e ' ( I , S )

e a ( i , S ) . The r e m a i n i n g p a r t of t h e p r o o f of t h e o r . 2 . 1 r e m a i n s unchanged. T h i s p r o v e s , b e c a u s e of t h e a r b i t r a r i n e s s of

is defined f o r every t

t h e s o l u t i o n v(t) o f ( 2 . 4 ) 1st Remark. Suppose f

o n l y d e p e n d i n g on

?

t

and

I

,

that

A.

V

and s a t i s f y -

i n g t h e above s t a t e d c o n d i t i o n s . 1) and 2 ) . Assume q ( t ) r vj , where L ;

i s a v e c t o r of V

.

Then t h e i n t e g r a l e q u a t i o n t

.i(tI= w . ,+

f[i,v(~)]d~ o

i s e q u i v a l e n t t o t h e Cauchy problem where t h e s o l u t i o n ~ ( ti s ) s o u g h t i n t h e c l a s s of t h e f u n c t i o n s which, i n a s u b i n t e r v a l

I

of A c o n t a i n i n g t, , c a n be r e p r e s e n t e d

a s follows:

I v ( t ) = u(t,l+ j w ( r

dr

,

to

where

w(T)

i s g e n e r a l l y c o n t i n u o u s a n d summable i n 1

~ ( t c)o i n c i d e s

.

The function

w i t h t h e d e r i v a t i v e w l ( t ) o f ' d t ) i n a n y p o i n t of 1

where ~ ( t i)s c o n t i n u o u s , i . e. e x c e p t i n a f i n i t e s e t of p o i n t s , where t h e d e r i v a t i v e of w C ~ ) c o u l d f a i l t o e x i s t .

(')

When we s a y t h a t v'(t) i s t h e d e r i v a t i v e of

(3)

V(t)

i n the point t

The d i f f e r e n t i a l e q u a t i o n ( 2 . 9 ) must b e s a t i s f i e d i n e v e r y p o i n t

t where

2r1[f)

is c o n t i n u o u s .

Theorems 2 . 1 and 2 - 1 1 p r o v i d e e x i s t e n c e and u n i q u e n e s s t h e o r e m s f o r t h e Cauchy problem ( 2 . 9 ) , ( 2 . 1 0 ) 2nd Remark.

.

The r e a s o n why we have n o t assumed p l t ) a I i n d e f i n ( 2 . 8 ) i s b e c a u s e , o t h e r w i s e , f o r p r o v i n g t h e con-

i n g t h e norm

t r a c t i v e c h a r a c t e r of t h e mapping T we had t o impose a f u r t h e r r e s t r i c t i o n on

I

.

T h i s would h a v e p r e v e n t e d u s from g e t t i n g a n

immediate proof of t h e theorem " i n t h e l a r g e " 2 . 1 1 and a l e s s e l e g a n t p r o o f f o r t h e e x i s t e n c e i n t h e l a r g e of v ( t ) h a d t o b e p r w i d e d ( s e e f o r i n s t a n c e 1131

,

p.292-291,theor.10.6.1).

By % (5)we d e n o t e t h e s p a c e of t h e l i n e a r c o n t i n u o u s mappings

into S

of 5

. &(S) i s

K

a complex Banach s p a c e when endowed w i t h the

norm

L e t K(+,r) be a f u n c t i o n d e f i n e d i n A x A

with values i n % ( S ) .

Suppose t h a t t h e f o l l o w i n g h y p o t h e s e s a r e s a t i s f i e d .

, k(t;t)

I ) For every t e A

c'

is a g e n e r a l l y c o n t i n u o u s f u n c t i o n d

i n A ;

1 1 ) II K (t,t)llL

t

L (7) w i t h L ( T )2 o

and l o c a l l y summable i n A

cA

;

t

and v(+)€eo(I,5) imply w ( t ) = k ( t , r ) u ( r ) d r ~ e ' ( I , S ) . t I t i s e a s i l y s e e n t h a t h y p o t h e s e s I , I I , I I I on k ( t , t ) i m p l y t h a t I

)

.

t h e h y p o t h e s e s 1 ) , 2 ) ,3) above s t a t e d f o r E ( t , ~ , v )a r e s a t i s f i e d when we assume F ( t , ~ , v ) :k

(t,r)v

, Vs S .

Hence from t h e o r e m 2.11 we deduce 2.111.

Given q(t] e

C'(A,S)?

u n d e r t h e h y p o t h e s e s I ) , 11) ,111)for

K ( t , 2 ) t h e r e e x i s t s one and o n l y one s o l u t i o n v(t)6

we mean t h a t

e0( A ,

S)

of t h e V o l t e r r a l i n e a r i n t e g r a l e q u a t i o n

~ ( t= )q ( t ) +

(2.11)

1;

K(~,T).L~(T d)

~

.

I t is e a s y t o s e e t h a t H y p o t h e s e s I), 11) ,111) f o r

2. IV.

compact i n t e r v a l A' c A

K (t,T) a r e

s a t i s f i e ? i n any

if K ( t , z ) ~~ ' [ A x A ,&,(s)] ,

hence t h e con-

c l u s i o n s of t h e o r e m 2.111 h o l d t r u e u n d e r t h i s h y p o t h e s i s on K(t,7)

3. The Peano-Gronwall lemma.

Let u ( t ) c e o ( l , ~and +)

3.1.

able function i n

If f o r

toh

I

I

l e t L ( t ) b e a n o n - n e g a t i v e and summ-

.

and f o r any t e

I

1

t

u(t,

(3.1) where

c

(3.2)

c r

5

I L ( ~ ) ~ (, ~ ) ~ ~ I

is a non-negative c o n s t a n t , then

~ ( ti) c exp

1 JT~(T)dil

( t ~ )1.

+*

A l t h o u g h t h e p r o o f of t h i s lemma, which c a n be s t a t e d i n a s l i g h t l y more g e n e r a l f o r m , c a n b e f o u n d i n many t e x t - b o o k s ,

we

s u p p l y h e r e , f o r t h e c o n v e n i e n c e of t h e r e a d e r , a s h o r t p r o o f . L e t u s f i r s t assume t

>t, . From (3.11, by m u l t i p l y i n g b o t h s i d e s t

for

we g e t

2- [ e r p (- / ci t

t

/

t

~ ( ~ ) d i L) ( ? ) a ( x , d r t o t*

]

t

t c l(t)ex,(- J~(.r)cir) t,

and, a f t e r i n t e g r a t i n g both s i d e s i n t h e i n t e r v a l

(3.3)

L

L -etcexp

S u b s t i t u t i n g t h e r i g h t hand s i d e of s i d e of

J+:

It.,

t3

l.(~)di.

(3.3) i n t o t h e r i g h t hand

(3.1) ,we g e t (3.2).

The p r o c e d u r e i s a n a l o g o u s f o r

t

cto.

Under t h e h y p o t h e s e s I), 2) ,3) on f ( t , t , v ), assumed i n S e c t . 2 ,

and f o r c Q ( t )e~o ( ~ , S ,) y . ( t ) € e o ( ~ , s ) , l e t u s s u p p o s e t h a t f o r t ~ I

(to E I

C A

)

v c t ) = q t t ) +f l i t , t , v ( i ) l d T

,

/

;

to

Hence

t o

t

I j ' L(7)I I W ) - u ( ~ ) I I ~. T / P

i t - t i L

I I L P ( O - Y ( ~t) I I

t

F o r t h e Peano-Gronwall

.

lemma, we g e t

I I V ( ~ ) - U ( + )-1I I mar I I ? ( ~ ) - Y ( + ) exp II

(3.4)

t

~ ( tt ) p[t,r, g ( T ) l d ~.

u(t)

I

t

1 j~ ( r ) d r l to

which e x p r e s s e s t h e c o n t i n u o u s dependence of t h e s o l u t i o n v ( t ) of

.

( 2 . 4 ) on t h e datum 4'Ct)

I f # ( t , ~ , v ) = ~ ( t , f ) v( l i n e a r c a s e ) and

K(~,~)E~~[AXA,%(S)I,

we have ,Iv(t)ll 5

w a x llQ(t)ll

I

exyilt-t.i

max ~ l ~ ( t , r ) l l ]

IxI which i s a n "a p r i o r i " e s t i m a t e of t h e s o l u t i o n vlt) of t h e l i n e a r i n t e g r a l e q u a t i o n (2.11) i n t e r m s of t h e d a t a

q ( t ) and

K

(t,f ) .

I t must be remarked t h a t from ( 3 . 4 ) one g e t s a g a i n t h e p r o o f of t h e u n i q u e n e s s s i n c e CP(t)r Y ( t ) i m p l i e s VCt) f u ( t )

.

However we have

s e e n t h a t t h e u n i q u e n e s s i s , i n o u r a p p r o a c h , a n i m m e d i a t e by- p r o d u c t of t h e same p r o c e d u r e which g i v e s t h e e x i s t e n c e of a solution.

4. Some a p p l i c a t i o n s of t h e a b s t r a c t t h e o r y . L e t A b e a bounded domain of t h e C a r t e s i a n s p a c e re21 variables defined i n

A

, xl

A,,...

(

5 is

.

t h a t f o r any p a i r x

h

We s a y t h a t t h e r e a l v a l u e d f u n c t i o n v(x)

t h e c l o s u r e of t h e domain A ) is d -H8lder-

,?

of p o i n t s of

Ivt*)- v ( J ) ! G H

A

Ix-3 I d

where d i s a g i v e n r e a l c o n s t a n t s u c h t h a t o c

d

L i

d-~glder-continuous functions defined A

s h a l l d e n o t e by

of t h e

whenever a n o n - n e g a t i v e c o n s t a n t H e x i s t s s u c h

-continuous i n A

of t h e r e a l

xt

e

a

(6) ,

. The ,

space

which we

i s a Banach s p a c e i f we i n t r o d u c e t h e

f o l l o w i n g norm:

Iv(x)-

l l v l l = wax l ~ ( x ) +l bup d A x,JtA

~ ( 7I )

Ix-ytd

x+7

This p a r t i c u l a r s p a c e w i l l be c a l l e d a Schauder s p a c e . L e t u s suppose t h a t A

e3

a r y of c l a s s

(8)

h a s a smooth boundary 3 A

, s a y a bound-

and c o n s i d e r t h e f u n c t i o n u which is a so-

l u t i o n of t h e D i r i c h l e t problem

A,u =

(4.1)

Suppose t h a t v t (4.3) where

in A

.IY

ed( A )

I1 u/;L ll,

(4.2)

, ( 0

4

d 4 i )

.

.

2A

The f o l l o w i n g e s t i m a t e s hold

c t l Itd~

5

on

u =G

( i , L = 4 , ..., t )

c i s a c o n s t a n t only depending on A and

d

. These

are the

c e l e b r a t e d Schauder e s t i m a t e s f o r t h e s o l u t i o n u of t h e problem (4. I), (4.2)

( s e e C14 1 , p . 3 3 5 , CIOI, S e c t . 3 4 ) .

We s h a l l denote by class to

in

u s denote by G t h e l i n e a r o p e r a t o r

a function wc

Hence

ril

s p a c e of f u n c t i o n s which a r e of

/? and s u c h t h a t e a c h n - t h p a r t i a l d e r i v a t i v e k l m g s

ed(aI L e t

( 4 . 1 ) , (4.2).

entd(E)t h e

C * (3) i n t o t h e s o l u t i o n

The range of

G

which t r a n s f o m

u of t h e D i r i c h l e t problem

belongs t o

e

.3.+6

(i).

is a bounded l i n e a r o p e r a t o r from

Moreover i f we put

eb(i) into

L e t u s now c o n s i d e r t h e V o z t e r r a i n t e g r o - d i f f e r e n t i a l

itself equation

which we s h a l l w r i t e i n t h i s more g e n e r a l form (4.5)

b 2 u = $ ( r , t 1-

jtZ,(Xp*.T)

ulLk( x J T ) 4%

( t lo)

0

We assume t h a t

, fLL a r e r e a l v a l u e d f u n c t i o n s such t h a t

1) + ( k , t ) c e O ( r \ x R + ) and, moreover, f o r e v e r y t > o : + ( x , t ) € C a ( b } ; 2)

f.

( x , ~ , )z E LL

eO(ax R + X R'

) and, moreover, f o r e v e r y ( t , )~E R+x RT :

e"

When we s a y t h a t t h e boundary Q A of A i s of c l a s s we inean t h a t i n some neighborhood of any p o i n t of t h e boundary, QA can be p a r a m e t r i c a l l y r e p r e s e n t e d by f u n c t i o n s of c l a s s e n .

("

E;R(x,t,t )

r Cd(i).

L e t u s a s s o c i a t e t o (4.5) t h e boundary c o n d i t i o n u (x,t

(4.6)

: o

( x , t ) ~2 A x

R'.

W e look f o r a s o l u t i o n of ( 4 . 5 ) , (4.6) which

i ) is c o n t i n u o u s i n A i i ) f o r e v e r y t r R+

,

x

,

'R

u(x>+)E

Set Azu = V(x,t

(4.7)

1.

W e may i n t e r p r e t V ( x,t)a s a f u n c t i o n d e f i n e d i n RT and w i t h valug

in

ed(A). W e

s h a l l d e n o t e by v(t) t h i s f u n c t i o n . Analogously

d e n o t e by q ( t ) t h e

ed(A)v a l u e d

function t

4

we

H(x,t).

Set

-!;c,(~.t,f)rLCI = k(t,t).

(4.8)

Because of (4.4) and of t h e assumed h y p o t h e s e s on f t h e o p e r a t o r K ( t , ? ) i s a bounded l i n e a r o p e r a t o r of

K ( t , r ) E C*{R'X RT, & [ ~ * ( A J I

i t s e l f . Moreover

F o r (4.7) and (4.8)

(y.t,C

ea(h1

)

,

into

1.

we deduce t h a t t h e problem ( 4 . 5 ) , ( 4 . 6 ) is

equivalent t o t h e Volterra l i n e a r i n t e g r a l equation

v ( t ) = q ( t ) +(

t

J

K(~,T)W{.;)~T

c o n s i d e r e d f o r f u n c t i o n s va:ued

i n the

C d( A )

space.

From theorem 2. I V we deduce 4.1.

Under t h e assumed h y p o t h e s e s on A

, 4 , PiL , t h e problem

( 4 . 5 1 , (4.6) h a s one and o n l y one s o l u t i o n u ( r , t ) s a t i s f y i n g t h e above s t a t e d c o n d i t i o n s ij ,i i ) . I t must be remarked t h a t , a l t h o u g h t h e o r . 4 . 1 p r o v i d e s an e x i s t e n c e and u n i q u e n e s s theorem, an u n i q u e n e s s theorem u n d e r less r e s t r i c t i v e h y p o t h e s e s on h and on t h e f u n c t i o n E a p ( x , t , t )c a n be 1 F.

o b t a i n e d by employing t h e o r i g i n a l p r o c e d u r e of V o l t e r r a , s i m p l i f i e d by t h e u s e of t h e Peano-Gronwall lemma. To t h i s end l e t u s assume t h a t A

h a s a piece-wise smooth boundary and i s s u c h t h a t

t h e c l a s s i c a l Gauss-Green f o r m u l a s h o l d i n r e g u l a r domain)

.

A

(briefly, A

is a

L e t u s assume t h a t f i t (0,T)

(T 7

0

pi R 1; ( x , t . z ) is

f o r e v e r y t , i s continuous

ir,

A

x

, a r b i t r a r i l y f i x e d ) and t h e s p a t i a l d e r i v a t i v e piece-wise continuous i n A x ( O , T )

Let a c o n s t a n t C, e x i s t (4.9)

( x , t r t 1,

If'i,(x.t,7 ' I

+

(3)

.

such t h a t

lfiR,i(x,t,r)l

5

CT

(x,t,~)

4.I I. Under t h e assumed hypotheses on A

c A x (b.T)x(O,T).

and Pik t h e problem e4(A) and

(4.6) h a s a t most one s o l u t i o n , w h i c h belongs t o

(4.5), has

piece-wise c o n t i n u o u s second d e r i v a t i v e s i n A . We have t o prove t h a t i f we assume i n (4.5)

( ~ , t ) 0r , Eq. (4.5)

,

(4.6) imply u ( x . t ) s 0 . Let u s multiply

b o t h s i d e s of (4.5) f o r u ( x , t )and i n t e g r a t e

over A . A f t e r an i n t e g r a t i o n by p a r t s and a f t e r applying Cauchy i n e q u a l i t y , we g e t

Hence,by u s i n g Schwarz and P o i n c a r 6 i n e q u a l i t i e s ,

C(T,A)

depends only on T and A

and from (4.6) we deduce

. From

u(x,t)i0

for

t h e Peano-Gronwall

(x,t)gAx

lemma

[O,Tl.

The e x t e n s i o n of theorems 4.1, 4.11 t o t h e problem (1.5), (1.6) p r e s e n t s only formal d i f f i c u l t i e s .

Of c o u r s e one has t o assume

s u i t a b l e smoothness hypotheses on t h e e l a s t i c i t i e s aibjk( x )

and

i n a d d i t i o n some " e l l i p t i c i t y h y p o t h e s i s " , f o r i n s t a n c e t h a t t h e

(9) A f u n c t i o n $ d e f i n e d i n a r e g u l a r domain 3 is s a i d t o be piecewise c o n t i n u o u s i n f5 i f

B = B f u 8 , u ... ~ 4 , . , . . . , B-... a r e mutually d i s j o i n t r e g u l a r domains; i i ) $ c o i n c i d e s i n Bk w i t h a f u n c t i o n continuous i n & .

i)

where B,

q u a d r a t i c form i n 6 v a r i a b l e s a;Rjkfx)&i;l, & .

( ELL : EL; )

Jk is p o s i t i v e d e f i n i t e f o r every x .

We s h a l l n o t d e v e l o p h e r e t h i s s u b j e c t and r e f e r t h e r e a d e r t o p a p e r s [ 151, i161, where t h e a n a l o g o u s of theorems 4 . I , 4. I I have been e x t a b l i s h e d , a l t h o u g h by somewhat more c o m p l i c a t e d p r o o f s , i n t h e c a s e of a homogeneous i s o t r o p i c body. We p r e f e r t o . c o n s i d e r i n t h e n e x t S e c t i o n t h e more i n t e r e s t i n g c a s e when on t h e boundary of t h e body, i n s t e a d of t h e d i s p l a c e ments, t h e s u r f a c e . f o r c e s a r e g i v e n . We wish now t o show, a s a u s e f u l example, how t h e a b s t r a c t t h e o r y of S e c t . 3 a p p l i e s t o a n o n - l i n e a r i n t e g r o - d i f f e r e n t i a l Let F(x,u,w,z ) for

XE

be a r e a l f u n c t i o n o f f o u r r e a l v a r i a b l e s defined

, u L V , w cW , P E Z ,

[o,.i]

problem.

where V

,W , Z

a r e closed in-

t e r v a l s of t h e r e a l a x i s . W e assume t h e f o l l o w i n g h y p o t h e s e s :

2) f o r

(X,U,W,Z)E

have 3) f o r

[o,ii

F(x,u,w,2 )

X E

XU x

x

Z

,

and f o r e v e r y

3

Z

, we

('O)

+ ~

[0,-i] and f o r u,;

W

€ ;2 E

U

;

w,GEW;

2,s E

Z

~ F ( x , u , w , z ) -F ( x , G , G , ~ 1 )G c , ( ~ u - ~ I + I w - ~ ~ + I z - ~ I ) ,

where c, is a c o n s t a n t i n d e p e n d e n t of x L e t H (x,t,'t, u , W , 2 )

.

be a r e a l f u n c t i o n of s i x r e a l v a r i a b l e s

s a t i s f y i n g t h e following hypotheses: 4 ) H(x,t,z,u,w,

5) f o r

f ) 6

, o [ ~ ~ , x4 R ~

( x , t , r ) t [0,-13xRXR

x R x ux W x Z

and f o r

3

u , ~ E U , w , ~ z t, $~E, Z

)I

I ~ ( x , t , t , ~ p ~ , 2 ) - H ( X , t , f , ; , ~ , ~

C4O)

;

H y p o t h e s i s 21 is e q u i v a l e n t t o t h e f o l l o w i n g ones: i) F r 0 i f Z i s bounded, ii) Ffo if inPZ =-C- , sup2 r +-, i i i ) F z o i f inPZ > - m , s u p z - + - .

,

where

C,

i s a c o n s t a n t i n d e p e n d e n t of

( x , t , ~

.

L e t L, and L,be t w o l i n e a r r e a l v a l u e d f u n c t i o n a l s d e f i n e d f o r every r e a l valued f u n c t i o n belonging. t o every p a i r of r e a l c o n s t a n t s a , b

e 4[ O , j j

and s u c h t h a t f o r

the following hypothesis

is

satisfied:

imply a r b : 0 . H y p o t h e s i s 6) a s s u r e s t h a t t h e s o l u t i o n of t h e problem

I,"'= .v.

L, ( u )

= L ~ ( = ~o ,)

~

r

~

~, lV C ~~ ' [ ,O , I~I

l

c a n be r e p r e s e n t e d a s f o l l o w s ~ ( x =,

(4.10)

I'G(X.~)Y(~)~-( 0

and moreover 1

Gx(x,)

u1(x) =

(4.11)

0

L e t u s d e n o t e by

~ () d1j

,

and by G d u t h e o p e r a t o r s g i v e n by (4.10)

Gu

and (4.11), r e s p e c t i v e l y . W e s h a l l c o n s i d e r t h e s e o p e r a t o r s a s o p e r a t o r s from t h e Banach s p a c e

e O [ o , i li n t o i t s e l f and assume

these f u r t h e r hypotheses

7)

c , ( l l ~11 t 114, 11 t i ) = 6 c2

8) F o r e v e r y

4

i

[(IIGIJ + I I G ~ I I ) ( . ( - s ) - ' i+ I

v ( x )

4

4.I 1 1 Given q ( x , t )

E

3) , 4 ) . 5 ) , 6 ) ,7),8), a a

i.

e* 1 [ 0 , i J , 2 ]

e'{[ o , { ] ~ R , Z-321, u n d e r )

o

c o n d i t i o n s 1) , 2 ) ,

e x i s t s s u c h t h a t one and o n l y

s o l u t i o n u ( x , t ) of t h e i n t e g r o - d i f f e r e n t i a l problem

one

exists for

-a

_L

t

5 a

large ( i . e . f o r g

6

.

[ - a , o.]

,

the solution e x i s t s i n the

R 1. The s o l u t i o n

e0{ [o, i]x

of f u n c t i o n s belonging t o every t

= R

If

,u ( x , t )

E

u (x,t)

[-a, a

i s sought i n t h e class

I] and such t h a t , f o r

C2[0,41.

Let u s f i r s t c o n s i d e r t h e a u x i l i a r y problem

where

u

E

e210,AJ

and (P(*) i s a given f u n c t i o n belonging t o

.

eo(Co,dl,Z),Set uH=2

If (4.15) a r e s a t i s f i e d we have u = G 1 .

Problem (4.14), (4.15) is e q u i v a l e n t t o t h e f o l l o w i n g one Z=Tz+q

(4.16)

2 = eo([o,l],z), which, being a c l o s e d s p a c e eo[0,4 ] , can be c o n s i d e r e d a s a

considered i n the space s u b s e t of t h e Banach

complete m e t r i c space. L e t II II

denote t h e norm i n

e0r 0 , d 1 i . e .

IIPII = * n a x l z ( x ) l . l0,il

Because of t h e assumed hypotheses we have T Z + q

E

z

for 1 6 C

and, moreover,

Hence t h e mapping $: T z + ~is a c o n t r a c t i o n of L e t u s denote by

2.a q

2

into itself.

t h e unique s o l u t i o n of (4.16). L e t

9 :a;.

We have Hence

II r (4-3 I-' I I ~ - $11

1 1 6 1- ~ L e t u s now s e t ~ ( x , t )=

ux J

a

( x , t ). Problem (4.12), (4.13) is equiv-

a l e n t t o t h e f o l l o w i n g one (4.11)

I-Te

= L P ( x , t ) + l t H [ x , ~ , T ~ G . z ,~z I d i .

I f we s e t 2-TZ =ir(x,t),we have t h a t (4.17) i s e q u i v a l e n t t o

Assuming

$ ( t , r , ~= )H [ x , ~ , T GGv,G,GLY,v] , ,

where $ ( x , t , v ) m u s t be viewed a s a f u n c t i o n v a l u e d i n t h e s p a c e

e0[o, { ] , a n d

g i v i n g t o ~ ( t and ) Q ( t ) a n o b v i o u s meaning, we may

w r i t e (4.18) a s follows:

v(+)=q ( t ) + [

f [ t , r , v ( T ) ] d ~ .

From theorems 2 . 1 , 2 . 1 1 we deduce t h e a c t u a l one. A l t h o u g h t h i s theorem must be c o n s i d e r e d n o t h i n g more t h a n a mere e x e r c i s e , i t is p e r h a p s u s e f u l p o i n t i n g o u t t h a t s e v e r a l p a p e r s a r e nowadays produced i n mathematics by t h e same procedure, which c o n s i s t s i n c o n s t r u c t i n g a r t i f i c i a l problems where t h e hyp o t h e s e s a r e a d a p t e d i n s u c h a way t o match t h e r e q u i r e m e n t s of some p r e - e x i s t i n g

general theory.

5. E l a s t i c body w i t h memory s u b j e c t e d t o g i v e n body- and surface-forces. L e t u s a s s o c i a t e t o t h e i n t e g r a l - d i f f e r e n t i a 1 s y s t e m (1.5)

the

f o l l o w i n g boundary c o n d i t i o n s on 9 A

v(*)t

I\),(*)

point

x

,\?L(x),

Q ~ ( X ) {

i n the

i s t h e u n i t inward normal t o Q A

.

The i n t e g r o - d i f f e r e n t i a l boundary v a l u e - p r o b l e m (1.5)

(5.1)

c o n c e r n s a n e l a s t i c body, which h a s a memory i n t h e s e n s e of V o l t e r r a and which i s s u b j e c t e d t o g i v e n body-forces t o given surface-forces

2(x,t)

F(x,t )

and

on '2A.

Although t h e r e s u l t s we a r e g o i n g t o p r o v e h o l d i n s l i g h t l y more g e n e r a l h y p o t h e s e s , we s h a l l assume, f o r s i m p l i c i t y , t h a t

e"'(x3)and v

c+A i s of c l a s s Coo , a L L j k ( ~ '

~ ~ ~ ~ ( ~ , ~ , T ) L

B e s i d e s ( 1 . 1 ) , we s h a l l suppose t h a t t h e q u a d r a t i c form

a ; L j t (E~ L L)E jk i n t h e s i x r e a l v a r i a b l e s &;A ( l 5 i f o r any x E

(Eik=

&Li )

3 ) i s positive definite

6

X3.

From ( 1 . 5 ) , (5.1) we deduce t h a t t h e g i v e n f o r c e s must s a t i s f y f o r any t 2 o

t h e "equilibrium conditions"

+

x A F(r,t)dx

IQAxAq(x~t)dG=O.

Hence we s h a l l assume t h a t

F(x,t)

, q ( x , t ) a r e smooth v e c t o r valued

functions, f o r i n s t a n c e belonging t o (5.2)

em(xLR t , x 3 )

and s a t i s f y i n g

.

L e t u s c o n s i d e r t h e problem (5.3) (5.4) Assume

(x>

4 ,y

c

4(x)

=

u. ) J l k I,

( G ; ~ , ~ ( X )

~ ~ ( Ux )

'lc

r(

=

in

A,

on ? A

XI

em(x3,X3)

and s u c h t h a t

J . A P ~ X

+

r ~ y d a = ~ . L A

A

T h i s problem h a s one and o n l y one s o l u t i o n

ue(x)

E

eoo(A )

such t h a t

L e t u s d e n o t e by L*O(X)

t h i s solution;

G

= G*+

HY

and H a r e l i n e a r o p e r a t o r s .

The most g e n e r a l s o l u t i o n of (5.31, (5.4) is g i v e n by u(x)=

q?'+H

r t a t b

~

x

where a and b a r e a r b i t r a r y c o n s t a n t 3 - v e c t o r s Let o c a

1

1

.

(see [ 171).

With t h e same meaning g i v e n t o t h e norm II

II,

t h e f o l l o w i n g Schauder e s t i m a t e h o l d s f o r u o ( s e e 1183,

i n Sect.4, p. 74) :

where 1 I

means t h a t t h e Schauder norm of t h e r e l e v a n t func-

I QA,d

t i o n is taken over 9 A

i s t h e g r a d i e n t of y o v e r a A ; c is

;

a c o n s t a n t o n l y depending on A and on t h e e l a s t i c i t i e s . I f we a s sume

4

6

ed(A)and

vf E ed(aA),

y such t h a t

(5.4) h a s a u n i q u e s o l u t i o n

LL'

t h e n t h e problem ( 5 . 3 )

,

s a t i s f y i n g ( 5 . 5 ) and ( 5 . 6 ) .

Set

e

L e t us d e n o t e by

I td

( 3) ~ t h e s p a c e of t h e ( v e c t o r v a l u e d )

f u n c t i o n s w(x) d e f i n e d on '2A C o n s i d e r t h e Banach s p a c e

and s u c h t h a t Ow

e b ( A ) x e'td(3,4),where

6

e d (aA ) .

t h e norm II Lr I1 of

t h e v e c t o r ZTi{y(~),~(x is) ! llv 11 = 11 'J;

(Id+ Il v-(x) ll

/3A ,d

t It V <(x)

llOA>

and c o n s i d e r ?r(t)ijv,(x,t),v 2 ( x , t ) fa s a f u n c t i o n , v a l u e d i n t h i s s p a c e L e t us i n t r o d u c e t h e o p e r a t o r s K i j ( t , ~by ) setting

K

K

( t , ~V,) t 24

Denote by

22

it.r 1%

=

uif.,&

t h e s u b s p a c e of

S

(x,~,T)

%(*) [ 4 5 t HV-1jlh.

ed(i) x eitd(2A)

v e c t o r s U ; v,(xl,v;(x){such t h a t

I ,Av,(rldx

+

A

lxhq('0d6=

0.

aA

I f we c o n s i d e r t h e m a t r i x o p e r a t o r K,, ( t , z ) k,, (t,z) KCt,t)

r

k t

kaz f t , ~ )

formed by t h e

we see t h a t i t maps, f o r any t ? o ,2. > v e c t o r of S

.

Denote by

$ ( x , t ) ] . Because of

G

,

a v e c t o r of

S into

t h e v e c t o r valued function

ip(t)

( 5 . 2 ) we have 9 (*) 6

.

5

a

1F[x,:

The problem (1.5),(5.1).

(5.5) is e q u i v a l e n t t o t h e l i n e a r V o l t e r r a i n t e g r a l equation functions valued i n the space

vci,

(5.7)

+ jtk

(t,r, V

.,

for

5 ( T )

nr

=

it).

0

(5.6) we have

Because of

and o n l y one s o l u t i o n

Zr(t)i

k ( t , z ) ~ e O [ ~ + x ~ + , ~ ( ~ ) ]one . ~ e n c e

Iy

3

(x,t),~~(~,t)

C'(R~,S) of Eq. (5.7)e*ts

( t h e o r . 2 . IV) .

is t h e

The most g e n e r a l s o l u t i o n of problem ( 1 . 5 1 , ( 5 . 1 )

follow-

ing: u(x,t)r

~ v ; ( j , t ) i +i u 2 ( i f , t ) + a ( + ) +b i t ) / \ h ,

where a(*) and b ( t ) a r e a r b i t r a r y 3 - v e c t o r v a l u e d f u n c t i o n s of t

6 . I n v a r i a n c e of t h e s p e c t r u m w i t h r e s p e c t t o p e r t u r b a t i o n s

due t o V o l t e r r a ' s l i n e a r i n t e g r a l t r a n s f o r m s . Let

L

be a l i n e a r t r a n s f o r m a t i o n ( o p e r a t o r ) w i t h domain

t h e complex Banach s p a c e s u b v a r i e t y of

DL . We

SL

in

5 and r a n g e i n 5 . L e t U be a l i n e a r

s h a l l c o n s i d e r t h e r e s t r i c t i o n of L t o L'.

