Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

733 Frederick Bloom

Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations

Springer-Verlag Berlin Heidelberg New York 1979

Author Frederick Bloom Department of Mathematics, Computer Science and Statistics University of South Carolina Columbia, S.C. 29208 USA

A M S Subject Classifications (1970): Primary: 7 3 S 0 5 Secondary: 53 C10 ISBN 3 - 5 4 0 - 0 9 5 2 8 - 4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 7 - 0 9 5 2 8 - 4 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data Bloom, Frederick, 1944 Modern differential geometric techniques in the theory of continuous distributions of dislocations. (Lecture notes in mathematics ; 733) Bbiliography: p. Includes index. 1. Dislocations in crystals. 2. Geometry, Differential. 3. G-structures. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 733. OD921.B56 548'.842 79-9374 ISBN 0-387-09528-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisheJ © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

FOR

HARRY

AND

MORDECHAI

Preface Among research workers

in mechanics

ematics there has been great interest, decades,

and applied math-

in the past two

in the area of continuum theories

More recently,

of dislocations.

attention has turned to the more difficult

problem connected with the motion of dislocations a continuum and its relation to various plasticity

formulations

theory for a body possessing

effort to formulate a continuous

dislocations was made by Kondo prescribing,

characterize

distribution

on the basis of certain heuristic

geometric

structures

a geometric

of the dislocation

was any serious

given to the types of constitutive

such as

which then served to

similar efforts were made by Bilby and by Kroner in none these theories

in

arguments,

on the body manifold,

certain properties

of

([ I ],[ 2 ]) and consisted

a metric and an affine connection,

however,

of

theory.

The first comprehensive

various

through

equations

distribution;

([3 ], [4 ]), consideration

which may be

associated with the body manifold. A new approach to the problem was made by Noll

[ 5 ] in

the early sixties and was later extended by Noll [ 6 ] and by Wang [ 7 ]. constitutive manifold

Here one starts with the prescription equation for particles

of a

belonging to the body

and, using the concept of a uniform reference,

VI

develops

a geometric theory which in many ways

to those considered by Kondo, work differs

Bilby,

is isomorphic

and KrOner.

Wang's

from that of Noll in that the body need only

admit a uniform reference

locally;

their work represents

the first known use of concepts belonging to the realm of modern differential

geometry

in formulating

a theory in

continuum mechanics. An application of the concepts Wang has been made by Toupin tions in crystalline media) have both examined, among the approaches and by themselves

developed by Noll and

[ 8 ] (to a theory of dislocaand Bilby

in detail,

[ 8 ] and KrOner

the relationships

taken by Noll and Wang,

and Kondo,

on the other%

[10]

which exist

on the one hand,

due to considera-

tions of space we will have to ignore such developments just as we shall pass over recent work of Wang, on wave behavior in inhomogeneous

elastic bodies

[13],

solutions

and on classes of universal

here

et. al, [ii],

[12],

[14].

A theory of dislocation motion in a continuum was formulated by Eckart "anelasticity". containing

[15] in 1848 in a proposal he dubbed

Eckart suggested as a model for a body

a continuous

distribution

of dislocations,

may be moving within the body manifold,

which

a continuum in

which the Cauchy stress arises in response to deformations from natural particles,

states which may be different,

and, perhaps,

for different

also varying w i t b t i m e .

proposal was examined by Truesdell

Eckart's

in [16] and by Truesdell

VII

and Toupin in [17] and in attempt was then made by Bloom

[18]

to fi

the basic tenets of Eckart's proposal into the framework d e v e l o p e d by Wang for static d i s l o c a t i o n distributions;

a correct formulation

of Eckart's a n e l a s t i c i t y proposal, w i t h i n th{s d i f f e r e n t i a l geometric setting, was given by Wang and Bloom [19] and has been extended by them in [20] to allow for t h e r m o d y n a m i c influences. More recently Wang [39] has sought to formulate the connection between anelastic response and recent ideas concerning m a t e r i a l s with elastic range. Our aim in preparing this m o n o g r a p h has been not only to try to present an accurate picture of the current status of dislocation theory, as

a

branch of c o n t i n u u m mechanics, but also

to illustrate an important a p p l i c a t i o n of modern d i f f e r e n t i a l geometric ideas in physics.

This is the proper place to acknow-

ledge a debt of gratitude to Professor C. C. Wang who has been the o u t s t a n d i n g major c o n t r i b u t o r to this important new area of continum physics.

Finally the author would like to thank

Mrs. Margaret Robinson,

for the excellent job of typing she has

done, and the college of Science and M a t h e m a t i c s at USC for a grant during the summer of 1975 which enabled me to complete the greater part of the work p r e s e n t e d here.

TABLE

OF C O N T E N T S

PREFACE

I.

MATHEMATICAL i.

INTRODUCTION

2.

DIFFERENTIABLE

MANIFOLDS

3.

FIBRE

ASSOCIATED

AND 4.

SOME

EXAMPLES

LIE ALGEBRAS, ON

5.

"G"

6.

COVARIANT AND

E(M)

INTRODUCTION

2.

BODY

AND

EXPONENTIAL

MAP,

AND

ON

E(M)

AND

PARALLEL

TRANSPORT

....

11 13

TORSION

15

IN E L A S T I C I T Y

. . . . . . . . . . . . . . . . . . . . . MOTIONS,

AND

IN C O N T I N U U M

18

DEFORMATION

. . . . . . . . . . . . . . . . . . . . . . STRESS

2

FUNDAMENTAL

. . . . . . . . . . . . . . . . . . . . .

MANIFOLDS,

I

BUNDLES,

. . . . . . . . . . . . . . . . . . . .

UNIFORMITY

i.

FORCE

PRINCIPAL

DERIVATIVES, CURVATURE,

FLATNESS

GRADIENTS

. . . . . . . . . . . . . . .

P

. . . . . . . . . . . . . . . . . .

THE

CONNECTIONS

MATERIAL

3.

. . . . . . . . . . . . . . . . . . . . .

BUNDLES,

FIELDS

II.

PRELIMINARIES

MECHANICS

. . . . . . .

19 22

4.

THE

CONSTITUTIVE

ELASTIC

POINT;

MATERIALLY 5.

THE

6.

III.

GROUPS

ELASTIC

ISOMORPHISMS

CHARTS AND

MATERIAL

TANGENT

7.

MATERIAL

8.

HOMOGENEITY,

AND

BODIES ...............

OF A M A T E R I A L L Y

MATERIAL ATLASES;

BUNDLE

FRAMES

T(B,~)

A N D THE

ON S I M P L E

30

BUNDLE

ELASTIC

LOCAL HOMOGENEITY,

25

THE

E(B,~) . . . . . . . . . . . . . . . . . . . . . .

CONNECTIONS

p

UNIFORM

BODY .............................

MATERIAL

OF R E F E R E N C E

9.