We s h a l l s a y t h a t t h e complex v a l u e 1 is a r e g u l a r v a l u e f o r L (restricted t o into

u

U

) i f a bounded o p e r a t o r

and i s s u c h t h a t

(L-1 ) G

z

C 1 e x i s t s , which maps 5

-

)

i

:

I

z

identity

operator). The s e t of a l l t h e s p e c t r u m of

L.

1 ' s which a r e n o t r e g u l a r f o r L form

the

(restricted t o U ).

L e t k l ( t , ~ )be a f u n c t i o n d e f i n e d i n A x A of t h e r e a l a x i s ) and w i t h v a l u e s i n Z ( S )

(

,

A compact i n t e r v a l s a t i s f y i n g t h e con-

d i t i o n s of theorem 2 . 1 1 1 . L e t ~ t ( t ) b ea c o n t i n u o u s f u n c t i o n d e f i n ed i n A with values i n the operator

U

.

Let us f i x

tu i n A and l e t u s c o n s i d e r

Lu(t)-l u(t)

+

1,H ( t , c ) , ( t ) d s t

t

5

t i n t h e v a r i e t y CO(A,V ) of t h e Banach s p a c e

Ou-lu(t)

C o(A,S)

where t h e f o l -

l o w i n g norm h a s b e e n assumed:

Ill vlll = max llw(t)ll The v a l u e 1 i s r e g u l a r f o r

.

O t i f a bounded o p e r a t o r

r,t e x i s t s

which maps e ' l ( ~ , S i n)t o e " ( A , U ) and i s s u c h t h a t f o r t~ A

.

( o t - 1 l )Xr t rAt ( Q ~ - X I 1 - 1 . se t

The

of a l l t h e

t h e s p e c t r u m of

t

@

3

' s which a r e n o t r e g u l a r f o r

C O ( ~ , u1.)

[restricted t o

From now on w e s h a l l c o n s i d e r L to

?' ( A , u ) L

6. I.

Let

where

h

Qt form

and

ot

restricted t o

and

U

respectively.

a n d O t have t h e same s p e c t r u m . be a r e g u l a r v a l u e f o r

L

.

Consider t h e equation

t.

qi*)tea(A.S).

Set

L u ( t ) - Xu(+) = u(t) W e have

u jt) =

is e q u i v a l e n t t o

G l ~ ( t ) Eq. . (6.1) r t

where

KA ( t , ~= )-!-I ( t , ~Gi ) Since f o r any

Kx

s a t i s f i e s t h e h y p o t h e s e s of t h e o r . 2 . I11 we have

,

c p ( t ) E C 3 ( ~ , ~ )one , and o n l y one s o l u t i o n w ( t ) 6 e 3 ( ~ , s )

Let us represent

vit) a s

~ ( t = ,v1 '9 i

.

By a c l a s s i c a l t h e o r e m of Banach ( s e e 1191 p . 1 0 2 ) bounded l i n e a r o p e r a t o r f r o m Assuming

r Xt = G XvXt

1 is regul.ar f o r

1 is r e g u l a r f o r L

.

is

a

itself.

we s e e t h a t 1 is a r e g u l a r v a l u e f o r

L e t u s now s u p p o s e t h a t we s e e t h a t

e ' ( ~S ,) i n t o

V :

@t

t

0.

. Assuming(;l= rt' a

7. The V o l t e r r a i n t e g r a l e q u a t i o n i n a i n f i n i t e i n t e r v a l .

The t h e o r y of t h e i n t e g r a l e q u a t i o n ( 2 . 4 ) becomes much more

-

d i f f i c u l t when we assume t, =

.

00

A l t h o u g h t h i s c a s e i s of l a r g e r

p h y s i c a l i n t e r e s t , t h e a n a l y t i c a l r e s u l t s which c a n be g i v e n

are

much more r e s t r i c t e d w i t h r e s p e c t t o t h e c a s e when, f o l l o w i n g Vol-

t,

t e r r a , w e assume t h e " c o n v e n i e n c e h y p o t h e s i s " L e t f ( t , r , v ) be a f u n c t i o n d e f i n e d i n i n t h e Banach s p a c e

R

x

R

> - a .

and w i t h v a l u e s

x S

S , s a t i s f y i n g t h e following hypotheses

1) a n o n - n e g a t i v e f u n c t i o n L ( t , r ) d e f i n e d i n Rx 2 e x i s t s t h a t , f i x e d t h e r e a l number c

i s a f u n c t i o n of for

(t,t)

E

t

E

, f o r e v e r y t ~ 1 , :(-=.z) , L

summable i n t h e i n t e r v a l

I Z x I Z,

Moreover, f o r where

7

1%

U,u

I,.L(t,s)is

::,-!

such t h a t

€ 5

,t

-

Lit,t)d.c

c, ,

5

0

c,, i s a p o s i t i v e c o n s t a n t o n l y d e p e n d i n g on z ; -,

2) l e t C 0 ( I , , S ) be t h e s p a c e of t h e f u n c t i o n s of

-t h a t

such

lj tr(t ) 11 is a bounded f u n c t i o n of t i n

e(I,.S)and

f o r e v e r y t c I t t h e f u n c t i o n of

t:

f

1,

.

ko(I,,5

!

such

For every v i z )

[ t , ~v (, r ) ] i s

6

summable ini;

3) t h e f u n c t i o n

(7.1) belongs t o

(L, S) .

7. I . Under t h e assumed h y p o t h e s e s on f

0

,

where

~

6

) t h e, i n t e g r a l e q u a t i o n

1 i s a complex p a r a m e t e r , h a s one and o n l y one s o l u t i e n

the space Set

( t , ' ; , ~, )g i v e n

E O ( I ~ ,if~ )I X I < T v = .P(t)+

-4 C

-00

Z

f

,

[ ~ , T , u ( T ~ ]d i .

T is a mapping from E a ( ~ , , ~ ) i n t iot s e l f . L e t u s assume i n ( ? " ( I , , S ) t h e norm

-

1%

which makes e m ( I , , S ) t o become a Banach s p a c e . We have

hence t h e proof f o l l o w s from t h e c o n t r a c t i o n p r i n c i p l e . L e t u s now assume t h a t f t

uniformly with respect t o t r 1

Z

.

I n t h e a p p l i c a t i o n s i t i s n a t u r a l t o assume t h a t t h e " p a s t h i s t o r y " of t h e m a t e r i a l s y s t e m u n d e r c o n s i d e r a t i o n h a s n o i n f l u e n c e

eJ(Ic

w h a t s o e v e r f o r v e r y remote p a s t t i m e , i n o t h e r words t h a t a e x i s t s such t h a t f o r every v ( T )E the integral

/*-LF

,S )

k>0

t h e c o n t r i b u t i o n of

( t 1%1

[ t , r , w ( r ) ld i

-m

c a n be n e g l e c t e d . We e x p r e s s t h i s f a c t by s a y i n g t h a t t h e l e n g t h of t h e memory i s

!I . I n h y p p t h e s i s 3) we k u s t r e p l a c e t h e i n t e g r a l

on t h e r i g h t hand s i d e of

(7.1) by

t

(t -C $ [ t , t , v ( z ) ~ d r .

7 . 1 1 Under t h e assumed h y p o t h e s e s , i f t h e l e n g t h of t h e memory is s h o r t , w e have e x i s t e n c e and u n i q u e n e s s .

We have now t o c o n s i d e r t h e i n t e g r a l e q u a t i o n t

u(t) = * ( t ) + I f w e set

jt-t, f l t , t ,

W ( ~ ) ] A T .

A

W e see that T

maps

- (It,S )

e

IIITv-T2111

O

,L

i n t o i t s e l f and moreover

Illw-2lll

I*

L(t,t)dr.

t-R

I f R is s u c h t h a t

l t-R L ( t , r ) d ~

d i ,

t

we have t h e proof of t h e theorem.

I t must be remarked t h a t i f

i s n o t s u i t a b l y s m a l l , theorem

k

7 . 1 1 c o u l d f a i l t o be t r u e . I f we c o n s i d e r , f o r i n s t a n c e , t h e v e r y simple l i n e a r i n t e g r a l equation. t

(7.3)

vlt): 9(t)t

We s e e t h a t t h e c o r r e s p o n d i n g tion for

0 4

c 1

,

v(7)dt.

L-t.

T

,

g i v e n by ( 7 . 2 ) , i s a c o n t r a c -

a n d , i n . thiS c a s e , w e have e x i s t e n c e and unique-

n e s s . However f o r d = 4

u n i q u e n e s s f a i l s t o h o l d s i n c e t h e homo-

geneous e q u a t i o n a s s o c i a t e d t o (7.3) i s s a t i s f y e d assuming arbitrary constant

%-!+)?

(I+)

.

L e t u s now c o n s i d e r t h e l i n e a r c a s e , i . e f(t,t,w)= K ( t , ~ ) u

where K ( t , ~i )s a f u n c t i o n d e f i n e d i n

.

1,

x

1 % and w i t h v a l u e s i n

L e t u s assume t h a t

I ) f o r every

I, and

+E

1%t h e f u n c t i o n of

J~ li K ( t , t ) u -

d r 6 c,

Z : Ill<(t,s)llis summable i n

:

Cr,

%

11) f o r e v e r y

w ( t ) ~e,

(l,,S)

Let us consider t h e l i n e a r i n t e g r a l equation (7.4)

~ ( t ) r=p l t ) t

i

a - *~ ( t , t ) u Cd~ t)

r q +X T u ,

we have

aterial rials

w i t h memory a r e s i m i l a r t o human b e i n g s : problems a r e s i m p l e r when t h e memory is s h o r t , more c o m p l i c a t e d when it is long: e d e i d i che furono l ' a s s a l s e il s o v v e n i r ! ( A . Manzoni : I1 5 Maggio )

..........

1

t

111

III T V 111 =

K ( t , ~vcr ) dr

ct III~III.

Ill r

-OD

Hence T c

& [ EO(Ic,S)]

and Eq. (7.4)

h a s one and o n l y one s o l u -

t i o n f o r 1 1 1 s m a l l enough. Observe t h a t Eq. ( 7 . 3 ) is a p a r t i c u l a r c a s e of ( 7 . 4 ) by assuming, g i v e n

K(t,~)

#!t? 0

, I=t

and

=I

for

t - k L t C

:0

for

r

r

t-R

.

The problem now a r i s e s : C h a r a c t e r i z e T s u c h t h a t (7.4) h a s one and o n l y one s o l u t i o n f o r any f i x e d X

i s a s e q u e n c e of l i n e a r bounded o p e r a t o r s of & ( 5 ),we

{ Ak

If

.

s h a l l c o n s i d e r t h e power series

where X i s a complex p a r a m e t e r . A w e l l known e x t e n s i o n of t h e c l a s s i c a l Cauchy-Hadamard

theorem s t a t e s t h a t i f w e s e t '1 L i f max l i m H A 1 I = tm.

f=O J

:[ma.

-

l i m 11 ~

k -7

E

h->m

~

t ] -, I ~

~ i f* a 5 mix l i m / I A k l L-7-

flit

r t w,

i f max l i i IA ll'lk 0 , k

9 is t h e convergence r a d i u s of t h e power series (7.5) ,

t h e s e r i e s ( 7 . 5 ) is t o t a l l y c o n v e r g e n t i n t h e d i s c

Pz0 I >t I 6

i .e . i f

f o r any

9'

9's

p

.

If

D Y' .-

t h e s e r i e s ( 7 . 5 ) i s n o t con-

Y L too

v e r g i n g i f I l l > ? . The convergence must be u n d e r s t o o d i n t h e s t r o n g t o p o l o g y of

If T

6

L(S)

( s e e [ 9 ] p.115-116).

c ( S ) a n d i f we c o n s i d e r t h e e q u a t i o n

(7.6)

V =X T u

+

rP

i t is w e l l known t h a t i f t h e power s e r i e s (7.5), where A h = 7 1 c o n v e r g e s , f o r some

# 0

, t h e n (7.6) h a s t h e u n i q u e s o l u t i o n &

L

v = z X T q For any XJ7.6)

k-o h a s one and o n l y one s o l u t i o n i f and o n l y i f t h e QO

convergence r a d i u s of k - o 1 ' ~ '

is t

=

( s e e E91 p. 117-1251.

On t h e o t h e r hand i t c a n be shown t h a t t h e f o l l o w i n g l i m i t 1, 'Ik

R;m I, T h-7-

e x i s t s and is f i n i t e ( s e e C91 p. 1 2 8 ) . T h i s l i m i t is known a s t h e s p e c t r a l r a d i u s of t h e o p e r a t o r T 7.111.

.

Hence

h a s e x a c t l y one s o l u t i o n , f o r a n y f i x e d 3.

Eq. (7.6)

,when

and o n l y when t h e s p e c t r a l r a d i u s of T v a n i s h e s . L e t u s c o n s i d e r a s a n example t h e i n t e g r a l e q u a t i o n (7.7)

q.r(t):

1 \fe-(t-r)P(-c)dl

+ LP(t)

-0

where

v ( t ) , cp (t l c

E" ( I,,

@ ) i. e . zr and

cP

a r e complex v a l u e d

f u n c t i o n s bounded and c o n t i n u o u s i n 1,. Set V ( t ) = etw(t)

, &(t)=et'f'(t)

Set

w ( * )=

/-

.

Eq.(7.7)

becomes

t l f ( ~d)i QI

From ( 7 . 8 ) we deduce

Hence

w(+,=ye1'

+

I

t

e

A ( t -t

*

(f )dZ

0

where y is a c o n s t a n t . W e h a v e

0

v l t ) i s a s o l u t i o n of ( 7 . 7 ) b e l o n g i n g t o

We have t h a t

i f and o n l y i f t h e r i g h t hand s i d e of

(7.9)

-

e3(Ic,&?)

r e p r e s e n t s a function

of t h i s s p a c e . I t i s e a s y t o s e e t h a t : ' 1) I f @& 1 2 .i , %he f u n c t i o n zrCi) = i v e n by ( 7 . 9 ) b e l o n g s t o -eU(Ic , 6?), no m a t t e r how t h e complex c o n s t a n t i s chosen.Hence C

E q . ( 7 . 7 ) h a s i n f i n i t e l y many s o l u t i o n s . 2 ) I f 6?e

1=

1

, t h e f u n c t i o n v(t) b e l o n g s t o

o n l y i f t h e "datum"

cQ(*)

E Y ( ~ @)t , i f

i s s u c h t h a t t h e f u n c t i o n of t

and

is bounded i n

I,

.

I f t h i s c o n d i t i o n is s a t i s f i e d t h e problem has

i n f i n i t e l y many s o l u t i o n s s i n c e c can be chosen a r b i t r a r i l y . 3) I f

Re 1

1

v ( t )belongs t o

,

E D( I,,

6? ) i f and o n l y i f t h e

c o n s t a n t e is g i v e n by

hence

and Eq. (7.7) h a s one and only one s o l u t i o n .

-

From t h i s a n a l y s i s we deduce t h a t t h e spectrum connected w i t h t h e i n t e g r a l e q u a t i o n (7.71, c o n s i d e r e d i n

e'(I,

,$1,

fills

the

whole h a l f -plane (ii.e 1 2 4.. Before e n d i n g t h i s s e c t i o n we propose t h e f o l l o w i n g two exerc i s e s t o the reader: 4 .

1) Consider t h e l i n e a r o p e r a t o r from

eo(j,,5 ) i n t o

i t s e l f given

by t h e V o l t e r r a l i n e a r i n t e g r a l t r a n s f o r m Tw = /k(t,r)vh)dr,

t

0

where k (t,t) s a t i s f i e s t h e hypotheses of theor.2.111.

Prove,

by

computation, t h a t t h e s p e c t r a l r a d i u s of T is z e r o . 2) Let

& ( 5) for

3 (5 )

be a f u n c t i o n d e f i n e d i n

and summable i n

LO,+

00).

LO,

+w

) with values i n

Consider i n t h e s p a c e

3 ( t ) c E"(I, ,S), t h e i n t e g r a l e q u a t i o n

Eo(1,, S )

t

w(t)=

Prove t h a t i f (7.10)

1

I,

% ( t - z ) v ( t ) d +~ 9 ( t ) .

111 4 ( ( l ~ ~ ( 5 l l l d ~ ) ~

t h i s e q u a t i o n h a s one and o n l y one s o l u t i o n . The example g i v e n by Eq. (7.7) shows t h a t , i f c o n d i t i o n ( 7 . 1 0 ) i s

143 v i o l a t e d , e i t h e r e x i s t e n c e o r u n i q u e n e s s c o u l d f a i l t o be t r u e .

8. Summary on Sobolev s p a c e s . Let A {v$(x)

be a domain (open s e t ) of t h e c a r t e s i a n s p a c e be a s e q u e n c e of f u n c t i o n s be'longing t o

them w i t h a compact s u p p o r t c o n t a i n e d i n A s p a c e of

('2)

(

X?

Let

er(A,en)e a c h

e nis

of

the vector

n - v e c t o r s w i t h complex components).

The f o l l o w i n g theorem due t o H.Weyl is of f u n d a m e n t a l importance i n t h e t h e o r y of f u n c t i o n s of r e a l v a r i a b l e s .

8.1. Suppose t h a t f o r e v e r y &

7 0

a

and f o r e v e r y k

3 &

e x i s t s such t h a t f o r

s>+

i

! ~ ~ + ~ ( x 5) - r ( ~ ) <~ d& .x

Then a subsequence

IY, ( x )

of t h e g i v e n s e q u e n c e e x i s t s which

k

c o n v e r g e s a l m o s t everywhere i n A .

(I3)

The proof of t h i s theorem c a n be found i n any text-book t h e o r y of f u n c t i o n s of r e a l v a r i a b l e s .

on

the

(44

I f we s e t we have a f u n c t i o n d e f i n e d a l m o s t everywhere i n A

.

The v e c t o r

s p a c e of a l l t h e f u n c t i o n s w ( x ) d e f i n e d i n s u c h a way i s t h e Lebesgue s p a c e

L'

(A).

We assume, by d e f i n i t i o n ,

(") The s u p p o r t of a f u n c t i o n t h e s e t where I v ( x ) i 7 0 .

V(X)

def.ined i n A is t h e c l o s u r e of

Almost everywhere i n A means t h a t t h e s e q u e n c e j v , ( x ) ] c o n v e r h g.es i n A e x c e p t , e v e n t u a l l y , i n t h e p o i n t s of a s e t N having Lebesgue measure z e r o . The s e t N i s s a i d t o have Lebesgue measure z e r o i f i t c a n be c o v e r e d by a sequence of i n t e r v a l s ( i . e . r e c t a n g u l a r domains u p i t h s i d e s p a r a l l e l t o t h e c o o r d i n a t e a x e s ) s u c h t h a t thesum of t h e i r measures i s l e s s t h a n any a r b i t r a r i l y g i v e n 6-70. (43)

(j4)

S e e , f o r i n s t a n c e , 1191, p. 321,464,466; 191, p. 76-82.

I t is p o s s i b l e t o prove t h a t t h e i n t e g r a l d o e s n o t depend on the

a l l the

p a r t i c u l a r s e q u e n c e employed f o r d e f i n i n g v ( x ) and e n j o y s

p r o p e r t i e s of a n i n t e g r a l ( a d d i t i v i t y , l i n e a r i t y , modulus theorem, e t c . ) . The new d e f i n e d i n t e g r a l is t h e Lebesgue i n t e g r a l

of

the

f u n c t i o n w(x). The s p a c e

L ' ( A ) is a Banach s p a c e i f endowed w i t h t h e f o l l o w i n g

norm:

t

More i n g e n e r a l , g i v e n t h e r e a l p z i 6

L'(A) , we d e f i n e t h e s p a c e L

If

p

and

and u E

i s s u c h t h a t lw(xj

by assuming t h e norm

(A)

-i + -i = i

L~ ( A ) ,

V E

L '(A),

luvdx

I f we d e n o t e by

1w i t h

W(x)

q a r e such t h a t

1 W(*)

, if

values

1

P

5

t h e ~ c h w a r z - ~ S l d e ri n e q u a l i t y h o l d s :

(1

A /VIPdX)!

(/,laIqdx)'

e m ( A ) t h e v e c t o r s p a c e of a l l t h e f u n c t i o n s i n en ] which a r e of c l a s s e w i n X "and have

-, we

a bounded s u p p o r t c o n t a i n e d i n A

I n the case p =2

(P>*)

9

have f o r any p z i

L P ( ~ )= i " ( ~ ) . , L ' ( A ) is a H i l b e r t s p a c e

where t h e s c a l a r prod-

u c t is d e f i n e d a s f o l l o w s :

L e t u s d e n o t e by

o(

t h e multi-index

a n o r d e r e d s e t of t i n t e g e r s

d,

(or

,... , d ,

%

. As

-index) (d,,.-.. , d , ) , i . e . u s u a l , we s e t

T h i s n o t a t i o n , a l t h o u g h u n i v e r s a l l y a c c e p t e d , c o u l d be misleading s i n c e by t h e same symbol Iul we i n d i c a t e t h e norm of a v e c t o r

However from t h e c o n t e x t i t w i l l be c l e a r i f we a r e d e a l i n g e i t h e r with a multi-index o r with a vector. s e c t o r of

X'

assuming

*i = l

If

E

17, ,..., T c

is a

we s e t

if

Ti

.

= d ;= o

Moreover we s e t

C o n s i d e r t h e s p a c e of n - v e c t o r v a l u e d f u n c t i o n s w ( x ) b e l o n g i n g

em( 2 )

to

and w i t h a bounded s u p p o r t . W e d e n o t e t h i s s p a c e by ?T!i)

S e t f o r each function

er(A)

u(x) h

4

The Banach s p a c e o b t a i n e d by f u n c t i o n a l c o m p l e t i o n theorem) t h r o u g h t h e norm (8.1) w i l l be d e n o t e d by H Hmz,,(A) =

-, P

(A)

"n,P

(A)

, i.e.

e;(/i).

L e t { v , ( * ) ]be a s e q u e n c e of f u n c t i o n s of

H

(H. W e y l ' s

C r ( &)

converging

in

t o the function u ( x ) . Set

u d i x ) = -t;m

.Ddvtx,

5-(U

where t h e l i m i t must be u n d e r s t o o d i n t h e

L~

-norm.

We s e t

D ~ U : LCd; d U

is the generalized derivative

Dd

of

on t h e p a r t i c u l a r s e q u e n c e [v, ( x ) 1 . F o r any w c

Hence f o r 5-Y

em( A ) w e 0

have

m

/ a D*W d.

.

(-+)IL'

/u

U-doc

u

and d o e s n o t depend

and

u"

is t h e

D

d

- d e r x v a t i v e of u i n t h e s e n s e of d i s t r i b u t i o n s

o r t h e Da-weak d e r i v a t i v e of u

.

There is an a l t e r n a t i v e approach f o r i n t r o d u c i n g t h e s p a c e s

-. P

(A)

.

L e t u s c o n s i d e r t h e Banach s p a c e of

tions ( p > l

)

LP ( A ) n - v e c t o r

valued

func-

which p o s s e s s weak d e r i v a t i v e s up t o t h e o r d e r

e a c h of them b e l o n g i n g t o L ' c A )

.

m

The norm i n t h e s p a c e i s t h e

one g i v e n by (8.1). L e t t h i s s p a c e be d e n o t e d by W" ( A ) . P s p a c e , i n t r o d u c e d by S o b o l e v , i s s u c h t h a t

However, u n d e r r a t h e r g e n e r a l hypotheseson A

This

, we have

These h y p o t h e s e s a r e s a t i s f i e d i f we suppose t h a t a A = a A

and

aA of c l a s s em . These h y p o t h e s e s w i l l be assumed from now on. Hence w e have n o t t o d i s t i n g u i s h between H

( A ) and W % ( A ) and P P s p a c e s , a s i t is u s u z l nowadays, ml

we s h a l l d e n o t e t h e s e f u n c t i o n a s Sobolev s p a c e s .

('j)

("1

L e t u s s u p p o s e t h a t A is t h e c i r c u l a r r i n g of t h e x, I < x:+x:44

x,

plane

.

c u t a l o n g t h e segment - 2 *, * - r , X , . O I n t h i s c a s e H & ) i s strictl y c o n t a i n e d i n W F ( A ) (m > i ) . I n f a c t i t is e a s y t c ' b r o v e t h a t I= the function a , t x ) (-r 4 a r g i X ) b e l o n g s t o ?"(A) b u t d o e s n o t b e l o n g t o H _ , ~(A ) . (IL) These s p a c e s , a t l e a s t i n t h e p a r t i c u l a r c a s e p - 2 ,were known s i n c e t h e v e r y b e g i n n i n g of t h i s c e n t u r y , t o t h e I t a l i a n mathemat i c i a n s Beppo L e v i r201 and Guido . F u b i n i r217 who i n v e s t i g a t e d t h e D i r i c h l e t minimum p r i n c i p l e f o r e l l i p t i c e q u a t i o n s . L a t e r on many m a t h e m a t i c i a n s have used t h e s e s p a c e s i n t h e i r work. Some French m a t h e m a t i c i a n s , a t t h e begi-nning of t h e f i f t i e s , d e c i d e d t o i n v e n t a name f o r s u c h s p a c e s a s , v e r y o f t e n , F r e n c h m a t h e m a t i c i a n s l i k e t o do. They proposed t h e name Beppo L e v i s p a c e s . Although t h i s name is n o t v e r y e x c i t i n g i n t h e I t a l i a n language a c d i-?sc,xnds because of t h e name "Beppo",somewhat p e a s a n t , t h e outcome i n F r e n c h must be gorgeous s i n c e t h e s p e c i a l French p r o n u n c i a t i o n of t h e names makes i t t o sound v e r y i m p r e s s i v e . U n f o r t u n a t e l y t h i s c h ~ i c e

W e list h e r e some of t h e most i m p o r t a n t r e s u l t s

on

Sobolev

s p a c e s . We r e s t r i c t o u r s e l v e s t o t h e p a r t i c u l a r c a s e p = Z . t h i s c a s e we s h a l l u s e t h e s h o r t h a n d n o t a t i o n Hm,2

(A )

.

Suppose from now on t h a t

mapping which transform: restriction )

'tv

to

A

a function(")

3 A

H,

is bounded. W X )

In

i n s t e a d of L e t r be t h e

i n t o its t r a c e

( or

. The f o l l o w i n g i n e q u a l i t y h o l d s ( s e e

1173, p.353) :

and

IIU !Im

i s t h e norm i n

be e x t e n d e d t o t h e s p a c e of

DPi*

for

H, .

Through ( 8 . 2 ) t h e o p e r a t o r

t

can

and we c a n d e f i n e t h e b o u n d a r y v a l u e s

H,

D r 1 p 1 4.n1-1.

I t must be remarked t h a t ( 8 . 2 ) h o l d s u n d e r much more h y p o t h e s e s on

( s e e r223p.52-64)

QA

general

and t h a t , u n d e r t h e assumed

s m o o t h n e s s h y p o t h e s e s on 3 A , an e s t i m a t e s h a r p e r t h a n ( 8 . 2 ) c a n be p r o v e d . The f o l l o w i n g t h e o r e m s h o l d ( s e e [23]p. 1 1 1 , C171 p . 3 5 2 , 3 5 4 ) : 8.I.

I f we assume i n t h e r a n g e 7 ( H m ) of t h e o p e r a t o r r t h e norm

( 8 . 3 ) , t i s a compact o p e r a t o r , i . e . gent i n t h e space

H ,,

maps a s e q u e n c e weakly c o n v e r -

i n t o a sequence s t r o n g l y convergent i n t h e

s p a c e T (/-I,,).

was d e e p l y d i s l i k e d by Beppo Levi,who a t t h a t t i m e was still a l i v e , and as many e l d e r l y people w a s s t r o n g l y a g a i n s t t h e modern way of v i e w i n g m a t h e m a t i c s . I n a r e v i e w of a p a p e r of a n I t a l i a n mathematician,who,imitating t h e Frenchmen,had w r i t t e n s o m e t h i n g on "Beppo L e v i spaces',' he p r a c t i c a l l y s a i d t h a t h e d i d n o t want t o l e a v e h i s name mixed u p w i t h t h i s k i n d of t h i n g s . Thus t h e name had t o be changed. A good c h o i c e was t o name t h e s p a c e s a f t e r S. L - S o b o l e v . S o b o l e v d i d n o t o b j e c t and t h e name S o b o l e v s p a c e s is nowadays u n i v e r s a l l y a c c e p t e d . (47) The t e r m f u n c t i o n must b e u n d e r s t o o d a s f u n c t i o n w i t h v a l u e s in and b e l o n g i n g t o e-(A).

-

en

-

8.11. L e t u s c o n s i d e r t h e mapping which makes t o correspond t o t h e same f u n c t i o n viewed as a f u n c t i o n of

H,,,

U ( * ) E

(embedding of

into HI 1.

H,

T h i s mapping is compact

H

withbcm 1. (Rellich

s e l e c t i o n p r i n c i p l e 1. 8.111. I f

err,

-,t , u ( x ) e H , i m p l i e s lul L

-_ax A

C

u(x)eeO(A)

and

IIuII~

)

where c is a p o s i t i v e c o n s t a n t only depending on A

and on m

(Sobolev theorem).

9. E l l i p t i c and s t r o n g l y e l l i p t i c d i f f e r e n t i a l o p e r a t o r s .

=

If

a;;1 ( i = 4 ,..., rn

; j = 4 ,...,n

m a t r i x w i t h complex

is a m x n

)

e n t r i e s and u t {u,, ...,unfa

n - v e c t o r w i t h complex components, by au

w e denote t h e m - v e c t o r

la,, u; ,.-- .-,a m j u j

n

{ dj; 1 where

matrix

x In

d-. r

Eij

. We

1 % ,...., ~ ~ 1 , i . ue ~. = i . i ~ ~ %u~is. I af

- v e c t o r , we s e t a u w = W e have

L e t a,

auv (x)

1

(OU)V

=

- . .; ~ Q . -Vu &J

1 and

m-vector and v a m vau

L.

:t

r ( a u ) = q &..&. . Lj

1

.

eO"(Xe).Set

x ( i n t h e s e n s e of petrowski)

L

J

be a n x n (complex) m a t r i x which, f o r s i m p l i c i t y ,we

The m a t r i x d i f f e r e n t i a l o p e r a t o r

drt

u, - - . . -,U,

u z

uhv.

assume belonging t o

point

J

We denote by c? t h e

have a l r e a d y used t h e symbol

u w f o r i n d i c a t i n g t h e s c a l a r p r o d u c t of '4:

3'

a,

15I:V

(x)

'#

0

L ( x , D ) is

e l l i p t i c i n the

if

f o r every r e a l

i-vector J # o .

(48)

The o p e r a t o r

('8)

F o r a more g e n e r a l d e f i n i t i o n of e l l i p t i c i t y s e e E I g l y - l . B - S S , L ? ~ s l l p . 2 i o

i s c a l l e d t h e ( f o r m a l ) a d j o i n t of t h e o p e r a t o r L(x,D).This o p e r a t o r is (formally) s e l f - a d j o i n t i f

operator

L

(x,D) = a

c a n n o t be e l l i p t i c i f t h e a

1,

2.2

L (x,

D ) = L * ( ~ , ~ ) . Af i r s t o r d e r

scalar

a

L

(x)

'3Xb

are real or i f

(x)

.

r>2

I n t h e case

t h e most c e l e b r a t e d f i r s t o r d e r e l l i p t i c s c a l a r o p e r a t o r i s

t h e Cauchy-Riemann o p e r a t o r

a -

.

F o r t h e t h e o r y of

o

+b----

ax,

0 XI

-

e l a s t i c i t y a r e p a r t i c u l a r l y important

the

2nd o r d e r m a t r i x o p e r a t o r s which we w r i t e a s f o l l o w s : (9.2)

QeL.k (*)ujlrk + b .