MATERIAL

OF A S I M P L E

UNIFORM ELASTIC

SYMMETRY

SIMPLE

EQUATION

BODIES...

34 ~J

AND MATERIAL

CONNECTIONS .....................................

50

FIELD EQUATIONS

56

GENERALIZED

ELASTIC

OF M O T I O N . . . . . . . . . . . . . . . . . . . . . . .

BODIES

i.

INTRODUCTION ....................................

63

2.

INDEX

63

3.

LOCAL MATERIAL AUTOMORPHISMS,

SETS A N D G E N E R A L I Z E D

TRANSITION

ELASTICITY ......................................

70

5.

THE M A T E R I A L - I N D E X

74

6.

MATERIAL

ISOMORPHISMS

TANGENT

THE M A T E R I A L A N D AND

PHASE

PHASE AND 68

MATERIAL

THE

B O D I E S .......

G R O U P .....

4.

POINTS,

ELASTIC

ISOTROPY

IN G E N E R A L I Z E D

ATLAS ........................

BUNDLES

AND

INDEX

INDEX ATLASES,

BUNDLES;

HOMOGENEITY

LOCAL HOMOGENEITY ...........................

84

Xl

7.

MATERIAL

8.

FIELD

AND

INDEX

EQUATIONS

CONNECTIONS .................

OF MOTION

IN

GENERALIZED

ELASTICITY .....................................

IV.

ANELASTIC

BEHAVIOR

AND

DISLOCATION

INTRODUCTION ...................................

2.

ELASTIC

3.

ANELASTIC

RESPONSE

ANELASTIC

TRANSFORMATIONS

ANELASTIC

SYMMETRY

AND

7KNELASTIC

FLOW

RULES;

UNIQUENESS

TRANSFORMATION 5.

V.

MATERIAL

107

OF THE ANELASTIC

FUNCTION ........................

UNIFORMITY

102

INNER

PRODUCTS ....................................... 4.

101

FUNCTIONS;

......................

GROUPS

89

MOTION

i.

AND

8~

IN T H E

THEORY

112

OF

ANELASTICITY

...................................

J22

6.

ELASTIC

ANELASTIC

133

7.

ANELASTIC

SOLID

8.

EQUATIONS

OF MOTION

AND

THERMODYNAMICS

AND

MATERIAL

BODIES;

C O N N E C T I O N S .....

DISLOCATION

FOR ANELASTIC

DISLOCATION

M O T I O N S ....

143

BODIES .......

152

MOTION

i.

INTRODUCTION

...................................

2.

THE

OF A T H E R M O E L A S T I C

CONCEPT

CLAUSIUb-DUHEM

POINT;

156

THE

INEQUALITY ......................

157

Xll

3.

GEOMETRIC

WITH

STRUCTURES

UNIFORM

THERMODYNAMICS

5.

SYMMETRY

AND

ANELASTIC

ISOMORPHISM

THERMO-ANELASTICITY STRUCTURAL

THERMOELASTIC

BODIES

SYMMETRY ..........................

4.

6.

ON

AND

RESPONSE ...........

SYMMETRY

GROUPS

ON

FIELD

EQUATIONS

FOR

THERM0-ANELASTIC VI.

SOME

RECENT

BIBLIOGRAPHY

DIRECTIONS

THERMOELASTIC

CURRENT

Ig7

AND

BODIES ......................... IN

172

THERM0-ANELASTIC

BODIES .......................................... 7.

168

IN

.............................

CONNECTIONS

-. 1 6 2

RESEARCH

179

199 2O3

Chapter I.

i.

M a t h e m a t i c a l Preliminaries

Introduction We w i s h to outline here those elements of d i f f e r e n t i a l

geometry w i t h w h i c h the reader should be conversant in order to u n d e r s t a n d the text.

As in past volumes in this

series, we shall assume that the r e a d e r is familiar with those basic concepts w h i c h underlie the d i f f e r e n t i a l manifold a p p r o a c h to differential geometry and the theory of Lie groups. Thus we aim, essentially, at setting the notation which we shall use in what follows. The a p p r o a c h to manifold theory which has been employed in most of the recent c o n t i n u u m mechanics literature on d i s l o c a t i o n theory is that of Kobayashi and Nomizu and the main r e f e r e n c e here would be [21].

Alternatively, the reader

may consult the excellent e x p o s i t i o n of differential geometry that is to be found in [22] and [23]; these later volumes have strongly influenced the author's viewpoint of m a n i f o l d theory and we shall rely on them as we present the definitions and theorems below.

2.

Differentiable

Definition dimension

I-i.

Manifolds

A differentiable

n is a pair c o n s i s t i n g

manifold

of class

of a H a u s d o r f f

F of real valued

k and

space M w i t h

a countable

base and a set

functions

which

are d e f i n e d

on open sets of M and w h i c h have the f o l l o w i n g

properties: (i)

if feF is d e f i n e d

on U (an open set in M) and

in U then fIv is in F; if f is d e f i n e d where

U :

U

U

(U , ~el,

V is open

on U (open set in M)

open in M) then feF if flu

is in

F for each ~el. (ii)

for each peM,

containing

there

exists

p and a h o m e o m o r p h i s m

an open n e i g h b o r h o o d

%: U ~ ~(U)cR n such that

V is open in U, the set of all feF w h i c h are d e f i n e d identical

with

Ck(~(V))

The f u n c t i o n s and the H a u s d o r f f manifold; above

homeomorphism then ~(q)

differentiable

space M is the u n d e r l y i n g

a coordinate

~ is called

are the c o o r d i n a t e of %).

feF are called

= (xl(q),

The pair

on V is

a coordinate

functions

where

charts

{(U

open c o v e r i n g

of M, is called

that an atlas

completely

of the

satisfies

(ii)

of p and the

map near p.

the xi(q),

If qeU

i = l,...,n

of ~ (or the local c o o r d i n a t e s

is a c o o r d i n a t e

of c o o r d i n a t e

functions

space

peM w h i c h

neighborhood

... xn(q))

(U,~)

if

0 ~.

an open set U c o n t a i n i n g

is called

U

, ~ ), ~el}, an atlas

determines

chart. where

A collection

{U , ~el}

is an

and it can be p r o v e n

a differentiable

manifold

if the maps

are d i f f e o m o r p h i s m s

of class

k.

Now, let M be a d i f f e r e n t i a b l e let U be an open set in M. F (k,U)

differentiable If feF

(k,p)

(k,p)

which

f and g agree.

induces

then there

functions

If i:

near

corresponds

to a s u b s p a c e

(k,p)

invariant

M

F (k,p)

Under

classes

÷ F (k,~(p)), of f via ¢).

consisting

Z (k,p)

is i n d e p e n d e n t

Z (k,p)

gives

of p on

i.e.,

zero f i r s t - o r d e r

partial

i, Z (k,%(p)) in such a way

order p a r t i a l

transformations

of the chart

near p; if

Let Z*(k,~(p))

at %(p) and it is easy to show that

under coordinate

of

of all d i f f e r e n t i a b l e

of F (k,p)

zero first

on U

for peM then

the i s o m o r p h i s m

iff fo% -I has

The s u b s p a c e

defined

by

are open n e i g h b o r h o o d s

is a chart

~(p) w h i c h have

at ¢(p).

derivatives

domains

(U,})

of F (k,¢(p))

defined

functions

is some open n e i g h b o r h o o d

derivatives

that feZ

whose

(the r e p r e s e n t a t i o n

be the s u b s p a c e

[22] we denote

then f is said to be d e f i n e d

an i s o m o r p h i s m

= fo#-i

I (k,p)

Wang

of class k and

the set of all e q u i v a l e n c e

functions

f,geF

i(f)

Following

the set of all d i f f e r e n t i a b l e

and by F (k,p), peU,

of p.

manifold

Z (k,p)

is

near p, i.e.,

(U,~).

rise to the q u o t i e n t

space

~ F (k,p)/Z 6k,p) w h i c h we call the c o t a n g e n t space of M P at p. Let d , peM, denote the n a t u r a l p r o j e c t i o n (a l i n e a r P

map)

f r o m F (k,p)

function vector

into F (k,p)/Z

in F (k,p) t h e n dpf,

in M

.

If

(U,~)

(k,p).

If f is a d i f f e r e n t i a b l e

its d i f f e r e n t i a l ,

is a c o t a n s e n t

is a chart of peM and feF

(k,p) then,

P as the c o o r d i n a t e

functions

xi(p),

i = l,...,n,

are in F*(k,p)

it is easy to see that dpf = ~(f°~ -i) dx i so that the set ~x I .p {dpX i ' i = l,...n} forms a basis for M*. The space dual to p * Mp is d e n o t e d by Mp and is c a l l e d the t a n g e n t space to M at p; its e l e m e n t s the chart

are c a l l e d t a n s e n t

(U,~) the basis

by { ~., i = 1 , 2 , . . . n } . ~x I is s u s c e p t i b l e

vectors

in M

w h i c h is dual in M* is d e n o t e d P P If M is a m a n i f o l d and peM, t h e n M P

of a r a t h e r

concrete

interpretation,

Let a and b be r e a l n u m b e r s

and let ~:

~(c)

Let

= p for some ee(a,b).

~(~(t))

and r e l a t i v e to

~ (xl(t),...,~n(t))

(a,b) ÷ M such that

(U,~) be a chart of p so that

for a ~ t ~ b.

are d i f f e r e n t i a b l e

functions

curve w h i c h passes

t h r o u g h p.

on

as follows:

If the ~ i , i = l , . . . n ,

(a,b) t h e n ~ is a d i f f e r e n t i a b l e A l i n e a r map ~

: F (k,p)÷R P

can t h e n be d e f i n e d via

-~p(f)

d ~ ~-~ f(~(t))It=c

_ ~(fo~-l) ~x I

P d~i d-t--Ic

7 * So that ~ (f) = 0 if feZ (k,p). F a c t o r i n g the l i n e a r map P t h r o u g h d we get ~ = ~ o d where, c l e a r l y , ~ is a P P P P P P l i n e a r map f r o m M into R so that ~ (the t a n g e n t v e c t o r of P P at p) is an e l e m e n t of M . P

As = ~(fo~-l)ll dX i the c o m p o n e n t s p ~X 1 P dt o

p' relative

to the basis

If v = v i ~

~

{~ - ~ , i = l,...,n} ~x I p

is in M

~--~p

then the curve

functions

li(t ) = ¢i(p)

If we d e f i n e

= v. ~

In this way we may

{~--~ , i = l,...,n} ~x I p

system

~

P

+ (t-c)6];

is the n a t u r a l therefore,

as P

curves

i,j

space

= 1,2,

I. l relative

Tr'S(p)

..n

at peM by

O M* @ ... @ M * and let

P ~ P

(U,~) be a

s

chart of p then the p r o d u c t

"

of M

¢ via the r e p r e s e n t a t i o n s

r

has,

p

i = 1,2 ..... n

to the c o o r d i n a t e

the tensor

O ... O M

11

i -- l,...,n.

(a,b) ÷ M whose

w h i c h pass t h r o u g h p; they are d e f i n e d

lJ.(t) = @J(p)

= M

~

basis

vectors

to the local c o o r d i n a t e

Tr'S(p)

+ (t_c)vi;

the n a t u r a l

the set of t a n g e n t (i=l,...,n)

Ic

via @ are

t h r o u g h p and satisfies

characterize

P

P

representation

passes

~:

are dt

of

i

~xr

basis

basis

@ d x

p

Jl

8...8

P

Js,

i,j

= 1,2,...n}

P

of ¢ for Tr'S(p);

the c o m p o n e n t

d x

any t e n s o r teTr'S(p) iI i Js from t = t. ....r ~ ~...@d x 31 3s il P ~x

Further

information

w i t h the tangent

concerning

space

the tensor

algebra

associated

we assume

will be injected as we require it; P that the reader is already familiar with the concepts

of tensor

product,

exterior

While we shall the properties manifolds

This

not require

of d i f f e r e n t i a b l e

much

etc. information

maps

between

concerning

two d i f f e r e n t i a b l e

the gradient

If (M,FI) , (N,F 2) are two d i f f e r e n t i a b l e

k and dimensions

continuous every

product,

we do need to know how to define

such a map. of class

M

m and n, respectively,

f2o~eFl(Or

later r e l a t i o n

if we define

F2(k,~(p))o~CFl(k,p)

induces

9p(f2 ) : f2o~

easy to verify

a linear

map 9p:

manifolds

then a

map @: M ÷ N is said to be d i f f e r e n t i a b l e

f2eF2,

of

if for

for all peM). F2(k,~(p))+Fl(k,p)

for all f2eF2(k,~(p)).

It is

that

~p(7N(k,9(p)) ) c 7M(k,p )

so that

there

such that linear

exists

dp((f2))

map ~p i s

gradient

Op

the linear

of ~ at p, which

map ~ , p : is defined

w eNd(p).

in terms of local

and ~(p) we r e f e r

linear

= ~p(d~(p)f2).

for all VeMp and a l l

~,p,

an induced

map ~p:

N~(p)

The transpose

of the

Mp + N~(p), c a l l e d

the

by =

of

systems defined near p

t h e r e a d e r t o Wang [ 2 2 ] .

3.

Fibre Bundles, Examples.