L;u

'

1

L

(x) J

ujlk+ c . . ( x ) u '1

~

( i , j = f ,..., m

) .

T h i s m a t r i x o p e r a t o r is e l l i p t i c i f f o r any r e a l

7 ' q,,... ,Fa 3 #

6

S & ' F ~ #] O .

det { a i r i a ~ x )

F o r a n i s o t r o p i c homogeneous body we have t h e 2nd o r d e r m a t r i x o p e r a t o r of c l a s s i c a l e l a s t i c i t y (9.3)

L.u

-.

Ui/aL

(t = w

=

3)

,

+

where p and X a r e r e a l c o n s t a n t s (Lam6 c o n s t a n t s ) . I t i s e a s y t o s e e t h a t t h e o p e r a t o r is e l l i p t i c i f b o t h t h e c o n d i t i o n s

p # O

,

1+2p # o

are satisfied. The o p e r a t o r (9.1) i s s t r o n g l y e l l i p t i c ( i n t h e p o i n t x ) i f

f o r any r e a l

t-vector J # o

and any r e a l n - v e c t o r

2

#

0

.

I t is

e v i d e n t t h a t s t r o n g e l l i p t i c i t y i m p l i e s e l l i p t i c i t y . The c o n v e r s e however, is f a l s e a s t h e example of t h e Cauchy-Riemann proves. I n f a c t

operator

vanishes f o r

7,

=

0

,

y1

7

and

arbitrary.

In t h e c a s e of t h e o p e r a t o r (9.2) t h e s t r o n g e l l i p t i c i t y condit i o n is

Qe

L'x))~

TL T ;7j + 0

f o r e v e r y ~ ~ { ~ , , . - . , ~ ~ e~v + e r yo ra ena ld Suppose

7 : { ?,,-...,r n j

and t h a t - t h e c o e f f i c i e n t s a i L j

.n=z

(x)

+O.

s a t i s f y condi-

t i o n s (1.1). Moreover suppose t h a t t h e q u a d r a t i c form W ( x , E ) = ( EiL = EL; ) i n t i L€ j ktive definite We have

a iLj

(x)

ir ( c + * )

r e a l v a r i a b l e s be p o s i -

.

,

x

i

t ( ~ ~ ? ~ + ) ;(jk.rj ~ ~ 'Ti) l [rk)l ! 'O .

7;

~;t.~tt(~'[

Hence p o s i t i v e n e s s of W ( r , E ) i m p l i e s s t r o n g e l l i p t i c i t y .

In the

c a s e of the o p e r a t o r (9.3) w e have P o s i t i v e n e s s of W : f o r Strong E l l i p t i c i t y : f o r Ellipticity

: for

p 7 O

31 t2p > 0 ;

t~ ( A + 2r+) 7 0 ; $ 0 ,X t l p $ : O .

~

10. Boundary v a l u e problems f o r s t r o n g l y e l l i p t i c o p e r a t o r s and

r e l a t e d integro-dif f e r e n t i a l equations. L e t a,,?(*) be a n x n m a t r i x belonging t o 0 5

Iq 1

m )

.

em(X')( 0 s Ipl 5 m ,

Consider t h e matrix d i f f e r e n t i a l o p e r a t o r of o r d e r

Rnt :

(10.1)

L(X.D)U

~

~

o

~D ~~ .(U x

)

I f A is connected, t h e s t r o n g e l l i p t i c i t y h y p o t h e s i s i n A can be

written

We a s s o c i a t e t o t h e o p e r a t o r (10.1) t h e b i l i n e a r form

I t i s o b v i o u s t h e i m p o r t a n c e of t h e q u a d r a t i c form Q ( u , u ) i n p h y s i c a l problems, s i n c e t h i s q u a d r a t i c form is a n e n e r g y integral.

6 ( u , v ) i s d e f i n e d and c o n t i n u o u s i n H, x H, . L e t u s d e n o t e by H , t h e s u b s p a c e of H, formed by a l l t h e f u n c t i o n s u of H , v a n i s h i n g on Q A w i t h t h e i r d e r i v a t i v e s up 0

t o the order

m-4

,

i.e. such t h a t

?U 5 0 .

I n t h i s s e c t i o n and i n t h e n e x t we s h a l l s u p p o s e t h a t A bounded, c o n n e c t e d and s a t i s f i e s t h e smoothness h y p o t h e s e s

. However,

sumed i n Sect8

is

as-

i t must be remarked t h a t some of t h e

r e s u l t s , we a r e g o i n g t o c o n s i d e r , h o l d u n d e r more g e n e r a l

hy-

.

p o t h e s e s on A

The f o l l o w i n g theorem due t o ~ & r d i [251 n ~ r e l a t e s the algekaic c o n d i t i o n (10.2) t o a n "a p r i o r i " e s t i m a t e s and h a s been

the

s t a r t i n g p o i n t of t h e modern approach t o boundary v a l u e problems f o r e l l i p t i c operators. 10. I . The o p e r a t o r (10.1) i s s t r o n g l y e l l i p t i c f o r e v e r y x c

and lo *, o

i f and o n l y i f (-4 )m

f o r every u r

@c

5

e x i s t such t h a t

B ( k , ~ )2 yo llv 11' m - 3ollvllG2

H,,, .

('0)

L e t u s now i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s : i)

V i s a l i n e a r s u b s p a c e of i,cVc

H ,

such t h a t

H,;

(49) A c t u a l l y B ( u , v ) i s l i n e a r w i t h r e s p e c t t o u and p s e u d o - l i n e a r r e s p e c t t o w , i . e . 0 ( L L , Q V +b w ) = E B(u,.v) + I 3 (u,Iu). Some a u t h o r s , i n s t e a d of b i l i n e a r , s a y s e s q u i l i n e a r . ('0)

F o r t h e proof s e e 117Jp.366-367.

is a b i l i n e a r f u n c t i o n a l d e f i n e d i n

ii) D ( w , z )

which is c o n t i n u o u s w i t h r e s p e c t t o t h e norm i n

T(H,)xC

~ ( H r n ) ,

(H,)

i.e.

the

norm (8.3) ; iii) f

is a l i n e a r bounded o p e r a t o r which maps into

H, i v ) w e set f o r

PROBLEM

)

;

H,

u,v a

. Given

z(H,

f c Ha ( s L'(A) ) , find u c V

(u,v) = ( f , w ) ,

(10.3)

such t h a t

f o r every

VEV.

We s h a l l make t h e f o l l o w i n g : MAIN ASSUMPTION

.

For every v c V

5

e x i s t s such t h a t

&>O

(ye7 0 ) . Znder this h y p o t h e s i s t h e f o l l o w i n g e x i s t e n c e and u n i q u e n e s s theorem h o l d s f o r problem (10.3) : 1 0 . 1 . -One and o n l y one s o l u t i o n a e x i s t s of problem ( 1 0 . 3 I . l f we d e n o t e by G t h e t r a n s f o r m a t i o n which maps f~ H a solution u e

1) G 2)

i n t o the

V of t h e problem (10.3) we have

is a bounded l i n e a r t r a n s f o r m a t i o n of

G viewed a s a mapping from

H~

into

V ;

W, i s compact.

F o r t h e proof of t h i s theorem we r e f e r t o C171p.368-371.

We only

o b s e r v e t h a t t h e p r o p e r t y 2) of G is a n o b v i o u s consequence of 1 ) and of t h e o r . 8. I I . I n a d d i t i o n t o e x i s t e n c e and u n i q u e n e s s w e have t h e f o l l o w i n g regularization results. 10.11. If

such t h g by V (10.4)

f c H, c A.

then

b e l o n g s t o HVt2,(4)

f o r any 8

Moreover, i f s u i t a b l e h y p o t h e s e s a r e s a t i s f i e d

D , u c HgtZ, 11

u:Gf

11

r+zm

(A)

and

6 C

\ I $11.

where c o n l y depends on v

,L ,A ,D ,& $ ' V.

F o r t h e proof of t h i s theorem and f o r t h e h y p o t h e s e s t o be s a t i s f i e d by V and p.355-365

D

i n o r d e r ( 1 0 . 4 ) t o h o l d , we r e f e r t o El71 ( s e e

and p . 3 7 1 ) . We o n l y mention t h a t (10.4) h o l d s i f we a s -

sume e i t h e r V r

; ,

o r VI H,,,

, DE0

,DPO.

From t h e o r . 8 . I11 we deduce t h a t , i f ( 1 0 . 4 ) h o l d s , t h e n $ E implies u E

eaD(A)

1.

Assume t h a t h y p o t h e s e s f o r r e g u l a r i z a t i o n h o l d , t h e n integration by p a r t s is f e a s i b l e and we have

B(U.Y) =

I

( L L I ) V ~ +X

1

M(u,v)~~,

A 9A where M ( u , v ) i s a b i l i n e a r d i f f e r e n t i a l o p e r a t o r of o r d e r 2m-1 i n u and of o r d e r m-1 Hence (10.3)

i n U.

can be writtten

+ If

D(h+u,~~)=

0

C"(A) we have

which i m p l i e s

L u Let WE

UV

=#'.

be t h e v e c t o r s p a c e of a l l t h e u E V

V: M ( u , ~dcr )

s u c h t h a t f o r any

D (pu,'iu)= 0 .

t

The problem ( 1 0 . 3 ) i s e q u i v a l e n t t o t h e f o l l o w i n g o n e : Lurf

,

ueUv

I n s e v e r a l p a r t i c u l a r cases the condition ucuV is expressed by a s e t of s t a n d a r d boundary c o n d i t i o n s on > A . L e t u s now c o n s i d e r t h e i n t e g r o - d i f f e r e n t i a l problem (10.5)

L U ( X , ~ )J=( x , + ) +

(10.6)

u

(x,

t

e u,

We assume t h a t

f,

and, f o r f i x e d x € A

t d ( x , t , r ) $ u ( x , T ) ~ T in A x [ t o , + - ) ( o r 1oc1 L 2 m ) for

(x,t,T )

to& t L+OO.

i s d e f i n e d i n A x [to,+-)x [to ,+

)

, i s a c o n t i n u o u s f u n c t i o n of ( t , ' t ) a n d , f o r

any f i x e d ( t , z ) , b e l o n g s t o H , ( A )

(9L 0 ) .

f u n c t i o n of t is continuous and, f o r

The f u n c t i o n cP(x,t)as a

fixed

t

,

belongs t o H,(A).

We have from (10.4) 5 C

Ilp li* .

Hence t h e o p e r a t o r

: D

fd(x.t.r)

f o r fixed

H,

.

t and

7

G w,

, is a l i n e a r bounded o p e r a t o r from

If we s e t La(*& ) = ~ ( x , t

i.e. u(x,t) =

G

H,

into

v(x,t),problem ( 1 0 . 5 ) ,

(10.6) is e q u i v a l e n t t o t h e V o l t e r r a e q u a t i o n i n t h e space

and t h e theory of S e c t . 2

-

i n p a r t i c u l a r theor.2.N

-

a p p l i e s . Thus

we have e x i s t e n c e and uniqueness f o r problem (10.51, ( 1 0 . 6 ) .

The

r e s u l t is a l s o a consequence of t h e o r . 6 . 1 . Before ending t h i s s e c t i o n we would l i k e t o propose t o t h e reade r , a s a n e x e r c i s e , t o apply t h e t h e o r y developed i n t h i s section

t o t h e p a r t i c u l a r c a s e of t h e c l a s s i c a l s c a l a r Laplace o p e r a t o r .

assuming

1)

Dso

( a,A

,

0

V z H , ;

s u b s e t of 9 A

;

I n each c a s e t h e main h y p o t h e s i s is a consequence of t h e known P o i n c a r e i n e q u a l i t i e s ( s e e t17J p.350-351,p.379).

well

11. G e n e r a l e l l i p t i c boundary v a l u e problems and r e l a t e d

integro-differential equations. The r e s u l t s on t h e i n t e g r o - d i f f e r e n t i a l e q u a t i o n s c o n s i d e r e d i n S e c t . 1 0 , a l t h o u g h c o n c e r n i n g a l a r g e c l a s s of e l l i p t i c o p e r a t o r s , a r e of r e s t r i c t e d i n t e r e s t f o r h e r e d i t a r y problems, s i n c e i n these problems t h e i n t e g r o - d i f f e r e n t i a l i n t h e domain A

,

operators a r e only considered

b u t n o t on t h e boundary. On t h e o t h e r hand, i n

a p p l i c a t i o n s we have problems where t h e i n t e g r o - d i f f e r e n t i a l

.

e r a t o r s must be c o n s i d e r e d b o t h i n A and i n Q A

op-

See, f o r instance,

t h e problem c o n s i d e r e d i n S e c t . 5. L e t u s expound h e r e a more g e n e r a l approach t o e l l i p t i c boundary v a l u e problems which p e r m i t s t o h a n d l e a b r o a d e r c l a s s of integro- d i f f e r e n t i a l problems of h e r e d i t a r y phenomena. We suppose t h a t t h e bounded domain A s a t i s f i e s t h e smoothness h y p o t h e s e s assumed i n t h e l a s t s e c t i o n s . C o n s i d e r i n A t h e l i n e a r d i f f e r e n t i a l system

where

L (x,D) is

a

nw n

matrix d i f f e r e n t i a l operator, e l l i p t i c i n

X ' i n t h e s e n s e of P e t r o w s k i ( s e e S e c t . 9) . $ is a g i v e n n-vector valued function.

Jfe a s s o c i a t e t o (11.1) t h e boundary c o n d i t i o n

where

B(x,D)

Bij (x,b)

(

is a

m x n m a t r i x d i f f e r e n t i a l o p e r a t o r whose e n t r i e s

i =4,---,m ; j = 4,--,n

)

are linear differential operators d

9;.(~,9)= ~ . . ( x ) D ~ d

whose c o e f f i c i e n t s b.. ( x ) Y

a r e d e f i n e d on Q A

p e r m i t t e d t o be of o r d e r p i suppose t h a t

4(x)

'1

J

.

(06ldll

pi)

These o p e r a t o r s am

with

pi a n a r b i t r a r y i n t e g e r . We c e*(xk) and t h a t b; ( x ) 6 e m ( 3 A ) ~ . ~~j,,..,j,,,]

i s a g i v e n m - v e c t o r v a l u e d f u n c t i o n d e f i n e d on Q A

. We

shall

d e n o t e by B; ( * . D ) ( i

=

)

4

the

--vector

,al,(,,~)].

operator{~~,(x,i)),..

I n c o n n e c t i o n w i t h t h e boundary v a l u e problem (11. I ) , ( 1 1 . 2 ) w e s h a l l consider the i n t e g r o - d i f f e r e n t i a l system

L(x,3)uscp(x,t)+

(11.3)

(~_LI~ILP,,,)

m a t r i c e s and

where f (x,t,?) a r e smooth n *lt A

- v e c t o r s , c p ( x , t ) and 9( x . + 1: ?I

[~,(v,t), ...,%

Pi

P

(x,t,=)

smooth n-

b,t){given f u n c t i o n s , r e s p e c t i v e l y

m

- v e c t o r v a l u e d and m - v e c t o r v a l u e d , and e i t h e r

o r , even,

= too r

f ~-ao :

-

= t A (where 4 is "the l e n g t h of t h e memory").

The a p ~ r o a c his t h e s t a n d a r d one w e have f o l l o w e d t h r o u g h o u t t h e s e l e c t u r e s which c a n be o u t l i n e d as f o l l o w s : 1 ) To assume t h a t t h e r e e x i s t s one and o n l y one s o l u t i o n .u problem ( 1 1 . 1 ) , ( 1 1 . 2 ) when

S and

q ~ {,,..., t gm]

2 ) Represented u

t o prove t h a t

f

is g i v e n i n a s u i t a b l e Banach s p a c e

i n a s u i t a b l e Banach s p a c e z : Z , x - - - x Z m . a s follows

d

(x,t,9)

o p e r a t o r from S x Z

:D ( G f'

In

C

t

tg; > (05

~ d j ~ _ z m )ai sbounded m

i . 4

into

5

and i i p ( r , t . = ) ! D

is a bounded o p e r a t o r from S x z t i n u o u s l y on

of

i n t o Zi

,

(G f + Z i:r

(Oi / : + P S I

b o t h d e p e n d i n g con-

(t,r).

When 1 ) and 2) have been a c h i e v e d , w e a r e i n p o s i t i o n t o r e d u c e

-

t h e problem ( 1 1 . 3 ) , ( 1 1 . 4 ) t o a n e q u i v a l e n t V o l t e r r a i n t e g r a l equat i o n , e i t h e r i n the space

$x

2 1. T h i s

(?'[I,,

Sr 2 ]

o r i n the space

e0 [: I,

,

i n t e g r a l e q u a t i o n i s one o f t h e types which have been

c o n s i d e r e d i n S e c t i o n s 2 and 7. The c o n d i t i o n s t o be s a t i s f i e d by t h e o p e r a t o r s L ( w , B ) a n d B ( x , D j f o r g e t t i n g 2 ) a r e w e l l known and c o n s i s t i n some a l g e b r a i c hypo t h e s e s t o be assumed on t h e s e o p e r a t o r s ( s e e ( 1 8 1 S e c t i o n s l,2). F o r c o n v e n i e n c e of t h e r e a d e r w e list h e r e t h e s e h y p o t h e s e s

a r e c a l l e d Sup-

which, a c c o r d i n g t o Agmon, D o u g l i s and N i r e n b e r g , p l e m e n t a r y c o n d i t i o n and Complementing c o n d i t i o n . Suwwlementarv c o n d i t i o n .

3

Set

, of e v e n d e g r e e Zm . F o r e v e r y p a i r of l i n e a r l y i n d e p e n d e n t r e a l v e c t o r s 7 ' and 7 , t h e polynomial L ( x ,'f + r 5 ' ) i n t h e complex v a r i a b l e r h a s e x a c t l y n x T h i s polynomial i n

i s , f o r every x

r o o t s with p o s i t i v e imaginary p a r t . T h i s c o n d i t i o n is s a t i s f i e d by any e l l i p t i c m a t r i x o p e r a t o r f o r 2

_*

3

,

b u t i t n e e d s t o be p o s t u l a t e d f o r 2 = 2

.

Complementing c o n d i t i o n . F o r any X E ~ and A any r e a l vector

f

tangent t o 9 A a t x

L,(x,'f where

+TV) =

set

Lt(x,t,~)L,(x,t,r),

is t h e e x t e r i o r u n i t normal t o 'JA

p o l y n o m i a l of d e g r e e

m

, non-zero

at

having a s z e r o e s t h e

x

m

and

L'

(w,),z)

a

z e r o e s of L o ( x , y + t * )

w i t h p o s i t i v e imaginary p a r t . S e t

a;. and d e n o t e by

LoL lt ( x,p

=

(x.5 J

z I ~ ! Z

p;

b:j

(.)

J~

1 t h e c o - f a c t o r of t h e e n t r y of i n d i c e s h , b

i n the matrix

L a,(x)'$. I5i=V

4

I t is assumed (complementing c o n d i t i o n ) t h a t t h e rows of t h e matrix

jk ((~'.(x.ftT~)L~(K,~ttr.))) &J

c o n s i d e r e d as p o l y n o m i a l s i n

~ f ( ,* , y , t ) , with c,

't

,

(L:,,--.,m

;

k:, ,..., +I)

a r e l i n e a r l y i n d e p e n d e n t modulo

i.e. that

,- -- ,c,,,

i m p l i e s c, : ...

complex c o n s t a n t s

.

c,,,,

c

and

Rk

polynomialsin

r,

0.

The c h o i c e of t h e s p a c e s

S

ways.We c o u l d assume as S and

and 2

C

c a n be done i n s e v e r a l

some Schauder s p a c e s a s wedidin

S e c t . 5 . However w e p r e f e r t o u s e now d i f f e r e n t s p a c e s i n o r d e r t o show t h e r e a d e r s e v e r a l ways of employing t h e t h e o r y of e l l i p t i c systems i n i n v e s t i g a t i n g t h e r e l e v a n t i n t e g r o - d i f f e r e n t i a l

equa-

tions. I n t h e spacer(ki,,,)which

we i n t r o d u c e d i n S e c t . 8 we now d e f i n e

a new norm d i f f e r e n t from (8.3). W e d e n o t e t h i s norm by II

Ilm-F L

and d e f i n e i t a s

I\

,=

Ii

m-Z where

is a f u n c t i o n of

u

by H_-

t h e s p a c e Z (H,.

in9 1 I u\Im

,

UEH,,,

such t h a t

H,

) w i t h t h i s new norm.

l e t u s now p r o v e

~ c c o r d i nt o ~ t h e theory developed i n Sect.6, t h a t , when t h e problem ( 1 1 . 1 )

, (11.2)

c a n be s o l v e d , a l s o t h e prob-

lem ( 1 1 . 3 ) , ( 1 1 . 4 ) c a n be s o l v e d ( a t l e a s t f o r Hence w e assume t h a t , s e t t i n g p = max given

, ji

$'r H,-t,

on ? A . We d e n o t e

ru=W

c 14

-r;-

(2nb

r = t , >1.- ~

, p,+i , ... ,vlllti), we have:

.

, t h e problem (11.1) (11.2) has

+

one and o n l y one s o l u t i o n b e l o n g i n g t o H p . The f o l l o w i n g t h e o r e m h o l d s ( s e e C171 ,p.78) :

11. I . Under t h e assumed h y p o t h e s e s , i f f E H+-2m j r ;r ) , t h e

s o l u t i o n of

a c (11

1I

(11.5)

( 1 1 . 1 ) . (11.2) belongs t o

where c d e p e n d s o n l y on If and

H ,-2,

3; c

9

,L ,

A

and

, t h e above i n t r o d u c e d o p e r a t o r s 2,

.

H,

Z

L u ( x , +=) v ( r , + )

a n d w e have

4.

4 f +; :L r.gi b e l o n g s t o 4

H,-:.

9; c HJ - t t ; -

F 11 ,-,,

H,-pi-r

a r e such t h a t

I ~ e l o n g st o Set

,

H,

and

0.a(x,t)

= w. (s,t ) .

Let us c o n s i d e r t h e V o l t e r r a i n t e g r a l system

and T ( G $ + ! , T ~ ; )

Fixed v z that

p , assume 5 = H ,-,, , Z ;

:

H9-p.-

where i t i s e v i d e n t

+

?

t-i+-3mmust be u n d e r s t o o d l i k e a s p a c e of

- ;l i k e

f u n c t i o n s and e a c h H ,-p,

n-vector

valued

a s p a c e of s c a l a r (complex)

val-

ued f u n c t i o n s . We s u p p o s e t h a t { q ( r , t ) ,y,( x , t ) , ... ,?v,(x,t )]can be viewed a s a t i o n $ ( t ) of CG[ I t , S x Z J i f

5

x Z ] i n other cases.

function V ( t ) r

y = t d - c - o r a s a f u n c t i o n of

where

L

jt,t)

2'1

I,,

The same must be assumed on t h e "unknown" w,(x,+)J.

(~(x,t),w,(x,t),.--,

Suppose f o r s i m p l i c i t y t h a t f d ( x , t , t ) , and m o r e o v e r , i f

func-

y=-rn ,

that for

0 5

e.

(~,t,t t

Ipl

5

e- ( x

%,

R

R )

p,

is t h e f u n c t i o n c o n s i d e r e d i n c o n d i t i o n 1) of Sect.7.

Set

\

K

t

mtr z

-.. K

(t,~)

mt4

,

( t ~ I)

mi4

W e can w r i t e t h e system (11.6) a s a s i n g l e V o l t e r r a equation r

*

which is e q u i v a l e n t t o t h e problem ( 1 1 . 3 ) If

5x2

y = t, > - -

3.

integral

, (11.4)

.

, ( 1 1 . 8 ) h a s one and o n l y one s o l u t i o n i n e'{[tb,+-j,

Hence ( f 1 . 3 ) , ( 1 1 . 4 ) h a s one and o n l y one s o l u t i o n U ( t ) s

u ( r , t ) b e l o n g i n g t o @ " { [ t, +o DO), W

,

1 . We

do n o t need t o s a y

more

on t h i s c a s e . Much more i n t e r e s t i n g i s t h e c a s e when i n the integro-

- d i f f e r e n t i a l s y s t e m we assume e i t h e r

y-- o r -00

F

t -L

.

(24)

L e t u s c o n s i d e r t h e l a t t e r c a s e , s i n c e t h e arguments developed when

y-= t - R

c a n be e a s i l y a d a p t e d t o t h e c a s e when

o t h e r hand w e b e l i e v e t h a t t h e c a s e y = t - k

g = --.On

the

i s t h e one t h a t , i n a

l i n e a r theory, has a r e a l physical significance. W e w r i t e 6q.s

(11.3), (11.4) i n t r o d u c i n g a complex p a r a m e t e r 1

The e q u i v a l e n t e q u a t i o n (11.8) is w r i t t e n t (11.9) V(t) = ( t ) + ~(t,-t)V(r)dr.

4

t -f,

L e t u s assume, i n s t e a d of (11.7), t h a t f o r any 0 i Iri L p

where

x

, t , r and f o r

L i s a p o s i t i v e c o n s t a n t . A f t e r somewhat t e d i o u s b u t com-

1:ietely e l e m e n t a r y c o m p u t a t i o n s , by a p p l y i n g t h e p r o c e d u r e

of

theorem 7 . 1 1 , w e s e e t h a t Eq. (11.9) h a s one and o n l y one s o l u t i o n if

(11.11) where N

13.1

L

(LNLC)-'

is a n u m e r i c a l c o n s t a n t , e x p l i c i t l y computable, and L

and c t h e c o n s t a n t s c o n s i d e r e d i n t h e estimates (11.10), (11.5).

I n o t h e r words t h e s p e c t r u m of (11.9) h a s n o p o i n t s i n t h e d i s c d e f i n e d by (11.11). I n p a r t i c u l a r f o r X = i we see t h a t

we

have

[^I) Leitman and F i s h e r i n t h e i r r e c e n t Monograph l26J w r i t e "E t h e v i s c o e l a s t i c i t y problem ( i .e . l i n e a r i n t e g r o - d i f f e r e n t i a l problem) i s n o t of i n i t i a l p a s t - h i s t o r y t y p e ( b r i e f l y i f y + t , ) t h e problem of e x i s t e n c e i s somewhat d i f f e r e n t " ( s e e p . 6 4 ) . And t h e y v e r y w i s e l y add " W e w i l l n o t p u r s u e t h i s problem f u r t h e r s i n c e cozside r a b l e t e c h n i c a l d e t a i l must be p r o v i d e d .

e x a c t l y one s o l u t i o n of t h e h e r e d i t a r y problem i f t h e bound f o r t h e " h e r e d i t y c o e f f i c i e n t s " and t h e l e n g t h

are s u c h t h a t

L

k of t h e memory

i

(11.12)

RLL

,c3

I n t h e e l a s t i c i t y problems t h e r i g h t hand s i d e of (11.12) must be u n d e r s t o o d a s a c o n s t a n t depending on t h e e l a s t i c n a t u r e

and

on t h e geometry of t h e body u n d e r c o n s i d e r a t i o n . I f (11.12)

is v i o l a t e d ,

c o u l d b e l o n g t o t h e s p e c t r u m of

I =l

( 1 1 . 9 ) . What is t h e p h y s i c a l s i g n i f i c a n c e of t h i s o c c u r r e n c e ?

1 2 . Z e g r o - d i f f e r e n t i a l problems i n domains w i t h s i n g u l a r boundaries. I n the preceeding S e c t i o n s , i n d e a l i n g with integro-dif ferential problems, we have supposed t h a t t h e r e l e v a n t domain ary

QA

A

h a s a bound-

which i s smooth. Although some of o u r r e s u l t s h o l d u n d e r

l e s s r e s t r i c t i v e h y p o t h e s e s on sumed by u s ( 9 A

€

QA

w i t h r e s p e c t t o t h e ones

as-

C w ) , t h e main r e s u l t s of t h e t h e o r y of e l l i p -

t i c o p e r a t o r s which we have u s e d , f a i l t o be t r u e i f 2 A

contains

s i n g u l a r i t i e s l i k e edges, v e r t i c e s , etc. I n f a c t i n t h e s e c a s e s t h e b a s i c i n e q u a l i t i e s (10.4)

, (11.5) a r e , i n g e n e r a l , f a l s e .

R e s e a r c h work i n t h i s a r e a of t h e t h e o r y of e l l i p t i c e q u a t i o n s is nowadays i n p r o g r e s s . However t h e r e s u l t s a v a i l a b l e a t p r e s e n t a r e n o t a b l e t o p r o v i d e a l l t h e t o o l s needed f o r e x t e n d i n g

the

t h e o r y of i n t e g r o - d i f f e r e n t i a l p r o b l e m s , which we have expounded i n t h e s e l e c t u r e s , t o a domain w i t h a s i n g u l a r boundary. The o n l y e q u a t i o n which h a s been i n v e s t i g a t e d t o a s a t i s f a c t o r y extent

is t h e c l a s s i c a l . V o l t e r r a

(1.11)

w i t h t h e boundary c o n d i t i o n

been c a r r i e d o u t a method

which

i n a paper c a n be

,

by

i n t e g r o - d i f f e r e n t i a l equation 1.12).

L.De V i t o

very l i k e l y

,

This analysis

has

1271 o f 1 9 6 1 , by

extended

t o more g e n e r a l

(22)

situations. The r e s u l t of De V i t o is founded on an a b s t r a c t e x i s t e n c e p r i n c i p l e due t o t h e w r i t e r (:')

This p r i n c i p l e has shown t o be a v e r y

u s e f u l t o o l i n many a p p l i c a t i o n s and must

be c o n s i d e r e d

like a

s y n t h e s i s and a g e n e r a l i z a t i o n of a l l t h e e x i s t e n c e methods founded on Banach s p a c e s , which s t a r t i n g from Caccioppoli C311 have been used i n t h e l i n e a r t h e o r y of 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 . L e t V be a complex v e c t o r s p a c e and M; (i=r,z)a l i n e a r mapping of V i n t o t h e complex Banach s p a c e 9; . L e t 0; d u a l of

B;

.

Denote by

<, >

be t h e topological

t h e d u a l i t y between a Banach

space

and i t s t o p o l o g i c a l dual. L e t cp be a given co-vector of B:. us consider f o r -every v r V

Let

t h e system of e q u a t i o n s

where " t h e unknown" is t h e co-vector

I\y

c

8;.

Denote by V, t h e k e r n e l of M, i .e.

V2 E {w, , V , L V , M , ? = O ] . L e t Q be t h e Banach f a c t o r - s p a c e

-

Q=

6 4

M,(V,)

and denote by M, t h e l i n e a r mapping which maps t h e v e c t o r v e v i n t o the equivalence class[M,v] t h e norm 11 Cw3jl g i v e n by

Q

. We

recall that

t h e subspace of

v.

of

Q

is

.

in? i w t M,v2Il ?a€

Yo be

Q

of an element (equivalence c l a s s ) [ w ]

Iliwlllg = Let

of t h e s p a c e

9,

9: d e f i n e d by t h e c o n d i t i o n

(29 The p a p e r 1271 by De V i t o was c r i t i c i z e d by E.Magenes (Mathe m a t i c a l Reviews,vo1.25,1963,p.452-453).Futility of Magenes sriti c i s m has been shown by De V i t o i n h i s paper C281. (23)

See [29] p. 174-178 and [30] p.11-16.

and d e n o t e by 3 t h e f a c t o r s p a c e

-

0

The above mentioned e x i s t e n c e p r i n c i p l e i s e x p r e s s e d by

the

f o l l o w i n g theorem: 1 3 . 1 . N e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of a .U s o l u t i o n of

( 1 3 . 1 ) f o r any a r b i t r a r i l y g i v e n rp

= .o

( 4 v,

v,

E

B:

such t h a t

)

is t h a t a p o s i t i v e constant c e x i s t s such t h a t

If y -

is any s o l u t i o n of ( 1 3 . 1 ) ,

t h e most g e n e r a l s o l u t i o n

g i v e n by Y + y , where y., i s an a r b i t r a r y c o - v e c t o r

of

Y . 0

f o l l o w i n g " d u a l " i n e q u a l i t y of

is The -

(13.2) holds:

We r e f e r f o r t h e p r o o f t o 1301 ( p . 1 1 - 1 6 ) . The theorem 1 3 . 1 c a n be u s e d i n two d i f f e r e n t ways: a ) i f

(13.2)

i s known, one c a n deduce a n e x i s t e n c e t h e o r e m and t h e d u a l i n e q u a l i t y ( 1 3 . 3 ) ; b) i f t h e e x i s t e n c e of a s o l u t i o n of

, (13.3) .

known, one c a n deduce t h e i n e q u a l i t i e s ( 1 3 . 2 ) L e t u s c o n s i d e r a n example. L e t

(x)

( 1 3 . 1 ) is

be t h e " e l a s t i c i t i e s "

c o n s i d e r e d i n S e c t . 1 and s a t i s f y i n g a l l t h e h y p o t h e s e s s t a t e d i n Sect.5.