We recall,

Definition structure

first

I-2.

of a C ~ m a n i f o l d

and we also

Definition group

by

(x,y)

L(e,m)

(ii)

L(g,

for all

1-3.

group

L(~,m))

Wang

[223 we w r i t e

on M = R n.

to s t a t e

is C

of

.

manifold

and G is

L: GxM ÷ M d e f i n e s

identity L(g,m)

of l e f t - m u l t i p l i c a t i o n

as the

L

g

G

linear Also,

each

group

now collected

= L m = gm and call g

of G on M.

Clearly

of G on M is said

to be e f f e c t i v e

map

group

acting

on M"

that

GL(n,R)

Lie

the

of G.

of M for e a c h

we r e c a l l group

element

is a d i f f e o m o r p h i s m

identity

example

transformation We h a v e

map

e is the

Left-multiplication

general

the m a p p i n g

= L(g~,m)

Following

a simple

the

= m

where

acts

that

has

on M if L s a t i s f i e s

mEM,

g

G which

+ xy -I , x,yeG,

g,~eG,

gEG.

the

is such

a differentiable

(Lg) -I = Lg -I so that

As

is a g r o u p

and

and Some

following

If M is a d i f f e r e n t i a b l e

then

L the o p e r a t i o n

if "L

the

Bundles,

need

as a Lie t r a n s f o r m a t i o n (i)

Principal

of all,

A Lie g r o u p

GxG ÷ G d e f i n e d

a Lie

Associated

the m a t r i x

that

g = e.

product

defines

as a Lie t r a n s f o r m a t i o n can be c o n s i d e r e d

on i t s e l f

basic

implies

facts

via

group

as a Lie

left-multiplication.

that we need

in o r d e r

Definition

1-4.

collection

consisting

bundle

space

Lie group

G called

(i)

the elements

open

sets

field

of three

acting

called

we call (ii)

gaB:

to G and,

atlas

N, a

is a Lie

of charts

which

charts,

satisfies consist

~ : U~×N ÷ ~-~(U Each

~

on U s via #e,p:

of

) such that

then defines N ÷ -l(p)

a

e Lp and

at p.

U nU~

the m a p p i n g s

furthermore,

÷ G are smooth;

g~(p)

a #-I ~,P°¢8,p:

fields

are the coordinate

The set ¢ is not a proper

subset

of any other

collection

Finally,

coincides

U nU~.

of charts

is m a x i m a l

Definition

on

which

relative

to

we state

I-5.

N÷N

the fields

these

transformations (iii)

space

the

on N, a smooth map

bundle

.

L is a

manifolds,

and a collection

for all peU

Lp the fibre

UenUB

effectively

U sCM and d i f f e o m o r p h i s m s

On the overlaps

belong

group which

the bundle

of d i f f e o m o r p h i s m s

bundle

M, and the fibre

(U ,~ ),called

= ~-l(p)

A fibre

differentiable

the projection,

,~ ), ~el}

#e6{p}xN)

Chp VI)

the structure

group

i ÷ M called = {(U

[22],

L, the base space

transformation ~:

(Wang

w i t h the fibre

exists

a principal

group,

and coordinate

(i) and

(i) and

(ii) above,

i.e.,

(ii).

the f o l l o w i n g

A fibre

It can be shown that

satisfies

bundle space

whose

structure

is called

if i is an a r b i t r a r y

bundle

say P, whose

transformations

group

a principal fibre

base

bundle.

bundle

space,

are identical

there

structure to those

of L.

(for a proof we refer the reader to Wang

such a bundle

is called the associated

principal

[22],

Chp. VI);

bundle

of L.

Examples I.

The Tangent

Bundle

T(M)

The base space is M and the bundle

w(p,v)

=

LJM • pcMp

w is a map 7: T(M) ~ M such that for

Thus the projection peM,

space T(M)

= p where

If we set T(U)

veM . ~ P

= w-l(u)

= - l ( p ) is called the fibre at p. P The bundle atlas {(U ,@ ), ~el} consists

=

U M peU p

then M

charts

such that ~ : U xR 3 + T(U

(U , ~ )

~ : {p}xR 3 . - l ( p )

).

and there exist maps

R 3 is called the fibre space.

linear group, Let {(U

that, GL(3),

~peM,

G B(p)eGL(3).

is the structure

We define

~ (p,vl,v2,v 3) = (p,v) ~ where If {(U

,~ ), ~ I }

transformations Thus the general

group of T(M).

for M such that

local coordinates

= (xl(p),x2(p),x3(p)).

(x i) on M, i.e. 9 (p)

the @e above by:

v = v i ~--~p"

is maximized

w.r.t,

all atlases

{(U

,ge), ~el} for M, we get the bundle atlas

@.

Then elements

systems

for T(M),

say,

(U ,~ )e@ give rise to local coordinate

(xl,x2,x3,vl,v2,v3)

coordinate

R3÷Mp(=~-l(p));

On U hUB, GaB( p ) ~ @ ~ p ° @ ~ , 6 :

,~ ), ~el} be an atlas

~ : M + R 3 induces

Hence

~,p:

R 3 + R 3, peU nU$, are called the coordinate and we require

of bundle

systems.

If ~

on T(U induces

), called the lifted local coordinates

(~i)

10 on U~ and pEU nU6 then the coordinate

transformations

are

bundle of T(M).

A

given by G 8(p) = det[~xl/3x-]]. II.

The Bundle of Linear Frames

E(M) is the associated

E(M)

principal

linear frame at p is an ordered basis for Mp, i.e., ep = {ep,l,." i=1,2,3}.

Set Ep = {ep} then the base space is

again M and the bundle

space is E(M) =

~: E(M) ÷ M such that z(p,ep) E(U) =

E peU

= ~-I(u)

~ = {(U

~ : U xGL(3) + E(U ).

Thus,

GL(3) + Ep(=z-l(p)).

is the fibre at p.

,~ ), eel} consists

has the r e p r e s e n t a t i o n

of right m u l t i p l i c a t i o n

by G on

]

E(M) is defined by R e

.G]; i=1,2,3}. P,] l Let ¢ = {(U ,¢ ), eel} be a bundle atlas

i = {(i,0,0), Define ep(e) ~,p:

of maps

as with T(M), there exist maps If GeGL(3)

G = [G~] then the operation ~

= -l(p)

If we set

P

The bundle atlas

~e,p:

= p, VpeM.

then E

P

~ Ep; the projection peM

P

(0,i,0), = Ce,p(i),

= {e

(0,0,i)}

$ (p,G)~ = RG(ep(e)) coordinate

the standard basis for R 3.

then ep(~)

R 3 ÷ Mp is an isomorphism.

for T(M) and

is a frame at p since We now define the map

~e by

and note that it is easy to show that the

transformations

on T(M) and E(M) coincide.

11

4.

Lie A l g e b r a s , the E x p o n e n t i a l Map, Fields on E(M).

Every a vector

and F u n d a m e n t a l

Lie g r o u p G has an a s s o c i a t e d

Lie a l g e b r a g, i.e.,

space w h i c h is e q u i p p e d w i t h a b r a c k e t o p e r a t i o n .

To d e f i n e this Lie a l g e b r a , and let LxY : xy, transformation

Vx,yeG

let v be a v e c t o r f i e l d on G, (i.e. we c o n s i d e r G as a Lie

g r o u p a c t i n g on i t s e l f via l e f t - m u l t i p l i c a t i o n

so t h a t Lx: G ÷ G , V x s G ; i f L x , y ( V ( y ) )

= v(xy)

for all

x , y e G t h e n v is said to be a l e f t - i n v a r i a n t

v e c t o r f i e l d and

the c o l l e c t i o n g of all such l e f t - i n v a r i a n t

vector

G t h e n forms a v e c t o r space the b r a c k e t

To d e f i n e

o p e r a t i o n on g we first d e f i n e the Lie d e r i v a t i v e

of one v e c t o r

field u w i t h r e s p e c t to a n o t h e r v as follows:

if M is a d i f f e r e n t i a b l e coordinate

in the o b v i o u s way.

fields on

manifold,

peM,

s y s t e m n e a r p so that u =

and ¢ a local

ui ~

~

•

~x I

, v = v p

i~_~

~

3x I

p

then i

•

°

[L u](p) = (~-~(p)v](p) v

~

3x J

3x J

The Lie d e r i v a t i v e respect

- ~V~(p)u](p))

in a c o o r d i n a t e - f r e e

we r e g a r d v as the i n f i n i t e s i m a l f a m i l y of d i f f e r e n t i a b l e [22],

Chp.

fields

shall s u f f i c e

define

Iv,u]

maps

~

with

manner

if

g e n e r a t o r of a o n e - p a r a m e t e r

~ of a n e i g h b o r h o o d

III) but the a b o v e d e f i n i t i o n for our p u r p o s e s .

= Lu and this b r a c k e t V

p

of any a r b i Z r a r y t e n s o r y c T r ' S ( p )

to v can be d e f i n e d

see W a n g

~ ~

If u,vEg

of p (i.e., for v e c t o r t h e n we

o p e r a t i o n endows

g with

12

the s t r u c t u r e of a Lie algebra. there

It can also be shown that

le : g~ ÷ G e such that

exists an i s o m o r p h i s m

le: v~ = v(e) ~

if veg. ~

~

If G is a Lie group and vcg t h e n v induces subgroup

Iv(t)

of G such that

~tlv(t)It=0

a one-parameter

= v(e).

In fact we

~

may take ~ (t) as the s o l u t i o n of ~(t)

= v(l(t))

satisfying

V ~

~(0)

= e so that i(0)

= v(e),

i.e.,

~

is an i n t e g r a l c u r v e V ~

of the v e c t o r f i e l d v.

The e x p o n e n t i a l map,

exp:

g ÷ G, is

~

t h e n d e f i n e d by exp(v)

= I (i) for all veg. V

i 2 ~ = i + Vt~ + ~Vt + ..., w h e r e

lv(t)

For G - GL(3),

~

exp V = i + V + ~V 2 + . . . .

Veg£(3),

For Vegl(3)

so that

and GeGL(3)

it is

G e x p ( V ) G -I and,

also,

L ~

possible

to p r o v e that e x p ( G V G -I)

=

~

that d e t [ e x p ( V ) ]

= exp(trV)

w h e r e trV d e n o t e s the t r a c e of

~

V. Finally, =

let G denote

{(u e e ,~(~), I }

the s t r u c t u r e

the b u n d l e

and h e n c e t h e r e exist maps

atlas

~,p,:

•

Then

chart;

~,p

for

E(M) (I) and

:

~

G

G x ÷ (Ep) x, i.e.,

then ~,p~.~(v) : 5e(p) w h i c h is a v e c t o r be s h o w n that t h e s e fields

group

field on Ep.

E

P if vcg It can

are i n d e p e n d e n t of the b u n d l e

~ is c a l l e d a f u n d a m e n t a l

the set of all such ~ by g.

f i e l d on E(M) and we d e n o t e

Now let ~:

E(M) + M so that

~

~,:

E(M) x ÷ M (x).

Then it is p o s s i b l e to show that ~.(v)=O,~ ~

VvE@[ and we say that the v lie in the fibre d i r e c t i o n s ; fact it is p o s s i b l e

to show that g is i s o m o r p h i c

(i) in this case, of course, a l g e b r a g = gl(3).

G = GL(3)

in

to ker ~ .

and the a s s o c i a t e d

Lie

13

5.

on E(M) and Parallel

"G" Connections Let xeE(M).

space

Then the vertical

= V

E(M)x

subspace

x

on E(M)

~ H , x

~,xlH

V x of the tangent V x = gl x •

is then a map H: x ÷ HxCE(M) x such that

xeE(M).

at x, i.e.

we have

subspace

to be V x = ker z~"x , i.e.,

E(M) x is defined

A connection

Transport

Hx,

We note

that the h o r i z o n t a l

is not unique.

~ nx: H x ÷ M~(x)

Since

Vx = ker ~ x

is an isomorphism.

Thus

X

VveM~(x), point

there

exists

~ is called

~~ on Ep such that ~(x)~ = nil(v).~

the h o r i z o n t a l

lift of v relative

The

to H.

~

Let % be a smooth

curve

in M.

The h o r i z o n t a l

lift of 1

~

is a curve

le[(M)

such that

~(l(t))

= l(t),

~te[a,b],

say,

~

and such that

I is horizontal,

i.e.,

the tangent

vectors

~

~tl(t)

are the h o r i z o n t a l

~t~(t), called

~te[a,b]. the [arallel

connection

let

%eU

Then,

.

@t,a~eG (2)

E(M)I(0)÷E(M)A(t) to the

since

a linear

isomorphism.

frame,

such I(0).

that H is a "G"

of I so that there

of the tangent

at peM,

Pt to ~t: ~I(0) Thus

for E(M)

~ ~-i ~,%(t) oPt(~) o ~

It can be shown ~

transports

chart

on E(M) if, for all

is independent

parallel

for Mp, we may extend linear

vectors

pt(~):

I relative

bundle

a '{G" connection

iff pt(1)

defined

~, i.e.

along

Pt,e : G + G by Pt,~

Define

~eM,

connection well

transports

exist maps

(U ,~ ) be a lifted

H is called

smooth

there

of the tangent

H.

Now, that

Thus

lifts

spaces

is an ordered ÷ Ml(t)'

"G" connections

which

exist along

basis is a

on E(M) c o r r e s p o n d

(2) thus, a "G" connection H on E(M) is a connection for which the m a p s P t e e GL(3); "G" connections on arbitrary principal bundles a r @ ' d e f i n e d in an analogous way.

14 to the classical affine connections

on T(M).

Once again, let (U ,~ ) be a lifted chart for ECM) corresponding M.