L i k e i n S e c t . 1 we set

& ;p, ( x * ) f o r any

ur

coo(fi),where A

4

;( v ; ,+~y,; )

i s t h e bounded domain c o n s i d e r e d i n S e c t .

5. S e t

= a t L j k ( x )r . ( v ) . Jk

L e t ?,A

be a s u b s e t of 'SA ( e v e n t u a l l y c o i n c i d i n g w i t h ? A ) c o n

s i s t i n g of smooth s u r f a c e s . S e t Q,A

V r j v ; YcC-(A), Denote by Jf

= 2A

- 9, A

and

v( = o ] . 3P,

t h e H i l b e r t s p a c e o b t a i n e d by c o m p l e t i o n from V

through t h e s c a l a r product

-

((UP)) = G L k ( u ) Cik ( v )d x

If

.

is t h e o p e r a t o r which maps v s Ca(A)into i t s b o u n d a r y

-c,

v a l u e s on a Z A , we assume l i k e spaceB,: ~ ' ( ~ ) x ~ ~ ( ? ) a lnidk e mapping

-

I x t., j . We assume 8, ;X and l i k e M, t h e embedding o f V i n t o 3 . Hence we. g e t a s p a r t i c u l a r c a s e of ( 1 3 . 1 ) t h e f o l l o w i n g problem: g i v e n g € ' L ( A , 9 6 L 2( I ; ) Z A ) , t h e mapping M , z

M,

find

IA

c

3

s u c h t h a t f o r any v c V

w h e r e t h e meaning t o b e g i v e n t o G i k ( u )

r e s u l t s from t h e functicn-

a 1 completion procedure. A c t u a l l y o u r e x i s t e n c e problem h a s t h e p h y s i c a l meaning of t h e one c o n n e c t e d w i t h t h e d e t e r m i n a t i o n of stress d i s t r i b u t i o n i n t h e e q u i l i b r i u m c o n f i g u r a t i o n o f a n e l a s t i c body A s u h !er:ted t o g i v e n s u r f a c e f o r c e s on 2 , A on A

.

fixed

a l o n g a,A,

and t o g i v e n body f o r c e s

I t is o b v i o u s t h e u n i q u e n e s s o f t h e s o l u t i o n . S i n c e

Et1, E L h

Qiejk'

i s p o s i t i v e d e f i n i t e , t h e e x i s t e n c e is e q u i v a l e n t t o t h e

f o l l o w i n g i n e q u a l i t y ( s e e t h e o r . 13. I )

which is a c o n s e q u e n c e of t h e c e l e b r a t e d 2nd Korn i n e q u a l i t y ( [ I 7 1 p. 381-385)

.

The " d u a l " i n e q u a l i t y of ( 1 3 . 5 ) i s

which is o f paramount p h y s i c a l i n t e r e s t s i n c e i t p e r m i t s t o g i v e an e s t i m a t e t o t h e "energy i n t e g r a l " corresponding t o t h e e q u i l i b r i u m c o n f i g u r a t i o n i n terms o f t h e g i v e n f o r c e s a c t i n g on thebody. L e t u s now r e t u r n t o t h e c l a s s i c a l V o l t e r r a i n t e g r o - d i f f e r e n t i a l e q u a t i o n ( 1 . 1 1 ) a n d assume t h a t A is a bounded domain s u c h t h a t 3A

:

a i

a n d w i t h a p i e c e - w i s e smooth boundary. Suppose t h a t A

s a t i s f i e s a "cone h y p o t h e s i s " ( s e e [301 p. 2 6 .

)

If

to i s a f i x e d

p o s i t i v e number we assume t h a t fi ( t , ~();:4,2,3 ) a r e L e b e s g u e ineasu r a b l e a n d bounded f u n c t i o n s i n

[ O , t , ] x [o,t,]:.

L e t V be t h e v e c t o r s p a c e of t h e r e a l v a l u e d f u n c t i o n s belonging t o

eoo{A x [O,to

a n d v a n i s h i n g on Q A

1J

V t h e s c a l a r product (13.6)

t

((u,Y#

f o,t, 1 . I n t r o d u c e i n

!

0

dt

~ , ~ ( ~ , t ) v , , () xd x, .f

t h e H i l b e r t s p a c e o b t a i n e d by f u n c t i o n a l comple-

3f

and d e n o t e by

=

%

t i o n through t h e s c a l a r product (13.6). S e t f o r

Assume

$(x,t

)6

L' { A x C O , t 0 1 3

B(Y,Y)+[.dt

(13.7)

and c o n s i d e r t h e problem

%(

)

(

x

3

.

Hence

((u,Tv)).

denotes t h e s c a l a r product i n

,

3

e x i s t s such t h a t

B ( U , V )= If

(*v€V ).

~ ( * , t ) ~ ( ~ , =+o ) du~c

The b i l i n e a r f o r m B ( u , v ) i s c o n t i n u o u s i n

T E

3

u,v

!A x

1 ] , Eq. ( 1 3 . 7 )

[ Q , ~ o

becomes L e t u s assume o p e r a t o r ) , M,= T

-

(

T =

B,: L'

{ A x [o,t.

(13.8)

.

(

3

(YyeV)

1 ,

B2 =

,

Jt

M, =

I

(identity

W e a r e i n p o s i t i o n t o a p p l y t h e o r . 13. I a n d t o

prove t h a t a s o l u t i o n of (13.8) e x i s t s i f and o n l y i f t h e f o l l o w i n g i n e q u a l i t y b o l d s f o r any v c V

i w , ~ L) c 2 (( T W ,Tv))

(13.9) If

LU b

(Y

,t ) ]

i n the space

(

3

k=

, De V i t o p r o v e s t h a t ( W . 9 ) is e q u i v a l e n t t o t h e

following inequality:

ht1

(13.10)

O

A

(C 7 0 )

i, 2 , . - . ) i s a n o r t h o n o r m a l a n d c o m p l e t e s y s t e m

OD

/ ~ i , t l l ' d d~

C, fi:4

{il t*

dtlULli

(.it)

[-I;

( ~ ~* t )

w h i c h h e p r o v e s by a v e r y r e f i n e d t e c h n i q u e . The i n e q u a l i t y ( 1 3 -10) o f De V i t o c a n b e v i e w e d a s a , g e n e r a l i z a t i o n of t h e c l a s s i c a l P o i n c a r e i n e q u a l i t y , which can be deduced f r o m (13.10) by a s s u m i n g q, ( t , t ) a 0 ( ;= 4,2,3 ) a n d v i n d e p e n d e n t of t.

D e V i t o proves a l s o t h a t t h e s o l u t i o n of

I f cp; (+,r) and

f (x,t)

(13.7) is u n i q u e .

s a t i s f y f u r t h e r r e g u l a r i t y r e q u i r e m e n t s , De

V i t o p r o v e s t h a t t h e s o l u t i o n of (13.7) is a s o l u t i o n of ( 1 . 1 1 ) , (1.12) i n t h e c l a s s i c a l s e n s e , e x c e p t , e v e n t u a l l y , i n t h e singular p o i n t s of ? A . We r e f e r f o r d e t a i l s t o t h e o u t s t a n d i n g p a p e r [27]by D e V i t o .

1 4 . Concluding remarks.

I n t h e s e l e c t u r e s w e have t r i e d t o g i v e a n a c c o u n t o f t h e anal y t i c a l problems c o n n e c t e d w i t h h e r e d i t a r y phenomena. Many problems a r e s t i l l u n s o l v e d b o t h i n t h e l i n e a r and i n t h e n o n - l i n e a r c a s e s . We r e f e r i n p a r t i c u l a r t o problems w i t h a l o n g memory o r t o problems i n domains w i t h s i n g u l a r b o u n d a r i e s , where t h e De V i t o t h e o r y s h o u l d be e x t e n d e d t o more g e n e r a l o p e r a t o r s and more g e n e r a l boundary c o n d i t i o n s . On t h e o t h e r hand w e have o n l y c o n s i d e r e d q u a s i - s t a t i c problems and no dynamic problems. However i n t h i s f i e l d v e r y few r e s u l t s a r e known t o t h e w r i t e r , e s p e c i a l l y a s f a r a s t h e e x i s t e n c e t h e o r y is c o n c e r n e d . B e s i d e s some i n i t i a l v e r y s i m p l e r e s u l t s by V o l t e r r a ( s e e [ 7 1 ) we r e f e r t h e r e a d e r f o r u n i q u e n e s s r e s u l t s t o t h e Monograph 1263 and t o two o u t s t a n d i n g p a p e r s C321 and 1331 by D a r i o G r a f f i ( s e e a l s o t h e annexed r e f e r e n c e s )

.

R e f e r e n c e s

I11 L.BOLTZMANN, Zur Theorie der elastischen Nachwirkung, Sitzber. Kaiserl.Akad.Wiss. Wien, Math. -Naturw.Kl.70,Sect.11,1874; p. [2! L.BOLTZMANN, Zur Theorie der elastischen Nachwirkung,

Ann.Phys.u.Chem.5,1878; p.430-432. f31 V.VOLTERRA, Sulle equazioni integro-differenziali, Rend.Acc. Naz.Lincei,v.XVIII,No.1,1909; p.167. [43 V.VOLTERRA , Sulle equazioni integro-differenziali della teoria della elasticitit, Rend.Acc.Naz.Lincei,~.XVIII,N0.2, 1909, p. 295-301.

151 V.VOLTERRA, Equazioni integro-differenziali della elasticit2 nel caso della isotropia, Rend.Acc.Naz.Lincei,~.XVIII,No.2, f9D9, p.

577- 5 8 6 .

[61 V-VOLTERRA, Sur les equations int6gro-differentielles etleurs applications, Acta Mathem.t.XXXV,1912,p.295-356. 171 V.VOLTERRA, Leqons sur les fonctions de lignes, GauthierVillars, par=, 1913. 181 M.PICONE, Appunti di Analisi superiore, Rondinella,Napoli,l940,

191 G.FICHERA, Integrale di Lebesgue ed equazioni integrali di Fredholm, Corso di Analisi Matem. IV, Istituto Matem. "Guido Castelnuovo", Univ. di Roma, 1976-77. i101 C. MIRANDA, Partial Differential Equations of Elliptic Type, 2nd.revised ed., Springer Verlag, Heidelberg,l970. [Ill G.FICHERA-L.DE VITO, Funzioni analitiche di una variabile complessa, 3a ed., Veschi, Roma, 1963. 1121 M.PICONE-G.FICHERA, Corso di Analisi Matematica,vol.I, ediz. Veschi, Roma, 1972. 113I J.DIEUWNN~, Foundations of modern analysis, Academic Press, New York, 1969. 1141 R.COURANT-D.HILBERT, Methods of Mathematical Physics,vol. 11, Interscience, New York, 1962. [I51 W.S.EDELSTEIN, Existence of Solutions to the Displacement Problem for quasistatic Viscoelasticity, Archive for Rat. Mechanics & Analysis, vo1.22,1966,p.121-128.

C16 I A . M.ANCONELL1, Vn teorema d i u n i c i t a p e r l ' e l a s t l c i t t t e r e d i t a r i a , A t t i Acc.Sci.Ist.Bologna,s.XIII,t.l,p.260-264, 1974. 1171 G.FICHERA, E x i s t e n c e Theorems i n E l a s t i c i t y , Handbuch d e r P h y s i k , v o l . V I a / 2 , Springer,1972,p.347-389. [I81 S.AGMON-A.DOUGLIS-L.NIRENBERG, E s t i m a t e s n e a r t h e Soundary f o r S o l u t i o n s of E l l i p t i c 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 s a t i s f y i n g G e n e r a l Boundary C o n d i t i o n s 1 1 , Comm.on P u r e & Appl Mathem. v . XVII, No. 1 , 1 9 6 4 , p.35-92. [19 1 G. FICHERA, L e z i o n i s u l l e t r a s f o r m a z i o n i l i n e a r i , 1st .Matem. U n i v . T r i e s t e 1953,ediz.Veschi,Roma. [201 B.LEV1, S u l p r i n c i p i o d i D i r i c h l e t , Rend.Circ.Matem.Palermo ~.22,1906,p.293-359. [211 G.FUBIN1, I1 p r i n c i p i o d i minimo e i t e o r e m i d i e s i s t e n z a p e r i problemi d i contorno r e l a t i v i a l l e equazioni a l l e d e r i v a t e p a r z i a l i d i o r d i n e p a r i , Rend.Circ.Matem.Palermo,v.23,1907, p. 58-84. 1221 G-FICHERA, Premesse a d una t e o r i a g e n e r a l e d e i problemi a 1 c o n t o r n o p e r l e e q u a z i o n i d i f f e r e n z i a l i , Corso INDAM 1956-57, Veschi , Roma, 1957. 1231 G.FICHERA, Problemi e l a s t o s t a t i c i con v i n c o l i u n i l a t e r a l i : i l problema d i S i g n o r i n i con ambigue c o n d i z i o n i a 1 c o n t o r n o , Memorle Ac~.iYaz.Llncel,v.~,sez.l,1Yb4,p.Y1-14U. [24

1 C.B. MORREY, J r , M u l t i p l e I n t e g r a l s i n t h e C a l c u l u s of V a r i a -

tions ,

S p r i n g e r , H e i d e l b e r g , 1960.

1251 L.GARDING; D i r i c h l e t ' s Problem f o r L i n e a r E l l i p t i c P a r t i a l D i f f e r e n t i a l Equations, Math. S c a n d . , ~ I. , 1 9 5 3 , p. 55-72.

-

[26 1 M . J. LEITMAN-G. M. C. FISHER, The L i n e a r Theory of V i s c o e l a s t i c i t y , Handbuch d e r P h y s i k , v o l , V I a / 3 , S p r i n g e r , 1 9 7 3 , p . l - 1 2 3 . [27J L.DE VITO, S u l l a e q u a z i o n e i n t e g r o - d i f f e r e n z i a l e d i t i p 0 e l l i t t i c o d i V o l t e r r a , Mem.Acc.Sci.Torino,s.III,t.4,parte l a n. 6 , 1 9 6 1 , p. 1-45. [ 2 8 ] L.DE VITO, S u l l e i p o t e s i d i un teorema d ' e s i s t e n z a p e r l'equaz i one i n t e g r o - d i f f e r e n z i a l e d i t i p 0 e l l i t t i c o d i V o l t e r r a ,

Rend.Acc.Naz.Lincei,v.XLIV,1968,p.633-638. [29 1 G. FICHERA, Alcuni r e c e n t i s v i l u p p i d e l l a t e o r i a d e i problemi a 1 contorno per l e equazioni a l l e d e r i v a t e p a r z i a l i l i n e a r i , A t t i d e l Convegno - s u l l e Equazioni a l l e d e r i v a t e p a r z i a l i , T r i e s t e , a g o s t o 1 9 5 4 , Ed.Cremonese, Roma,p..174-227 E301 G.FICHERA, L i n e a r e l l i p t i c d i f f e r e n t i a l s y s t e m s and e i g e n v a l u e p r o b l e m s , L e c t u r e Notes i n Mathem. 8 , S p r i n g e r , ~ e itiT1b e r g , 1965.

[31] R. CACCIOPPOLI , Sui teoremi di esistenza di Riemann, Ann. Sc Norm.Sup.Pisa,v.6,1937,p.177-187. [32J D.GRAFFI, Sul teorema di unicita nella dinamica dei corpi visco-elastici, Rev-Roumaine des Mathem.pures et appl.,v.XIII, [33J D. GRAFFI, Sulla termodinamica dei materiali con memoria, Rend. Sem.Matem. e Fis. Milano,v.XLIV,1974,p. 155-170.

CENTRO INTERNAZIONALE MATEMATICO E S T I M ( c . 1 . ~ ~1 ~ .

THE BLACK BOX APPROACH AND SYSTEMS WITH MEMORY

M. R I B A R I C

Corso t e n u t o a Bressanone dal 2 all'll

giugno 1977

THE BLACK BOX APPROACH AND SYSTEMS WITH MEMORY

M. RibariE Institute J.Stefan, University of Ljubljana, Ljubljana, Yugoslavia

I. The black box approach to general systems theory

Any kind of approach to the description of a real system has some inherent advantages over other kinds in the wealth of conclusions and the easiness of their deduction, as well in what it does or does not come natural to say about 'the studied system. This lecture is meant as a short sight-seeing tour intended to convey an idea of what the black-box approach is all about. Suppose we have a collection of identical systems, say T, which we are observing during the time interval t E [Or=) from the outside. We notice that our inputs. say = ,i iin(t), t E [O,-), to T are transformed into uniquely determined outputs, say ,i = i,,,(t), t E [O,-1; a fact we choose to represent as follows:

where A is the response operator of T and q is the zero-input response of T, i.e.,

,,i

= q

when

ii, = 0 V t

> 0,

and so represents a directly observable part of the memory of T. The directly unobservable part of the.memory of the system T is hidden in the changed properties of its response operator and can be ascertained only if we have an unchanged reference system T. Note that all our direct experience with the physical world around us is collected by observing pairs of associated inputs and outputs, e.g. when looking at things we are registering their response to light falling on them. Needless to say, we are always outside any system we observe. To apply the black-box approach to explaining a system T, we interpret it as a composite system T, composed of N parts T,; so to speak, we split T into subsystems T,, as shown schematically in the following figure.

N

Now we assume that T, has been composed of known parts T, , so to say T, = U T, K=1 so that: a) there are N specified relations between inputs of TKrs and their outputs, say

,

iin =

CKK, ioUt ,, ,

K

= 1'2, ...,

N,

(1.3)

where C ,,, are assumed to be known and are often some kind of projection operators; and b) input-utput relations

of all N parts TK are known, i.e., we consider also nonlinear operators A,{ 1's and zeroinput responses q,s' as given. N Equations (1.3) and (1.4) mathematically completely determine the system U TK; solving them we can express and, therefore explain, the properties A and q of T inK&ms of the properties of its parts, say

and

For theoretical considerations it is sometimes useful to collect Equations (1.3) and (1.4) into a compact form called the Basic Equation of U T, which reads as follows:

.,

i, = (C, + A,)

,

K= 1

I i, i + qu ,

(1.7)

where i, a (iinl ,io,, ; ... , ii, iOut ),. q, = (O,q,; ...;0,q,), and C, and A, are direct sums of operators CKK, and A,, respectively, cf. RibariE 3976 §§ 2,3 and 4 for details and RibariE 1973, Ch. X V l l for modifications of the Basic Equation. In operator A, are collected the reflection properties of all T,'s, whereas C tells us how TK1shave been put together to effect the composed system T,.

Solving the Basic Equation (1.7) and studying the properties of relations (1.5) and (1.6) characterizes the black-box approach to studying system T by interpreting it as a composed system U TK whose parts TK determine i t s properties. To give an idea of the characteristics of the byaik-box approach, let us give a few examples of the questions one may ask and problems one may pose within this conceptual framework. First we note that it is not necessary that the division of T into subsystems TK should be geometrical; it need be only conceptual. When it is an actual geometrical division, then any real part T, has non-zero volume and therefore memory. Then we may note that though the Basic Equation (1.7) is universal in the sense that we may use it for the description of any physical system U TK in terms of the properties of its parts TK, it still has a specific form, often used as :=starting point for theoretical studies of mathematical equations. The simplest problem to pose in the black-box approach is about inheritable properties: assuming that the reflection operators A, K = 1,2, ..., N, of parts T, have a certain property 9 , so that

fi

we examine under what conditions this property f' is inherited by T = TK, i.e., what are K=l the conditions such that the relations (1.8) imply

In the following example, we will identify some inheritable properties. On the other hand, certain quantities, say Q,may be additive functions of properties AKfs. q, 's, of parts T, 's, so to say

Examples of such quantities are contents, capacities and various thermodynamic functions; the black-box approach may be used to establish relation (1.10) for specific cases, cf. RibariE 1975 for an example of how the black-box approach has been used to establish the existence and properties of the thermodynamics of linear transport processes. A new set of problems arises from considering a system T(Q1,!Z2)whose geometric shape depends on two parameters P I , Q2 e [0,1] as indicated in the following drawing:

On assuming that the response operator of T(QI.P2), say A(Q1,P2), is a differentiable linear operator valued function of Q2, we can derive the following Riccati -type differential equation aA"1#P2) = [Pr a 9-2

+

AP, ] A ( Q ~ ) [ P ~ + A~d ,

(1.1 1)

PL and P, being projection operators associated with directions P2- and Q2+ , and I + (Q2 - Q1)A(Q2), R, > e,, the reflection operator of an infinitely thin slice T(f2-,P2+ ); the detailed derivation of (1.1 1 ) is given in ~ i b a r i c1973, SecXXll.a. We note that Eq.(l.ll) consists actually of four coupled differential equations which can be solved succesively, with only the first one being nonlinear. Using certain transformations, we can obtain an associated system of two first order linear equations, equivalent to a second order linear equation which can be interpreted as the underlying field equation, 6. RibariE 1973, Table (XXll.c.15) for connections between various concepts. These connections enable us to solve a problem posed in terms of field equations within some domain T = U TK in two steps by first obtaining local solutions for small regions TK and then piecingKfhemtogether by solving the associated Basic Equation. These procedures usually go under the name of invariant imbedding and an extensive literature exists on the subject, cf. Scott 1974.

II. An e x a u

We now abandon general considerations and will use an extremely simple composed system to provide concrete examples of the kind of concepts, problems and results aswciated with composed systems in general. In this way we hope to be able better to convey the scope, spirit and flavour of the black-box approach to the study of systems by interpreting them in terms of their subsystems. We will thus be able to illustrate many general ideas without danger of being side-tracked by technicalities. The system T, considered consists of two parts, say TI and TII. The first part TI has an input and an output terminal with ilin(t), ilOut(t)' E C(0.m). being the associated inputs and outputs, respectively; they are related as follows ilout = A, ilin

with

A,

=

f ,

where f E (- -, -) is a parameter characterizing TI = TI (f). The second part TII also has an output and an input terminal, the associated outputs and inputs being related as follows: illout = All ill in + qll , illout(t), illin(t), qI1 (t)

E

C(O,-) ,

with

being a convolution transform, where 9 characterizing part TII = TI1(9, A).

E

and A

E

(--.-I

are parameten

(11.2)

Equations (11.1) and (11.2) completely characterize parts TI and TII. I n order to define system T, = TI U TI[, we specify that parts TI and TII, and their outside are interconnected as follows

in = i ~ o u t

+

iuin

and iuout =

i

ill out

(11.5)

where iuin and iuOutare input and output of T, (i.e. output and input of the outside of T, cf. RibariE 1976, § 6). I n the following diagram we have schematically shown the system of Eqs. (11.1) t o (11.5) identified a$ a composed system T, = TII U TII, where Equations (11.1) and (11.2) are identified as characterizing parts TI and TIl, respectively, whereas Eq. (11.4), and Eqs(11.3) and (11.5) are identified as connecting relations, describing how TI and TII are put together t o effect TI :ITlI.

Equations (11.1) to (11.5) represent the Basic Equation of TI U Tll written in terms of its components. On combining them, we obtain a reduced form of the Basic Equation

which completely determines the outward behaviour of the composed system T, = T,(f,S.X), i.e. the relation between its output iuOut and input iUin. The reduced form of the Basic Equation together with relations

gives implicitly a complete mathematical description of the system T, = TI UTII . In this simpie case the Basic Equation (11.6) can be solved explicitly, and we write the relation between iuOut and iUinin analogy with (11.2) as follows:

where

with 6,

is the response operator of T ,

d/(Z-fa)

and h,

r

Ail-ffb).

(11.10)

and

the zero-input response of T,. Relations (11.9). (11.10) and (11.11) are simple examples of relations (1.5) and (1.6) telling us how the reflection properties A, and q, of T, depend on the reflection properties Al,ql and AI1,ql1 of i t s parts TI and TI,, respectively, cf. RibariE 1973, 55 V.c.7 and XXll.a.5 for more general examples.

Solvability of the Basic Equation. In the case considered, the Basic Equation has always a unique solution. And also in general, the Basic Equation corresponding to a real composed system always has to have a solution which is unique, since that i s the way real systems behave. A non-unique solution of the Basic Equation means that the description of the corresponding system is not complete, whereas non--existence of the solution of the Basic Equation means that the considered description of the corresponding system is not correct. For instance, had we taken All = 2/f, then the associated Basic Equation (11.6) would not be solvable, which reminds us that the assumption Al = f is an idealisation, since no real physical system has an instantaneous response, and some care is required when treating systems having such idealised parts.

Decomposition of a given system. Inverting relations (11.10) we obtain relations

a = 29,(7+f9,)-'

and h = hu(l+ffiu)f

(11.12)

implying that knowledge of the reflection properties of the composed system T, = TI UTII generally does not suffice to determine the reflection properties of i t s parts TI and TII, i.e., knowing fully the outward behaviour of T, we have no way of being quite sure what is going on inside T.,

Causality. We note that the operator relation

A,

of the composed system is causal, i.e.,

it satisfies

where [Oft) is the characteristic function of the time-interval [O,t). We may ctloose to interpret this fact as being a consequence of causality of operators Al and All. If it is, then causality would be an inheritable characteristic as defined by relations (1.8) and (1.9). And indeed causality is an inheritable property even in a general non-linear case, as is easy to show by multiplying the Basic Equation (1.7) with [O,t) and taking account of (11.13).

Time-invariance.

Further, we note that the operator A,

of T,

is time-invariant,

i.e.

-

where T,f(t) = f ( t 7) if t 2 7 and = 0 if t e [O,T). Since also parts TI and TI( are time invariant, we may again surmise that time-invariance i s in general an inheritable property, as indeed it turns out t o be.

Linearity. Likewise we observe that A, as well as Al and All are all linear operators; and again it is not difficult to show that linearity is an inheritable property. Even more, one can show that under quite general conditions linearizability (i.e. the existence of the Frechet derivative) is an inheritable property, and that a uniformly bounded unique solvability of all linear approximations l o a given Basic qua ti on implies the unique solvability of the original Basic Equation, cf. RibariE 1976, 1 13.

Singularities of admittance operators. On applying the Laplace transform, say Lp, to relations (11.1), (11.2). (11.8) and (11.9) we obtain Lp ilout = Alp

L, il,,,,

L, ilin

= Allp L,, illin +

with Alp

L, qll

=

with Allp

f

,

=

6 X/(P + X) ,

and Lp ,i out = A,

L, ,i

in

+ L, q,

with Aup

=

l ~ d / +(h(1 ~

2

-i

f 8 ) ).

(11.17)

Operators Al , All and Aupare admittance operators of TI, TII and T, = TI UTII , respectively. Operators Allp and AUp as functions of the complex variable p, have poles of the f i m order at, say pII =

-X

and p,

=

- h ( l - -21 f 9 ) ,

(11.18)

respectively. Constant f determines the strength of the interaction between TI and TII ; when f = 0 there i s none. Comparing pll and p, we notice that p, -. pll as f + 0, and so we may say that on composing parts TI and TII into TI UTll the mutual interaction moved the singularity of the admittance operators of disjoint parts from - X to

- a ( l - 2 f 9 ) . This observation elicits the qvenion or now the singularities (sometimes called resonances for obvious physical reasom) of admittance operators of parts change on putting them together to effect a composed system; in particular, if there are special kinds of singularities which are immovable. Such immovable singularities of the system's admittance operators are of particular interest, since they would not move or disappeai also on decomposing the system into arbitrarily small subsystems, and are, therefore, a property of infinitesimally small parts of matter and not affected by the geometrical shape, physical matter is put into. For instance, we have such a case with an electric resonator whose resonance in the radio-frequency range is influenced by i t s shape, whereas the absorption lines of its atoms in the visual spectrum are uninfluenced. A mathematical characterization of immovable singularities for the compact operator-valued analytic functions Ap of the complex variable p has been given by RibariE & Vidav 1969 and RibariE 1973, 5 Vl.a.11.

I

Positivity, passivity and losslessness. For a moment let us restrict ourselves to nonnegative inputs il in, ill and iuin.When f, 9, X > 0, then the reflection operator A, of T, preserves non-negativity and so also do reflection operators Al and All of parts TI and TIIS respectively. Let us call such operators positive, and again it can be shown. that positivity is an inheritable property; d. RibariE 1976, (8.1 1). When 0 < f, 19 < 1 and qll = 0, then

and also

Relations (11.19) and (11.20) tell us that in this case parts TI , Tll , as well as the composed system T, always produce less than has been put into them; all of them are strictly passive. When f = 9 = 1, then we have equality in relations (11.19) and (11.20). i.e., parts TI. TII and the composed system T, are lossless. It can be shown in general, that strict passivity as well as losslessness are inheritable properties, 6.RibariE 1976, 5 11.

Fading memory. System TII has fading memory when X

>

0. We note that

in the norm topology of C(0,-). Constant h > 0 determining how fast the memory of TII fades, we see how TI can be interpreted as a system with an infinitely quickly. fading memory, i.e. as a limiting case X + 0. Results (11.9) and (11.10) show that the composed system Tu has a fading memory iff

implying that fading memory is generally not an inheritable property. Moreover, wmoosed

may also display fading memory (Xu > 0) in the case when h < 0 but 1, i.e., also when one of its parts, TI, definitely does not have a fading memory. Relation (11.221, however, points out that when parts TI and TII are passive and have fading memory, i.e. when f, 9 E (0,l) and X > 0, then also the composed system T, has a fading memory. This example suggests that fading memory combined with passivity may be inheritable. And indeed in the theory of nuclear reactors it is known that absorbing parts which always have a decaying memory, when put together again effect an absorbing part with decaying memory; for a mathematical proof see Ribari; 1973, § Vlll.b.3. system T,

1 f9 > 2

State of a system at the time instant 7. Any quantity associated with a system which has the property that i t s value at some time instant uniquely determines its value at any later time instant, may be considered as a state of a system. In order to define a state of T, with an arbitrary input iUinit turns out to be convenient to consider T, and its complement. say as a new system. say T, ~ fwhere , the output of the complement is defined as follows:

7

equals zero, and the zero-input

i.e. the reflection operator of iUin.We define the state of T,

where stTu(71.

+ T_?[7, =)qU

T,[r,-)A,[O,r)iUin

' 8,,XUf

-1 t

=e

eh'"'-"i

=

urn . (r')dr8+q,,(t+r)

end qu(s;t)

5

,i

response of T,

uT, as a two-dimensional column vector

+ r ) with

>

t,~

0

.

Then we can verify that

where

is the time--evolution operator, forming a semi-group, i-e.,

equals

and having the following infinitesimal generator, say

cf. RibariE 1973 [A.XI] for general properties of such semigroups. Relation (11.27) verifies that stT UT as defined by (11.241, (11.25) and (11.26) is indeed a state of VUuTu. U u It makes sense to interpret stTU(r,t) as a state of Pu (cf. RibariE 1973, XX.a.1); and we note that at the time instant T the state of Tu equals the output of Tu when its input iui" = 0 V t > 7, i.e., the state s\(T) of TU equals the observable memory of Tu at the time instant P. In an analogous way, it is possible. to define the state of a general causal system with memory, which is neither linear nor time-invariant, cf. Ribaric* 1976, § 10.