2

to the local coordinate

Then (p,ep)eM(U~)

relative to (U ,~ )

,

system (xl,x ,x 3) on

has local coordinates where e

= { e ~ ( p ) ~~x

p

(xl(p), e (p)) }.

p

Then E(M) x is spanned by the natural basis:

Put x=(p,ep).

{ ~

~

}

~-~x

x

If we compute the matrix ~, and apply it to this set we find that V x is spanned by { ~

} and thus H x is spanned by X

{ a___~. - r ~ . ( p , e

p

) 2. x

where

the

are

symbols of H.

Now, let ~(t) = (~(t),e

representation

of the horizontal

the

connection

(t)a/~xJlk(t))

lift of ~(t)cM.

to the lifted chart (U ,$ ), ~(t) = (xi(t),e~(t)), ~t~(t)

= (~t~l(t),St e (t)).

of ~t~(t)

be the

Relative and thus

If we set the vertical components

= 0 then we find the equations of parallel transport,

namely ' Ste~(t)

+ F kj i ~t ~k = 0.

with appropriate ~(t) = (~(t),e

The solution of this equation,

initial conditions,

(t)~/~x31~(t))

is such as to render

a horizontal

curve.

Finally

we can show, by virtue of the fact that we are dealing with a "G" connection that F ~ i ( P , e p ) =

F~i(P)e~,

~peM.

15

6.

Convariant

Derivatives,

Curvature,

Torsion,

It can be shown that the parallel #t: Mk(0) spaces:

÷ Ml(t)

#~,s:

induce

Tr,s

-I(Q)

linear

Tr,s

+ -l(t)'

y(t" ~r,s ~ Js T r's l(t) and y(O) ell(O).

Here,

r,s

TI(O)

denotes

operation DZdt :

y(t)

: Pt

transports

(y(O)) where

of course, ~

0 ... @_~MI(o) @ ~ ( 0 )

,~c

"'" @ MI(O)'

where

MX(O)

--~ Now let ~z(t)sT[ 's (t); the

the dual space of MI(O).

parallel

of the tensor

Ar,s

~'~

= MI(O)

r

--

transports,

isomorphisms

i.e.,

and Flatness

of the tensor

of covariant

l i m i/At{z(t)-p t ~ At+o

spaces

then induce the

differention,

Ar,s

z(t-At)},

transport

which

says that ~z(t) is

iff Dz ~-~ = O.

obtained

via parallel

possible

to show that if z is a smooth tensor

on some open set in M, l(t) s Dom

It is then field defined

(z), and ~(t)

= z(~(t))

is a smooth field, then there exists a tensor field Dz such that D_~z D~z. For instance, if z is actually a dt ~ ~ ~ vector field v~ then, in terms of local coordinates,

Dv = [vi'k dt l(t)

~(t) + £ ~ k ( ~ ( t ) ) v J ( t ) ] ~ k ( t ) ~ ICt)

= ([vi,k + Fi. vJ] ~ • I 0 d x k) o ( ~ k _ ~ 3~ 3xl IP P ~x

:

" (vl,k~----~I ax~Ip

0 d x k) P

= Dvll(t ) 0 ~IP"

o

(i k

~

a--~ p

)

) p

18

F i n a l l y we have the f o l l o w i n g

definitions which will

be u t i l i z e d w h e n we come to c h a r a c t e r i z e nections tions

w h i c h arise

in the study of c o n t i n u o u s

of d i s l o c a t i o n s

affine

in a continua:

c o n n e c t i o n on T(M)

if for all psM there such that r e l a t i v e

if it admits

local c o o r d i n a t e s An a f f i n e

that t h e r e e x i s t s connections

a one-to-one

is c o m p l e t e l y

in the n e i g h b o r -

concept p r e c i s e

correspondence

on T(M) and G c o n n e c t i o n s

map Now,

on E(M) so that a conare h o r i z o n t a l

If H is a c o n n e c t i o n on T(M) and xeT(M)

g i v e n any f i e l d P of r - d i m e n s i o n a l

defines

i(N)

spaces

subspaces

t h e n the w(x)

= p.

(r~n) of

on M let i:N ÷ M be an i m m e r s i o n w h i c h

as a s u b m a n i f o l d

integral manifold (if i(q)

recall

between affine

I : H + M is an i s o m o r p h i s m w h e r e ~x E ~*x H x x p

the t a n g e n t

(x l)

the F symbols

n e c t i o n H on T(M) is a s m o o t h f i e l d w h o s e values subspaces.

flat if

system

connection

an i n t e g r a l m a n i f o l d

h o o d of p for each p e M ; t o make this

distribu-

(loeall~)

a local c o o r d i n a t e

to t h e s e

con-

first of all, an

is said to be

exists

of the c o n n e c t i o n vanish. integrable

the m a t e r i a l

of M.

Then i(N) is c a l l e d an

of p if for e v e r y point qsN,

i,q(Nq)cP(i(q))

= peM t h e n P ( p ) c M

and has d i m e n s i o n r~n). P The c o n d i t i o n of i n t e g r a b i l i t y for a G c o n n e c t i o n H on

E(M)(equivalently,

affine

a c t e r i z e d by the c u r v a t u r e to a local c o o r d i n a t e Rj mri

=

~r j .

ml sxr

~r j

mr ~x i

connections

on T(M)) is char-

tensor R whose

system

(x l) are

+ Fi Fs _ F j. F s sr ml sl mr

components

relative

17

where

the F's are the c o n n e c t i o n

theorem

of Frobenius,

iff R = O.

a connection

cross-section

over

= {(p,fi(p));

the i d e n t i t y map on M); horizontal parallel

transport,

exists

a local

fi : ~yl ~ - which,

[fi,f.] k 3

H is flat.

(a cross-

the e q u a t i o n s

:

is

of

It then follows

~f~ fm. ~x m ±

~f~i ~x m

~

f°

=

3

T k fmfn mn i ]

of the t o r s i o n

If T = 0 then

[fi,fj]k

= 0

can be used again to infer that

coordinate

in turn,

P

in saying that the c r o s s - s e c t i o n

with H.

theorem

connection

such that ~o~ = id M

Tk ~ Fk - F k are the c o m p o n e n t s mn mn nm T associated

integrable

be a h o r i z o n t a l

= f!1 ~

+ FO i.e.,-i 0 ~x m im fk : "

brackets

and the F r o b e n i u s there

fi(p)

we imply that the f! satisfy ~f~ 3

that the P o i s s o n

tensor

i : 1,2,3}

U~cM such that

By the famous

integrable

over UcM is a map o: U ÷ [(U)

section

of H.

H is c o m p l e t e l y

Now let H be a c o m p l e t e l y

on [(M) and let ~(p)

where

symbols

implies

system y that

i

on M such that

F~k(y)

: O, i.e. , that

Chapter i.

!I.

Material

Uniformity

in Elasticity

Introduction We present,

formulation

body

in a simple

his early w o r k

in this

tise by T r u e s d e l l and Noll ment,

chapter.

to be d e r i v a b l e

tant p a r a l l e l i s m ,

was

it here,

paper

[7 ].

of C. C. Wang

of a simple

can be found

elaborate

smooth

which

at the same time as [ 6 ], has the a d v a n t a g e

moving

the u n n e c e s s a r i l y

treatment probably

of a m u c h w i d e r

smoothness

and thus

ap-

of re-

assumption

allows

of simple bodies;

for the it is

also the first w o r k of its kind to d e m o n s t r a t e

effectively tools

class

dis-

on a truly r e m a r k a b l e

peared

above,

is

The theory,

This w o r k of Wang's,

of Noll, w h i c h we m e n t i o n e d

treat-

structure

from a g l o b a l l y

restrictive

of

in the trea-

geometric

is b a s e d

com-

an account

given by Noll in [ 6 ].

as we shall p r e s e n t

of

of the p a r t i c l e s

[ 5 ] and a more

of

materials

The p r o b l e m

structure

but one in which the m a t e r i a l

still r e q u i r e d

an e x t e n s i o n

t r e a t e d by Noll;

direction

distributions

of n o n - s i m p l e

equations

the body was first

mathematical

static

body;

classes

geometric

from the c o n s t i t u t i v e

prising

elastic

in the f o l l o w i n g

the m a t e r i a l

a concrete

of c o n t i n u o u s

to cover certain

will be given determining

chapter,

of the t h e o r y

of d i s l o c a t i o n s this theory

in this

the p o w e r of m o d e r n

differential

geometric

in c o n t i n u u m mechanics. W~ile we shall

continue

t h e o r y to be p r e s e n t e d

here,

b e y o n d the

static

dislocation

to treat n o n - s i m p l e

materials

19

and anelastie response accounts

(dislocation motion),

of the material

[24] and [25].

some other

in this chapter may be found in

In addition,

reprints

of the foundation

papers by Noll and Wang together with several other papers which treat

classes of universal

uniform elastic bodies,

solutions

for materially

as well as wave propagation

materials, are to be found in the collection

[26].

in such Truesdell

has included a lucid summary of Noll's basic ideas in this area in his Lectures interpretations

on Natural Philosophy

of the concepts

[27] and other

introduced by Noll,

as well

as comparisons with their own work in dislocation theory, have been given by Bilby

[ 9 ] and KrOner

collection which contains these may find two brief expositions

latter works the reader by Noll and Wang,

ly, of the basic ideas which underlie sented below.

[i0]; in the same

respective-

the theory to be pre-

As far as possible we shall retain the nota-

tion of the original papers. 2.

Body Manifolds,

Motions

and Deformation Gradients

We begin with the following, Definition

II-i

three-dimensional

(Wang) A bo__qJ~manifold differentiable

B is an oriented

manifold which is connected

and has the property that there exist diffeomorphisms, ~, 4, X,..-

(which we shall call confisurations

B) which map B into R 3, i.e., If pcB, a body manifold,

say

of the b o d ~

%: B ÷ R 3. then a linear isomorphism

20

r ~p : B p ÷ R 3 is called as the t a n g e n t vector

space

space

a local c o n f i g u r a t i o n B

is an o r i e n t e d t h r e e - d i m e n s i o n a l P it must be a l g e b r a c i a l l y i s o m o r p h i c to R3; we

note m o r e o v e r that b o t h c o n f i g u r a t i o n s tions

are r e q u i r e d to be o r i e n t a t i o n

the g r a d i e n t

of p; of course,

of a c o n f i g u r a t i o n

f i e l d of local c o n f i g u r a t i o n s ~: B ÷ R 3 is a c o n f i g u r a t i o n

and local c o n f i g u r a -

preserving

~ of B gives

of points of B t h e n

local c o n f i g u r a t i o n

of p for each psB.

true as, in g e n e r a l ,

a given

can not be o b t a i n e d

as the g r a d i e n t

psB, ~p:

and that

rises to a i.e.,

Bp ÷ R 3 is a

The c o n v e r s e

of a c o n f i g u r a t i o n

of B, w h e r e

r~p (t) of local c o n f i g u r a t i o n s

of p.

family

family

If we choose to p i c k

local c o n f i g u r a t i o n

call r a local r e f e r e n c e ~p

of B.

t is a time v a r i a b l e ,

and a local m o t i o n of peB to be a o n e - p a r a m e t e r

out a p a r t i c u l a r

is not

f i e l d of local c o n f i g u r a t i o n s

We now d e f i n e a m o t i o n of B to be a o n e - p a r a m e t e r ~(t) of c o n f i g u r a t i o n s

if

r ~p of psB then we w i l l

configuration;

of B and r is such a local r e f e r e n c e ~p

if ~(t)

is a m o t i o n

configuration

of p

then we can d e f i n e the t e n s o r

~PF(t) --- @~p(t)

w h i c h we t e r m the Now,

o r~p-i ,

local d e f o r m a t i o n

t > O

at p ( r e l a t i v e to r ~p ).

even t h o u g h we may not be able to find c o n f i g u r a t i o n s

of B such that rp = ~ p ly such a r e l a t i o n s h i p The chain rule

, V P EB, if we fix PsB then c e r t a i n ~

can be s a t i s f i e d at this one point.

for g r a d i e n t s

then y i e l d s

21

~PF(t) = 9~p(t)

o ~

= [9(t)

and as 9(t)

o ~-i:

~(B)

tion of the

(open)

domain

deformation

@radient

orientation

preserving

o ~-l],~(p)

÷ [9(t)](B) ~(B)

represents

in R 3 we also

at the point

call F (t) the ~p

at time t.

isomorphism

a deforma-

As F is an

of R 3 we have

det F(t)>O

for all t>O. If 9: B + R 3 is a c o n f i g u r a t i o n characterized

by three

smooth

of B then 9 can be

functions

xi(p),

peB,

i = 1,2,3,

such that 9(P)

where

= (xl(p),

the x

Now let

i

are,

x2(p),

of course,

{ha' a = 1,2,3}

(0,0,i)}

which

x3(p)),

comprise

the

denote

psB

coordinate

the vectors

the s t a n d a r d

r : B ÷ R 3 is a local c o n f i g u r a t i o n ~P P a c t e r i z e d by a basis {@a' a = 1,2,3} ~p(~a ) = ~a' erence

basis

a = 1,2,3.