Collective memory. Relations (11.9). (11.10) and (11.11) tell us that qu 9 qI1, unless qll = 1 so that the memory of TII about i t s past t < 0 changes when TI1 is part of the system TII UTI , i.e. the collective memory qu of TII UTI at t = 0 differs from the memories ql = 0 and qll of its individual members TI and TII due to mutual interaction. This difference can be quite striking, as witnessed by the followi~gexample. When qll = qI1(0) e-ht, then qu= 2 qI1( ~ ) e - ~ " - ? ~by ~ '(11.9). ~ , (11.10), (11.11); and taking h > 0 Pnd f9 = 1 we have qu(t)=$q,l(0) b't > 0. So due to interaction between part TI having no memory, and part TII with a fading memory and whose memory qll(t) about i t s past t E (---,0) is fading too (qI1 (t -. -) + O), the composed system TI UTll is unable to forget. When f 9 < 0, this example demonstrates that it is also possible for TI to help TII to forget its past by 'negative reinforcement. The example considered again brings forth the most fascinating and useful fact in systems theory, that putting together parts of the same kind, we may succeed in creating a completely new kind of system.

References

RibariE, M.: Functional-Analytic Concepts and Structures of Neutron Transport Theory, Vof. I and II. Slovene Academy of Sciences and Arts, Ljubljana, 1973. RibariE, M.: Basic equation of input-output models and some related topics,in Control Theory andfopics in Functional Analysis, Vol. II, pp. 257-279, Int. Atomic Energy Agency, Vienna, 1976. RibariE, M. & I. Vidav: Analytic properties of the inverse A(z)-' of an analytic linear operator valued function. Arch. Rational Mech. Anal. 32, pp. 298-310 (1969). Scott, M.R.: A bibliography on invariant imbedding and related topics. SLA-744284, Sandia Laboratories Albuquerque, N.M. 1974.

CENTRO INTERNAZIONALE MATFMATICO ESTIVO (c.I~M~E~)

NOTES ON THE THEORY OF CONSTITUTIVE EQUATIONS

R o n a l d S. RIVLIN

C o r s o tenuto a B r e s s a n o n e d a l 2 a l l ' l l giugno 1977

NOTES ON THE THEORY OF CONSTITUTIVE EQUATIONS Ronald S. Rivlin Lehigh University, Bethlehem, Pa., U.S.A.

Contents Chapter 1.

One-Dimensional Constitutive Equations

Introduction Types of viscoelastic behavior A

representation for the stress.

An alternative procedure Additive functionals Hereditary materials Implicit constitutive equations Alternative constitutive assumptions Strain expressed as Taylor series 10.

Non-linear integral representations

11.

Polynomial approximation

Chapter 2.

The Effect of Superposed Rotation

1.

Conservative materials

2.

Cauchy elastic materials

3.

Stress dependent on the deformation gradient matrix and its time derivatives

4.

Functional constitutive equations

5.

Fluids

Chapter 3 .

Restrictions on Constitutive Equations Due to Material Symmetry

1. Description of material symmetry 2.

Restrictions on the constitutive equation imposed by material symmetry in implicit form Some invariant-theoretical concepts Canonical forms for constitutive equations Function basis Constitutive equations of the function type Constitutive equations of the functional Type Invariant functionals Peano's theorem Tables of typical invariants Canonical form for invariant functionals Canonical form for tensor functionals

Chapter 4.

Integrity Bases for Finite Groups

1.

The basic theorems

2.

Determination of the integrity basis in a simple case

3.

Some results from group representation theory

4.

Transformation of the carrier space

5.

An example

6.

Irreducibility

7.

Determination of linearly-independent invariants

8.

Historical note

Chapter 5.

Integrity Bases for the Full and Proper Orthogonal Groups

1.

Introduction

2.

Isotropic tensors

3.

Isotropic tensor polynomials

4.

The integrity basis for N second-order symmetric tensors

5.

The integrity basis for six or fewer second-order symmetric tensors

6.

Irreducibility

Chapter 1 One-Dimensional Constitutive Equations 1.

Introduction The object of these lectures is to present the manner in

which constitutive equations in mechanics and other branches of continuum physics can be formulated in a systematic manner on the basis of clearly stated concepts of the type of physical behavior that it is intended to model.

In order to do this, we have first

to decide on the variables which it is appropriate to relate to each other, in view of the physical situations to which it is intended to apply the constitutive equations.

Secondly, we have

also to consider such questions as the smoothness of the relations between these variables, since different assumptions regarding continuity or differentiability of the constitutive relations can radically alter the physical behavior which they model. Both of these matters can be conveniently illustrated by studying one-dimensional constitutive equations, without introducing the complexities which arise in the three-dimensional case. In this chapter, these aspects of the problem of formulating constitutive equations are discussed in the context of the mechanics of viscoelastic materials, in which a tensile force is assumed to depend on a tensile strain, or rate-of-strain, or on the history of these.

It will be evident that similar situations apply in a

wide variety of other physical contexts. Once these one-dimensional aspects of formulating constitutive equations have been understood, there remain two further

aspects which arise only in the three-dimensional case*.

The

first of these involves the effect of a superposed rigid motion in restricting the manner in which the dependent variable in a constitutive equation can depend on the independent variables. These aspects of the subject are discussed in Chapter 2.

The re-

maining chapters are devoted to the further restrictions which may be imposed on a constitutive equation in view of any specified

**

symmetry that the material which is modelled by it may possess 2.

.

Types of viscoelastic behavior Consider that a thin rod of perfectly elastic material of

uniform cross-section is loaded by a time-dependent load a(t), per unit area measured in the undeformed state. We suppose that, as a result of this loading, the rod undergoes a simple extension E(Z]

per unit initial length. We shall call ~ ( t )and ~ ( tthe )

stress and strain respectively at.time t.

Since the material of

the rod is perfectly elastic, the value of the stress at time t depends only on the strain at time t and not on the strain at any other time.

We shall assume that the material is such that the

relation between ~ ( t )and ~ ( t ) is linear, thus; rlt) = EeCt),

(2.1)

where E is a constant, which is called the tensile modulus for the material of the rod.

*

**

One might also be concerned with the formulation of twodimensional constitutive equations. The problems which then arise are essentially the same as those which arise in the three-dimensional case, but are usually much more easily dealt with from a mathematical standpoint.

I am grateful to Dr. G.F. Smith for many valuable discussions in connection with this nart of the work.

There are many materials for which the assumption of perfect elasticity is not valid and for which the stress at time t depends not only on the strain at time t, but on the strain at all previous times.

Such materials are said to have memory.

In mathematical

terms we may write

where

f

is a functional of the strain hZstory

the range (-m,t].

E(T)

defined over

We will see later that there may exist mater-

ials with memory for which the stress at time t is not strictly expressible as a functional of the strain-history, i.e. by the constitutive relation (2.2).

However, for the moment, we will

exclude such materials from our discussion. We consider only strain histories for which

E(T)

= 0

up to

and including some specified time to. Then we may replace the relation (2.2) by

We shall now describe two typical kinds of material behavior. These do not exhaust all the possible kinds of material behavior which are observed, but are characteristic of many materials. (i)

At time ?, say, greater than to, a very rapid - idealized as

instantaneous

-

increase in the strain takes place from zero to

some finite value and the strain is subsequently held constant, as shown in Fig. l(a).

Correspondingly, the stress increases

instantaneously by a finite amount and then decays with increase of time, either to zero as shown in curve I of Fig. l(b) or to some finite non-zero value as shown in curve I1 of Fig. l(b). The material is said to exhibit inszantaneous elasticity and

Fig. 1 ( a )

Fig. l ( b )

Fig. 2 ( a )

Fig. Z(b)

stress rezaxation.

Again, if we increase the stress instan-

taneously at time t, say, by a finite amount and then hold it constant, as shown in Fig. 2(a),

the strain increases instan-

taneously by a finite amount and then continues to increase more slowly with time either to a constant finite level, as shown in curve I of Fig. 2(b),

or indefinitely (i.e. until the onset of

rupture), as shown in curve I1 of Fig. 2(b).

The material is

said to exhibit instantaneous elasticity and creep. (ii)

In the second type of material which we consider here, an

instantaneous change in the stress at time

t does not result in a

corresponding instantaneous change in the strain.

If the stress

is increased instantaneously from zero to a finite value and then held constant, as shown in Fig. 3(a),

the strain increases con-

tinously from zero, either to a finite value, as shown in curve I of Fig. 3(b),

or indefinitely, as shown ih curve I1 of Fig. 3(b).

In such a material, it is not possible to increase the strain instantaneously by applying a finite stress. The essential difference between the two types of material we have considered is that, while both of them are materials with memory for which the stress may be regarded as a functional of the strain history, the first material exhibits instantaneous elasticity and the second does not. 3.

A representation for the stress There are many materials of the type described for which

at any rate for small enough values of

E(T)

-

-

the relation bet-

ween the stress and strain history is linear; i.e. if al(t)

and

-n2(t) are the stresses at time t corresponding to strain his-

tories

E~(T)

and

E~(T)

respectively, then the stress at time t

Fig. 3 ( a )

Fig. 3 ( b )

Fig. 4 ( a )

Fig. 4C'b)

corresponding to the strain history

.

n2 (t)

linear

*

is nl(t)

E~(T) + E*(T)

+

For such materials, the functional f in (2.3) is a functional of the strain history.

We shall first consider materials for which'the stress is a linear functional of the strain history.

To emphasize this we

write (2.3) as

*(%I

t

L

=

[E(T)~,

T'=t

where

Our first objective wiil be to obtain expressions for ~ ( t ) in useful analytical form.

We shall first do so in the case when

the material exhibits instantaneous elasticity. Consider the strain history E(T), funct.ion of

T,

space C[tO,t]),

as shown in Fig. 4(a), or

E(T)

where (i.e.

is a continuous

E(T) E(T)

lies in the

is a piecewise continuous function of

T

having a countable numb-er of salti, as shown in Fig. 4 ( b ) , (i.e. E(T)

lies in the space M[tO,t]). We shall assume that the linear functional dependence of

~ ( t )on E~(T)

E(T)

is c o n t i n u o u s in the following sense.

E~(T)

and

be two strain histories such that

Let vl(t)

and n2(t)

the strain histories

*

Let

be the stresses at time t (T)

and

E~ (T)

corresponding to

respectively.

We assume

We remark here that in functional analysis, functionals which have the property we describe here as linearity are often called a d d i t i v e functionals, while the term l i n e a r is reserved for functionals uhich have an additional property which we call c o n t i n u i t y .

that the functional L is such that [al(t)

-

~ ~ ( t ) l+ 0

as

6

+

0;

(3.4)

i.e. L is a continuous functional of ~ ( r )with respect to the supremum norm.

of

We now consider the rod to undergo a strain-history E(T) the type illustrated in Fig. 4.

We divide the interval [tO,t]

into n sub-intervals [tO,tl), [tl,t2),...,[tn-l,tn=t],

the salti

if they exist occurring at one or more of the times tl, ...,tn. Let E(r)

be a strain-history which is constant over each of the

intervals and is equal to ~ ( r )at the initial point of each interval, i.e.

. ..,n) .

(i=O,l,

B(ti) = €(ti)

(3 5)

This strain-history is shown schematically in Fig. 4. It follows from the assumed continuity of L that

1

as n is increased in such a way that sup. t.-t. 1 1-11

Let u(6,r)

+

0.

be a unit step-function defined by 0 for

uCS,r)

=

11

for

to

(

5 < r

r 5 E

and let

We assume that the material is such that for any specified value of t, g(r, t) is of bounded variation,

This condition is certainly

satisfied for a material with instantaneous elasticity. The strain-history c ( r ) may be regarded as the superposition of a number of step-functions with appropriate amplitudes, as illustrated in Fig. 5.

Thus, noting that e ( t O )

=

0,

Fig. 5

Fig. 6

=1 2'

Fig. 7(a)

'C

Fig. 7 ( b )

Fig. 8

5

From the linearity of the functional L, together with (3.9),

it

follows that n

L[E(T)I

=

1 {€(ti) i=l

-

~(t~-~)~g(t~,t).

(3.10)

Now, let n tend to infinity in such a way that sup.lti-t i-11 We obtain, with (3.6), a representation for L[E(T)]

+

0.

in the form

of a Stieltjes integral thus

I

t

n(t)

= L[E(T)]

=

(3.11)

g(~,t)d~(~). 0

The function g(r,t) for the material.

in (3.11)

is called the memory function

From its definition in (3.8),

we see that it

is the stress at time t resulting from a strain-history which is zero up to time

T

and unity in the interval [~,t].

It may be argued that if the material is such that it is not possible to produce in it a saltus in

E(T)

of finite magnitude by

means of a finite stress, then the procedure we have adopted here is not valid.

This argument can be met by invoking the Hahn-

Banach theorem.

It follows from this theorem that if .the func-

tional (3.1), with E(T)

in C[tO,t], is continuous in the sense of

the supremum norm, we can construct a functional nf(t), E(T)

n'(t)

say, for

in M[tO,t], continuous in the same sense, such that = n (t)

when E(T)

is in C[tO,t]

.

The argument given above

*

then goes through when applied to the functional n (t), and hence to ~ ( t ) if we restrict the argument functions to C[tO,t]. 4.

An alternative procedure In this section we outline an alternative procedure for

arriving at a representation for the linear functional in (3.1) equivalent to (3.11). E(T)

Instead of approximating the function

by means of step functions, we approximate it by a number of

adjacent flat-topped pulses, as shown in Fig. 6.

Consider now a

pulse in the interval [ T ~ , T ~as ) shown in Fig. 7(a).

For this

pulse

1 for rl

( T

< r2

(4 .I)

=

E(T)

0

for

T < T

and

T 2 T ~ .

We can construct such a pulse by superposing two step functions, in which the steps have height -1 and take place at times T

T~

and

2'

Let 1 u(5,~)

=

E < T

for

C?T

0 for

and let h(~,t)

t L [u(S,~)l.

=

T=-m

We note that u(~,T) is the particular choice of ~ ( 6 )shown in

Fig. 7(b).

The pulse (4.1) is then u(S,'r2)

-

u(S,T~).

As in 53, we divide the interval [tO,t] into n sub-intervals [tO,tl) , [t,,t2),

..., [tn-,,tn=t] , the

salti in E(T),

exist, occurring at one or more of the times tl,

E(T)

if they

...,tn.

Let

be a strain-history which is the superposition of pulses of

height

E (ti-l)

. ..,n)

(i=l,

occurring in the intervals [ti-l,ti]

Then, we may write

With (4.3),

it follows that

.

We now let n

+ m

in such a way that sup.lti-ti-l

1

-+

0.

It k '

follows, with the assumed continuity of the functional L,that t ~ ( t )= L[~(r)l = ~t L[i(~)l = ~(~)dh(r,t) + g(t,t)~(t). n-

j

(4.6)

The relation between the representations (4.6) and (3.11) can be established in the following manner.

We note from the

definitions of h(~,t) and g (T,t) given in (4.3) and (3.8), bearing in mind the different definitions of u(5,~) in the two relations, that h(T,t)

+

gCr,t) = g(-m,t)

-

-

(4 7 )

Using this relation to substitute for h(~,t) in (4.6), we have, with €(to) = 0,

5.

Additive functionals The procedure used in 14 to arrive at a representation for

the stress, in the case when it is a linear functional of the

*

The result given in (4.6) could, in principle, be read off from Riesz's theorem, if the latter is properly interpreted. However, the proof of Riesz's theorem given in many of the standard texts on functional analysis is somewhat incorrect and leads to the result (4.6) with g(t ,t) = 0.

strain history, can also be used to obtain a representation for the stress in certain cases in which it is not a linear function-

al of the strain-history. In Fig. 8, we show a strain-history ~ ( r )consisting of two disjoint strain-histories E~(T) and E~(T).

Let al (t) and a2 (t)

be the stresses at time t corresponding to the strain histories E~(T)

and

E~(T)

taken separately and let a(t)

be the stress at

time t corresponding to the strain-history E(T).

for all disjoint

E~(T)

If

and E~(T), we say that the stress is an

additive functionaZ of the strain history E(T).

We now suppose that t = F [ECT)I, T=tO

where F is an additive functional.

We divide the time interval

[tO,t] into n sub-intervals and approximate the strain-history by a strain history E(r) consisting of n adjacent, but dis-

E(T)

joint, rectangular pulses.

The strain history u ( ~ , t ~ - ~ , t ~ )

corresponding to the pulse in the interval [ti-l,ti) is for , ' ( u

ti-l 5 T < ti

...,n)

(i=l,

ti-l,ti)=

(5.3) 0

We define

otherwise.

~ I (ti-l), E ti-l,t}

by

(ti-ti-l)kI~(ti-l),ti-l,tl

=

F[u(~,t~-~,t~)l ;

(5 - 4 )

i.e. kI~(t~,~),t~-~,t) is the stress at time t associated with the pulse-like strain history (5.3), per unit duration of the pulse.

It follows from the additivity of F that

where g{~(t),t,t)

is the stress at time t associated with a step

in the strain from zero to ~ ( t )at time t. Now, allowing n

with sup. ] ti-ti-ll

+

that F is a continuous functional of

E(T)

-+

0, and assuming

with respect to the

supremum norm, we obtain

6.

Hereditary materials For many of the materials with which we are concerned, it is

a good approximation to assume that their properties do not change purely as a result of the passage of time.

Such materials are

called h e r e d i t a r y materials. Let cl(r)

and

E~(T)

be two strain histories in a hereditary

material, such that E1(d

=

c2(T+T),

where T is some constant positive time.

Let nl(t)

and r2(t)

be the corresponding stresses at an arbitrary time t.

Then, if

the material is hereditary,

Also, from ( 2 . 3 ) ,

Since the material considered is hereditary, the time to in each of equations (6.3) may be arbitrarily chosen prior to the time at which the strain becomes non-zero. We can accordingly

r e p l a c e (6.3)

by t+T

For each o f t h e s t r a i n h i s t o r i e s E ~ ( T and ) E ~ ( T ) l, e t s d e n o t e t i m e p r i o r t o t h e i n s t a n t a t which t h e s t r e s s i s measured. We c a l l s t h e l a p s e d t i m e .

Then, f o r e q u a l v a l u e s of s t h e

s t r a i n s a r e e q u a l and we may w r i t e

el ( T ) = E ~ ( T + T =) 2 ( s ) Then, from (6.3)

and (6.4)

, we

, say.

(6.5)

have

and

i is

where

a f u n c t i o n a l o f t ( s ) and an o r d i n a r y f u n c t i o n o f t i n

(6.6)1 and of t + T i n (6.6) 2 .

Since t h e r e l a t i o n ( 6 . 2 ) i s v a l i d

f o r a r b i t r a r y p o s i t i v e T , i t f o l l o w s from (6.6) t h a t

?

i n (6.6).,

must be i n d e p e n d e n t o f t f o r a h e r e d i t a r y m a t e r i a l . We c o n c l u d e t h a t i f t h e s t r e s s s ( t ) i n a h e r e d i t a r y m a t e r i a l , c o r r e s p o n d i n g t o a s t r a i n h i s t o r y E ( T ) , i s g i v e n by ( 2 . 3 ) ,

then

i t may be e x p r e s s e d i n t h e form

where ;(s)

= E(T),

S

=

t-T,

and t o i s any time such t h a t E ( T ) = 0 f o r T L t o . We now a p p l y t h i s r e s u l t t o t h e c a s e , e x p r e s s e d by ( 3 . 1 ) , when ~ ( t i) s a l i n e a r f u n c t i o n a l of ~ ( r ) . Then,

where

Writing g(r,t)

=

g(s),

and using (6.8), we find that for a

hereditary material, the constitutive equation (4.8) may be written as n(t)

(s)di(s)

=

+

(0) i(0)

,

(6.11)

0 where so=t-to. Similarly, the constitutive equation (3.11) may be written as

Equations (6.11) and (6.12) may, of course, also be written in the forms

-/

t

n(t)

=

~(~)di(t-T) + ;(o)E(~),

(6.13)

0 n(t) = lt;(t-~)dp (r) 0 respectively. 7.

Implicit constitutii*e equations We take as our starting point the constitutive equation

(6.14) with to

=

and write a =a(t),

--.

For convenience, we omit the roof over g

thus: n

=

/

t g(t-r)d~(-c).

-0

We now assume that

E(T)

is differentiable, so that (7.1) may be

re-written as

We approximate the kernel g(t-r) by

where Ck and ak are constants. Differentiating (7.2) i times with respect to t, we obtain

Taking i=O,...,n we obtain from (7.4) of which is (7.2)

.

n+l equations, the first

With (7.3), we obtain

-

J

w

We can eliminate the integrals in (7.5) from the (n+l) equations obtained by taking i=O,l, ...,n.

We obtain

where S1,S2,,..,Sn are the sums of products of the a's taken 1,2,

...,n

at a time and SO=l. This equation has the form

where Pn is the operator

and Qn is an operator of similar form with different coefficients. When n=l, we have

and the relation (7.6) becomes

The material modelled by (7.10) is called a MaxweZZian fluid. Of course, the differential equation (7.6) for n (which may equally be considered to be a differential equation for

E

if r is

given) does not provide a complete constitutive equation unless n appropriate initial conditions are specified. We may generalize (7.6) to the non-linear ease heuristically as

8.

Alternative constitutive assumptions It was remarked earlier in 52 that there may exist materials

with memory for which the stress a(t)

is not strictly expressible

as a functional of the strain history up to and including time t, i.e. by a relation of the type ( 2 . 2 1 .

This is the case if the

stress at time t is a function of the instantaneous value, at time t, of the rate of change of strain, which may have a saltus at time t.

For example, in the case of an incompressible New-

tonian fluid, we have ~ ( t )= 3Q&(t)/{l+E(t) where

Q

1,

(8.11

is the viscosity of the fluid and i(t) denotes

I

[ d (~~ ) / d ~ lT=t.

It is evident that z(t) and consequently ~ { i '

cannot be expressed as a functional of

E(T)

~ i t hsupport

(-=,TI,

if :(T)

has a saltus at time t, i.e. if ;(T)

= O

(~
and

;(T)

f 0

(~)t).

(8.2)

However, this limitation can be removed, in a variety of ways, by modifying the constitutive assumption expressed by (2.2).

For example, we may introduce the dependence of ~ ( t )on

i(t)/{l+~(tj

)

into the constitutive assumption explicitly, thus:

Alternatively we may extend the support of E(T)

in the

constitutive assumption and write

To see that this accomplishes our purpose, we note that

where 6(

) denotes the Dirac delta function and :(

its time derivative d6 (

) denotes

)/d~.

Again, we may replace the constitutive assumption (2.2) by

9.

Strain expressed as Taylor series Now suppose E(T)

can be expressed as a Taylor series, i.e.

If a is a linear functional of E(T), thus

then, with (9.1),

i.e.

we have

,... .

is a linear function of E,;,:

x

We now define E~(T)

by EV(T)

=

! 1 J 1 (T-t)a E Ca) (t).

a=o

Let a

!J

be the stress corresponding to the strain history

E (T).

U

Then, from (9.2),

is a linear function of E,: ,..., NOW suppose P is a u continuous functional of E(T) in the sense of the supremum norm. i.e. n

Then,

T

IJ

+

a as

SUP.IE(T)-E~(T)I

+

n b y a linear function of E,;,.

Thus, we can approximate

0.

.. ,E(')

as closely as we please by

taking IJ large enough.

A somewhat similar argument can evidently be applied if a non-linear functional of

If

E(T)

E (T)

B

is

, thus:

is expressible in the form (9.1),

it follows that

P

must

be expressible as a function of the countably infinite set of quantities E,;,;,. of

E(T)

.. .

Moreover, if

T

is a continuous functional

in the 'sense of the supremum norm, it can be approximated

with any desired accuracy by a function of E,;,E,. vided that

!J

is taken large enough.

. .,E(')

pro-

We will see later, in SIX,

that this function may be taken to be polynomial.

10. Non-linear integral representations We have defined linear viscoelastic behavior in the following manner. Let E~(T) and n

1

E~(T)

be two strain histories and let

and n2 be corresponding stresses measured at time t.

The

behavior is linear if the stress at time t, corresponding to the strain-history c1 (T)

+

E~ (T) ,

is n1

4

T ~ .We

shall now discuss

the representations for the stress which are possible when this is not the case. Notwithstanding the limitations on the constitutive assumption (2.2), which we have noted in 5 8 , we shall take this as our starting point.

Accordingly, we write

By making appropriate assumptions regarding the space of functions in which

E(T)

lies and regarding the smoothness of the

dependence of n on E(T), we can give more explicit form to the functional F

.

Let us suppose that the material is hereditary.

Then we may

write (10.1) as

Now let s = f(t-T) be a smooth monotonically decreasing function of t-T which maps the interval

T =

(-m,t] onto [0,1].

Then, we

may replace (10.2) by n(t)

1 = G IE(s),tl, s=o

where

We now restrict ourselves to strain histories E(T),

which

are such that E(s)

can be expressed as a cosine Fourier series

with period 2; i.e. we define a function E*(s) such that E*(-S) and suppose E*(s)

= E*(s) =

E(s),

s

=

[0,1],

(10.5)

can be expressed in the form

With (10.5), we have B,

= 2

I'

0

E (s) cosans. ds (a 2 1) ,

Plainly, a(t) may be regarded as an ordinary function of the countably infinite set of quantities B0,B19B2,..., n(t] Nu*,

=

thus:

G(BO,Bl,.. .;t).

(10.8)

consider a strain history E (s) defined by 1.1

From (10.31, the corresponding stress

IT

U

(t) is given by

Now, if G , in (10.3), is a continuous functional of E(s)

in the

sense of the supremum norm, we see that Ir(t)-n,,(t)l

+

0

as sup.lB(s)-Z,,(s)l

+

0.

(10.11)

s3E0711

In both 59 and the present section, we have seen that under appropriate conditions on the space of strain histories considered, functional dependence of the stress on the strain history may be

replaced by functional dependence on a countably infinite set of

*.

linear functionals of the strain history

The conditions imposed on the strain history in 99 and the present section are different.

By imposing yet other conditions

on the strain history, a wide variety of representations in terms of other countably infinite sets of linear functionals of the strain history can be obtained. 11.

Polynomial approximation In each of the cases discussed in 559 and 10, we have point-

ed

out that the stress may be approximated with any desired

accuracy by a function of a sufficiently large finite set of linear functionals of the strain history, provided that the functional dependence of the stress on the strain history is assumed to be continuous.

By invoking the Stone-Weierstrass theorem, we

can assert that this functional dependence may be taken as polynomial. In order to apply the Stone-Weierstrass theorem, we consider strain histories

E(s)

= E(T)

which lie in a compact Hausdorff

space and assume that ~ ( t )is a continuous functional of these with respect to some norm N , say.

Suppose we can construct an

infinite sequence of linear functionals of ~(r), say L1,L2,..., which separate points of the space of the set of functions L1,L2, ponding functions

E(T)

E(T);

i.e. if two values of

... are different, then the corres-

are different in the sense that the

"distance" between them, measured by the norm N, is not zero.

* Although, as in

58, the time derivatives of the strain at time t are not, in general, functionals of the strain history with support (-m,t], they are so in the case when the strain at time t is expressible as a Taylor series about time t.

It then follows from the Stone-Weierstrass theorem that we can construct a sequence of polynomials P (L 1rL27...sLv), such that P

P approximates s(t)

as closely as we please if

1-1

p

is taken large

enough. We apply this result in the case we discussed in 59 in which the space of strain histories consists of those which can be expressed as a Taylor series, as indicated in (9.1). of linear funceionals L1,L2,

... is the

all orders, evaluated at time t, i.e.

Then, the set

set of time derivatives of

. .

(t) (a=O,l,Z,. .)

It

follows that we can approximate ~ ( t )as closely as we please by a polynomial in

E(")

(t) (a=0,1,2,.

..,u)

provided that u is large

enough. Again, consider the case discussed in 510, in which the space of S ( s ) consists of those strain historjes which can be expressed as a Fourier series in the manner indicated in (10.6). The set of linear functionals L1,L 27... is now B0,B1,B2, ..., given by (10.7).

It follows that ~ ( t )may be approximated, with

any desired accuracy, by a polynomial in BO,B1, that u is large enough.

...,Bu , provided

Since the B's are given by (10.7), we

may with any desired accuracy express ~ ( t ) in the form of the sum o f multiple integrals thus:

This relation may be re-written as n(t1

=

1

It It.. .

a

-m - m

-

jtg(t-rl,t-rp,. . ,~-T~)E(T~)E(T~). ..E(T") -m

d ~ ~ d ~ ~ . . . d r ~(11.2) , where

.. .

g ( t - ~ ~ , t - ~ ,t-+,) ~,

= (-1)"

f' (t-rl)

.. .f' (t-~,)E(s~,s~,. . . ,sa) (11.3)

with sg

=

f ( t - ~ ~ ) (8=1,2,.

.. , ) ,

and where the prime denotes

differentiation with respect to the argument. It will, of course, be apparent that representations for

T

analogous to those derived in 559-11 can be obtained if any of the alternative constitutive assumptions in 58 is taken as starting point.

Chapter 2 The Effect of Superposed Rotation 1.

Conservative materials In this chapter we discuss the manner in which three-

dimensional continuum theories for materials with memory can be formulated.

In order to introduce the methods involved, we shall

first consider simpler cases when the materials considered do not possess memory.

As a first example, we consider briefly the case

when the material is elastic and we wish to construct a constitutive equation adequate for the description of its mechanical behavior when subjected to isothermal deformations by the application of external forces. The material is assumed to undergo a deformation in which a particle which has vector position

5

at some reference time T,

say, is at vector position E(T) at time T, i.e.

E(T) I f the dependence of

Z(T)

completely described.

on

g(z,r)-

=

and

T

(1.1)

is known, the deformation is

We shall, for brevity, use the notation

x=z(t) and show how an expression for the Cauchy stress at time t can be calculated.

Let xi(r)

and XA be the components of Z(T)

and & respectively in a rectangular cartesian coordinate system x and let x(r)

ponents.

and X be the column matrices formed from these com-

We define the deformation gradient matrix g ( ~ by ) ) ' ( g

=

1

1 lgiA(')1

=

1 lxi,A(r)l

1,

(I.2]

where we have used the notation ,A to denote differentiation with respect to X A .

Since det g(r)

of a material element at times

is the ratio between the volu~~~es T

and T respectively, it follows

that for deformations possible in a real material

det g(r) > 0.

(1-3)

If the material is elastic, its isothermal mechanical properties are characterized by a strain-energy function W, say, per unit mass, the value of which, at time t, is assumed to depend only on the instantaneous value g, at time t, of the deformation gradient matrix, thus

w

= F(&?).

Now, the dependence of W on

is not coinpletely arbitrary.

To see this we superpose on the assumed deformation rotation., x

+

X+x

a rigid

a x = 5 say, where a is a constant proper orthogonal

matrix, so that ata = aat = 6,

detaEl,

where the dagger denotes the transpose and 6 is the unit matrix. As a result of this superposed rotation, g changes to ag, while the strain-energy is unaltered.

Accordingly, we have

for all proper orthogonal a. The restriction on W implied by (1.6) can be made explicit in the following manner.

Since (1.6) is valid for all proper

*

orthogonal a, it is, in particular, valid for

where C, called the Cauchy strain matrix, is defined by

*

The square root of a real symmetric matrix, m say, is defined as a matrix which has the same eigen-vectors as m and eigenvalues which are the square roots of those of m. We note that the matrix C defined by (1.8). has positive eigen-values so that c - $ is a real matrix. It can easily be verified that the matrix a, defined by (1.7), satisfies the relations (1.5) and is, consequently, a proper orthogonal matrix.

It then follows from (1.6) that

Thus, W must be expressible as a function of the symmetric matrix C, thus:

w From (1.8) and (1.5),

=

W(C).

(1.10)

it is evident that replacement of g by

ag, where a is an arbitrary proper orthogonal matrix, leaves C un-

altered.

It follows that if W has the form (1.10) it satisfies

the condition (1.6) for arbitrary proper orthogonal a. From (1.lo), an expression for the Cauchy stress matrix

a,

at time t, can be readily obtained from thermodynamic considerations as

where

p,,

*

and

p

are the densities of the material at times T and t

respectively. For non-isothermal deformations, the strain-energy function W depends on the temperature 8, which is a scalar quantity, and the relation (1.10) is accordingly replaced by

the Cauchy stress being given by (1.11). If an electric displacement field D , or a magnetic induction field

*

B,

or both exist in the material, then the constitutive

By aW/ag we mean the matrix with element a W / a ~ ~ in, the ~ ith row and A t h column.

assumption (1.4) is replaced

*

by

respectively. Paralleling the passage from (1.4) to (1.10), we consider that the deformed body and the fields D and B are subjected to a rigid rotation, such that x + a x and correspondingly D + a D and B + a B , where a is an arbitrary proper orthogonal matrix. Since W is unaltered by this rotation, we have F(g,D,B)

= W(ag,aD,aB),

(1.14)

for arbitrary proper orthogonal a.

Proceeding as before, we see that W must be expressible as a function of C, gt E , g t B thus:

More generally, it is evident that if we start with the constitutive assumption

where V 1 ,

...,V ?J are

p

column matrices, formed from the components

in the system x of a number of vectors which rotate with the body, then W must be expressible in the form

2.

Cauchy elastic materials A Cauchy elastic material is defined as one for which the

appropriate constitutive assumption for isothermal deformations is 0 =

f(91,

(2.1)

but a strain-energy function W, from which o can be calculated

*

We regard D and B as column matrices formed by the components of the fields in the system x.

by (1.11), does not exist.

Here a and 9 are the Cauchy stress

and deformation gradient matrices respectively in the system x .

*

While, as has been pointed out elsewhere

, it follows from thermo-

dynamic considerations that such an assumption is not a realistic one

**

, it is nevertheless instructive to consider the restrictions

which can be imposed on the manner in which a depends on g if this assumption is made. As in dealing with conservative systems, we superpose a rigid rotation on the assumed deformation, so that g + a g , where a is an arbitrary proper orthogonal matrix.

Then, the Cauchy

stress matrix in the system x becomes aaat and we have aoat

=

f(ag),

From (2.1) and (2. 2), we obtain

Now, making the particular choice of a given by (1.7), we obtain

Thus, a must be expressible in the form

*

R.S. Rivlin, "An introduction to non-linear continuum mechanics'' in Non-Linear Continuum Theories in Mechanics and Physics and Their Applications, ed: R.S. Rivlin, Edizioni Cremonese, Rome (1970); A.E. Green and P.M. Naghdi, Arch. Rational Mech. Anal. 40, 37-49 (1971).

**

It has been shown (R.S. Rivlin, Q. Appl. Math. 14, 83-89 (1956)) that the assumption (2.1) is useful in the discussion of stress-relaxation in materials with niemory.

Since replacement of g by ag, where a is an arbitrary proper orthogonal matrix, leaves

C

unaltered, no further restriction is

imposed by (2.3) on the form of f. Paralleling the case when the material possesses a strainenergy function, if we wish to include non-isothermal deformations among those covered by our initial constitutive assumption, we introduce the temperature B as an additional argument in (2.1) and it appears correspondingly as an additional argument of the function

f in

(2.5).

Analogous considerations may be applied if we take as our starting point the constitutive assumption that the stress matrix u is a function of the deformation gradient matrix g and a number

of column matrices Vl, ...,Vp, which are the components in the system x of a number of vectors which rotate with the body. = f(g9V11.-

,VU).

Then, 12.6)

We find, in a manner similar to that previously employed, that a must be expressible in the form =

sf(C,gt V1,.

- - ,gtvU)g + ...,v )

As an example, the column matrices Va (a= 1,

(2.7) may be the

electric field E, the magnetic field H and the temperature gradient VB, or any one or two of these. We also consider the manner in which the electric displacement D, magnetic induction B, and heat flux & depend on E, H , and 78.

We take U = D , B, or & and assume that it is a function

of the instantaneous values of g , E, H and 08, thus:

Then, we find that f must satisfy the relation

for all proper orthogonal a.

From this it follows, by considera-

tions similar to those previously used, that U must be expressible in the form

u 3.

=

f(ggE,a,ve) = gf(c,gi~,gta,g

t

ve).

(2.10)

Stress dependent on the deformation gradient matrix and its time derivatives We shall now assume that the Cauchy stress depends not only

on the deformation gradient matrix g at time t, but also on its time derivatives g (a) (=dag (r)/d.ra 1 T=t) (a = 1,. ,p) at time t,

..

thus : (a=l, ...,FI).

a = f C g . g (a))

(3.1)

We now assume that a time-dependent rotation is superposed on the assumed deformation, so that g(r)

+

a(T)g(T).

Then,

and

Analogously with the derivation of ( 2 . 3 ) , we find that the matrix function f in (3.1) must satisfy the condition

f(9,s(a)) =

atffag, (ag) ( a ) ~ a

(3.3)

for all time-dependent proper orthogonal transformations a(~). In particular, it must be satisfied for the particular choice of a (TI U(T)

Thus, from (3.3),

=

CC(T)I

-4gt (T).

(3.41

Now, (c+)(~) is a function of

C

and its first a time derivatives.

It follows that a must be expressible in the form

Since C(T) is unchanged when an arbitrary time-dependent rotation is superposed on the assumed deformation, no further restriction is imposed on the form of a by the fact that (3.5) is satisfied for aZZ time-dependent proper orthogonal a(~). o is a symmetric matrix,

7

Of course, since

must be a symmetric matrix function of

its arguments. It is sometimes convenient to express c'")

in a somewhat

different form, particularly when the material considered is fluid-like in character.

We define the symmetric matrices Aa by

so that

We may cali the matrices Aa, so defined, the RivZin-Ericksen matrices.

They are the 3x3 matrices formed from the components

in the system x of tensors *A

which are called the RivZin-Ericksen

f ensors.

Differentiating (3.7) with respect to time, we obtain

Using (3.7) with a replaced by a+l, we obtain from (3.9)

whence

Let vi be the components of velocity, in the system x, of the particle which is located at

x

at time t.

Then, defining the

velocity gradient matrix d by d =

Ild..ll 1~ = Ilvi ,j

II

(3.12)

= Ilavi/axjlls

we note that t

and

% = a g

Introducing these relations into (3.11),

4.

(3.13)

=gtdt.

we obtain

Functional constitutive equations We now make the constitutive assumption that the Cauchy

stress is a functional of the deformation gradient matrix, thus:

As before, we find that the functional F must satisfy the relation

where a(r) is an arbitrary time-dependent proper orthogonal transfor~nationfor which a(T)

= 6.

In the particular case when

equation ( 4 . 2 ) yields

F [g (T) ]

=

[IC(r) 1'1

g~-'f

C-'gt .

Thus, a must be expressible in the form a = F[~(T)I

Since

C(T)

=

<.;.sj

~F[cc(T)II~~.

is unchanged when g ( T ) is replaced by a (r)g

(T)

, it is

easily seen by substituting (4.5) in (4.2) that the relation (4.2) is satisfied for arbitrary proper orthogonal a(T). Analogous results can evidently be obtained if we replace the constitutive assumption (4.1) by a constitutive assumption which parallels, in the three-dimensional context, one or other of the one-dimensional constitutive assumptions of the functional type discussed in 58 of Chapter 1.

For example, if paralleling

the one-dimensional constitutive assumption (1.8.31, we assume that

then by arguments similar to those used in arriving at (4.5) and (3.6), we arrive at a =

s~[c(T) ;I=(cd

11.ct

lg

.

(4.7)

The constitutive equations (4.5) and (3.6) can be related in a manner similar to that used in 59 of Chapter 1 in the parallel one-dimensional situation.

We consider deformations for which

C(T) may be expressed as a Taylor series about ~ = t ,thus:

where

It follows that i in (4.5) may be expressed as an ordinary matrix function of the countably infinite set of matrices C ( a ) We define C,,(r)

Let cr

P

.

by

be the.corresponding value of a.

We assume that a is a

continuous functional of g(~) in the sense of the supremum norm

defined in the following way.

Let gl(r)

and g2(~) be the defor-

mation gradient matrices associated with two deformation histories. Then, we define the supremum norm as sup-tr[(g1(')

- 92(T))f-91(') - g2(~)1

t

1

(4.11)

and note that this vanishes if and only if gl (r) = g2 (T) .

-

suppose that a is a continuous functional of g(r) that tr ( u -~ u2)

0 as the expression in (4.11)

+

in the sense 0, where ul and

o2 are the values of o corresponding to the values gl(r) g2(r) of g(~). t ional of C(r)

It follows that

and

in (4.5) is a continuous func-

in the sense of the supremum norm

- C2(~l 2 ,

(4.12)

are the values of C(r)

corresponding to the

sup.tr{C1(~) where Cl ( T ) and C2(r) values gl(s)

We now

and g2(r)

of g(r).

It then follows that we can

approximate F as closely as we please by a matrix function of c ( ~ )(a = 0,1,2,.

..,u) provided

we take u large enough.

Indeed,

hy invoking the Stone-Weierstrass theorem, we can make this

dependence polynomial. Paralleling the discussion in 5510 and 11 of Chapter 1 for the one-dimensional case, we can obtain a representation for

0

for a wider class of deformations than that for which C ( r ) can be

expressed by the Taylor series (4.8).

variable s = f(t-T),

We define a time-like

which is a smooth monotonically decreasing

Function of t-r and which maps the interval r = (-m,t] onto [O , I ] . Then, we may replace (4.5) by

where

We now limit ourselves to the class of deformations for which C(s) can be expressed as a cosine Fourier series with half-period unity, thus

where

Plainly, G may be regarded as an ordinary function of the countably infinite set of linear matrix functionals B~ ( a = O,l, ...). Furthermore, if o is a continuous matrix functional of

g ( ~ )

in the sense of the supremum norm (4.11) and hence G is a continuous matrix functional of ?(s)

in the sense of the supremum norm

sup.trlEl(s)

- ~~(s))~,

(4.17)

we can approximate G as closely as we please by a matrix function of Ba (a= O,l,

...,p )

provided that we take p large enough.

In-

deed, by invoking the Stone-Weierstrass theorem, we can make this dependence polynomial. 5.

Fluids The constitutive equations we have so far discussed are, in

principle, applicable both to solids and fluids. In the area of fluid mechanics, their most important application is to fluids which, to a high degree of approximation, may be treated as incompressible.

Accordingly, in considering the modifications which

may be made in the various constitutive equations, we shall consider only the case of incompressible fluids, thereby avoiding certain complexities which would otherwise arise.

For an incom-

pressible material, whether solid or fluid, the Cauchy stress is

undetermined to the extent of an arbitrary hydrostatic pressure, if the deformation is specified, and the deformation itself must satisfy the constraint that each element of the material remains unchanged in volume as a result of the deformation. Accordingly, for an incompressible material, the constitutive equation (3.6) is replaced by

a

=

*

gf(~,~(~))g~- p6

(5.1)

and the constitutive equation (4.5) is replaced by

a

=

~F[CC(T)II~~- p6,

(5.2)

where p may be arbitrarily chosen if the deformation is specified and 6 denotes the unit matrix. T\e distinction between a solid and a fluid is heuristic in character.

However, there would probably be general agreement

that one feature of a fluid, which distinguishes it from a solid, is that it can undergo deformations which are indefinitely large

while the Cauchy stress remains finite. Accordingly, it becomes convenient to take the constitutive equation for a fluid in a form in which the gradients of the velocity, acceleration, etc. (if they occur in the constitutive equation) are taken with respect to the particle position at time t, rather than with respect to their positions at some fixed reference time. This is achieved by using the relation (3.7) to substitute for c'")

(a~l). Equation (5.1) then takes the f6rm =

sf(C,sfaas)gf -

with an evident change in the meaning of f.

* For

(5.3)

PA,

We now consider a

convenience, we omit the bars over f and F.

*

**

class of materials , which are called simple fluids, for which t gf( )g is independent of g - and hence of C - and accordingly depends only on Aa (a~l). That it is indeed possible to choose f, in the constitutive equation (5.3),

from the following example.

in such a way, is evident

It is apparent that a particular

choice of f in (5.3) is

where the K's are constants.

Noting that

and introducing this relation into (5.4), we obtain, from (5.3),

Thus, for an incompressible simple fluid, the constitutive equation (5.3) takes the form, originally pro.posed by Rivlin and

rickse en' , a = f(A,)

-

p6.

(5.7)

The term simple material implies that the constitutive equation for the stress involves no spatial derivatives, of higher order than the first, of the displacement, velocity, acceleration, etc; nor does it involve other vector or tensor fields.

The

terminology is unfortunate in that one can construct constitutive equations, which

* J. G. Oldroyd, Proc. Roy. ** The terms simple material

Soc. A 2 0 0 , 523-541 (1950).

and simple fluid are due to W. Noll, Arch. Rational Mech. Anal. 2, 197-226 (1958).

+

R.S. Rivlin and J.L. Ericksen, J. Rational Mech. Anal. 4 , 323-425(1955).

(i)

are fluid-like in their ability to undergo indefinitely

large deformations, while the stress remains finite, (ii) (iii)

are simple materials, are not simple fluids, since the stress d o e s depend on the

deformation gradients with respect to a fixed reference configuration. As an example, we may write

where A, B, P, Q are constants. We turn now to the functional relation (5.2),in

which the

gradients of the velocity, acceleration, etc. do not occur explicitly.

In this case, we can avoid the presence of a fixed refer-

ence configuration in the constitutive equation by choosing as the configuration, with respect to which the deformation gradients are measured, the current configuration, i.e. the configuration at time t at which the stress is measured.

and noting that gt(t)=6,

With the notation

the constitutive equation (5.2) takes

the form a = FECtC~)l

-

(5.10)

~ 6 ,

where the matrix functional F does not depend on a fixed configuration.

This form for the constitutive equation for a simple

fluid with memory was originally proposed.by ~oll*. However, the argument advanced by No11 to justify it is incorrect.

He defined

an incompressible simple fluid as one in which the constitutive equation is invariant with respect to arbitrary unimodular trans-

* W. Noll. Arch. Rational Mech. Anal.

2, 197-226 (1958).

formations of the reference configuration.

In essence, Nollls

argument then proceeds in the following manner.

Modifying the

constitutive assumption (4.1) to the case when the material is incompressible, we have

No11 then argues that, for a simple fluid, the functional F should be invariant under unimodular transformations of the reference configuration; thus f

[g (TI I =

f

[ g (TISI

,

(5.12)

for all unimodular S. He concludes that F[g(~)l

F[gt(~)l-

=

*

It was recently pointed out

(5.13)

that, while this is indeed the

case, the restriction on F implied by (5.13) does not exhaust the restrictions implied by (5.12).

Indeed, if the full restrictions

on F implied by (5.12) are developed, it becomes evident that they provide undesirable constraints on the class of fluids covered.

* E.

Fahy and G.F. Smith, pending publication.

Chapter 3 Restrictions on Constitutive Equations Due to Material Symmetry 1.

Description of material symmetry In the last chapter, we have seen that, starting from a

constitutive assumption, we can express the constitutive equation as a relation between tensors in which a dependent tensor is a function of one or more tensor arguments, or a functional of one or more tensor functions.

The tensors involved in these relations

may be either absolute tensors or relative tensors. We shall consider the case when they are absolute tensors.

The modifica-

tions which result if one or more of the tensors involved are relative tensors can then be made with ease. We take as our starting point the situation when a tensor,

2

say, of order P, is a function of N tensor$ E ( A ) (A = 1 , . . . ,N) of orders PA:

-T = -F ( -v ( ~ ) ) . We shall discuss the restrictions which can be placed on the tensor function

11

by any symmetry which the material considered

may possess. Let x and

x

be two rectangular cartesian coordinate systems.

Let T il.. .i and Ti .i P be the components of the tensor ;in P and p i A ) j the systems x and 2 respectively and let V ( A 1 jl. - 3

,. . .

-

be the corresponding components of the tensor the coordinate systems x and that S =

/

x

P~

E ( ~ ) . We shall take

to have the same origin and suppose

ISijl I is the transformation matrix which relates coor-

dinates referred to the systems 2 and x.

Thus, if x = (xi) and

3: = (Xi) are the column matrices formed by the coordinates of a point in the systems x and ? respectively, we have

2

= sx,

or, in cartesian notation, xi=s x ij

(1.3)

3'

S is, of course, a three-dimensional orthogonal matrix and, accordingly, satisfies the relations

where 6 is the unit matrix. The law for the transformation of tensors yields

Referred to the coordinate system x, the relation (1.1) may be written as

and, referred to the system g, it may be written as

where F i,.. .ip ( ) are, in general, different tenP sor functions of their indicated arguments. However, if Pi, .. . P

is the same function of v ( ~ )

as is Fil..mi of dj,A..) .j ' j,.. .j P P~ then the relation (1.6) is said to be f o r m . - i n v a r i a n t with respect to the transformation S and the coordinate systems x and

x

are

said to be e q u i v a l e n t coordinate systems. We now consider two different sets of values of the argument tensors

say

'

and

corresponding values of the tensor

and let

z.

r'

and

T*

Let ~1")' l...i P~

be the

and T;l...i P

2' in the system x and let and f * il. - . i pbe the components of 1( A ) * and g* in the

be the components of -(A)* vil...i

1( A ) '

and

P. A

system 2.

Then, if x and

x are equivalent coordinate systems,

we have

and

Now, if we choose

and v ( ~ ) so * that

A

(A)'

v ~ l -. .j then

P~

In physical terms, this means that if the argument tensors

v ( ~ ) have * the same relation to the system x as do the argument tensors ( " I ' to the system x, then the dependent tensors and

z*

T' correspondingly have the same relation to the systems -

2 and x

respectively. Let 2 and 7 be two rectangular cartesian coordinate systems, each of which is equivalent to the system x.

Then, it is evident

that they are equivalent to each other. In order to describe the symmetry of a material in the neighborhood of a particle P, say, we proceed in the following manner.

We choose a rectangular cartesian coordinate system x,

say, with its origin at P.

Let 2 and

be any rectangular car-

tesian coordinate systems equivalent to x. is, of course, equivalent to itself.) tion matrix relating S

to 2.

(We note here that x

Let S be the transforma-

The set (Sl of all transformations

obtained by making all possible choices of x and 2 form a group,

which provides a description of the material symmetry in the neighborhood of P, referred to the frame x (or any of the frames equivalent to x). Now let us suppose that a different choice is made for the coordinate system x - a coordinate system y, say, inequivalent to x.

Let y and x be related by y = Rx.

Let

x be

a frame equivalent to x and related to it by

-

(1.12)

x = sx.

Then, it can easily be seen that the frame f , related to 2 and y by

-y

-

= R; = RSR 1y,

is equivalent to y. The set {RSR-'1

of all transformation matrices relating

pairs of franies equivalent to y form a group which provides a description of the material symmetry in the neighborhood of P, referred to the frame y (or any of the frames equivalent to y). It is easily seen that the sets of matrices IS1 and I R S R - ~ I have the same multiplication table; i.e. if S1 and S 2 are two matrices of the set is1 and S1S2 = S 3 , then (RS~R-') ( R S ~ R ' ~ )= RS~R-'. Thus, the sets of matrices IS] and IRSR-'1 t i o n s of the same a b s t r a c t g r o u p .

are m a t r i x r e p r e s e n t a -

This abstract group is said to

be the symmetry group o f t h e m a t e r i a l in the neighborhood of P. 2. -

Restrictions on the constitutive equation imposed by material symmetry in implicit form If x and 2 are equivalent coordinate systems, then (1.6) and

(1.7) become

With (1.5)1, we obtain from (2.1)

where o ( ~ )

jl-.- J

and

jl.

- -3

are related by (1.5)*.

The relation

(2.2) must, of course, be valid for all transformations of the group ( S l which describes the symmetry of the material.

It pro-

vides, in implicit form, the restrictions imposed by material symmetry on the manner in which sors

can depend on the argument ten-

-v ( ~ ) .Our next task is to make this restriction explicit. In order to do this, we convert the relation (2.2), which is Let

a tensor relation, into a scalar relation. the same order as

2. Let ll,il...i and P

2 be a tensor of

Gil...i P be

of ~ + n the systems x and 2 respectively.

the components

Then,

In inverse form, this relation can be written as

We multiply (2.2) throughout by 'il..

F

. i p il..

il.. .i

.

.iP ('J~.- ( A ) .j

.

= i . . i

F

and use (2.4) to obtain P

)

(A) i . . iP (Vjl.. .j

.

) = F

(say).

(2.5)

P~

The relation (2.5) expresses the fact that F is a scalar invariant function of the tensors

and 2, linear in the latter, for t b c

group of transformations {S). We can, accordingly, use results

in the theory of invariants to make statements regarding the form which F must take and then recover corresponding statements about the manner in which

may depend on the argument tensors

by

using the relation

If

',

and hence g, has some symmetry with respect to inter-

change of indices on its components, it is usually advantageous, although not essential, to choose the tensor J, with the same symmetry with respect to interchange of indices on its components, i.e. to let suppose

'

J,

be a tensor of the same k i n d as g.

For example,

is symmetric with respect to interchange of the Kth and

Lth indices on its components, then we choose 9 to be symmetric with respect to interchange of the Ktli and Lth indices on its components. We may then replace (2.6) by

3.

Some invariant-theoretical concepts Before progressing further, we shall digress to introduce

some of the basic concepts of the theory of invariants. Let I be a scalar polynomial in a number of tensors which is invariant under a group of transformations { S l . Noether

*

It was shown by

many years ago that if the group IS) is finite there

exists a f i n i t e set of polynomial scalar invariants J1,

...,J A ,

say, such that I can be expressed as a polynomial in these. analogous theorem was later proven by

* See, for example, H. Weyl, Univ. Press (1946).

* Hilbert ,

An

subject only to

The CZassicaZ Groups, Princeton

certain technical restrictions which will not concern us here, in the case when the group IS) is a Lie group.

The set of invar-

iants J1, ...,J X , in terms of which an arbitrary polynomial invariant can be expressed as a polynomial, is called an i n t e g r i t y basis.

It depends, of course, on the group considered and on the

number and orders of the argument tensors.

If we eliminate from an integrity basis any element which can be expressed as a polynomial in the remaining ones, we are left with an i r r e d u c i b l e or m i n i m a t integrity basis. Let J1, ...,JX be an irreducible integrity basis for some set of tensors and some group.

Although none of the elements of the

set Ji, ...,J A is expressible as a polynomial in the remaining elements, there may nevertheless exist implicit polynomial relations between them of the type

Such relations are called s y z y g i e s .

It follows from a theorem

due to Hilbert (the Second Main Theorem)

*

that all of these

syzygies are algebraic consequences of a finite number of them. It is, of course, evident that an irreducible integrity basis is not unique.

For, if J1, ...,J A form an irreducible in-

tegrity basis, so do any X linearly independent linear combinations of these. We now consider that I is an invariant scalar function of the argument tensors, rather than an invariant scalar polynomial. It will be shown in 5 5 that if J1,

...,JA is

for the set of tensors and the group

*

See footnote on previous page.

{S},

an integrity basis

then I may be expressed

as a function of J1,...,JX.

Any set of invariants in terms of

which an arbitrary invariant may be expressed as a function is called a function basis. function basis.

Thus, an integrity basis is also a

A function basis is said to be irreducible if

none of its elements is expressible as a function of the others It is evident that although an irreducible integrity basis is a function basis, it is not necessarily an irreducible function basis, since it may be possible, by the use of a syzygy, to express one of its elements as a function of the others. 4.

Canonical forms for constitutive equations Let us assume that, in our initial constitutive assumption

(1.1),

2 is a tensor polynomial in the argument tensors

Then, in (2.5),

-v ( * ) ,

.

F is an invariant polynqmial in the tensors 9 and

linear in the former.

Now, suppose that J1,.

irreducible integrity basis for the tensors group (5'1.

[( A )

9

. . ,JX is

and

an

and the

Let 11, ..., I v be those elements of this integrity

basis which are independent of ments which are linear in form

9.

9 and

let K1, ...,K

?J

be those ele-

Then, F must be expressible in the

!J

F

1 Aa(I1,..-,IpO, a=1

(4.1)

I

where the A's are invariant polynomials in 11, ...,Iv.

From (2.6),

we obtain

We note that the tensors aKa/a$il...ip, which are of course independent of

9, are

also independent of the polynomial F.

They do,

however, depend on the choice of the irreducible integrity basis for the M + 1 tensors 2 and I J ( A ) , which is not unique.

Similarly,

the arguments 11, ...,I v of Aa also depend on the choice of this irreducible integrity basis. It will be shown in 56, that if, in (l.l), function of

2

is a tensor

rather than a tensor polynomial, the relation

(4.2) is still valid, if the A's are invariant functions of 11,

...,I V

rather than invariant polynomials.

We have reduced the problem of determining a canonical form for the tensor g in (1.1),

which expresses the restrictions im-

posed on it by material symmetry, to that of determining an integrity basis, preferably irreducible, for an appropriate set of tensors and an appropriate group of transformations.

The

techniques for finding such integrity bases differ significantly accordingly as the group is finite or not.

Some of the techniques

which have been developed for finite groups will be outlined in Chapter 4 and for non-finite groups in Chapter 5. 5.

Function basis In this section we shall prove the theorem, due to Pipkin

*

and Wineman, that an integrity basis is necessarily a function basis.

The proof given here is that for a non-finite group.

With a slight modification, which is indicated, it also provides a proof in the case when the symmetry group is finite.

For finite

symmetry groups, a simpler proof than that given here is available

*

. Let 11,

v"!. -

..

...,I y

be an integrity basis for the N tensors

and the group of transformations { S h ] , where 1 is a

set of parameters which identify the individual transformations

* A.C. Pipkin and A.S. Wineman, Arch. Rational Mech. Anal.

.iZ,

420-426 (1963); A.S. Wineman and A,C. Pipkin, Arch. Rational Nech. Anal. 1 7 , 184-214 (1964).

We may, without loss of generality, take X to be

of the group. such that

Let vO be a column matrix whose elements are the independent components of the N tensors

(A = 1,.

.. ,N)

in the reference

frame x0,, say, with respect to which the group of transformations is defined.

Let xX be the reference frame equivalent to xo, such

that ZA =

SXxO.

Let v,, be the column matrix formed by the components in the system x,, of the N tensors

5 , arranged

in the same order as

their components in the system xo are arranged

*

in v 0 '

The column matrices vX and vo are related by a transformation matrix s,, thus:

ca

=

S~vo,

(5.3)

where s h is a square matrix of dimension equal to the total number of independent components in the set of tensors

The

set of matrices {sh) form a group which has the same multiplication table as the group of transformations if

SaSg = S

Y'

then

s s

a 8

isX); i.e. = s

Y

.

We define the o r b i t of v o as the set of column matrices { v , , )

*

As an example, suppose that N = Z and v ( ~ and ) are firstorder and symmetric second-order tensors respectively. Let i and v ! ~be ) their components in the system x and let 911) 1J and t j 2 )be their components in the system rX. We could take A 3 as vo the nine-rowed column matrix

Then Y,, would be the nine-rowed column matrix obtained from this by placing bars over the V's.

given by ( 5 . 3 ) . Let v be any column matrix in the orbit of vo. 11, ..., I

Since

is an integrity basis, we have

Ia (a = 1 , ...,v ) will be a function basis if and only if the set of equations I,(v>

=

I,(v0)

( a = l ,...,v )

(5.6)

has as solutions only the column matrices v given by

...,v )

i.e. only if Ia (a = 1,

determine the argument tensors com-

pletely apart from a transformation of the symmetry group. In order to prove that (5.6)

does imply (5.7),

we first de-

fine the distance D * ( V ; V ~ ) of v o from the orbit of v by

t

(5.8) D R ( v ; v O ) = min. { ( s A v - v o ) ( s X v - v O ) } . X We note that D* so defined is an invariant of v and is also a con-

tinuous function of v.

From the Weierstrass theorem, it can be

approximated by a polynomial in v , say P ( v ; v o , ~ ) , such that for any E > O IP(v;v~,E) - D * ( v ; v ~ )1 <

in a bounded region. iant.

E

(5.9)

Of course, P is not necessarily an invar-

We can, however, construct from it an invariant polynomial

P* which approximates D* equally well.

This is achieved by inte-

gration over the group (Hurwitz integration), thus

We have thus constructed an invariant polynomial P* in v , such t ha t

/P*(V;V~,E)- D*(V;V~)/ <

E.

Now suppose v does not lie on the orbit of vo. D*(v;v0)

=

We have

D, say, $ 0,

and, evidently,

D* (vO;vO) = 0. We choose E < D / ~ . Then,

Accordingly, we have constructed an invariant polynomial which takes different values on the orbits of v and vo.

Since P* is an

invariant polynomial in v , it must be expressible as a polynomial in the elements of the integrity basis, I& (a= 1,...,v ) .

It

follows that at least one of these elements must take different values on the orbits of v and vo; i.e. equations (5.6) have as solutions for v only (5.7) and we have proven our theorem: an integrity basis is also a function basis. The result can also be obtained for finite groups if, in the argument given above, we replace the definition of P*(v;v~,E) in (5.10) by summation over the group, thus: l!J-l

P*(v;v~,E) = where

- 1 P(S~V;V~,E),

(5.15)

vx=o

u is the order of the group and s A (X=O,l, ...,p- 1) are the

transformations which transform vo into vA (A = O,l,..'.,p-1). course, in this case the orbit of vo consists of the

!J

Of

discrete

column matrices vA. 6.

Constitutive equations of the function type We now suppose that, in the constitutive equation (1.1),

is a function, not necessarily polynomial, of v ( * ) (A = 1,. .. ,N) .

As in 55, let v X be a column matrix whose elements are the independent components of all of the tensors

in the reference

frame xi and let t X be the column matrix Eormed from the independent components of

2

in this reference frame. Then, in the frame

xX, we may write the constitutive equation (1.1) as

where the function f is the same for all frames equivalent to a frame xo with respect to which the symmetry is defined. Let to and vo be the values of t X and vA in the frame xo.

Then, as in

55, we see that vi and vo and t A and to are linearly related thus: vA

=

shvO; t X

= .

rXtO.

(6.2)

.

We have seen in 54 that if, in (1.1),

P is a polynomial, the

restriction on the form of the equation implied by material symmetry can be expressed by (4.2).

In our present notation, this

can be rewritten in the coordinate system xX as

.

where the column matrices f(a) (a = 1,. . ,u) are polynomials in vX, which are independent of X and of the particular form of F_. Also, 11, ...,Iv are invariants of v A under transformations of the group CsX1 and Aa (a= 1, ...,u) are polynomials in these. We shall now show that if

g

in (1.1), and hence f in (6.1), is a

function of its arguments, rather than a polynomial, the canonical form (6.3) is still valid, where Aa (a = 1,. . . ,u) are now functions of 11,...,IV. In order to do this, we note that f varies continuously with v on an orbit and invoke the Weierstrass theorem to assert that

for any specific orbit there exists a column matrix p , which is a

polynomial in v and approximates f as closely as we please; i.e. for any E>O, there exists a polynomial column matrix p(v;e) such that {(P-f)'(P-fIl*

< E.

Of course, this column matrix is not necessarily form-invariant under the transformation sA. We can, however, construct from p a polynomial column matrix p* (v;c) which is form-invariant and approximates f with any desired accuracy on a specified orbit, thus : p*(v;~)

=

I'

r A p(sAv;c/m 2)dh,

(6.5)

where m is the number of rows in the column matrix p.