We call

of r and note that ~p

B then the r e f e r e n c e

basis

basis of psB

functions {(I,0,0), of R 3.

of 9. (0,i,0),

If

it can be char-

in Bp such that

{~a' a = 1,2,3}

the ref-

if 9 is a c o n f i g u r a t i o n

of 9,p is just the n a t u r a l

of

basis

{~----i i : 1,2,3}. ~x p' Now let K: B + R 3 be a p a r t i c u l a r w h i c h we shall tion;

we denote

single

out and use as a r e f e r e n c e

the c o o r d i n a t e

If 9 is any other

configuration

configuration

from K to 9 is a d i f f e o m o r p h i s m

functions

of B

configura-

of < by X A, A = 1,2,3.

of B then the d e f o r m a t i o n

22

0 < which,

as K(B)

:

:

and @(B)

be c h a r a c t e r i z e d ' b y i

-I

O;

furthermore, K.

of ~, i.e., collection

is i d e n t i c a l

K: R 3 ÷ R 3 is all r e f e r e n c e There may

isomorphisms of all such

w i t h the i s o t r o p y

is d e f i n e d

below;

group

this follows

such isomorphisms.

Groups

of ~ M a t e r i a l l y

Uniform

Simple

Bod Z.

Material a c c o r d i n Z to w h i c h must ference

esl} where

relation

characterizes

The

, K0r~),

~ = GO and the

It can be

if ~ is a r e f e r e n c e

in this way by some

to ~, w h i c h

from the simple

[ 7 ]) that

isomorphisms

isomorphisms

relative

on B are not unique.

bodies

(i) their d e f i n i n g

satisfy

and

in c o n t i n u u m

these c o n s t i t u t i v e e q u a t i o n s formulation mechanics

was

we give b e l o w Definition isomorphism

11-4

+ B P

[28],

essentially,

Let pEB,

i: B

(or isotropy)

a simple

is a m e m b e r q

equations, frame-indif@roups

which

The first truly r i g o r o u s

of the i s o t r o p y

given by Noll are due,

of m a t e r i a l

admit.

of the concept

are c l a s s i f i e d

constitutive

the p r i n c i p l e

(ii) the s z m m e t r y

mechanics

group

in c o n t i n u u m

[29], and the d e f i n i t i o n s to him. elastic

body.

of the i s o t r o p y

Then an group

31 g(p) iff E(¢,p, p) ~ E(%,p o i, p)

for all configurations

(II-5)

@: B ÷ R 3, i.e., iff i is a material

isomorphism of p with itself. We require

(Noll [29]) that g(p) be a subgroup of

SL(B ), the special linear group on B ; SL(B ) is that subP P P group of GL(B ) consisting of isomorphisms i of B with P P determinant equal to one. We also make Definition

11-5

An isomorphism G: R 3 + R 3 belongs to G(p), ~

the isotropy group relative to the local reference configuration r~ p -

-

of peB if for all FeGL(3) S([, p) = S([G,p)

(II-6)

From these last two definitions,

and the relationships

which exist between the response functions E and S, it is a ~

straightforward matter to deduce

(Noll, [ 6 ]) that the isot-

ropy groups G(p) and G(q), p,qeB are related via G(p) = rpog(p)or-l~p

(II-7)

g(q) = ~(p,q)og(p)o~(p,q)-I

(II-8)

and that

if r(p,q):~ Bp + Bq is a material isomorphism. r~pB are both local reference A G:

R3 ÷ R3 is

configurations

If re~p and

of peB and

defined by GA = r@or~_ I then G (p) and Gs(p), ~

p

P

32

the i s o t r o p y figurations

groups

relative

if r(p,q)

(11-7)

and

erence

r

A

(p)~

(II-9)

-i . r~p is a m a t e r i a l

easily y i e l d

the i s o t r o p y

G(p)

groups

isotropy

group

to

relative

= G(q)

of p.

group by G

(U , re).

isomorphism so that,

G (p), r e l a t i v e

on U cB, are i n d e p e n d e n t

we denote this

GaG

con-

A_l

GO

= r~q o

(11-8)

particular,

local r e f e r e n c e

satisfy

GB(p) : Also,

to these

From

then in

to a ref-

Following

Wang

[7]

and call it the i s o t r o p y (II-6)

it follows

iff S (F) = S (FG) for all FeGL(3).

If

that

(U , r~)s~,

a

~

meference this

atlas,

then the G

case, we set G

group

relative

~ G(}), ~

to ~.

for all FsGL(3);

are i n d e p e n d e n t

this

of ~ and,

in

w h i c h we call the i s o t r o p z

Clearly,

GaG(i)

iff S~([)

confirms

our remarks

= S%([9)

regarding

ref-

~

erence

isomorphisms

which

appear

at the end of

not too d i f f i c u l t

to show now that the i s o t r o p y

introduced

must

G(K~)

above,

= K G ( ~ ) K -I where

morphism Remark sociated

satisfy

§3.

groups

the t r a n s f o r m a t i o n

K is an o r i e n t a t i o n

It is G(}),

law

preserving

iso-

of R 3. If U , U B are two r e f e r e n c e reference

maps

neighborhoods

w i t h as-

r ~, r B then on the overlaps

the

fields ~or B-I

G s(p) = rp are smooth

P

,

peU

and f u r t h e r m o r e ,

Ges:

(II-10)

U~ + G(~).~

To see this

33

recall that the compatability of the charts

(U~, r~)~ , (U B, rB)~

in ¢ implies that ~

E(For e, p) : E(For 8, p) ~ ~ ~p - ~ ~p

'

peU hUB

for all ~ FeGL(3); replacing F + F G ~

E(FG .r , p) ~~~~p ~

thus G ~ ( p ) e G ( ¢ ) G(p)

= I, Y p s U

=

E(F~;

P)

and, hence, so does G 8(p).

Obviously

and it is also trivial to show that

G ~(p) : G8 (p)-l, V p s U

VpsU nUsn Uy.

we get

hUB,and G~B(P)GBy(P ) : G y(p),

The fields ~e8 will appear again as the

coordinate transformations

on the material tangent bundle

T(B,~) and the bundle of reference frames

E(B,}) which

we

will introduce in 54. Remark

Certain particular types of symmetry will be of

interest to us in this chapter as well as in the chapters to follow.

Once again, the basic definitions here are due

to Noll ([28] and [29]). Definition 11-6 (i)

A simple elastic particle p is called a

solid particle,

if there exists a local reference

configuration r of p such that G(p)c0(3), ~p

the orthogonal

34

3 group over R , and (ii)

a fluid crystal particle, Special

subclasses

of the above categories~

elastic fluid pamtieles be considered

and isotropie

as they arise.

elastic particles groups of SL(S)~

the isotropy groups

i.e. ~ simple

solid particles,

from the fact that

is a Lie subgroup

group G(@) is closed

and that

(Wan Z [7 ]) if the re-

the continuity

condition

lim S~([Gn ) = S~(FG)

for all [sGL($)

(II-ll)

and every convergent

such that lim G

will

are closed Lie sub-

that this is so follows

sponse function S@ satisfies

particle

For all these types of simple

every closed subgroup of SL(S) the isotropy

if it is a non-solid

sequence

{gn}