With p *

defined in this way, we have {

(p*-f)i(p*-f)l*

on the orbit considered.

Since p

*

< E

is form-invariant under the

transformations of the group IsA] and since 11, ..., I

lJ

are con-

stants on an orbit, we have, on each orbit,

where the A's are constants which depend on.the orbit considered and on E.

The column matrices f(a) (a) span a complete real vec-

tor space R, say, of finite dimension 3.

(The dimension of the

space will be less than 1.1 if the f's are linearly dependent.)

In

view of (6.6) we can construct a Cauchy sequence of polynomials P*(v;~n) (n=1,2,...) which has f as its limit and, from ( 6 . 7 ) , the elements in this sequence lie in R.

Since R is complete, it

follows that f must lie in R, i.e. f must be expressible in the form

on each orbit, where the A's are real constants which depend on the orbit considered and may accordingly be regarded as functions of 11,

..., I,.

This is the desired result.

This result is also valid for finite groups and may be obtained by the argument given above by replacing (6.5) by

with the notation of (5.15). i

e We now consider that the argument tensors

are functions of time

T

(T)

. .. ,N)

(A=l,

and that the value at time t, say, of the

dependent tensor g in a constitutive equation depends on the values of ll(A)(~) for all r ~ t . Then we may say that E(t) tensor functional of the tensor functions

is a

v ( * ) (T) with support

-m<~lt, thus:

Effectively, we are regarding T(t)

as a tensor function of the

infinity of argument tensors obtained from V ( A ) ( r )by giving

T

all possible values in the range (--,t]. We can obtain, in implicit form, the restrictions imposed by material symmetry on the form of the functional

in a manner

similar to that used in the case when the dependence of

2 on the

argument tensors is of the function, rather than the functional, type.

We find that

must satisfy the relation

with

for all transformations I S 1 of the group describing the material -(A) symmetry. Vjl.. .

and V

(T) are, of course, the comP~ ponents of I(T)in equivalent coordinate systems 5 and x related (T)

jl.. - 3

by the transformation S. As before, we introduce an auxiliary tensor the same kind at 2, with components

9, preferably

of

qi

.ip and qil.. .ip in the coordinate systems 5 and x respectively, so that

Then, multiplying (7.2) throughout by

5il.. .ip , we

obtain, with

(7.41,

This relation expresses the fact that the scalar functional 0 is invariant under the group of transformations (s}. When canonical form has been given to for

@,

we may recover a canonical expression

by means of the relation

This expression for Til.. .ip will, of course, be modified in an evident manner, paralleling that for the case of function dependence discussed in 92, if Til.. .i possesses some symmetry with P respect to interchange of subscripts. In order to obtain a canonical form for the scalar functional @

and hence for the tensor functional

2, we

digress in 58 to

discuss in general terms invariant scalar functionals. 8.

Invariant functionals Let I

= 1 [v ( * )( T ) ]

functions

be a scalar functional of the M tensor

( r ) ( A = 1,. . . ,M) which is invariant with respect to

the transformations of a group C S X l , where h is a set of parameters, the values of which identify the individual transformations of the group. Let v 0 ( r ) be the column matrix whose elements are the inde-

pendent components, in a rectangular cartesian coordinate system x

0'

of the set of tensor functions V ( A ) (r)

.

Let v X( r ) be the

analogous column matrix whose elements are the corresponding independent components of the set of tensor functions the equivalent coordinate system xA.

in

We suppose that the systems

x X and xo are related by x x = sX x 0

and that v X ( = )and v o ( ~ are ) related by

The elements of

V ~ ( T )and

v X ( r ) may be regarded as the components,

in two different reference frames related by the transformations sX, of the vector function

X(T).

We may regard I as a functional of ~ ( r which ) is invariant under the group of transformations ( s X ) , thus

We may also regard 1 as an invariant function of the infinite set of values of ~ ( r obtained ) by giving

T

all possible values in the

support range of ~ ( r ) . We are accordingly led to consider in the next section the

integrity basis for scalar invariants of N vectors E(T=), t((~~),

..., v(rN), 9.

where N is arbitrarily large.

Peano's theorem We consider N tensors I(A) (A = 1,.

. . ,N)

of the same kind.

Let vLA) be the column matrix whose elements are the independent components of

in some rectangular cartesian coordinate sys -

ten xo and let viA) be the column matrix whose elements are the corresponding components in an equivalent rectangular cartesian system xh. Let Sh be the transformation relating the systems x; and xh and letshbe the corresponding transformation relating the column matrices vLA) and v:"),

so that

Let n be the number of rows in each of the column matrices vjA).

Then the elements of vLA) and vjA) can be regarded as the

compohents in n-dimensional reference frames, related by the transformation sly of an n-dimensional vector (A)

.

It is evident that a polynomial invariant of the N tensors Ir(A)

-

(A= 1,...,N), with respect to the group of transformations

isl}, may also be regarded as a polynomial invariant, P say, of the N vectors 2(A), with respect to the group of transformations Ish}.

Then,

P

=

p(vlA))

=

P(vo( A ) )

It is easily shown that

Thus, the polynomial on the right-hand side of (9.3) is an invariant.

The process by which this invariant is formed from P

is called poZarization. Peano's theorem states that an integrity basis for (A = 1,. .. ,N) - and hence for 41(A)

(A)

(A = 1,. . .N) - where N is arbi-

trarily large, can be formed from that for any n of the vectors v(A), say u (1),.. ,v(n), by polarization. Mo-reover, if the Gram determinant [u(1),. .. 91.' (n)] is an invariant, then the integrity

.

basis can be formed, by polarization, from this and the integrity basis for n-1 of the vectors.

If

[v(1).,...,r(n)]

can be expressed

as a polynomial in the elements of an integrity basis for sets of n-1 vectors selected from the n vectors, then it can be omitted from the set of invariants from which an integrity basis for an arbitrary number of the vectors can be obtained by polarization. We consider the integrity basis for an arbitrary number of first-order tensors.

Since a first-order tensor has three inde-

pendent components, it follows from Peano's theorem that the integrity basis for an arbitrary number of first-order tensors can be obtained, by polarization, from that for three.

Further,

in the case of the proper orthogonal group, or a sub-group of it, since the scalar triple product of three first-order tensors is an invariant, an integrity basis for an arbitrary number of first-order tensors can be obtained by polarization from that for two tensors and the scalar triple product of three tensors. An irreducible integrity basis for two first-order tensors v ( ~ )(A = 1,2), with respect to the proper orthogonal group, is provided by

*

v ( ~ ) . ~ ( ~(A,B ) = 1,2;A
*

We use the notation I(A)-i(B)

=

v(~)~o(~).

(9.41

We denote the scalar triple product of three first-order tensors

) 1,2,3) -v ( ~ (A=

by

111 ( 1 )9 1 ( 2 ) v ( 3 ) ] .

-

(9 5)

9-

(A = 1,. . . ,N) an integrity

Then, for N first-order tensors

basis can be obtained from (9.4) and ( 9 . 5 ) , by polarization, in the form

It is trivially evident that this is irreducible.

In the case of the full orthogonal group, an irreducible integrity basis for three first-order tensors

(A = 1,2,3) is

provided by and, from Peano's theorem, it follows that an integrity basis for

N first-order tensors (9.7)

(A = 1,.

.. ,N)

can be obtained from

, by polarization, in the form

.

v ( ~ ) - ~(A,B ( ~= l,Z,. ) . ,N;A
(9.8)

If v ( ~ (A ) = 1,. . . ,N) are second-order tensors, it follows from Peano's theorem that an integrity basis may be obtained, by polarization, from that for nine of them.

If they are symmetric

tensors, the integrity basis may be obtained from that for six of them.

However, a somewhat more subtle application of the

theorem enables us to reduce these numbers to eight and five respectively provided that the group considered is the full orthogonal group or a sub-group of it. Let us consider the case when the tensors are symmetric. Let

v ( ~ be) the symmetric matrices formed by the components of

v ( ~ in ) some rectangular cartesian coordinate system. We note that tr v ( ~ is ) an invariant. We can therefore replace our problem by that of determining an integrity basis for the N traceless tensors defined by

This, together with tr v ( * ) (A = 1,. basis for the N tensors.

..,N) , will provide an

integrity

Since a three-dimensional traceless

tensor has only five independent components, we can determine an integrity basis for N such tensors, by polarization, from an integrity basis for only five of them. 10.

Tables of typical invariants We have seen, from Peano's theorem, that an integrity basis

for an arbitrary number of tensors of the same kind may be obtained, by polarization, from the integrity basis for n, or in certain circumstances n-1, of the tensors. integrity basis by J. highest degree be basis for N(>&)

n.

We denote this

Let the degree of the invariants in J of Then, it can be seen that an integrity

tensors of the. same kind can be obtained from

that for 6 of the tensors by s u b s t i t u t i o n of all possible selections of 5 tensors from the N tensors. Moreover, if the integrity basis for

n

tensors is irreducible, then the integrity basis for

N tensors, so obtained, will also be irreducible. It is, of course, evident from Peano1s theorem that an integrity basis for 6 tensors can be obtained from the integrity basis J for n tensors by polarization.

However, even if J is

irreducible, the integrity basis so obtained is not necessarily irreducible.

An irreducible integrity basis can be obtained from

it by omitting redundant tensors. We note that the irreducible integrity basis for the 5 tensors will itself consist of irreducible integrity bases for each selection of n-1 tensors from the

n tensors, together with

invar-

iants involving all ii tensors. Each of the integrity bases for n-1 tensors will consist of irreducible integrity bases for each selection of 6-2'tensors from the 5-1 tensors, together with invariants involving the n-1 tensors, and so on. We can construct a table, which is called a table of typicaZ invariants, consisting of an irreducible integrity basis for one tensor, the invariants in an irreducible integrity basis for two tensors which involve both of the tensors, the invariants in an irreducible integrity basis for there tensors which involve all three of the tensors, tegrity basis for

...,

the invariants in an irreducible in-

n tensors which

involve all

i of the tensors.

The table of typical invariants will, of course, depend on the kind of tensors and the symmetry group considered.

From such a

table of typical invariants, we can construct an irreducible integrity basis for an arbitrary number, N say, of tensors of the same kind.

This is achieved by substituting in the table of

typical invariants all possible selections of 1,2,. . . ,ii tensors from the N tensors. We now consider NI tensors kind, N2 tensors !:")(A= tensors

(A = 1,. . . ,N1) of the same

1 ,...,N2) of the same kind,

v ( ~ (A ) = 1,. . .,NX) -x

of the same kind.

... , Nx

In this case also

we can construct a table of typical invariants from which we can obtain, by substitution, an irreducible integrity basis for a set of tensors consisting of arbitrary numbers of tensors of each kind.

The argument j u s t i f y i n g t h i s proceeds i n t h e f o l l o w i n g manner.

...,nX

Let nl,n2,

be the.number of independent components

i n a t y p i c a l t e n s o r o f t h e f i r s t , second, tively.

..., x t h .

kinds respec-

Then, analogous.ly w i t h Peano's theorem, a s s t a t e d i n 99,

we can c o n s t r u c t , by p o l a r i z a t i o n , an i n t e g r i t y b a s i s f o r t h e

...,N X

c a s e when N1,N2,

b a s i s f o r nl,n2,...,nX kinds respectively.

a r e a r b i t r a r i l y l a r g e , from an i n t e g r i t y t e n s o r s of t h e f i r s t , second,

..., x t h .

( I n p a r t i c u l a r c a s e s , one o r more of t h e s e

numbers may b e reduced by u n i t y . )

Let ;a

( a = l , ...,x)

be t h e

degree i n t e n s o r s of t h e a t h . kind of t h e i n v a r i a n t , i n the l a t t e r i n t e g r i t y b a s i s , o f h i g h e s t d e g r e e i n t e n s o r s of t h e a t h . kind.

Then, an i r r e d u c i b l e i n t e g r i t y b a s i s f o r a r b i t r a r y numbers

N1,N2,...,N

X

of t e n s o r s of t h e f i r s t , second,

...,

x t h . k i n d s can

be o b t a i n e d by s u b s t i t u t i o n , i n an i r r e d u c i b l e i n t e g r i t y b a s i s

-

-

for nl,n2,...,hensors X

of t h e f i r s t , second,

..., x t h .

kinds,

a l l p o s s i b l e s e l e c t i o n s o f i i l , ~ , . . . , ~ e n s o r sfrom t h e N1,N2, X N tensors. X By an argument analogous t o t h a t p r e s e n t e d i n d i s c u s s i n g t h e

...,

c a s e when X-1, i t f o l l o w s t h a t we c a n c o n s t r u c t o u r d e s i r e d t a b l e of t y p i c a l i n v a r i a n t s by o m i t t i n g from o u r i n t e g r i t y b a s i s t h o s e i n v a r i a n t s which c a n be o b t a i n e d from t h e remaining ones by substitutions. 11.

Canonical form f o r i n v a r i a n t f u n c t i o n a l s We r e t u r n now t o t h e d i s c u s s i o n of i n v a r i a n t f u n c t i o n a l s

commenced i n 58.

In o r d e r t o s i m p l i f y t h e d i s c u s s i o n , we w i l l

suppose t h a t I i s an i n v a r i a n t f u n c t i o n a l of a s i n g Z e t e n s o r function

E(r)

w i t h r e s p e c t t o a group of t r a n s f o r m a t i o n s S = iSA19

where X i s a s e t of p a r a m e t e r s i d e n t i f y i n g t h e i n d i v i d u a l

transformations of the group.

Thus,

7 = r[I(T)].

Let vo(~) be the column matrix whose elements are the independent components of

in some rectangular cartesian coordi-

!(T)

nate system xc and let vA(~) be the corresponding column matrix formed from the components of

I(T)in

the equivalent coordinate

system.xA,related to xo by XI =

Shxo.

(11.2)

We then have (cf. 08) vA (TI = sAvc( T I 7 where n

=

(11.3)

isA] is a representation of the group S.

Let 11, ...,,I, be the elements in a table of typical invar-

...,

iants for the arbitrarily large set of tensors V(T~),~(T~),

...,

V(-rN), where T ~ , T~ are arbitrarily chosen values of T in the support range of

! ( T )

in (11.1).

We suppose that this table of

typical invariants can be written in terms of only p of the set We may then write

of tensors, say V(T~),~(T~)~.. .,I(T~). Ia

=

Ia<~I(~l),...,vI (T P)I= Ia
.

(a= I,.. ¶v).

*

It has been shown

(11.4)

that the invariant functional I must be

expressible as a functional of Ia(cc=l, ...,v) if these are regarded as functionals of the set of p functions v ~ ( T ~ ).,. . , V ~ ( T ~ ) , where the arguments T ~... ,

JP

are considered as independent

variables each with the same range as

T

in (11.1).

From relations of the type

* A.S.

Wineman and A.C. Pipkin, Arch. Rational Mech. Anal. 17, 183-214 (1964).

where 6 ( ) is the Dirac delta function, it is evident that any of the invariants 11,...,I", which is nonlinear in any of the p argument functions v A (rl),

. .. ,vA (rP) , can be

expressed as a

functional of an invariant of the set which is not.

Accordingly,

we may express 5 as a functional of the elements of the set of

...,IV which are of first degree in each of the column matrices vA (T~),...,v (r ) on which they depend. A P invariants 11,

A similar result applies if I is assumed to be an invariant functional of a number of tensor functions.

Then, I must be

expressible as a functional of the invariants in a table of typical invariants for arbitrary numbers of tensors of the same kinds as each of the tensor functions. 12.

Canonical form for tensor functionals We consider a constitutive equation in which ~ ( t )is a tensor

functional of the tensor function I(') with support - m < ~ l t ,thus: t

We have seen in 57 that the restrictions imposed on this constitutive equation by material symmetry, with respect to the group of transformations (SA.l, can be reduced to the problem of determining a canonical form for a scalar invariant, @ say, which is a func-

tional of v(r)

and an ordinary linear function of a tensor

of

the same kind as ;(t). As in 511, let Ia(a

=

1,

...,o )

be the multilinear invariants

in a table of typical invariants for an arbitrarily large set of tensors

I(T~)

,. .. .

Also, let Ja (a=].,.

. ., p )

be the multi-

linear invariants in a table of typical invariants for

and the

set of tensors !(T~),~(T~)

?.

. ..

*

It has been shown that Q may be

expressed as a functional of 11, ...,. I of J1,...,Jp, where

... s

and a linear functional

are considered as independent var-

T ~

iables each with the same range as

T

in (12.1).

Also, the T'S

occurring in the expressions for the various 1's and J's are all treated as independent variables. From this canonical expression for Q, a canonical expression for g(t)

can, of course, be obtained by differentiation with

respect to

(cf. (7.6)).

Analogous results are valid if z(t) is a functional of a number of tensor functions, rather than of a single tensor function. Similar results were obtained earlier the tensor functional

**

in the case when

can be expresseil as a polynomial in linear

tensor functionals of the argument functions (cf. 54 of Chapter 2).

*

A.S. Wineman and A.C. Pipkin, Arch. Rational Mech. Anal. 17, 184-214 (1964).

* * A.E.

Green and R.S. Rivlin, Arch. Rational Mech. Anal., I, 1-21 (1957).

Chapter 4 Integrity Bases for Finite Groups 1.

The basic theorems In the case when the symmetry group is finite, the determina-

tion of an irreducible integrity basis for any number of tensors of any orders depends essentially on the application of three simple theorems on integrity bases for n-tuples under permutation groups.

These are:

T h e o r e m 1:

tuples.

Let y = (yl '.

.. ,y,)

.

and z = (z17.. ,zn) be two n-

be a polynomial in y and z which is symmetric

Let F(y,z)

under interchange of y and

Then F must be expressible as a

z.

polynomial in Yi Theorem 2 :

z i , yizj

+

Let y i

=

+

yjzi

(i,j=17...,n ; iLj).

(Y1(i) ,Y2(i),Y3 ( i ) ) (i-1,. . . ,n) be n triples.

Any polynomial in these which is invariant under interchange of any pair of subscripts is expressible as a polynomial in

where T h ~ o r e m3 :

(i,j,k = 1

,.-.,n).

Let yi = (yli) ,yii) ,yii))

(i=l,. .. ,n) be n triples.

Any polynomial in these which is invariant under cyclic permutation of the subscripts is expressible as a polynomial in the quantities listed in theorem 2, together with

where

(i,j,k

=

1 ,..., n).

In determining the irreducible integrity basis for a finite number of tensors and a finite group, it is comforting to be able to set an upper bound on the number of elements in this integrity basis.

This can be done by means of Noether's theorem which

states that if p is the order of the group and n is the number of independent components in the set of tensors considered, the degree of each element in an irreducible integrity basis is not greater than p and the number of elements in the basis is not greater than (p+l) (p+2).

. . (p+n)/n!

In practice, the actual number

is usually much less than this. For the simpler crystallographic groups, integrity bases can be obtained by direct application of theorems 1, 2 and 3 provided that the number of tensors involved is not too large, nor their order too high.

In cases where direct application of theorems 1,

2 and 3 becomes too cumbersome, theorems in group theory can be used to recast the problem in a form in which the algebraic manipulations involved in using theorems 1, 2 and 3 become less cumbersome. We shall first discuss a simple case in which theorem 1 can be used to obtain the desired integrity basis and shall then give, as an illustration, a case in which, although theorems 1 and 2 can be used to give us the desired integrity basis, some economy emerges from the use of group-theoretical considerations to recast the problem in a form in which the algebraic manipulations involved in using theorems 1 and 2 are somewhat simpler.

2.

Determination of the integrity basis in a simple case We shall, as a first example, obtain an irreducible integrity

basis for a first-order tensor tensor

and a second-order symmetric

under the group of transformations I S a ) of order 4 de-

fined with respect to a frame xl by the diagonal matrices

Let y ( l ) be the column matrix formed by the components of the first-order tensor in the frame xl.

Let T ( ' )

v:')

be the 3 x 3

matrix formed by the components ~ ( 1 of ) the tensor 2 in this frame. 13

Let (

1 =

T(E)

5ar7(1)St a'

;

(2.2)

Let v C a ) be the nine-rowed column matrix whose elements are the independent elements of e.g.

,( a )

,

a , v a , v a

(vl

and T C a ) arranged in some order; (a) T(a) 5T22 , 3 3 (2.3)

For brevity we will write vr)= vA ' We then have

We note that

S4 = S 2 S 3 ,

so S 2 and S 3 form a set of generators

for the group and invariance under these transformations guarantees invariance under the group. It is seen that

V1?V4,V5,V6

(2.51

are invariant under S2 and S3 and hence are elements of the integrity basis. Also, we note that Sly S2 forms a sub-group of

is,}.

We first determine an irreducible integrity basis for this

sub-group. Then, applying theorem 1 with the y's and z's identified as

respectively, we see that any polynomial in )'(v

and T(~), in-

variant under the sub-group S1, S2, is expressible as a polynomial in (2.5) and (2.7)

v 3Yv82v2 2' v2 7' v2 9,V 2- V7Yv2Vg,vpv9.

Here we have omitted the evidently redundant terms v2 vi and 3' v v which are also generated by application of theorem 1. 3 8. We now examine the effect of the transformation S on (2.7). 3 he terms

are unaltered and are accordingly elements of the integrity basis.

The remaining terms v 3,v8,v2vT'v7v9

be come -v3,-v8,-v2V7,-vv 7 9'

(2.10)

Again, identifying the y's and 2's in theorem 1 with (2.9) and (2.10) respectively, we generate the invariants

where we have drawn a slash through those elements which are redundant.

The remaining elements plainly cannot be expressed as

, (2.8) and (2.11).

polynomials in the remaining elements of (2.5) Hence ( 2 . 5 ) ,

(2.8) and the unslashed elements in (2.11) form an

irreducible integrity basis for the tensors

1

and

T

under the

group considered. Some results from group representation theory

3.

While the procedure adopted in the previous section for determining an irreducible integrity basis can i n p r i n c i p l e be used to find the integrity basis for any number of tensors (whether symmetric or not) of any orders, for any finite group, it becomes prohibitively cumbersome when the number of indepen-

dent components in the set of tensors becomes large and when the group has high order. In such cases somewhat more sophisticated methods are desirable.

These depend, for the most part, on theorems in the theory

of matrix representations of groups.

We shall accordingly recall

some of these theorems in the present section. An abstract group E is a set of abstract elements C1, ... ,=,J for which an operation called m u l t i p Z i c a t i o n is envisaged, which has the property that if any two elements of the group, Ca and C

8

say, (which may be the same element), are subjected to this

operation, their p r o d u c t , denoted E3CB, is an element of the group.

C contains an i d e n t i t y e l e m e n t El s a y , such that

CIC, = ZaCl = 2, and to each element C, there corresponds a unique element called its inverse, the product of which with Cay in either order, is the identity element.

The number of elements

in the group, i.e. p, is called the order of the group. An abstract group is completely specified if its multiplication table is given. A set of matrices S whose elements S,(a=l,

...,p)

can be set

into correspondence with the elements of the abstract group Z.in the sense that it has the same multiplication table as C is called a matrix rep~esentatiokof C. Multiplication of the elements of the set S is understood-to be the usual matrix multiplication.

~videntlythe elements of S must be square matrices

of the same dimensionality.

It is also evident that the set of p

unit matrices of any dimensionality form a matrix representation of any abstract group of order

p.

Such a representation is called

an identity representation. We note that while the correspondence between the elements of S and C may be one-to-many, as for example in the case of the identity representation, the converse cannot be the case; i.e. no two elements of an abstract group are the same.

If the corres-

pondence between the elements of S and C is one-to-one, then S is said to be a faithfuZ representation of C. representation of C and not, we may describe

6

6

If S is a faithful

is another repr.esentation, faithful or

as a matrix representation of S.

is,) be a faithful representaand let A = {s(S,)) and d = {;(s a ) )

Let the set of matrices S tion of some abstract group C

=

be two representations of C and hence of S. matrix t, say, exists such that

If a non-singular

;(sat

=

t-ls(sci)t,

then the representations A and

(3.1)

6 are said to be equivalent. If

no such matrix exists they are said to be inequivatent.

It is

evident that if two representations are each equivalent to a third representation, they are equivalent to each other.

A matrix representation A is said to be reducible, if there exists an equivalent representation d of the form

where m(S ci) and n(S ci) are square matrices whose dimensions are independent of a.

If this is not the case then the representa-

tion A is said to be irreducibte. It is evident that the multiplication tables for m ( S a ) (a=l,.

. . ,p)

and for n(Sci) (a=l,.

plication table for s(Sa)(a=l rices {m(s,)}

and {n(S,)}

. ., p )

,...,p),

are the same as the multiso that the sets of mat-

are representations of the group S. We

shall denote these representations by m and n respectively.

The

representation b is said to be the direct sum of the representations m and n.

It is clear that, by means of a transformation

matrix which permutes rows and columns in

s(sa)

in an appropriate

manner, we can obtain a representation, equivalent to the representation (3.2),

in which the matrices m(S,)

and n(Sa)

are

interchanged. If the representations m or n are reducible, then there evidently exists a representation equivalent to 6 , and hence to

b ,

which is the direct sum of more (counting repetitions) than two representations.

It follows that we can. always find a

representation, equivalent to A , which is the direct sum of irreducible representations and in which the individual irreducible representations may occur more than once.

Such a representation

will be called a c a n o n ~ c arepresentation. ~ For each of the finite groups with which we are concerned in the theory of constitutive equations, the irreducible representations, which are finite in number, are known*.

For certain

groups, some of the irreducible representations may be complex. It is, of course, evident that the'one-dimensional identity representation is an irreducible representation of any group. Let rl, r2,

...,rh, with r,

=

Ira(S,)},

be the irreducible rep-

resentations of the group S. Then, given any representation 6, we can obtain an equivalent canonical representation 6

=

{ i(S,) 1,

where

We shall suppose that the representations rl,.

. . ,rh are repeated

vl, ...,vh times respectively in the canonical representation (3.3).

It is worth noting that the canonical representation

(3.3) is not unique, since a canonical representation equivalent to 6 , and hence to A , can be found in which the order of the terms in the direct sum is changed in any desired manner. It can be proven that va, the number of times the irreducible representation

ra

occurs in a canonical representation of

6 , is given by

where the tilde denotes the complex conjugate.

* See, for example, F.D. Murnaghan, The

Theory o f Group R e p r e s e ~ t a t i o n s , The Johns Hopkins Press, Baltimore (1938).

Thus, if we have any representation

6

of a group and we know its

irreducible representations Tl,...,rh,

we can construct an equi-

valent canonical representation by using the formula (3.4) to calculate the number of times each irreducible representation occurs in it. The set of quantities CtrsCS,)} the representation

A

is called the c h a r a c t e r of

and the set of quantities Itrra(S,))

is

called the character of the irreducible representation Fa.

It is,

of course, evident that the characters of two equivalent representations are the same.

Also, it is of interest to note that the

characters of the irreducible representations satisfy the orthogonality conditions

We note that if d is the dimensionality of the representation

6

and dl,

...,dh

are the dimensions of the irreducible rep-

resentations Tl,...,rh

then d

=

v1d1

+

...

+

vhdh.

It can be proven that for a group of order 1-1

We now introduce the concept of the classes of a group of order 1-1. group.

Let S be a faithful representation of an abstract We take any element of S, say S,,

S - ~ SS for all values of 8 = 1,. B a B

..,p.

and form the products The set of matrices so

obtained are all elements of S and are said to form a c Z a s s . We proceed in this way taking, as the element S,, S in turn.

each element o f

By this procedure, we divide up the elements of the

group into a number of disjoint classes.

The number of elements

in a class is called its order.

It is evident that one of the

classes always consists of the identity element and this is accordingly a class or order unity.

It is not difficult to show

that the order of each class in S is a submultiple of the order of S.

In can also be -shown that the number h of inequivalent

irreducible representations of any finite group is precisely the same as the number of classes in the group.

It is clearly a

trivial matter to determine the latter for any given finite group. 4.

Transformation of the carrier space Let xa(a=l, ...,u)

be a set of rectangular cartesian reference

frames equivalent to xl, such that Xa

= Sax1,

(4.1)

where Sa is an element of a faithful three-dimensional matrix representation S of some finite abstract group. (A) .i (A=l,... N ) be N tensors with components Vil.

Let v

P~ in the system xl.

Let vl be a d-rowed column matrix whose

elements are formed from the independent components in xl of the N tensors v(~) in some way. Let va be the d-rowed column matrix whose elements are formed from the components of v(*) in xa in the same way. Let

d =

(s(S,)I

be the d-dimensional representation of the

group S which transforms v1 into va(a=l,

...,u ) ,

thus

va = s (Sa)vl. Let

a

=

{s(S,)l

(4.2)

be a canonical representation equivalent to A ,

so that

i(sa)

=

t-ls(sa)t,

where t is a non-singular matrix.

From (4.2) and (4.3)

(4.3)

2 66

it follows that

where

v

1

and i1 define d-dimensional vector spaces which are called

the carrier spaces for the representations

6

It can be shown that the elements of

and

respectively.

can be determined by

the formula

provided that this does not yield the result zero.

If it does,

we replace the subscript 1 in (4.6) by 2, or 3, ..., or d until we obtain a non-zero result. We note that the carrier space for the canonical representa-

tion

i consists of a number of sub-spaces, corresponding to the (In particular the one-dimensional

irreducible representations.

sub-spaces corresponding to the one-dimensional identity representations in d are invariants.) The manner in which the replacement of the representation A by an equivalent canonical representation

z,

and the concomitant

replacement of the carrier space by a number of sub-spaces, eases the problem of determining integrity bases will be shown in the illustration given in the next section. 5.

An example Let the group of transformations S 0

0

1

=

Isa) be

Jf

-z -2-

s1 0

0

It can easily be verified that the multiplication table for this group is

We consider the determination of an integrity basis for a single second-order symmetric tensor T.

1j

under this group of transfor-

mati.ons. Taking v l as the column matrix V1

the matrices s(S,)

(TllYT12YT22YT13,T23YT33)

are the 6x6 matrices

9

(5.1)

From the multiplication table for S we see, quite easily, that

there are three classes in this group, IS1),

{S2,S31, IS ,S ,S I . 4

5

6

We note that the orders of these are 1, 2 and 3 respectively and that these are submultiples of 6, the order of the group (cf .53). Also, the sum of their orders is the order of the group.

Since (cf.53) the number of irreducible representations must be equal to the number of classes, there are three irreducible

representations. given

* by

These are

Irl(sa) I,

{r2(Sa) I and (r3(Sa) I

* See, for example, F.D. Murnaghan, The Theory of Group

Representations, The Johns Hopkins Press, Baltimore (19383.

Ql"

"'I"

m

c

m

d

a

c

w

0 ) F - 0

.

E a

a

L

C (d

U

rl

vi

D

w

+

@

0

.

n

In

c

m

. m

o g +

A

O U r C c

m

Ci al v i

c

0)

3

C al b a C cd

3

E

+

cd m

'A

a

r

C

o l 1

w

5 C

C c

i d

C

l

0 0 -

E

X . d W

VI

.

M

Q %

. h

r.

w

C

C +

0

l

a al vi al

.ri

a

C

b

w,

.ri rl

r

0

X

y

fn

.ri F: .rl ce Ci C, . s m ~ c W % l f n

d

We can find the number of times, v1,v2,v3, the irreducible representations, I'1 ,r 2,r3, occur in a canonical representation equivalent to 4 , by using the formula (3.4). We obtain v1 = 2, v2 = 0, v3 = 2.

Accordingly d =

o

(5.2)

{6(sa)1 may be taken as

Correspondingly, from (4.6), we find the carrier space

for

this canonical representation as v =

[6v69

3 31vl+v31, 3V2, Z(v1-~3)

9

v4, v51.

(5.4)

We see immediately that v6 and vl

+

v3

(5.5)

are elements of the integrity basis, since they span the onedimensional carrier spaces for the irreducible identity representations.

We see also that G3,

Gq

transform. independently of the

remaining elements of f a and so do ii5, v6. -

We have

The required integrity basis is given by (5.5), integrity basis for (c3,ch,c5,c6). using Theorem 2.

together with the

The latter may be obtained by

In order to do so, we introduce the new

variables yil), y$2) (i=1,2,3) defined by

(i=1,2,3). We note that

yk2)

+

,+ and ,,v6 - are transformed by I. 3 (Sa. ) , y$l) and 3 4 (i=1,2,3) are transformed by the elements of the full permutation

As ir

group on the subscripts 1,2,3.

Theorem 2 then yields, with

-

-

(5.6), the following integrity basis for (v3,v4,v5,c6) : i2+,2 -2 - 2 + (3v2--2 - 2 -2 3 49 v 5 +v6, 4 3 ~ 4 9 ) "6(3v5-v6)

ii 3 5 o (i 6

+

+

C4",

"(+4"-2+

-2i i ) 6 4 3 5

i

)

3 5

-

i

+2

6 3'

9

(5.9)

- i4i$

With (5.4) and (5.1) , we obtain from (5.9) the elements of this integrity basis

in terms of Tij (i,j=1,2,3) and together with

T33 and T1l+ T22, obtained from (5.5) and (5. I), we then have the elements of the required integrity basis. It is evident that this is irreducible. For the group considered, the problem of determining an irreducible integrity basis for N second-order symmetric tensors is not significantly more difficult than that of determining an irreducible integrity basis for one such tensor. representation

The canonical

;(s a ) , given by (5.3), is replaced by the direct

sum of N such representations and the carrier space, which now has dimensionality 6N, consists of N 6-dimensional sub-spaces given by the 6-rowed column matrices

vl

G2,

...,vN,

say, which

are defined in terms of the elements of the individual tensors

by equations analogous to (5.4) and (5.1). P =

(PI 11 11 ii

(P.1,.

..,N) , we

see that

With the notation

iiP) and

;iP)

are elements

of the integrity basis and the remaining elements may be obtained by defining yjZP-')

We

and ylZP) in a manner analogous to (5.7).

can then proceed, as before, to apply theorem 2 to the determiriation of an integrity basis. In both the case of a single second-order tensor or N such tensors, we could, without significant increase in difficulty, have discussed the problem along rather similar lines, without replacing the representation tation.

of A by a canonical represena For example, in the case of a single tensor, we see S(S )

immediately that v6 is an invariant.

rather than by (5.7),

We then define yl'),

yi2)

and again apply Theorem 2 to obtain an

integrity basis. We have

chosen not to adopt this procedure, purely for

illustrative purposes.

The advantage of the procedure we have

adopted lies in its generality. matrix representation of

6 ,

It can be applied even when the

under which o transforms, is not the

direct sum of a number of representations of small dimensions. We bear in mind here that, for all of the crystal symmetries, the irreducible representations always have dimensionality no greater than three. 6.

Irreducibility In the problem discussed in 55, when the elements cf an

integrity basis are generated by Theorem 2, it is quite evident which of these elements is redundant.

For other groups and other

systems of tensors, this is not always the case.

Consequently

it is important to have a systematic procedure for doing this. We shall outline such a procedure for a set of tensors (A=1,...,N) and a finite group S of order p. Let 1 be an integrity basis for the tensors I(A), whose irreducibility is under consideration.

Let I(al,

set of invariants in I of partial degrees al,. Let v(al,

(A=,.,N) K(al,

...,aN)

...,aN)

...,aN)

be the

. . ,aN in

be the number of these.

Let

be the set of invariants in 1 which have partial

degrees less than or equal to al, ...,aN and total degree less than al+ ... +aN. We adopt the following iterative procedure. (i)

We suppose that the set o'f invariants K(al,.

.. ,aN)

is

irreducible. (ii) We determine the number X(al,

...,aN)

of linearly-indepen-

dent invariants which have partial degrees al, ...,aN. L(al,

...,aN)

be this set of invariants.

ted from the formula i =

f

Let

X can be calcula-

trs(sa). a=1 (iii) We form all monomials in the elements of the set K which are of partial degrees al,..., aN.

...,aN) be R(al, ...,aN).

Let p(al,

number of these and let them form the set

the

p can, of course, be readily calculated by means of simple

combinatorial considerations.

The elements of R are not

linearly-independent if syzygies of partial degrees alp...,aN exist between the elements of K . a(al,

...,aN)

Suppose

such linearly-independent syzygies exist.

Then only

p-a

elements of R are linearly independent.

Let

us denote this set of invariants b-y 8. We construct from the set L another set p-s

L

of h linearly-independent invariants,

elements of which are the elements of 8. The remaining

A-(0-0)

elements of the set i provide the elements of

partial degrees a ,...,aN. in an irreducible integrity basis. (iv) If we find that

then hone of the elements in the set ?(al,

...,aN)

is

reducible; i.e. none of these elements can be omitted from the desired irreducible integrity basis. IV)

I f X - p < v , then the following procedure must be adopted.

We must identify, 'by examination, those elements of T(al,

...,aN )

which can be expressed as linear combinations

of other elements of the set I(al,

.. . ,aN)

and omit them.

Then we find as many elements as we can in the remaining set which can be expressed as polyn.omials in the elements of K , i.e. which are linear combinations of the elements

of R.

We omit these.

have ;(a1,..

Let the remaining set i(al,

.,aN) elements.

...,aN)

We now search-for linearly-

independent relations between the elements of the set R. Suppose we can find :(al,

...,a N )

of them.

We continue

these searches until

~ - ( ~ - a=) v When this has been achieved the invariants in 1 provide the elements of an irreducible integrity basis of partial degrees al,...,aN.

For the elements of I of lowest degree it is, of course, relatively easy to omit by inspection the redundant elements and in this way provide a starting point for our iterative procedure. 7.

Determination of linearly-independent invariants In the procedure for obtaining an integrity basis for a set

of tensors I(A) (A=l,.

..,N)

and a group S, we first write the

independent components, in some rectangular cartesian coordinate system, as a column matrix.

We then calculate the matrix rep-

resentation.4 of S under which this column matrix transforms. Then, we obtain a canonical representation d equivalent to

A.

We noted that the one-dimensional carrier spaces for the onedimensional identity representation in 6 are invariants. Similar considerations enable us tp determine, rather easily, all invariants of

-

*:

'"N

(A=l,.

. . ,N)

of partial degrees

which are linearly-independent. We proceed in the

following manner.

We form all monomials in the components of

v(*) (A=l,.. . ,N) , referred to a rectangular cartesian coordinate system x, which have partial degrees al,

. . .,aN in

(A.1,.

. . ,N) .

We then arrange these monomials as a column matrix v , say. We then calculate the matrix representation

b

of S under which v

transforms. We now obtain a canonical representation d equivalent to

4.

The one-dimensional carrier spaces for the one-dimensional

identity representations in d provide a set of linearly-independent

.

invariants of partial degrees ol,. .,aN in _V(A1(A=l,.

.. ,N).

We

note that their linear independence is ensured by the fact that these carrier spaces are mutually orthogonal. lJ

lated by using the formula (cf. (4.6))

They can be calcu-

1 s. (sa)v;l), a=l =q

where i

is the row of 6.in which the identity representation considered

occurs. The number of these linearly-independent invariants, denoted h(al,

...,aN)

in 56, is, of course, the number of times the iden-

tity representation occurs in the canonical representation

i.,

This is given by (3.4) with trra(s ) = I. a The above analysis provides an alternative procedure to that adopted so far, for determining an irreducible integrity basis. The procedure is similar to 'that outlined in 56, except that we now take as the invariants,-which-arecandidates for inclusion in the irreducible integrity basis, the sets of linearly-independent invariants of various partial degrees.

The sets of linearly-

independent invariants of lowest degree must necessarily be included in the irreducible integrity basis. iteratively, as in §6(v),

We then proceed

to eliminate, from the linearly-

independent sets of invariants of successively higher degrees, those which are irreducible.

We can systematize this step in the

following way.

. .,aN) linearlyindependent invariants of partial degrees al, ...,aN. Let R(a l?...,aN) be the set of p(al, ...,aN) monomials of partial Let L(al,.

. . ,aN) be

the set of h(al,.

degrees al, ...,aN in the tensors which can be formed from the irreducible invariants of partial degrees no greater than al,...,aN and total degree less than al +...+

a*.

We express

each element of R as a linear combination of the elements of L. Since L is a linearly-independent set this can be done uniquely. In this way, we obtain a set of p linear equations relating the elements, R17...,Rp say, of R to some or all of the elements, Ll,.

.. ,LA say, of

L.

Let us suppose that

X

of the elements of L

occur in these equations and let us denote them by L1

,...,Li.

Then, we have

where the A's are constants.

Plainly, the invariants LX+~,...,L~

are elements of the irreducible integrity basis. rank of the matrix

11 A BY 11.

Let r be the

Then, h-r of the invariants L

.,Li are elements of the irreducible integrity basis, while p-r linear

...,R P '

relations, i.e. syzygies, exist between R1,, find which X-r of the

h

invariants L1,.

. .,Li

In order to

are elements of the

irreducible integrity basis, we proceed to eliminate from (7.1) i-r of the invariants L,- ,.

. . ,Li .

These eliminated invariants

together with LX+~,...,L~are the elements of the irreducible integrity basis of partial degrees al, 8.

..., aN.

Historical note The procedures described in this chapter have been used to

obtain irreducible integrity bases for sets of tensors of various orders with respect to the crystallographic point groups. The three theorems given in 51 were used by G.F. Smith and Rivlin

*

,

in the manner described in 52, to obtain integrity bases

for a single symmetric tensor for each of these groups and by

G.F. Smith, M.M. Smith and Rivlin

**

to obtain integrity bases for

a single second-order symmetric tensor and' a single first-order tensor.

They demonstrated the irreducibility of the integrity

bases obtained by using the procedure outlined in 96.

*

G.F. Smith and R.S. Rivlin, Trans. Amer. Math. Soc. 88, 175-193 (1958).

** G.F.

Smith, M.M. Smith and R.S. Rivlin, Arch. Rational Mech. Anal. 12, 93-133 (1963).

Integrity bases for an arbitrary number of first-order tensors were obtained by G.F. Smith and Rivlin

*

for all but one

(the gyroidal class of cubic symmetry) of the crystallographic groups.

For some of the groups-themethod used was that des-

cribed in 52 and for others it was substantially that mentioned in 57.

Again, the irreducibility of the integrity bases obtained

was demonstrated by the method described in 56.

G.F. Smith and Kiral

**

have obtained irreducible integrity

bases for an arbitrary number of second-order symmetric tensors for each of the crystallographic groups and for arbitrary numbers of tensors of any orders for 26 of these groups+, the omitted groups being the cubic groups and the hexagonal group of highest symmetry.

In their work, they use the full range of procedures

outlined in this chapter.

* G.F. Smith and R.S. Rivlin, Arch. Rational Mech. Anal.

15,

169-221 (1964).

* * G.F. Smith and E. Kiral, Rend. Circolo Matematico Palermo

18,

5-22 (1969).

+

E. Kiral and G.F. Smith, Int. J. Eng. Sci. 12, 471-490

(1973).

Chapter 5 Integrity Bases for the Full and Proper Orthogonal Groups 1.

Introduction In this chapter, we discuss the manner in which an irre-

ducible integrity basis can be determined for an arbitrary number of symmetric second-order tensors with respect to the full orthogonal group.

(The corresponding result for the proper orthogonal

group is the same as that for the full orthogonal group.)

This

was first achieved in a series of papers by Riv1,in and Spencerx. Here we follow closely the presentation given in a review paper

**

by Spencer papers.

, which streamlines the argument given in the eariler

The irreducibility of the integrity basis obtained was

demonstrated by smith'. Somewhat similar methods have been used to obtain irreducible integrity bases for an arbitrary number of second-order symmetric tensors and an arbitrary number of absolute or relative first++ order tensors with respect to the proper orthogonal group . From these results the corresponding results for the full orthogonal group have been obtained by Smitht .

*

R.S. Rivlin, J. Rational Mech. Anal. 4', 681-702 (1955); A.J.M. Spencer and R.S. Rivlin, Arch. Rational Mech. Anal. 2, 309-336 (1959); i b i d . 2, 435-446 (1959); z b i d . 4, 214-230 (1960); A.J.M. Spencer, Arch. Rational Mech. Anal. 7, 64-77 (1961).

**

A.J.M. Spencer, T h e o r y of I n v a r i a n t s . Continuum Mechanics; ed. A.C. Eringen, publ. Academic Press., New York 1, 239-353 (1971).

+

G.F. Smith, Arch. Rational Mech. Anal. 5, 382-389 (1960);

i b i d . 28, 282-292 (1965).

++

A.J.M. Spencer and R.S. Rivlin, Arch. Rational Mech. Anal. 9, 45-63 (1962); A.J.M. Spencer, Arch. Rational Mech. Anal. 18, 51-82 (1965).

-t

G.F. Smith, Arch. Rational Mech. Anal. 18, 282-292 (1965).

2.

Isotropic tensors A tensor is said to be centrosymmetric isotropic if its COm-

ponents in all rectangular cartesian coordinate systems are the same.

It is said to be non-centrosymmetrie isotropic if its com-

ponents in all rectangular cartesian coordinate systems of the same hand (i.e. left-handed or right-handed) are the same. Let

a_ be a tensor of order P.

be its comLet Ail.. . i p and Ail. .. P ponents in rectangular cartesian coordinate systems. x and 5 respectively.

~f the systems x and

x are related by (2.1)

x = Sx,

then

If A is centrosymmetric isotropic

for a22 orthogonal S.

If it is non-centrosymmetric isotropic,

(2.3) is valid only for proper orthogonal S.

.

and f(B) = ( ~ 1 ~ ) (B ) = l,.. . ,P) be the

Let Y ( ~ ) @lB))

column matrices formed by the components in the systems x and 2 respectively of p first-order tensors

V[ B).

Then,

Multiplying (2.2) throughout by Vi -(1)~(2)...~IP) and using (2.3) 1 i2 P and (2.4), w e obtain

~y.. . -(PI

1

'ip Ail.

..i p

=

~1:'...vi )A (P

il.. .ip

= P

(say).

(2.5)

A

Equation (2.5) states that P is a multilinear invariant of the ? first-order tensors -

with respect to the full or proper

orthogonal group, accordingly as (2.3) is considered to be salid

for all orthogonal or only proper orthogonal S. We have seen in Chapter 3 that the set of invariants

forms an irreducible integrity basis for an arbitrary number of

.

(A = 1,. . ,P) , with respect to the full

first-order tensors orthogonal group.

It follows that if P is odd, then the A's must

be zero and if.P is even, P must be expressible in the form

where the C's are constants and B1...Bp

is a permutation of

l...P, the summation being taken over all such permutations. Now, from (2.5), we have

aP P

Ail...ip - a V ~ l i . + . a y ~ P.) 1 P Introducing ( 2 . 7 ) into ( 2 . 8 ) , we obtain

where the 6's are three-dimensional Kronecker deltas.

...i

i

P is a permutation of il...i P and the summation is taken over all such permutations. The expression (2.9) provides a canonical 1

expression for an arbitrary centrosymmetric isotropic tensor of even order.

There are no centrosymmetric isotropic tensors of

odd order. We have seen in Chapter 3 that an irreducible integrity basis for an arbitrary number of first-order tensors

1( A )

(A= 1,..., P), with respect to the proper orthogonal group, is given by

We note also that the product of any two scalar triple products may be expressed as a polynomial in inner products. if P, the order of

A

A,

Accordingly,

is even, we arrive at the conclusion that

must be expressible in the form (2.9).

If P is odd, then P

must be expressible in the form (B1)

P =

zcB1.. .BprE

*-v

(Be)

91

(B3)

1( E

(B4)

(B5)

(Bp-l)

(Bp)

01 ).--(I *l 1, (2.11)

where, as before, B1, ...,Bp is a permutation of 1,. ..,P and the summation is taken over all such permutations.

Then, from

(2.$1, we obtain

where the

E'S

are three-dimensional alternating symbols.

Thus,

equations (2.9) and (2.12) provide canonical expressions for arbitrary non-centrosymmetric isotropic tensors of even and odd orders respectively. 3.

Isotropic tensor polynomials Let P be a polynomial in the components, in the rectangular

cartesian coordinate system x, of N second-order symmetric tensors

v ( ~ ) (B = 1,. -

where B1....Bp

. .,N).

Denoting these components by v(~), we can write ij

...,N,

is a selection of P numbers from 1,

inc1udj.n.q

repetitions, and the summation is taken over all such selections and over all P.

It is easily shown that if P is an invariant with respect to the full or proper orthogonal groups, the A's in (3.1) must be isotropic tensors.

Since they are of even order, they can in

either case be expressible in the canonical form (2.9).

Intro-

ducing this into (3.1) it is evident that P must be expressible as a polynomial in traces of products formed from the matrices v ( ~ )=

I

1~1:) / I , with scalar coefficients.

As a particular case we may consider that P is a polynomial in a single second-order symmetric tensor

1. Then, P must be

expressible as a polynomial in traces of powers of V with scalar coefficients. According to the Hamilton-Cayley theorem, any second-order matrix satisfies its own characteristic equation. Thus,

v3 -

(trv)v2 +'i{(tr~)~- trv21v

-

(detV)6

=

0.

(3.3)

It follows immediately that

Multiplying (3.3) throughout by V, taking the trace of the resulting equation, and using (3.4), y e find that tr 'V expressed as a polynomial in tr V, tr v2 and tr v3. plying (3.3) throughout by

v ~ and - ~taking

may be

Again, multi-

the trace of the

resulting equation, we see that tr V" may be expressed as a polynomial in tr v " - ~ , tr v"-~,etc.

Accordingly, it may be expressed

as a polynomial in tr V, tr v2, tr v3.

It .follows that these three

invariants form an integrity basis, which is evidently irreducible, for the symmetric second-oraer tensor and the full or proper orthogonal group. In the next section, we will see how a corresponding result, albeit a much more complicated one, for an arbitrarily large

number of symmetric second-order tensors can be obtained, essentially as a deduction from the ~amilton-cayleytheorem. 4.

The integrity basis for N second-order symmetric tensors In the previous section we have shown that any isotropic

polynomial invariant of any number of symmetric second-order .tensors can be expressed as a polynomial in traces of products formed from the matrices of their components in an arbitrary rectangular cartesian coordinate system.

In the present section

we will see that we may limit these products to be of degree no To do this we prove that if a,b, ...,g are seven

higher than six. 3x3

matrices, then trab ...g may be expressed as a polynomial in

traces of products of the matrices of degree 6 or less.

This

resalt is a consequence of the Hamilton-Cayley theorem, which states that any matrix satisfies its own characteristic equation. Thus, if m is any 3x3 matrix [cf. (3.3) and (3.4)] m3

-

1 (trm)m2 + T{(trm)2

-

2 trm 3m + (detm)6

= 0

and

(4.1) detm

=

&1 ( t ~ m ) ~ - 3(trm)(trm2)

+

2trr33.

Let m = a + Ab + PC,

where A and

are arbitrary scalars.

Introducing (4.2) into

(4.1) , the coefficient of Ap yields

- tr b tr c) + b (tr ca - tr c tr a) c(trab - tra trb) + (bc+cb)tra

Cabc = a (tr bc +

+ (ca + ac) tr b +

+

(ab + ba) tr c

6(tratrb t.rc- t r a t r b c - trb trca

- tr c tr ab + tr abc + tr cba) ,

where the notation Zabc = abc is used.

+

bca

cab

+

+

bac

acb + cba

+

We introduce the following terminology.

(4.4)

If a matrix

polynomial can- be expressed as a matrix polynomial of lower degree

*

, we

say that it is reducible.

Thus, (4.3) expresses the

fact that Cabc is reducible and we may express this by writing Cabc

0.

(4.5)

If the difference PI.-P2 of two matrix polynomials P1,P2 of equal degree is reducible, we say they are equivalent and write

P 1 Z P2 '

(4.6)

Again, if the trace of a product of matrices is expressible as a polynomial in traces of products of lower degree, we shall say that it is reducible.

Accordingly, we may state the result

we shall prove in this section as "trab . . . g

is reducible" and we

shall write this as trab . . . g E 0.

(4.7)

We now proceed to prove this statement. From (4.3), we have

Taking b=a in (4.8), we have tr(a 2c +aca-+ca2) d

=

tr d(a 2c + a c a + ca2)

E

0.

(4.9)

Replacing d by ad in (4.9), we have tr(a 2ca+aca2.+ c a3)d

* By

E

0.

(4.10)

the degree of a matrix polynomial we shall mean the degree of the matrix product in it of highest degree. Thus, the left-hand side-of (4.3) has degree 3 and the right-hand side has degree 2.

S i n c e , from t h e Hamilton-Cayley theorem, t r ca3d E 0 , we s e e from (4.10) t h a t

From ( 4 . 9 ) we o b t a i n , on r e p l a c i n g c by b and d b y c b d , tr(aba)cbd E

-

2

2

t r ( a b + b a )cbd

2 2 2 2 2 2 2 E t r a (b c + c b ) d + t r t b a c + a cb ) d .

(4.13)

I t a l s o f o l l o w s from ( 4 . 9 ) t h a t

S u b t r a c t i n g ( 4 . 1 4 ) from ( 4 . 1 3 ) ,

we o b t a i n

t r a 2 c b 2 d 'E 0 . I n ( 4 . 1 5 ) we r e p l a c e a b y ' A a + p e , where A and p a r e a r b i t r a r y

scalars.

Then, t h e c o e f f i c i e n t o f Ap y i e l d s tr(ae

+

2

ea)cb d

Again, r e p l a c i n g b by Ab+pf i n ( 4 . 1 6 ) ,

Z

0.

(4.16)

t h e c o e f f i c i e n t of

Xu

yields tr(ae

+

ea)c(bf + fb)d

2

0.

R e p l a c i n g d b y g d i n ( 4 . 1 7 ) , we o b t a i n

R e p l a c i n g b by bg i n ( 4 . 1 7 ) , we o b t a i n

t r ( a e + e a ) c (bgf

+

f b g ) d : 0.

We now s u b t r a c t (4.19) from ( 4 . 1 8 ) t o o b t a i n

R e p l a c i n g c by cb and b b y g i n ( 4 . 1 7 ) , we o b t a i n

(4.19)

We now add (4.20) and (4.21) to obtain tr aecbfgd r -tr eacbfgd..

(4.22)

A similar result can be obtained for interchange of any pair of adjacent matrices in the product on the left-hand side of (4.22). Now, in ( 4 . 8 ) ,

we replace a, b, c, d by ab, cd, ef, g re-

spectively. We note that each of the six resulting terms can be obtained from trabcdefg by an even permutation of the factors in the product abcdefg.

Thus, from (4.22) each of these six terms

is equivalent to trabcdefg and hence we obtain the result (4.7). 5.

The integrity basis for six or fewer second-order symmetric tensors We have seen in the previous section that in order to find

an integrity basis for an arbitrary number N, say, of symmetric second-order tensors with respect to the orthogonal group, we need only find the integrity basis for six such tensors.

An in-

tegrity basis for the N tensors is then provided by the bases for every selection of six tensors from the N tensors.

Indeed, if

the integrity basis for the six tensors is irreducible, this procedure will provide an irreducible integrity basis for the N tensors. Accordingly, in the present section we will derive an irreducible integrity basis for six tensors whose component matrices are a, b, c, d, e, f.

It has been seen that such an integrity

basis may have as its elements traces of products formed from these matrices.

Also, from the reducibility relation (4.7) it

is evident that we need consider as candidates for inclusion in

this basis only traces of products of degree six or less. The required irreducible integrity basis must include irreducible integrity bases for every set of five tensors selected from the six. tensors; the irreducible integrity basis for five tensors must include irreducible integrity bases for every selection of four tensors from the ffve, and so on.

Accordingly, we

shall first derive an irreducible integrity basis for one tensor. We then derive those elements in an irreducible integrity basis for two tensors which involve both of them, those elements in an integrity basis for three tensors which involve all three of them, and so on, until we have derived those elements in an integrity basis for six tensors which involve all six of them.

In

presenting our results, we shall draw a box around those sets of invariants which actually appear in our final irreducible integrity basis. (i) One tensor, a. Taking m=a in (4.1), it is easy to show that

Thus, tra, t r a 2 , tra 3

form an integrity basis for one tensor

and this is evidently irreducible. (ii) Two tensors, a and b. We consider invariants of the form

From the Hamilton-Cayley theorem it is evident that this is reducible if any of the m t s or n t s is greater than 2.

Also, from

(4.9) it follows that if in (5.1) any two of the m's or any twc

of the n's are the same, then it is equivalent to an expression

or the sum of expressions of this form in which this is not the case.

It follows that we need consider as candidates for inclu-

sion in the irreducible integrity basis only trab, trab2 , t r a2b, t r a2b 2 and tr aba2b 2

.

We have omitted invariants obtained from these by cyclic permutation of the factors in the matrix products.

Now from (4.12)

w e have tr aba2b2

-

E

Hence, tr aba2b2

Z

tr a 2bab2 =

-

tr b 2a 2ba

=

- tr aba2b 2 .

0.

(iii) Three tensors, a, b and c. From considerations similar to those employed in discussing the invariants of two tensors, together with (4.15), it follows that we need consider as candidates for inclusion in the desired irreducible integrity basis only trabc, t r a2be, t r a2b 2 c , t r a2bac, t r a 2. b2 c2 , t r a2b 2 ac and terms obzained from these by permutation of a, b, c. Since tr abc is unaltered by permutation of a, b, c , the only trilinear term we retain is trabc. Also, since t r a2be = t r a2 cb, the only terms we retain, from among those provided by permutation of a, b, c in tr a 2be are tr a 2bc and

r t r b ca, t r c ab.

Similar considerations applied to t r a2b 2 c lead to retention of tr a2b2c and

trb 2c2a, trc 2a2b. From (4.12) it follows that tr

n2h-a = -

+v

nhn2a

=

- +, nn2hn

=

-

+r n'hnn

-so- that - -- - -

tra2 bac r 0. Similarly t r a 2b 2ac E 0. Therefore, t r a 2bac,tra 2b 2ac and terms obtained from them by ~ermutationof a - b. c can be omitted from the inteeritv basis. Again 2 tr a2b2c2 = tr a2(b2c2

+

c2b2).

With (4.9) and (4.15), it follows that r<

7>

I'hus, tra-b-c- and terms obtained trom it by permutatlon ot a , b.

c

can be - . omitted from the inte~ritv - - ~ -basis. -- -, --

-

~

- - - -

~-

( i v ) Four tensors a, b, c, d.

From the fact that invariants of degree greater than six are reducible, together with (4.9) and (4.15), it follows that

-

we need consider as candidates for inclusion in the desired integrity basis only

tr abcd, tr a 2bed, tr a2b2cd, tr a2bacd and Invariants obtalned trom these by permutatlon or a, b, c,

d.

From the fact that trabcd is unaltered under certain permutations of a, b, c, d, it follows that we may omit from the invariants tr abed and its permutations all except tr abcd, tr abdc ,

tr acbd.

Since tr adcb = tr abcd, tr adbc

=

tr - a d d ,tr acdb

=

tr abdc,

we obtain from (4.8) trabcd + trabdc

+

tracbd

5

0.

We can accordingly, omit tracbd from the integrity basis.

Now,

replacing a by a2 in the above agrument,, we are led to retain f ~ o mtr a2bcd and invariarits obtained from it by permutation of b, c, d only t r a 2bed, tra 2bdc. By analogy, we are led to retain from tra2 bcd and invariants obtained from kt by permutation of a, b, c, d only" t r a 2bed, tr a2bdc, and invariants obtained by cyclic permutation of a, b, c, d. From (4.8), we have tr(a 2Cb 2cd) With (4.15),

-

0.

this yields t r a2b 2 (cd+dc)

5

0.

Accordingly, we retain the six invariants

Replacing b by b+a in (5.3), we obtain, with (5.3) and the Hamilton-Cayley theorem, tra 2baed s - t r a2badc. Also, from (4.16), we have tra 2baed Accordingly, we retain

5

- t r a2bead

=

-

t r a 2dacb.

t r a 2 baed and invariants obtained from it by cyclic permutation of a , b, c, d.

(v) Five tensors: a, b, c, d , e. Again, since the trace of any product of degree greater than six is reducible, we need consider as candidates for inclusion in the integrity basis only trabcde, t r a 2 bcde and invariants obtained from these by permutation of a , b, c, d , e. Of the 120 invariants trabcde and those obtained from it by permutation of a, b, c , d, e, all but 12 may be excluded by the fact that trabcde is unaltered by cyclic permutation of a , b , c , d, e and by reversal of the order of these.

Of these 12, 6 may

be excluded by.the fact that we can find 6 independent relations of the type (4.5) by making appropriate substitutions for a , b , c.

We thus conclude that we need to retain only six of the mul-

tilinear invariants. These may be taken as

From (4.16) , we have t r a 2 b(cd +dc)e E 0.

(5.4)

Also, replacing b by b+Xc and c by d+ue in (5.2), we obtain from the coefficient of Xu t r a2 (bc+cb)(de +ed) r 0.

(5.5)

There are six relations of the type (5.4), corresponding to the six ways in which two matrices can be selected from four.

Also

there are three relations of the type (5.5) corresponding to the three ways in which two pairs of matrices can be chosen from four. Of these nine relations, only eight are independent. We consider the 24 invariants, t r a 2bcde and those which can be obtained from it by permutation of b, c, d, e.

Of these, we

may omit from consideration 12, since they can be obtained from the remaining ones by relations of the type tr a 2 ebcd = tr ebcda2

=

tr a2dcbe.

The eight independent relations indicated above then imply that we need retain only four of the invariants.

These may be taken

as 2

2

t r a bcde, t r a bced, t r a 2cbde, t r a 2bedc. 2

Similar considerations applied to the invariants t r b cdea, tr c2deab, etc. and invariants obtained from these by permuting the last four factors in each case, lead to the conclusion that we retain in our integrity basis the invariants

and invariants obtained from these by cyclic permutation of a, b, c, d, e. (vi) Six tensors: a , b, c, d, e, f. We need to consider as candidates for inclusion in the desired integrity basis trabcdef and the invariants formed from it by permutation of the factors.

There are 6! such invariants,

of which all but 60 may be excluded by the fact that trabcdef is unaltered by cyclic permutation of the factors a, . . . , f and by reversal of the order of these.

*

A.J.31.

*

It was shown by Spencer

that

Spencer, Arch. Rational Mech. Anal. 7, 64-77 (1961).

fifty independent relations can be formed from (4.8) by replacing a, b, c, d by appropriate products formed from the six matrices a, f

.

It follows that we need include in the integrity basis

only ten of the invariants under consideration.

These may be

taken as tracfebd, tradcbfe, tradcfbe, tradfbce, tradfcbe, traebdcf, traecbdf, traecdbf, traedbcf, traedcbf. 6.

Irreducibility In the previous section we have derived an isotropic integ-

rity basis for six symmetric second-order tensors and have, as far as possible, eliminated redundant invariants.

Our hope is

that the integrity basis so derived is irreducible. indeea the case can be proven by

2

That this is

procedure similar to that des-

cribed in 96 of Chapter 4 for establishing the irreducibility of integrity bases for finite groups. lated the number uf plrtial degrees

In that procedure, we calcu-

...,an) of linearly-independent invariants al,.. .,aN in the tensors considered and'then

X(al,

omitted those which can be expressed as polynomials in elements of lower degrees in an integrity basis, which have already been established as non-redundant. In the present case, we must replace the formula, given in 56 of Chapter 4, for the calculation of

X(al,

...,a.N)

by a dif-

ferent formula, in which summation over the group is replaced by integration over the group.

Apart from this change, the procedure

is precisely that given in the case of finite groups, except that for the problem discussed in this chapter, it is found that no syzygies of relevant degrees exist and accordingly a is, in fact, zero in the formula for the number of elements in the irreducible integrity basis of specified partial degrees.